Logic, Rationality, and Interaction - LORI 2011

Lecture Notes in Artificial Intelligence 6953 Edited by R. Goebel, J. Siekmann, and W. Wahlster Subseries of Lecture...

Author: Hans van Ditmarsch | Jerome Lang | Shier Ju

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Lecture Notes in Artificial Intelligence

6953

Edited by R. Goebel, J. Siekmann, and W. Wahlster

Subseries of Lecture Notes in Computer Science FoLLI Publications on Logic, Language and Information Editors-in-Chief Luigia Carlucci Aiello, University of Rome "La Sapienza", Italy Michael Moortgat, University of Utrecht, The Netherlands Maarten de Rijke, University of Amsterdam, The Netherlands

Editorial Board Carlos Areces, INRIA Lorraine, France Nicholas Asher, University of Texas at Austin, TX, USA Johan van Benthem, University of Amsterdam, The Netherlands Raffaella Bernardi, Free University of Bozen-Bolzano, Italy Antal van den Bosch, Tilburg University, The Netherlands Paul Buitelaar, DFKI, Saarbrücken, Germany Diego Calvanese, Free University of Bozen-Bolzano, Italy Ann Copestake, University of Cambridge, United Kingdom Robert Dale, Macquarie University, Sydney, Australia Luis Fariñas, IRIT, Toulouse, France Claire Gardent, INRIA Lorraine, France Rajeev Goré, Australian National University, Canberra, Australia Reiner Hähnle, Chalmers University of Technology, Göteborg, Sweden Wilfrid Hodges, Queen Mary, University of London, United Kingdom Carsten Lutz, Dresden University of Technology, Germany Christopher Manning, Stanford University, CA, USA Valeria de Paiva, Palo Alto Research Center, CA, USA Martha Palmer, University of Pennsylvania, PA, USA Alberto Policriti, University of Udine, Italy James Rogers, Earlham College, Richmond, IN, USA Francesca Rossi, University of Padua, Italy Yde Venema, University of Amsterdam, The Netherlands Bonnie Webber, University of Edinburgh, Scotland, United Kingdom Ian H. Witten, University of Waikato, New Zealand

Hans van Ditmarsch Jérôme Lang Shier Ju (Eds.)

Logic, Rationality, and Interaction Third International Workshop, LORI 2011 Guangzhou, China, October 10-13, 2011 Proceedings

13

Series Editors Randy Goebel, University of Alberta, Edmonton, Canada Jörg Siekmann, University of Saarland, Saarbrücken, Germany Wolfgang Wahlster, DFKI and University of Saarland, Saarbrücken, Germany Volume Editors Hans van Ditmarsch University of Sevilla Camilo José Cela s/n, 41018 Sevilla, Spain E-mail: [email protected] Jérôme Lang Université Paul Sabatier, IRIT 118 Route de Narbonne, 31062 Toulouse Cedex 04, France E-mail: [email protected] Shier Ju Sun Yat-sen University Institute of Logic and Cognition Department of Philosophy Guangzhou, 510275, China E-mail: [email protected]

ISSN 0302-9743 e-ISSN 1611-3349 ISBN 978-3-642-24129-1 e-ISBN 978-3-642-24130-7 DOI 10.1007/978-3-642-24130-7 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011936231 CR Subject Classiﬁcation (1998): F.4, G.2, I.2.6, F.3, I.2.3 LNCS Sublibrary: SL 7 – Artiﬁcial Intelligence

© Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by author, data conversion by Scientiﬁc Publishing Services, Chennai, India Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This volume contains the papers presented at LORI-3, the Third International Workshop on Logic, Rationality and Interaction held during October 10–14, 2011 in Guangzhou, P.R. China. There were 52 submissions. Each submission was reviewed by at least two programme committee members. The committee decided to accept 25 full papers and 12 abstracts for poster presentation. The First International Workshop on Logic, Rationality and Interaction (LORI-I) was initially proposed by Dutch and Chinese logicians; it took place in Beijing in August 2007, with participation by researchers from the ﬁelds of artiﬁcial intelligence, game theory, linguistics, logic, philosophy, and cognitive science. The workshop led to great advances in mutual understanding, both academically and culturally, between Chinese and non-Chinese logicians. The Second International Workshop on Logic, Rationality and Interaction (LORI-II) took place in Chongqing, China, during October 6–11, 2009. The programme of these previous LORI workshops can be found at http://www.golori.org/, a web portal for the research community on logic and rational interaction. The LORI-3 workshop followed the theme of previous LORI events and mainly centered on logical approaches to knowledge representation, game theory, linguistics, and cognitive science. More than half of the papers focused on modelling and reasoning about knowledge and belief, another one-third covered game theory and related matters. LORI-3 took place on the south campus of Sun Yat-sen University, Guangzhou, P.R. China, and was hosted by the Institute of Logic and Cognition (ILC). The ILC is one of the key research institutes of the Ministry of Education of China, and is dedicated to exploring the intersection between logic, cognitive science and computer science. The institute also contributed a lot to promoting cooperations and academic studies between Chinese and non-Chinese logicians, gaining increasing reputation in the international communities. The programme chairs of LORI-3 are greatly in debt to the support of ILC in making this workshop happen. In particular, we are indebted to ILC member Yuping Shen, who single-handedly produced these proceedings. We further wish to acknowledge the continuous support of LORI standing committee members Fenrong Liu and Johan van Benthem. Finally, we acknowledge the use of EasyChair, with its wonderful facility to check LNCS style ﬁle compliance and assist in the production of the proceedings. This greatly reduced our work in publishing the programme. July 2011

Hans van Ditmarsch J´erˆome Lang Shier Ju

Organization

Programme Committee Guillaume Aucher Patrick Blackburn Richard Booth Mihir Chakraborty James Delgrande C´edric D´egremont Hans van Ditmarsch Jan van Eijck Ulle Endriss Nina Gierasimczuk Sven Ove Hansson Andreas Herzig Brian Hill John Horty David Janin Shier Ju Mamoru Kaneko Willem Labuschagne J´erˆome Lang Fangzhen Lin Fenrong Liu Weiru Liu Emiliano Lorini Pierre Marquis Guo Meiyun Eric Pacuit Gabriella Pigozzi Francesca Poggiolesi Hans Rott Jeremy Seligman Yuping Shen Sonja Smets Leon van der Torre

University of Rennes 1/INRIA, France INRIA, Lorraine, France ILIAS, University of Luxembourg Jadavpur University, India Simon Fraser University, Canada University of Groningen, The Netherlands University of Seville, Spain CWI, The Netherlands ILLC, University of Amsterdam, The Netherlands University of Groningen, The Netherlands Royal Institute of Technology, Stockholm, Sweden IRIT-CNRS, France HEC Paris, France University of Maryland, USA LaBRI, Universit´e de Bordeaux I, ENSEIRB, France Sun Yat-sen University, China University of Tsukuba, Japan University of Otago, New Zealand LAMSADE-CNRS, France Hong Kong University of Science and Technology Tsinghua University, China Queen’s University Belfast, UK IRIT-CNRS, France CRIL-CNRS and Universit´e d’Artois, France South-West University, China Tilburg University, The Netherlands LAMSADE - Universit´e Paris-Dauphine, France VUB, Belgium University of Regensburg, Germany The University of Auckland, New Zealand Sun Yat-sen University, China University of Groningen, The Netherlands ILIAS, University of Luxembourg

VIII

Organization

Minghui Xiong Tomoyuki Yamada Mingyi Zhang Beihai Zhou

Sun Yat-sen University, China Hokkaido University, Japan Guizhou Academy of Science, China Peking University, China

Additional Reviewers Enqvist, Sebastian Horty, John Kooi, Barteld Liu, Hu Ma, Jianbing Ma, Minghui Parent, Xavier Rodenh¨auser, Ben

Schulte, Oliver Simon, Sunil Easaw Van Benthem, Johan Vesic, Srdjan Wang, Yanjing Wang, Yisong Wen, Xuefeng Wu, Maonian

Table of Contents

Logical Dynamics of Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Johan van Benthem and Eric Pacuit

1

Dynamic Restriction of Choices: Synthesis of Societal Rules . . . . . . . . . . . Soumya Paul and R. Ramanujam

28

Agreeing to Disagree with Limit Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . Christian W. Bach and J´er´emie Cabessa

51

A Semantic Model for Vague Quantiﬁers Combining Fuzzy Theory and Supervaluation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ka Fat Chow

61

An Application of Model Checking Games to Abstract Argumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Davide Grossi

74

Schematic Validity in Dynamic Epistemic Logic: Decidability . . . . . . . . . . Wesley H. Holliday, Tomohiro Hoshi, and Thomas F. Icard III

87

Knowledge and Action in Semi-public Environments . . . . . . . . . . . . . . . . . . Wiebe van der Hoek, Petar Iliev, and Michael Wooldridge

97

Taking Mistakes Seriously: Equivalence Notions for Game Scenarios with Oﬀ Equilibrium Play . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alistair Isaac and Tomohiro Hoshi

111

Update Semantics for Imperatives with Priorities . . . . . . . . . . . . . . . . . . . . Fengkui Ju and Fenrong Liu

125

A Measure of Logical Inference and Its Game Theoretical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mamoru Kaneko and Nobu-Yuki Suzuki

139

Partial Semantics of Argumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beishui Liao and Huaxin Huang

151

A Dynamic Logic of Knowledge, Graded Beliefs and Graded Goals and Its Application to Emotion Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emiliano Lorini

165

DEL Planning and Some Tractable Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . Benedikt L¨ owe, Eric Pacuit, and Andreas Witzel

179

X

Table of Contents

Mathematics of Public Announcements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minghui Ma

193

Logics of Belief over Weighted Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . Minghui Ma and Meiyun Guo

206

Game Semantics for the Geiger-Paz-Pearl Axioms of Independence . . . . . Pavel Naumov and Brittany Nicholls

220

Algebraic Foundations for Inquisitive Semantics . . . . . . . . . . . . . . . . . . . . . . Floris Roelofsen

233

A Dynamic Analysis of Interactive Rationality . . . . . . . . . . . . . . . . . . . . . . . Eric Pacuit and Olivier Roy

244

Seeing, Knowledge and Common Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . Fran¸cois Schwarzentruber

258

Measurement-Theoretic Foundations of Probabilistic Model of JND-Based Vague Predicate Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Satoru Suzuki

272

An Epistemic Logic with Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Levan Uridia and Dirk Walther

286

Minimal Revision and Classical Kripke Models: First Results . . . . . . . . . . Jonas De Vuyst

300

On Axiomatizations of PAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yanjing Wang

314

Public Announcement Logic with Distributed Knowledge . . . . . . . . . . . . . Y`ı N. W´ ang and Thomas ˚ Agotnes

328

An Alternative Logic for Knowability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xuefeng Wen, Hu Liu, and Fan Huang

342

Conditional Ought, a Game Theoretical Perspective . . . . . . . . . . . . . . . . . . Xin Sun

356

The Categorial Logic of Vacuous Components in Natural Language . . . . . Chongli Zou, Kesheng Li, and Lu Zhang

370

A Logic for Strategy Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Can Ba¸skent

382

Eﬃcient Action Extraction with Many-to-Many Relationship between Actions and Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jianfeng Du, Yong Hu, Charles X. Ling, Ming Fan, and Mei Liu

384

Table of Contents

XI

Reﬂections on Vote Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jan van Eijck, Floor Sietsma, and Sunil Simon

386

Playing Extensive Form Negotiation Games: A Tool-Based Analysis (Abstract) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sujata Ghosh, Sumit Sourabh, and Rineke Verbrugge

388

The Dynamics of Peer Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhen Liang and Jeremy Seligman

390

On Logic of Belief-Disagreement among Agents . . . . . . . . . . . . . . . . . . . . . . Tian-Qun Pan

392

Algebraic Semantics and Model Completeness for Intuitionistic Public Announcement Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mehrnoosh Sadrzadeh, Alessandra Palmigiano, and Minghui Ma

394

Bayesianism, Elimination Induction and Logical Reliability . . . . . . . . . . . . Renjie Yang and Min Tang

396

A Logic of Questions for Rational Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . Zuojun Xiong and Jeremy Seligman

398

Capturing Lewis’s “Elusive Knowledge” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhaoqing Xu

400

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

403

Logical Dynamics of Evidence Johan van Benthem1 and Eric Pacuit2 1

ILLC, University of Amsterdam and Stanford University [email protected] 2 Tilburg Institute for Logic and Philosophy of Science [email protected]

Abstract. Evidence is the underpinning of beliefs and knowledge. Modeling evidence for an agent requires a more ﬁne-grained semantics than possible worlds models. We do this in the form of “neighbourhood models”, originally proposed for weak modal logics. We show how these models support natural actions of “evidence management”, ranging from update with external new information to internal rearrangement. This perspective leads to richer languages for neighborhood semantics, including modalities for new kinds of conditional evidence and conditional belief. Using these, we indicate how one can obtain relative completeness theorems for the dynamic logic of evidence-changing actions.1

1

Introduction

Logical studies of information-driven agency tend to use standard possible-worlds models as the vehicle for information change. Thus, knowledge is often based on what is true in the set of epistemically accessible worlds, the current information range. This set presumably resulted from some process of investigation, say, as an intersection of various information ranges, but these details have disappeared. Now, in a number of areas, the need has been recognized for more ﬁnely-grained notions of information structure, where we keep track of the “reasons”, or the evidence for beliefs and other cognitive attitudes.2 One might take reasons or evidence to have linguistic structure, that can be manipulated through deliberation, inference and argumentation. In this paper, however, we explore an intermediate level, viz. that of neighborhood semantics, where evidence is recorded as a family of sets of worlds. Neighborhood models have long been a technical tool for studying weak modal logics. But here, we show how they support a notion of evidence with matching languages for attitudes based on it, as well as an array of natural actions that transform evidence. 1

2

For further technical details and more elaborate proofs of deﬁnability and completeness results, we refer to J. van Benthem & E. Pacuit, “Dynamic Logics of EvidenceBased Beliefs”, to appear in Studia Logica. An extended preprint is available at http://www.illc.uva.nl/Research/Reports/PP-2011-19.text.pdf. Such more ﬁne-grained ideas are found in the study of belief revision ([29], [18], [12]), conditionals ([17,40]), scientiﬁc theories ([35]), topological models for knowledge ([22]), and sensor-based models of information-driven agency in AI ( [33]).

H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 1–27, 2011. c Springer-Verlag Berlin Heidelberg 2011

2

J. van Benthem and E. Pacuit

Our paper is a programmatic ﬁrst step. We state the basics of neighborhood models for evidence and belief, and the logics that they support. Then we move to our main theme, showing how these models support natural actions of “evidence management”, from dealing with external new information to internal rearrangement. A dynamic analysis of recursion laws for such actions then feeds back into the design of new static neighborhood logics. For further perspective, we compare this picture with that provided by current plausibility models for belief. We indicate some further directions at the end.

2

Evidence Models

Our semantics of evidence is based on neighborhood models (cf. [6, Chapter 7], [14,24,15]). We use ﬁnite models, though most of our results generalize to inﬁnite settings. For convenience, we will discuss single-agent models only, though evidence coming from many sources has a clear “social” character. 2.1

Neighborhood Models

Let W be a set of possible worlds, one of which represents the actual situation. An agent gathers evidence about the actual world from a variety of sources. To simplify things, we assume these sources provide evidence in the form of subsets of W , which may, but need not, contain the actual world. We impose a few intuitive constraints: – No evidence set is empty (evidence per se is never contradictory), – The whole universe W is an evidence set (agents know their ‘space’).3 Definition 1 (Evidence Model). An evidence model is a tuple M = W, E, V with W a non-empty set of worlds, E ⊆ W × ℘(W ) an evidence relation, and V : At → ℘(W ) a valuation function. A pointed evidence model is a pair M, w with “actual world” w. When E is a constant function, we get a uniform evidence model M = W, E, V , w with E the ﬁxed family of subsets of W related to each state by E. We write E(w) for the set {X | wEX}. The above two constraints on the evidence function then become: (Cons) For each state w, ∅ ∈ E(w). (Triv) For each state w, W ∈ E(w). Example 1. To illustrate this structure, consider two worlds W = {w, v}, with p true at w and not at v. The following might be evidential states: 3

An additional often-found property is monotonicity: “If an agent i has evidence X and X ⊆ Y , then i has evidence Y .” To us, this is a property of propositions supported by evidence, not of evidence itself. We will model this feature diﬀerently later on, as a valid principle of our logic.

Logical Dynamics of Evidence

w

v

There is no evidence for or against p. w

w

v

There is evidence that supports p.

v

There is evidence that rejects p.

3

w

v

There is evidence that supports p and also evidence that rejects p.

In what follows, we shall mainly work with uniform evidence models. While this may seem very restrictive, the reader will soon see how much relevant structure can be found even at this level. Note that, even though evidence pieces are non-empty, their combination through the obvious operation of taking intersections need not yield consistent evidence: we allow for disjoint sets. But even though an agent may not be able to consistently combine all of her evidence, there will be maximal collections that she can safely put together: Definition 2 (Maximal consistent evidence). A family X of subsets of W has the finite intersection property (f.i.p.) if X = ∅. X has the maximal f.i.p. if X has the f.i.p. but no proper extension of X does. We will now develop the logic of this framework. Clearly, families of sets give us more detail than information states with just sets of (accessible) worlds. 2.2

A Static Logic of Evidence and Belief

To make a connection with familiar systems of reasoning, we ﬁrst introduce a basic logic for reasoning about evidence and beliefs. Language of evidence and belief Definition 3 (Evidence and Belief Language). Let At be a set of atomic propositions. L0 is the smallest set of formulas generated by the grammar p | ¬ϕ | ϕ ∧ ψ | Bϕ | ϕ | Aϕ where p ∈ At. Additional propositional connectives (∧, →, ↔) are deﬁned as usual, and the existential modality Eϕ is deﬁned as ¬A¬ϕ.

4

J. van Benthem and E. Pacuit

The interpretation of ϕ is “the agent has evidence that implies ϕ” (the agent has “evidence for” ϕ”) and Bϕ says that “the agents believes that ϕ”. We include the universal modality (Aϕ: “ϕ is true in all states”) for convenience. One can also think of this as a form of knowledge. Having evidence for ϕ need not imply belief. In order to believe a proposition ϕ, an agent must consider all her evidence for or against ϕ. To model the latter scenario, we will make use of Deﬁnition 2. Semantics We now interpret this language on neighborhood models. Definition 4 (Truth). Let M = W, E, V be an evidence model. Truth of a formula ϕ ∈ L0 is deﬁned inductively as follows: M, w |= p iﬀ w ∈ V (p) (for all p ∈ At) M, w |= ¬ϕ iﬀ M, w |= ϕ M, w |= ϕ ∧ ψ iﬀ M, w |= ϕ and M, w |= ψ M, w |= ϕ iﬀ there is an X with wEX and for all v ∈ X, M, v |= ϕ M, w |= Bϕ iﬀ for each maximal f.i.p. family X ⊆ E(w) and for all worlds v ∈ X , M, v |= ϕ – M, w |= Aϕ iﬀ for all v ∈ W , M, v |= ϕ – – – – –

The truth set of ϕ is the set of worlds [[ϕ]]M = {w | M, w |= ϕ}. The standard logical notions of satisfiability and validity are deﬁned as usual. Various extensions to the above modal language make sense. For instance, our notion of belief is cautious, quantifying over all maximal f.i.p’s. But we might also say that an agent “boldly believes ϕ” if there is some maximal f.i.p. X in the current evidence set with X ⊆ [[ϕ]]. We will discuss such extensions below. 2.3

Conditional Belief and Conditional Evidence

Our language still lacks some basic features of many logics of belief. Anticipating the evidence dynamics of Section 4, we now introduce conditional belief and evidence: B ϕ ψ and ϕ ψ to obtain the language L1 .4 Conditional evidence. The interpretation of ϕ ψ is “the agent has evidence that ψ is true conditional on ϕ being true”. Now, when conditioning on ϕ one may have evidence X inconsistent with ϕ. Thus, we cannot simply intersect each piece of evidence with the truth set of ϕ. We say that X ⊆ W is consistent with ϕ if X ∩ [[ϕ]]M = ∅. Then we deﬁne: – M, w |= ϕ ψ iﬀ there is an evidence set X ∈ E(w) which is consistent with ϕ such that for all worlds v ∈ X ∩ [[ϕ]]M , M, v |= ϕ. It is easy to see that ϕ ψ is not equivalent to (ϕ → ψ). No deﬁnition with absolute evidence modalities works, as can be shown by bisimulation methods. 4

We can deﬁne absolute belief and evidence: Bϕ := B ϕ and ϕ := ϕ.

Logical Dynamics of Evidence

5

Conditional belief. Conditional belief (B ϕ ψ) involves “relativizing” an evidence model to the formula ϕ. Some of the agent’s current evidence may be inconsistent with ϕ (i.e., disjoint with [[ϕ]]M ). Such evidence must be “ignored”: Definition 5 (Relativized maximal overlapping evidence). Let X ⊆ W . Given a family X of subsets of W , the relativization X X is the set {Y ∩ X | Y ∈ X }. We say thata family X has the finite intersection property relative to X (X-f.i.p.) if X X = ∅. X has the maximal X-f.i.p. if X has X-f.i.p. and no proper extension X of X has the X-f.i.p. When X is the truth set of formula ϕ, we write “maximal ϕ-f.i.p.” for “maximal [[ϕ]]M -f.i.p.” and so on. Now we deﬁne conditional belief: – M, w |= B ϕ ψ iﬀ foreach maximal ϕ-f.i.p. family X ⊆ E(w), for each world v ∈ X ϕ , M, v |= ψ While this base language of evidence models looks rich already, it follows familiar patterns. However, there are further natural evidence modalities, and they will come to light through our later analysis of operations that change current evidence. The latter dynamics is in fact the main topic of this paper, but we ﬁrst explore its static base logic a bit further.

3

Some Logical Theory: Axiomatization and Definability

Axiomatizing valid inference While complete logics for reasoning about evidence are not our main concern, we do note a few facts. Fact 1. (i) A satisﬁes all laws of modal S5, B satisﬁes all laws of KD, and satisﬁes only the principles of the minimal “classical” modal logic: the rule of upward monotonicity holds (“from a theorem ϕ → ψ, infer ϕ → ψ”.), but conjunction under the modality: (ϕ ∧ ψ) → (ϕ ∧ ψ) fails. (ii) The following operator connections are valid, but no other implications hold: Bϕ Aϕ

Eϕ ϕ

Verifying these assertions is straightforward. Over our special class of uniform evidence models, we can say much more. First note that the following are now valid: Bϕ → ABϕ

and

ϕ → Aϕ.

It follows easily that belief introspection is trivially true, as reﬂected in: ϕ ↔ Bϕ

and

¬ϕ ↔ B¬ϕ

These observations suggest the following more general observation:

6

J. van Benthem and E. Pacuit

Proposition 1. On uniform evidence models, each formula of L0 is equivalent to a formula with modal operator depth 1. Axiomatizing the complete logic of our models seems quite feasible, though the combination of a standard modality B and a neighborhood modality poses some interesting problems. As for the conditional variants, their logic shows analogies with logics of conditionals, and indeed, some of our later recursion axioms for eﬀects of evidence change suggest interesting analogies (and dis-analogies) with principles of conditional logic. Model theory and deﬁnability. Moving from deductive power to expressive power, analyzing deﬁnability in our language requires a matching notion of bisimulation. A natural notion to start with is “monotonic bisimulation” for neighbourhood semantics [15,14] and game logics [25]. Definition 6 (Monotonic bisimulation). Let M1 = W1 , E1 , V1 and M2 = W2 , E2 , V2 be two evidence models. A non-empty relation Z ⊆ W1 × W2 is a bisimulation if, for all worlds w1 ∈ W1 and w2 ∈ W2 : Prop If w1 Zw2 , then for all p ∈ At, p ∈ V1 (w1 ) iﬀ p ∈ V2 (w2 ). Forth If w1 Zw2 , then for each X ∈ E1sup (w1 ) there is a X ∈ E2sup (w2 ) such that for all x ∈ X , there is a x ∈ X such that xZx . Back If w1 Zw2 , then for each X ∈ E2sup (w2 ) there is a X ∈ E1sup (w1 ) such that for all x ∈ X , there is a x ∈ X such that xZx . We write M1 , w1 ↔ M2 , w2 if there is a bisimulation Z between M1 and M2 with w1 Zw2 . A bisimulation Z is total if every world in W1 is related to at least one world in W2 , and vice versa. The sublanguage of L0 without belief modalities is invariant under total bisimulations. Thus, with respect to statements about evidential states, two evidence models are the “same” if they are neighborhood bisimilar. But interestingly, beliefs are not invariant under this notion of bisimulation. Fact 2. The belief modality is not deﬁnable with only evidence modalities. Proof. Consider the following two evidence models:

q X

p E1 = {X, Y }

q

p Y

E1 = {X}

q Z

Logical Dynamics of Evidence

7

The dashed line is a total bisimulation between the two models. Still, Bp is true in the model on the left, but not in that on the right. Finding a notion of bisimulation respecting the whole language of evidence and belief, and their later conditionalized variants, seems a natural open problem for neighborhood modal logic.

4

Evidence Dynamics

Evidence is continually aﬀected by new incoming information, and also by processes of internal re-evaluation. Our main new contribution is to show how this dynamics can be made visible on neighborhood models. Our methodology in doing so comes from recent dynamic logics of knowledge update [9,37] and belief revision [34,3], which model informational actions driving agency. Formally, these actions change current models, viewed as snapshots of an agent’s information and attitudes in some relevant process over time.5 Our neighborhood models of evidence and belief suggest a new scope for these methods in dealing with more ﬁnely-structured evidence dynamics.6 Deconstructing public announcement. For a start, consider the well-known operation of “public announcement” for a formula ϕ in a model M = W, E, V . Deﬁning this is straightforward: remove all ¬ϕ-worlds, and intersect the old evidence sets with truthϕ when consistently possible. But from the more ﬁnegrained perspective of evidence, the event !ϕ can be naturally “deconstructed” into a combination of three distinct actions: 1. Evidence addition: the agent accepts that ϕ is an “admissible” piece of evidence (perhaps on par with the other available evidence). 2. Evidence removal: the agent removes any evidence for ¬ϕ. 3. Evidence modification: the agent incorporates ϕ into each piece of evidence gathered so far, making ϕ the most important piece of evidence. Our richer evidence models allows us to study these operations individually. 4.1

Public Announcements

Definition 7 (Public Announcement). Let M = W, E, V be an evidence model and ϕ a formula. The model M!ϕ = W !ϕ , E !ϕ , V !ϕ has W !ϕ = [[ϕ]]M , for each p ∈ At, V !ϕ (p) = V (p) ∩ W !ϕ , and for all w ∈ W , E !ϕ (w) = {X | ∅ = X = Y ∩ [[ϕ]]M for some Y ∈ E(w)}. 5

6

Examples range from “hard” information provided by public announcements or public observations [26,11] to softer signals encoding diﬀerent policies of belief revision (cf. [28]) by radical or conservative upgrades of plausibility orderings. Other dynamic logics describe acts of inference or introspection that raise “awareness” [36,39], and of questions that modify the focus of a current process of inquiry [38]. Dynamic neighborhood methods have been used in game scenarios: [7,41].

8

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There is a natural matching dynamic modality [!ϕ]ψ stating that “ψ is true after the public announcement of ϕ”: (PA)

M, w |= [!ϕ]ψ iﬀ M, w |= ϕ implies M!ϕ , w |= ψ.

On evidence models, the standard recursion axioms for public announcement remain valid, yielding dynamic equations for evidence change under hard information. Here is the result, stated as a form of “relative completeness”: Theorem 3. The dynamic logic of evidence and belief under public announcement is axiomatized completely over the chosen static base logic, given the usual rulse of Necessitation and Replacement of Provable Equivalents, by (a) the minimal modal logic for the separate dynamic modalities, (b) the following set of recursion axioms: Table 1. Public Announcement Recursion Axioms (P A1)

[!ϕ]p

↔ (ϕ → p)

(p ∈ At)

(P A2)

[!ϕ](ψ ∧ χ) ↔ ([!ϕ]ψ ∧ [!ϕ]χ)

(P A3)

[!ϕ]¬ψ

↔ (ϕ → ¬[!ϕ]ψ)

(P A4)

[!ϕ]ψ

↔ (ϕ → ϕ [!ϕ]ψ)

(P A5)

[!ϕ]Bψ

↔ (ϕ → B ϕ [!ϕ]ψ)

(P A6)

[!ϕ]α ψ

↔ (ϕ → ϕ∧[!ϕ]α [!ϕ]ψ)

(P A7)

[!ϕ]B α ψ

↔ (ϕ → B ϕ∧[!ϕ]α [!ϕ]ψ)

(P A8)

[!ϕ]Aψ

↔ (ϕ → A[!ϕ]ψ)

Proof. We only verify P A6 as a typical example. Let M = W, E, V be an evidence model. Suppose for simplicity that M, w |= ϕ. Then we get M, w |= [!ϕ]α ϕ iﬀ M!ϕ , w |= α ϕ iﬀ there is X ∈ E !ϕ (w) compatible with [[α]]M!ϕ such that X ∩ [[α]]M!ϕ ⊆ [[ψ]]M!ϕ (note [[ψ]]M!ϕ = [[[!ϕ]ψ]]M and [[α]]M!ϕ = [[[!ϕ]α]]M )

iﬀ there is X ∈ E !ϕ (w) compatible with [[[!ϕ]α]]M such that X ∩ [[[!ϕ]α]]M ⊆ [[[!ϕ]ψ]]M (note that X = Y ∩ [[ϕ]]M for some Y ∈ E(w))

iﬀ there is Y ∈ E(w) compatible with [[ϕ ∧ [!ϕ]α]]M such that X ∩ [[ϕ ∧ [!ϕ]α]]M ⊆ [[[!ϕ]ψ]]M iﬀ M, w |= ϕ∧[!ϕ]α[!ϕ]ψ.

Logical Dynamics of Evidence

4.2

9

Evidence Addition

Next consider the ﬁrst component in our earlier deconstruction. Definition 8 (Evidence Addition). Let M = W, E, V be an evidence model, and ϕ a formula in L1 .7 The model M+ϕ = W +ϕ , E +ϕ , V +ϕ has W +ϕ = W , V +ϕ = V and for all w ∈ W , E +ϕ (w) = E(w) ∪ {[[ϕ]]M }.

This operation can be described explicitly with a dynamic modality [+ϕ]ψ stating that “ψ is true after ϕ is accepted as an admissible piece of evidence”: (EA)

M, w |= [+ϕ]ψ iﬀ M, w |= Eϕ implies M+ϕ , w |= ψ.

Here, since evidence sets are non-empty, the precondition is that ϕ is true at some state. By contrast, public announcement required that ϕ be true. To capture evidence change, we want to ﬁnd “dynamic equations” that describe the eﬀect of its action on models. Here are a few preliminary notions: Definition 9 (Compatible/Incompatible). Let M = W, E, V be an evidence model, X ⊆ E(w) a family of evidence sets, and ϕ a formula: 1. X is maximally ϕ-compatible provided ∩X ∩ [[ϕ]]M = ∅ and no proper extension X of X has this property; and 2. X is incompatible with ϕ if there are X1 , . . . , Xn ∈ X such that X1 ∩ · · · ∩ Xn ⊆ [[¬ϕ]]M . Maximal ¬ϕ-compatibility need not imply incompatibility with ϕ. Next, we rephrase our deﬁnition of conditional belief, in a new notation: M, w |= B +ϕ ψ iﬀ foreach maximally ϕ-compatible X ⊆ E(w), X ∩ [[ϕ]]M ⊆ [[ψ]]M But we also need a new conditional belief operator, based on incompatibility: M, w |= B −ϕ ψ iﬀ for all maximal f.i.p., if X is incompatible with ϕ then X ⊆ [[ψ]]M Now, here is the axiom for belief after evidence addition that we are after: Lemma 1. [+ϕ]Bψ ↔ Eϕ → (B +ϕ [+ϕ]ψ ∧ B −ϕ [+ϕ]ψ) is valid. Proof. Let M = W, E, V be an evidence model and ϕ a formula with [[ϕ]]M = ∅. We ﬁrst note the following facts: 1. X ⊆ E(w) is maximally ϕ-compatible iﬀ X ∪{[[ϕ]]M } ⊆ E +ϕ (w) is a maximal f.i.p. family of sets. 7

Eventually, we can even allow formulas from our dynamic evidence logics themselves.

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J. van Benthem and E. Pacuit

2. X ⊆ E(w) is a maximal f.i.p. that is incompatible with ϕ iﬀ X ⊆ E +ϕ (w) is a maximal f.i.p. that does not contain [[ϕ]]M . The proof of both facts follows by noting that E(w) ⊆ E +ϕ (w), while any X that is a maximal f.i.p. in E +ϕ (w) but not in E(w) must contain [[ϕ]]M . Now suppose that M, w |= [+ϕ]Bψ. Then, (∗) for all maximal f.i.p. X ⊆ E +ϕ (w), we have X ⊆ [[ψ]]M+ϕ We must show M, w |= B +ϕ [+ϕ]ψ ∧ B −ϕ [+ϕ]ψ. To see that the left conjunct is true, let X ⊆ E(w) be any maximally ϕ-compatible collection of evidence. By (1), X ∪ {[[ϕ]]M } ⊆ E +ϕ (w) is a maximal f.i.p. set. Then, we have X ∩ [[ϕ]]M = (X ∪ {[[ϕ]]M }) ⊆ [[ψ]]M+ϕ = [[[+ϕ]ψ]]M where the inclusion comes from (∗). Since X was an arbitrary maximally ϕcompatible set, we have M, w |= B +ϕ [+ϕ]ψ. For the right conjunct, let X ⊆ E(w) be any maximal f.i.p. set incompatible with ϕ. By (2), X ⊆ E +ϕ (w) is a maximal f.i.p. (not containing [[ϕ]]M ). Again by (∗), X ⊆ [[ψ]]M+ϕ = [[[+ϕ]ψ]]M Hence, since X was an arbitrary maximal f.i.p. subset of E(w) incompatible with ϕ, we have M, w |= B −ϕ [+ϕ]ψ. This shows that [+ϕ]Bψ → B +ϕ [+ϕ]ψ ∧ B +ϕ [+ϕ]ψ is valid. Suppose now that M, w |= B +ϕ [+ϕ]ψ ∧ B +ϕ [+ϕ]ψ. Then A. For all maximally ϕ-compatible X ⊆ E(w), we have X ∩[[ϕ]] M ⊆ [[[+ϕ]ψ]]M ; B. For all maximally f.i.p. X ⊆ E(w) incompatible with ϕ, X ⊆ [[[+ϕ]ψ]]M . We must show M+ϕ , w |= Bψ. Let X ⊆ E +ϕ (w) be a maximal f.i.p. set. There are two cases to consider. First, [[ϕ]]M ∈ X . Then, by (1), X − {[[ϕ]]M } ⊆ E(w) is maximally ϕ-compatible. Furthermore, by (A) we have X = (X − {[[ϕ]]M }) ∩ [[ϕ]]M ⊆ [[[+ϕ]ψ]]M = [[ψ]]M+ϕ The second case is [[ϕ]]M ∈ X . Then by (2), X ⊆ E(w) is a maximal f.i.p. that is incompatible with ϕ. By (B), we have X ⊆ [[[+ϕ]ψ]]M = [[ψ]]M+ϕ In either case,

X ⊆ [[ψ]]M+ϕ ; hence, M+ϕ , w |= Bψ, as desired.

This proof will suﬃce to show that analyzing evidence changes is non-trivial. We had to come up with a new notion of conditional belief.8 8

In particular, the reader may verify that the new B −ϕ ψ is not the same as the conditional belief B +¬ϕψ.

Logical Dynamics of Evidence

11

Language Extension. But we are not yet done. We have now extended the base language, and hence, we need complete recursion axioms for the new conditional beliefs after evidence addition – hopefully, avoiding an inﬁnite regress. Let L2 be the smallest set of formulas generated by the following grammar: p | ¬ϕ | ϕ ∧ ψ | ϕ | B ϕ,ψ χ | Aϕ where p ∈ At and ϕ is any ﬁnite sequence of formulas from the language.9 Definition 10 (Truth for L2 ). We only deﬁne the new modal operator: ϕ,ψ M, w |= B χ iﬀ for all maximally ϕ-compatible sets X ⊆ E(w), if X ∩ [[ϕ]]M ⊆ [[ψ]]M , then X ∩ [[ϕ]]M ⊆ [[χ]]M

Note that we can deﬁne B +ϕ as B ϕ, and B −ϕ as B ,¬ϕ . Theorem 4. The dynamic logic of evidence addition is axiomatized completely by (a) the static base logic of evidence models for the extended language, (b) the minimal modal logic for each separate dynamic modality, and (c) the following set of recursion axioms: Table 2. Evidence Addition Recursion Axioms ↔ (Eϕ → p)

(p ∈ At)

(EA1)

[+ϕ]p

(EA2)

[+ϕ](ψ ∧ χ) ↔ ([+ϕ]ψ ∧ [+ϕ]χ)

(EA3)

[+ϕ]¬ψ

↔ (Eϕ → ¬[+ϕ]ψ)

(EA4)

[+ϕ]ψ

↔ (Eϕ → ([+ϕ]ψ ∨ A(ϕ → [+ϕ]ψ)))

(EA5)

[+ϕ]Bψ

↔ (Eϕ → (B +ϕ [+ϕ]ψ ∧ B −ϕ [+ϕ]ψ))

(EA6)

[+ϕ]α ψ

↔ (Eϕ → ([+ϕ]α [+ϕ]ψ ∨ (E(ϕ ∧ [+ϕ]α)∧ A((ϕ ∧ [+ϕ]α) → [+ϕ]ψ))))

(EA7)

[+ϕ]B ψ,α χ ↔ (Eϕ → (B ϕ∧[+ϕ]ψ,[+ϕ]α[+ϕ]χ∧ B [+ϕ]ψ,¬ϕ∧[+ϕ]α[+ϕ]χ))

(EA8)

[+ϕ]Aψ

↔ (Eϕ → A[+ϕ]ψ)

This result shows that the static and dynamic language of evidence addition are now in “harmony”. A proof is found in the extended version of this paper. Our dynamic logic of evidence addition with its natural modalities of conditional belief is an interesting extension of standard neighborhood logic. It also ﬁts with our earlier analysis of public announcement: Fact 5. The following principle suﬃces for obtaining a complete dynamic logic of evidence addition plus public announcement: [!ϕ]B ψ,α χ ↔ B ϕ∧[!ϕ]ψ,ϕ→[!ϕ]α[!ϕ]χ 9

Absolute belief and evidence versions again arise by setting some parameters to .

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J. van Benthem and E. Pacuit

4.3

Evidence Removal

With a public announcement of ϕ, the agent also agrees to ignore states inconsistent with ϕ. The latter attitude suggests an act of evidence removal as a natural converse to addition. While “removal” has been a challenge to dynamic-epistemic logics, our richer setting suggests a natural logic. Definition 11 (Evidence Removal). Let M = W, E, V be an evidence model, and ϕ a formula in L1 . The model M−ϕ = W −ϕ , E −ϕ , V −ϕ has W −ϕ = W , V −ϕ = V and for all w ∈ W , E −ϕ (w) = E(w) − {X | X ⊆ [[ϕ]]M }. This time, the corresponding dynamic modality is [−ϕ]ψ (“after removing the evidence that ϕ, ψ is true”), deﬁned as follows: (ER)

M, w |= [−ϕ]ψ iﬀ M, w |= ¬Aϕ implies M−ϕ , w |= ψ 10

Again, we look for a dynamic recursion axiom. As with evidence addition, the analysis is not purely a passive imposition of action superstructure. Finding a total dynamic language that is in harmony again aﬀects the choice of the base language itself, and hence it is an instrument for discovering new logical structure concerning evidence. L For a start, let L− 1 extend the language L1 with the operator [−ϕ]. Proposition 2. L− 1 is strictly more expressive than L1 . Proof. Consider the two evidence models M1 = W, E1 , V and M2 = W, E2 , V : r

r

q

r

r

p

p

E1 10

q

E2

Removing the evidence for ϕ is weaker than the usual notion of contracting one’s beliefs by ϕ in the theory of belief revision [27]. It is possible to remove the evidence for ϕ and yet the agent maintains her belief in ϕ. Formally, [−ϕ]¬Bϕ is not valid. To see this, let W = {w1 , w2 , w3 } with p true only at w3 . Consider an evidence model with two pieces of evidence: E = {{w1 , w3 }, {w2 , w3 }}. The agent believes p and, since the model does not change when removing the evidence for p, [−p]Bp is true. The same is true for the model with explicit evidence for p, i.e., E = {{w1 , w3 }, {w2 , w3 }, {w3 }}.

Logical Dynamics of Evidence

13

The formula [−p](p ∨ q) of L− 1 is true in M1 but not in M2 . But no formula of L1 can distinguish M1 from M2 . To see this, note that E1sup = E2sup , while the agent has the same beliefs in both models. Adding compatibility. So far, we have looked at conditional evidence and beliefs, generalizing the usual notion to restriction and incompatibility versions. This time, we also need to look at evidence that is merely “compatible” with some relevant proposition. An agent had evidence that ψ conditional on ϕ if there is evidence consistent with ϕ such that restriction to the worlds where ϕ is true entails ψ. Our next conditional operator ϕ ψ drops the latter condition: it is true if the agent has evidence compatible with ϕ that entails ψ. In general, we include operators ϕ ψ where ϕ is a sequence of formulas. The intended interpretation is that “ψ is entailed by some admissible evidence compatible with each of ϕ”. Definition 12 (Compatible evidence). Let M = W, E, V be an evidence model and ϕ = (ϕ1 , . . . , ϕn ) a ﬁnite sequence of formulas. A subset X ⊆ W is compatible with ϕ if, for each ϕi , X ∩ [[ϕi ]]M = ∅. Truth of a matching new formula ϕ ψ is then deﬁned as follows: M, w |= ϕ ψ iﬀ some X ∈ E(w) compatible with ϕ has X ⊆ [[ψ]]M This new operator gives us a very natural reduction axiom for :

11

Fact 6. The formula [−ϕ]ψ ↔ (¬Aϕ → ¬ϕ [−ϕ]ψ) is valid. Proof. Let M = W, E, V be an evidence model with [[ϕ]]M = W (otherwise, for all w, E −ϕ (w) = ∅). We show that [−ϕ]ψ ↔ ¬ϕ [−ϕ]ψ is valid on M. Let w ∈ W . The key observation is that for all X ⊆ W , X ∈ E −ϕ (w) iﬀ X ∈ E(w) and X is compatible with ¬ϕ. Then we get M, w |= [−ϕ]ϕ iﬀ M−ϕ , w |= ϕ iﬀ there is a X ∈ E −ϕ (w) such that X ⊆ [[ψ]]M−ϕ (note that [[ψ]]M−ϕ = [[[−ϕ]ψ]]M ) iﬀ there is a X ∈ E(w) compatible with ¬ϕ such that X ⊆ [[[−ϕ]ψ]]M iﬀ M, w |= ¬ϕ [−ϕ]ψ. Note how this principle captures the logical essence of evidence removal. But as before, we are not done yet. We also need a reduction axiom for our new operator ϕ . This can be stated in the same style. But we are not done even then. With the earlier conditional evidence present as well, we need an operator α ϕ ψ saying there is evidence compatible with ϕ and α such that the restriction of that evidence to α entails ψ. We also need one more adjustment: 11

The precondition is needed because the set of all worlds W is an evidence set.

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J. van Benthem and E. Pacuit

Definition 13 (Compatibility evidence - set version). A maximal f.i.p. set X is compatible with a sequence of formulas ϕ provided for each X ∈ X , X is compatible with ϕ. Language and dynamic logic We are now ready to prceed. Let L3 be the set of formulas generated by the following grammar: p | ¬ϕ | ϕ ∧ ψ | Bϕα ψ | α ϕ ψ | Aϕ where p ∈ At and ϕ is any ﬁnite sequence of formulas from the language.12 Definition 14 (Truth of L3 ). We only deﬁne the new modal operators: – M, w |= α ϕ ψ iﬀ there exists a set X ∈ E(w) compatible with ϕ, α such that X ∩ [[α]]M ⊆ [[ψ]]M . – M, |= Bϕα ψ iﬀ for each maximal family α-f.i.p. X compatible with ϕ, w α X ⊆ [[ψ]]M . α We write α ϕ1 ,...,ϕn for (ϕ1 ,...,ϕn ) and ϕ, α for (ϕ1 , . . . , ϕn , α). Also, if ϕ = (ϕ1 , . . . , ϕn ), then we write [−ϕ]ϕ for ([−ϕ]ϕ1 , . . . , [−ϕ]ϕn ).

Theorem 7. The complete dynamic logic of evidence removal is axiomatized, over the complete logic of the static base language for evidence models as enriched above, by the following recursion axioms: Table 3. Evidence Removal Recursion Axioms

↔ (¬Aϕ → p)

(p ∈ At)

(ER1)

[−ϕ]p

(ER2)

[−ϕ](ψ ∧ χ) ↔ ([−ϕ]ψ ∧ [−ϕ]χ)

(ER3)

[−ϕ]¬ψ

↔ (¬Aϕ → ¬[−ϕ]ψ)

(ER4)

[−ϕ]α χ ψ

↔ (¬Aϕ → [−ϕ]ψ,¬ϕ [−ϕ]χ)

(ER5)

[−ϕ]Bψα χ

↔ (¬Aϕ → B[−ϕ]ψ,¬ϕ [−ϕ]χ)

(ER6)

[−ϕ]Aψ

↔ (¬Aϕ → A[−ϕ]ψ)

[−ϕ]α

[−ϕ]α

Proof. We only do axiom ER5. Let M = W, E, V be an evidence model, w ∈ W and [[ϕ]]M = W . First of all, the key observation in the proof of Fact 6 extends to sets of evidence sets (cf. Deﬁnition 13). That is, for all worlds w, X ⊆ E −ϕ (w) is compatible with ψ iﬀ X ⊆ E(w) is compatible with [−ϕ]ψ, ¬ϕ. Next, for all states w, X ⊆ E −ϕ (w) is a maximal α-f.i.p. iﬀ X ⊆ E(w) is a maximal [−ϕ]α-f.i.p. compatible with ¬ϕ.13 Then we calculate: 12 13

Absolute belief and evidence versions again arise by setting some parameters to . The compatibility with ¬ϕ is crucial: it is not true that every X ⊆ E −ϕ (w) that is a maximal α-f.i.p. corresponds to a maximal [−ϕ]α-f.i.p. subset of E(w).

Logical Dynamics of Evidence

15

M, w |= [−ϕ]Bψα χ iﬀ M−ϕ , w |= Bψα χ −ϕ iﬀ for each maximal α-f.i.p. α X ⊆ E (w) compatible with ϕ, X ⊆ [[χ]]M−ϕ = [[[−ϕ]χ]]M iﬀ for each maximal [−ϕ]α-f.i.p. X ⊆ E(w) compatible with [−ϕ]ϕ and ¬ϕ, [−ϕ]α X ⊆ [[[−ϕ]χ]]M [−ϕ]α iﬀ M, w |= Bψ,¬ϕ [−ϕ]χ. The above principles state the essence of evidence removal, as well as the beliefs one can still have after such an event. The additional insight is that removal essentially involves compatibility as well as implication between propositions – something of independent logical interest. Logics for evidence once more. This is a beginning rather than an end. Extending the base language in this manner will have repercussions for our earlier analyses. Using our style of analysis, it is possible to also ﬁnd reduction axioms for our new evidence and belief operators under actions of evidence addition and public announcement. For example, for the compatible evidence operator ψ with ψ = (ψ1 , . . . , ψn ), we have the following validities: [+ϕ]ψ χ ↔ [Eϕ → ([+ϕ]ψ [+ϕ]χ ∨ (

E(ϕ ∧ ψi ) ∧ A(ϕ → [+ϕ]ψ)))]

i=1,...,n

[!ϕ]ψ χ ↔ (ϕ → ϕ [!ϕ]χ) [!ϕ]ψ We do not include all combinations here. The key point is that the analysis is in harmony, it does not lead to further extensions of the base language. Perhaps more challenging further problems have to do with the “action algebra” of combining our three basic actions on evidence so far. What happens when we compose them? Our guess is that we need to move to an “event model” version of our logics in the style of dynamic-epistemic logic. 4.4

Evidence Modification

We have analyzed the two major operations on evidence that we can see. Nevertheless, the space of potential operations on neighborhood models is much larger, even if we impose conditions of bisimulation invariance as in process algebra (cf. [23] and [15]). Instead of exploring this wide realm, we show one new operation that might make sense. So far, we added or removed evidence. But one could also modify the existing pieces of evidence. To see, the diﬀerence, here is a new way of making some proposition ϕ highly important: Definition 15 (Evidence Upgrade). 14 Let M = W, E, V be an evidence model and ϕ a formula in L1 . The model M⇑ϕ = W ⇑ϕ , E ⇑ϕ , V ⇑ϕ has W ⇑ϕ = W , V ⇑ϕ = V , and for all w ∈ W , 14

This operation is a bit like “radical upgrade” in dynamic logics of belief change.

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J. van Benthem and E. Pacuit

E ⇑ϕ (w) = {X ∪ [[ϕ]]M | X ∈ E(w)} ∪ [[ϕ]]M . This is stronger than simply adding [[ϕ]]M as evidence, since one also modiﬁes each admissible evidence set. But it is still weaker than publicly announcing ϕ, since the agent retains the ability to consistently condition on ¬ϕ. Fact 8. The following recursion principles are valid: 1. [⇑ϕ]ψ ↔ (Eϕ → A(ϕ → [⇑ϕ]ψ)) 2. [⇑ϕ]Bψ ↔ (Eϕ → A(ϕ → [⇑ϕ]ψ)) Proof. For the second law, note that in E ⇑ϕ (w), there is only one maximal f.i.p. whose intersection is [[ϕ]]M . The ﬁrst law goes as with Fact 9 below. As these principles show, ⇑ϕ gives a very special status to the incoming information ϕ, blurring the distinction between evidence and belief. This suggests a weaker operation that modiﬁes the evidence sets in favor of ϕ, but does not add explicit support for ϕ. Deﬁne M⇑w ϕ as in Deﬁnition 15 except for setting E ⇑w ϕ (w) = {X ∪ [[ϕ]]M | X ∈ E(w)}. A simple modiﬁcation to Principle 2 in the above fact gives us a valid principle for our evidence operator. However, the case of belief poses some problems.15 Fact 9. The formula [⇑wϕ]ψ ↔ ([⇑wϕ]ψ ∧ A(ϕ → [⇑wϕ]ψ)) is valid. Proof. Let M = W, E, V be an evidence model with w ∈ W . Then, M, w |= [⇑wϕ]ψ iﬀ M⇑ϕ , w |= ψ iﬀ there is a X ∈ E ⇑ϕ (w) such that X ⊆ [[ψ]]M−ϕ (note that [[ψ]]M⇑ϕ = [[[⇑ϕ]ψ]]M ) iﬀ there is X ∈ E(w) with X ∪ [[ϕ]]M = X ⊆ [[[⇑ϕ]ψ]]M iﬀ there is X ∈ E(w) with X ⊆ [[[⇑ϕ]ψ]]M and [[ϕ]]M ⊆ [[[⇑ϕ]ψ]]M iﬀ M, w |= [⇑ϕ]ψ ∧ A(ϕ → [⇑ϕ]ψ) 4.5

From External to Internal Actions: Evidence Combination

We have now brought to light a rich repertoire of evidence-modifying actions. Still, the operations discussed above all exemplify “external evidence dynamics” responding to some outside source, where the agent reacts appropriately, either by incorporating ϕ or removing ϕ from consideration. But our neighborhood models also suggest internal operations that arise from pondering the evidence, without external triggers. We will discuss only one such internal operation in this paper, be it a basic one. One natural operation available to an agent is to combine her evidence. Of course, as we have noted, an agent’s evidence may be contradictory, so she can only combine evidence that is not inconsistent. 15

The new complication is that, without adding ϕ to the evidence sets, intersections of maximal f.i.p. sets in the upgraded model may contain more than just ϕ states.

Logical Dynamics of Evidence

17

Definition 16 (Evidence combination). Let M = W, E, V be an evidence model. The model M# = W # , E # , V # has W # = W , V # = V and for all w ∈ W , E # (w) is the smallest family of sets of worls closed under (non-empty) intersection and containing E(w). The corresponding dynamic modal operator is deﬁned as M, w |= [#]ϕ iﬀ M# , w |= ϕ. A complete study of this operation will be left for future work, since it poses some challenges to our recursive style of analysis so far.16 Nevertheless, we can observe the following interesting facts: Fact 10. The following formulas are valid on evidence models: 1. 2. 3. 4.

[#]ϕ → [#]ϕ (combining evidence does not remove old evidence17 ) B[#]ϕ ↔ [#]Bϕ (beliefs are immune to evidence combination) Bϕ → [#]ϕ (beliefs are explicitly supported after combining evidence18 For factual ϕ, Bϕ → ¬[#]¬ϕ (if an agent believes ϕ then the agent cannot combine her evidence so that there is evidence for ¬ϕ)

Proof. The proof that the ﬁrst three items are valid is left to the reader. For the fourth, note that ¬ϕ → ¬Bϕ is valid. The proof is as follows: First of all, in any evidence model M = W, E, V , every piece of evidence in X ∈ E(w) is contained in a maximal f.i.p. X ⊆ E(w) (models are ﬁnite, so simply ﬁnd the maximal f.i.p. containing X which may be {X, W }). Suppose that ¬ϕ is true at a state w, then there is an X ∈ E(w) such that X ⊆ [[¬ϕ]]M . Let X be the maximal f.i.p. containing X. Hence, X ⊆ X ⊆ [[¬ϕ]]M . Therefore, Bϕ is not true at w. This shows that ¬ϕ → ¬Bϕ is valid, as desired. We can then derive Principle 3 by noting the following series of implications: Bϕ → [#]Bϕ → [#]¬¬ϕ → ¬[#]¬ϕ Here the ﬁrst implication follows from the second principle applied to factual formulas ϕ (for which ϕ ↔ [#]ϕ is valid), the second implication follows from the fact that Bϕ → ¬¬ϕ is valid (as argued above) while [#] is a normal modal operator and the third implication follows from the fact that the evidence combination operation is functional. Finally, we note that a full account of combination dynamics seems to require an additional modality of “having evidence that ϕ”, but we forego details here. 16 17

18

The problem may be that standard modal languages are too poor, forcing us upward in expressive power to hybrid or ﬁrst-order logics – but we suspend judgment here. Deﬁnition 16 assumed that always E(w) ⊆ E # (w). Thus, in the process of combination, an agent does not notice inconsistencies present in her evidential state. A deeper analysis would include acts of removing observed inconsistencies. The converse is not valid. In fact, one can read the combination [#] as an existential version of our belief operator. It is true if there is some maximal collection of evidence whose intersection implies ϕ. In plausibility models for doxastic logic, this says that ϕ is true throughout some maximal cluster. This notion of belief is much riskier then Bϕ, and again we encounter the variety of agent attitudes mentioned in Section 2.

18

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Comparison with Plausibility Models

In this section, we will contrast our neighborhood models with another modal framework for belief change. This excursion (which can be skipped without loss of coherence) throws new light on our proposals. We merely state some main notions and results, referring to the extended version of this paper for proofs. Plausibility models. Originally used as a semantics for conditionals (cf. [19]), the following idea is wide-spread in modal logics of belief [34,37,3,12]. One endows epistemic ranges with an ordering w v of relative plausibility on worlds (usually uniform across epistemic equivalence classes): “according to the agent, world v is at least as plausible as w”.19 Plausibility orders are typically assumed to be reﬂexive and transitive, and often also connected, making every two worlds comparable. In our discussion, we will allow pre-orders with incomparable worlds. Definition 17 (Plausibility model). A plausibility model is a tuple M = W, , V where W is a ﬁnite nonempty set, ⊆ W × W is a reﬂexive and transitive ordering on W , and V : At → ℘(W ) is a valuation function. If is also connected (for each w, v ∈ W , either w v or v w) then we say M is a connected plausibility model. A pair M, w where w is a state is called a pointed (connected) plausibility model. Language and logic. Plausibility models interpret a standard doxastic language. Let L be the smallest set of formulas generated by the following language p | ¬ϕ | ϕ ∧ ψ | B ϕ ψ | []ϕ | Aϕ As usual, Bϕ is deﬁned as B ϕ. For X ⊆ W , let M in (X) = {v ∈ X | v w for all w ∈ X } Given a set X, M in (X) is the set of most plausible worlds in X (minimal elements of X in the plausibility order). We only deﬁne the modal operators – M, w |= B ϕ ψ iﬀ M in ([[ϕ]]M ) ⊆ [[ψ]]M – M, w |= []ϕ iﬀ for all v ∈ W , if v w then M, v |= ϕ – M, w |= Aϕ iﬀ for all v ∈ W , M, v |= ϕ. In particular, any pre-order forms a partial order of “clusters”, maximal subsets where the relation is universal. A ﬁnite pre-order has one or more ﬁnal clusters, not having any proper successors. (Connected order have only one ﬁnal cluster.) Belief means truth in all ﬁnal clusters. The logic of this system is basically the minimal conditional logic over pre-orders that we encountered before. Instead of pursuing it, we make some comments on deﬁnability. Plausibility orders are binary relations supporting a standard modal language. Indeed, as was noted by [5], on ﬁnite models, belief and conditional belief are deﬁnable in the language with A and [] only: 19

In conditional logic, plausibility or “similarity” is a world-dependent ternary order.

Logical Dynamics of Evidence

19

Fact 11. Belief and conditional belief can be explicitly deﬁned as follows: – Bϕ := A[]ϕ – B ϕ ψ := A(ϕ → (ϕ ∧ [](ϕ → ψ))) While the plausibility modality may look like a technical device, [3] interpret []ϕ as “a safe belief in ϕ”. Following [32], they show that this amounts to the beliefs the agent retains under all new true information about the actual world.20 This simple modal language over plausibility models will turn out to be a natural limit of expressive power. Dynamics on plausibility models. Plausibility models support a dynamics of informational action through model change. Belief change under hard information. One paradigmatic action was discussed in Section 4.1. “Hard information” reduces current models to deﬁnable submodels: Definition 18 (Public announcement - plausibility models). Let M = W, , V be a plausibility model. The model M!ϕ = W !ϕ , !ϕ , V !ϕ has W = [[ϕ]]M , for all p ∈ At, V !ϕ (p) = V (p) ∩ W !ϕ and !ϕ = ∩(W !ϕ × W !ϕ ). Dynamic logics exist that describe belief change under such events of new hard information, i.e., the logical laws governing [!ϕ]Bψ. The crucial recursion axioms for belief are the same as those for evidence models in Section 4.1: [!ϕ]Bψ ↔ (ϕ → B ϕ [!ϕ]ψ) ϕ∧[!ϕ]ψ

[!ϕ]Biψ χ ↔ (ϕ → Bi

[!ϕ]χ)

Public announcement assumes that agents treat the source of the new information as infallible. But in many scenarios, agents trust the source of the information only up to a point. This calls for softer announcements. Here are a few examples: [37, Chapter 7] and [4] have much more extensive discussion. Belief change under soft information. How to incorporate evidence that ϕ is true into some (epistemic-)doxastic model M? Soft announcements of a formula ϕ do not eliminate worlds, but rather modify the plausibility ordering that structures the current information state. The goal is to rearrange all states in such a way that ϕ is believed, and perhaps other desiderata are met. There are many “policies” for doing this (cf. [28]) – we only mention two basic ones. Example 2. The following picture illustrates soft update as plausibility change: B

E D

A

ϕ

C 20

For the same notion in the computational literature on agency (cf. [31]).

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One policy that has been extensively studied is radical upgrade where all ϕ worlds are moved ahead of all other worlds, while keeping the order inside these two zones the same. In the above example, the radical upgrade by ϕ would result in the ordering A ≺ B ≺ C ≺ D ≺ E. More formally, the model transformation here is relation change: Definition 12 (Radical Upgrade.) Given an epistemic-doxastic model M = W, , V and a formula ϕ, the radical upgrade of M with ϕ is the model M⇑ϕ = W ⇑ϕ , ⇑ϕ , V ⇑ϕ with W ⇑ϕ = W , V ⇑ϕ = V , where ⇑ϕ is deﬁned as follows: 1. for all x ∈ [[ϕ]]M and y ∈ [[¬ϕ]]M , set x ≺⇑ϕ y, 2. for all x, y ∈ [[ϕ]]M , set x ⇑ϕ y iﬀ x y, and 3. for all x, y ∈ [[¬ϕ]]M , set x ⇑ϕ y iﬀ x y. A logical analysis of this type of information change uses modalities [⇑ ϕ]ψ meaning “after radical upgrade of ϕ, ψ is true”, interpreted as follows: M, w |= [⇑ϕ]ψ iﬀ M⇑ϕ , w |= ψ. The crucial recursion axiom for belief change under soft information is ([34]): [⇑ϕ]B ψ χ ↔ (E(ϕ ∧ [⇑ϕ]ψ) ∧ B ϕ∧[⇑ϕ]ψ [⇑ϕ]χ) ∨ (¬E(ϕ ∧ [⇑ϕ]ψ) ∧ B [⇑ϕ]ψ [⇑ϕ]χ) This shows how revision policies as plausibility transformations give agents not just new beliefs, but also new conditional beliefs. But radical upgrade is not the only way for an agent to accept incoming information. Equally important is conservative upgrade, which lets the agent only tentatively accept the incoming information ϕ by making the best ϕ the new minimal set and keeping the old plausibility ordering the same on all other worlds. In the above picture a conservative upgrade with ϕ results in the new ordering A ≺ C ≺ D ≺ B ∪ E. This, and many other revision policies can be analyzed in the same dynamic logic style. From plausibility models to evidence models. Here is an intuitive connection. Let M = W, , V be a plausibility model: the appropriate evidence sets are the downward -closed sets of worlds. To be more precise, we ﬁx some notation: – Given a X ⊆ W , let X↓ = {v ∈ W | ∃x ∈ X and v x} (we write X↓ when it is clear which plausibility ordering is being used). – A set X ⊆ W is -closed if X↓ ⊆ X. Here is the formal deﬁnition for the above simple idea:

Logical Dynamics of Evidence

21

Definition 19 (Plausibility-Based Evidence Model). Let M = W, , V be a plausibility model. The evidence model generated from M is21 EV (M) = W, E , V with E as follows: E = {X | ∅ = X is -closed } Given a plausibility model M, the evidence model generated by the plausibility order of M satisﬁes the basic properties of Section 2: the sets are non-empty, and the whole universe is among them. But more can be said: Fact 13. The family of evidence sets of any generated model EV (M) is closed under taking intersections. Example 3. The following three plausibility models – with their induced evidence sets drawn in gray – highlight three key situations that can occur:

w

p

p

w

p

q

EV (M1 ), w |= ¬B(p ∧ q) EV (M1 ), w |= p ∧ q

q

EV (M2 ), w |= ¬B(p ∧ q) EV (M2 ), w |= ¬p ∧ ¬q

w p

p

p

p, q

p, q

EV (M3 ), w |= Bp ∧ ¬Bq EV (M3 ), w |= (p ∧ q) But not every evidence model comes from a plausibility model. Example 4. Let M be an evidence model with W = {w, v, x}, a constant evidence function with range E = {{w, v}, {v, x}} and the valuation function deﬁned by V (p) = {w, v} and V (q) = {v, x}. Note that we have M, w |= B(p ∧ q) 21

Here the set of worlds and valuation function remain as in the model M.

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(the agent believes p and q) but M, w |= p ∧ q ∧ ¬(p ∧ q) (even though there is evidence for p and evidence for q, there is no evidence for p ∧ q). Plausibility models represent a situation where the agent has already “combined” all of her evidence (cf. # in Section 3.4), as reﬂected in this property: If X, Y ∈ E and X ∩ Y = ∅ then X ∩ Y ∈ E . This connection between plausibility models and evidence models can be extended to a translation between their languages: Definition 20 (P -translation). The translation (·)P : L → L is deﬁned as: – – – –

pP = p, (¬ϕ)P = ¬ϕP , (ϕ ∧ ψ)P = ϕP ∧ ψ P , (Aϕ)P = AϕP , (ϕ)P = E[]ϕP , (ϕ ψ)P = E(ϕP ∧ [](ϕP → ψ P )), P P P P P (ϕ γ ψ) = E( i γi ∧ (ϕ ∧ [](ϕ → ψ ))),

– (B ϕ ψ)P = A(ϕP → (ϕP ∧ [](ϕP → ψ P ))), – (B ϕ,α ψ)P = A(([]αP ∧ []ϕP ) → (ϕP ∧ [](ϕP → ψ P ))), and – (Bγϕ ψ)P = A((ϕP ∧ i γiP ) → ((ϕP ∧ i γiP ) ∧ []((ϕP ∧ i γiP ) → ψ P ))). Lemma 2. Let M = W, , V be a plausibility model. For any ϕ ∈ L1 and world w ∈ W , M, w |= ϕP iﬀ EV (M), w |= ϕ From evidence models to plausibility models. Going in the opposite direction, we start with a family of evidence sets, and need to induce a natural ordering. Here one can use a ubiquitous idea, occurring in point-set topology, but also in theories of relation merge (cf. [2,20]), the so-called specialization (pre)-order: Definition 21 (Plausibility based evidence model). Suppose that M = W, E, V is an evidence model (with constant evidence function E whose range is E). The plausibility model generated by M is the structure ORD(M) = W, E , V where E is an ordering on W deﬁned as follows: w E v iﬀ ∀X ∈ E, v ∈ X implies w ∈ X 22 To make this deﬁnition more concrete, here is a simple illustration. 22

E is reﬂexive and transitive, so ORD(M) is indeed a plausibility model.

Logical Dynamics of Evidence

w1

w2

w3

w1

w4

w2

w3

w4

ORD(M)

M

w1 w1

w2

w3

23

w3

w4

w4 w2

M

ORD(M)

Our two representations are related as follows: Fact 14. (i) For all models plausibility models M, ORD(EV ((M)) = M, (ii) The identity EV (ORD(M)) = M does not hold for all evidence models M. (iii) For all evidence models M, EV (ORD(M)) = M# , where # is the combination operation of Deﬁnition 16. Translations and languages. The preceding connection again comes with a translation for modal languages, in particular for (conditional) beliefs on evidence neighborhood models and their induced plausibility models. But other notions are less easily reduced. The extended version of this paper will show how dealing with “safe belief” on plausibility orders requires a new notion of reliable evidence that extends our earlier modal evidence languages. This concludes our brief comparison of relational and neighborhood semantics for belief and evidence. We have clariﬁed their relationship as one of generalization, where neighborhood models describe one more level of detail: the internal combination stages for evidence. Even so, many of the new operations that we have found in earlier sections would also make sense as deﬁnable operators in the natural modal logic of plausibility models, and we have shown how various interesting new questions arise at this interface.

6

Conclusion and Further Directions

We have shown that evidence dynamics on neighborhood models oﬀers a rich environment for modeling information ﬂow and pursuing logical studies. Here are some avenues for further research. Some are more technical, some increase coverage. We start with the former.

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Exploring the static logic. We have found quite a few new evidence-based modalities of conditional belief. What is the complete logic of this system? This is a new question of axiomatization, that can be appreciated also outside of our dynamic perspective. One reason for its complexity may be that we are mixing a language of neighborhood-based modalities with normal operators of belief with a matching relational semantics. What is the right notion of bisimulation? Designing logical languages invites matching up with notions of structural invariance for evidence models. We have seen that standard bisimulation for neighborhood models matches modal logics with only evidence operators. But Fact 2 showed that this does not extend to the modal language with belief referring to intersections of maximally consistent families of evidence sets. And we introduced even stronger modal evidence languages in the dynamics of Section 4. What would be well-motivated stronger notions of bisimulation, respecting more evidence structure? Finally, there are some obvious technical generalizations to be made, to inﬁnite models, and also to DEL-style product update mechanisms for rich input. Reliable evidence and its sources. But one can also use our setting for modeling further phenomena. For instance, there is a natural notion of “reliable” evidence, based only on sets containing the actual world. What is the complete logic of this operator? This suggests a broader study of types of belief based on reliable evidence, in line with current trends in epistemology. But eventually, we also want explicit modeling of sources of evidence and what agents know or believe about their reliability. Social notions. We have seen that interesting evidence structure arises in the single agent case. But multi-agent scenarios are also natural: e.g., argumentation is social confrontation of evidence, which may result in new group attitudes among participants. This raises a number of interesting issues of its own. The most pressing is to ﬁnd natural notions of group evidence and belief. Here evidence structure soon takes us beyond the usual notions of group beliefs or group knowledge in the epistemic literature based on relational models. Priority structures. The evidence dynamics in this paper treats evidence sets on a par. As a consequence, removal may seem arbitrary and non-deterministic, since there is nothing in the structure of the evidence itself which directs the process. A next reasonable step would be to model levels of reliability of evidence. One natural format for this are the “priority graphs” of [2], which have already been used extensively in dynamic-epistemic logic [20,12]. These graphs provide much richer input to evidence management, and can break stalemates between conﬂicting pieces of evidence. It should be possible to extend the above framework to one with ordered evidence sets – and conversely, then, our logics may help provide something that has been missing so far: modal logics working directly on priority graphs.

Logical Dynamics of Evidence

25

Other logics of evidence. “Evidence” is a notion with many diﬀerent aspects. Our proposal has been set-theoretic and semantic, while there are many other treatments of evidence for a proposition ϕ, in terms of proofs for ϕ, or using the balance of probability for ϕ versus ¬ϕ. What we ﬁnd particularly pressing is a junction with more syntactic approaches making evidence something coded that can be operated on in terms of inference and computation. If ﬁner operational aspects of inference and introspection enter one’s notion of evidence, then the methods of this paper should be extended to include dynamic logics of awareness and inference [10,1,36,39]. Related frameworks. But also, the style of analysis in this paper should, and can, be linked up with other traditions, including the seminal work by [8] and [30] on evidence, probabilistic logics of evidence [13], or the “topologic” of [22]. And one can add more, such as the “priority graphs” inducing preference orders in [21], or the “belief base” account of belief revision in (see [16] and references therein). We intend to clarify these connections in future work.

7

Conclusion

We have made a pilot proposal for using neighborhood models as ﬁne-grained evidence structures that allow for richer representation of information than current relational models of belief. We have shown how these structures support a rich dynamics of evidence change that goes beyond current logics of belief revision. A number of relative completeness theorems identiﬁed the key dynamic equations governing this process, while also suggesting new static languages of evidence and belief. Finally, we discussed some of the interesting new issues that lie ahead, such as ﬁnding logics with priority structure and group evidence that exploit the more ﬁnely-grained neighborhood setting.

References ˚gotnes, T., Alechina, N.: The dynamics of syntactic knowledge. Journal of Logic 1. A and Computation 17(1), 83–116 (2007) 2. Andreka, H., Ryan, M., Schobbens, P.Y.: Operators and laws for combining preference relations. Journal of Logic and Computation 12(1), 13–53 (2002) 3. Baltag, A., Smets, S.: Conditional doxastic models: A qualitative approach to dynamic belief revision. In: Mints, G., de Queiroz, R. (eds.) Proceedings of WOLLIC 2006. LNCS, vol. 165, pp. 5–21 (2006) 4. Baltag, A., Smets, S.: ESSLLI (2009) course: Dynamic logics for interactive belief revision (2009), Slides available at http://alexandru.tiddlyspot.com/#%5B%5BESSLLI09%20COURSE%5D%5D 5. Boutilier, C.: Conditional Logics for Default Reasoning and Belief Revision. Ph.D. thesis, University of Toronto (1992) 6. Chellas, B.: Modal Logic: An Introduction. Cambridge University Press, Cambridge (1980) 7. Demey, L.: Agreeing to Disagree in Probabilistic Dynamic Epistemic Logic. Master’s thesis, ILLC University of Amsterdam, LDC 2010-14 (2010)

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8. Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics 38(2), 325–339 (1967) 9. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Synthese Library. Springer, Heidelberg (2007) 10. Fagin, R., Halpern, J.: Belief, awareness and limited reasoning. Artiﬁcial Intelligence 34, 39–76 (1988) 11. Gerbrandy, J.: Bisimulations on Planet Kripke. Ph.D. thesis, Institute for Logic, Language and Computation, DS-1999-01 (1999) 12. Girard, P.: Modal Logic for Belief and Preference Change. Ph.D. thesis, ILLC University of Amsterdam Dissertation Series DS-2008-04 (2008) 13. Halpern, J., Pucella, R.: A logic for reasoning about evidence. Journal of AI Research 26, 1–34 (2006) 14. Hansen, H.H.: Monotonic Modal Logic. Master’s thesis, Universiteit van Amsterdam (ILLC technical report: PP-2003-24) (2003) 15. Hansen, H.H., Kupke, C., Pacuit, E.: Neighbourhood structures: Bisimilarity and basic model theory. Logical Methods in Computer Science 5(2), 1–38 (2009) 16. Hansson, S.O.: A Textbook of Belief Dynamics. Theory Change and Database Updating. Kluwer, Dordrecht (1999) 17. Kratzer, A.: What must and can must and can mean. Linguistics and Philosophy 1, 337–355 (1977) 18. Leitgeb, H., Segerberg, K.: Dynamic doxastic logic: why, how and where to? Synthese 155(2), 167–190 (2007) 19. Lewis, D.: Counterfactuals. Blackwell Publishers, Oxford (1973) 20. Liu, F.: Reasoning about Preference Dynamics. Synthese Library, vol. 354. Springer, Heidelberg (2011) 21. Liu, F.: A two-level perspective on preference. Journal of Philosophical Logic (to appear, 2011) 22. Moss, L., Parikh, R.: Topological reasoning and the logic of knowledge. In: Moses, Y. (ed.) Proceedings of TARK IV. Morgan Kaufmann, San Francisco (1992) 23. Nicola, R.D.: Extensional equivalences for transition systems. Acta Informatica 24, 211–237 (1987), http://dx.doi.org/10.1007/BF00264365 24. Pacuit, E.: Neighborhood semantics for modal logic: An introduction (2007), ESSLLI 2007 course notes, http://ai.stanford.edu/~ epacuit/classes/ 25. Pauly, M.: Logic for Social Software. Ph.D. thesis, ILLC University of Amsterdam Dissertation Series DS 2001-10 (2001) 26. Plaza, J.: Logics of public communications. Synthese: Knowledge, Rationality, and Action 158(2), 165–179 (2007) 27. Rott, H.: Change, Choice and Inference: A Study in Belief Revision and Nonmonotonic Reasoning. Oxford University Press, Oxford (2001) 28. Rott, H.: Shifting priorities: Simple representations for 27 iterated theory change operators. In: Lagerlund, H., Lindstr¨ om, S., Sliwinski, R. (eds.) Modality Matters: Twenty-Five Essays in Honor of Krister Segerberg. Uppsala Philosophical Studies, vol. 53, pp. 359–384 (2006) 29. Segerberg, K.: Belief revision from the point of view of doxastic logic. Journal of the IGPL 3(4), 535–553 (1995) 30. Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976) 31. Shoham, Y., Leyton-Brown, K.: Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. Cambridge University Press, Cambridge (2009) 32. Stalnaker, R.: Knowledge, belief and counterfactual reasoning in games. Economics and Philosophy 12(02), 133–163 (1996)

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Dynamic Restriction of Choices: Synthesis of Societal Rules Soumya Paul and R. Ramanujam The Institute of Mathematical Sciences CIT Campus, Taramani Chennai - 600 113, India {soumya,jam}@imsc.res.in

Abstract. We study a game model to highlight the mutual recursiveness of individual rationality and societal rationality. These are games that change intrinsically based on the actions / strategies played by the players. There is an implicit player - the society, who makes actions available to players and incurs certain costs in doing so. If and when it feels that an action a is being played by a small number of players and/or it becomes too expensive for it to maintain the action a, it removes a from the set of available actions. This results in a change in the game and the players strategise afresh taking this change into account. We study the question: which actions of the players should the society restrict and how should it restrict them so that the social cost is minimised in the eventuality? We address two variations of the question: when the players are maximisers, can society choose an order of their moves so that social cost is minimised, and which actions may be restricted when players play according to given strategy speciﬁcations.1

1

Motivation

In India, which is dependent greatly on oil imports for its energy needs, petrol prices are regulated but increase periodically. Every increase leads to a round of price rise of many commodities, and hence every announced oil price increase meets with great opposition and varied systemic reaction. In response, the government rolls back prices, not to the level before the announced increase but one a little higher, and another cycle of behaviour is initiated. We see this as an example, not of a game, but a dynamic game form: the structure of the game is preserved over time, but the set of options available to players changes dynamically, and the rationale for such change is dependent on player behaviour. Social choice theory concerns itself with aggregating individual choices. On the other hand, theories of mechanism design and market design seek to drive individual choices (based on players’ preferences over outcomes) towards desired social goals. This works well for one-shot games, but there are situations 1

This paper is a follow-up of [19].

H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 28–50, 2011. c Springer-Verlag Berlin Heidelberg 2011

Dynamic Restriction of Choices: Synthesis of Societal Rules

29

where societal interventions are not designed a priori but are required during play. Many social mechanisms (especially ﬁnancial ones such as interest rates and taxes) depend on making available an initial set of options, and based on what strategies are selected by players, revising these options in the long run. Such revision of available moves is, in turn, anticipated by players, and this mutual interdependence of individual and social action inﬂuences game dynamics signiﬁcantly. In principle, such dynamic game forms are unnecessary, since all possible changes can themselves be incorporated into one big game whereby these gamechanging moves are merely additional moves, and players strategize taking these possibilities into account. However, in the context of resource bounded players such reasoning may be impossible. In the societal contexts referred to above, it would be considered wiser to not rely on computations that undertake to include all possible eﬀects but instead seek to observe player behaviour and apply course corrections. Economists are, of course, well aware of such phenomena. The availability of individual choices is, in general, determined by choices by the society as a whole, and in turn, social choices are inﬂuenced by patterns of individual choices. In this process, the set of choices may expand or contract over time. However, there is a political or philosophical value attached to availability of individual choices. A strategy sa may be justiﬁed by the presence of another option sb but if eventually sb is forced out, the rationale for sa may disappear, though sa is the only one present. In a world where all possible eventual consequences can be computed, the cost of such disappearance of choices can also be taken into account, but (as we see in the case of environment conservation) realisation typically comes post-facto. The general situation is as follows. At every stage, an individual has certain choices to make. But making a choice also comes with a cost which is associated with that choice and which the individual has to incur in making the choice. On the other hand, society also incurs a certain cost in making these choices available to individuals. This cost is a function of the choices being provided as well as the proﬁle of choices made by individuals. From time to time, based on the history of choice proﬁles and predictions of the future, society revises the choices it provides to individuals as well as the cost individuals have to incur to make these choices. This in turn has an eﬀect on individuals’ strategies, who switch between available choices. The dynamics of this back and forth process can be quite interesting and complicated. In game theoretic models of such social phenomena, social rules are considered as game forms, and individual behaviour is regulated using payoﬀs. Rule changes are considered to be exogenous, and correspond to change of payoﬀ matrices. In evolutionary game theory, rules are considered as game equilibria: individuals following rules are players, and the desired properties of rules are given by equilibrium strategies, thus describing enforced rules. However what we discuss here is endogenous dynamics of these rules that takes into account the fact that individual behaviour and rules operate mutually and concurrently. In this sense,

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individual rationality and social rationality are mutually dependent, and what we seek to study are the patterns of reasoning that inform such dependence. In [19], we studied game forms that change dynamically according to prespeciﬁed rules stated in a formal logic, and asked which action choices are eventually stable (in the sense that no further game changes will eliminate them), and under what conditions. We showed that these questions are algorithmically solvable. In this paper, we address a dual question: if players were to play according to some strategy speciﬁcations, when should society intervene and in what form? We look at the quantitative aspect of the choice-restriction phenomenon. Since these are games with a large number of players, societal decisions are not inﬂuenced directly by individual choices but by distribution of strategies in the player population. Thus we consider anonymous games where the cost incurred by society in a particular round is given by a function of the action distribution of the players. The cumulative cost is the limit-average (mean-payoﬀ) of these costs. We then ask whether it is possible for the society to synthesise rules for removal of actions of the players so that the eventual social cost is less than a certain threshold. We show that such synthesis is possible and that the rules require only ﬁnite memory. Related Work Dynamic game forms have been studied extensively in the context of evolutionary game theory ([23]). [18] studies a model where actions of players depend on the forecast outcome and behaviour switching of the players in eﬀect causes the game itself to change in a dynamic fashion. In [20] Young deﬁnes and studies the long run dynamics of a model of how innovations spread in a social network. [5] looks at equilibrium selection by players who revise strategies by a learning process. They note that the stable behaviour of agents depend on the dynamics of the game itself and argue that it is important to incorporate these changes into the model. Switching behaviour of players has also been studied in dynamical system models of social interaction ([22], [16]). Going further, Hashimoto and Kumagai ([13]) even propose a model in which interaction rules of replicator equations change dynamically, and oﬀer computer simulations of dynamic game changes. While this paper studies quantitative rule synthesis, our work is broadly situated in qualitative reasoning about games and is thus related to logical formalisms. Modal logics have been used in various ways to reason about games and strategies. Notable among these is the work on alternating temporal logic (ATL) and its extensions ([1], [15], [14], [17]): assertions are made on outcomes a coalition of players can ensure, and what strategy a player plays may depend on her intensions and epistemic attitudes. In [2,3] van Benthem uses dynamic logic to describe games as well as strategies. [12] presents a complete axiomatisation of a logic describing both games and strategies in a dynamic logic framework where assertions are made about atomic strategies. [21] studies a logic in which not only are games structured, but so also are strategies. [4] lists a range of issues to be studied in reasoning about strategies.

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2

31

Dynamic Game Restriction

We study concurrent games on ﬁnite graphs. N = {1, 2, . . . , n} is the set of players and Ai is the set of actions of player i. An arena A = (V, E) is a ﬁnite graph with the vertex set V and edge relation E. For v ∈ V , let vE = {(v, u) ∈ E}, i.e., the set of edges outgoing from v. The edges of the arena are labelled with labels from A. For an edge label a = (a1 , . . . , an ) we let a(i) denote the ith component of a, i.e., a(i) = ai . Thus edges are labelled with tuples from the set A ∪ {}. The components give the default transitions in the arena. If the i i∈N ith component of an edge label is then that edge deﬁnes the transition of the game in a situation where player i plays an action a ∈ Ai but a is not available to her anymore, in that, the society has removed a from her available set of actions. Note that ∈ / Ai , that is is not a strategic choice for the players. We assume that for every a ∈ i∈N Ai ∪ {} and every v ∈ V , there exists an edge (v, v ) which is labelled with a. That is, the labelling is complete with respect to the labels at every vertex. An initial vertex v0 ∈ V is distinguished and the game G = (A, v0 ) consists of an arena A and the initial vertex v0 . A sub-arena A of the arena A is a graph (V , E ) such that V ⊂ V and E is the set of edges induced by V . The game proceeds as follows. Initially a token is placed at v0 . If the token is at some vertex v, then players 1 to n simultaneously choose actions a1 , . . . , an from their action sets A1 , . . . , An respectively. This deﬁnes a tuple a = (a1 , . . . , an ). If a is the label of the edge (v, u) then the token is moved to u. If a is not present among the labels of the outgoing edges then for all i : 1 ≤ i ≤ n such that the action a(i) is not available to player i, a(i) is replaced by in a to get a . If (v, u) is the edge with label a then the token is moved to u. This deﬁnes a path a a ρ = v0 →0 v1 →1 . . . in the arena. Such a path is called a play. A ﬁnite play is also called a history. The tree unfolding of the arena A at a node v0 is a subset TA ⊂ A∗ such that a a ∈ TA is the root and for all t = a0 a1 . . . ak ∈ TA such that v0 →0 . . . →k vk is the corresponding path in A, ta ∈ TA for all (vk , u) ∈ vk E such that (vk , u) is a a labelled with a. For a node t = a0 a1 . . . ak ∈ TA such that v0 →0 . . . →k vk is the corresponding path in A, we let λ(t) = vk . We also use the notation TG to denote the tree unfolding of the game G = (A, v0 ). A strategy of a player tells her how to play the game. In other words, it prescribes at every position which move to make. Formally a strategy si of player i is a function si : A∗ → Ai . Note that the codomain of si is Ai and not Ai ∪ {}. The empty action is not a strategic choice for a player; rather it is forced when the action she plays is not available. A strategy si can equivalently be thought of as a subtree Tsi , the strategy-tree, of TA with root corresponding to the position v0 such that: – For any node t = a0 a1 . . . ak if si (t) = a then the children of t in Tsi are exactly those nodes ta ∈ TA such that a(i) is equal to a. A strategy s is said to be bounded memory if there exists a ﬁnite state machine M = (M, g, h, mI ) where M is a ﬁnite set denoting the memory of the strategy,

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mI is the initial memory, G : A × M → M is the memory update function, and h : A × M → Ai the output function which speciﬁes the choice of the player such that if a0 . . . ak is a play and m0 . . . mk+1 is a sequence determined by m0 = mI and mi+i = g(ai , mi ) then s(a0 . . . ak ) = h(ak , mk+1 ). The strategy s is said to be memoryless if M is a singleton. The crucial elements for deﬁning game restrictions are: when a restriction is to be carried out in the course of play, and what the eﬀects of a restriction are. We choose a very simple answer to the latter, namely to eliminate a subset of choices at selected game positions, that is, to restrict the set of actions available to a player. The former is treated logically, to be deﬁned below, by tests for logical conditions. Formally the restriction is triggered by a rule of the form r = pre ⊃ A where pre is a precondition which is interpreted on partial plays and A is a restriction of the arena. For an arena A and a partial (ﬁnite) play ρ ∈ A, we say that the rule r = pre ⊃ A is enabled at (A, ρ) if the following conditions hold. – The partial play ρ conforms to the precondition pre. – The arena A = (V , E ) is a sub-arena of A. – last(ρ) ∈ V . When the rule r = pre ⊃ A is applied to a partial play ρ, the game proceeds to the new arena A starting at the node last(ρ). 2.1

Induced Game Tree

The restriction rules are speciﬁed along with the initial game arena. Let R = {r1 , . . . , rm } be a ﬁnite set of restriction rules. For an arena A, let SA(A) denote the set of all subarenas of A. Given an initialised arena (A, v0 ) and a ﬁnite set of rules R, the extensive form game tree is the (inﬁnite) tree TA (R) = (T, E) where T ⊆ (V × A∗ × SA(A)) and is deﬁned inductively as follows: – t0 = (v0 , , A) is the root. – At any node t = (v, u, A ) of the tree, check if for a rule (rj = pre j ⊃ Aj ) ∈ R, it is the case that t |= rj . If more than one rule is enabled at t then choose any one of them, say pre j ⊃ Aj . Let Aj = (Vj , Ej ). • If pre j ⊃ Aj , then the subtree starting from t is the unfolding of the arena Aj from the vertex v. Note that v ∈ Vj since we have ensured that v = last(ρ(v0 , u)) ∈ Vj . • If there is no such rule then the children of t are the same as those of t in the unfolding of A and the edge labels are also the same. 2.2

Strategising by Players

In the presence of dynamic game restriction operations, the players can keep track of the restriction rules which are triggered by observing the history of play and adapt their strategies based on this information. A strategy speciﬁcation for a player i would therefore be of the form pre ⊃ a where, as earlier, pre is a

Dynamic Restriction of Choices: Synthesis of Societal Rules

33

precondition which is interpreted on partial plays and a ∈ Ai . The speciﬁcation asserts that if a play ρ conforms to the precondition pre, then the action a is taken by the player. To formally deﬁne the structure of preconditions, we let P be a set of atomic propositions and bool (P) be the set of boolean formulas over P (i.e. built using the syntax p ∈ P | ¬β | β1 ∨ β2 ). We also use the following abbreviations:

≡ p∨¬p and ⊥ ≡ p∧¬p. Let val : V → 2P be a valuation function be given on the game arena. val can be lifted to TA in the natural way, i.e., val (t) = val (λ(t)). The strategy of players can in general depend on properties of the history of the play. These can therefore be speciﬁed as a collection of formulae of the form ϕ ⊃ a where ϕ ∈ Φ is given by the following syntax: − - ψ ϕ ∈ Φ ::= P ∈ P | ¬ψ | ψ1 ∨ ψ2 | a ψ |

A formula ϕ of player i is evaluated on the game tree TG . Then the truth of ψ at a node t of TG , denoted TG , t |= ψ is deﬁned inductively in the standard manner: – – – – –

TG , t |= P iﬀ p ∈ val (t). TG , t |= ¬ϕ iﬀ TG , t ϕ . TG , t |= ϕ1 ∨ ϕ2 iﬀ TG , t |= ϕ1 or TG , t |= ϕ2 . − TG , t |= a ϕ iﬀ t = t a and TG , t |= ϕ . - ϕ iﬀ for all preﬁxes t of t, TG , t |= ϕ . TG , t |=

Note that a strategy speciﬁcation of this form is partial, since it does not constrain game positions at which the precondition does not hold; the player is free to choose any enabled action. Syntax and semantics of restriction precondition. A restriction precondition ψ comes from the syntax ϕ using which we can specify properties of the indeﬁnite past and bounded future. ψ is evaluated on TA (R) as usual. - ψ makes assertion about the unbounded past, it speciﬁes the The modality transitive closure of the one step past operator. We can deﬁne the corresponding construct for future, ψ which makes assertions about unbounded future. The technical results go through even with the addition of this construct. However, for the applications we have in mind, this construct is not required. 2.3

Capturing Costs in the Logical Formalism

Following a strategy induces a certain cost for the player. The distribution of strategies chosen by players carry a social cost. We ﬁrst take an abstract view of costs associated with individual players and social costs associated with providing facilities. The social cost typically depends on the history of the choices made by players in the past. When the social cost crosses some pre-deﬁned threshold, it might be socially optimal to make certain facilities part of the common infrastructure which reduces the individual costs.

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When the costs arise from a ﬁxed ﬁnite set, they can be coded up using propositions in the logical framework on the lines of [8]. The cost c (say) can be represented using the proposition pc and orderings are inherited from the implication available in the logic. Furthermore, costs can be dependent on the actions enabled at a game position. This can also be easily represented in the logical formalism by making use of the one step future modality. Let (A, v0 ) be an initialised arena, R be a ﬁnite set of game restriction rules, {Σi }i∈N be a ﬁnite set of strategy speciﬁcations or each player i ∈ N . Let α be a formula from the syntax: α ::= α ∈ bool (P) | a+ α We say α is stable in (A, R, {Σi }i∈N ) if there exists a sub-arena A such that for all game positions t ∈ TA , we have: t |= α. Thus stability with respect to an observable property captures the existence of a subarena to which the game stabilises under the dynamics speciﬁed by R and {Σi }i∈N . For the applications we consider, we do not require the full power of temporal logic for α. In [19], we proved the following theorem. Theorem 1. Given an initialised arena (A, v0 ), a ﬁnite set of restriction rules R, a ﬁnite set of strategy speciﬁcations {Σi }i∈N and a formula α, the following question is decidable: – Is α stable in (A, R, {Σi }i∈N )?

3

Quantitative Objectives

In this section we change our model to one where the costs (both social and individual) are given by certain functions instead of being coded up as propositions as before. We then ask whether it is possible for the society to restrict the actions of the players in such a way that the social cost stays within a certain threshold. We ﬁrst develop some preliminaries. As before we let N = {1, 2, . . . , n} be the set of players. However we assume that the players have a common action set A, that is, A1 = . . . = An = A. We study anonymous games [6,7,10,9] because in large games, the payoﬀs are usually dependent on the ‘distribution’ of the actions played by the players rather than the action proﬁles themselves. Moreover, in such games the payoﬀs are independent of the identities of the players. An action distribution is a tuple |A| y = (y1 , y2 , . . . , y|A| ) such that ∀i, yi ≥ 0 and i=1 yi ≤ n. Let Y be the set of all action distributions. Given an action proﬁle a, we let y(a) be its corresponding action distribution, that is, y(a)(k) gives the number of players playing the kth action in A. Now as the payoﬀs are dependent on the action-distribution of the players, we convert the arena A to a new arena A[Y] so that the payoﬀs can be assigned to the vertices of the arena. A[Y] is deﬁned as A[Y] = (V [Y], E[Y]) as follows:

Dynamic Restriction of Choices: Synthesis of Societal Rules

35

– V [Y] = V × Y a a – E[Y] ⊆ V [Y] × A × V [Y] such that (v1 , y1 ) → (v2 , y2 ) iﬀ v1 → v2 and y(a) = y2 . – Delete all vertices in V [Y] that do not have any incoming or outgoing edges. As we shall exclusively deal with the expanded arena A[Y] in the entire development, we denote A[Y] = (V [Y], E[Y]) by just A = (V, E) and assure that it will not result in any confusion. A tuple (A, v0 ) where A is an arena and v0 is a distinguished vertex is called an initialised arena. Every player i has a function fi : Y → Q which can be seen as the payoﬀ of i for a particular distribution. There is also a function f : Y → Q which can be viewed as the cost incurred by the society for maintaining the actions. These functions can be lifted to the vertices of V [Y] as f (v, y) = f (y). We now investigate if it is possible for the society to impose restrictions in such a way that the social cost stays within a certain threshold. We look at two variations of our model: a. At the beginning of each round the society chooses an order for the n players and makes it known to them. The players then choose their actions according to this order. b. The players play according to strategy speciﬁcations (as in the previous sections). The society, at any point, can restrict some action a ∈ A of the players, in that, it can make the action a unavailable. In (a), we wish to investigate if it is possible for the society to pick the orders of actions of the players in such a way that the eventual social cost is within a certain threshold. In (b), we want to ﬁnd out if the society can restrict the action of the players based on certain rules so that the same eﬀect is obtained. 3.1

Restriction of Order

The game proceeds in rounds. At the beginning of every round, the society chooses an order for the players to play and makes it known to them. The players choose actions in that particular order. These actions deﬁne a tuple a ∈ A and the play moves along the edge labelled a to the next vertex. This process goes on a a forever. Given an initial vertex v0 this deﬁnes an inﬁnite play ρ = v0 →1 v1 →2 . . . in the arena. We study the discounted-payoﬀ Player i gets: n

pi (ρ) = lim inf n→∞

1 fi (vj ). n j=1

Similarly the society incurs a cost of: n

1 c(ρ) = lim inf f (vj ). n→∞ n j=1 There is a threshold cost θ. The aim of each player i is to play in such a way that the quantity pi is maximised and the aim of the society is to choose the

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orders in such a way that the quantity c(ρ) is always less than θ for every play ρ no matter what actions the players play according to the order it selects. We are interested in the following question: Question. What orders can the society choose so that the social cost c always remains less than the threshold θ? We ﬁrst deﬁne a normalised version of the game where we subtract θ from the cost associated with every vertex of the arena A. In other words, we deﬁne a new function f˜ from f such that f˜(v) = f (v) − θ for every v ∈ V . For a play ρ in A we let: n

c˜(ρ) = lim inf n→∞

1˜ f(ρ(j)). n j=1

Note that for any play ρ in A, c(ρ) < θ iﬀ c˜(ρ) < 0. Now, to answer the above question, we ﬁrst deﬁne a tree unfolding TA of the initialised arena (A, v0 ). The unfolding takes into account the order in which the players choose their actions. TA is constructed inductively, the set of nodes being: T ⊆ (V × {soc}) ∪ (V × N × π(N )) where π(N ) is the set of permutations of 2N (the subsets of N ) such that (v, j, w) ∈ TA only if j = w(1).

Fig. 1. The unfolding

We now present the construction, see Figure 1 for an illustration. The root node is (v0 , soc). Suppose TA has been constructed till level i. Consider an unprocessed node at level i. – If this node is of the form (v, soc) and if (v, soc) has an identical ancestor already present in TA constructed so far, then we call (v, soc) a leaf node

Dynamic Restriction of Choices: Synthesis of Societal Rules

37

and do not process it any further. Otherwise the set of children of (v, soc) correspond to all the permutations (orders) for the following round. In other words, the set of children of (v, soc) are of the form (v, j, w) ∈ T where w ∈ π(N ). – If the node is of the form (v, j, w) and |w| > 1 then its children correspond to all the possible actions that j can choose. That is, it has |A| many children of the form (v, k, w2 ) ∈ T . The edge from (v, j, w) to the th child is labelled with a ∈ A. If |w| = 1 then again (v, j, w) has |A| children, the th edge being labelled with a ∈ A such that the following holds. (v , soc) is a child of (v, j, w) if and only if the actions corresponding to the path from (v, soc) to (v , soc) in the tree give the action tuple a and v is the neighbour of v along the edge labelled a. The above procedure is repeated until all the branches have seen a leaf node and there are no more nodes to process. Note that as the set T is ﬁnite, the procedure does terminate. We then deﬁne a backward induction procedure on TA as follows. In the process, we construct another tree TA∗ which is a subtree of TA and which gives the orders that the society can choose so that the social cost always remains less than the threshold. Procedure 1. – Label the leaf nodes with tuples from Qn+1 as follows. For every leaf node (v, soc) there exists an identical ancestor. This means that on this branch, the game has settled down to a simple cycle involving the vertex v. Let a a a C = v0 →1 v1 →2 . . . →k vk where v0 = vk = v be this cycle. Label (v, soc) with (p0 (C), p1 (C), . . . , pn+1 (C)) where: p0 (C) =

k

f˜(vj )

j=1

and pi (C) =

k

fi (vj ), i ∈ N.

j=1

– For a non-leaf node, suppose all its children have been labelled. • If the non-leaf node is of the form (v, j, w) let L be the set of labels of its children. Let Lj = {pj | (p0 , . . . , pn ) ∈ L}. Let mj = max Lj and let (p0 , . . . , pn ) ∈ L be such that pj = mj . Label (v, soc) with (p0 , . . . , pn ). • If the non-leaf node is of the form (v, soc) let L be the set of labels of its children. Let L<0 = {p0 | (p0 , . . . , pn ) ∈ L and p0 < 0} and let L≥0 = L \ L<0 . If L<0 = ∅ then label (v, soc) with (p0 , . . . , pn ) ∈ L<0 . Otherwise label (v, soc) with (p0 , . . . , pn ) ∈ L≥0 . Delete the subtrees rooted at every child of (v, soc) whose label is in the set L≥0 .

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The above backward induction procedure generates a subtree TA∗ which is the strategy of the society, in that, TA∗ gives all the orders that the society can choose so that the social cost always remains less than the threshold. When a play reaches a vertex v ∈ V , the society chooses a permutation w of the players which is dictated by TA∗ , in that, there is a node (v, soc) ∈ TA∗ such that it has a child of the form (v, j, w). Proposition 1. TA∗ gives all the orders that the society can choose so that the social cost c always remains less than the threshold θ. Proof. The fact that it is enough to unfold the arena till a cycle is completed and that it is enough for the players to play memoryless strategies follows from [11]. Then it is clear that if the society enforces the orders as given by TA∗ and the players play to maximise their own respective payoﬀs, then the social cost remains less than θ. Conversely, suppose the society enforces an order that is not given by TA∗ . Then again the correctness of the backward induction procedure and the fact that the players play memoryless strategies implies that since the players play to maximise their own respective payoﬀs, the game settles down to a cycle such that the social cost is greater than θ. The following corollary is immediate from the above proposition. Corollary 2. When the society can specify the order in which the players choose their actions, it has a ﬁnite memory strategy to ensure that the social cost remains within a given threshold. 3.2

Restriction of Action

In the analysis above the society assumes that all players are maximizers, and hence regulates only the order of their choices. What happens when players are resource bounded and hence limited in their strategic ability? In this case players play according to strategy speciﬁcations. We now study if and how the society can restrict the actions of such players so as to keep the social cost within a certain given threshold. Let A be the common set of actions for the players. The speciﬁcations of player i is a set Σi whose elements are of the form ϕi ⊃ a where a ∈ A and ϕi is a temporal formula from a syntax similar to Φ except that the atomic formulas also have propositions of the form pi (x) < d. A formula ϕ is evaluated on ﬁnite paths (plays/histories) which are nodes of the unfolding of the arena TA . The truth of the atomic proposition pi < d at a node t is deﬁned as: – t |= pi < d iﬀ • there exists a preﬁx t of t such that last(ρ(t )) = last (ρ(t)) = v and • let t be the longest preﬁx of t such that last(ρ(t )) = v and let C be the cycle from t to t. Then we have |C| j=1 fi (C(j)) < d.

Dynamic Restriction of Choices: Synthesis of Societal Rules

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That is, pi < d holds at a node t if the play has settled down to a cycle C on that branch and the cumulative reward to player i from that cycle is less than d. Among other possible propositions, the set P has propositions of the form pa× which if true at a vertex v says that the action a is not available at v. Using these propositions, players can form observables like - (pi (x) ≥ d ∧ pa× ) ⊃ a ♦ which says that if in the past the play settled down to a loop with utility greater than d and the society restricted the action a then play a . Each player i has a set Σi of strategy speciﬁcations of the form ϕ ⊃ a. The players play according to their strategy speciﬁcations. For a history ρ and for a player i, if there exists a speciﬁcation ϕ ⊃ a ∈ Σi such that ρ |= ϕ then she plays action a. Otherwise she plays any action from her set of actions A non-deterministically. The society can remove certain actions from the available set of actions of the players at a particular vertex v. The removal of an action a results in the following. For every edge in the arena, if the edge has a label a such that a(i) = a for some i ∈ N then the label a is removed. If this results in an edge without a label, then the edge itself is removed. Like in the previous case, the aim of the society is to restrict the actions of the players in such a way as to make the game eventually settle down so that the social cost is less than a certain threshold. We thus wish to generate a set of rules (a ﬁnite state automaton) M for the society, such that if it restricts actions of the players as prescribed by M then the game always settles down so that the social cost is less than a threshold even though the players play according to their speciﬁcations. Given a formula ϕ ∈ Φ, let SF (ϕ) denote the set of all subformulas of ϕ closed under negation. Let AT (ϕ) denote the set of atoms of ϕ, where an atom is a propositionally consistent subset of SF (ϕ). We ﬁrst construct a set AT up (ϕ) from AT (ϕ) as follows. For every D ∈ AT (ϕ) we do the following: – Let D = D1 D2 where D1 consists only of formulae of the form a− ϕ . If D1 = ∅ then we let D ∈ AT up (ϕ). Otherwise: – For every a− ϕ ∈ D1 we deﬁne the following sets: • D a − ϕ = {a− ϕ }. − − • D a

− ϕ = {¬b ϕ | b ϕ ∈ D1 , b = a}. • D ⊆ D2 such that D is a maximal subset of D2 that is consistent with D a − ϕ ∪ D a

− ϕ . We let D a − ϕ ∪ D a

∈ AT up (ϕ). − ϕ ∪ D Every set D ∈ AT up (ϕ) contains at most one formula of the form a− ϕ . We call such a set a unique-past atom. We construct another set AT up−c (ϕ) which is the set of unique past atoms without the (cost) formulae of the form pi < d. Formally, for every D ∈ AT up−c (ϕ), let D = D\({pi < d | i ∈ N, pi ∈ D}∪{¬(pi < d) | i ∈ N, pi ∈ D}).

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For all maximal subsets D of D such that D is propositionally consistent, let D ∈ AT up−c (ϕ). Let the modal depth of a formula ϕ, md (ϕ) be deﬁned inductively as: – – – – –

md (p) = 0. md (¬ϕ ) = md (ϕ ). md (ϕ1 ∨ ϕ2 ) = max{md(ϕ1 ), md (ϕ2 )}. md (a− ϕ ) = md (ϕ ) + 1. - ϕ ) = md (ϕ ) + 1. md (♦

Let Mi = maxϕ⊃a∈Σi {md (ϕ)} and M = maxi Mi . We now describe an unfolding of the initialised arena (A, v0 ) combined with the unique-past atoms of the strategy speciﬁcations. The unfolding is a tree TA which is built in stages and is described as below. Intuitively, the purpose of Stage 1 is to unfold the arena according to the speciﬁcations of the players so as to reach a cycle on every branch. Once a cycle C is reached on a branch, the formulas of the form pi < d will start to hold true or false. We then move to Stage 2 to unfold the game again till the modalities involving the formulas of the form pi < d come into eﬀect. Stage 1 and 2 are repeated alternatively till cycles are reached on every branch. Note that it is not enough to stop at the cycles of Stage 1 since the modalities involving the formulas pi < d have not yet come into eﬀect. Also note that, it is enough to stop when a cycle is reached involving nodes of both Stage 1 and Stage 2 since we have considered all the possible ways that any modal formula can or cannot be satisﬁed. This is because, the number of possible nodes of the tree is ﬁnite and also each formula is of bounded modal-depth. Below let C denote the tuple (C1 , . . . , Cn ), and similarly for C , C etc. Stage 1: The vertices of TA in stage 1 is the set T1 = (V × 2AT up−c (ϕ) × {plr , soc} × (A ∪ )) ∪ . i∈N ϕ⊃a∈Σi

1. The root node is (level 0). 2. The nodes at level 1 are of the form (v0 , C , soc, ) such that for every i ∈ N , for every restriction ϕ ⊃ a ∈ Σi and for every atom D ∈ AT up−c (ϕ), D is a component of Ci iﬀ D doesn’t have any formula of the form ϕ and D ∩ P = val (v0 ). 3. Every node (v0 , C , soc, ) of level 1 has k + 1 children where the ith child, i ∈ [k], is (v0 , C , plr , ai ) and the k +1th child is (v0 , C , plr , ). Intuitively, the ith child represents the situation where the society has banned the action ai from the set of available actions of the players. The k + 1th child represents the situation where the society hasn’t applied any restriction. 4. Let (v0 , C , plr , a ) be a node at level 1. It’s children are determined as follows. For every player i construct a set Ai ⊆ A as: if there exists ϕ ⊃ a such that ϕ ∈ D ∈ Ci then let Ai be the set of all such a’s. If a ∈ Ai then let Ai = (Ai \ {a }) ∪ {}. Otherwise let Ai = Ai . Finally, if there does not exist ϕ ⊃ a such that ϕ ∈ D ∈ Ci then let Ai = A. Now for every a ∈ i∈N Ai , (v0 , C , plr , a ) has a child (v1 , C , soc, ) such that

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a

– v0 → v1 , – For every i for every D ∈ Ci and D ∈ Ci and for every ϕ ⊃ a ∈ Σj , a− ϕ ∈ D iﬀ ϕ ∈ D, – For every D ∈ Ci , D ∩ P = val (v1 ). Note that since the set T1 is ﬁnite, along every branch some node has to repeat. Steps 3 and 4 are repeated till on every branch some node repeats. This indicates that the game has settled down to a cycle on every branch. This completes the description of Stage 1. See Figure 2 for an illustration.

Fig. 2. Stage 1

Stage 2: We now unfold the game starting from every leaf node of Stage 0 enough number of times so that any modality involving formulas of the form pi < d can take into eﬀect. For that purpose, we mimic the cycle already reached on every branch in Stage1. As and when some new modal formula is satisﬁed, the subtree changes according to the actions speciﬁed by the modal formula. The vertices of TA in this stage are in the set: T2 = (V ×

2AT up (ϕ) × {plr } × (A ∪ )) ∪ .

ı∈N ϕ⊃a∈Σi

The leaf nodes of Stage 1 are modiﬁed to constitute the level 0 nodes of Stage 2 (because now we have to keep track of formulas of the form pi < d). Every leaf node (v, C , soc, ) represents a cycle that the game has settled down to. Suppose

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the cycle for the leaf (v, C , soc, ) is C. We denote by C(v, C , soc, ) the identical ancestor of (v, C , soc, ) with the greatest depth. Let (u, C , plr , a) be the parent of (v, C , soc, ) and suppose the edge from (u, C , plr , a) to (v, C , soc, ) was labelled a. Replace (v, C , soc, ) with another vertex (v, C , soc, ) such that for every i ∈ N and for every ϕ ⊃ a ∈ Σi , for every D ∈ AT up (ϕ), D ∈ Ci iﬀ – D ∩ P = val (v), – For every pi < d in SF (ϕ), pi < d ∈ D iﬀ pi (C) < d, and – For every D ∈ Ci , a− ϕ ∈ D iﬀ ϕ ∈ D . These constitute the Level 0 nodes of stage 2. See Figure 3 for an illustration.

Fig. 3. Stage 2, Level 0

Level 1: Every node (v, C , soc, ) of level 0 has a single child determined as follows. Let (v, C , plr , a) be the unique ancestor of (v, C , plr , a) which is a child of C(v, C , soc, ). Modify every set Ci to Ci as described above. The child of (v, C , soc, ) is the node (v, C , plr , a). This constitutes level 1. Level 2 and above: For the nodes of Level 2 and above, we ﬁrst check if any new modal formula is satisﬁed. If not, then we let the players play the same action as was played by them on the cycle involving that branch and wait. The children of every node (v, C , plr , a ) at level 1 (the Level 2 nodes) are determined as follows. Let I ⊆ N be the set of indices such that for every player i ∈ I, there exists ϕ ⊃ a such that ϕ ∈ D ∈ Ci . Let Ai be the set of all such a’s. For i ∈ N \ I, let Ai = A. If I = ∅ then for every a ∈ i∈N Ai , (v, C , plr , a ) has a child (v , C , soc, ) ∈ T1 such that a

– v → v , – For every i for every D ∈ Ci and D ∈ Ci and for every ϕ ⊃ a ∈ Σi , a− ϕ ∈ D iﬀ ϕ ∈ D, – For every D ∈ Cj , D ∩ P = val (v ). Call each such child a leaf node of stage 2.

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If no new restriction is applicable, then we wait. In other words, on every such branch, we copy the actions played previously by the players in the corresponding cycle. Formally, if I = ∅, let a be the label of the outgoing edge from the node C(v, C , plr , a ) to the ancestor of (v, C , plr , a ). Then (v , C , soc, ) is a child of (v, C , plr , a ) iﬀ a

– v → v , – For every i for every D ∈ Ci and D ∈ Ci , a− ϕ ∈ D iﬀ ϕ ∈ D, and – For every D ∈ Ci , D ∩ P = val (v ). For every non-leaf node, the above process is repeated for 2M steps.2 This constitutes stage 2. Stages 1 and 2 are repeated in alternation till along each branch, a node repeats. This completes our description of the tree TA . Let θ be the threshold value such that the society wishes that the game eventually settles down so that the social cost is less than θ. We now construct an automaton M from TA which is the strategy automaton for the society as follows. First, we construct a subtree TA∗ of TA using the following backward induction procedure. Procedure 2. – We label the leaf nodes with either 0 or 1 as follows. Let t be a leaf node. Let C(t) be its identical ancestor and let C be the cycle from C(t) to t. Label t with 1 only if c˜(C) < 0. Otherwise label it with 0. – Suppose all the children of a node t has been labelled. Delete all the children with label 0. If there is no remaining child of t then label it with 0. Otherwise label it with 1. The above procedure thus generates a subtree TA∗ of TA . We then construct the strategy automaton M from TA∗ . Note that every leaf node of TA∗ is part of a cycle. Thus to construct M, for every leaf node t of TA∗ such that t is the parent of t, a a and t → t we make a loop t → C(t) in M. The initial state of M is the root of TA∗ and its transition relation is given by the parent-child structure of TA∗ . Note that the subtree TA∗ obtained from Procedure 2 may be empty, in which case the society cannot force the play to settle down to a social cost less than θ by removing one action at a time from the vertices. But if TA∗ is nonempty, then we have: Proposition 2. By playing the bounded memory strategy M, the society can make the game eventually settle down to a social cost less than the threshold θ given that the players play according to their strategy speciﬁcations. 2

M is the maximum modal depth of any formula in the strategy speciﬁcations. The factor of 2 is due to the alternation between a society node and a player node in the unfolding.

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Proof. For every play ρ in the arena that is consistent with the strategy speciﬁcations we can associate a node t(ρ) in TA . Note that then it is suﬃcient to prove: Claim. For all i, for all ϕ ⊃ a ∈ Σi and for all ϕ in the subformula closure of ϕ, ρ |= ϕ iﬀ ϕ ∈ t(ρ). That is because all the actions that a player i can play by testing the truth of the speciﬁcation pre-conditions ϕ in Σi is present in TA and all such actions that lead to unfavourable cycles are removed in the construction of M from TA . To prove Claim 3.2 we proceed by induction on the structure of a subformula ϕ . The base cases ϕ ≡ p ∈ P and ϕ ≡ pi (x) < d are immediate from the construction of TA . ρ |= ϕ1 ∨ ϕ2 iﬀ ρ |= ϕ1 or ρ |= ϕ2 iﬀ ϕ1 ∈ t(ρ) or ϕ2 ∈ t(ρ) iﬀ ϕ1 ∨ ϕ2 in t(ρ), the corresponding component being an atom. ρ |= ¬ϕ iﬀ ρ ϕ iﬀ ϕ ∈ / t(ρ) iﬀ ¬ϕ ∈ t(ρ), since the corresponding component is an atom. Suppose there exists a play ρ in the arena such that ρ |= a− ϕ . We know that the modal depth of a− ϕ is at most M and hence the modal depth of ϕ is at most M −1. Since we have unfolded the cycles in the even stages 2M , the even stages are eﬀectively of length M (since each vertex along a branch is repeated twice, one for the society and the other for the players). Now, ρ |= a− ϕ iﬀ a ρ = ρ → v and ρ |= ϕ [by induction hypothesis which can be applied because a of the above observation] iﬀ ϕ ∈ t(ρ ) iﬀ t(ρ ) → t(ρ) in TA (by construction) iﬀ − t(ρ) |= a ϕ (by construction). - ϕ then there exists a preﬁx ρ of ρ such that ρ |= ϕ . Finally, suppose ρ |= ♦ We do a second induction on = |ρ| − |ρ |. If = 0 then ρ |= ϕ iﬀ ϕ ∈ t(ρ) - ϕ ∈ t(ρ). Now suppose > 0 in that case |ρ | < |ρ|. Then ρ |= ♦ - ϕ iﬀ iﬀ ♦ - ϕ where ρ is a preﬁx of ρ and |ρ | = |ρ| − 1. Then by the second ρ |= ♦ - ϕ ∈ t(ρ) since t(ρ) is a child of t(ρ ) by - ϕ ∈ t(ρ ) iﬀ ♦ induction hypothesis, ♦ construction of TA . As an immediate corollary to the above proposition we have: Corollary 3. Let (A, v0 ) be an initialised arena where A = (V, E) and A is the common set of actions of the players. Given strategy speciﬁcations {Σi }i∈N for the players and given a function f : V → Q for the social cost, if the society can force the game to eventually settle down so that the social cost is less than θ then it can do so using a ﬁnite memory strategy, the memory being: V × 2AT up (ϕ) × {plr , soc} × (A ∪ ) ∪ ı∈N ϕ⊃a∈Σi

We thus see that if the society imposes restrictions, on the order of play or the availability of actions, based on the ﬁnite state automata derived in the above proofs, then it can ensure that the social cost always remains within a certain threshold. These automata are ﬁnite memory strategies which can be seen as rules for the society for applying the restrictions. Note that although we carried out our analysis for limit-average payoﬀs, a similar analysis also goes through in the setting of discounted payoﬀs.

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4

45

Reducing the Number of Players

Large games lack some of the desirable qualities of the traditional games: viz., perfect information, bounded number of players, unbounded computational resource etc. However, there is usually one feature of most of these games that may make them amenable to tractable analysis. In such games there are usually a small number t of types such that t << n where n is the number of players. Hence, it would be nice if one could carry out all the analysis using only the t types and then lift the results to the entire game. Why should such an analysis be possible? Since outcomes are determined by player type distributions rather than strategy proﬁles, we can conﬁne our attention to types. When types describe ﬁnite memory strategies (as in the case of modal strategy speciﬁcations) we can consider them to be ﬁnite state transducers that observe play, make boundedly many observations and output the moves to be played. For n players, the strategy space is the n-fold product of these memory states; what we wish to do is to map this space into a t-fold product, whereby we wish to identify two players of the same type. Suppose two players 1 and 2 are of identical type deﬁned by a transducer with m states; their product space unfortunately has m2 states. If 500 players in a population of 1000 play an identical strategy we have m500 states in the product. We show below, that in the case of deterministic transducers, that such a blow-up is avoidable, since the product of a type with itself is isomorphic to the type. Thus a population of 1000 players with only two types needs to be represented only by pairs of states and not 1000-tuples. However, there is no free lunch: a price has to be paid for determinacy and hence this analysis works only when the number of players is signiﬁcantly larger than the number of types. In this section, we give some pointers in that direction. We study situations when such an analysis can be carried out when the types of the players are speciﬁed using ﬁnite state transducers. Determinisation of Finite State Transducers We wish to deﬁne the product of a strategy transducer Q with itself. And as we shall notice presently, we need the states of the product transducer to be of the form (q, q). One way to guarantee this is to ensure that the transducer Q itself is deterministic. We ﬁrst see how to determinise such a transducer. As usual let N = {1, 2, . . . , n} be the set of players and assume that the players have a common action set A. Let A = An . Let Q = (Q, δ, I, f ) be an FST with input alphabet A and output alphabet A. We determinise Q to obtain another FST Q = (Q , δ , q0 , f ) with input and output alphabets A and 2A respectively as follows. (Q , δ , q0 ) is the determinisation of the automaton (Q, δ, I) and f : Q → 2A where f (X) = {f (q) | q ∈ X}.

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Definition 1. Let Q1 = (Q1 , δ1 , qo1 , f1 ) and Q2 = (Q2 , δ2 , q02 , f2 ) be deterministic FSTs. Let Q1 × Q2 = (Q, δ, q0 ) be the product of the automata (Q1 , δ1 , q01 ) and (Q2 , δ2 , q02 ), that is: – Q = Q1 × Q 2 , – q0 = (q01 , q02 ), – (q1 , q2 ), a, (q1 , q2 ) ∈ δ iﬀ (q1 , a, q1 ) ∈ δ1 and (q2 , a, q2 ) ∈ δ2 . Delete all the unreachable states of Q1 × Q2 and call the resulting automaton Q . Then Q1 ⊗ Q2 = (Q , f ) where f : Q → 2A × 2A such that f (q1 , q2 ) = (f (q1 ), f (q2 )). Definition 2. Let Q = (Q, δ, q0 , f ) be a deterministic FST. Let Q × Q be the product of (Q, δ, q0 ) with itself. Delete all the unreachable states of Q × Q and call the resulting automaton Q = (Q , δ , q0 ). Then Q ⊗ Q = (Q , f ) where f : Q → 2A such that f (q, q) = f (q). This is well-deﬁned because of the following claim. Claim. The only states in Q ⊗ Q are of the form (q, q). Proof. Suppose not and suppose there exists a state in Q ⊗ Q of the form (q, q ) such that q = q . Since (q, q ) is reachable from (q0 , q0 ), there exists a path from (q0 , q0 ) to (q, q ) labelled by u say. Suppose (q, q ) is the ﬁrst such state along this path (otherwise repeat the argument for the ﬁrst such state) and suppose |u| = k. u

Then (q0 , q0 ) → (q, q ) and hence (q0 , q0 ) u(k)

u(1)...u(k−1)

u(k)

→

u(k)

(qk−1 , qk−1 ) → (q, q ). But

this means qk−1 → q and qk−1 → q contradicting the determinacy of Q. We next develop notions of when an FST captures the behaviour of another and when two FSTs are equivalent. Definition 3. Let Q1 = (Q1 , δ1 , q01 , f1 ) and Q2 = (Q2 , δ2 , q02 , f2 ) be deterministic FST’s. Q1 Q2 if there exists functions h : Q2 → Q1 and g : A2 → A1 where A2 and A1 are output alphabets of Q2 and Q1 respectively, such that: – h(q02 ) = q01 , – (q2 , a, q2 ) ∈ δ2 iﬀ (h(q2 ), a, h(q2 )) ∈ δ1 , – f2 (q) = g(f1 (h(q))). Q1 ≡ Q2 if Q1 Q2 and Q2 Q1 . Definition 4. Let (G1 , v01 ) and (G2 , v02 ) be two initialised directed edge labelled graphs where G1 = (V1 , E1 ) and G2 = (V2 , E2 ). G1 G2 if there exists h : V2 → a a V1 such that h(v02 ) = v01 and v2 → v2 iﬀ h(v2 ) → h(v2 ). G1 ≡ G2 if G1 G2 and G2 G1 . Claim. Q1 ⊗ Q2 ≡ Q2 ⊗ Q1 .

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Proof. Let Q1 = (Q1 , δ1 , q01 , f1 ) and Q2 = (Q2 , δ2 , q02 , f2 ) be deterministic FST’s. Let Q1 ⊗ Q2 = (Q, δ, q0 , f ) and let Q2 ⊗ Q1 = (Q , δ , q0 , f ). We show Q1 ⊗ Q2 Q2 ⊗ Q1 . Let h(q2 , q1 ) = (q1 , q2 ) and g(X2 , X1 ) = (X1 , X2 ). Now, we have: (q2 , q1 ), a, (q2 , q1 ) ∈ δ iﬀ (q2 , a, q2 ) ∈ δ1 and (q1 , a, q1 ) ∈ δ2 iﬀ (q1 , q2 ), a, (q1 , q2 ) ∈ δ , that is, h(q2 , q1 ), a, h(q2 , q1 ) ∈ δ . Let f (q2 , q1 ) = (X2 , X1 ), i.e., f2 (q2 ) = X2 , f1 (q1 ) = X1 . Thus, by deﬁnition, f (q1 , q2 ) = (X1 , X2 ), that is, f (q2 , q1 ) = g(f (h(q2 , q1 ))). The other direction is symmetrical where the function h (say) is given as h (q2 , q1 ) = (q1 , q2 ). Note that in the above proof, since h = h−1 we have that Q1 ⊗ Q2 is, in fact, isomorphic to Q2 ⊗ Q1 , the isomorphism being h. Claim. Q ⊗ Q ≡ Q. Proof. Let Q = (Q, δ, q0 , f ) and Q ⊗ Q = (Q , δ , q0 , f ). By earlier claim the states of Q ⊗ Q are only of the form (q, q). To see that Q ⊗ Q Q let h(q) = (q, q) and g(X) = X. Then we have (q, a, q ) ∈ δ iﬀ (q, q), a, (q , q ) ∈ δ , i.e., h(q), a, h(q ) ∈ δ . And f (q) = X iﬀ f (q, q) = X, i.e., f (q) = g(f (h(q))). To see that Q Q ⊗ Q, let h (q, q) = q and g (X) = X. The argument is then similar to above. Once more note that in the above proof, since h = h−1 we have that Q ⊗ Q is isomorphic to Q, the isomorphism being h. Let (A, v0 ) be an initialised arena where A = (V, E). Suppose every player is one of t types where each type is speciﬁed as an FST, Q say. We deﬁne the restriction of A with respect to Q, denoted by A ⊗ Q, as follows: Definition 5. Let Q = (Q, δ, q0 , f ) be an FST of type i. Let A ⊗ Q = (A , v0 ) such that – V = V × Q, – v0 = (v0 , q0 ), a a – (v, q) → (v , q ) iﬀ v → v , (q, a, q ) ∈ δ and a(i) ∈ f (q). Definition 6. Let Q1 ⊗ · · · ⊗ Qn = (Q, δ, q0 , f ). A ⊗ (Q1 ⊗ · · · ⊗ Qn ) = (A v0 ) such that A = (V , E ) – V = V × Q, – v0 = (v0 , q0 ), a a – (v, q) → (v , q ), where v is of type i, iﬀ v → v , (q, a, q ) ∈ δ and a(i) ∈ (f (q))(i), for all i ∈ N . It is then easy to see that: Claim. A ⊗ (Q1 ⊗ · · · ⊗ Qn ) ≡ (· · · ((A ⊗ Q1 ) ⊗ Q2 ) · · · ⊗ Qk ) Let A = (V, E) be an arena and P be a set of atomic propositions and val : V → 2P be a valuation function. Let α be a formula from the syntax Φ+ . We say α is stable in a subarena Z = (VZ , EZ ) of A if t |= ϕ for all nodes t ∈ TZ for unfoldings TZ starting at every node z ∈ VZ .

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Proposition 3. Let Q, Q1 and Q2 be FSTs for types of players. We have (i) α is stable in A ⊗ Q iﬀ α is stable in A ⊗ (Q ⊗ Q). (ii) α is stable in A ⊗ Q1 ⊗ Q2 iﬀ α is stable in A ⊗ Q2 ⊗ Q1 . Proof. (i) Let A ⊗ Q = (V1 , E1 ) and let A ⊗ (Q ⊗ Q) = (V2 , E2 ). Since we know that Q is isomorphic to Q ⊗ Q, we have, by construction, that A ⊗ Q is isomorphic to A⊗ (Q ⊗ Q). Let h be this isomorphism. Let TA⊗Q (v0 , q0 ) denote the unfolding of A ⊗ Q at any node (v0 , q0 ) ∈ V1 and let TA⊗(Q⊗Q) (v0 , q0 , q0 ) denote the unfolding of A ⊗ (Q ⊗ Q) at any node (v0 , q0 , q0 ) ∈ V2 . Suppose α is stable in A⊗Q. Then, by deﬁnition, for any node t = (v, q, u) ∈ TA⊗Q (v0 , q0 ), (v0 , q0 ) ∈ V1 , we have t |= α. We then have to show that h−1 (v, q), u |= α for the node h−1 (v, q), u ∈ TA⊗(Q⊗Q) (h−1 (v0 , q0 )). The only interesting case is when α ≡ a+ α. (v, q, u) |= a+ α iﬀ there exists a (v , q , ua) such that (v, q) → (v , q ) and (v , q , ua) |= α iﬀ (v , q , q , ua) |= α, that is, iﬀ h−1 (v, q), u |= α. The other direction is similar. (ii) Let A ⊗ Q1 ⊗ Q2 = (V1 , E1 ) and let A ⊗ Q2 ⊗ Q1 = (V2 , E2 ). Since we know that Q1 ⊗ Q2 is isomorphic to Q2 ⊗ Q1 , we have, by construction, that A ⊗ Q1 ⊗ Q2 is isomorphic to A ⊗ Q2 ⊗ Q1 . Let h be this isomorphism. Let TA⊗Q1 ⊗Q2 (v0 , q01 , q02 ) denote the unfolding of A ⊗ Q1 ⊗ Q2 at any node (v0 , q01 , q02 ) ∈ V1 and let TA⊗Q2 ⊗Q1 (v0 , q02 , q01 ) denote the unfolding of A ⊗ Q2 ⊗ Q1 at any node (v0 , q02 , q01 ) ∈ V2 . Suppose α is stable in A ⊗ Q1 ⊗ Q2 . Then, by deﬁnition, for any node t = (v, q1 , q2 , u) ∈ TA⊗Q1 ⊗Q2 (v0 , q01 , q02 ), (v0 , q01 , q02 ) ∈ V1 , we have t |= α. We have to show that h−1 (v, q1 , q2 ), u |= α for the node h−1 (v, q1 , q2 ), u ∈ TA⊗Q2 ⊗Q1 (h−1 (v0 , q01 , q02 )). Once again, the only interesting case is when α ≡ a+ α. (v, q1 , q2 , u) |= a+ α iﬀ there exists (v , q1 , q2 , ua) such that a (v, q1 , q2 ) → (v , q1 , q2 ) and (v , q1 , q2 , ua) |= α iﬀ (v , q2 , q1 , ua) |= α, that is, iﬀ h−1 (v, q1 , q2 ), u |= α. The other direction is symmetrical. When Is the Construction Worthwhile? We thus see from Proposition 3 that when the players are of a ﬁxed number t of types, we can carry out our stability analyses using the FST speciﬁcations of only these types and infer correctly about the outcome of the entire game; instead of dealing with the strategy FSTs of all of the n players. This might be helpful in situations when t << n which is usually the case. However, there is a price to pay. One can construct simple counterexamples to show that the type FSTs necessarily need to be deterministic for the above analysis to go through. Hence there is an exponential blowup in the size of the FSTs that are used in the analysis. So a natural question to ask is when is it worthwhile to carry out the above construction. Let A = (V, E) be an arena with |V | = m. Let there be a total of n players divided into t types and let χ be the mapping of players to their types. Let the ith type be given by the nondeterministic FST Ri , having state set Ri . Let p = maxi |Ri |. Let Qi be the determinisation of Ri . Qi has a state set of size at most 2p .

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49

The number of nodes in the graph A ⊗ Rχ(1) ⊗ . . . Rχ(n) is at most m · p · . . . · p = m · pn. On the other hand, the number of nodes in the graph A ⊗ n

p Q1 ⊗ . . . ⊗ Qt is at most m · 2 . . · 2p = m · 2tp . Hence the construction is · . t

worthwhile only when m · 2tp < m · pn . That is when n log2 p > tp. That is when n > 0.693 · t · π(p), where π(p) is the number of primes less than or equal to p.

5

Discussion

We have presented a simple quantitative model for reasoning about endogenous societal interventions based on player strategies. This should be seen as the analogue of mechanism design in our framework. While we have conﬁned our study here to removal of actions (and players), introduction of new actions and players is also interesting, and needs considerable changes in the framework. Another line of work relates to hierarchies: there is no reason to limit the interaction studied here to one level of social aggregation, except that of technical convenience. At this preliminary level, we have conﬁned our analysis to nondeterministic rules and strategy speciﬁcations. The model would be better developed when it is based on expectations both on the part of players and societal interventions. On the other hand, a logical study of the mutual interdependence of player and social behavior would be very interesting.

References 1. Alur, R., Henzinger, T., Kupferman, O.: Alternating-time temporal logic. Journal of ACM 49(5), 672–713 (2002) 2. van Benthem, J.: Games in dynamic epistemic logic. Bulletin of Economic Research 53(4), 219–248 (2001) 3. van Benthem, J.: Extensive games as process models. Journal of Logic Language and Information 11, 289–313 (2002) 4. van Benthem, J.: In praise of strategies. In: van Eijck, J., Verbrugge, R. (eds.) Foundations of Social Software. Studies in Logic, pp. 283–317. College Publications (2007) 5. Binmore, K., Samuelson, L.: Evolutionary drift and equilibrium selection. Review of Economic Studies 66, 363–393 (1999) 6. Blonski, M.: Anonymous games with binary actions. Games and Economic Behaviour 28, 171–180 (1999) 7. Blonski, M.: Characterisation of pure strategy equilibria in ﬁnite anonymous games. Journal of Mathematical Economics 34, 225–233 (2000) 8. Bonano, G.: Branching time logic, perfect information games and backward induction. Games and Economic Behaviour 36(1), 57–73 (2001) 9. Brandt, F., Fischer, F., Holzer, M.: Symmetries and the complexity of pure nash equilibrium. Journal of Computer and System Sciences 75(3), 163–177 (2009)

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10. Daskalakis, C., Papadimitriou, C.H.: Computing equilibria in anonymous games. In: Proceedings of the 48th Symposium on Foundations of Computer Science (FOCS), pp. 83–93. IEEE Computer Society Press, Los Alamitos (2007) 11. Ehrenfeucht, A., Mycielski, J.: Position strategies for mean payoﬀ games. International Journal of Game Theory 8, 109–113 (1979) 12. Ghosh, S.: Strategies made explicit in dynamic game logic. In: Logic and the Foundations of Game and Decision Theory (2008) 13. Hashimoto, T., Kumagai, Y.: Meta-evolutionary game dynamics for mathematical modelling of rules dynamics. In: Banzhaf, W., Ziegler, J., Christaller, T., Dittrich, P., Kim, J.T. (eds.) ECAL 2003. LNCS (LNAI), vol. 2801, pp. 107–117. Springer, Heidelberg (2003) 14. van der Hoek, W., Jamroga, W., Wooldridge, M.: A logic for strategic reasoning. In: Proceedings of the Fourth International Joint Conference on Autonomous Agents and Multi-Agent Systems, pp. 157–164 (2005) 15. van der Hoek, W., Wooldridge, M.: Cooperation, knowledge, and time: Alternatingtime temporal epistemic logic and its applications. Studia Logica 75(1), 125–157 (2003) 16. Horst, U.: Dynamic systems of social interactions. In: NSF/CEME Mathematical Economics Conference at Berkeley (2005) 17. Jamroga, W., van der Hoek, W., Wooldridge, M.: Intentions and strategies in game-like scenarios. In: Bento, C., Cardoso, A., Dias, G. (eds.) EPIA 2005. LNCS (LNAI), vol. 3808, pp. 512–523. Springer, Heidelberg (2005) 18. Parke, W., Waters, G.: An evolutionary game theory explanation of ARCH eﬀects. Journal of Economic Dynamic and Control 31, 2234–2262 (2007) 19. Paul, S., Ramanujam, R., Simon, S.: Dynamic restriction of choices: A preliminary logical report. In: Proceedings of the 12th Conference on Theoretical Aspects of Rationality and Knowledge (TARK), pp. 218–226 (2009) 20. Peyton Young, H.: The diﬀusion of innovations in social networks. Economics Working Paper Archive 437, The Johns Hopkins University, Department of Economics (May 2000) 21. Ramanujam, R., Simon, S.: Dynamic logic on games with structured strategies. In: Proceedings of the Conference on Principles of Knowledge Representation and Reasoning, pp. 49–58 (2008) 22. Skyrms, B., Pemantle, R.: A dynamic model of social network formation. Proceedings of the National Academy of Sciences 97(16), 9340–9346 (2000) 23. Weibull, J.W.: Evolutionary Game Theory. MIT Press, Cambridge (1997)

Agreeing to Disagree with Limit Knowledge Christian W. Bach1 and J´er´emie Cabessa2, 1

Department of Quantitative Economics Maastricht University 6200 MD Maastricht, The Netherlands [email protected] 2 Department of Computer Science University of Massachusetts Amherst Amherst, MA 01003-9264, United States [email protected]

Abstract. The possibility for agents to agree to disagree is considered in an extended epistemic-topological framework. In such an enriched context, Aumann’s impossibility theorem is shown to no longer hold. More precisely, agents with a common prior belief satisfying limit knowledge instead of common knowledge of their posterior beliefs may actually entertain distinct posterior beliefs. Hence, agents can actually agree to disagree. In particular, agreeing to disagree with limit knowledge is illustrated within a representative epistemic-topological situation. Keywords: agreeing to disagree, agreement theorems, limit knowledge, interactive epistemology.

1

Introduction

The so-called agreement theorem by Aumann [1] establishes the impossibility for two Bayesian agents with a common prior belief to entertain common knowledge of posterior beliefs that are distinct. Understanding two individuals as likeminded if they are both Bayesian and equipped with exactly the same prior information, Aumann’s seminal result states that two like-minded individuals that get access to diﬀering information cannot entertain opposing opinions in the case of their opinions being common knowledge. In other words, the agents cannot agree to disagree. Along these lines Milgrom and Stokey [11] establish an impossibility theorem of speculative trade. Intuitively, their result states that if two traders agree on a prior eﬃcient allocation of goods, then upon receiving private information it cannot be common knowledge that both traders have an incentive to trade. From an empirical or quasi-empirical point of view, the agreement theorem seems quite startling since real world agents do frequently disagree on a large variety of issues. It is then natural to scrutinize whether Aumann’s basic

Research support from the Swiss National Science Foundation (SNSF) under grant PBLAP2-132975 is gratefully acknowledged.

H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 51–60, 2011. c Springer-Verlag Berlin Heidelberg 2011

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result still holds with weakened or slightly modiﬁed assumptions. In this spirit, Geanakoplos and Polemarchakis [10] drop the assumption of common knowledge of posteriors and show that two Bayesian agents with common priors and ﬁnite information partitions already agree on their posterior beliefs after ﬁnitely many rounds of communicating their respectively updated posteriors back and forth. Note that, although common knowledge of the posteriors is not needed prior to the communication procedure, it does actually hold after sameness of the posteriors has been established. Moreover, Monderer and Samet [12] replace common knowledge by the weaker concept of common p-belief and establish an agreement theorem with such an approximation of common knowledge. Indeed, they show that if the posteriors of Bayesian agents equipped with a common prior are common p-belief for large enough p, then these posteriors cannot diﬀer signiﬁcantly. Besides, Samet [13] drops the implicit negative introspection assumption – which states that agents know what they do not know – and shows that Aumann’s agreement theorem remains valid with agents ignorant of their own ignorance. Further works on Aumann’s agreement theorem are surveyed in Bonanno and Nehring [9]. Here, Aumann’s result is revisited in an extended epistemic-topological framework. The epistemic operator common knowledge is replaced by the epistemictopological operator limit knowledge introduced and studied by Bach and Cabessa [7,8]. Assuming common priors, Bayesian agents, and limit knowledge of posteriors, we show that the agents’ posteriors may diﬀer. Thus, agents can indeed agree to disagree.

2

An Epistemic-Topological Approach to Agreeing to Disagree

Set-based interactive epistemology provides the formal framework in which the agreement theorem is modelled. Having been introduced and notably developed by Aumann [1,2,3,4,5,6], the discipline provides tools to formalize epistemic notions in interactive situations. A so-called Aumann structure A = (Ω, (Ii )i∈I , p) consists of a countable set Ω of possible worlds, which are complete descriptions of the way the world might be, a ﬁnite set of agents I, a possibility partition Ii of Ω for each agent i ∈ I representing his information, and a common prior belief function p : Ω → [0, 1] such that ω∈Ω p(ω) = 1. The cell of Ii containing the world ω is denoted by Ii (ω) and consists of all worlds considered possible by i at world ω. In other words, agent i cannot distinguish between any two worlds ω and ω that are in the same cell of his partition Ii . Moreover, an event E ⊆ Ω is deﬁned as a set of possible worlds. For example, the event of it raining in London contains all worlds in which it does rain in London. Note that the common prior belief function p can naturally be extended to a common prior belief measure on the event space p : P(Ω) → [0, 1] by setting p(E) = ω∈E p(ω). In this context, it is supposed that each information set of each agent has non-zero prior probability, i.e. p(Ii (ω)) > 0 for all i ∈ I and ω ∈ Ω. Moreover, all agents are assumed to

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be Bayesians and to hence update the common prior belief given their private information according to Bayes’s rule. More precisely, given some event E and some world ω, the posterior belief of agent i in E at ω is given by p(E | Ii (ω)) = p(E∩Ii (ω)) p(Ii (ω)) . Farther, an Aumann structure A = (Ω, (Ii )i∈I ) is called ﬁnite if Ω is ﬁnite and inﬁnite otherwise. In Aumann’s epistemic framework, knowledge is formalized in terms of events. More precisely, the event of agent i knowing E, denoted by Ki (E), is deﬁned as Ki (E) := {ω ∈ Ω : Ii (ω) ⊆ E}. If ω ∈ Ki (E), then i is said to know E at world ω. Intuitively, i knows some event Eif in all worlds he considers possible E holds. Naturally, the event K(E) = i∈I Ki (E) then denotes mutual knowledge of E among the set I of agents. Letting K 0 (E) := E, m-order mutual knowledge of the event E among the set I of agents is inductively deﬁned by K m+1 (E) := K(K m (E)) for all m ≥ 0. Accordingly, mutual knowledge can also be denoted as 1-order mutual knowledge. Furthermore, an event is said to be common knowledge among a set I of agents whenever all m-order mutual knowledge of it simultaneously hold. Formally, it is standard to deﬁne the event that E is common knowledge among the set I of agents as CK(E) := m>0 K m (E). Aumann’s agreement theorem states that if two agents entertain a common prior belief function and their posterior beliefs in some event are common knowledge, then these posterior beliefs must coincide. In other words, if two agents with common prior beliefs hold distinct posterior beliefs, then these posterior beliefs cannot be common knowledge among them. Intuitively, it is impossible for agents to consent to distinct beliefs. Thus, agents cannot agree to disagree. Now, the impossibility for agents to agree to disagree is considered from a topologically enriched epistemic perspective. In fact, the standard set-based approach to interactive epistemology lacks a general framework providing some formal notion of closeness between events. An amended topological dimension could be capable of introducing an agent perception of closeness of events. In such a more general epistemic-topological framework, the reasoning of agents may thus also depend on topological instead of mere epistemic features of the underlying interactive situation. In this context, Bach and Cabessa [7,8] consider Aumann structures equipped with topologies on the event space and introduce the operator limit knowledge, which is linked to epistemic features as well as topological aspects of the event space. More precisely, limit knowledge is deﬁned as the topological limit of higher-order mutual knowledge. Definition 1. Let (Ω, (Ii )i∈I , p) be an Aumann structure, T a topology on P(Ω), and E an event. If the limit point of the sequence (K m (E))m>0 is unique, then LK(E) := limm→∞ K m (E) is the event that E is limit knowledge among the set I of agents. Accordingly, limit knowledge of an event E is constituted by – whenever unique – the limit point of the sequence of iterated mutual knowledge, and thus linked to both epistemic as well as topological aspects of the event space.

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Limit knowledge can be understood as the event which is approached by the sequence of iterated mutual knowledge, according to some notion of closeness between events furnished by a topology on the event space. Thus, the higher the iterated mutual knowledge, the closer this latter epistemic event is to limit knowledge. Note that limit knowledge should not be amalgamated with common knowledge. Indeed, both operators can be perceived as sharing distinct implicative properties with regards to highest iterated mutual knowledge claims. While common knowledge bears a standard implicative relation in terms of set inclusion to highest iterated mutual knowledge, limit knowledge entertains an implicative relation in terms of set proximity with highest iterated mutual knowledge. Besides, limit knowledge also diﬀers from Monderer and Samet’s [12] notion of common p-belief. Indeed, common p-belief – as an approximation of common knowledge in the sense of common almost-knowledge – is implied by common knowledge, whereas limit knowledge is not. Actually, it is possible to link limit knowledge to topological reasoning patterns of agents based on closeness of events. Indeed, agents satisfying limit knowledge of some event are in a limit situation arbitrarily close to entertaining all highest iterated mutual knowledge of this event, and the agents’ reasoning may be inﬂuenced accordingly. Note that a reasoning pattern associated with limit knowledge depends on the particular topology on the event space, which ﬁxes the closeness relation between events. The operator limit knowledge is shown by Bach and Cabessa [7,8] to be able to provide relevant epistemic-topological characterizations of solution concepts in games. Despite being based on the same sequence of higher-order mutual knowledge claims, the distinguished interest of limit knowledge resides in its capacity to potentially diﬀer from the purely epistemic operator common knowledge. Notably, it can be proven that such diﬀering situations necessarily require an inﬁnite event space as well as sequences of higher-order mutual knowledge that are strictly shrinking.1 In fact, the topologically amended epistemic framework enables agents with a common prior belief to agree to disagree on their posterior beliefs. Theorem 1. There exist an Aumann structure A = (Ω, (Ii )i∈I , p) equipped with a topology T on the ˆ ∈Ω event space P(Ω), an event E ⊆ Ω, and worlds ω, ω such that ω ∈ LK( i∈I {ω ∈ Ω : p(E | Ii (ω )) = p(E | Ii (ˆ ω ))}), as well as both p(E | Ii (ˆ ω )) = p(E | Ij (ˆ ω )) and p(E | Ii (ω)) = p(E | Ij (ω)) for some agents i, j ∈ I. Proof. Consider the Aumann structure A = (Ω, (Ii )i∈I , p), where Ω = {ωn : n ≥ 0}, I = {Alice, Bob}, IAlice = {{ω2n, ω2n+1 } : n ≥ 0}, IBob = {{ω0 }} ∪ 1 {{ω2n+1 , ω2n+2 } : n ≥ 0}, and p : Ω → R is given by p(ωn ) = 2n+1 for all1 n ≥ 0. Note that the common prior belief function p is well deﬁned since n≥0 2n+1 = 1. Now, consider the event E = {ω2n : n ≥ 1}, and the world ω ∈ Ω. Besides, for 2 sake of notational convenience, let the event i∈I {ω ∈ Ω : p(E | Ii (ω )) = 1

Given some event E, the sequence of higher-order mutual knowledge (K m (E))m>0 is called strictly shrinking if K m+1 (E) K m (E) for all m ≥ 0.

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p(E | Ii (ω2 ))} be denoted by E . First of all, observe that p(E | IAlice (ω2 )) = 23 and p(E | IBob (ω2 )) = 13 . Moreover, {ω ∈ Ω : p(E | IAlice (ω )) = p(E | IAlice (ω2 )) = 23 } = Ω \ {ω0 , ω1 } and {ω ∈ Ω : p(E | IBob (ω )) = p(E | IBob (ω2 )) = 13 } = Ω \ {ω0 }, whence E = (Ω \ {ω0 , ω1 }) ∩ (Ω \ {ω0 }) = Ω \ {ω0 , ω1 }. Farther, the deﬁnitions of the possibility partitions of Alice and Bob ensure that K m (E ) = K m (Ω \ {ω0 , ω1 }) = Ω \ {ω0 , ω1 , . . . , ωm+1 }, for all m > 0. Consequently, (K m (E ))m>0 is strictly shrinking and the sequence CK(E ) = {ω ∈ Ω : i∈I Ii (ω) ⊆ E } = ∅. Now, consider the topology T on P(Ω) deﬁned by T = {O ⊆ P(Ω) : {ω0 , ω1 , ω2 } ∈ O} ∪ {P(Ω)}. Then, the only open neighbourhood of the event {ω0 , ω1 , ω2 } is P(Ω), and all terms of the sequence (K m (E ))m>0 are contained in P(Ω). Thus (K m (E ))m>0 converges to {ω0 , ω1 , ω2 }. Moreover, for every event F ∈ P(Ω) such that F = {ω0 , ω1 , ω2 }, the singleton {F } is open, and since K m+1 (E ) K m (E ) for all m > 0, the strictly shrinking sequence (K m (E ))m>0 will never remain in the open neighbourhood {F } of F from some index onwards. Hence (K m (E ))m>0 does not converge to any such event F . Therefore the limit point {ω0 , ω1 , ω2 } of the strictly shrinking sequence (K m (E ))m>0 is unique, and LK(E ) = limm→∞ K m (E ) = {ω0 , ω1 , ω2 }. Next, consider the world ω1 . Note that ω1 ∈ LK(E ). Also, observe that p(E | IAlice (ω2 )) = 23 = 13 = p(E | IBob (ω2 )) as well as p(E | IAlice (ω1 )) = 0 = 13 = p(E | IBob (ω1 )). Finally, taking ω = ω1 and ω ˆ = ω2 concludes the proof.

The preceding possibility result counters Aumann’s impossibility theorem in the sense of showing that agents actually can agree to disagree. More precisely, agents may hold distinct actual posterior beliefs, while at the same time satisfying limit knowledge of their posteriors. Hence, agents may agree in the sense of satisfying limit knowledge of their posteriors, while at the same time disagree in the sense of actually entertaining diﬀerent posterior beliefs. Generally speaking, the mere fact of topologically enriching the event space concurrently with replacing the purely epistemic operator common knowledge by the epistemic-topological operator limit knowledge enables our possibility result. In such an amended perspective, agents can now be seen to have access to a further dimension in their reasoning that remarkably permits them to agree to disagree on their posterior beliefs. In fact, the agents are in a limit situation of entertaining higher-order mutual knowledge of their posteriors, which, in connection with the particular notion of closeness furnished by the topology, leads them to actually possess diﬀerent posterior beliefs.

3

A Representative Example

The extension of the standard set-based approach to interactive epistemology with a topological dimension has been shown to enable the possibility for agents to agree to disagree on their posterior beliefs. The question then arises whether agents can still agree to disagree in interactive situations furnished with topologies based on epistemic features. A topology describing a speciﬁc agents’ perception of the event space is now presented and is then shown to enable agreeing to disagree with limit knowledge.

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Towards this purpose, suppose an Aumann structure A = (Ω, (Ii )i∈I , p) and an event E. Farther, for any world ω ∈ Ω, let Eω denote the event consisting of that induce the same agents’ posterior beliefs in E as ω, i.e. Eω = all worlds i∈I {ω ∈ Ω : p(E | Ii (ω )) = p(E | Ii (ω))}. Note that constancy of the agents’ posterior beliefs in E yields an equivalence relation on the set of possible worlds, and hence every Eω represents an equivalence class of worlds. Consequently, the collection C = {Eω : ω ∈ Ω} of all equivalence classes of worlds that induce a same posterior belief proﬁle forms a partition of Ω. Given some event E and some index m∗ > 0, the epistemically-based topology TE,m∗ is deﬁned as the topology on the event space P(Ω) generated by the subbase {{K m (Eω ) : m ≥ 0} : ω ∈ Ω} ∪ {P(Ω) \ {K m (Eω ) : m ≥ 0 and ω ∈ Ω}}

∪ {{K m (Eω )} : 0 ≤ m < m∗ and ω ∈ Ω} ∗ ∪ {{K m +j (Eω ) : 0 < j ≤ n} : n > 0 and ω ∈ Ω}. The topology TE,m∗ is illustrated in Figure 1, where the inﬁnite sequence (K m (Eω ))m≥0 is represented by a horizontal sequence of points for each ω ∈ Ω, and open sets of the subbase by circle-type shapes around these points.

P(Ω)

Eω

K(Eω )

K 2 (Eω )

∗

K m (Eω )

Km

∗

+1

(Eω )

∗

Km

+2

(Eω )

∗

Km

+3

(Eω )

Fig. 1. Illustration of the topology TE,m∗

The topology TE,m∗ reveals a speciﬁc agent perception of the event space, according to which the agents entertain a more reﬁned distinction between the m∗ ﬁrst iterated mutual knowledge of their posterior beliefs in E than between the remaining ones. This speciﬁc perception is formally reﬂected by two separation properties satisﬁed by the topology TE,m∗ .

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Firstly, given two events X and Y , if X and Y are two distinct terms of a same sequence (K m (Eω ))m>0 for some ω ∈ Ω, and both are iterated mutual knowledge of order strictly smaller than m∗ in this sequence, then X and Y are T2 -separable, and therefore also T0 -separable.2 Secondly, if X and Y are two diﬀerent elements of a same sequence (K m (Eω ))m>0 for some ω ∈ Ω, and both are iterated mutual knowledge of order strictly larger than m∗ in this sequence, then X and Y are T0 -separable, yet not T2 -separable. According to these two separation properties, agents have access to a more reﬁned distinction between the m∗ ﬁrst iterated knowledge claims of their posterior beliefs in E than between the iterated mutual knowledge claims of order strictly larger than m∗ . In other words, iterated mutual knowledge claims are only precisely discerned up to a given amount of iterations, and thereafter the higher iterations become less distinguishable for the agents. Also, from a bounded rationality point of view, the agent perception of higher-order mutual knowledge furnished by the topology TE,m∗ reﬂects that people typically lose track from some iteration level onwards when reasoning about higher-order mutual knowledge. Farther, the topology TE,m∗ notably satisﬁes the following epistemic-topological property: for any event Eω , if the sequence (K m (Eω ))m>0 is strictly shrink∗ ing, then LK(Eω ) = K m (Eω ). Indeed, suppose that the sequence (K m (Eω ))m>0 is strictly shrinking. Then, by deﬁnition of TE,m∗ , the only open neighbour∗ hoods of K m (Eω ) are P(Ω) and {K m (Eω ) : m ≥ 0}. Since both sets contain ∗ all terms of the sequence (K m (Eω ))m>0 , it follows that K m (Eω ) is a limit point of the sequence (K m (Eω ))m>0 . To see that this limit point is actually ∗ unique, consider F ∈ P(Ω) such that F = K m (Eω ). Then either F = K m (Eω ) for some m < m∗ and some ω ∈ Ω, or F = K m (Eω ) for some m > m∗ ∗ and some ω ∈ Ω, or F = K m (Eω ) for some ω = ω, or F = K m (Eω ) for all m ≥ 0 and all ω ∈ Ω. These four mutually exclusive cases are now considered in turn. First of all, if F = K m (Eω ) for some m < m∗ and some ω ∈ Ω, then {K m (Eω )} is an open neighbourhood of F . Since the sequence (K m (Eω ))m>0 is strictly shrinking, it can then not be the case that the singleton open neighbourhood {K m (Eω )} of F contains all terms of the sequence (K m (Eω ))m>0 from some index onwards. Therefore F is not a limit point of the sequence (K m (Eω ))m>0 . Secondly, if F = K m (Eω ) for some m > m∗ and ∗ some ω ∈ Ω, then {K m +j (Eω ) : 0 < j ≤ m − m∗ } is an open neighbourhood m∗ +j of F . Since the set {K (Eω ) : 0 < j ≤ m − m∗ } is ﬁnite, F cannot be a ∗ limit point of the sequence (K m (Eω ))m>0 . Thirdly, if F = K m (Eω ) for some n ω = ω, then {K (Eω ) : n ≥ 0} is an open neighbourhood of F . Moreover, ∗ ∗ since K m (Eω ) = K m (Eω ) = F , it directly follows that Eω = Eω . Yet since C = {Eω : ω ∈ Ω} is a partition of Ω, it holds that Eω ∩ Eω = ∅. Moreover, as K m (Eω ) ⊆ Eω for all m ≥ 0, and K n (Eω ) ⊆ Eω for all n ≥ 0, as well as Eω ∩ Eω = ∅, it follows that K m (Eω ) = K n (Eω ) for all m, n ≥ 0. Thus the 2

Given a topological space (A, T ), two points in A are called T2 -separable if there exist two disjoint T -open neighbourhoods of these two points. Moreover, two points in A are called T0 -separable if there exists a T -open set containing precisely one of these two points. Note that T2 -separability implies T0 -separability.

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open neighbourhood {K n(Eω ) : n ≥ 0} of F contains no term of the sequence (K m (Eω ))m>0 whatsoever. Therefore, F is not a limit point of the sequence (K m (Eω ))m>0 . Fourthly, if F = K m (Eω ) for all m ≥ 0 and all ω ∈ Ω, then P(Ω) \ {K m (Eω ) : m ≥ 0 and ω ∈ Ω} is an open neighbourhood of F . Yet this set contains no term of the sequence (K m (Eω ))m>0 . Thus F is not a limit point of the sequence (K m (Eω ))m>0 . To summarize, there consequently exists no ∗ F = K m (Eω ) which is a limit point of the sequence (K m (Eω ))m>0 . Therefore, ∗ the limit point K m (Eω ) of the sequence (K m (Eω ))m>0 is unique, and thence ∗ LK(Eω ) = limm→∞ K m (Eω ) = K m (Eω ). Furthermore, since the sequence ∗ (K m (Eω ))m>0 is strictly shrinking, CK(Eω ) = m>0 K m (Eω ) K m (Eω ), and hence CK(Eω ) = LK(Eω ). Finally, the following example describes an interactive situation, in which the epistemically-based topology TE,m∗ provides a possibility for the agents to agree to disagree on their posterior beliefs with limit knowledge. Example 1. Consider the Aumann structure A = (Ω, (Ii )i∈I , p), where Ω = {ωn : n ≥ 0}, I = {Alice, Bob}, IAlice = {{ω0 }, {ω1 , ω2 }, {ω3 , ω4 , ω5 , ω6 }, {ω7 , ω8 , ω9 }} ∪ {{ω2n, ω2n+1 } : n ≥ 5}, IBob = {{ω0 , ω1 , ω2 , ω3 , ω4 }, {ω5 , ω6 , ω7 , ω8 }} ∪ {{ω2n+1 , ω2n+2 } : n ≥ 4}, and p : Ω → R is given by p(ωn ) = 1 for all n ≥ 0. Also, consider the event E = {ω1 , ω5 } ∪ {ω2n : n ≥ 1} 2n+1 and the world ω10 . Besides, for sake of notational convenience, let the event {ω ∈ Ω : p(E | Ii (ω )) = p(E | Ii (ω10 ))} be denoted by E . First of all, i∈I observe that the computation of the posterior beliefs of Alice and Bob gives a variety of distinct values for the ﬁrst ten worlds {ω0 , ω1 , . . . , ω9 }, as well as p(E | IAlice (ωn )) = 23 and p(E | IBob (ωn )) = 13 , for all n ≥ 10. It follows that {ω ∈ Ω : p(E | IAlice (ω )) = p(E | IAlice (ω10 ))} = Ω \ {ω0 , ω1 , . . . , ω9 } and {ω ∈ Ω : p(E | IBob (ω )) = p(E | IBob (ω10 ))} = Ω \ {ω0 , ω1 , . . . , ω8 }, thus E = (Ω\{ω0 , ω1 , . . . , ω9 })∩(Ω\{ω0 , ω1 , . . . , ω8 }) = Ω\{ω0 , ω1 , . . . , ω9 }. Moreover, the deﬁnitions of the possibility partitions of Alice and Bob ensure that K m (E ) = Ω \ {ω0 , ω1 , . . . , ωm+9 }, for all m > 0. Consequently, the sequence (K m (E ))m>0 is strictly shrinking and CK(E ) = m>0 K m (E ) = ∅. Now, let m∗ > 0 be some index and suppose that P(Ω) is equipped with the topology TE,m∗ . Since the sequence (K m (E ))m>0 is strictly shrinking, the deﬁnition of this topology ∗ ensures that LK(E ) = K m (E ) = Ω \ {ω0 , ω1 , . . . , ωm∗ +9 }. Consequently, the computations of the posterior beliefs of Alice and Bob give p(E | IAlice (ω)) = 23 and p(E | IBob (ω)) = 13 , for all ω ∈ LK(E ). In other words, for all ω ∈ LK(E ), it holds that p(E | IAlice (ω)) = p(E | IBob (ω)).

4

Conclusion

In an epistemic-topoloigcal framework, agents have been shown to be able to agree to disagree. More precisely, if Bayesian agents entertain a common prior belief in a given event as well as limit knowledge of their posterior beliefs in the event, then their actual posterior beliefs may indeed diﬀer. This possibility result also holds in interactive situations enriched by a particular epistemically-based topology revealing a cogent agent perception of the event space.

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The topological approach to set-based interactive epistemology, in which topologies model agent closeness perceptions of events, can be used to describe various agent reasoning patterns that do not only depend on mere epistemic but also on topological features of the underlying interactive situation. For instance, the event It is cloudy in London seems to be closer to the event It is raining in London than the event It is sunny in London. Now, agents may make identical decisions only being informed of the truth of some event within a class of close events. Indeed, Alice might decide to stay at home not only in the case of it raining outside, but also in the case of events perceived by her to be similar such as it being cloudy outside. Moreover, we envision the construction of a more general epistemic-topological framework – topological Aumann structures – comprising topologies not only on the event space but also on the state space. Such an extension permits an explicit consideration of a notion of closeness between events as well as between worlds, enabling to model common agent perceptions of the event and state spaces. In particular, it might be of distinguished interest to base topologies on ﬁrst principles such as epistemic axioms or natural closeness properties. In line with this perspective, the topology provided in Section 3 reﬂects the natural agent perception for which iterated mutual knowledge becomes imprecise from some level onwards. Besides, in order to model subjective rather than common agent perceptions of the event and state spaces, the epistemic-topological framework envisioned here could be amended by assigning speciﬁc and potentially distinct topologies to every agent. A collective topology reﬂecting a common closeness perception could then be constructed on the basis of the particular agent topologies, and limit knowledge be deﬁned in such a global topological context. For instance, by providing a topology that is coarser than each agent’s one, the meet topology could be used as a representative collective topology. Alternatively, an agent speciﬁc operator limit knowledge could be deﬁned with respect to each particular topology, and mutual limit knowledge as their intersection then be considered. Finally, in a general epistemic-topological framework, various issues can be addressed. For example, the possibility of agents to agree to disagree with limit knowledge can be further analyzed for other epistemically-based as well as agent speciﬁc topologies. Furthermore, analogously to the epistemic program in game theory that attempts to provide epistemic foundations for solution concepts, an epistemic-topological approach could generate epistemic-topological foundations for solution concepts. In addition, it could be attempted to develop a theory of counterfactuals in set-based interactive epistemology founded on a notion of similarity of worlds or events furnished by topologies on the state or event space, respectively. Acknowledgement. We are highly grateful to Richard Bradley, Adam Brandenburger, Jacques Duparc and Andr´es Perea for illuminating discussions and invaluable comments.

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References 1. Aumann, R.J.: Agreeing to disagree. Annals of Statistics 4(6), 1236–1239 (1976) 2. Aumann, R.J.: Correlated equilibrium as an expression of bayesian rationality. Econometrica 55(1), 1–18 (1987) 3. Aumann, R.J.: Backward induction and common knowledge of rationality. Games and Economic Behavior 8(1), 6–19 (1995) 4. Aumann, R.J.: Interactive epistemology i: Knowledge. International Journal of Game Theory 28(3), 263–300 (1999) 5. Aumann, R.J.: Interactive epistemology ii: Probability. International Journal of Game Theory 28(3), 301–314 (1999) 6. Aumann, R.J.: Musings on information and knowledge. Econ. Journal Watch 2(1), 88–96 (2005) 7. Bach, C.W., Cabessa, J.: Limit knowledge of rationality. In: Proceedings of the 12th Conference on Theoretical Aspects of Rationality and Knowledge, TARK 2009, pp. 34–40. ACM, New York (2009) 8. Bach, C.W., Cabessa, J.: Common knowledge and limit knowledge. Theory and Decision (to appear) 9. Bonanno, G., Nehring, K.: Agreeing to disagree: A survey. Working paper series no. 97-18, Department of Economics, University of California, Davis (1997) 10. Geanakoplos, J.D., Polemarchakis, H.M.: We can’t disagree forever. Journal of Economic Theory 28(1), 192–200 (1982) 11. Milgrom, P., Stokey, N.: Information, trade and common knowledge. Journal of Economic Theory 26(1), 17–27 (1982) 12. Monderer, D., Samet, D.: Approximating common knowledge with common beliefs. Games and Economic Behavior 1(2), 170–190 (1989) 13. Samet, D.: Ignoring ignorance and agreeing to disagree. Journal of Economic Theory 52(1), 190–207 (1990)

A Semantic Model for Vague Quantifiers Combining Fuzzy Theory and Supervaluation Theory Ka Fat Chow The Hong Kong Polytechnic University [email protected]

Abstract. This paper introduces a semantic model for vague quantifiers (VQs) combining Fuzzy Theory (FT) and Supervaluation Theory (ST), which are the two main theories on vagueness, a common source of uncertainty in natural language. After comparing FT and ST, I will develop the desired model and a numerical method for evaluating truth values of vague quantified statements, called the Modified Glöckner’s Method, that combines the merits and overcomes the demerits of the two theories. I will also show how the model can be applied to evaluate truth values of complex quantified statements with iterated VQs. Keywords: vague quantifiers, Generalized Quantifier Theory, Fuzzy Theory, Supervaluation Theory, Modified Glöckner’s Method.

1

Introduction

Vagueness is a common source of uncertainty in natural language. No doubt vague quantifiers (VQs) constitute an important type of quantifiers, the target of study of the Generalized Quantifier Theory (GQT). However, since it is difficult to model vagueness under standard Set Theory, the study on VQs has remained a weak point of GQT. In GQT, the most typical approach of representing the truth condition of a VQ is to represent it as a comparison between an expression consisting of the VQ’s arguments and a context-dependent standard. For example, according to [11], there are three interpretations of “many”. The truth condition of “many2” is as follows: many2(A)(B) ↔ |A ∩ B| ≥ k|A|

(1)

where k ∈ (0, 1) is a context-dependent constant. This condition says that “Many As are B” is true whenever the proportion of those As that are B among all As is at least as great as a standard, i.e. k, representing the threshold of “many”. Since k is dependent on context, the above condition may yield different truth values for two different quantified statements “Many A1s are B1” and “Many A2s are B2” even if |A1| = |A2| and |A1 ∩ B1| = |A2 ∩ B2|. While this approach is most straightforward, what it genuinely reflects is the context dependence rather than the vagueness of VQs. In this paper, I will leave aside the issue of context dependence and concentrate on the vagueness of VQs. Moreover, I will only deal with VQs in a general manner and will not work out the detailed H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 61–73, 2011. © Springer-Verlag Berlin Heidelberg 2011

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semantics of any particular VQ. Since vague concepts are characterized by blurred boundaries and uncertain membership, we need to invoke theories that deal with such phenomena. In the next section, I will introduce two such theories: Fuzzy Theory and Supervaluation Theory. The former is further divided into two approaches: the Fuzzy Set Cardinality Approach and the Quantifier Fuzzification Mechanism Approach.

2

Basic Theories for VQs

2.1

Fuzzy Theory (Fuzzy Set Cardinality Approach)

Fuzzy Theory (FT) is a cover term for all those theories that are based on or derived from the Fuzzy Set Theory developed by [14]. Ever since [14], FT has become a new paradigm and is widely applied in many areas. Under FT, vague concepts are modeled by fuzzy sets, which differ from crisp sets (i.e. non-fuzzy sets) in one important aspect: instead of having sharp boundaries between members and non-members, every individual in the universe belongs to a fuzzy set to a certain degree ranging from absolute membership to absolute non-membership. By using ║p║ to denote the truth value of a proposition p, we can represent this degree by a membership degree function (MDF), ║x ∈ S║, which outputs a numerical value in [0, 1] representing the degree to which an individual x belongs to a fuzzy set S1. For example, ║j ∈ TALL║ = 0.7 means that John is tall to the degree 0.7. Sometimes, the MDF may take the form of a mathematical function that depends on a numerical value (henceforth called the “input” of the MDF). For example, as the tallness of a person depends on the person’s height, the aforesaid MDF for TALL may take the alternative form ║h ∈ TALL║, where h represents the height of a person. Fuzzy theorists have also defined certain crisp sets corresponding to each fuzzy set. Let X be a fuzzy set and α be a real number in [0, 1]. Then the α-cut (denoted X≥α), and strict α-cut (denoted X>α) of X are defined as follows (in what follows, U represents the universe): X≥α = {x ∈ U: ║x ∈ X║ ≥ α}

(2)

X>α = {x ∈ U: ║x ∈ X║ > α}

(3)

Another characteristic of FT is that it treats Boolean operators (BOs) as truth functions such as2: ║p ∧ q║ = min({║p║, ║q║})

(4)

║p ∨ q║ = max({║p║, ║q║})

(5)

║¬p║ = 1 – ║p║

1

2

(6)

In the literature, the MDF is often expressed as μS(x). In this paper I use ║x ∈ S║ instead for convenience. In the literature, there is a whole range of possible definitions of BOs. What follows are the “standard” definitions of the most commonly used BOs.

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Inspired by GQT, fuzzy theorists also tried to formalize theories about VQs, with [15] and [12-13] being the earlier attempts. Since VQs can be seen as fuzzy sets of numbers, they can also be modeled by MDFs. For example, borrowing ideas from [1], we may represent the VQ “(about 10)” by the following MDF: ║(about 10)(A)(B)║ = T–4, –1, 1, 4(|A ∩ B| – 10)

(7)

There are several points to note concerning the above formula. First, I have adopted [8]’s notation that represents a quantified statement in the form of a tripartite structure “Q(A)(B)” where Q, A and B represent the quantifier and its two arguments, respectively3. Syntactically, these two arguments correspond to the subject (excluding the quantifier) and the predicate of the quantified statement. Second, the above formula makes use of a piecewise-defined function Ta, b, c, d(x) with the following definition:

Ta, b, c, d(x) =

0, (x – a) / (b – a),

if x < a if a ≤ x < b

1

if b ≤ x ≤ c

(d – x) / (d – c), 0,

if c < x ≤ d if x > d

(8)

The above function is named “T”, standing for “trapezoid”, because its graph has a trapezoidal shape. Figure 1 shows the graph of T–4, –1, 1, 4:

Fig. 1. T–4, –1, 1, 4

When the parameters are such that a = b or c = d, since there is no x such that a ≤ x < a or c ≤ x < c, the 2nd or 4th piece of (8) would disappear. In these cases, T becomes degenerate and its graph is shaped like half of a trapezoid. For example, Figure 2 shows the graph of T–0.25, –0.1, ∞, ∞: 3

In this paper, I only consider VQs that have two arguments. Using standard GQT notation, such kind of VQs belongs to type <1,1> quantifiers, also called “determiners”.

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Fig. 2. T–0.25, –0.1, ∞, ∞

Note that the function T given in (8) gives just one possible example of MDFs that may be used for representing VQs. In fact, any function whose general shape is similar to T can also serve the same purpose. More specifically, this function should be a function whose domain can be partitioned into 5 parts such that the values at the 1st, 2nd, 3rd, 4th and 5th parts are constantly 0, increasing, constantly 1, decreasing and constantly 0, respectively4. Using the MDFs for VQs, one can then evaluate the truth values of sentences containing VQs. However, the evaluation of truth values of these sentences sometimes may involve some complications. For example, consider the following sentence: About 10 tall girls sang.

(9)

This sentence contains the VQ “(about 10)”. According to (7), the input of the MDF for “(about 10)(TALL-GIRL)(SING)” is the number |TALL-GIRL ∩ SING| – 10. However, since TALL-GIRL ∩ SING is fuzzy, its cardinality is not well defined. We now encounter the following problem: how can we evaluate the truth value of (9) if we cannot say for sure how many “tall girls” there are? The solution of the early fuzzy theorists is to generalize the notion of crisp set cardinality to fuzzy set cardinality, which may have different definitions. One definition (called the Sigma Count) is the sum of the membership degrees of all individuals in the universe with respect to the fuzzy set. For example, if the fuzzy set TALL-GIRL ∩ SING = {1/a, 0.7/b, 0.5/c, 0.2/d, 0.1/e}5, then the Sigma Count of this set is 1 + 0.7 + 0.5 + 0.2 + 0.1 = 2.5. Using this cardinality, the truth value of (9) is then equal to ║2.5 ∈ (about 10)║, which is equal to 0 according to (7). This shows that (9) is absolutely false with respect to the aforesaid fuzzy set TALL-GIRL ∩ SING. This is in accord with our intuition because according to that fuzzy set, there are only 2 members (i.e. a and b) who may be counted as singing tall girls with a relatively high certainty, and 2 absolutely falls short of being “about 10”. 4 5

In case the function becomes degenerate, then some of the aforesaid parts would disappear. Here I adopt a notation used by fuzzy theorists under which a fuzzy set S is represented in the form {r1/x1, r2/x2, …} where xis are individuals and ris are their respective membership degrees, i.e. ri = ║xi ∈ S║. In case the membership degree of an individual is 0, it is not listed.

A Semantic Model for Vague Quantifiers Combining Fuzzy Theory

2.2

65

Fuzzy Theory (Quantifier Fuzzification Mechanism Approach)

Later, some scholars (e.g. [1], [4-5], [10]) realized the demerits of the old approach, which was able to treat only certain types of VQs and could not be applied to more general types of VQs. Moreover, since different notions of fuzzy set cardinality were used for different VQs, there was not a uniform treatment for various types of VQs. Instead of using the concept of fuzzy set cardinality, they proposed the concept of quantifier fuzzification mechanisms (QFMs). This approach distinguishes two types of VQs: semi-fuzzy and fuzzy quantifiers. Semi-fuzzy quantifiers are those VQs that only take crisp sets as arguments; while fuzzy quantifiers are those VQs that may take either crisp or fuzzy sets as arguments. Note that the distinction between semi-fuzzy and fuzzy quantifiers has nothing to do with the meaning of the VQs. Thus, the same linguistic quantifier such as “(about 10)” may manifest either as a semi-fuzzy or a fuzzy quantifier, depending on the types of its arguments6. Under this approach, all VQs are initially modeled as semi-fuzzy quantifiers. This has the advantage of greatly simplifying the semantics of VQs. We only need to formulate an appropriate MDF or truth condition for each VQ without worrying about its inputs because all inputs are crisp. The evaluation of truth values of sentences involving semi-fuzzy quantifiers is easy: we only need to plug the crisp inputs into the appropriate MDFs or truth conditions. When it comes to a sentence involving fuzzy quantifiers with fuzzy inputs (such as (9)), we have to make use of a QFM, which is in fact a mapping that transforms a semi-fuzzy quantifier to a fuzzy quantifier. Among the QFM approach, [4]’s framework has certain merits compared with its competitors in that it proposes a number of axioms that an adequate QFM should satisfy 7 . These axioms guarantee that the QFM will preserve crisp arguments, the identity truth function and monotonicities of a VQ as well as its arguments, and that the QFM will commute with the operations of argument transposition, argument insertion, external negation, internal negation, internal meet (as well as other Boolean) operators and functional application. Note that the aforesaid properties / operations are crucial to the study of quantifiers under GQT. Next I introduce a QFM proposed in [4]8. First let X be a fuzzy set and γ be a real number in [0, 1] which is called the “cut level”. We can reduce X into two crisp sets Xγmin and Xγmax at the cut level γ using the following formulae: for γ > 0 9, Xγmin = X≥ 0.5 + 0.5γ; 6

7

8

9

Xγmax = X> 0.5 – 0.5γ

(10)

Since crispness can be seen as a special case of fuzziness, any crisp quantifier such as “every” can be seen as a semi-fuzzy or fuzzy quantifier, depending on the types of its arguments. Actually, Glöckner used the term “determiner fuzzification schemes” (DFSs) in [4]. In [5], he used QFMs as a general term for all mappings that map a semi-fuzzy quantifier to a fuzzy quantifier and used DFSs to refer to those QFMs that satisfy his axioms. To simplify notation, in what follows I will just use the umbrella term QFM. Glöckner has proposed a number of QFMs that satisfy all his axioms. This paper only discusses the simplest one. There are in fact separate definitions for Xγmin and Xγmax at γ = 0. But since a single point will only contribute the value 0 to a definite integral to be introduced below, we do not need to consider the case γ = 0 for computational purpose.

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Based on the above, we can then define a family of crisp sets associated with X: Tγ(X) = {Y: Xγmin ⊆ Y ⊆ Xγmax}

(11)

Then let Q be a semi-fuzzy quantifier and X1, … Xn be n fuzzy sets. Now for each of X1, … Xn we can define Tγ(X1), … Tγ(Xn). For each possible combination of Y1 ∈ Tγ(X1), … Yn ∈ Tγ(Xn), we can evaluate ║Q(Y1, … Yn)║ by using a suitable MDF or truth condition because Y1, … Yn are crisp sets. Then we aggregate the various values of ║Q(Y1, … Yn)║ for all possible combinations of Y1, … Yn into ║Qγ(X1, … Xn)║10 by the following formula: ║Qγ(X1, … Xn)║ = m0.5({║Q(Y1, … Yn)║: Y1 ∈ Tγ(X1), … Yn ∈ Tγ(Xn)})

(12)

where m0.5, called the “generalized fuzzy median”, is defined as follows. Let Z be a set of real numbers, then

m0.5(Z) =

inf(Z), sup(Z), 0.5, r,

if |Z| ≥ 2 ∧ inf(Z) > 0.5 if |Z| ≥ 2 ∧ sup(Z) < 0.5 if (|Z| ≥ 2 ∧ inf(Z) ≤ 0.5 ∧ sup(Z) ≥ 0.5) ∨ (Z = ∅) if Z = {r}

(13)

Now for each cut level γ, we have a corresponding value ║Qγ(X1, … Xn)║. Finally we need to combine all these values into one value. According to [4], there are various methods of combination, one such method (which is denoted by “M” in [4]) is to use the standard definite integral11: ║M(Q)(X1, … Xn)║ =

1



0

║Qγ(X1, … Xn)║dγ

(14)

Although the above formula appears as an integral, in practical calculation of linguistic applications involving finite universes, we often only need to consider a finite number of variations of γ and ║Qγ(X1, … Xn)║ is constant at each such γ, and so the integral above often reduces to a sum, which can be seen as a “weighted average” of ║Qγ(X1, … Xn)║ at the various γs. 2.3

Supervaluation Theory

The Supervaluation Theory (ST) for vagueness is a keen competitor of the FT. Some supervaluation theorists, such as [3] and [6-7], pointed out certain flaws of FT. The most serious one is that FT cannot correctly predict the truth values of certain statements that must be true / false by virtue of traditional logical laws or intuition with respect to a model (such statements are called “penumbral connections” in [3]). Consider the following model: M1

10 11

U = {j, m}; TALL = {0.5/j, 0.3/m}

Note that here Qγ should be seen as a fuzzy quantifier evaluated at the cut level γ. In the following formula, “M” should be seen as a QFM that transforms the semi-fuzzy quantifier Q to a fuzzy quantifier M(Q).

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67

Intuitively, according to this model, the truth values of the following sentences should both be absolutely false (where John and Mary are represented by j and m above): John is tall and John is not tall.

(15)

Mary is tall and John is not tall.

(16)

But using the truth functions for BOs (4) – (6), the calculation results show that the above sentences are both true to a certain degree under FT: ║(15)║ = ║j ∈ TALL║ ∧ ║j ∉ TALL║ = min({0.5, 1 – 0.5}) = 0.5 ║(16)║ = ║m ∈ TALL║ ∧ ║j ∉ TALL║ = min({0.3, 1 – 0.5}) = 0.3 Supervaluation theorists point out that the above wrong predictions arise from the wrong assumption that BOs are truth functional when applied to vague concepts. Note that the aforesaid flaw does not hinge on the particular definitions of BOs. It is argued in [7] that the definitions of BOs are subject to various plausible constraints. For example, one may hope that the definitions will preserve p → q ≡ ¬p ∨ q, or that p → p is always true. But unfortunately, no set of definitions can satisfy all these plausible constraints under FT. Supervaluation theorists view vague concepts as truth value gaps and evaluate the truth values of vague sentences by means of complete specifications. A complete specification is an assignment of the truth value 1 or 0 to every individual with respect to the relevant vague sets in a sentence. In other words, a complete specification eliminates the truth value gaps and makes a vague sentence precise. Thus, this process is called “precisification”. If a sentence is true (false) on all admissible complete specifications, then we say that it is true (false)12. Otherwise, it has no truth value. The concept of “admissible” is very important in ST. Let’s use model M1 to illustrate this point. This model contains two individuals: j and m such that both are borderline cases of the vague set TALL with j taller than m. Here is a list of all admissible complete specifications for M1: (i) ║j ∈ TALL║ = 1, ║m ∈ TALL║ = 1; (ii) ║j ∈ TALL║ = 1, ║m ∈ TALL║ = 0; (iii) ║j ∈ TALL║ = 0, ║m ∈ TALL║ = 0. The above list does not include ║j ∈ TALL║ = 0, ║m ∈ TALL║ = 1 because it is inadmissible to assign a person to the set of TALL without at the same time assigning another person who is even taller to TALL. Having identified the admissible specifications, we can then evaluate ║(15)║ and ║(16)║. Since (15) and (16) are both false on all of (i) – (iii) above, we obtain ║(15)║ = ║(16)║ = 0, in conformity with our intuition. Thus, ST provides an alternative method that can deal with penumbral connections correctly. The same method can also be used to evaluate truth values of sentences containing VQs, although the precisification process may be more complicated. Using (9) as an example, the precisification process will involve two levels. At the first level, the vague concept “tall girl” will be precisified, after which we obtain a set 12

In [3] the terms “super-true” (“super-false”) were used to denote propositions that are true (false) on all admissible complete specifications. To simplify notation, I will just call such propositions “true” (“false”).

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TALL-GIRL ∩ SING whose cardinality is known. Then, at the second level, the VQ “about 10” will be precisified based on the aforesaid cardinality. The main weakness of ST is that it cannot distinguish different degrees of vagueness because it treats all borderline cases alike as truth value gaps. The evaluation of truth values of vague sentences under ST is uninteresting because all those vague sentences other than penumbral connections have no truth values. Moreover, in applied studies such as Control Theory, Artificial Intelligence, etc., the concept of membership degrees is of great use. That is why while FT has become very popular in applied studies, ST is only popular in theoretical studies. As a matter of fact, [6] has discussed how to develop a version of ST that incorporates the notion of degrees. More recently, [2] even showed that FT and ST, though often seen to be incompatible with each other, can in fact be combined. In the next section, I will propose such a combined theory.

3 3.1

Combining FT and ST The Modified Glöckner’s Method

Although all borderline cases can be treated as truth value gaps, they may behave differently in the process of precisification. For example, among all admissible complete specifications in which individuals are assigned to the set TALL, a taller person x is more likely to be assigned full membership of TALL than a shorter person y, because whenever y is assigned full membership of TALL in an admissible specification, x must also be so, but not vice versa. An individual x’s membership degree with respect to a vague set S may thus be seen as representing the likelihood of x being assigned to S in an admissible specification. By reinterpreting membership degrees in this way, we have established a link between FT and ST and the semantic model for VQs developed below will follow the tradition of FT by using MDFs as a measure of truth values of VQs. How are we to evaluate the truth values of vague sentences such as (15) and (16)? As mentioned ahove, the traditional FT approach of treating BOs as truth functions like (4) – (6) has to be abandoned. Neither can we use ST’s method because we now want to distinguish an infinite number of truth values. Fortunately, Glöckner’s method in [4] as introduced in Subsection 2.2 can meet our requirements. The essence of Glöckner’s method in [4] is to reduce a sentence with vague arguments to sentences with crisp arguments at different cut levels. The truth values of these sentences with crisp arguments are then evaluated using the MDFs or truth conditions and aggregated into the truth values of a vague quantified sentence at a cut level. Finally, the truth values at all cut levels are combined into a “weighted average”, which is then taken to be the truth value of the original sentence. Using the aforesaid method, there is no need to invoke (4) – (6). Moreover, the aforesaid reduction process can be seen as a precisification process and the family of crisp sets Tγ(X) as defined in (11) can be seen as a set of complete specifications of X. To guarantee that these are also admissible specifications, we need to modify the definition of Tγ(X) as shown below:

A Semantic Model for Vague Quantifiers Combining Fuzzy Theory

69

Tγ(X) = {Y: Xγmin ⊆ Y ⊆ Xγmax ∧ Y represents an admissible complete specification of X}

(17)

Glöckner’s method with the above modification will henceforth be called the Modified Glöckner’s Method (MGM). With MGM, we can evaluate ║(15)║ and ║(16)║ with respect to M1. Since the result of ║(15)║ is obvious, I only show the evaluation of ║(16)║. In order to use MGM, we first need to express (16) as a conjoined quantified statement. One way is to make use of the quantifier “every” satisfying the truth condition every(A)(B) ↔ A ⊆ B: every({m})(TALL) ∧ ¬every({j})(TALL)

(18)

Now, for 0 < γ ≤ 0.4, we have by (10), TALLγ = ∅, TALLγ = {j}, By (17), Tγ(TALL) = {∅, {j}} since both ∅ and {j} represent admissible complete specifications. Then, we have min

max

║everyγ({m})(TALL) ∧ ¬everyγ({j})(TALL)║ =

m0.5({║every({m})(Y) ∧ ¬every({j})(Y)║: Y ∈ Tγ(TALL)})

=

m0.5({║every({m})(∅) ∧ ¬every({j})(∅)║, ║every({m})({j}) ∧ ¬every({j})({j})║})

=

m0.5({0})

=

0

by (12)

by (13)

For 0.4 < γ ≤ 1, TALLγmin = ∅, TALLγmax = {j, m}. By (17), Tγ(TALL) = {∅, {j}, {j, m}} since ∅, {j} and {j, m} represent admissible complete specifications. Note that although ∅ ⊆ {m} ⊆ {j, m}, {m} is not included in Tγ(TALL) because {m} represents the inadmissible complete specification ║j ∈ TALL║ = 0, ║m ∈ TALL║ = 1. Then, we have ║everyγ({m})(TALL) ∧ ¬everyγ({j})(TALL)║ =

m0.5({║every({m})(Y) ∧ ¬every({j})(Y)║: Y ∈ Tγ(TALL)})

=

m0.5({║every({m})(∅) ∧ ¬every({j})(∅)║, ║every({m})({j}) ∧ ¬every({j})({j})║, ║every({m})({j, m}) ∧ ¬every({j})({j, m})║})

=

m0.5({0})

=

0

by (12)

by (13)

Finally, by (14), ║(16)║ = ║(18)║

1

=



0

║everyγ({m})(TALL) ∧ ¬everyγ({j})(TALL)║dγ

= 0 × (0.4 – 0) + 0 × (1 – 0.4) =0

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which is as desired. Note that if we had included {m} as an admissible complete specification for 0.4 < γ ≤ 1, then we would have got ║(16)║ = 0.3, contrary to our intuition. The above computation shows that MGM is able to correct the flaw of FT. 3.2

Some Properties of MGM

The modification of the definition of Tγ(X) as shown in (17) may incur a cost in that some nice properties of Glöckner’s original theory may be lost. By scrutinizing the proofs of the various lemmas and theorems in [4], one can find that the important properties of the orginal theory introduced in Subsection 2.2 are not affected by the modification with two exceptions, namely under MGM the QFM represented by M does not commute with internal meet and functional application. This means, for example, that when we evaluate13 ║Qγ(X1 ∩ X2)║

(19)

for a particular γ, the result of first precisifying X1 and X2 and then intersecting the resultant crisp sets, i.e. m0.5({║Q(Y1 ∩ Y2)║: Y1 ∈ Tγ(X1), Y2 ∈ Tγ(X2)})

(20)

may be different from the result of first intersecting X1 and X2 and then precisifying the resultant fuzzy set, i.e. m0.5({║Q(Y)║: Y ∈ Tγ(X1 ∩ X2)})

(21)

because {Y1 ∩ Y2: Y1 ∈ Tγ(X1), Y2 ∈ Tγ(X2)} may not be equal to {Y: Y ∈ Tγ(X1 ∩ X2)}. The crux of the problem is that the intersection of two sets each representing an admissible complete specification may not be a set representing an admissible complete specification. For instance, while {a, b} represents an admissible complete specification for the set X1 = {1/a, 0.9/b, 0.8/c} and {b, c} represents an admissible complete specification for the set X2 = {0.5/a, 0.6/b, 0.7/c}, {a, b} ∩ {b, c} = {b} represents an inadmissible complete specification for X1 ∩ X2 = {0.5/a, 0.6/b, 0.7/c}. The same problem can be said of functional application for an arbitrary function. Is this a defect of MGM? Not necessarily. The essence of MGM is to deny the truth functionality of BOs and other arbitrary functions when applied to vague sets. Under MGM, when evaluating the truth value of a vague statement involving BOs or other arbitrary functions, we never apply the BOs or functions to the vague arguments directly because such application is undefined. Instead, we always proceed by first precisifying the vague arguments and then applying the BOs or functions to the resultant crisp arguments. This means, for example, that when evaluating (19) we always do (20), never (21), and so the problem that (20) ≠ (21) simply does not arise. Thus, we may say that MGM has preserved the essential nice properties of Glöckner’s original theory. Note that MGM also has another nice property. Suppose the membership degrees with respect to the vague sets X1, ... Xn in a model are restricted to {0, 1, 0.5} and the 13

To simplify notation, in what follows I use the same symbol “∩” to denote the intersection operation of crisp sets and vague sets. Under FT, the vague version of “∩” may be defined based on the BO “∧”.

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truth values of a semi-fuzzy quantifier Q applied to any n crisp arguments are also restricted to {0, 1, 0.5}. Then for 0 < γ ≤ 1, we must have {║Q(Y1, … Yn)║: Y1 ∈ Tγ(X1), … Yn ∈ Tγ(Xn)} equal to any one of the following: {0}, {1}, {0.5}, {0, 1}, {0, 0.5}, {1, 0.5}, {0, 1, 0.5}. By (12) and (13), we have ║Qγ(X1, … Xn)║ equal to 0, 1 or 0.5 according as {║Q(Y1, … Yn)║: Y1 ∈ Tγ(X1), … Yn ∈ Tγ(Xn)} contains only 0, only 1 or otherwise. Then by (14), we have ║M(Q)(X1, … Xn)║ also restricted to {0, 1, 0.5}. So in this case MGM gives us the same result as that obtained by the supervaluation method if we use 0.5 to represent the truth value gap. MGM is thus indeed a generalization of the supervaluation method and provides us with the flexibility in determining how we should model vagueness.

4

Iterated VQs

According to [9], a sentence containing both subject and object(s) can be viewed as containing a polyadic quantifier. There is an important type of polyadic quantifiers, called iterated quantifiers, that can be represented by a tripartite structure with one of its arguments containing another tripartite structure. For example, the sentence Every boy loves every girl.

(22)

may be seen as containing the iterated quantifier “(every … every)” and can be represented by the following tripartite structure: every(BOY)({x: every(GIRL)({y: LOVE(x, y)})})

(23)

which, in daily language, means “Every boy x is such that for every girl y, x loves y”. Based on the above expression, one can then evaluate the truth value of (22) with respect to any model according to the truth condition of “every”. MGM is readily applicable to iterated VQs14. Consider the following sentence: Almost every boy met about 10 girls.

(24)

with respect to the following model: M2

BOY = {a, b, c, d, e} x |GIRL ∩ {y: MEET(x, y)}|

a 10

b 9

c 11

d 13

e 8

For computational purpose, suppose we use (7) as the MDF for “(about 10)” and the following MDF for “(almost every)” (where ε represents an infinitesimal positive magnitude): ║(almost every)(A)(B)║ = T–0.4, –0.2, –ε, –ε(|A ∩ B| / |A| – 1)

(25)

To evaluate ║(24)║, we first write (24) as the following tripartite structure: (almost every)(BOY)({x: (about 10)(GIRL)({y: MEET(x, y)})})

14

(26)

For simplicity, here I only consider iterated VQs composed of 2 VQs. It is not difficult to generalize the theory to iterated VQs composed of more than 2 VQs.

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In the above, {x: (about 10)(GIRL)({y: MEET(x, y)})} denotes the set of those who met about 10 girls. For convenience, let’s call this set X. Since X is a vague set, we cannot evaluate ║(26)║ directly. To facilitate further computation, we need to determine this vague set first. According to (7), for each x, ║(about 10)(GIRL)({y: MEET(x, y)})║ depends on the input |GIRL ∩ {y: MEET(x, y)}| – 10. By substituting the data given in M2 into (7) for each x, we can determine the following vague set: X = {1/a, 1/b, 1/c, 0.33/d, 0.67/e}. We then use MGM to evaluate ║(26)║. For 0 < γ ≤ 0.33, we have Xγmin = {a, b, c, e}, Xγmax = {a, b, c, e} and Tγ(X) = {{a, b, c, e}} because {a, b, c, e} represents an admissible complete specification. Then, we have ║(almost every)γ(BOY)(X)║ = m0.5({║(almost every)γ(BOY)({a, b, c, e})║}) = m0.5({1}) = 1. For 0.33 < γ ≤ 1, we have Xγmin = {a, b, c}, Xγmax = {a, b, c, d, e} and Tγ(X) = {{a, b, c}, {a, b, c, e}, {a, b, c, d, e}} because {a, b, c}, {a, b, c, e} and {a, b, c, d, e} all represent admissible complete specifications (Note that {a, b, c, d} represents an inadmissible complete specification and is thus excluded). Then, we have ║(almost every)γ(BOY)(X)║ = m0.5({║(almost every)γ(BOY)({a, b, c})║, ║(almost every)γ(BOY)({a, b, c, e})║, ║(almost every)γ(BOY)({a, b, c, d, e})║}) = m0.5({0, 1}) = 0.5. Finally, by (14), ║(24)║ = ║(26)║

1

=



0

║(almost every)γ(BOY)(X)║dγ

= 1 × (0.33 – 0) + 0.5 × (1 – 0.33) = 0.67 Note that the above calculation has been greatly simplified because in (24), “boy”, “girls” and “met” are all represented by crisp predicates. In general, given a sentence with iterated VQs, we first express it in the following form: Q1(A1)({x: Q2(A2)({y: B(x, y)})})

(27)

Then for each possible x, we determine {y: B(x, y)}, which may be a vague set, and ||Q2(A2)({y: B(x, y)})|| by MGM. By doing so, we will obtain the following set: {x: Q2(A2)({y: B(x, y)})} = {||Q2(A2)({y: B(xi, y)})||/xi, …}, where xi ranges over all possible xs. Finally, we can evaluate ||Q1(A1)({x: Q2(A2)({y: B(x, y)})})|| by MGM.

5

Conclusion

In this paper, I have discussed the merits and demerits of the FT and ST approaches to vagueness and have proposed MGM as a model for VQs. This model inherits certain desirable properties of Glöckner’s framework in [4]. It is also able to distinguish different degrees of vagueness and is thus useful for practical applications. Moreover, this model has overcome a demerit commonly found in FT frameworks, i.e. it yields correct results for penumbral connections. I have also shown that Glöckner’s original method in fact includes a process reminiscent of the precisification process of ST. This provides a plausible way to combine FT and ST, the two main competing theories for vagueness.

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Nevertheless, this paper has just concentrated on one particular QFM represented by M. As a matter of fact, Glöckner and other scholars have proposed other possible QFMs in [1] and [4-5] which I have not had the chance to discuss in this paper. It would be instructive to consider how these QFMs can be modified to suit the requirement of ST and what properties of Glöckner’s original theory are preserved under the modification and would thus be a possible direction for future studies.

References 1. Díaz-Hermida, F., Bugarín, A., Barro, S.: Definition and classification of semi-fuzzy quantifiers for the evaluation of fuzzy quantified sentences. International Journal of Approximate Reasoning 34, 49–88 (2003) 2. Fermüller, C.G., Kosik, R.: Combining supervaluation and degree based reasoning under vagueness. In: Hermann, M., Voronkov, A. (eds.) LPAR 2006. LNCS (LNAI), vol. 4246, pp. 212–226. Springer, Heidelberg (2006) 3. Fine, K.: Vagueness, Truth and Logic. Synthese 30, 265–300 (1975) 4. Glöckner, I.: DFS – An Axiomatic Approach to Fuzzy Quantification. Report TR97-06, Technical Faculty, University Bielefeld (1997) 5. Glöckner, I.: Fuzzy Quantifiers: A Computational Theory. Springer, Berlin (2006) 6. Kamp, H.: Two Theories about Adjectives. In: Keenan, E.L. (ed.) Formal Semantics of Natural Language, pp. 123–155. Cambridge University Press, Cambridge (1975) 7. Keefe, R.: Theories of Vagueness. Cambridge University Press, Cambridge (2000) 8. Keenan, E.L.: Some Properties of Natural Language Quantifiers: Generalized Quantifier Theory. Linguistics and Philosophy 25, 627–654 (2002) 9. Keenan, E.L., Westerståhl, D.: Generalized Quantifiers in Linguistics and Logic. In: van Benthem, J., ter Meulen, A. (eds.) Handbook of Logic and Language, pp. 837–893. Elsevier Science, Amsterdam (1997) 10. Losada, D.E., Díaz-Hermida, F., Bugarin, A.: Semi-fuzzy quantifiers for information retrieval. In: Herrera-Viedma, E., Pasi, G., Crestani, F. (eds.) Soft Computing in Web Information Retrieval: Models and Applications. Springer, Heidelberg (2006) 11. Westerståhl, D.: Quantifiers in Formal and Natural Language. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. IV, pp. 1–131. Reidel Publishing Company, Dordrecht (1989) 12. Yager, R.R.: Reasoning with Fuzzy Quantified Statements: Part I. Kybernetes 14, 233–240 (1985a) 13. Yager, R.R.: Reasoning with Fuzzy Quantified Statements: Part II. Kybernetes 15, 111–120 (1985b) 14. Zadeh, L.A.: Fuzzy sets. Information and Control 8(3), 338–353 (1965) 15. Zadeh, L.A.: A Computational Approach to Fuzzy Quantifiers in Natural Languages. Computers and Mathematics with Applications 9(1), 149–184 (1983)

An Application of Model Checking Games to Abstract Argumentation Davide Grossi ILLC, University of Amsterdam Amsterdam, The Netherlands [email protected]

Abstract. The paper presents a logical study of abstract argumentation theory. It introduces a second-order modal logic, within which all main known semantics for abstract argumentation can be formalized, and studies the model checking game of this logic. The application of the game to the formalized semantics yields adequate game-theoretic proof procedures for all known extension-based semantics, in both their skeptical and credulous versions.

1

Introduction

Structures A = (A, ), where A is a non-empty set and ⊆ A2 is a binary relation on A, are the building blocks of abstract argumentation theory. Once A is taken to represent a set of arguments, and an ‘attack’ relation between arguments (so that a b means “a attacks b”), the study of these structures—called argumentation frameworks [1]—provides very general insights on how competing arguments interact and how collections of them form ‘tenable’ or ‘justiﬁable’ positions in an argumentation (cf. [2] for a recent overview). While the study of diﬀerent formal deﬁnitions of the notion of ‘justiﬁability’— called extensions—constitutes the main body of abstract argumentation theory, many researchers in the last two decades have focused on ‘proof procedures’ for argumentation, i.e., procedures able to adequately establish whether a given argument belongs or not to a given extension. Many of such proof procedures have resorted to abstractions coming from game theory (cf. [3] for a recent overview) and have given rise to a number of diﬀerent types of games, called dialogue or argument games. The present paper contributes to this line of research by showing that games commonly studied in logic—model checking games (cf. [4])—can be successfully used for this purpose. Outline. The paper builds on the simple idea—ﬁrst put forth in [5]—that argumentation frameworks are, in fact, the sort of structures that modal logic [6] calls Kripke frames: arguments are modal states, and the accessibility relation is given by the attack relation. The paper formalizes all main Dung-style extension-based semantics in a second-order modal logic (Section 2). This logic is proven to have an adequate model checking game, which is then used to provide adequate games for all known extension-based semantics and, in general, for H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 74–86, 2011. c Springer-Verlag Berlin Heidelberg 2011

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Table 1. Basic notions of abstract argumentation theory cA characteristic function of A iﬀ X is acceptable w.r.t. Y in A iﬀ X conﬂict-free in A iﬀ X admissible set of A iﬀ X complete extension of A iﬀ X stable extension of A iﬀ X grounded extension of A iﬀ X preferred extension of A iﬀ X semi-stable extension of A iﬀ

cA (X) = {a | ∀b : [b a ⇒ ∃c ∈ X : c b]} X ⊆ cA (Y ) ∃a, b ∈ X s.t. a b X is conﬂict-free and X ⊆ cA (X) X is conﬂict-free and X = cA (X) X = {a ∈ A | ∃b ∈ X : b a} X is the minimal complete extension of A X is a maximal complete extension of A X is a complete extension and X ∪ {a | ∃b ∈ X : b a} is maximal

all such semantics (Section 3). To the best of our knowledge, no game-theoretic proof procedure for argumentation has ever achieved this level of generality. The paper concludes by brieﬂy comparing our ﬁndings with the literature on dialogue games and indicates directions for future research at the interface of logic and argumentation (Section 4). The paper presupposes familiarity with both modal logic and abstract argumentation theory.1

2

Logical Languages for Argumentation Theory

The present section introduces a modal logic view on abstract argumentation theory—ﬁrst proposed in [5]—and extends it to a rich logical language for the formalization of all known extension-based semantics. 2.1

Argumentation Models

Being Dung argumentation frameworks nothing but Kripke frames, the addition of a labeling (or valuation function) to an argumentation framework yields a Kripke model. Definition 1 (Argumentation models). Let P be a set of propositional atoms. An argumentation model, or labeled argumentation framework, M = (A, V) is a structure such that: A = (A, ) is an argumentation framework; V : P −→ 2A is an assignment from P to subsets of A.2 The class of all argumentation models is A. A pointed argumentation model is a pair (M, a) where M is a model and a an argument. Argumentation models are nothing but argumentation frameworks together with a way of ‘naming’ sets of arguments or, to put it otherwise, of ‘labeling’ arguments.3 So, the fact that an argument a belongs to the set V(p) in a given model 1 2 3

Some of the main argumentation-theoretic notions to which we will refer throughout the paper have been recapitulated in Table 1. Note that, although often assumed in researches on abstract argumentation, we do not impose a ﬁniteness constraint on A. It might be worth noticing that this is a generalization of the sort of labeling functions studied in argumentation theory (cf. [2]).

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b

d

c

p

a

p

e

a

A

B

Fig. 1. Examples of labeled argumentation frameworks. Argument a is labeled with proposition p.

M reads in logical notation as (A, V), a |= p. By using the language of propositional logic we can then form ‘complex’ labels ϕ for sets of arguments stating, for instance, that “a belongs to both the sets called p and q”: (A, V), a |= p ∧ q. In order to formalize argumentation-theoretic statements one more linguistic ingredient is needed. Let us mention a couple of examples: “there exists an argument in a set named ϕ attacking argument a” or “for all attackers of argument a there exist some attackers in a set named ϕ”. These are statements involving a bounded quantiﬁcation and they can be naturally formalized by a modal operator ♦ whose reading is: “there exists an attacking argument such that . . . ”. This takes us to modal languages. 2.2

Argumentation in Modal Logic

We turn to logic KU , an extension of the minimal modal logic K with the universal modality.4 As shown in [5], this logic, despite its simplicity, allows one to express a number of key notions of abstract argumentation theory. Syntax and semantics. Language LU is a standard modal language with two modalities: ♦ and U (the universal modality). It is built on the set of atoms P by the following BNF: LU (P) : ϕ ::= p | ⊥ | ¬ϕ | ϕ ∧ ϕ | ♦ϕ | Uϕ where p ranges over P. Standard deﬁnitions for the remaining Boolean operators and the duals and [U] are assumed. Definition 2 (Satisfaction for LU ). Let ϕ ∈ LU . The satisfaction of ϕ by a pointed argumentation model (M, a) is inductively deﬁned as follows: M, a |= ♦ϕ ⇐⇒ ∃b ∈ A : a b and M, b |= ϕ 4

This logic is well-studied and well-behaved: it has a simple strongly complete axiomatics, a P-complete model checking problem and an EXPTIME-complete satisﬁability problem (cf. [6, Ch. 7]).

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M, a |= Uϕ ⇐⇒ ∃b ∈ A : M, b |= ϕ Boolean clauses are omitted. As usual, ϕ is valid in an argumentation model M iﬀ it is satisﬁed in all pointed models of M, i.e., M |= ϕ. The truth-set of ϕ is denoted |ϕ|M . The set of LU -formulae which are true in the class A of all argumentation models is called (logic) KU . These are fully standard clauses for modal logic semantics, but let us see what their intuitive reading is in argumentation-theoretic terms. The ﬁrst clause states that argument a belongs to the set called ♦ϕ iﬀ some argument b is reachable via the inverse −1 of the attack relation and b belongs to ϕ or, more simply, iﬀ a is attacked by some argument in ϕ. The second clause states that argument a belongs to the set called Uϕ iﬀ there exists some argument b in ϕ, in other words, iﬀ the set called ϕ is non-empty. So, to put it shortly, logic KU is endowed with modal operators of the type “there exists an argument attacking the current one such that . . . ”, i.e. ♦, and “there exists an argument such that . . . ”, i.e. U, together with their duals. Example 1. Consider the labeled frameworks A and B in Figure 1. Here are a few illustrative modal statements: MA , a |= ¬⊥ MA , a |= ♦p

MB , a |= U(♦♦p ∧ ♦♦♦p) MB , a |= U(p ∨ ♦p).

The two on the left state that argument a in framework A is not ‘unattacked’ and, respectively, that all its attackers are attacked by some argument in p (in this case a itself). The one at the top right corner states that in framework B there exists an argument (namely d) which has both a chain of two and three attackers ending in p. Finally, the one at the bottom right corner states that there exists an argument (namely c) such that all its attackers are either in p or are attacked by some argument in p. Argumentation theory in KU . Logic KU can express many key argumentationtheoretic notions such as: that a given set p is acceptable w.r.t. another set q, that it is conﬂict-free, that it is admissible, that it is a complete extension, and that it is a stable extension: Acc(p, q) := [U](p → ♦q)

(1)

CFr (p) := [U](p → ¬p) Adm(p) := CFr (p) ∧ Acc(p, p)

(2) (3)

Cmp(p) := CFr (p) ∧ [U](p ↔ ♦p))

(4)

Stb(p) := [U](p ↔ ¬p)

(5)

The adequacy of these deﬁnitions with respect to the ones in Table 1 is easily checked. Let us rather give an intuitive reading of some of these formulas. First, a set of arguments p is acceptable with respect to the set of arguments q

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if and only if all p-arguments are such that for all their attackers there exists a defender in q (Formula 1). A set of arguments p is conﬂict free if and only if all p-arguments are such that none of their attackers is in p (Formula 2). A set p is a complete extension if and only if it is conﬂict free and it is equivalent to the set of arguments all the attackers of which are attacked by some p-argument (Formula 4). Finally, a set p is a stable extension if and only if it is equivalent to the set of arguments whose attackers are not in p (Formula 5). Remark 1. It is worth noticing that only two modal patterns occur in Formulae 1-5: ¬ (“all attackers are not such that . . . ”) and ♦ (“all attackers are attacked by at least an argument such that . . . ”). By the semantics in Deﬁnition 2, both patterns specify functions from sets of arguments to sets of arguments. The ﬁrst one yields, given ϕ, the set ¬ϕ of arguments that are not attacked by any argument in ϕ. The second one was shown in [5] to be the modal logic formulation of what in argumentation theory is called the characteristic function of a given framework (see Table 1). It yields, given ϕ, the set ♦ϕ of arguments defended by ϕ. 2.3

Argumentation in Second-Order Modal Logic

A quick inspection of Table 1 suﬃces to show that KU is not expressive enough to capture notions such as grounded, preferred, semi-stable extensions, or skeptical and credulous membership to extensions.5 In this section we add monadic second order quantiﬁcation to KU . In doing this we are clearly not aiming at ﬁnding the right expressive power needed to express each of the notions in Table 1 in isolation, but we are rather ﬁxing a language that can accommodate all at once and which, as we will see in Section 3, can still provide us with the sort of games we are after. Syntax and semantics. We expand language LU (P) by allowing quantiﬁcation over monadic predicates (i.e., propositional variables). The resulting secondorder language is built by the following BNF: L2U (P) : ϕ ::= p | ⊥ | ¬ϕ | ϕ ∧ ϕ | ♦ϕ | Uϕ | ∃p.ψ(p) where by ψ(p) we indicate that p occurs free in ψ. In what follows we will make extensive use of the auxiliary symbol denoting inclusion between truth-sets: ϕ ψ := [U](ϕ → ψ). Strict inclusion is deﬁned in the obvious way. 5

It is worth recalling that, given an extension type E (e.g., stable, grounded, etc.) and a framework A, an argument a is credulously (resp., skeptically) included in E in A iﬀ there exists one extension (resp., for all extensions) of type E for A s.t. a is included in that extension (resp., those extensions).

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Definition 3 (Satisfaction for L2U ). Let ϕ ∈ L2U . The satisfaction of ϕ by a pointed argumentation model (M, a) is deﬁned as follows (the clauses for LU formulae are omitted): M, a |= ∃p.ϕ(p) ⇐⇒ ∃X ⊆ A : Mp:=X , a |= ϕ(p) where Mp:=X denotes model M where V(p) is set to be X. The set of all L2U formulae which are satisﬁed by all argumentation model is called (logic) K2U . Intuitively, we are simply adding to KU the necessary expressivity to talk about properties involving quantiﬁcation over sets of arguments. As the following remark shows in some detail, this takes us to the binary fragment of monadic second-order logic. Remark 2 (K2U , SOPML and MSO). Logic K2U is an extension of SOPML (secondorder propositional modal logic, [7]) with the universal modality. It coincides with the (one free variable) binary fragment of MSO (monadic second-order logic). To appreciate this, note that by a simple extension of the standard translation [6, Ch. 2.4] K2U can be easily proven to be a fragment of MSO. The translation in the other direction is less obvious, and is brieﬂy sketched here. First let us recall the (constant-free) language L of the binary fragment of MSO: L : ϕ ::= x = y | Rxy | P x | ¬ϕ | ϕ ∧ ϕ | ∃x.ϕ | ∃P.ϕ The key idea of the translation is to treat ﬁrst-order variables x, y as propositional variables which are interpreted as singletons. Now, that a propositional variable x is a singleton can be expressed in K2U by the formula: sing(x) := ∀q. (q x → (∀r.(q r) ∨ x q)). So, the desired translation is recursively deﬁned as follows (Boolean cases omitted): tr(x = y) = x y ∧ sing(x) ∧ sing(y) tr(Rxy) = x ♦y ∧ sing(y) ∧ sing(x) tr(P x) = x p ∧ sing(x) tr(∃x.ϕ) = Ux ∧ sing(x) ∧ tr(ϕ) tr(∃P.ϕ) = ∃p.tr(ϕ) To give an example, here is a translation of the formula ∃P.∃x.(Rxy ∧ P (y)): tr(∃P.∃x.(Rxy ∧ P (y))) = ∃p.Ux ∧ sing(x) ∧ x ♦y ∧ sing(y) ∧ y p. It is then instructive to note that, as K2U coincides with (the binary fragment of) MSO, it includes the μ-calculus, which is known to be the bisimulation-invariant fragment of MSO.6 6

For a good presentation of the μ-calculus, the interested reader is referred to [8]. The μ-calculus will be of relevance later also in Remarks 4 and 5.

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Extensions formalized. We are now in the position to complete the formalization of the whole of Table 1 covering the cases of grounded, preferred and semi-stable extensions. Grn(p) := Cmp(p) ∧ ∀q.(Cmp(q) → p q)

(6)

Prf (p) := Cmp(p) ∧ ¬∃q.(Cmp(q) ∧ p q) SStb(p) := Cmp(p) ∧ ¬∃q.((p ∨ ♦p) (q ∨ ♦q))

(7) (8)

The adequacy of the above deﬁnitions w.r.t. Table 1 is easily checked, as the formalization is ‘literally’ following the wording of the table. Intuitively, an atom p is said to be: a grounded extension iﬀ it is the smallest complete extension (Formula 6); a preferred extension iﬀ it is a maximal complete extension (Formula 7); a semi-stable extension iﬀ it is a complete extension which maximizes its union with the set of arguments it attacks, i.e., its range (Formula 8). Remark 3. One aspect worth noticing in the above formalization concerns the order theory of complete extensions. Formulae 6 and 7 express a minimality and, respectively, a maximality requirement. However, while Formula 6 expresses the minimality of p by stating that all other possible complete extensions lie above p, Formula 7 expresses the maximality of p by stating that no other complete extension exists which lies above p, rather than saying that all other complete extensions lie below p. This has to do with the fact that the set of complete extensions always form a pre-order with one minimal element—the grounded extension— and with a set of maximal elements—the preferred extensions—which is not necessarily a singleton (cf. [1]). Remark 4. As observed in [5], the grounded extension can be expressed by a simple formula of the μ-calculus: μp.♦p. This formula denotes the smallest ﬁxpoint of the characteristic function, which, modally, corresponds to the modal pattern ♦ (Remark 1).7 The K2U formula Grn(p) (Formula 6) expresses something slightly diﬀerent, namely that a given atom p denotes the grounded extension. The relation between the two is, formally, that for any argumentation model M: |q|M = |μp.♦p|M iﬀ M |= Grn(q). In addition, K2U can express the two argumentation-theoretic variants of membership with respect to a given semantics: the skeptical and, respectively, the credulous membership. Here below we provide examples for the skeptically and credulously stable membership, and for the skeptically and credulously preferred membership.

7

SkStb := ∀p.Stb(p) → p CrStb := ∃p.Stb(p) ∧ p

(9) (10)

SkPrf := ∀p.Prf (p) → p CrPrf := ∃p.Prf (p) ∧ p

(11) (12)

Recall that the smallest ﬁxpoint of the characteristic function coincides with the minimal complete extension [1].

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So, in general, for any extension E the skeptical membership w.r.t. E requires the argument at the point of evaluation to be included in some set which is an E-extension. Similarly, the credulous membership w.r.t. E requires that if some set is an E-extension, then it includes the argument at the point of evaluation in the argumentation model. It is ﬁnally worth stressing that other notions (e.g., ideal [9] extensions) are also formalizable in K2U since they are expressible in MSO (Remark 2).

3

Model-Checking Games for Argumentation

Model-checking games (sometimes also called evaluation games or Hintikka games) are two-player adversarial procedures for checking whether a formula from a given logical language is satisﬁed in a given structure (e.g., a pointed model). The proponent or veriﬁer (∃ve) tries to prove that the formula holds at the given pointed model, while the opponent or falsiﬁer (∀dam) tries to disprove it. The idea behind this section is very simple. As K2U allows one to formalize all key semantic notions of argumentation theory, a model checking game that can be proven adequate for K2U would automatically provide adequate game-theoretic proof procedures for all those notions. So we turn now to the presentation of the K2U -model checking game. 3.1

The Model Checking Game for K2U

Definition 4 (Model-checking game for K2U ). Let ϕ ∈ L2U (P) be in positive normal form,8 A be an argumentation framework and V a valuation. The K2U model checking game of ϕ on A is a tuple EU2 (ϕ, A) = (N, S, turn, move, win) where: – N := {∃, ∀}. An element of N , a player, is denoted P . – S := L2U × A × (2A )P , that is, the set of positions consists of the set of triples formula-argument-labeling.9 Sequences of elements of S are denoted s. – turn : S −→ N assigns players to positions as Table 2. – move : S −→ 2S assigns to each position a set of accessible positions (moves) as in Table 2. The set Play(EU2 (ϕ, A)) denotes the set of all ﬁnite sequences s (plays) which are consistent with function move. – win : Play (EU2 (ϕ, A)) −→ N is a partial function assigning winners to plays as in Table 3. A game EU2 (ϕ, A) is instantiated by pairing it with an initial position (ϕ, a, V) s.t. a ∈ A and V : P −→ 2A , in symbols: EU2 (ϕ, A)@(ϕ, a, V). 8 9

It must be clear that any formula of L2U can be easily translated into an equivalent one in positive normal form. Equivalently, positions could be pairs formula-model.

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D. Grossi Table 2. Rules of the model checking game for K2U Position Turn Moves (ϕ1 ∨ ϕ2 , a, V) ∃ {(ϕ1 , a, V), (ϕ2 , a, V)} (ϕ1 ∧ ϕ2 , a, V) ∀ {(ϕ1 , a, V), (ϕ2 , a, V)} (♦ϕ, a, V) ∃ {(ϕ, b, V) | (a, b) ∈} (ϕ, a, V) ∀ {(ϕ, b, V) | (a, b) ∈} (U ϕ, a, V) ∃ {(ϕ, b, V) | b ∈ A} ([U]ϕ, a, V) ∀ {(ϕ, b, V) | b ∈ A} (∃p.ϕ(p), a, V) ∃ {(ϕ, a, Vp:=X ) | X ⊆ A} (∀p.ϕ(p), a, V) ∀ {(ϕ, a, Vp:=X ) | X ⊆ A}

Position Turn Moves (⊥, a, V) ∃ ∅ (, a, V) ∀ ∅ (p, a, V) and a ∈ V(p) ∃ ∅ (p, a, V) and a ∈ V(p) ∀ ∅ (¬p, a, V) and a ∈ V(p) ∃ ∅ (¬p, a, V) and a ∈ V(p) ∀ ∅

So a model checking game is a ﬁnite two-player zero-sum extensive game with perfect information. The following features of the game are worth stressing. Positions consist of a formula, an argument, and a valuation. Each move reduces the syntactic complexity of the formula—hence the game is ﬁnite—and, apart from the Boolean positions, moves happen by either selecting another argument (possibly along the attack relation), or by picking an interpretation for a given atom. Finally, it should be noted that turn-taking, as well as the sort of moves available at a given position, are fully determined by the formula in that position. In other words, it is the logical structure of the to-be-checked formula which, together with the model at hands, dictates the rules of the game. The deﬁnition of winning strategy and position follows. Definition 5 (Winning strategies and positions). A strategy for player P in EU2 (ϕ, A)@(ϕ, a, V) is a function telling P what to do in any play from position (ϕ, a, V). Such a strategy is winning for P if and only if, for all plays in accordance with the strategy, P wins. A position (ϕ, a, V) in EU2 (ϕ, A) is winning for P if and only if P has a winning strategy in EU2 (ϕ, A)@(ϕ, a, V). The set of winning positions for P in EU2 (ϕ, A) is denoted Win P (EU2 (ϕ, A)). This brings us to the technical underpinnings of our contribution, consisting of the following simple theorem. Theorem 1 (Adequacy of the K2U -model checking game). Let ϕ ∈ L2U (P), and let A be an argumentation framework. Then, for all a ∈ A, and valuations V: (ϕ, a, V) ∈ Win ∃ (EU2 (ϕ, A)) ⇐⇒ (A, V), a |= ϕ. Table 3. Winning conditions for the K2U -model checking game Matches ∃ve wins ∀dam wins s ∈ Play(EU2 (ϕ, M)) turn(s) = ∀ and move(s) = ∅ turn(s) = ∃ and move(s) = ∅

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Proof (Sketch of proof ). The proof is by induction on the syntax of ϕ. The base as well as the Boolean and modal cases in the induction step are taken care of by the adequacy of the model checking game for KU [5]. Here we focus on the two remaining cases. [ϕ = ∃p.ψ(p)] Left to right. Assume (∃p.ψ(p), a, V) ∈ Win ∃ (EU2 (∃p.ψ(p), A)). As, according to Deﬁnition 4 it is ∃ve’s turn to play, by Deﬁnition 5 it follows that there exists X ⊆ A s.t. (ψ, a, Vp:=X ) ∈ Win ∃ (EU2 (ψ, A)), from which, by IH, we conclude that (A, Vp:=X ), a |= ψ and, by Deﬁnition 3, that (A, V), a |= ∃p.ψ. Right to left. Assume (A, V), a |= ∃p.ψ. This means, by Deﬁnition 3, that there exists X ⊆ A s.t. (A, Vp:=X ), a |= ψ. By IH it follows that (ψ, a, Vp:=X ) ∈ Win ∃ (EU2 (ψ, A)) for some X, and from the fact that it is ∃ve’s turn to play, by Deﬁnition 5 we can thus conclude that (∃p.ψ(p), a, V) ∈ Win ∃ (EU2 (∃p.ψ(p), A)). [ϕ = ∀p.ψ(p)] Left to right. Assume (∀p.ψ(p), a, V) ∈ Win ∃ (EU2 (∀p.ψ(p), A)). As, according to Deﬁnition 4 it is ∀dam’s turn to play, by Deﬁnition 5 it follows that for all X ⊆ A we have that (ψ, a, Vp:=X ) ∈ Win ∃ (EU2 (ψ, A)). From this, by IH we obtain that for all X ⊆ A (A, Vp:=X ), a |= ψ and by Deﬁnition 3 we can conclude that (A, V), a |= ∀p.ψ. Right to left. Assume (A, V), a |= ∀p.ψ. This means, by Deﬁnition 3, that for all X ⊆ A s.t. (A, Vp:=X ), a |= ψ. By IH it follows that (ψ, a, Vp:=X ) ∈ Win ∃ (EU2 (ψ, A)) for all X, and from the fact that it is ∀dam’s turn to play, by Deﬁnition 5 we can conclude that (∀p.ψ(p), a, V) ∈ Win ∃ (EU2 (∀p.ψ(p), A)). The K2U -model checking game in argumentation. Theorem 1 bears immediate relevance for argumentation. It states that for any ϕ ∈ L2U and any framework A, the K2U -model checking game for ϕ on A is adequate. But, as shown in Section 2.3, L2U can express all known argumentation semantics (e.g., Formulae 9-12) and, more generally, any MSO-deﬁnable semantics. It follows that Theorem 1 actually guarantees that all extension-based semantics get an adequate game, namely their K2U -model checking game. We discuss now an example of deﬁnite relevance for argumentation. Example 2 (The game for skeptical preferred). For a given framework A, the game for skeptically preferred extensions is EU2 (SkPrf , A). Take A to be the argumentation framework B in Figure 1, and let the game be played from argument d, i.e., be initialized at position (SkPrf , d, V) with V arbitrary. Part of the resulting game tree is depicted in Figure 2. By Theorem 1 ∃ve has a winning strategy in this game, as {d} is skeptically preferred since it belongs to both the preferred extensions of the framework, namely: {a, d} and {b, d}. In fact, a visual inspection of the tree in Figure 2 reveals that ∃ve can force the game to end up in a position where she wins independently of ∀dam’s choices. It is now important to stress the natural argumentation-theoretic reading of these games. To do so, consider again the game in Figure 2 and the play indicated by ‘∗’. In this branch ∃ve plays wrongly—not in accordance to her winning strategy—and ∀dam wins. The game starts with ∃ve claiming that d is skeptically preferred. ∀dam challenges the claim, and to do so, he selects a valuation for p (in this case the set {e}). Now it is up to ∃ve to show that if {e} is a preferred

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(Prf (p) → p, Vp:={a,d} , d) (p, Vp:={a,d} , d)

∀

(Prf (p) → p, Vp:={e} , d)

∃

(¬Prf (p), Vp:={e} , d)

∃ (p, Vp:={e} , d)

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∃ wins (∃q.(Cmp(q) ∧ p q), Vp:={e} , d)

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(¬Cmp(p), Vp:={e} , d)

(U((p ∧ ♦¬p) ∨ (¬p ∧ ♦p)), Vp:={e} , d)

((p ∧ ♦¬p) ∨ (¬p ∧ ♦p), Vp:={e} , e) (p ∧ ♦¬p, Vp:={e} , e) (p, Vp:={e} , e)

∃ wins

(♦¬p, Vp:={e} , e)

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(¬p, Vp:={e} , e)

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∃

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(p ∧ ♦p, Vp:={e} , e)

∃ ∀

(♦p, Vp:={e} , e) ∃

∗ (p, V

p:={e} , d)

∀ wins

∃ wins

Fig. 2. Partial game tree of the KU -model checking game for CrPrf run at d in framework B of Figure 1. Turn-taking is given through the ∃ and ∀ labels.

extension, then d belongs to it. Rightly, she chooses to show that {e} is not a preferred extension, rather then trying to show that d belongs to {e}, which would lead her to sure defeat. So she goes on proving that {e} is not even a complete extension. But here she makes a mistake, and she chooses to prove that {e} is not conﬂict-free by claiming that e belongs to p and is attacked by some argument in p. ∀dam capitalizes on this mistake and challenges her to show which argument belonging to p attacks e. To this ∃ve cannot reply and loses. Remark 5 (Alternative games for grounded). As observed in Remark 4, the fact that an argument belongs to the grounded extension can be expressed with either a K2U -formula, i.e., Cmp(p) ∧ ∀q.(Cmp(q) → p q)), or with a μ-formula, i.e., μp.♦p. Now, also the μ-calculus has an adequate model checking game (cf. [4] or [8]). It turns so out that the grounded extension has two adequate logicbased proof procedures. Interestingly, these model checking games are radically diﬀerent, and to appreciate this consider the 2-cycle depicted in part A of Figure 1, whose grounded extension is ∅. The μ-calculus game run on that framework at argument a generates an inﬁnite game consisting of one single play won by ∀dam. On the contrary, the K2U model-checking game generates a ﬁnite game whose tree has maximum branching factor equal to |2A | = 4 (i.e., the cardinality of the set of all possible valuations of variable q).

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Discussion and Conclusions

4.1

Model-Checking Games vs. Dialogue Games

The choice of Example 2 to illustrate our games has not been casual, as a game theoretic proof procedure for skeptical preferred is still, to the best of our knowledge, an open problem in the literature (cf. [3]). In general, while adequate games for some extensions are known (typically grounded and credulous preferred), many are still lacking (e.g., skeptical preferred, skeptical stable, semi-stable, etc.) even in highly systematized settings such as [10]. The K2U -model checking game provides adequate games for all of them. What are the key diﬀerences between our logic-based approach, and the dialogue games studied in the argumentation theory literature, (which we will call here dialogue games)? A systematic comparison falls out of the scope of the present paper, however, there are key diﬀerences worth stressing here. First, the structure of the games is, at least at ﬁrst sight, very diﬀerent. Positions in dialogue games consist of arguments alone and the game proceeds along the (inverse of) the attack relation (cf. [3]). In our games, positions consist of arguments plus a formula and a valuation. This allows the game to host much more structure as players can ‘move’ in the game not only by navigating along the attack relation but also by selecting sets of arguments.10 Second, while in our games the rules of play follow directly from the logical structure of the formula formalizing the extension at hands, in dialogue games the rules are exogenously determined. This might be related to the fact that adequate dialogue games seem harder to ﬁnd for some semantics, while in our case we could easily obtain, via logic, a general adequacy theorem. 4.2

Future Work

The results presented, we hope, show the feasibility and usefulness of the importation of methods and techniques from logic to argumentation. At the same time, they open the door to research questions at the interface of the two disciplines. Let us mention two of them. First, grounded extensions have two types of adequate model checking games (Remark 5), and one adequate dialogue game (with some variants). The question is whether any signiﬁcant structural relations exist between these diﬀerent types of games that could provide us with some form of inter-deﬁnability between them and, in particular, between dialogue and model checking games. This would provide a ﬁrst systematic comparison of the two types of games at which we hinted above. Another way to systematically approach such comparison could be by analyzing the complexity of each model checking game with respect to its dialogue game counterparts. Second, the paper has used a very expressive logic (essentially, the binary fragment of MSO) to provide a common language for the formalization of all 10

In the literature on dialogue games, this latter feature can be found only in the type of games studied in [10].

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main Dung-style skeptical and credulous semantics. The question now arises of the extent to which weaker languages can deal with, if not all, at least some of those semantics. For instance, in the paper we have constantly referred to the μ-calculus (Remarks 2, 3 and 5). However, this logic turns out to be unable to express even credulous conﬂict-freeness, ∃p(CFr (p) ∧ p), which is clearly not invariant under bisimulation (consider a reﬂexive point vs. its unraveling). In short, what extensions of the μ-calculus are needed to cope with Formulae 9-12? Answering this question would provide an accurate picture of the sort of logical languages needed to handle signiﬁcant fragments of abstract argumentation theory, without explicitly resorting to second-order quantiﬁcation. Acknowledgments. We are grateful to the anonymous reviewers of LORI3 for their helpful comments. This work has been supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek under the NWO VENI grant 639.021.816.

References 1. Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artiﬁcial Intelligence 77(2), 321–358 (1995) 2. Baroni, P., Giacomin, M.: Semantics of abstract argument systems. In: Rahwan, I., Simari, G.R. (eds.) Argumentation in Artiﬁcal Intelligence. Springer, Heidelberg (2009) 3. Modgil, S., Caminada, M.: Proof theories and algorithms for abstract argumentation frameworks. In: Rahwan, I., Simari, G. (eds.) Argumentation in AI, pp. 105–132. Springer, Heidelberg (2009) 4. Gr¨ adel, E.: Model checking games. In: de Queiroz, R., Pereira, L.C., Haeusler, E.H. (eds.) Proceedings of WOLLIC 2002. Electronic Notes in Theoretical Computer Science, vol. 67, pp. 15–34. Elsevier, Amsterdam (2002) 5. Grossi, D.: On the logic of argumentation theory. In: van der Hoek, W., Kaminka, G., Lesp´erance, Y., Sen, S. (eds.) Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2010), IFAAMAS, pp. 409–416 (2010) 6. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001) 7. Fine, K.: Propositional quantiﬁers in modal logic. Theoria 36, 336–346 (1970) 8. Venema, Y.: Lectures on the modal μ-calculus. Renmin University in Beijing, China (2008) 9. Dung, P.M., Mancarella, P., Toni, F.: A dialectic procedure for sceptical assumption-based argumentation. In: Proceedings of the 1st International Conference on Computational Models of Argument (COMMA 2006), pp. 145–156. IOS Press, Amsterdam (2006) 10. Dung, P.M., Thang, P.M.: A uniﬁed framework for representation and development of dialectical proof procedures in argumentation. In: Proceedings of the TwentyFirst International Joint Conference on Artiﬁcial Intelligence (IJCAI-2009), pp. 746–751 (2009)

Schematic Validity in Dynamic Epistemic Logic: Decidability Wesley H. Holliday1,2 , Tomohiro Hoshi1,2,3 , and Thomas F. Icard, III1,2 1

Logical Dynamics Lab, Center for the Study of Language and Information, Stanford, CA, USA 2 Department of Philosophy, Stanford University 3 Educational Program for Gifted Youth, Stanford University {wesholliday,thoshi,icard}@stanford.edu

Abstract. Unlike standard modal logics, many dynamic epistemic logics are not closed under uniform substitution. The classic example is Public Announcement Logic (PAL), an extension of epistemic logic based on the idea of information acquisition as elimination of possibilities. In this paper, we address the open question of whether the set of schematic validities of PAL, the set of formulas all of whose substitution instances are valid, is decidable. We obtain positive answers for multi-agent PAL, as well as its extension with relativized common knowledge, PAL-RC. The conceptual signiﬁcance of substitution failure is also discussed. Keywords: modal logic, dynamic epistemic logic, Public Announcement Logic, schematic validity, substitution core, decidability.

1

Introduction

The schematic validities of a logic are those formulas all of whose substitution instances are valid [3]. Typically the set of schematic validities of a logic, its substitution core, coincides with the set of validities, in which case the logic is closed under uniform substitution. However, many dynamic epistemic logics axiomatized using reduction axioms [8,1,4,16] are not substitution-closed.1 The classic example is Public Announcement Logic (PAL) [17,10]. In this paper, we consider the schematic validity problem for PAL and its extension PAL-RC with relativized common knowledge [4]. We answer positively the open question [3,2,4] of whether the substitution cores of multi-agent PAL and PAL-RC are decidable. The conceptual signiﬁcance of substitution failure is also discussed. 1

Dynamic epistemic logics are not the only modal logics to have been proposed that are not closed under substitution. Other examples include the modal logic of “pure provability” [6], ˚ Aqvist’s two-dimensional modal logic as discussed by Segerberg [18], and an epistemic-doxastic logic proposed by Halpern [11]. For each of these logics there is an axiomatization in which non-schematically valid axioms appear.

H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 87–96, 2011. c Springer-Verlag Berlin Heidelberg 2011

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1.1

Review of Public Announcement Logic

Let us brieﬂy recall the details of PAL. The language LPAL is deﬁned as follows, for a countable set At of atomic sentences and a ﬁnite set Agt of agent symbols: ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | Ki ϕ | ϕϕ, where p ∈ At and i ∈ Agt. We denote the set of atoms in ϕ by At(ϕ) and deﬁne [ϕ]ψ as ¬ϕ¬ψ. As in epistemic logic, we take Ki ϕ to mean that agent i knows or has the information ϕ. For the “announcement” operator, we take ϕψ to mean that ψ is the case after all agents publicly receive the true information ϕ. We interpret LPAL using standard relational structures of the form M = M, {∼i | i ∈ Agt}, V , where each ∼i is an equivalence relation on M . We use the notation ∼i (w) = {v ∈ W | w ∼i v} to denote the set of possibilities consistent with the knowledge or information of agent i in world w. Each Ki is the universal modality for the associated ∼i relation, and each ϕ is a dynamic modality corresponding to a model-relativization, with the following truth deﬁnitions: M, w Ki ϕ iﬀ ∀v ∈ W : if w ∼i v then M, v ϕ; M, w ϕψ iﬀ M, w ϕ and M|ϕ , w ψ, where M|ϕ = M|ϕ , {∼i|ϕ | i ∈ Agt}, V|ϕ is the model obtained by eliminating from M all worlds in which ϕ was false, i.e., M|ϕ = {v ∈ M | M, v ϕ}, each relation ∼i|ϕ is the restriction of ∼i to M|ϕ , and V|ϕ (p) = V (p) ∩ M|ϕ for all p ∈ At. We denote the extension of ϕ in M by ϕM = {v ∈ M | M, v ϕ}. In essence, the semantics of PAL is based on the intuitive idea of information acquisition as elimination of possibilities, as illustrated by Example 2 below. An axiomatization of PAL is given by the S5 axioms for each Ki modality, the rule of replacement of logical equivalents (from α ↔ β, derive ϕ(α/p) ↔ ϕ(β/p)), and the following reduction axioms [17]: (i) (ii) (iii) (iv)

ϕp ↔ (ϕ ∧ p); ϕ¬ψ ↔ (ϕ ∧ ¬ϕψ); ϕ(ψ ∧ χ) ↔ (ϕψ ∧ ϕχ); ϕKi ψ ↔ (ϕ ∧ Ki (ϕ → ϕψ)).

Using (i) - (iv) and replacement, any LPAL formula can be reduced to an equivalent formula in the basic modal language. Completeness and decidability for PAL are therefore corollaries of completeness and decidability for multi-agent S5. The language of PAL-RC [4], LPAL-RC , extends LPAL with relativized common knowledge operators C ϕ ψ with the truth deﬁnition: M, w C ϕ ψ iﬀ every path from w through ϕM along any ∼i relations ends in ψM . The standard notion of common knowledge, that everyone knows ψ, and everyone knows that everyone knows that ψ, etc., is deﬁned as Cψ := C ψ. Using the reduction axiom (v) ϕC ψ χ ↔ (ϕ ∧ C ϕψ ϕχ), every LPAL-RC formula can be reduced to an equivalent formula without dynamic operators. Therefore, an axiomatization for PAL-RC may be obtained from (i) (v) plus an axiomatization for multi-agent S5 with relativized common knowledge [4]. Since the latter system is decidable, so is PAL-RC by the reduction.

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Conceptual Significance of Substitution Failure

Reduction axiom (i) reﬂects an important assumption of PAL: the truth values of atomic sentences p, q, r, . . . are taken to be unaﬀected by informational events. It is implicitly assumed that no atomic sentence is about the epistemic or informational states of agents. Hence an atomic sentence in PAL is not a propositional variable in the ordinary sense of something that stands in for any proposition. For there is an implicit restriction on the atomic sentence’s subject matter. Purists may protest that the atomic sentences of a real logic are supposed to be “topic-neutral.” Our reply is practical: it is useful for certain applications to use the atomic p, q, r, . . . to describe stable states of the external world, unaﬀected by informational events, while using modal formulas to describe the changeable states of agents’ knowledge or information. As we show in other work [14], it is possible to develop a variant of PAL, which we call Uniform Public Announcement Logic (UPAL), in which atomic sentences are treated as genuine propositional variables. Which way one goes is a modeling choice. Given the special treatment of atomic sentences in PAL, it is perhaps unsurprising that uniform substitution should fail. For example, the substitution instance pKi p ↔ (p∧Ki p) of reduction axiom (i) is not valid. Since we take Ki p to mean that agent i knows or has the information p, if pKi p ↔ (p ∧ Ki p) were valid, it would mean that an agent could learn p only if the agent already knew p. Since PAL is designed to reason about information change, the non-schematic validity of reduction axiom (i) is a feature of the system, not a bug. Although substitution failures are to be expected in PAL, the speciﬁc failures illuminate subtleties of information change. Example 1 provides the classic example. Example 2 shows that some substitution failures are not at all obvious. Example 1 (Moore Sentence). The formula [p]Ki p is valid, for when agent i acquires the information p, agent i comes to know p. Yet this formula is not schematically valid, and neither is the valid formula [p]p. Simply substitute the famous Moore sentence p ∧ ¬Ki p for p. The non-schematic validity of [p]p is the well-known issue of “unsuccessful formulas” [9,8,15], which is also at the heart of the Muddy Children puzzle [9, §4]. In these cases, the failure of schematic validity for a valid PAL principle shows that the principle does not hold for all types of information—in particular, for information about agents’ own information. Not only is the substitution instance [p∧¬Ki p](p∧¬Ki p) of [p]p invalid, but also [p ∧ ¬Ki p]¬(p ∧ ¬Ki p) is valid. Is the latter also schematically valid? Informally, is there a ϕ such that if you receive the true information that “ϕ but you don’t know ϕ,” it can remain true afterward that ϕ but you don’t know ϕ? As Hintikka [12] remarks about sentences of the Moorean form, “If you know that I am well informed and if I address the words . . . to you, these words have a curious eﬀect which may perhaps be called anti-performatory. You may come to know that what I say was true, but saying it in so many words has the eﬀect of making what is being said false” (p. 68f). Surprisingly, this is not always so. Example 2 (Puzzle of the Gifts [13]). Holding her hands behind her back, agent i walks into a room where a friend j is sitting. Agent j did not see what if anything

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i put in her hands, and i knows this. In fact, i has gifts for j in both hands. Instead of the usual game of asking j to “pick a hand, any hand,” i (deviously but) truthfully announces: (G) Either I have a gift in my right hand and you don’t know that, or I have gifts in both hands and you don’t know I have a gift in my left hand. Let us suppose that j knows i to be an infallible source of information on such matters, so j accepts G. Question 1: After i’s announcement, does j know whether i has a gift in her left/right/both hand(s)? Question 2: After i’s announcement, is G true? Question 3: After i’s announcement, does j know G? Finally, Question 4: If ‘yes’ to Q2, then what happens if i announces G again? Let l stand for ‘a gift is in i’s left hand’ and r stand for ‘a gift is in i’s right hand’. Before i’s announcement, j has not eliminated any of the four possibilities represented by the model M in Fig. 1. (Reﬂexive arrows are not displayed.) w1 l, r

M

w2

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Fig. 1. models for the Puzzle of the Gifts

We can translate G into our language as (G) (r ∧ ¬Kj r) ∨ (l ∧ r ∧ ¬Kj l). Clearly GM = {w1 , w2 }. Hence after i’s announcement of G, j can eliminate possibilities w3 and w4 , reducing j’s uncertainty to that represented by the model M|G in Fig. 1. Inspection of M|G shows that the answer to Question 1 is that M|G , w1 Kj r ∧ ¬(Kj l ∨ Kj ¬l). Observe that GM|G = {w1 }, which answers Questions 2 (‘yes’) and 3 (‘no’). It follows that the principle ϕϕ → ϕKj ϕ is not schematically valid. One can fail to come to know what is (true and remains true after being) announced by a source whom one knows to be infallible! Suppose that instead of initially announcing G, i announces (H) G ∧ ¬Kj G.2 2

“The following is true but you don’t know it: either I have a gift in my right hand and you don’t know that, or I have gifts in both hands and you don’t know I have a gift in my left hand.”

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Given GM = {w1 , w2 } above, clearly HM = {w1 , w2 }. It follows that M|G = M|H , so given GM|G = {w1 } above, clearly HM|H = {w1 }. It follows that M, w1 HH. Hence [p ∧ ¬Kp]¬(p ∧ ¬Kp) is valid but not schematically valid. Announcements of Moore sentences are not always self-refuting! We leave the answer to Question 4 to the reader (see M|G|G in Fig. 1). There are many other examples of valid but not schematically valid PAL principles. Noteworthy instances include Ki (p → q) → (qKi r → pKi r) and (pKi r ∧ qKi r) → p ∨ qKi r. Example 2 shows that discovering that there is an invalid substitution instance of a valid PAL formula can be a non-trivial task. A natural question is whether we can give an eﬀective procedure to make such discoveries. The rest of the paper addresses this technical question. 1.3

The Problem of the Substitution Core

Let us now state precisely the problem to be solved. For a language L whose set of atomic sentences is At, a substitution is any function σ : At → L, and σ ˆ : L → L is the extension such that σ ˆ (ϕ) is obtained from ϕ by replacing each p ∈ At(ϕ) by σ(p). Abusing notation, we write σ(ϕ) for σ ˆ (ϕ). A formula ϕ is schematically valid iﬀ for all such σ, σ(ϕ) is valid. The substitution core of PAL is the set {ϕ ∈ LPAL | ϕ schematically valid} and similarly for PAL-RC. In van Benthem’s list of “Open Problems in Logical Dynamics” [3], Question 1 is whether the substitution core of PAL-RC is decidable. We answer this question positively for PAL and PAL-RC in the following section.

2

Decidability

The idea of our proof is to provide a procedure for constructing a ﬁnite set of substitution instances for a given formula ϕ, such that if ϕ is not schematically valid, then there is a falsiﬁable substitution instance in the ﬁnite set. Suppose that for some substitution σ and model M, we have M, w σ(ϕ). From σ and M, we will construct a special substitution τ such that τ (ϕ) is false at w in a suitable extension (on the valuation function) of M. The construction reveals that τ is in a ﬁnite set of substitutions determined solely by the structure of ϕ. Therefore, to check whether ϕ is schematically valid, we need only check the validity of ﬁnitely many substitution instances of ϕ, which is a decidable problem for PAL and PAL-RC. We begin with a preliminary deﬁnition and result. Definition 1. The set of simple formulas is deﬁned as the smallest set such that: all p ∈ At are simple; if ϕ is simple, so are ¬ϕ, Ki ϕ, and ϕ ± p, where ±p is either p or ¬p for p ∈ At; if ϕ and ψ are simple, so are ϕ ∧ ψ and C ϕ ψ. Proposition 1. For every formula ϕ ∈ LPAL-RC , there is an equivalent simple formula ϕ . Proof : By induction on ϕ, using the schematic validities (ii) - (v) in §1 and the schematic validity pqr ↔ pqr [7].

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2.1

Transforming Substitutions

Fix a formula ϕ in LPAL or LPAL-RC . By Proposition 1, we may assume that ϕ is simple. Suppose that for some substitution σ and M = M, {∼i | i ∈ Agt}, V , we have M, w σ(ϕ). We will now provide a procedure to construct a special substitution τ from σ and a model N from M, as discussed above, such that N , w τ (ϕ). Whether ϕ is in LPAL or LPAL-RC , the resulting formula τ (ϕ) will be in LPAL-RC . However, in §2.2 we will obtain substitution instances in LPAL . To construct τ (p) for a given p ∈ At, let B1 , . . . , Bm be the sequence of all Bi such that [Bi ] ± p occurs in ϕ, and let B0 := . For 0 ≤ i, j ≤ m, if σ(Bi )M = σ(Bj )M , then delete one of Bi or Bj from the list (but never B0 ), until there is no such pair. Call the resulting sequence A0 , . . . , An , and deﬁne s(i) = {j | 0 ≤ j ≤ n and σ(Aj )M ⊂ σ(Ai )M }. Extend the language with new variables p0 , . . . , pn and a0 , . . . , an , and deﬁne τ (p) = κ1 ∧ · · · ∧ κn such that

κi := pi ∨

0≤j≤n, j=i

Caj ∧

¬Cak .

0≤k≤n, k∈s(j)

Without loss of generality, we assume that M is generated by {w} [5, Def. 2.5], so the C operator in κi functions as the global modality in M. Having extended the language for each p ∈ At(ϕ), extend the valuation V to V such that for each p ∈ At(ϕ), V (p) = V (p), and for the new variables: (a) V (pi ) = σ(p)M|σ(Ai ) ; (b) V (ai ) = σ(Ai )M . Let N = M, {∼i | i ∈ Agt}, V be the extension of M with the new V . We will show that τ (p) has the same extension as σ(p) after relativization by any σ(Ai ), which has the same extension as τ (Ai ). It will follow that N , w τ (ϕ) given M, w σ(ϕ). Fact 1. For p ∈ At(ϕ), σ([Ai ] ± p)M = ±pi N . Proof : By basic deﬁnitions, σ([Ai ] ± p)M = [σ(Ai )] ± σ(p)M = ±σ(p)M|σ(Ai ) = ±pi N , where the last equality holds by (a) and the deﬁnition of N .

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Lemma 1. For p ∈ At(ϕ) and 0 ≤ i ≤ n, τ (p)N|ai = pi N . Proof : We ﬁrst show that for 0 ≤ i, j ≤ n, i = j: 1. κi N|ai = pi N|ai ; 2. κj N|ai = ai N|ai (= M|ai ). For 1, we claim that given i = j, Caj ∧

¬Cak N|ai = ∅.

0≤k≤n, k∈s(j)

By construction of the sequence A0 , . . . , An for p and (b), aj N = ai N . If ai N ⊂ aj N , then Caj N|ai = ∅. If ai N ⊂ aj N , then by (b) and the deﬁnition of s, i ∈ s(j). Then since ai is propositional, ¬Cai N|ai = ∅. In either case the claim holds, so κi N|ai = pi N|ai given the structure of κi . For 2, κj contains as a disjunct:

Cai ∧

¬Cak .

0≤k≤n, k∈s(i)

Since ai is propositional, Cai N|ai = M|ai . By deﬁnition of s and (b), for all k ∈ s(i), ak N ⊂ ai N , which gives ¬Cak N|ai = M|ai . Hence κj N|ai = M|ai . Given the construction of σ, 1 and 2 imply: τ (p)N|ai = κi N|ai ∩ κj N|ai = pi N|ai ∩ ai N|ai = pi N . j=i

The last equality holds because pi N ⊆ ai N , which follows from (a) and (b). Lemma 2. For all simple subformulas χ of ϕ, τ (χ)N = σ(χ)M . Proof : By induction on χ. For the base case, we must show τ (p)N = σ(p)M . By construction of the sequence A0 , . . . , An for p ∈ At(ϕ), there is some Aj = , so σ(Aj )M = M . Then by (b), aj N = M , and hence N

τ (p)N = τ (p) |aj = pj N M = σ(p) |σ(Aj ) = σ(p)M .

by Lemma 1 by (a)

The boolean cases are straightforward. Next, we must show τ (Kk ϕ)N = σ(Kk ϕ)M . For the inductive hypothesis, we have τ (ϕ)N = σ(ϕ)M , so

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τ (Kk ϕ)N = Kk τ (ϕ)N = {w ∈ M | ∼k (w) ⊆ τ (ϕ)N } = {w ∈ M | ∼k (w) ⊆ σ(ϕ)M } = Kk σ(ϕ)M = σ(Kk ϕ)M . Similar reasoning applies in the case of C ϕ ψ. Finally, we must show τ ([Bi ] ± p)N = σ([Bi ] ± p)M . For the inductive hypothesis, τ (Bi )N = σ(Bi )M . By construction of the sequence A0 , . . . , An for p ∈ At(ϕ), there is some Aj such that σ(Bi )M = σ(Aj )M . Therefore, τ (Bi )N = σ(Aj )M = aj N

by (b),

and hence τ ([Bi ] ± p)N = [τ (Bi )] ± τ (p)N = [aj ] ± τ (p)N = ±τ (p)N|aj = ±pj N = σ([Aj ] ± p)M = σ([Bi ] ± p)M

by Lemma 1 by (a) given σ(Bi )M = σ(Aj )M .

The proof by induction is complete.

Fact 2. N , w τ (ϕ). Proof : Immediate from Lemma 2 given M, w σ(ϕ).

2.2

Proof of Decidability

Given M, w σ(ϕ), using the procedure of §2.1, we can construct a special substitution τ and an extended model N with N , w τ (ϕ). It is clear from the procedure that we need M, σ, and ϕ to construct τ . For each p ∈ At(ϕ), given the subformulas A0 , . . . , An of ϕ, we deﬁned τ (p) = κ1 ∧ · · · ∧ κn , where Caj ∧ κi := pi ∨ ¬Cak . 0≤j≤n, j=i

0≤k≤n, k∈s(j)

Since we deﬁned s(i) = {j | 0 ≤ j ≤ n and σ(Aj )M ⊂ σ(Ai )M } for i ≤ n, we required information from σ and M in order to construct τ . However, there are only ﬁnitely many functions s : n + 1 → ℘(n + 1), and n is bounded by |ϕ|. Hence ϕ induces a ﬁnite set of substitution instances, one for each s function (for each p ∈ At(ϕ)), in which at least one formula is falsiﬁable if ϕ is not schematically valid. This observation yields a decision procedure for the substitution core of PAL-RC. For a given ϕ, construct the ﬁnite set of substitution instances as described. Check the validity of each formula in the set

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by the standard decision procedure for PAL-RC. If ϕ is schematically valid, then all of its substitution instances in the set will be valid. If ϕ is not schematically valid, then one of the substitution instances will be falsiﬁable by Fact 2. Theorem 1 (Decidability for PAL-RC). The substitution core of multi-agent PAL-RC is decidable. Suppose that we have obtained from the PAL-RC procedure a substitution instance τ (ϕ) and a model M for which M, w τ (ϕ). Since the C operator appears in the deﬁnition of τ (p), we have τ (ϕ) ∈ LPAL-RC . If ϕ ∈ LPAL , we may now obtain a substitution τ with τ (ϕ) ∈ LPAL and a model M for which M , w τ (ϕ). If there is a Kj modality that does not occur in ϕ, we may modify τ to τ by replacing all occurrences of C in τ (ϕ) by Kj ; then modify M to M by setting the ∼j relation equal to the transitive closure of the union of all ∼i relations. It is straightforward to verify that M , w τ (ϕ) given M, w τ (ϕ). If all Kj modalities occur in ϕ, then we use the fact that for any ﬁnite model M, we can deﬁne the formula Cα in M by E |M| α, where Ki α and E n+1 α := EE n α. E 1 α := i∈Agt

By the ﬁnite model property for PAL-RC [4], we may assume that the model M, for which M, w τ (ϕ), is ﬁnite. Hence we modify τ to τ by replacing all occurrences of Cα in τ (ϕ) by E |M| α. It is straightforward to verify that M, w τ (ϕ) given M, w τ (ϕ). Theorem 2 (Decidability for PAL). The substitution core of multi-agent PAL is decidable.

3

Conclusion

In this paper, we have answered positively the open question [3,2,4] of whether the substitution cores of multi-agent PAL and PAL-RC are decidable. In a continuation of this work [14], we will show that our approach to proving decidability applies not only when interpreting the languages of PAL and PAL-RC in models with equivalence relations, but also when allowing models with arbitrary relations. We will also present axiomatizations of the substitution cores of PAL and PAL-RC in a system of Uniform Public Announcement Logic (UPAL). Acknowledgements. We wish to thank Johan van Benthem for stimulating our interest in the topic of this paper and two anonymous referees for comments.

References 1. Baltag, A., Moss, L., Solecki, S.: The Logic of Public Announcements, Common Knowledge and Private Suspicions. In: Gilboa, I. (ed.) Proceedings of the 7th Conference on Theoretical Aspects of Rationality and Knowledge (TARK 1998), pp. 43–56. Morgan Kaufmann, San Francisco (1998)

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2. van Benthem, J.: One is a Lonely Number: Logic and Communication. In: Chatzidakis, Z., Koepke, P., Pohlers, W. (eds.) Logic Colloquium 2002, pp. 96–129. ASL & A.K. Peters (2006) 3. van Benthem, J.: Open Problems in Logical Dynamics. In: Gabbay, D., Goncharov, S., Zakharyashev, M. (eds.) Mathematical Problems from Applied Logic I, pp. 137–192. Springer, Heidelberg (2006) 4. van Benthem, J., van Eijck, J., Kooi, B.J.: Logics of communication and change. Information and Computation 204(11), 1620–1662 (2006) 5. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001) 6. Buss, S.R.: The modal logic of pure provability. Notre Dame Journal of Formal Logic 31(2), 225–231 (1990) 7. van Ditmarsch, H.: The Russian cards problem. Studia Logica 75, 31–62 (2003) 8. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Springer, Heidelberg (2008) 9. van Ditmarsch, H., Kooi, B.: The Secret of My Success. Synthese 151, 201–232 (2006) 10. Gerbrandy, J., Groenevelt, W.: Reasoning about Information Change. Journal of Logic, Language and Information 6(2), 147–169 (1997) 11. Halpern, J.Y.: Should Knowledge Entail Belief? Journal of Philosophical Logic 25, 483–494 (1996) 12. Hintikka, J.: Knowledge and Belief: An Introduction to the Logic of the Two Notions. Cornell University Press, Ithica (1962) 13. Holliday, W.H.: Hintikka’s Anti-Performatory Eﬀect and Fitch’s Paradox of Knowability (2011) (manuscript) 14. Holliday, W.H., Hoshi, T., Icard, T.F.: A Uniform Logic of Information Update (2011) (manuscript) 15. Holliday, W.H., Icard III, T.F.: Moorean Phenomena in Epistemic Logic. In: Beklemishev, L., Goranko, V., Shehtman, V. (eds.) Advances in Modal Logic, vol. 8, pp. 178–199. College Publications (2010) 16. Kooi, B.: Expressivity and completeness for public update logics via reduction axioms. Journal of Applied Non-Classical Logics 17(2), 231–253 (2007) 17. Plaza, J.: Logics of public communications. In: Emrich, M., Pfeifer, M., Hadzikadic, M., Ras, Z. (eds.) Proceedings of the 4th International Symposium on Methodologies for Intelligent Systems, pp. 201–216. Oak Ridge National Laboratory (1989) 18. Segerberg, K.: Two-dimensional modal logic. Journal of Philosophical Logic 2(1), 77–96 (1973)

Knowledge and Action in Semi-public Environments Wiebe van der Hoek, Petar Iliev, and Michael Wooldridge University of Liverpool, United Kingdom {Wiebe.Van-Der-Hoek,pvi,mjw}@liverpool.ac.uk

Abstract. We introduce and study the notion of a Public Environment: a system in which a publicly known program is executed in an environment that is partially observable to agents in the system. Although agents do not directly have access to all variables in the system, they may come to know the values of unobserved variables because they know how the program is manipulating these variables. We develop a logic for reasoning about Public Environments, and an axiomatization of the logic.

1

Introduction

Our primary concern in the present paper is the following issue: Suppose that a number of agents are engaged in a commonly known algorithmic activity in some environment. What can be said about (the evolution of ) their knowledge if they cannot make a complete and correct observation of their environment? To investigate this issue, we introduce a computational model for epistemic logic known as Public Environments (pes), and we then investigate the notions of knowledge that arise in pes. pes build upon the well-known interpreted systems model of knowledge [6], in which the knowledge of an agent is characterised via an “epistemic indistinguishability” relation over system states, whereby two states s and t are said to be indistinguishable to an agent i if the variables visible to i have the same values in s and t . In pes, as in the interpreted systems model of knowledge [6], agents have access to a local set of variables. But, crucially, their knowledge can be based on a reﬁnement of the indistinguishability relation that derives from these variables: agents may observe the occurrence of an action, and from this be able to rule out some states, even though the local information in such states is the same as in the current state. Moreover, the protocol in a pe is a “standard” pdl program, and it is commonly known that the program under execution is “visible” to all agents in the pe. To better understand this idea, suppose there is some pdl program π that is being executed, and that π manipulates the values of the variables in the system. The program π is assumed to be public knowledge. Suppose agent i sees the variables y and z , and this is commonly known. So, it is public knowledge that i knows the value of y and z , at any stage of the program. There is a H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 97–110, 2011. c Springer-Verlag Berlin Heidelberg 2011

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third variable, x , which is not visible to i. Now, suppose that the program π is actually the assignment x := y. After this program is executed, it will be common knowledge that i knows the (new) value of x , even though the value of x is not visible to i, because i sees the value of y and knows that the program assigned this value to x . If the program y := x is executed instead, then i will come to learn the (old and current) value of x . Were the assignment y := x + z , then i would learn the value of x as well. Thus agents can ‘learn’ the values of variables through the execution of certain publicly known programs, even though those variables are not visible to the agents. The kind of questions that we want to investigate using pes then typically relate to, for example, how the process of executing a publicly known program in a pe will aﬀect the knowledge of agents who are able to see only some subset of the variables that the program is manipulating. Using pe, it is possible to analyse problems like the Russian Cards [4] and the Dining Cryptographers [3], and indeed many game-like scenarios, by incorporating the protocol (the outcome of the tossing of a coin, the passing of a card from one player to another) explicitly in the object language: we are not restricted to modelling every action by an information update, as in DEL. So, pe’s address the problem of Dynamic Epistemic Logic with factual change [9]. [5] also studies Dynamic Epistemic Logic with assignments, but where our assignments are restricted to program variables, assignments in [5] regard the assignment of the truth of an arbitrary formula to a propositional atom.

2

Motivation

Recall our motivating concern. Suppose that a number of agents are engaged in a commonly known algorithmic activity. What can be said about (the evolution of ) their knowledge if they cannot make a complete and correct observation of their environment? To motivate this question we present two well-known, but seemingly unrelated, scenarios and show that the solutions to the problems described in these scenarios require an answer to particular instances of our question above. We begin with a well known story, ﬁrst described in [3]. Three cryptographers are sitting down to diner at their favorite threestar restaurant. Their waiter informs them that arrangements have been made with the maitre d’hotel for the bill to be paid anonymously. One of the cryptographers might be paying for the dinner, or it might have been NSA. The three cryptographers respect each other’s right to make an anonymous payment, but they wonder if NSA is paying. They resolve their uncertainty fairly by carrying out the following protocol: Each cryptographer ﬂips an unbiased coin behind his menu, between him and the cryptographer on his right, so that only the two of them can see the outcome. Each cryptographer then states aloud whether the two coins

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he can see-the one he ﬂipped and the one his left-hand neighbor ﬂippedfell on the same side or on diﬀerent sides. If one of the cryptographers is the payer, he states the opposite of what he sees. An odd number of diﬀerences uttered at the table indicates that a cryptographer is paying; an even number indicates that NSA is paying (assuming that the dinner was paid for only once). Yet if a cryptographer is paying, neither of the other two learns anything from the utterances about which cryptographer it is. What are the essential features of the problem and its solution? The obvious answer is that we are looking for a cryptographic protocol that is provably correct with respect to some formal model of the task. The search for such a protocol is not related to our main question. However, we ﬁnd the proposed solution to be very relevant to the problem we study. Note that in order to be convinced of the correctness of this solution, we must be able to reason about the evolution of the knowledge of the cryptographers during the execution of a commonly known protocol; what is more, the essential feature of this protocol is that the cryptographers cannot make complete and accurate observations of their environment. Our second example comes from [2]. Before describing the problem, we would like to remind the reader about the main idea of this paper. Roughly, the authors want to formalise the following intuition. Suppose that we have a task that must be performed by a robot. Then we can decide if the robot can perform the task as follows. First, we try to determine the minimal knowledge required to perform the task and then we try to determine if the robot’s sensors and “reasoning abilities” allow it to gain that knowledge. If we have a match, then the robot is potentially able to perform the task. Their formal treatment is based on the following example. Two horizontal, perpendicular robotic arms must coordinate as follows. The ﬁrst arm must push a hot object lengthwise across a table until the second arm is able to push it sideways so that it falls into a cooling bin. The length of the table is marked in feet, from 0 through 10. The object is initially placed at position 0 on the table. The second arm is able to push the object if it is anywhere in the region [3, 7]. . . . We consider two variants of the problem: 1. The arms share a controller. The controller has access to a sensor reporting the position of the object with error no greater than 1, i.e., if the object’s current location is q then the reading can be anywhere in [q − 1, q + 1]; 2. Same as above, except that the error bound is 4 rather than 1. It is not hard to see that in the second case, there is no protocol that performs the task, whereas in the ﬁrst case there is. For example, a centralized protocol that deals with 1 is the following (where r is the current reading): If r ≤ 4 then Move(arm1 ) else Move(arm2 ).

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If we apply the authors’ intuition to this problem, we may say that the minimal knowledge necessary to perform the task is to know when you are in a certain location (namely the region [3, 7]) within some reasonable error bound. It is obvious that in the ﬁrst case the robot is able to aquire this knowledge while in the second it is not. And again, it is obvious that in order to see that this answer is correct, we must say something about (the evolution of) the knowledge of an agent engaged in an algorithmic activity while not being able to make complete and correct observations of its environment. We show that the protocols described in both examples can be modeled in an uniform way. Let’s ﬁrst model the solution to the dining cryptographers problem proposed by David Chaum. We name the three cryptographers 1, 2 and 3 respectively. Suppose that instead of coins each diﬀerent pair of cryptographers (i, j ), where i, j ∈ {1, 2, 3} and i = j , is assigned a diﬀerent variable c(i,j ) that can take the boolean values 0 and 1. These values model the two sides of the coin shared between i and j and the current value of the variable c(i,j ) is visible only to i and j . Let each cryptographer i be assigned a private variable pi that is not visible to the rest. These three variables can take only the values 0 and 1. If a certain variable is set to 1, then the respective cryptographer is paying. We will assume that at most one of the variables p1 , p2 , p3 is set to 1. Next we model the announcement made by each cryptographer in the following way. Let us associate a variable ai with each cryptographer i. Each ai is visible to all the cryptographers and holds the value c(i,j ) ⊕ c(i,k ) if pi = 0 or the value 1 − c(i,j ) ⊕ c(i,k ) otherwise, where ⊕ stands for the “exclusive or ” or xor of the two values and j = k . In this way we model the two types of announcement a cryptographer can make depending on the fact whether he is paying or not. Finally, we introduce a variable r which holds the result of a1 ⊕ a2 ⊕ a3 , i.e., the number of diﬀerences uttered at the table. Then r = 1 if and only if the number of diﬀerences is odd. Let us summarise. We have a set of variables V = {p1 , p2 , p3 , c(1,2) , c(1,3) , c(2,3) , a1 , a2 , a3 , r }. We associate a subset V (i) ⊆ V with every cryptographer i, where i ∈ {1, 2, 3}. Each set V (i) represents the variables visible to the respective cryptographer. Therefore, we have V (1) = {p1 , c(1,2) , c(1,3) , a1 , a2 , a3 , r }; V (2) = {p2 , c(1,2) , c(2,3) , a1 , a2 , a3 , r }; V (3) = {p3 , c(1,3) , c(2,3) , a1 , a2 , a3 , r }. All variables range over the set B = {0, 1}. Now we can represent the protocol described in [3] as an (non-deterministic) algorithm that changes the values of the variables in the appropriate way. For example, 1. if p1 := 0 then a1 := c(1,2) ⊕ c(1,3) ; else a1 := 1 − ( c(1,2) ⊕ c(1,3) ); 2. if p2 = 0 then a2 := c(1,2) ⊕ c(2,3) ; else a2 := 1 − ( c(1,2) ⊕ c(2,3) );

4. if p3 = 0 then a3 := c(1,3) ⊕ c(2,3) ; else a3 := 1 − (c(1,3) ⊕ c(2,3) ); 5. r := a1 ⊕ a2 ⊕ a3 ;

The ﬁrst 4 lines of the program say that if cryptographer i is not paying, i.e., the variable pi is set to 0, then i truthfully announces whether c(i,j ) and c(i,k )

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are diﬀerent or not, i.e., ai is set to c(i,j ) ⊕ c(i,k ) . If i is paying, i.e., pi is set to 1, then ai is set to the opposite value of c(i,j ) ⊕ c(i,k ) . Line 5 sets the value of r to xor of a1 , a2 , a3 . Note that we assume that not more than one of the variables p1 , p2 , p3 is set to 1. This is common knowledge among the agents. We do not assume that they have no other knowledge beside the knowledge obtained from the observable variables. For example, they must be convinced that an odd number of diﬀerences implies that one of them is paying, i.e., they must know some of the properties of the xor function and they implicitly know that not more than one of them is paying, no one is lying etc. In short, in order to model the dynamics of the relevant knowledge (in this case this means the knowledge obtained from observation), we may have to assume a lot of relevant background knowledge. The only requirement we have is: – The knowledge gained from observation does not contradict the background knowledge of the agents, i.e., in this case no one of the agents can consider it possible (based on the values of the observable variables only) that more than one of the variables pi can have the value 1. Now, we show how to model our second example in this setting. The essential problem here is that the robot’s sensor can detect its current position within some error bound. Therefore, the problem consists in the relation between the position read by the sensor, and the real position, i.e., between the current value of some observable variable that is within some error bound with respect to the value of some unobservable variable. So, let x be a variable observed by an agent (intuitively, this variable stores the sensor readings), y be a variable that is not observable (intuitively y stores the real position) and e be another visible variable (intuitively, e has some constant value that represents the error bound). This means that we assume that the robot “knows” what its error bound is. We have the following simple program. 1. x := y − k ∪ x := y + k ; where k ≤ e . That is, x is non-deterministically assigned some value that diﬀers from y within the error bound e. Now, we may ask the question what a robot observing x can know about the value of y. Having established that the essential features of the scenario from [2] can be modeled in our framework, we will concentrate on the dining cryptographers problem to motivate the assumption we make. Of course, these assumptions will be true for our second example. Let us address the evolution of the knowledge gained from observation of the three cryptographers during the execution of this program. Each step of the program induces some change in the knowledge of each cryptographer by changing the value of the relevant program variables. We can model the knowledge of all the cryptographers at a given point during the execution of the algorithm using the standard machinery of Kripke models. The underlying set of such a model consists of: – (W ) All the assignments of values to the program variables considered possible by at least one of the cryptographers at this particular program step.

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The epistemic relation modelling the knowledge of a cryptographer i must be such that – If i cannot distinguish between two valuations then they must coincide on the values assigned to the variables in V (i); – If a cryptographer i can distinguish two valuations at a particular step of the computation, then s(he) can distinguish their updated images at the next step of the computation. If M is a model for our description of the protocol, then we would like it to satisfy properties like the following. Let n = (¬p1 ∧ ¬p2 ∧ ¬p3 ) (i.e., the NSA paid) Ki n) ∧ (¬n → [π] Ki ¬n) M |= (n → [π] i=1,2,3

i=1,2,3

Saying that if the NSA paid, all cryptographers will know it afterwards, and if the NSA did not pay but a cryptographer did, this will become known as well. Of course, on top of this we need: M |= pi → [π] (¬Kj ¬pi ) j =i

Note that we do not say that if two valuations coincide on the values of the variables visible to i, then i cannot diﬀerentiate between them. This assumption would lead to counterintuitive scenarios. Let us summarise. A group of agents is engaged in an algorithmic activity. These agents can observe only a part of their environment which is aﬀected by this activity. Based on their knowledge of the algorithm and on the values of the observable variables, they can draw some conclusions and update their knowledge at each step of the program. We assume that each “variable” can be assigned only ﬁnitely many values. To make things easier, we further assume that the possible values can be just 0 and 1. During its execution, the algorithm can act on only ﬁnitely many variables of the agents’ environment. The basic algorithmic steps are assignments of the form x := t , where t is a term of the language to be deﬁned soon and tests. The agents’ knowledge at each particular step of the algorithm is modelled using Kripke models. The dynamics of the knowledge is modelled using suitably deﬁned updates of the Kripke models.

3

Language and Semantics

Let Ag = {a1 , . . . , am } be a set of agents and Var = {x1 , . . . xn } a set of variables. We deﬁne ϕ ∈ L: ϕ := ϕ0 | Vi x | ¬ϕ | ϕ ∧ ϕ | [τ ]ϕ | Ki ϕ | Oi ϕ | 2ϕ where i ∈ Ag. Vi x says that agent i sees the value of x . Oi ϕ denotes that i observes that ϕ holds. Ki is the knowledge operator, and 2 will be a universal

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modal operator: it facilitates us to express properties like 2[τ ](Ki (x = 0) ∧ ¬Kj (x = 0): ‘no matter what the actual valuation is, after execution of τ , agent i knows that the value of x is 0, while j does not know this. To deﬁne a Boolean expression ϕ0 , we ﬁrst deﬁne terms t . In this paper, terms will have values over {0, 1}, this is a rather arbitrary choice, but what matters here is that the domain is ﬁnite. Terms are deﬁned as t := 0 | 1 | x | t + t | t × t | −t VAR(t ) denotes the set of variables occurring in t . Boolean expressions over terms are: ϕ0 := t = t | t < t | ¬ϕ0 | ϕ0 ∧ ϕ0 Finally, we deﬁne programs: τ := ϕ0 ? | x := t | τ ∪ τ | τ ; τ where x ∈ Var and ϕ0 is a Boolean expression. A valuation θ : Var → {0.1} assigns a value to each variable. Let the set of valuations be Θ. We assume that v is extended to a function v : T er → {0, 1} in a standard way, i.e., we assume a standard semantics |=B for which v |=B ϕ0 is deﬁned in terms of giving a meaning to +, −, ×, <. We extend this semantics |=B to establish the truth of algebraic claims ϕ0 , in particular that enables us to reason about such claims: {ϕ1 , ϕ2 , . . .} |=B ϕ0 simply means that any valuation satisfying ϕ1 , . . ., also satisﬁes ϕ0 . Each agent i is able to read the value of some variables V (i) ⊂ Var . For two valuations θ1 , θ2 , we write θ1 ∼i θ2 if for all x ∈ V (i), θ1 (x ) = θ2 (x ). Definition 1 (Epistemic Models). Given Var and Ag, we ﬁrst deﬁne an epistemic S 5-model for Ag with universal modality, or epistemic model, for short, as M = W , R, V , f where 1. W is a non-empty set of states; 2. f : W → Θ assigns a valuation θ to each state; 3. R : Ag → 2(W ×W ) assigns an accessibility relation to each agent: we write Ri uv rather than (u, v ) ∈ R(i). Each R(i) is supposed to be an equivalence relation, and moreover, we assume that Ri uv implies that f (u) ∼i f (v ); 4. V : Ag → 2Var keeps track of the set of variables U that agent i ‘can see’. If M = W , R, V , f , we write w ∈ M for w ∈ W . For u, v ∈ M , we write u ∼i v for f (u) ∼i f (v ). A pair M , v (with v ∈ M ) is called a pointed model. Let EM denote the class of pointed epistemic models. We now deﬁne Rτ ⊆ EM× EM. This is a simple variation of pdl ([8]), for which we use assignments and tests as atomic programs, and sequential composition and choice as composites. In a setting like ours, where we have knowledge, the

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test will behave as a public announcement ([1]). First, given a valuation θ ∈ Θ, deﬁne (x := t )(θ) ∈ Θ as follows: θ(y) if y = x (1) ((x := t )(θ))(y) = θ(t ) else Definition 2 (Assignment). Given (M , v ) = ( W , R, V , f , v ), we deﬁne R(x :=t) (M , v )(M , v ) iﬀ (M , v ) = W , R , V , f , v ), where 1. 2. 3. 4.

W = {w | w ∈ W } Ri v u iﬀ v ∼i u and Ri vu V = V f (w ) = (x := t )(f (w )); i.e., if f (w ) = θ, then f (w ) = (x := t )(θ)

Definition 3 (Test). Given a pointed model M , v , we now deﬁne the eﬀect of a test ϕ0 ?. This time, we ﬁrst specify the result of applying the test to the whole model M = W , R, V , f : (ϕ0 ?)(M ) = M = W , R , V , f where 1. 2. 3. 4.

W = {v | v ∈ W & M , v |= ϕ0 } Ri v u iﬀ v ∼i u and Ri vu V = V f (v ) = f (v )

We then say that Rϕ0 ? (M , v )(M , v ) if (ϕ0 ?)(M , v ) = (M , v ) and M , v |= ϕ0 . Finally, we deﬁne the relations for sequential composition and choice ([8]): – Rτ1 ;τ2 (M , v )(M , v ) iﬀ ∃M , v such that Rτ1 (M , v )(M , v ) and Rτ2 (M , v )(M , v ). – Rτ1 ∪τ2 (M , v )(M , v ) iﬀ either Rτ1 (M , v )(M , v ) or Rτ2 (M , v )(M , v ). Definition 4 (Public Environment). Given a set of programs T , a Public Environment P = EM , T is a set of pointed epistemic models EM ⊆ EM closed under programs τ ∈ T . Let M = W , R, V , f ∈ EM . A triple P , M , v is called a state of the public environment, or a state. We deﬁne: – – – – – – – –

P, M , v P, M , v P, M , v P, M , v P, M , v P, M , v P, M , v P, M , v

|= ϕ0 iﬀ f (v ) |=B ϕ0 |= Vi x iﬀ x ∈ V (i) |= ¬ϕ iﬀ not P , M , v |= ϕ |= ϕ ∧ ψ iﬀ P , M , v |= ϕ and P , M , v |= ψ |= Ki ϕ iﬀ for all u ∈ W : Ri vu implies P , M , u |= ϕ |= Oi ϕ iﬀ for all w ∈ W : f (w ) ∼i f (v ) ⇒ P , M , w |= ϕ |= 2ϕ iﬀ for all w ∈ W : P , M , w |= ϕ |= [τ ]ϕ iﬀ (τ )(M , v )(M , v ) implies P , M , v |= ϕ

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ˆ i ϕ will denote the We write Mi ϕ for the dual of Ki ϕ, i.e., Mi ϕ = ¬Ki ¬ϕ. Also, O dual of Oi ϕ. Note that the operator Oi is deﬁnable in terms of valuation descriptions and boxes. Let x1 , . . . xm be the variables that agent i can observe (which can be expressed as Ψi = k ≤m Vi xk ∧ ¬ j >m Vi xj ). Then Oi ϕ is equivalent to the following schema, where each ci ranges over the set {0, 1}. Ψi → (x1 = c1 ∧ ×2 = c2 ∧ . . . ∧ xm = cm → 2(x1 = c1 ∧ x2 = c2 ∧ . . . ∧ xm = cm → ϕ))

Example 1. Figure 1 (left) shows a public environment for two variables x and y where V1 = {x } and V2 = {y}. The program that is executed is (x = 1 ∨ y = 1)?; x := 0; y := 1. Note that the ﬁnal epistemic model it is common knowledge that x = 0∧y = 1. Note however that we cannot identify the three states despite the fact that the valuations of the variables are the same.

01

11 2

1

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10

(x = 1 v y = 1)? 01

11 2 1

10

x:=0

00 01

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01 2

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2

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1, 3 10

11

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x := y y:=1

01

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2 1

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3 11

Fig. 1. Executing (x = 1 ∨ y = 1)?; x := 0; y := 1 (left) and x := y (right)

Example 2. Consider the model M from Figure 1 (right). Assume V1 = {x }, and V2 = V3 = ∅. The following table summarises the change in knowledge from M , 00 (ﬁrst row) to M , 00 (second row) while executing the program x := y. ¬K1 (y = 0) ¬K2 (x = 0) ¬K3 (x = y) K3 (x = 0) K1 (y = 0) K2 (x = 0) K3 (x = y) ¬K3 (x = 0) Through the assignment x := y agent 1 learns the value of y (because he reads x ), agent 2 learns the value of x (because he know the value of y) and agent 3, like all other agents, comes to know that x = y.

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Example 3. We show under which conditions an agent i can learn the value of a variable. Consider the following program, where x and y are diﬀerent variables: α = ((y = 0?; x := t ) ∪ (y = 1?; x := u)) The following are valid: 1. (Ki (y = 0) → [α]Ki (x = t )) ∧ (Ki (y = 1) → [α]Ki (x = u)) Knowing the condition for branching implies knowing which program is executed; 2. (Vi x ∧ Ki (t = u)) → ([α]Ki y = 0 ∨ [α]Ki y = 1) If an agent can read a variable x , and the value of the variable depends on a variable y, then the agent knows retrospectively what y was. 3. (¬Vi y ∧ ¬Vi x ∧ ¬Ki (y = 0) ∧ ¬Ki (y = 1)) → [α](¬Ki (y = 0) ∧ ¬Ki (y = 1)) An agent who cannot see x nor y cannot deduce y’s value from α. Example 4. We want to swap the value of two variables in a public environment, where the only spare variable is visible to an observer. Can we swap the variables without revealing the variables we want to swap? Formally, let Var = {x1 , x2 , x3 }, Ag = {1}, and V (1) = {x3 }. Informally, the designer of the program wants to ensure that agent 1 never learns the value of x1 or x2 . Formally, if i ∈ {1, 2}, we can capture this in the following scheme: χ = 2[π](¬K1 (xi = 1) ∧ ¬K1 (xi = 0)) Consider the following program π: x3 := x1 ; x1 := x2 ; x2 := x3 Clearly, if M is the epistemic model that formalises this, we have M |= ¬χ. But of course, π above is not the only solution to the problem of swapping variables. Now consider the following program π: x1 := x1 + x2 ; x2 := x1 − x2 ; x1 := x1 − x2 ; In this case, with M the epistemic model, we have M |= χ, as desired. Learning and Recall The principle of recall (wrt. a program τ ) is Ki [τ ]ϕ → [τ ]Ki ϕ

(2)

It is straightforward to verify that assignment and test satisfy (2), and moreover, that it is preserved by sequential composition and choice. However, now consider the converse of (2), which is often referred to as no-learning: [τ ]Ki ϕ → Ki [τ ]ϕ

(3)

This principle is not valid, as we can see from Example 2. In that example, we have P , M , 00 |= M1 x := y (x = 1), but not P , M , 00 |= x := y M1 (x = 1) (and hence [x := t ]Ki ϕ → Ki [x := t ]ϕ is not valid). Semantically, no learning is ∀w , t (Ri vw & Rτ wt ⇒ ∃s(Rτ vs ∧ Ri st ))

(4)

We can phrase failure of (4) in Example 2 as follows: In 00, agent 1 considers the state 01 possible, which with the assignment x := y would map to 11, however,

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after the assignment takes the state 00 to 00, since 1 sees variable x , he does not consider the state 11 as an alternative any longer. Loosely formulated: through the assignment x := y in 00, agent 1 learns that 01 was not the current state. From the deﬁnition of assignment, it is easy to see that the following is valid: ˆ i ϕ ∧ Mi x := t ϕ x := t Mi ϕ ↔ x := t O

(5)

Also, the test operator fails no learning: again, in Example 2, we have M , 00 |= M1 y = 1? , but we do not have P , M , 00 |= y = 1? M1 : by the fact that 00 does not ‘survive’ the test y = 1, agent 1 ‘learns’ that it was not the real state. Before revisiting the dining cryptographers, we mention some validities. • |= [x := t ]Ki (x = t ) ∧ [x = t ?]Ki (x = t ) ∧ x := t Ki (x = t ) Agents see the programs and are aware of their eﬀects. An assignment x := t can always be executed, which is not true for a test x = t ?. • |= Ki (t = u) → [x := t ]Ki (x = u) Knowing the value of a term implies knowing the value of a variable if it gets assigned that term. Dining cryptographers revisited. We will now indicate how to model this scenario in such a way that it is possible to reason about lying participants, a situation where more than one of the cryptographers is paying etc. We introduce just three new variables h1 , h2 , h3 and modify slightly our previous algorithm. 1. if h1 = 1 then else p1 := 1 − p1 2. if h2 = 1 then else p2 := 1 − p2 3. if h3 = 1 then else p3 := 1 − p3

p1 := p1 ; ; p2 := p2 ; ; p3 := p3 ; ;

4. if p1 = 0 then a1 := else a1 := 1 − ( c(1,2) ⊕ 5. if p2 = 0 then a2 := else a2 := 1 − ( c(1,2) ⊕ 6. if p3 = 0 then a3 := else a3 := 1 − ( c(1,3) ⊕ 7. r := a1 ⊕ a2 ⊕ a3

c(1,2) c(1,3) c(1,2) c(2,3) c(1,3) c(2,3)

⊕ c(1,3) ; ); ⊕ c(2,3) ; ); ⊕ c(2,3) ; );

Notice the way we have modelled lying in the ﬁrst three lines of the program. If hi = 0 then the agent Ai will, in eﬀect, behave contrary to what his paying variable indicates. In the original treatment of the problem, it is implicity assumed that all the cryptographers are honest and at most one of them is paying. This (extremely) strong assumption can be made explicit in our framework. Let ϕ stand for the formula: (h1 ∧h2 ∧h3 )∧{(p1 ∧¬p2 ∧¬p3 )∨(¬p1 ∧p2 ∧¬p3 )∨(¬p1 ∧¬p2 ∧p3 )∨(¬p1 ∧¬p2 ∧¬p3 )} If we stipulate that our initial epistemic models satisfy ϕ then the original assumptions of [3] will become common knowledge among the agents. And for the epistemic model M associated with our assumptions, we have the properties we wanted to conﬁrm in Example 1, namely: Ki ¬n) ∧ (¬a4 → [π] Ki n) M |= (a4 → [π] i=1,2,3

i=1,2,3

If we chose, however, to concentrate on the general case where the motivation and honesty of the participants is unknown then many of the shortcomings of

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this protocol can be easily seen. If two participants behave as if they have paid then we will have a collision and the ﬁnal result displayed in r will be 0 making this case indistinguishable from the case where no one is paying. Notice the words ’behave as if they have paid’. This behaviour can result from the fact that they may be lying, or only one of them is lying or both have paid. As long as the variable hi , is visible only to Ai , these cases cannot be distinguished by the other two agents. Similar observations can be made in the case where only one or all the cryptographers behave as if they have paid.

4

Axiomatization

Let L0 be the language without modal operators. The usual ([8]) dynamic axioms apply, although not to the full language: exceptions are the axioms Assign, and test (τ ). Propositional and Boolean Component Prop ϕ if ϕ is a prop. tautology if B ϕ0 Bool ϕ0

Rules of Inference Modus Ponens ϕ, (ϕ → ψ) ⇒ ψ Necessitation ϕ ⇒ 2ϕ

Epistemic and Universal Component IK the S 5 axioms for knowledge UB the S 5 axioms for 2 BO 2ϕ → Oi ϕ OK Oi ϕ → Ki ϕ KV (x = c ∧ Vi x ) → Ki (x = c)

Visibility VD1 VD2 VB1 VB2 VK

Dynamic Component Assign [x := t]ϕ0 ↔ ϕ0 ([t/x ]) Func [x := t]ϕ ↔ x := t ϕ K (τ ) [τ ](ϕ → ψ) → ([τ ]ϕ → [τ ]ψ) union(τ ) [τ ∪ τ ]ϕ ↔ ([τ ]ϕ ∧ [τ ]ϕ) comp(τ ) [τ ; τ ]ϕ ↔ [τ ][τ ]ϕ test(τ ) [ϕ?]ϕ0 ↔ (ϕ → ϕ0 )

Dynamic and Epistemic RL x := t Mi ϕ ↔ ˆ i ϕ ∧ Mi x := t ϕ x := t O Dynamic and Universal 3 τ 3ϕ → 3τ ϕ x := t 3 3x := t ϕ → x := t 3ϕ ϕ0 ? 3 3ϕ0 ? ϕ ↔ (ϕ ∧ ϕ0 ? 3ϕ)

Vi x → [τ ]Vi x ¬Vi x → [τ ]¬Vi x Vi x → 2Vi x ¬Vi x → 2¬Vi x (Vi x ∧ x = c) → Ki (x = c), c ∈ {0, 1}

Fig. 2. Axioms of kppe

Axiom Assign is not valid for arbitrary ϕ, as the following example shows. First of all, we need to exclude formulas of the form Vi x . It is obvious that [x := y]Vi x does not imply Vi y. But even if we made an exception for Vi formulas, we would leave undesired equivalences: Take y for t . If [x := y]K1 (y = 0) holds and Ass were valid, this would imply that K1 y = 0, which it should not (it might well be that agent 1 learned the value of y since he sees that x becomes 0). Note that VK is only valid for values 0 and 1, and not for arbitrary terms: we do not have |= (Vi x ∧ x = t ) → Ki (x = t ) (take for instance t = z for a counterexample). Our completeness proof uses Theorem 1, of which the proof immediately follows from the following equivalences:

Knowledge and Action in Semi-public Environments [x := t]Vi x [x := t]ϕ0 [α](ϕ ∧ ψ) [ϕ0 ?]¬ϕ [ϕ0 ?]2ϕ [ϕ0 ?]Ki ϕ

↔ Vi x ↔ ϕ0 [t/x ] ↔ ([α]ϕ ∧ [α]ψ) ↔ (ϕ0 → ¬[ϕ0 ?]ϕ ↔ (ϕ0 → 2[ϕ0 ?]ϕ ↔ (ϕ → Ki [ϕ0 ?]ϕ)

[ϕ0 ?]Vi x ↔ [ϕ0 ?]ψ0 ↔ [x := t]¬ϕ ↔ [x := t]2ϕ ↔ [x := t]Ki ϕ ↔

109

(ϕ0 → Vi x ) (ϕ0 → ψ0 ) ¬[x := t]ϕ 2[x := t]ϕ (Ki [x := t]ϕ ∨ [x := t]Oi ϕ)

Theorem 1. Every formula is equivalent to one without dynamic operators. Theorem 1 implies that every eﬀect of a program is completely determined by the ‘ﬁrst’ epistemic model. More precisely, it implies that for every formula ϕ there is an equivalent epistemic formula (using Vi and 2) ϕ which is provably equivalent. Theorem 2. The logic kppe is sound and complete with respect to public environments.

5

Related Work and Conclusions

We have introduced a framework for reasoning about programs and valuations where knowledge is based on a hybrid semantics using notions from interpreted systems and general S 5 axioms. This is only a ﬁrst step in doing so. Possible extensions are manyfold. First, we think it is possible to include repetition ( ) as an operator on programs and still obtain a well-behaved logical system, although the technical details for doing so can become involved. There are several restrictions in our framework that may be worthwhile relaxing, like allowing a richer language for tests, and not assuming that it is common knowledge which variables are seen by whom, or what the program under execution is. Those assumptions seem related, and removing them may well be a way to reason about Knowledge-based Programs ([7]), where the programs are distributed over the agents, and where it would be possible to branch in a program depending on the knowledge of certain agents.

References 1. Baltag, A., Moss, L.S., Solecki, S.: The logic of common knowledge, public announcements, and private suspicions. In: Gilboa, I. (ed.) Proceedings of the 7th Conference on Theoretical Aspects of Rationality and Knowledge (TARK 1998), pp. 43–56 (1998) 2. Brafman, R., Halpern, J.Y., Shoham, Y.: On the knowledge requirements of tasks. Artificial Intelligence 98, 317–349 (1998) 3. Chaum, D.: The dining cryptographers problem: Unconditional sender and recipient untraceability. Journal of Cryptology 1(1), 65–75 (1988) 4. van Ditmarsch, H.: The russian cards problem. Studia Logica 75, 31–62 (2003) 5. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic epistemic logic with assignments. In: AAMAS 2005, pp. 141–148 (2005)

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6. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. The MIT Press, Cambridge (1995) 7. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Knowledge-based programs. Distributed Computing 10(4), 199–225 (1997) 8. Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. The MIT Press, Cambridge (2000) 9. Sietsma, F.: Model checking for dynamic epistemic logic with factual change. Tech. rep., UvA and CWI, Amsterdam (2007), http://homepages.cwi.nl/~ sietsma/papers/mcdelfc.pdf

Taking Mistakes Seriously: Equivalence Notions for Game Scenarios with oﬀ Equilibrium Play Alistair Isaac1,3, and Tomohiro Hoshi2,3 1

2

Department of Philosophy, University of Michigan, Ann Arbor, MI [email protected] Educational Program for Gifted Youth, Stanford University, Stanford, CA [email protected] 3 Logical Dynamics Lab, Stanford University, Stanford, CA

Abstract. This paper investigates strategies for responding rationally to opponents who make mistakes. We identify two distinct interpretations of mistakes in the game theory literature: trembling hand and risk averse mistakes. We introduce the concept of an EFG Scenario, a game plus strategy proﬁle, in order to probe the properties of these diﬀerent types of mistake. An analysis of equivalence preserving transformations over EFG Scenarios reveals that risk averse mistakes are a form of rational play, while trembling hand mistakes are equivalent to moves by nature.

1

Introduction: Two Types of Mistake

People make mistakes, and these mistakes aﬀect the structure of strategic interactions. But what is the rational response to an opponent’s mistake? Suppose, for example, an agent knows an opponent has committed a mistake, but is uncertain which amongst a set of possible mistakes has been made. In such a situation, the agent needs to place a probability distribution over possible scenarios in order to calculate rational action. But this requires him to reason about the mistake: to “guess” what mistake has been made. How does rationality constrain the probability distribution over possible mistakes? At a ﬁrst pass, one might assume that mistakes are completely random. In this case, any probability distribution over possible mistakes may be rationally justiﬁable. Consider typos, for example. If I know you’ve mistyped a letter on your keyboard, but I don’t know which letter was substituted, what probability distribution should I place over letters? A ﬂat distribution? A distribution peaked around letters on keys close to the intended letter? But what if you are using a non-standard keyboard layout? Many diﬀerent probability distributions seem rationally justiﬁable here, and the mere fact that a mistake has occurred alone does not decide between them. We call mistakes for which any probability distribution over outcomes is justiﬁable, “trembling hand” mistakes.

Supported by a postdoctoral fellowship from the McDonnell Foundation Research Consortium on Causal Learning.

H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 111–124, 2011. c Springer-Verlag Berlin Heidelberg 2011

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A more sophisticated analysis of mistakes is possible if we know more about the agent. For example, you are much more likely to misspell a word while typing than to get into an accident while driving. Why is that? One possible explanation is that a car crash has a far greater cost for you than a misspelling, and consequently you are more careful while driving than you are while typing. If this is a general feature of mistakes, we might analyze it by treating the probability of a mistake as inversely proportional to the cost to the agent. We call mistakes for which only probability distributions over outcomes inversely proportional to cost are justiﬁable, “risk averse” mistakes. These two types of mistakes are relevant to the theory of games when one considers how to calculate oﬀ equilibrium play. In a strategic interaction, a (Nash) equilibrium is a pattern of moves such that no agent can do better against the others by changing his move. In the case of an extensive form game, we can think of an equilibrium as a path through the tree, or, more generally, as a probability distribution over branches. If an agent ﬁnds himself at a decision point assigned zero probability by an equilibrium, how should he play? Since the point is oﬀ equilibrium, he can only have reached it if his opponent made a mistake. Nash’s solution concept is unhelpful here because it allows the agent to make a completely arbitrary move; this is equivalent to abandoning all rationality when an opponent makes a mistake. [5] deﬁned the perfect equilibrium to characterize rational play against opponents who make trembling hand mistakes. [3] deﬁned the proper equilibrium to characterize rational play against opponents who make risk averse mistakes. But what exactly are we committed to when we adopt one of these analyses of mistake making? Section 2 argues that there is a close conceptual connection between the treatment of oﬀ equilibrium play as communicative (i.e. deliberate) and the risk aversion interpretation of mistakes. The remainder of the paper is devoted to demonstrating the conceptual connection between trembling hand mistakes and moves by nature. Section 3 reviews basic game deﬁnitions and Sect. 4 introduces our fundamental formal object, game scenarios. In Sect. 5, we extend equivalence relations for extensive form games to game scenarios and argue that trembling hand mistakes can best be studied by analyzing their behavior across the “coalescing of moves” transformation. Section 6 implements this project and proves our main result: trembling hand mistakes are equivalent to moves by nature.

2

Proper Equilibria and Wishful Thinking

Consider the extensive form games in Fig. 1. ΓA and ΓB both share the same reduced normal form representation. This is because in each game player I has three non-redundant strategies. In ΓA , I’s strategies are {a, c, d}; in ΓB his strategies are {ac, ad, bc, bd}, but ad and ac are equivalent, so removing redundancies leaves three strategies, {a, bc, bd}. Since ΓA and ΓB share reduced normal form, they are related by a Thompson transformation, in this case coalescing of moves ([6], see Sect. 5 for details). Since proper equilibria are preserved across Thompson transformations, both games have the same (in this case unique) proper equilibrium, which takes players to the payoﬀ 6,6.

Taking Mistakes Seriously: Equivalence Notions for Game Scenarios

ΓA

ΓB

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2,2.

Fig. 1. Two extensive form games which share a normal form, but not perfect equilibria

However, as discussed by [2], there is a perverse perfect equilibrium for ΓA which does not appear in ΓB . In this perverse scenario, I plays a while II plays y. A perfect equilibrium is the limiting best response to a sequence of completely probabilistic strategies. To see that (a, y) is such a limit, consider the perturbed strategy for player I which assigns probabilities (1 − 10, , 9) and the strategy for player II which assigns (, 1 − ). As shrinks, Player II’s rational probability assignment for the lower node of his information set converges on .9. Player II is thus rationally justiﬁed in playing y because his expected payoﬀ is then .9(2)+.1(0) = 1.8, while his expected payoﬀ for playing x is only .1(6)+.9(0) = .6. Consequently, the limiting strategy (a, y) is a perfect equilibrium. This perturbation of strategies assumes any deviation player I makes from his choice of a is brought about by a trembling hand error, i.e. assignment of small probabilities to c and d is completely unconstrained. This analysis is ruled out in the proper equilibrium case by the restriction to risk averse mistakes. If I intends to play a but makes a risk averse mistake, then I is much more likely to mistakenly play c than d (because I’s expected payoﬀ is greater after c than after d). This implies that no matter what I plans, II places higher probability at his top node, and consequently moves x. Once I reasons that II will move x, his best response move becomes c. The same reasoning applies to the equilibrium (bc, x) in ΓB (for full discussion, see [4], Ch. 5). There is another way to reason about oﬀ equilibrium play which produces this same result: [P]layer II knows that I will never choose d, which is strictly dominated by a . . . so if II sees he has to play, he should deduce that I, who was supposed to play a and was sure to get 4 in this way, certainly did not choose d, where he was sure to get less than 4; player II should thus infer that I had in fact played c, betting on a chance to get more than 4 (and on the fact that II would understand this signal); and so player II should play x. ([2], 1007, labels replaced appropriately throughout) Notice that in this pattern of reasoning, we started with two of the same assumptions (that I would play a and that II (against expectation) would ﬁnd himself

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able to make a move) and reasoned to the same conclusion (that II should play x and thus I should switch his intentions to c) as in the risk aversion analysis. An important diﬀerence, however, is that we dropped the interpretation of (supposed) oﬀ equilibrium play as produced by a mistake, and instead treated it as a form of communication. According to this “forward induction” style of reasoning, player II infers from his ability to play that player I has sent him a “signal” that he is at his top node (the one best for player I). Forward induction and risk averse reasoning won’t always coincide. However, we think the reason they coincide here is rather deep. Forward induction arguments assume players do not make mistakes at all. If a player ﬁnds himself at an unexpected information set, he can infer which node he is at by assuming that his very presence at that information set is a signal from an earlier player. This assumes a kind of “wishful thinking” on the part of the earlier player—he takes a risk under the (hopeful) assumption that the later player will understand his signal and respond accordingly. This (rational, but risky) play is mimicked by risk averse mistakes. The player who makes a risk averse mistake acts “rationally” even while making a mistake, since his mistaken act (probabilistically) minimizes risk (hence the name “risk averse” for these types of mistakes). This allows players that ﬁnd themselves at surprising information sets to reason as if they arrived there via a rational choice, since even the “errors” of a risk averse agent are self interested. This also explains why ΓA and ΓB are strategically equivalent for risk averse agents. In ΓA , the choice between c and d may be determined by mistake, whereas in ΓB it must be determined deliberately. Yet for the risk averse player, deliberate and accidental play coincide in relative probability (c is more likely than d, whether the choice is made by accident or deliberation). If these considerations are correct, then the risk averse interpretation of mistakes is deeply unsatisfying. Mistakes are not being taken seriously here, just treated as a domain to which rationality can be extended. The powerful concept of proper equilibrium which is thus produced assumes complete rationality. If we want to think seriously about how to play against an agent who makes mistakes, we’ll need to assume a weaker equilibrium concept. If that concept is perfect equilibrium, what exactly are we committing ourselves to in our assumptions about mistakes? The remainder of the paper addresses this question.

3

Review: Extensive Form Games and Their Equilibria

This section reviews basic deﬁnitions, and may be omitted without loss of continuity. Our presentation will closely follow [4], to which readers are referred for explanation and motivation. Definition 1 (Extensive Game). An extensive form game (EFG) is a tuple Γ = (T , N , {Ak }k∈N , {Ik }k∈N , {uk }k∈N ) where: 1. T = (W, ≺) is a ﬁnite tree with a disjoint union of vertices W = k∈N Vk ∪Z where Vk denotes the set of player k’s decision points and Z is the set of

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4. 5.

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terminal vertices. The set of immediate successors of w ∈ W is deﬁned by Succ(w) = {v | w ≺ v ∧ ¬∃x : w ≺ x ∧ x ≺ v}. N is a set of players. Ak maps (w, w ), where w ∈ Succ(w), to the action that k can play at w which leads to w . It is assumed that u = v implies Ak (w, u) = Ak (w, v). The set of available moves at w is deﬁned by Ak (w, ·) = {Ak (w, v)|v ∈ Succ(w)}. Ik partitions Vk and induces the function fk that maps w ∈ Vk to k’s information set fk (w) containing w. I denotes the set of all information sets. It is assumed that, if w, w are in the same information set, Ak (w, ·) = Ak (w , ·). The set of available moves at a point in an information set I may be thus denoted by AI . uk : Z → R is the payoﬀs for the player k on terminal nodes.

For the purposes of this paper, we assume that EFG’s satisfy perfect recall, i.e. players cannot be uncertain about their own past moves (see Sect. 5). Let Γ be an EFG. A pure strategy sk of a player k ∈ N in Γ is a function that assigns to every information set I ∈ Ik an action in A I . We deﬁne Sk to be the set of all pure strategies for k ∈ N so that Sk= I∈Ik AI . The set of pure strategy proﬁles for Γ is then deﬁned by S = k∈N Sk . The outcome O(s) of s ∈ S is the terminal node that results when each player k plays the corresponding game by following sk , i.e. O(s) = w1 . . . wn such that, for each m (1 ≤ m ≤ n), wm ∈ Vk for some k ∈ N and Ak (wm , wm+1 ) = sk (fk (wm )). Definition 2 (Strategic Form). The strategic form of an extensive form game Γ is a strategic game Γ s = (N , S, {vk }k∈N ), where vk : S → R is such that vk (s) = uk (O(s)) for s ∈ S. Definition 3 (Multiagent Form). The multiagentform of an EFG Γ is a strategic game Γ a = (I, I∈I AI , (vI )I∈I ), where vI : I∈I (AI ) → R is deﬁned in the following way: for any (aI )I∈I ∈ I∈I (AI ), if (sk )k∈N ∈ S is the strategy proﬁle such that si (J) = aJ for all i ∈ N and J ∈ Ik , then vI ((aI )I∈I ) = uk ((si )i∈N ) for k such that I ∈ Ik . Intuitively, the multiagent form of an EFG is a strategic game which treats every information set in Γ as a separate player. Each new agent receives the same payoﬀs as the player in the EFG who plays at that information set. A mixed strategy for k ∈ N in Γ is a probability distribution over Sk . Let Δ(X) be the set of probability distributions over X for a given set X. The set of all mixed strategies k ). The set of all mixed strategy for k is then denoted by Δ(S proﬁles of Γ is k∈N Δ(Sk ). Thus, each σ ∈ k∈N Δ(Sk ) can be written as σ = (σk )k∈N where each σk is a probability distribution in Δ(Sk ). We also denote the set of mixed strategy proﬁles that assign only non-zero probabilities to all elements in Sk by k∈N Δ0 (Sk ). Finally, for a given set X and a function f : X → R, we deﬁne argmax f (x) = {y ∈ X | f (y) = max f (x)}. x∈X

x∈X

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And for a sequence x = (xm )m∈X , we write x−n for (xm )m∈X−{n} . Definition 4 (Perfect Equilibrium). Let Γ a = (N, (Sk )k∈N , (u k )k∈N ) be the multiagent form of an extensive game Γ . A mixed strategy σ ∈ k∈N Δ(Sk ) is a perfect equilibrium of Γ iﬀ there exists a sequence (ˆ σ m )∞ m=1 such that 1. σ ˆ m ∈ k∈N Δ0 (Sk ) for all m ∈ N, 2. lim σ ˆkm (sk ) = σk (sk ) for all k ∈ N and sk ∈ Sk , and k→∞

m 3. σ ∈ argmax uk (ˆ σ−k , τk ) for all k ∈ N . τk ∈Δ(Sk )

A perfect equilibrium is a game solution in which each strategy is the best response to the limit of a sequence of purely probabilistic strategies. This ensures that there exists a small perturbation in an opponent’s strategy to which any oﬀ equilibrium move is a best response. Definition 5 (-Proper Equilibrium). Let Γ s = (N, (Sk )k∈N , (uk )k∈N ) be the strategic form of an extensive game Γ . For any positive number , a mixed strategy proﬁle σ is an -proper equilibrium iﬀ 1. σ ∈ k∈N Δ0 (Sk ) 2. for every k ∈ N and sk , tk ∈ Sk , uk (σ−k , [sk ]) < uk (σ−k , [tk ]) implies σk ([sk ]) < σk ([tk ]) Where (σ−k , [xk ]) is exactly like σ except that player k’s mixed strategy is replaced with the pure strategy xk ∈ Sk , i.e. k plays xk with probability 1. Definition 6 (Proper Equilibrium). Let Γ s = (N, (Sk )k∈N , (uk )k∈N ) be the strategic form of an extensive game Γ . A mixed-strategy proﬁle σ ¯ is a proper equilibrium iﬀ there is a sequence ((m), σ m )∞ such that k=1 1. for all k ∈ N and sk ∈ Sk , (a) lim (m) = 0 m→∞

(b) lim σkm (sk ) = σ ¯k (sk ) m→∞

2. for every m, σ m is an (m)-proper equilibrium. A proper equilibrium is a perfect equilibrium in which only strategies which assign move probabilities proportional (modulo ) to payoﬀ are permitted in the limiting sequence of purely probabilistic strategies.

4

Deﬁning Mistakes in EFG Scenarios

What is the appropriate representation of a game for analyzing mistakes? Mistakes occur in time—a mistake made before a move may aﬀect play, while a mistake made after a move may not. As demonstrated in Sects. 1 and 2, the beliefs of a rational player about what mistake has occurred inﬂuence his response to it. So, if we want to explore the conceptual commitments behind the trembling

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hand interpretation of mistakes, we’ll need to consider game structures which represent both temporal and epistemic structure. This motivates investigating EFG Scenarios: intuitively, EFG’s with probability distributions over actions at each information set. An EFG Scenario represents a player’s beliefs about how play progresses, including where, if at all, mistakes occur. Let Γ be an EFG. A behavioral strategy for an information set I is a probability distribution over AI . A behavioral strategy for a player k is an ele ment of Δ(A ). The I I∈I k set of behavioral strategy proﬁles in Γ is deﬁned by k∈N I∈Ik Δ(AI )(= I∈I Δ(AI )). Based on this deﬁnition, for a behavioral strategy proﬁle σ, we may write σ = (σk )k∈N = (σk,I )I∈I,k∈N . Definition 7 (EFG Scenario). An extensive form game scenario (EFGS) is a pair G = (Γ, σ), where Γ is an extensive form game and σ is a behavioral strategy proﬁle in Γ . EFGS’s induce a probability distribution over nodes in the tree. Let G = (Γ, σ). An action path A(w, v) is the sequence of actions (edges) which leads from w to v. Suppose that A(w, v) = a1 . . . an and let ki be the player of the action ai . Deﬁne Pσ (v | w) in G to be the probability that a node v is reached in Γ via σ, given w is reached, i.e. n σki (ai ). Pσ (v | w) = i=1

The evaluation node nI of an information set I is the “most recent” decision point in an EFG Γ such that all nodes in I fall under it: ∀w[∀v ∈ I(w ≺ v) → w nI ]. The evaluation node deﬁnes a subgame in the game tree with respect to which play at an information set may be evaluated for rationality. We deﬁne the conditional evaluation probability Pσ (w | I) of w ∈ I as Pσ (w | nI ) . w ∈I Pσ (w | nI )

Pσ (w | I) =

Suppose that a = AI (w, v) (a is played at I from w to v). Let Zv be the set of endpoints reachable from v and k ∈ N the agent who moves at I. The normalized expected value of a ∈ AI at w is ev(σ, w, a) = uk (z)Pσ (z | v). z∈Zv

The normalized expected value of action a weights the payoﬀs of reachable end nodes by the probabilities that they will be reached. Definition 8 (Rational Play at I). Given an EFGS G, a player k ∈ N plays rationally at I if 1. there exists w ∈ I such that Pσ (w | nI ) = 0 and σI = argmax Pσ (w | I) τ (a)ev(σ−I τ, w, a) τ ∈Δ(AI ) w∈I

a∈AI

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2. or for all w ∈ I, Pσ (w | nI ) = 0 and there exists some P ∈ Δ0 (I) such that P (w) τ (a)ev(σ−I τ, w, a) σI = argmax τ ∈Δ(AI ) w∈I

a∈AI

Definition 9 (Mistake at I). Given an EFGS G, a player k ∈ N makes a mistake at I if k does not play rationally at I. Intuitively, a mistake is a failure to maximize utility. For example, consider ΓB of Fig. 1; if σ takes play to any endpoint other than 6, 6 in the subgame below Player I’s second decision point, then either Player I makes a mistake, or Player II makes a mistake, even if σI (b) = 0. The only move for Player I which maximizes his expected payoﬀ at his second decision node is c, so if he plays d he makes a mistake. Likewise, since Player I’s second decision point is the evaluation node for Player II’s information set, a positive probability is assigned to at least one of the nodes in Player II’s information set, and clause 1 applies (unlike in ΓA , where clause 2 applies, and y is a rational move for Player II). A mixed strategy is a probability distribution over pure strategies. A behavioral strategy is a probability distribution over actions at each information set. Clearly, behavioral strategies over EFG’s are equivalent to mixed strategies over multiagent representations of EFG’s. This motivates the following deﬁnition. Definition 10 (In Perfect Equilibrium). An EFGS G = (Γ, σ) is in perfect equilibrium if σ is a perfect equilibrium of the EFG Γ . The following is a straightforward consequence of the above deﬁnitions. Proposition 1. If EFGS G = (Γ, σ) is in perfect equilibrium, then for all I ∈ I, play is rational at I. The relationship between mixed strategies and behavioral strategies is not unique, but many–many. Since EFGS’s begin with a behavioral strategy, we are interested here in only one direction, from behavioral strategies to mixed strategies. Given an EFGS G = (Γ, σ), we would like to identify a unique member σ ¯ ∈ k∈N Δ(Sk ) to use as the mixed strategy representation of σ = (σk )k∈N . For each k, deﬁne σ ¯k such that for all sk ∈ Sk , σ ¯k (sk ) = σk,I (sk (I)) I∈Ik

Then σ ¯ = (¯ σk )k∈N is the mixed strategy representation of σ. Definition 11 (In Proper Equilibrium). An EFGS G = (Γ, σ) is in proper equilibrium if the mixed representation σ ¯ of σ is a proper equilibrium of the normal form of EFG Γ . The following is an immediate consequence of Proposition 1. Proposition 2. If EFGS G = (Γ, σ) is in proper equilibrium, then for all I ∈ I, play is rational at I.

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5

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The Problem is Coalescing of Moves

EFGS’s allow us to probe the nature of trembling hand mistakes. One strategy for doing this is to examine the eﬀect on perfect equilibria of “strategically irrelevant” game transformations. The example of Fig. 1 demonstrates that perfect equilibria are not preserved over one such transformation, coalescing of moves. Coalescing of moves is one of four transformations proposed by [6] which are suﬃcient to take any EFG into any other EFG with the same strategic form. Unfortunately, they do not preserve perfect recall. [1] demonstrated that three transformations are suﬃcient to take any EFG with perfect recall into any other with the same strategic form, while preserving perfect recall. Since we assume perfect recall, we will examine the transformations of [1]. Two of these transformations, interchange of moves and coalescing of moves, are unchanged from those of [6]. The third transformation, addition of decision nodes, generalizes [6]’s addition of a superﬂuous move. Here, we will discuss interchange of moves and addition of a decision node informally, arguing that they preserve perfect equilibria. Consequently, coalescing of moves is the only transformation which does not preserve perfect equilibria, motivating a closer investigation of the relationship between coalescing of moves and mistakes in the following section. For the full deﬁnitions of these transformations, see [1].

I

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Fig. 2. Interchange of moves over EFG Scenarios

Interchange of moves is illustrated in Fig. 2. This transformation reverses the order of play of two contiguous information sets when the latter player is unaware of the former’s move. Extending interchange of moves to EFGS’s is trivial since the change in the ordering of moves does not aﬀect the actions available to each player, consequently, the behavioral strategy proﬁle with which the game is labeled may remain unchanged. Thus, we have the following result. Proposition 3. If G = (Γ, σ) and G = (Γ , σ ) are EFG Scenarios such that Γ is derived from Γ by interchange of moves and σ = σ, then G is in perfect equilibrium iﬀ G is in perfect equilibrium. Addition of decision nodes is illustrated in Fig. 3. The basic idea here is to add an information set for Player II which spans the entire subtree without aﬀecting

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Fig. 3. Addition of decision nodes over EFG Scenarios

perfect recall. In order to do this, copies of information sets below it will need to be duplicated. So, if Γ is derived from Γ by addition of decision nodes, it will not in general have the same number of information sets, and thus will not share a multiagent representation with Γ . Nevertheless, there is a natural transformation over strategy proﬁles that preserves perfect equilibrium across addition of decision nodes. We illustrate it with an example. Let Γ, Γ be the left and right games in Fig. 3. Let X and Y be the left and right information sets for player II in Γ , and X , Y1 , Y2 , his topmost and left and right lower information sets in Γ . Now, suppose that σ is a perfect equilibrium in Γ . Then there is a sequence (ˆ σm )∞ m=1 of strategies in the multiagent form of Γ satisfying the conditions in Deﬁnition 4. Construct a sequence (ˆ τ m )∞ ˆm in Γ in the follow way: m=1 of behavioral strategies τ m m ˆIm , if I ∈ {X , Y1 , Y2 }; τˆX ˆX ; and τˆYm1 = τˆYm2 = σ ˆYm . τˆIm = σ = σ

τ m )∞ Given that (ˆ σ m )∞ m=1 converges to σ, (ˆ m=1 clearly converges to τ deﬁned by: τI = σI , if I ∈ {X , Y1 , Y2 }; τX = σX ; and τY1 = τY2 = σY . It is easy to check that τ and (ˆ τ m )∞ m=1 satisfy the other conditions in Deﬁnition 4. The other direction is similar once we note that any sequence converging on τY1 will also converge on τY2 , so, without loss of generality, one of them may be chosen for constructing a sequence which converges in Γ . This strategy can easily be extended to the general case. Therefore we have the following. Proposition 4. Let G = (Γ, σ) and G = (Γ , τ ) be EFG scenarios. Suppose that Γ is derived from Γ by addition of decision nodes. Further suppose that τ is constructed from σ as above (and vice versa). Then G is in perfect equilibrium iﬀ G is in perfect equilibrium.

6

Trembling Hands and the Will of Nature

Any breakdown in the preservation of perfect equilibria across games which share strategic form must be due to coalescing of moves. The basic form of coalescing

Taking Mistakes Seriously: Equivalence Notions for Game Scenarios

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Fig. 4. Coalescing of moves over EFG Scenarios

of moves is to remove (introduce) a redundant decision node for a single player (Fig. 4). In this section, we show that equilibrium can always be preserved when coalescing of moves occurs on a positive path of play. When coalescing of moves occurs oﬀ the equilibrium path, we can preserve equilibrium by identifying the new node with a move by nature. This demonstrates a deep equivalence between trembling hand mistakes and moves by nature. COA We write Γ −−→ Γ if coalescing of moves transforms Γ into Γ ; we call Γ the pre-coalesce game and Γ the post-coalesce game. The deﬁnition of the coalesce transformation is relatively straightforward; essentially a node or nodes in Γ (J2 below) is dropped in Γ while ≺ stays the same. Definition 12 (Coalescing of Moves). Let Γ and Γ be EFG scenarios. We COA write Γ −−→ Γ , if there is a player l ∈ N who plays at information sets J1 = {w1 , . . . , wn } and J2 = {v1 , . . . , vn } that satisfy the following conditions, where Γ = (T , N , {Ak }k∈N , {Ik }k∈N , {uk }k∈N ): 1. vi ∈ Succ(wi ) for 1 ≤ i ≤ n (vi is an immediate successor of wi .) 2. Al (wi , vi ) = Al (wj , vj ) for 1 ≤ i, j ≤ n. 3. Γ = (T , N , {Ak }k∈N , {Ik }k∈N , {uk }k∈N ), where (a) N = N and {uk }k∈N = {uk }k∈N (b) T is the restriction of T to W \ J2 , where W is the domain of T . (c) Ak = Ak for all k ∈ N such that k = l. For all (x, y), Al is deﬁned by: i. if x = wi for all i, Al (x, y) = Al (x, y), and ii. if x = wi for some i, Al (x, y) = Al (vi , y) for all y ∈ Succ(x). (d) Ik = Ik for all k ∈ N such that k = l, and Il = Il \ {J2 }. We refer the reader to [6] or [1] for further details. COA Suppose G = (Γ, σ) and G = (Γ , σ ) are such that Γ −−→ Γ ; what trans formation between σ and σ might preserve equilibrium? In Fig. 4, an obvious demand is that α = ζδ and β = ζ. Unfortunately, this constraint produces a unique strategy assignment in only one direction, from left to right. So, from left to right we can show a preservation theorem.

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Definition 13. Let G = (Γ, σ) and G = (Γ , σ ) be EFGS’s such that Γ −−→ Γ . COA We deﬁne G −−→ G (using the notation in Deﬁnition 12) as follows: 1. for all players k in Γ such that k = l, σk = σk , and 2. σl (Al (wi , x)) = σl (Al (wi , vi )) · σl (Al (vi , x)) for 1 ≤ i ≤ n coa

We say σ coalesces σ and write σ −−→ σ when σ and σ satisfy 1 and 2. COA

Proposition 5. Given G = (Γ, σ) and G = (Γ , σ ) such that G −−→ G , if G is in perfect equilibrium, then G is in perfect equilibrium. Proof. Given σ is a perfect equilibrium in Γ (G is in perfect equilibrium), there is a sequence (ˆ σ m )∞ m=1 of strategies in the multi-agent form of Γ satisfying the conditions in Deﬁnition 4. Construct a sequence (ˆ τ m )∞ m=1 in the following way: – τˆIm = σ ˆIm , if I = J1 , m – τˆI (Al (wi , x)) = σ ˆJm1 (Al (wi , x)), if I = J1 and x ∈ Succ(vi ), and m ˆJm1 (Al (wi , vi )) · σ ˆJm2 (Al (vi , x)), if I = J1 and x ∈ Succ(vi ), – τˆI (Al (wi , x)) = σ where we use the same notation as in Deﬁnition 12. Given (ˆ σm )∞ m=1 converges to σ, (ˆ τ m )∞ converges to σ . Also it is straightforward to check that (ˆ σm )∞ m=1 m=1 satisﬁes conditions 1–3 in Deﬁnition 12. Hence G is in perfect equilibrium.

This demonstrates that rational play is preserved across coalescing of moves in the left to right direction. So what about the right to left direction? Consider again Fig. 4 and note that if either α or β are not equal to zero, the constraint coa that σ −−→ σ uniquely speciﬁes a probability distribution in the right to left direction. In the example of Fig. 4 it is ζ = α + β, δ = α/(α + β), and = β/(α + β). The same consideration applies in the general case. COA

Proposition 6. Given G = (Γ, σ) and G = (Γ , σ ) such that G −−→ G and G is in perfect equilibrium, if σI (AI (wi , x)) = 0 for some wi ∈ J1 and x ∈ Succ(vi ) in Γ (as deﬁned in Deﬁnition 12), then G is in perfect equilibrium. Proof. Given the (ˆ σ m )∞ m=1 which converges to σ by the assumption that σ is a perfect equilibrium, we can construct a (ˆ τ m )∞ which converges to σ by m=1 reversing the construction in the proof of Proposition 5. The constraint that σJ 1 (AJ1 (wi , x)) = 0 for some wi ∈ J1 ensures the uniqueness of σJ1 (AJ1 (vi , ·)), which in turn is needed to check that (ˆ τ m )∞

m=1 in fact converges to σ.

The real problem for the right to left direction arises when α and β are both zero. coa Then the constraint that σ −−→ σ does not uniquely determine the strategy in the pre-coalesce game. In particular, for a ﬁxed σ there may be members of coa {σ : σ −−→ σ } such that Player I or Player II (or both) play irrationally (as in Fig. 1). This blocks a general preservation theorem. However, we can show a preservation theorem over a stronger transformation. Reversible coalescing of moves ensures the rationality of play below the coalesced node.

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Definition 14. Let G = (Γ, σ) and G = (Γ , σ ) be EFG scenarios such that COA R-COA G −−→ G . Using the notation in Deﬁnition 12, we deﬁne G ←−−− G and say that G reversibly coalesces G , if for all vi ∈ J2 and all information sets J(⊆ Succ(vi )) reached from vi : τ (a) σJ (b)ev(σ−J2 τ, xa , b) σJ2 = argmax τ ∈Δ(AJ2 ) a∈A

J

b∈AJ

where AJ is the set of actions at J and xa is deﬁned by a = Al (vi , xa ). r-coa coa We say σ reverse coalesces σ , and write σ ←−−− σ , if σ −−→ σ and this condition is satisﬁed. Remark 1. By Propositions 1, 5, and 6, if G is in perfect equilibrium and coa r-coa σI (AI (wi , x)) = 0 for some wi ∈ J1 , then σ −−→ σ implies σ ←−−− σ . Now we can ask, when is Player I’s play at the added node in Γ rational? Of course, it is rational if ζ = 0, since we have already demonstrated that equilibrium is preserved in those cases, and play is rational at all nodes when games are in equilibrium. It will also be rational in situations where ζ = 0 and Player II’s move below constitutes a proper equilibrium. Remember our discussion in Sect. 2: the rationality of the risk averse player propagates through his mistakes, in the sense that players respond to him as if he plays rationally, even when he makes mistakes. Consequently, a move introduced like this to replace a mistake will also satisfy the constraints of rationality. Conversely, introducing a move to replace the mistake of a trembling hand player will in general look irrational, in particular, completely random. This is because we placed no constraint on the probability of diﬀerent errors for such a player. What does completely random mean here? Just that a trembling hand player is indistinguishable from a move by nature if we isolate his mistake node R-COA from the rest of his play. Let G = (Γ, σ) and G = (Γ , σ ) such that G ←−−−− G . Consider the precoalesced nodes in Γ , J2 = {v1 , . . . , vn } (by the notation in Deﬁnition 12). Deﬁne G J2 = (Γ J2 , σJ2 ) by – Γ J2 is the EFG with the nodes in J2 replaced with chance nodes in which nature take actions AJ2 according to the strategy σJ2 . – Use σ J2 is a behavioral strategy proﬁle of Γ J2 in which a strategy of J2 is dropped, i.e. σJ2 = σ−J2 R-COA

Proposition 7. Let G = (Γ, σ) and G = (Γ , σ ) such that G ←−−− G . If G is in perfect equilibrium, then G J2 is in perfect equilibrium. Proof. If σ is in perfect equilibrium in Γ , then there exists a sequence of strategies (ˆ σ m )∞ τ m )∞ m=1 as in Deﬁntion 4. Let (ˆ m=1 be a sequence of behavioral strateJ2 m gies in Γ such that each τˆI is identical to σ ˆIm for all I (= J1 , J2 ). For I = J1 , σ ˆIm (Al (wi , x)). τˆIm (Al (wi , vi )) = x∈Succ(vi ) It is clear that (ˆ τ m )∞ σm )∞

m=1 converges on σ−J2 , given (ˆ m=1 converges on σ .

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Remark 2. The results of Propositions 3–7 can be extended to the case of proper equilibrium with varying degrees of diﬃculty. In the case of Proposition 3, for example, it is straightforward to check that the conditions for -proper equilibrium are unchanged by the transformation. Not so for Proposition 7; the additive construction of the τˆIm ’s will not in general preserve the ratios necessary to ensure -proper convergence. A much more elaborate procedure is needed to construct the appropriate converging sequence. Note, however, that Proposition 7 is uninteresting in the case of proper equilibrium. As discussed above, if G is in proper equilibrium, play at the pre-coalesced node in G will always be rational. So, the “nature” move introduced in G J2 will replace a rational move by player l. We have shown that trembling hand mistakes are equivalent to moves by nature. More speciﬁcally, a player who reasons about an opponent as if he may make trembling hand mistakes produces the same rationalizable path of action as a player who faces a combination of rational opponents and moves by nature.

7

Conclusion: Taking Mistakes Seriously

In order to calculate a rational response to a player who makes mistakes, we need a model of mistake making. We discussed two such models implicit in solution concepts in the game theory literature: trembling hand mistakes and risk averse mistakes. We saw that players who make risk averse mistakes act rationally even as they make mistakes. By examining transformations over EFG Scenarios, we found that trembling hand mistakes are equivalent to moves by nature. These two analyses of mistaken play represent extremes: either mistakes are made rationally, or randomly. But do either of these approaches take mistakes seriously? Realistic mistake-making agents may lie somewhere between these extremes. Investigating the strategic consequences of more nuanced analyses of mistake making is a project for future research.

References 1. Elmes, S., Reny, P.J.: On the strategic equivalence of extensive form games. Journal of Economic Theory 62(1), 1–23 (1994) 2. Kohlberg, E., Mertens, J.F.: On the strategic stability of equilibria. Econometrica 54(5), 1003–1037 (1986) 3. Myerson, R.B.: Reﬁnements of the nash equilibrium concept. International Journal of Game Theory 7(2), 73–80 (1978) 4. Myerson, R.B.: Game Theory: Analysis and Conﬂict. Harverd University Press, Cambridge (1991) 5. Selten, R.: Reexamination of the perfectness concept for equilibrium points in extensive games. International Journal of Game Theory 4(1), 25–55 (1975) 6. Thompson, F.B.: Equivalence of games in extensive form. Rand Corporation Research Memorandum 789 (1952)

Update Semantics for Imperatives with Priorities Fengkui Ju1, and Fenrong Liu2 1

Department of Philosophy, Beijing Normal University, Beijing, China [email protected] 2 Department of Philosophy, Tsinghua University, Beijing, China [email protected]

Abstract. Imperatives occur ubiquitously in our social communications. In real life we often get conﬂicting orders issued by diﬀerent speakers whose authorities are ranked. We propose a new update semantics to interpret the meaning of imperatives with priorities and illustrate what changes they bring about in the addressee’s cognitive state. The general properties of the semantics, as well as its core philosophical ideas are discussed extensively in this paper. Keywords: imperatives, conﬂicts, update semantics, priorities.

1

Introduction

Imperatives occur ubiquitously in our social communications. To act successfully in a society, we have to fully understand their meaning, as imperatives regulate actions. Logical studies on imperatives have been carried out for some time, and deontic logics belong to such a tradition. From the 1990s, several prominent new frameworks have been proposed. Following the slogan “you know the meaning of a sentence if you know the change it brings about in the cognitive state of anyone who wants to incorporate the information conveyed by it”, update semantics ([9]) was proposed to deal with information update, and it was later applied to imperatives in [8], [6] and [10]. On the basis of deontic logics, [1] made a proposal to study actions that are typically expressed by STIT-sentences “see to it that...”, bringing actions with choices made by agents together. Other recent works in this line are [4], [2] and [3]. Adopting dynamic epistemic logic (DEL) approach, [11] and [12] introduced a new dynamic action of “commanding” to deontic logic, and dealt with imperatives in the framework of dynamic deontic logics. So far, the main purpose of those approaches has been to understand the meaning of one single imperative. Not much attention has been paid to conﬂicting orders, which were simply taken to be absurd, thus resulting very trivial facts in the existing frameworks. In addition, though agency was introduced to the research agenda, the focus has been always on the addressee, not on the addressors. However, in

The order of the authors’ names is alphabetical, and both authors contributed equally to this work.

H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 125–138, 2011. c Springer-Verlag Berlin Heidelberg 2011

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real life, we often get conﬂicting orders issued by diﬀerent authorities. Facing such situations, consciously taking the ranks of addressors into account, we form an ordering over possible actions and resolve the diﬃculties successfully. The ideas in this paper combine two sources: natural language semantics and logics of agency. We hope to show that this is a fruitful mixture of traditions. Let us start immediately with an example: Example 1. A general d, a captain e and a colonel f utter the following sentences, respectively, to a private. (1) The general: Do A! Do B ! (2) The captain: Do B ! Do C ! (3) The colonel: Don’t do A! Don’t do C ! Clearly, these are conﬂicting orders, w.r.t actions A and C. Intuitively, instead of getting stuck, the private will come up with the following plan after a deliberation: She should do A, do B, but not do C. Her reasoning rests on the following fact: The authorities of d, e and f are ranked as e < f < d. According to this, she can make her decision on which orders to obey, and which ones to disobey. 1 The aforementioned existing frameworks cannot systematically handle such examples. The aim of this paper is to take some recently developed ideas from preference logic, in particular, priority-based preference models for agency (cf. [5] and [7]) to deal with the problem. To simply phrase our ideas in the language of preference, agent’s preference over alternatives is derived from some ordered priorities (or better call it ranked authorities in this context). On our way of developing this idea, we will retain the tradition of update semantics and hope to make a new contribution to natural languages studies. The remaining sections are organized as follows. We will ﬁrst introduce the basic deﬁnitions and techniques of update semantics for imperatives in Section 2. In Section 3, we present our new proposal and study its general properties. In Section 4, we show that introducing ranks of authorities can solve the diﬃculties we had in Example 1. We also state our background philosophical ideas. We end the paper with some conclusions and possible future directions in Section 5.

2

Force Structures and Track Structures

An update system is a triple L , Σ, [·], where L is a language, Σ a set of information states, and [·] a function from L to Σ → Σ, which assigns each sentence φ an operation [φ]. For any φ, [φ] is called an update function, which is intended to interpret the meaning of φ. The meaning of a sentence lies in how it updates information states — the core idea of update semantics. In his recent work ([10]), Veltman has presented a new semantics for imperatives based on the update semantics, and argued that the meaning of imperatives is an update function on plans. Inspired by [10], [6] interpreted the meaning of 1

Note that [12] also discussed conﬂicting commands from diﬀerent authorities.

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imperatives as an update function on force structures. In this section, we introduce an equivalent version of the semantics given in [6] in a diﬀerent way, and then extend it in the next section and make it work for our problems. Deﬁnition 1 (languages). Let Φ be a set of propositional variables, and p ∈ Φ. The standard language of propositional logic Y is deﬁned as follows:2 φ := p | ¬φ | φ ∧ ψ | φ ∨ ψ The language L of imperatives is deﬁned as the set {!φ | φ ∈ Y }. Each ﬁnite set T of literals of Y is called a track. A track T is consistent if and only if it does not contain both p and ¬p for any p. Information states are identiﬁed with track structures, as deﬁned below. Deﬁnition 2 (track structures). A ﬁnite set L of tracks is a track structure iﬀ (1) Each T ∈ L is consistent; (2) Any T, T ∈ L contain the same variables. Example 2. The following picture represents a track structure. p4 , ¬p2 , p3 , p1 p4 , p2 , p3 , p1 p4 , p2 , p3 , ¬p1 ¬p4 , ¬p2 , p3 , p1

The reading of track structures is this: For any track structure L, the agent has to choose a track of L and make all literals in it true, but she may freely choose which one. If the agent makes all literals of some track of L true, we say that L is performed. There are two special track structures: {∅} and ∅. The former can always be trivially performed, which is called the minimal track structure. The later can never be performed, which is called the absurd track structure. In what follows we deﬁne a procedure which recursively outputs a track structure for any given imperative !φ. To do that, we ﬁrst introduce the notion of force structures. Deﬁnition 3 (force structures). Each ﬁnite set K of ﬁnite sets of literals of Y is called a force structure. Example 3. {{p4 , ¬p2 }, {p3 }, {p2 , p1 }} is a force structure. 2

We do not consider the connective → as a primitive symbol. The reason is that we will use the language Y to express propositional content of imperatives. In natural languages, imperatives do not take implications as propositional content.

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Deﬁnition 4 (tracks of force structures). Let K = {X1 , . . . , Xn } be any force structure. For any Xi , let Xi be the smallest set such that both p and ¬p are in Xi for any p occurring in Xi . T = X1 ∪ . . . ∪ Xn is a track for K iﬀ (1) Xi ⊆ Xi and Xi ∩ Xi = ∅; (2) For any p occurring in Xi , one and only one of p and ¬p is in Xi . Example 4. The picture in Example 2 represents the set of all consistent tracks of the force structure in Example 3. Let K be any force structure, we deﬁne functions T + and T − as follows: Deﬁnition 5 (T + and T − ). (a) T + (K, p) = T − (K, p) =

{{p}} {X∪{p} | X∈K}

{{¬p}} {X∪{¬p} | X∈K} −

if K = ∅ otherwise if K = ∅ otherwise

(b) T + (K, ¬φ) = T (K, φ) T − (K, ¬φ) = T + (K, φ) (c) T + (K, φ ∧ ψ) = T + (K, φ) ∪ T + (K, ψ) T − (K, φ ∧ ψ) = T − (T − (K, φ), ψ) (d) T + (K, φ ∨ ψ) = T + (T + (K, φ), ψ) T − (K, φ ∨ ψ) = T − (K, φ) ∪ T − (K, ψ)

For any imperative !φ, T + (∅, φ) is called the force structure of it. We see that imperatives correspond to force structures in a recursive way. Example 5. The force structure of the imperative !((p1 ∧ p2 ) ∨ (p3 ∧ p4 )) is {{p1 , p3 }, {p1 , p4 }, {p2 , p3 }, {p2, p4 }}. Let U (φ) be the set of all consistent track of T + (∅, φ).3 U (φ) is the track structure of !φ. Deﬁnition 6 (compatibility of track structures). Track structures L1 and L2 are compatible iﬀ (1) For any track T1 ∈ L1 , there is a track T2 ∈ L2 such that T1 ∪ T2 is consistent; (2) For any track T2 ∈ L2 , there is a track T1 ∈ L1 such that T1 ∪ T2 is consistent. Compatibility is used to characterize conﬂicts among imperatives. Example 6. Two speakers respectively utter these two imperatives to an agent: (a) Close the door or the window! (b) Close the window! 3

Readers may realize that T + (∅, φ) corresponds to a conjunctive normal form (CNF) of φ in propositional logic. That is true. However, please note that the new notions (e.g. compatibility, validity) to be deﬁned on the basis of track structures have very diﬀerent meaning in this context.

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Intuitively, there is some conﬂict between these two commands, although they are consistent from the propositional logic point of view. It is easy to verify that the track structures corresponding to the two imperatives are not compatible. Deﬁnition 7 (merge of track structures). 4 L1 L2 = {T1 ∪T2 |T1 ∈ L1 , T2 ∈ L2 , and T1 ∪ T2 is consistent} The semantics for imperatives is deﬁned as follows: Deﬁnition 8 (update of track structures with imperatives). L U (φ) if L and U (φ) are compatible L!φ = ∅ otherwise

5

Basically, updating a track structure L with an imperative !φ is the merge of L and the track structure of !φ. The exceptional cases are those that L and the track structure of !φ are not compatible. When this case takes place, the result of the update is the absurd track structure ∅. 6 At this point, we would like to return to Example 1. We take the minimal force structure {∅} as the beginning point. After the general’s and captain’s commands, the track structure of the private becomes {{A, B, C}}. This means that the agent has to make A, B and C true. The track structure of the imperative “don’t do A” is {{¬A}}, which is not compatible with {{A, B, C}}. Therefore, after the colonel’s ﬁrst command, the track structure of the private becomes {∅}. Intuitively, this means that the agent gets stuck, and he will be stuck forever. We see that the semantics given in Deﬁnition 8 does not work for Example 1. Similarly, regarding this example, no satisfying solution has been provided in [10] either. This is the starting point of the current work. In our view, to handle such diﬃculties, we should take the ranks of the speakers into account. Our attempt will follow in the next section.

3

Update with Priorities

3.1

Introducing Authorities

A new update system is a tuple L , Σ, [·], A, ≤, where A is a ﬁnite set of speakers, and ≤ is a preorder on A. For any a, b ∈ A, a ≤ b means that b has a rank at least high as what a has. Now we formulate the semantics based on the new update system, incorporating the authorities in the framework presented in the preceding. 4 5

6

This terminology is from [10]. [10] deﬁnes meaning of imperatives as an update function on plans. Intuitively, a plan is a set of free choices, and a track structure is also a set of free choices. In this sense, the update deﬁned here is similar to [10]. Their main diﬀerence lies in what are viewed as free choices. The notion of validity by the invariance of track structures can solve Ross’s paradox.

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Deﬁnition 9 (agent-oriented language). The language L of imperatives is deﬁned as the set {!aφ|φ ∈ Y , a ∈ A}, where Y is the language given in Deﬁnition 1. One can see that all imperatives are relative to speciﬁc speakers now. Let L be the set of literals of Y . Let L = {la |l ∈ L, a ∈ A}. Each ﬁnite set T of L is called a track. We deﬁne three properties of tracks below. Deﬁnition 10 (resolvability of tracks). A track T is resolvable iﬀ for any pa and pb , if both pa and ¬pb are in T , then either a < b or b < a. Deﬁnition 11 (succinctness of tracks). A track T is succinct iﬀ there are no pa and pb such that (1) a < b; (2) Either both pa and pb are in T or both ¬pa and ¬pb are in T . Deﬁnition 12 (consistency of tracks). A track T is consistent iﬀ (1) T is succinct; (2) There are no pa and pb such that both pa and ¬pb are in T . The property of consistency is not just stronger than succinctness, but also stronger than resolvability: For any track T , if T is consistent, then it is resolvable, but this might not be the case the other way around. Example 7. Suppose a < b. The track T1 = {pa , qc , rd , ¬pb } is resolvable and succinct, but not consistent. Compared with the ordinary notion of consistency in logic, the notion of consistency deﬁned here seems somewhat heavy, as it contains the notion of succinctness. We do this for technical reasons, which will be explained in Section 4. Intuitively, consistent tracks are “good” ones, while inconsistent tracks are not. Deﬁnition 13 (track structures with authorities). A ﬁnite set L of tracks is a track structure iﬀ (1) Each T ∈ L is resolvable; (2) For any T, T ∈ L, T and T contain the same variables. Deﬁnition 2 deﬁnes track structures without authorities, where each track of a track structure is required to be consistent. This requirement gets relaxed for track structures with authorities, namely, each track of a track structure is resolvable. Finally, if all tracks of a track structure are consistent, we call it a consistent one. In the previous section, we describe a procedure by which an imperative !φ corresponds to a track structure, where authorities are not considered. Similarly, we can build up a correspondence between an imperative !a φ and a track structures with authorities. Here we don’t go through the details. We simply use U (!a φ) to denote the track structure of !a φ. 3.2

Update Function and Some Properties

First, again, some technical notions. For any two tracks T and T , we call T a sub-track of T if T ⊆ T . If T is consistent, it is a consistent sub-track.

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Deﬁnition 14 (maximally consistent sub-tracks). A track T is a maximally consistent sub-track of a track T iﬀ T is a consistent sub-track of T and for any sub-track T of T , if T ⊂ T , then T is not consistent. Example 8. Suppose b < a. The track {pa , qc , rd } is a maximally consistent subtrack of the track {pa , pb , qc , rd , ¬pe }. Note that the track {pa , pb , qc , rd } is not a maximally consistent sub-track of {pa , pb , qc , rd , ¬pe }, as {pa , pb , qc , rd } is not succinct. Proposition 1. For any track T , all of its maximally consistent sub-tracks contain the same variables as what T has. Proof. Let T be any track. Let T1 be any maximally consistent sub-track of T . Suppose that T and T1 do not contain the same variables. Since T1 ⊆ T , there is a variable, say p, such that T contains p but T1 does not. Then there is a literal li containing p such that li ∈ T but li ∈ / T1 . Since T1 does not contain p, then T1 ∪ {li } ⊃ T1 is consistent and a sub-track of T . Therefore, T1 is not a maximally consistent sub-track of T , which is strange. Hence, T1 and T contain the same variables. Deﬁnition 15 (preorder on maximally consistent sub-tracks). Let T be any track. Let T and T be any maximally consistent sub-tracks of T . T T iﬀ for any la ∈ T , there is a lb ∈ T such that lb contains the same variable as what la has, moreover, a ≤ b. It is easy to see that is reﬂexive and transitive, so it’s a preorder. may not be antisymmetric. Here is a simple counter-example. Let T = {pa , ¬pa }, T = {pa} and T = {¬pa }. Both T and T are maximally consistent sub-tracks of T . T T and T T , but T = T . Hence, might not be a partial order. Deﬁnition 16 (strict partial order ≺ on maximally consistent subtracks). Let T be any track. Let T and T be any maximally consistent subtracks of T . T ≺ T iﬀ T T but T T . Lemma 1. Let T be any resolvable track containing only one variable. Let X be the set of its maximally consistent sub-tracks. X has a greatest element under the relation ≺. Proof. We consider two cases. First, we suppose that T is consistent. Then X is a singleton. Clearly, X has a greatest element. Next, we suppose that T is not consistent. Again, there are two possible cases: (1) There are no pa and ¬pb such that both of them are in T ; (2) there are such pa and ¬pb . In the ﬁrst case, T = {pa1 , . . . , pam } or T = {¬pb1 , . . . , pbm }.7 we can verify that X is a singleton, no matter whether T = {pa1 , . . . , pam } or {¬pb1 , . . . , pbm }. Therefore, X has a greatest element. We consider the second case. Let T = {pa1 , . . . , pam , 7

Note that {pa1 , . . . , pam } and {¬pb1 , . . . , ¬pbn } might not be consistent, because they might not be succinct.

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¬pb1 , . . ., ¬pbn }, where 1 ≤ m, n. T has two maximally consistent sub-tracks: T1 = {pam1 , . . . , pamk }, T2 = {¬pbn1 , . . . , ¬pbnl }, where k ≤ m and l ≤ n. Hence, X = {T1 , T2 }. Suppose that X does not have a greatest element under ≺, then T1 ≺ T2 and T2 ≺ T1 . We can get that for any bj , there is an ai such that ai < bj , because otherwise T1 ≺ T2 . Since T is resolvable, we have that for any bj , there is an ai such that bj < ai . Similarly, for any ai , there is a bj such that ai < bj . Now we can obtain an inﬁnite decreasing chain, say bh < ai < bj < . . .. This is impossible. Therefore, X has a greatest element. Proposition 2. Let T be any resolvable track. Let X be the set of its maximally consistent sub-tracks. X has a greatest element under the relation ≺. Proof. Suppose that T contains n diﬀerent variables. Let T = T1 ∪ . . . ∪ Tn such that for any Ti , all literals in it contain the same variables. In fact, T ∈ X if and only if T = T1 ∪ . . . ∪ Tn , where each Ti is a maximally consistent sub-track of Ti . By Lemma 1, each Ti has a greatest maximally consistent sub-track under ≺. Let T = T1 ∪ . . . ∪ Tn , where Ti is the greatest maximally consistent sub-track of Ti . We see that T ∈ X. It can be easily veriﬁed that T is a greatest element of X under ≺. Example 9. Suppose a < d. The track {¬pd , qb , rc } is a greatest maximally consistent sub-track of {pa , qb , rc , ¬pd } under the relation ≺. The conﬂict in the track {pa, qb , rc , ¬pd } can be resolved according to the given authority rank a < d and the conﬂict-free result is {¬pd , qb , rc }. Deﬁnition 17 (sub-structures). Let L be any track structure. A track structure L is a sub-structure of L iﬀ (1) For any T ∈ L , there is a T ∈ L such that T ⊆ T ; (2) For any T ∈ L, there is a T ∈ L such that T ⊆ T . If L is consistent, we say that L is a consistent sub-structure of L. Deﬁnition 18 (suﬃcient consistent sub-structures). Let L = {T1 , . . . , Tn} be any track structure. A track structure L is a suﬃcient consistent sub-structure of L iﬀ L = {T1 , . . . , Tn }, where each Ti is the greatest maximally consistent sub-track of Ti under the relation ≺. By Proposition 1, it can be veriﬁed that for any structure L = {T1 , . . . , Tn }, the set {T1 , . . . , Tn }, where each Ti is the greatest maximally consistent sub-track of Ti , is always a consistent sub-structure of L. Therefore, any track structure has a suﬃcient consistent sub-structure. Furthermore, any track structure has only one suﬃcient consistent sub-structure. Example 10. Suppose c < e. L is the suﬃcient consistent sub-structure of L.

Update Semantics for Imperatives with Priorities

pa , qb , rc , ¬sd , ¬re

pa , qb , ¬sd , ¬re

¬pa , qb , rc , sd , ¬re

¬pa , qb , sd , ¬re

pa , ¬qb , rc , ¬sd , ¬re

pa , ¬qb , ¬sd , ¬re

¬pa , ¬qb , rc , ¬sd , ¬re

¬pa , ¬qb , ¬sd , ¬re

L

133

L

Track structure L is not consistent. The suﬃcient consistent sub-structure of L is the result of making L consistent while respecting the authorities in L to the greatest extent. Deﬁnition 19 (merge of track structures). L1 L2 = {T1 ∪T2 |T1 ∈ L1 , T2 ∈ L2 , T1 ∪ T2 is resolvable} Deﬁnition 20 (compatibility of track structures). Track structures L1 and L2 are compatible iﬀ (1) For any T1 ∈ L1 , there is a T2 ∈ L2 such that T1 ∪ T2 is resolvable; (2) For any T2 ∈ L2 , there is a T1 ∈ L1 such that T1 ∪T2 is resolvable. We can verify that L1 and L2 are compatible if and only if both L1 and L2 are sub-structures of L1 L2 . For any track structure L, we use V (L) to denote the suﬃcient consistent sub-structure of L. The semantics for imperatives which takes authorities into account is deﬁned as follows. Deﬁnition 21 (update of track structures with authorities). V (L U (!a φ)) if L and U (!a φ) are compatible L!a φ = ∅ otherwise Meaning of imperatives is an update function on track structures. Let L be any track structure and !a φ be any imperative. If L and the track structure L corresponding to !a φ are compatible, the result of updating L with !a φ is the suﬃcient consistent sub-structure of the merge of L and L , otherwise the result is ∅, which is an absurd track structure.

4 4.1

Illustrations and Background Ideas Illustrations

We illustrate some properties of the semantics deﬁned above. First, let us look at Example 1 again. Recall that a general d, a captain e and a colonel f utter the following sentences, respectively, to a private. (1) The general: Do A! Do B ! (2) The captain: Do B ! Do C ! (3) The colonel: Don’t do A! Don’t do C !

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Suppose that the starting track structure of the private is L0 = {∅}, which means that the private does not bear any imperative force. According to our new semantics, these imperatives update the track structures of the private in the following way. !d A

→

L0 !e C

→

!d B

Ad

→

L1

Ad , Bd , Ce

!f ¬A

→

!e B

Ad , Bd

→

L2 !f ¬C

Ad , Bd , Ce

L4

→

L3 Ad , Bd , ¬Cf

L5

Ad , Bd

L6

Ad , Bd , Ce , ¬Af

Ad , Bd , Ce , ¬Cf

L5

L6

After the imperative !d A, the track structure L0 changes to L1 , and after !d B, L1 changes to L2 , and so on. L6 is the ﬁnal track structure, according to which the private should do A, B, but should not do C. This is what we expect. The track structures L5 and L6 in the dash rectangles are auxiliary for us to understand the update process, and they are not results of any update of this process. When !f ¬A is uttered, L4 is updated to L5 . Since L5 is not consistent, it changes to L5 after a deliberation of the private. Actually, L5 is equal to L4 . L5 is the suﬃcient consistent sub-structure of L5 and respects the authority of the general. The similar case also happens to L6 . This example shows how our semantics works in practice. Next, let us consider an example from [10] which involves free choices. Example 11. John is ill and goes to see doctors c and d respectively. (1) The doctor c: Drink milk or apple juice! (2) The doctor d : Don’t drink milk! Suppose that the original track structure of John is L0 = {∅}. First, we suppose that John trusts d more than c. The update process of John’s track structures is illustrated by the following picture. Mc , A c !c (M ∨ A)

→ L0

¬Mc , Ac

!d ¬M

→

¬Md , Ac

Mc , ¬Ac

¬Md , ¬Ac

L1

L2

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L2 is the ﬁnal result of this update process. According to L2 , John should not drink milk, and he may and may not drink apple juice. As d has a higher authority, this result is perfectly ﬁne. Now we suppose that c has a higher authority than d has. With this constraint, John’s track structures are updated in the following: Mc , A c !c (M ∨ A)

→

¬Mc , Ac

Mc , Ac !d ¬M

→

¬Mc , Ac

Mc , ¬Ac

Mc , ¬Ac

L1

L2

L0

L2 is the ﬁnal result of this update process. The imperative uttered by d does not essentially make much sense to John, because L1 = L2 . According to L2 , John should drink milk or apple juice, and he may only drink milk, only drink apple juice and drink both. We see that drinking milk is allowed. This result seems not plausible. It seems practically reasonable to think that John should drink apple juice but not drink milk, as if he does so, both of the imperatives could be performed. In other words, only drinking apple juice seems to be safer than only drinking milk or drinking both. However, we think that even not to drink milk is more practically reasonable than to drink milk, John is still allowed to drink milk in this case. We show this point by an example. Example 12. A general and a captain utter the following to a private respectively. (1) The general: You may have a rest. (2) The captain: Move! These two sentences are conﬂicting. Suppose that the private chooses to have a rest, then normally, he will not be punished. This implies that the private is allowed to stop, even to move is safer for him. Hence, we think that the result mentioned above is plausible. Actually, the following claim seems reasonable: An agent a has a higher authority than what b has in giving commands if and only if a has a higher authority than b in giving permissions. The semantics given in Deﬁnition 8 is a special case of the semantics given in Deﬁnition 21: When restricted to singleton of agents, the latter collapses to the former. Note that the semantics deﬁned in Deﬁnition 8 does not satisfy the property of commutativity.8 Therefore, the semantics given in Deﬁnition 21 does not have that property either. Previously, we have taken tracks which are succinct and do not contain any conﬂict as “good” tracks. The reason why we require succinctness is out of the following consideration: Without succinctness, some thing “bad” could happen, in which commutativity plays a role. Here is an example. 8

See [6] for examples.

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Example 13. Consider some agent a, his grandmother has a higher authority than his parents, and his father and mother have the same authority. Here are two sequences. (1) Grandmother: Stop! Father: Don’t stop! Mother: Stop! (2) Grandmother: Stop! Mother: Stop! Father: Don’t stop! If we keep all other things unchanged and just drop the succinctness requirement, the second sequence makes the agent get stuck, while the ﬁrst one does not. This is weird. 4.2

Imperative Forces and Authorities

In this subsection we are going to state our intuitions and main ideas in understanding imperatives in natural languages. In practice, any kind of sentences are uttered by speciﬁc speakers. We think that speakers contribute to the meaning of sentences. We ﬁrst explain our ideas in terms of indicatives. According to the meaning theory of dynamic semantics, the meaning of a sentence lies in how it changes an agent’s information state. However, one same indicative may change an agent’s information state very diﬀerently, if it is uttered by diﬀerent speakers. For an example, consider the situation in which the sentence uttered conﬂicts with the agent’s knowledge. Following the same philosophy, we think that meaning of an imperative also lies in how it changes an agent’s cognitive state, more speciﬁcally this time, imperative force state. Imperative force states are states of imperative forces which agents bear. Imperatives have propositional content. Imperatives produce imperative forces, which tend to “push” the agent to make their propositional content true. Uttering an imperative may change the imperative force state of the agent. Similarly, one same imperative may cause diﬀerent changes to an agent’s imperative force state, if the imperative is uttered by diﬀerent speakers. This is the reason that we introduce authorities into the semantics for imperatives. Technically, we use track structures to characterize states of imperative forces, as explained in Section 2. We do not distinguish state of imperative force and imperative force conceptually, and they are considered to be the same. Imperative forces produced by imperatives are also characterized as track structures. An imperative is in force if and only if (1) The agent has to make the propositional content of the imperative true; (2) The agent may make the propositional content of it true in any way. Particularly, the imperative “drink milk or apple juice” is in force if and only if the agent has to drink milk or apple juice, he may drink milk but not drink apple juice, he may drink apple juice but not drink milk, and he may drink both. About imperative force, there is one more thing which we want to emphasize. Whether an agent is bearing some imperative force is not objective, but determined by the agent’s mind. The word “imperative force” might be misleading, as it reminds us of physical forces. Instead of saying that an agent is bearing some imperative force, we should say that, the agent thinks that he is bearing some imperative force. Consider the following example.

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Example 14. A general and a captain utters the following to a private. (1) The general: Move! (2) The captain: Stop! The private would think that the imperative of the captain is not in force. Finally, some comments on authority order. First of all, authority order is relative to speciﬁc agents. Two speakers might be ranked diﬀerently from one agent to another. For example, two doctors might be in diﬀerent authority relations for diﬀerent patients. Secondly, authority order is not ﬁxed universally. More speciﬁcally, it depends on speciﬁc contexts. Speaker a might have a higher authority than speaker b for agent c in one context, but b might have a higher authority than a in another context. For instance, suppose that a is c’s father, and b is c’s mother. Suppose that a is a general, b is a colonel and c is a private in the same army. In army, a has a higher authority than b for c, but b might have a higher authority than a in family.

5

Conclusions and Future Work

Motivated by the realistic examples that involves conﬂicting orders, we have introduced authorities explicitly into the logical language, and proposed a new semantics for imperatives. It combined ideas from natural language semantics and logics for agency. We think that the resulting picture of “force” and “authority” is more realistic, both in understanding imperative discourse and commands driven human action. We have applied the new semantics to analyzing many examples in the paper. Some general properties of the new semantics have been proved. In addition, we have discussed extensively our background ideas. Nevertheless, our investigation is still on the side of semantics, next on our agenda are the following issues: (i) Thinking syntactically, we would like to see whether there is a complete logic for this update system with priorities. (ii) The issue studied in our paper seems to be closely related to normative conﬂicts in the literature of deontic logics, and we would like to compare our proposal and those in that tradition. (iii) Finally, we would like to extend our framework to the situation in which knowledge plays a role in understanding imperatives, shown by the example: Example 15. A general and a captain utter the following imperatives to a private respectively in some circumstance. (1) The general: Save the boy! (2) The captain: Don’t make him get hurt! This is considered conﬂict-free in our framework. But suppose the reality is that the private can not save the boy without making him get hurt to some extent. In this circumstance, the private can not perform these two imperatives, as there is

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a conﬂict after all. What is going on here? In fact what the agent knows about the world matters. We would like to take the agent’s knowledge into account, and model the interaction between the agent’s knowledge and imperative forces. Acknowledgement. The authors would like to thank Maria Aloni, Johan van Benthem, Frank Veltman, Tomoyuki Yamada, the anonymous LORI referees, and the audience of the Workshop on Modal Logic, Language and Logical Dynamics in June 2011 in Beijing, for their useful comments. Fengkui Ju is supported by The Major Bidding Project of the National Social Science Foundation of China (NO. 10&ZD073). Fenrong Liu is supported by the Project (NO. 09YJC7204001) and the Scientiﬁc Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

References 1. Belnap, N., Perloﬀ, M., Xu, M.: Facing the Future. Oxford University Press, Oxford (2001) 2. Broersen, J., Herzig, A., Troquard, N.: Embedding alternating-time temporal logic in strategic STIT logic of agency. Journal of Logic and Computation 16, 559–578 (2006) 3. Herzig, A., Troquard, N.: Knowing how to play: Uniform choices in logics of agency. In: Stone, P., Weiss, G. (eds.) Proceedings of the Fifth International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 209–216. ACM, New York (2006) 4. Horty, J.: Agency and Deontic Logic. Oxford University Press, Oxford (2001) 5. de Jongh, D., Liu, F.: Preference, priorities and belief. In: Grune-Yanoﬀ, T., Hansson, S. (eds.) Preference Change: Approaches from Philosophy, Economics and Psychology. Theory and Decision Library, pp. 85–108. Springer, Heidelberg (2009) 6. Ju, F.: Imperatives and logic. Studies in Logic 3(2), 361–379 (2010) 7. Liu, F.: A two-level perspective on preference. Journal of Philosophical Logic 40, 421–439 (2011) 8. Mastop, R.: What Can You Do? Imperative Mood in Semantic Theory. Ph.D. thesis, ILLC, University of Amsterdam (2005) 9. Veltman, F.: Defaults in update semantics. Journal of Philosophical Logic 25, 221–261 (1996) 10. Veltman, F.: Imperatives at the borderline of semantics and pragmatics (2010) (manuscript) 11. Yamada, T.: Acts of commands and changing obligations. In: Inoue, K., Satoh, K., Toni, F. (eds.) Proceedings of the 7th Workshop on Computational Logic in Multi-Agent Systems, pp. 1–19. Springer, Heidelberg (2006) 12. Yamada, T.: Logical dynamics of some speech acts that aﬀect obligations and preferences. Synthese 165(2), 295–315 (2008)

A Measure of Logical Inference and Its Game Theoretical Applications Mamoru Kaneko1 and Nobu-Yuki Suzuki2 1

Institute of Policy and Planning Sciences, University of Tsukuba, Japan [email protected] 2 Faculty of Science, Shizuoka University, Shizuoka, Japan [email protected]

Abstract. This paper presents a measure of inference in classical and intuitionistic logics in the Gentzen-style sequent calculus. The deﬁnition of the measure takes two steps: First, we measure the width of a given proof. Then the measure of inference assigns, to a given sequent, the minimum value of the widths of its possible proofs. It counts the indispensable cases for possible proofs of a sequent. This measure expresses the degree of diﬃculty in proving a given sequent. Although our problem is highly proof-theoretic, we are motivated by some general and speciﬁc problems in game theory/economics. In this paper, we will deﬁne a certain lower bound function, with which we may often obtain the exact value of the measure for a given sequent. We apply our theory a few game theoretical problems and calculate the exact values of the measure. Keywords: Classical Logic, Intuitionistic Logic, Gentzen-style Sequent calculus, Game Theoretic Decision Making.

1

Introduction

This paper presents a measure of inference in classical and intuitionistic logics in the Gentzen-style sequent calculus (Gentzen [2], [3]). The deﬁnition of the measure takes two steps: For each proof (tree) P , we measure the width, i.e., the number of leaves, of P. Then the measure of inference assigns, to a given sequent σ = Γ → Θ, the minimum value in the widths of possible proofs of σ, if σ is provable, and if not, the assigned value is +∞. It counts the indispensable cases for possible proofs of σ. By this measure, we represent the degree of diﬃculty in proving a given sequent. Although our problem is highly proof-theoretic, we are motivated by problems in game theory/economics. Here, we explain, ﬁrst, our motivation; and second, the contribution of this paper. Lastly, we present one game theoretical example, to which our theory will be applied in Section 5. The aim of game theory/economics is to study human behavior and decisionmaking in a game/social situation. It is more directly related to human activities than mathematics. The importance of bounded rationality has been emphasized in the economics literature since Simon [11]. Simon himself criticized the assumption of super-rationality for economic agents’ decision making, but touched only H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 139–150, 2011. c Springer-Verlag Berlin Heidelberg 2011

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a particular perceptual form of bounded rationality. Since then, only scattered approaches have been given. The aim of this paper is to provide a conceptual tool for a study bounded rationality from the viewpoint of logic. It is also related to the problem of logical omniscience/omnipotence, which will be mentioned below. One important aspect of bounded rationality is the logical inferential ability of a player (person). As soon as, however, we consider provability of a logical system, we eﬀectively assume super-rationality and logical omnipotence. To discuss bounded rationality, we should consider how complex inferences are required for a given problem. Game theoretical problems are mathematically formulated, and players involved there are implicitly assumed to have mathematical inferential abilities. Such mathematical inferences are explicitly studied in proof theory. In this sense, proof theory is a suitable ﬁeld for a study of bounded rationality. Our approach looks related to the computational complexity in computer sciences. This is formulated as a question of how the required time and memory size increase as the length of input data increases. The approach of proof complexity (the lengths of proofs) is along this line; the literature has focused on the size of a required algorithm - - see Krajiˇcek [9] and Pudl´ ak [10]. In these approaches, algorithms are compared by their limiting behaviors, while we focus on measuring inferences required for each single instance of a sequent but not on the performance of an algorithm. Our approach is well understood from the viewpoint of bounded rationality in game theory/economics. For this, we should mention two related literatures: epistemic logics of shallow (interpersonal) depths (Kaneko-Suzuki [6]); and inductive game theory (Kaneko-Kline [4]). We will discuss the ﬁrst in several places i n this paper , and the second only in Section 7. Let us see our approach brieﬂy. The measure of inference, denoted by η L∗ , is deﬁned in classical and intuitionistic logics L = CL or IL in Gentzen’s [2], [3] sequent calculus. We have four types of those measures, depending upon L = CL or IL, and with or without cuts, i.e., η CLw , η CLf , η ILw and η ILf . We are interested in giving a method to calculate the exact value η L∗ (σ) for an arbitrary given sequent σ = Γ → Θ. Finding a proof P of σ is not enough for the calculation of η L∗ (σ), since it gives only its upper bound for η L∗ (σ). We give the lower bound method (LB-method) to calculate η L∗ (σ). We present Theorem 2 that this function β L gives a lower bound β L (σ) of η L∗ (σ) for any sequent σ. This β L gives often the exact value of η L∗ (σ). We adopt classical and intuitionistic logics as the environments for our study. Classical logic is the basic reference point, but intuitionistic logic is also important for us since it is of constructive nature and game theoretical decision making is directly related to constructiveness of a choice. Also, proofs with cuts will be important for further developments of our study in various manners, though the cases of cut-free proofs are easier than those with cuts. In this paper, we do not directly touch epistemic logic of shallow depths of Kaneko-Suzuki [6], but the main result of this paper (Theorem 2) can be extended to it. Here, we provide a small game theoretic example to motivate ourselves more. Consider the situation where a large store, 1 (a supermarket), and a small store,

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2 (a minimart) are competing. Store 2 has the subjective understanding of the situation: Store 1 is large enough to ignore store 2, but store 2’s proﬁts are inﬂuenced by 1’s choice. 2’s understanding is described by Tables 1 and 2. Store 1 has only three alternative actions, and his payoﬀ is determined by his own choice. On the other hand, 2 has 10 alternative actions, and the resulting payoﬀs are determined by the choices of both 1 and 2. Table 1; g1 a1 6000 a2 2000 a3 1000

s1 1 5 5

Table 2; g2 s2 s3 · · · s9 s10 2 3 · · · 9 10 6 7 · · · 13 14 7 9 · · · 21 23

In Table 2, store 2 has a dominant action, s10 , which gives the highest payoﬀ whatever 1 chooses. To achieve this knowledge, he compares the payoﬀ from s10 with those from s1 , ..., s9 in all the three cases of a1 , a2 and a3 ; hence it needs at least 9 × 3 = 27 comparisons. In this consideration, he thinks only about Table 2 but not about Table 1. Store 2 has an alternative decision criterion: First, he predicts the choice by store 1, and, using his prediction he chooses an action. For this criterion, he needs 2 comparisons to predict that 1’s choice would be a1 , and then he needs to make at least 9 comparisons to verify that s10 is the best toward a1 . Here, the minimum number of required comparisons is 11. There is a trade-oﬀ: The concentration on his own payoﬀ matrix does not require interpersonal inferences, but interpersonal considerations may simplify his decision-making with respect to the number of payoﬀ comparisons. This argument will be described in terms of the measure ηL∗ of inference in Section 5. The above example shows two important aspects of bounded rationality. In the game theory literature, it is regarded as necessary to assume that the game structure is common knowledge between the players. However, no common knowledge is required, but only interpersonal thinking of very shallow depths are required, which is captured in epistemic logic of shallow (interpersonal) depths (KanekoSuzuki [6]). This paper takes one step further to measure intrapersonal complexity for decision making in such a situation. Thus, the trade-oﬀ mentioned above can be discussed in a meaningful manner. In this sense, the contribution of the paper is to put one step forward in the direction of a study of bounded rationality.

2

Classical and Intuitionistic Logics

We present classical and intuitionistic logics CL and IL. We adopt the following list of primitive symbols: countably infinite number of propositional variables: p0 , p1 , ...; logical connective symbols: ¬ (not), ⊃ (implies), ∧ (and), ∨ (or); parentheses: ( , ); comma: , ; and braces { , }.

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We deﬁne formulae inductively: (o): any propositional variable p is a formula; (i): if C, D are formulae, so are (C ⊃ D) and (¬C); (ii): if Φ is a ﬁnite set of formulae with its cardinality |Φ| ≥ 2, then (∧Φ) and (∨Φ) are formulae. We denote the set of all formulae by P. Note that (ii) is not standard; this simpliﬁes our game theoretical arguments since conjunctions and disjunctions consisting of many formulae appear often in game theoretical applications. This will change slightly the rules for ∧ and ∨ in the formulation of sequent calculus. Let Γ, Θ be ﬁnite (possibly empty) sets of formulae in P. Using auxiliary symbol →, we introduce a new expression Γ → Θ, which we call a sequent. We abbreviate (set-theoretical) braces, for example, {A} ∪ Γ → Θ ∪ {B} is written as A, Γ → Θ, B, and also, Γ ∪ Δ → Θ ∪ Λ is abbreviated as Γ, Δ → Θ, Λ. The logical inferences are governed by one axiom schema and various inference rules. Axiom Schema (Initial Sequents): A → A, where A is any formula. Structural Rules: The following inference rules are called the thinning and cut : Γ →Θ (th) Δ, Γ → Θ, Λ Γ → Θ, A A, Δ → Λ (cut) Γ, Δ → Θ, Λ In (th), the sets Δ and Λ may be empty. Operational Rules: Γ → Θ, A (¬ →) ¬A, Γ → Θ Γ → Θ, A B, Δ → Λ (⊃→) A ⊃ B, Γ, Δ → Θ, Λ A, Γ → Θ (∧ →) where A ∈ Φ ∧Φ, Γ → Θ {A, Γ → Θ : A ∈ Φ} (∨ →) ∨Φ, Γ → Θ

A, Γ → Θ (→ ¬) Γ → Θ, ¬A A, Γ → B, Θ (→⊃) Γ → A ⊃ B, Θ {Γ → Θ, A : A ∈ Φ} (→ ∧) Γ → Θ, ∧Φ Γ → Θ, A (→ ∨) where A ∈ Φ. Γ → Θ, ∨Φ

In (→ ∧) and (∨ →), the multiple upper sequents indexed with A ∈ Φ are assumed to be proved. This change from the standard formulation is needed by adopting the applications of ∧ and ∨ to a set of formulae Φ. Since we measure the complexity of a proof, we need an explicit deﬁnition of it. A proof P in CL is deﬁned as a triple (X, <; ψ) with the following properties: (i): (X, <) is a ﬁnite tree consisting of the set of nodes X and the immediate predecessor relation <;

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(ii): ψ is a function which associates a sequent ψ(x) = Δ → Λ to each node x ∈ X; (a): for any leaf (maximal node) x in (X, <), ψ(x) is an instance of the axiom; (b): for any non-leaf x ∈ X, {ψ(x ) : x < x } I ψ(x)

(1)

is an instance of one inference rule. We say that σ is provable in CL iﬀ there is a proof of σ in CL, denoted by

CL σ. Intuitionistic logic IL is obtained from CL by giving the restriction that the succedent of each sequent has cardinality at most 1. A proof and provability for IL are deﬁned in the parallel manner with this restriction. Since a proof in IL is a proof in CL, IL σ implies CL σ. The following theorem by Gentzen [2], [3] is important in this paper. In fact, we need some elaborated version of this theorem in order to prove Theorem 2. Theorem 1. (Cut-Elimination for CL and IL). Let L be CL or IL. If L σ, then there is a cut-free proof P of σ in L.

3

Measure of Inference η L∗

We deﬁne the measure of inference η L∗ for each σ = Γ → Θ. For each proof P = (X, <; ψ) in L = CL or IL, we deﬁne η(P ) = the number of the leaves of the tree (X, <).

(2)

Thus, η measures the width of P, and ignores its depth1 . Our ultimate goal is to study the measure for a sequent, rather than a proof. Definition 1. We define the measure ηLf of inference for a sequent σ = Γ → Θ in logic L = CL or IL as follows: ⎧ ⎨ min{η(P ) : P is a cut-free proof of σ in L} if L σ η Lf (σ) = (3) ⎩ +∞ otherwise. By eliminating “cut-free” in (3), we have the other measure η Lw (σ). The expression η L∗ (σ) denotes either η Lf (σ) or η Lw (σ). The measure η L∗ counts the indispensable contents of the sequent σ to be proved. In other words, by tracing upwards in a proof P , we would meet an initial sequent, and if this initial sequent occurs in any proof of σ, we regard it as indispensable for σ. Since L is CL or IL, we have four types of measures; η CLw (σ), η CLf (σ), ηILw (σ) and ηILf (σ). The following are straightforward observations. 1

Urquhart [12] counted the number of all sequents in a proof. Then, he studied the computational complexities of various speciﬁc sets of problem instances.

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Lemma 1. (1): η CLw (σ) ≤ η ILw (σ) and η CLf (σ) ≤ η ILf (σ); (2): η Lw (σ) ≤ η Lf (σ) for L = CL, IL. Measure η L∗ is system-speciﬁc. If we adopt a Hilbert-style formulation of classical or intuitionistic logics, then the value changes a lot. However, there are diverse formulations of Hilbert-style systems. In the case of classical logic, one system in Kaneko-Nagashima [5] may be well comparable with the sequent formulation of the present paper. A proof in a sequent calculus can be converted into the Hilbert-style formulation and vice versa. For this conversion, (cut) does play an important role. Since the LB-method works even for η CLw , we would be able to compare our measure with that for the Hilbert-style formulation. Measure η L∗ depends also upon the choice of a language such as a language with binary ∧ and ∨. Nevertheless, this dependence is, perhaps, less serious than that upon a logical system.

4

The Lower Bound Method

It is not easy to calculate the exact value η L∗ (σ) for a given sequent σ = Γ → Θ. Here, we provide one method of ﬁnding the exact value η L∗ (σ). The following lemma is a straightforward observation, but explains our motivation to introduce a lower bound function. Lemma 2. (LB-Method). Let β be a function assigning a natural number to every sequent σ. For any sequent σ, if (1) β(σ) ≤ η L∗ (σ) and (2) β(σ) = η(P ) for some proof P of σ, then β(σ) = η L∗ (σ). We would like to ﬁnd a meaningful lower bound function. Speciﬁcally, we deﬁne a lower bound function, denoted by β L , in the following. The basic idea for β L is to count the occurrences of subformulae ∧Φ, ∨Ψ and A ⊃ C in σ = Γ → Θ which are necessarily the principal formulae of some applications of (→ ∧), (∨ →) and (⊃→) in any proof of σ. However, we need to estimate the applications of (→ ∧), (∨ →) and (⊃→) by looking only at σ. For this estimation, ﬁrst, let us connect the width of a tree with the number of branches in the tree. Lemma 3. Let (X, <) be a finite tree with the nonterminal nodes 1, ..., k with mt branches at nonterminal node t. Let n1 , ..., n be the nonterminal nodes for which mnt > 1 for all t = 1, ..., . Then the number of terminal nodes of (X, <) is given as t=1 mnt − ( − 1). To deﬁne the lower bound function β L and to state the main theorem (Theorem 2), we use various proof-theoretic concepts. First, we use a linear order ≺ over the set of all formulae P. We stipulate that each ﬁnite nonempty set Φ of formulae is ordered by ≺ . We use also the concept of an occurrence ξ at three levels, which indicates the address of a formula B relative to (i) a formula A; (ii) a sequent σ; and (iii) a proof P. Only (i) requires a careful consideration, and (ii), (iii) are straightforward. Since (i) and (ii) are need for Theorem 2, we give only (i) and (ii). (iii) will be needed for a proof of Theorem 2.

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Consider one example: A = ¬(∨{p1 , p2 , B} ⊃ ∧{p1 , B ⊃ p2 }),

(4)

where B = ∧{p1 , p2 , p3 }. The subformulae in ∨{p1 , p2 , B} and ∧{p1 , B ⊃ p2 } are ordered by ≺ . This A has two occurrences of B, and we would like to separate each without ambiguity. The left occurrence of B is identiﬁed in the following manner: ξ = [A | · ¬ · (⊃, −1) · (∨, 3) : B]. (5) The ﬁrst A is the reference formula, denoted by ρ[ξ], and the last B is the target formula, τ [ξ]. Formally, we deﬁne an occurrence relative to a reference formula A by induction from the outermost viewpoint of A. We can also talk about an occurrence ξ of B in A in a sequent σ = {C1 , ..., Cj , ..., Cm } → {D1 , ..., Dk }, where A = Cj . The sign (positive or negative) of an occurrence ζ in a sequent σ or a formula A is unambiguously deﬁned. Roughly speaking, an occurrence is deﬁned ﬁrst relative to a formula - - the direct occurrences in the scope of ¬ and in the premise of ⊃ are negative, and the other direct occurrences are positive. For example, consider the occurrence of the 2nd B in (4): The second B is negative relative to the formula ∧{p1 , B ⊃ p2 } since it is the premise of B ⊃ p2 , but it is positive relative to entire A since B ⊃ p2 is in the scope of ¬. This assignment of a sign is extended to a sequent: The occurrence of the second B in A = Cj is negative relative to σ, since the occurrence of A is negative since it occurs in the “premise” {C1 , ..., Cj , ..., Cm } of σ. A strength of the sequent calculus is the sign-preserving property that once a formula occurs in a proof P , all descendants have the same sign. Now, we isolate some occurrences in a sequent σ using the concepts introduced above. We say that an occurrence ζ is a companion of another occurrence ζ in σ iﬀ τ [ζ] = τ [ζ ] and ζ, ζ have opposite signs. Then, we say that an occurrence ζ in σ is legitimate iﬀ (i): ζ has no companions; (ii): τ [ζ] of ζ is expressed as either ∧Φ, ∨Ψ or A ⊃ C; (iii): if τ [ζ] = ∧Φ, ζ is positive in σ, and if τ [ζ] = ∨Ψ or A ⊃ C, it is negative in σ. Consider a set μ of legitimate occurrences ζ 1 , ..., ζ m in σ. Let the target formulae τ [ζ 1 ], ..., τ [ζ m ] be ∧Φ1 , ..., ∧Φ1 , ∨Ψ1 , ..., ∨Ψ2 , A1 ⊃ B1 , ..., A3 ⊃ B3 .

(6)

Then we deﬁne w(μ) =

1

t=1

|Φt | +

2

t=1

|Ψt | + 2 3 − ( 1 + 2 + 3 − 1).

(7)

It is seen from Lemma 3 that this number is an estimation of the number of leaves of a proof of σ.

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We consider a set μ of legitimate occurrences ζ 1 , ..., ζ m in σ satisfying if ζ t is included in a legitimate occurrence ζ, then ζ ∈ μ.

(8)

Let ζ 1 , ..., ζ k be the other legitimate occurrences in σ each of which is maximal in the sense of nesting, i.e., each ζ t (t = 1, ..., k) includes no occurrences of ζ 1 , ..., ζ k but ζ t itself. Let q1 , ..., qk be new propositional variables not occurring in σ. Then, we deﬁne σ ∗ = Γ ∗ → Θ∗ to be the sequent obtained from σ by replacing ζ 1 , ..., ζ k by q1 , ..., qk . We say that a set of legitimate occurrences ζ 1 , ..., ζ m satisfying (8) is genuine iﬀ L Γ ∗ → Θ∗ . We denote the set of all genuine sets by ML (σ). We take the minimal value of w(μ) over ML (σ), i.e., β L (σ) =

min

μ∈ ML (σ)

w(μ).

(9)

Now, we state the main theorem: We have (1) of Lemma 2 for any sequent σ. Theorem 2. (Lower Bound Function β L ). β L (σ) ≤ η Lw (σ) ≤ η Lf (σ) for any sequent σ = Γ → Θ. To prove the inequality β L (σ) ≤ η Lw (σ), we need some elaboration of Theorem 1 (cut-elimination). Given a sequent σ, if we ﬁnd a proof P of σ in logic L so that β L (σ) = η(P ), then we obtain the exact value η L∗ (σ). Now, our problem is to evaluate how good the lower function β L (σ) is. This evaluation will be considered in the following two sections.

5

An Application to the Game with Small and Large Stores

Let us apply measure η L∗ to the game example in Section 1. Consider two decision-making criteria for store 2 : (i): the dominant-strategy criterion; and (ii): the prediction-decision criterion. To formulate (i) and (ii) in our propositional language, we introduce propositional variables as follows: p(al , al ), l, l = 1, 2, 3; and q(sr , sr | al ), l = 1, 2, 3 and r, r = 1, ..., 10. They are possible (strict) preferences by stores 1 and 2 : p(al , al ) means that 1 prefers al to al ; and q(sr , sr | al ) means that conditional upon the choice al by 1, store 2 prefers strictly sr to sr . Game (g1 , g2 ) : Tables 1 and 2 are expressed as the sets of preferences: gˆ1 = {p(al , al ), ¬p(al , al ) : l, l = 1, 2, 3 with l < l };

(10)

gˆ2 = {q(sr , sr | al ), ¬q(sr , sr | al ) : r > r and l = 1, 2, 3}.

(11)

The ﬁrst states that store 1 prefers action al to al if l < l , and the second states that when 1 chooses al , store 2 would prefer sr to sr with r > r .

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Dominant-Strategy (DS) Criterion: It is formulated as ∧{q(sr , sr | al ) : r = r and l = 1, 2, 3}, which is denoted by Dom2 (sr ). Looking at Table 2, we ﬁnd that s10 is a unique dominant strategy. This is formulated as the sequent and L gˆ2 → Dom2 (s10 ). We can ﬁnd a proof P of gˆ2 → Dom2 (s10 ) with η(P ) = 27. Also, we can calculate β L (ˆ g2 → Dom2 (s10 )) = 27, since Dom2 (s10 ) is a unique legitimate occurrence and constitutes a unique genuine set. Hence, we have η L∗ (ˆ g2 →Dom2 (s10 )) = 27 by the LB-method (Lemma 2 and Theorem 2). In fact, we can add gˆ1 to this statement, i.e., η L∗ (ˆ g1 , gˆ2 → Dom2 (s10 )) = 27. Prediciton-Decision (PD) Criterion: The prediction part is formulated as ∧{p(al , al ) : l = l}, denoted by Bt1 (al ), where l = 1, 2, 3, and also the best response of player 2 conditional upon al , i.e., ∧{q(sr , sr | al ) : r = r}, is denoted by Bt2 (sr | al ).. Then the prediction-decision criterion is formulated as PD2 (sr ) := ∨l [Bt1 (al ) ∧ Bt2 (sr | al )]. Store 2 with this criterion looks at 1’s preferences gˆ1 as well as his own preferences gˆ2 . He predicts that 1 would prefer a1 to a2 , a3 (so he would choose a1 ), and he ﬁnds that s10 is better than the other alternative actions. That is, L gˆ1 , gˆ2 → PD2 (s10 ), and we have β L (ˆ g1 , gˆ2 → PD2 (s10 )) = (3 + 9) − (2 − 1) = 11.

(12)

In fact, we ﬁnd a proof P of gˆ1 , gˆ2 →PD2 (s10 ) with η(P ) = 11, which is given as follows: p(a1 ,a2 ) → p(a1 ,a2 ) p(a1 , a3 ) → p(a1 , a3 ) (th) (th) gˆ1 → p(a1 , a2 ) gˆ1 → p(a1 , a3 ) (→ ∧) gˆ1 → Bt1 (a1 ) T1 : (th) gˆ1 , gˆ2 →Bt1 (a1 )

T2 :

q(s10 , sr | a1 )→ q(s10 , sr | a1 ) (th) gˆ2 → q(s10 , sr | a1 ) gˆ2 → Bt2 (s10 |a1 ) gˆ1 , gˆ2 →Bt2 (s10 |a1 )

r =10

(→ ∧) (th)

and these proofs are combined as follows: T1 T2 (th) (th) gˆ1 , gˆ2 →Bt1 (a1 ) gˆ1 , gˆ2 →Bt2 (s10 | a1 ) (→ ∧) gˆ1 , gˆ2 →Bt1 (a1 )∧Bt2 (s10 | a1 ) (→ ∨) gˆ1 , gˆ2 →PD2 (s10 ) Hence, by Lemma 2 and Theorem 2, we have ηL∗ (ˆ g1 , gˆ2 →PD2 (s10 )) = 11. This calculation is the same for L = CL or IL. This is also exactly the same as the calculation given in Section 1.

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Evaluations of Some Contradictory Statements

Contradiction-freeness is important in general, but it is speciﬁcally relevant for game theory/economics in that the beliefs owned by a player may be contradictory but he himself does not notice it. Here, we give some general considerations, and then a speciﬁc problem arising in economics. In logic L = CL or IL, a contradictory statement is formulated as either L Γ → or L Γ → ¬A ∧ A for some A. These are equivalent but diﬀer in general with respect to ηL∗ . Theorem 3. Suppose L Γ →

. Then, for any formula A,

η Lw (Γ → ¬A ∧ A) ≤ η Lw (Γ → ) ≤ η Lw (Γ → ¬A ∧ A) + 1.

(13)

Let At = ∧{pt, q} for t = 1, ..., m, and Bt = ∨{pt , q} for t = 1, ..., n, where p1 , ..., and q are all propositional variables. Then, consider the following sequents: σ 0 = ∨ {A1 , ..., Am } → ∧ {B1 , ..., Bn }; σ1 = ∨{A1 , ..., Am }, ¬ ∧ {B1 , ..., Bn } → ;

(14)

σ2 = ∨{A1 , ..., Am }, ¬ ∧ {B1 , ..., Bn } → ¬q ∧ q. Those are provable and deductively equivalent in L = CL, IL. Theorem 4. (0): β L (σ 0 ) = β L (σ 1 ) = β L (σ 2 ) = m + n − 1; (1): ηLw (σ 0 ) = ηLw (σ 1 ) = η Lw (σ 2 ) = m + n; (2): η Lf (σ 0 ) = ηLf (σ1 ) = m × n and ηLf (σ 2 ) = m + n. This theorem has various implications: The lower bound function β L does not capture the exact values of η L∗ (σ 0 ), η L∗ (σ 1 ) and η L∗ (σ 2 ), which diﬀer for η Lw and η Lf . Also, it shows some similarity between the role of the cut-formula and the role of the contradictory formula. Now, let us apply our measure to some economics problem: Suppose that a decision maker (DM) faces a problem to choose one from k alternatives a1 , ..., ak . He has two types of beliefs: (1) basic preferences comparing consecutive alternatives; (2) additional properties for preferences. We prepare propositional variables prs (1 ≤ r, s ≤ k, r = s); each prs is intended to mean that DM strictly prefers alternative ar to ak . For (1), DM makes direct comparisons only between ar to ar+1 (k+1 is understood as 1). These preferences are called the basic preferences over k alternatives: Then let Δk = {p12 , p23 , ..., p(k−1)k } ∪ {pk1 }. (15) He has a cycle of preferences over those alternatives. The set Δk constitutes raw data for him. This lacks preference comparisons between remote alternatives, ar and as (|r − s| > 1). DM may compensate for such missing parts by some beliefs of general properties for preferences. Here, we

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assume transitivity and asymmetry for his preferences: ΠkT = {prs ∧ pst ⊃ prt : 1 ≤ r, s, t ≤ k}; and ΠkA = {prs ⊃ (¬psr ) : 1 ≤ r, s ≤ k}. These are DM’s beliefs of basic properties. Indirect comparisons between nonconsecutive alternatives such as p13 are made from the raw data and his beliefs of basic properties. For example, he derives p13 from p12 , p23 using p12 ∧ p23 ⊃ p13 .The union of three sets Δk , ΠkT , ΠkA , yields a contradiction. The value of measure η L∗ is given in the following theorem. Theorem 5. (Cyclical Contradiction): Let k ≥ 2. Then we have: (1): β L (Δk , ΠkT , ΠkA → ) = 2k − 2; (2): η Lw (Δk , ΠkT , ΠkA → ) = η Lf (Δk , ΠkT , ΠkA → ) = 2k − 2. In this theorem, the LB-method gives the exact values of η Lw and η Lf .

7

Conclusion

We have developed a theory of the measure of inference for classical logic CL and intuitionistic logic IL. By our theory, we have certain important consequences both from the viewpoints of logic as well as game theory/economics. Now, we have a lot of diﬀerent aspects as well as a lot of applications to be considered. Here, we will discuss a few aspects and applications to be emphasized. (1): Refinement of the LB-Method: This works well in the examples in this paper in that β L (σ) gives the exact value η L∗ (σ) or approximates it. Only two exceptions are the sequents σ0 and σ 1 given in (14) in that β L (σ) is very diﬀerent from ηLf (σ). This is caused by the fact that the lower bound function β L counts legitimate occurrences as a summation form. However, in the examples σ 0 and σ1 , the ancestors of legitimate occurrences are nested, which requires us to count them as a multiplication form. The diﬀerence caused by nesting can be seen more severely in the application of the LB-method to the Pigeonhole Principle (see Buss [1]). A reﬁnement of the LB-method toward this direction is an important open question. (2): Computational Complexity and Proof Search: By (1), we may recall the literature of computational complexity and proof search. Once the measure of inference is well developed, we can use it for an analysis of computational complexity for various classes of problem instances as well as for proof search. This is a side problem along the line of our original motivation. Nevertheless, it would be important to think about this application. (3): Epistemic Logics of Shallow Depths: This is closely related to the present research in motivations, and was brieﬂy discussed in the game with large and small stores in Section 5. The extension of the measure η L∗ itself to those epistemic logics is straightforward, though we have diﬀerent possible ways of counting epistemic depths. Also, the function β L and Theorem 2 can be extended to those logics. Then, we can discuss the trade-oﬀ mentioned above in a more explicit manner.

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(4):Mechanical Method of Calculations: We have decomposition properties along the principal formula of an application of an inference. However, it gave only decompositions with inequalities, and does not help the calculation of η L∗ . Therefore, we have developed the LB-method. However, when we restrict our attention to some class of sequents, we can expect those decompositions with equalities, perhaps, for intuitionistic logic L = IL. In fact, this is partially done for intuitionistic-based epistemic logics. Then we can calculate η L∗ in a mechanical way, and expect to a mechanical construction of a proof. (5): Connections to Inductive Game Theory: From our research viewpoint, it is more direct to apply the theory to the induction process in inductive game theory of Kaneko-Kline [4]. For example, the inductive process itself was not formulated in [4], but it can be formulated as an algorithm from accumulated experiences to an individual view. This algorithm can be formulated as a set of beliefs of a player, and he infers his view based on this set of beliefs from his accumulated experiences. If our measure gives a large number, he would have a diﬃculty in constructing his view. The present authors have already developed some of those problems - - some papers [7] and [8] will be available. Yet, we have a lot of open problems about the theory presented here. We expect a lot of further contributions along the line of the research given in this paper.

References 1. Buss, S.R.: Polynomial size proofs of the propositional pigeonhole principle. J. Symbolic Logic 52(4), 916–927 (1987) 2. Gentzen, G.: Untersuchungen u ¨ber das logische Schließen. I. Math. Z. 39(1), 176–210 (1935); english translation, ibid 3. Gentzen, G.: Untersuchungen u ¨ber das logische Schließen. II. Math. Z. 39(1), 405–431 (1935); english translation, Investigations into Logical Deduction, The Collected Papers of Gerhard Gentzen (1969) 4. Kaneko, M., Kline, J.J.: Inductive game theory: a basic scenario. J. Math. Econom. 44(12), 1332–1363 (2008) 5. Kaneko, M., Nagashima, T.: Game logic and its applications. I. Studia Logica 57(23), 325–354 (1996) 6. Kaneko, M., Suzuki, N.Y.: Epistemic models of shallow depths and decision making in games: Horticulture. J. Symbolic Logic 68(1), 163–186 (2003) 7. Kaneko, M., Suzuki, N.Y.: Contentwise complexity of inferences in epistemic logics of shallow depths ii: Ec-sequents (2005); mimeo 8. Kaneko, M., Suzuki, N.Y.: Contentwise complexity: An evaluation of arrow’s impossibility theorem (2008); mimeo 9. Kraj´ıˇcek, J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory. In: Encyclopedia of Mathematics and its Applications, vol. 60. Cambridge University Press, Cambridge (1995) 10. Pudl´ ak, P.: The lengths of proofs. In: Handbook of Proof Theory, Stud. Logic Found. Math., vol. 137, pp. 547–637. North-Holland, Amsterdam (1998) 11. Simon, H.A.: A behavioral model of rational choice. The Quarterly Journal of Economics 69(1), 99–118 (1955) 12. Urquhart, A.: The complexity of Gentzen systems for propositional logic. Theoret. Comput. Sci. 66(1), 87–97 (1989)

Partial Semantics of Argumentation Beishui Liao and Huaxin Huang Center for the Study of Language and Cognition, Zhejiang University,Hangzhou 310028, China {baiseliao,rw211}@zju.edu.cn

Abstract. In various argumentation systems, under most of situations, only the justiﬁcation status of some arguments of the systems should be evaluated, while that of other arguments is not necessary to be ﬁgured out. Based on this observation, we introduce an eﬃcient method to evaluate the status of a part of arguments in an argumentation framework. This method is based on the notion of unattacked sets of an argumentation framework and the directionality criterion of argumentation semantics. Given an argumentation framework and a subset of arguments within it, we ﬁrstly identify the minimal set of arguments that are relevant to the arguments in this subset (called the minimal unattacked set). Then, under an argumentation semantics satisfying the directionality criterion, the set of extensions of the sub-framework induced by the minimal unattacked set (called a partial semantics of the original argumentation framework) can be evaluated independently. Then, we analyze two basic properties of the partial semantics of argumentation: monotonicity and combinability. Keywords: Argumentation, semantics, partial extension, directionality, unattacked sets.

1

Introduction

In recent years, argumentation has become a core study area of artiﬁcial intelligence [7], due to its ability to handle incomplete, uncertain, and conﬂicting information. So far, various aspects of argumentation, including semantics [10, 6, 11, 4, 5], proof theories and algorithms [12, 15, 23, 18, 25, 20], and applications [19, 1, 13, 2, 3, 9, 24, 21] have been widely studied. However, according to current literature, little attention has been paid to the following phenomenon: In a variety of argumentation systems [19, 1, 13, 2, 3, 9, 24, 21], under most of situations, only the justiﬁcation status of some arguments is concerned, while that of others is unnecessary to be ﬁgured out. According to the existing proof theories and algorithms [15, 12, 16, 18, 23, 25, 20,17], as well as the deﬁnitions of various argumentation semantics [10,11,4,5], they have not considered how to partially evaluate the justiﬁcation status of a part of arguments of a system. Based on this observation, a basic problem arises: H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 151–164, 2011. c Springer-Verlag Berlin Heidelberg 2011

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How to evaluate the justiﬁcation status of a subset of arguments in an argumentation system, without evaluating the justiﬁcation status of the whole set of arguments in the system? In order to resolve the above problem, intuitively, we may take advantage of the idea of local computation that has been applied in some existing work, such as [6], [8], and [22], etc. First, in [8], Boella et al. studied the dynamics of argumentation by exploring the principles according to which the extensions do not change when the set of arguments or the attacks between them are changed. Second, in [6], Baroni et al. proposed an SCC-recursive scheme for argumentation semantics, based on decomposition along the strongly connected components of an argumentation framework. So, according to the deﬁnitions of SCC-recursive semantics, it is possible to compute the extensions of an argumentation framework based on local computation at the level of strongly connected components. Third, in [22], in order to eﬃciently compute the dynamic of argumentation, an argumentation framework is divided into two sub-frameworks, in which one is related to the set of aﬀected arguments and the other is related to the set of unaﬀected arguments. Then, only the status of aﬀected arguments is recomputed, while the status of unaﬀected arguments remains untouched. According to these diﬀerent kinds of theories, although they addressed different problems in the area of argumentation and used diﬀerent methods, they all made use of the idea of local computation, which is obviously an eﬀective approach to improve the eﬃciency of computation. Motivated by the above problem, and inspired by the idea of local computation, in this paper we propose a basic theory to eﬃciently evaluate the justiﬁcation status of a part of arguments in an argumentation system. The rest of this paper is organized as follows: First, some basic notions that will be used in this paper are introduced. Second, the deﬁnitions and basic properties of the partial semantics of argumentation are presented. Third, we conclude the paper and point out some future work.

2

Preliminaries

In this paper, we only treat with abstract argumentation framework, which is based on Dung’s seminal work in [14]. Meanwhile, as presented in [6], the feasibility of local computation is closely related to a basic property of argumentation semantics: directionality. So, in this section, some basic notions related to these two aspects are introduced. 2.1

Dung’s Theory of Argumentation

According to [14], an abstract argumentation framework (or brieﬂy, argumentation framework) is deﬁned as follows. Definition 1. An argumentation framework is a tuple AF = A, R, where A is a set of arguments, and R ⊆ A × A a set of attacks.

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In Deﬁnition 1, each argument is an abstract entity (i.e., the internal structure of each argument is left unspeciﬁed), while an argument α attacks another argument β is denoted as (α, β). For simplicity, throughout this paper, we assume that the set of arguments A is generated by a reasoner at a given time point, and therefore is ﬁnite. Given an argumentation framework, a fundamental problem is to determine which arguments can be considered justiﬁed. According to [14], extension-based argumentation semantics is a formal way to resolve this problem. Here, an extension represents a set of arguments that are considered to be acceptable together, which is based on the following three important concepts: conﬂict-free, acceptability, and admissible set. Definition 2. Let AF = A, R be an argumentation framework. – A set B ⊆ A of arguments is conﬂict-free if and only if α, β ∈ B, such that (α, β) ∈ R. – An argument α ∈ A is acceptable with respect to a set B ⊆ A of arguments, if and only if ∀(β, α) ∈ R, ∃γ ∈ B, such that (γ, β) ∈ R. – A conﬂict-free set of arguments B ⊆ A is admissible if and only if each argument in B is acceptable with respect to B. With the notion of admissible set, the extensions of an argumentation framework under diﬀerent argumentation semantics can be deﬁned as follows: Definition 3. Let AF = A, R be an argumentation framework, and B ⊆ A an admissible set of arguments. – B is a preferred extension if and only if B is a maximal (with respect to set-inclusion) admissible set of arguments. – B is a complete extension if and only if each argument in A that is acceptable with respect to B is in B. – B is a grounded extension if and only if B is the the minimal (with respect to set-inclusion) complete extension. – B is ideal if and only if B is admissible and it is contained in every preferred set of arguments. The ideal extension is the maximal (with respect to setinclusion) ideal set. – B is stable if and only if B is a preferred extension that defeats (w.r.t. R) all arguments in A \ B. We use ECO (AF ), EPR (AF ), EGR (AF ), EID (AF ), and EST (AF ) to denote the set of complete, preferred, grounded, ideal, and stable extensions of AF , respectively. Based on the set of extensions of an argumentation framework, we diﬀerentiate three justiﬁcation statuses of arguments: sceptically justiﬁed, credulously justiﬁed, and indefensible. Formally, we have the following deﬁnition: Definition 4. Given an argumentation framework AF = A, R and a semantics S ∈ {CO, PR, GR, ID, ST }, an argument α ∈ A is credulously justiﬁed if and only if ∃E ∈ ES (AF ), s.t. α ∈ E; it is skeptically justiﬁed if and only if ∀E ∈ ES (AF ), α ∈ E; and it is indefensible if and only if E ∈ ES (AF ) s.t. α ∈ E.

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Directionality of Argumentation Semantics

The basic idea of directionality is that under some argumentation semantics, the status of an argument α is aﬀected only by the status of its defeaters (which in turn are aﬀected by their defeaters and so on), while the arguments which only receive an attack from α (and in turn those which are attacked by them and so on) do not have any eﬀect on the status of α. Currently, most argumentation semantics have been proved to satisfy the directionality criterion, such as grounded, complete, preferred, ideal, CF2 , and the prudent version of grounded semantics [4]. In this paper, for simplicity, we only consider grounded, complete, preferred, and ideal semantics. According to [4], some basic notions related to directionality of argumentation semantics are as follows: Definition 5. Given an argumentation framework AF = A, R, a set U ⊆ A is unattacked if and only if α ∈ (A \ U ), s.t. α attacks an argument in U . The set of unattacked sets of AF is denoted as US(AF ). Definition 6. Let AF = A, R be an argumentation framework, and B ⊆ A a subset of A. A sub-framework induced by B (also called the restriction of AF to B) is deﬁned as follows: AF↓B = B, R ∩ (B × B)

(1)

The directionality criterion is then deﬁned by requiring an unattacked set to be unaﬀected by the remaining parts of the argumentation framework as far as extensions are concerned. Definition 7. A semantics S satisﬁes the directionality criterion if and only if ∀AF = A, R, ∀U ∈ US(AF ), ES (AF↓U ) = {(E ∩ U ) | E ∈ ES (AF )}. Based on the above notions, let us see the following example: Example 1. Let AF1 = A1 , R1 be an argumentation framework as shown in Figure 1, in which A1 = {α1 , α2 , α3 } and R1 = {(α1 , α1 ), (α2 , α3 )}. Under an argumentation semantics (say, preferred semantics) that satisﬁes the directionality criterion, the justiﬁcation status of a part of arguments in AF1 can be evaluated locally. Let us consider two subsets C1 = {α1 } and C2 = {α3 } of A1 , with respect to the set of preferred extension of AF1 . First, we compute the set of preferred extensions of AF1 , i.e., EPR (AF1 ) = {E1 } in which E1 = {α2 }. Then, we restrict the set of preferred extensions of AF1 to an unattacked set with respect to C1 (respectively, C2 ). On the one hand, since C1 is itself an unattacked set, according to Deﬁnition 7, we have EPR (AF1↓B1 ) = {E1 ∩ B1 } = {E1,1 }, in which E1,1 = ∅. On the other hand, since C2 is not an unattacked set, we ﬁrstly get a minimal unattacked set that contains C2 , which is C2 = {α2 , α3 }. Then, according to Deﬁnition 7, EPR (AF1↓C2 ) = {E1 ∩ C2 } = {E1,2 }, in which E1,2 = {α2 }.

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It is obvious that the justiﬁcation status of arguments in C1 and C2 can be evaluated with respect to EPR (AF1 ↓C1 ) and EPR (AF1 ↓C2 ), respectively, i.e., both α1 and α3 are indefensible. However, under an argumentation semantics that does not satisfy the directionality criterion (for instance, stable semantics), we may not able to evaluate the justiﬁcation status of a part of arguments in an argumentation framework with respect to the set of extensions of a sub-framework induced by the related unattacked set. In this example, since AF1 has no stable extension, all arguments in A1 are indefensible. However, since EST (AF1↓C2 ) = {{α2 }}, it turns out that α2 is skeptically justiﬁed. Contradiction.

α1

α2

/ α3

Fig. 1. Argumentation framework AF1

3 3.1

Definitions and Properties of Partial Semantics of Argumentation Definitions

As illustrated in Example 1, given an argumentation framework AF = A, R and a subset B ⊆ A of arguments, we may evaluate the justiﬁcation status of arguments in B in an unattacked set, which contains B and is minimal (called the minimal unattacked set with respect to B). Formally, we have the following deﬁnition: Definition 8. Let AF = A, R be an argumentation framework, and B ⊆ A a subset of A. The minimal unattacked set with respect to B is a subset B ⊆ A, such that: (i) B ⊇ B; (ii) B is an unattacked set; and (iii) for all α ∈ (B \ B), B \ {α} is not an unattacked set. Example 2. According to Example 1, the subset C2 is the minimal unattacked set with respect to C2 , while C2 ∪ {α1 } = A1 is not. Now, given a subset B of arguments in an argumentation framework AF = A, R, according to the deﬁnition of directionality of argumentation semantics [4], only those arguments, each of which is in B or has a path to the arguments in B, may be relevant to the status of arguments in B. For convenience, we call them the set of relevant arguments of B, denoted as rlvtAF (B). In terms of [22], if an argument α has a path to another argument β (β and α can be the same argument), then we say that β is reachable from α. Formally, we have the following deﬁnitions: Definition 9. Let AF = A, R be an argumentation framework. For all α, β ∈ A, the reachability of these two arguments w.r.t. R is recursively deﬁned as follows:

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– If (α, β) ∈ R, then β is reachable from α; – If ∃γ ∈ A, s.t. γ is reachable from α w.r.t. R and (γ, β) ∈ R, then β is reachable from α. Definition 10. Given an argumentation framework AF = A, R and a subset B ⊆ A of arguments, the set of relevant arguments of B is deﬁned as follows: {β ∈ A \ B : α is reachable from β w.r.t. R} (2) rlvtAF (B) = B ∪ α∈B

According to Deﬁnitions 9 and 10, it is easy to verify that the set of relevant arguments of B is the minimal unattacked set with respect to B. Formally, we have the following proposition: Proposition 1. Let AF = A, R be an argumentation framework, B ⊆ A a subset of arguments in AF , and rlvtAF (B) the set of relevant arguments of B. It holds that rlvtAF (B) is the minimal unattacked set with respect to B. Proof. Firstly, we verify that rlvtAF (B) is an unattacked set. Assume the contrary. Then, according to Deﬁnition 5, ∃γ ∈ (A \ rlvtAF (B)), such that γ attacks an argument α in rlvtAF (B). If α ∈ B, then: according to Deﬁnition 9, α is reachable from γ. According to Deﬁnition 10, γ ∈ rlvtAF (B). Contradiction. Otherwise, if α ∈ rlvtAF (B) \ B, then: according to Deﬁnition 10, ∃α ∈ B, such that α is reachable from α. Then, according to Deﬁnition 9, α ∈ B is reachable from γ, and therefore γ ∈ rlvtAF (B). Contradiction. Secondly, we verify that rlvtAF (B) is minimal. Assume the contrary. Then, there exists Φ ⊆ A, such that Φ is the minimal unattacked set with respect to B, and Φ ⊂ rlvtAF (B). It follows that Φ ⊇ B (according Deﬁnition 8), and for all α ∈ (rlvtAF (B) \ Φ), β ∈ Φ, such that (α, β) ∈ R (according to Deﬁnition 5). As a result, α is not reachable to any arguments in B ⊆ Φ, and therefore α∈ / rlvtAF (B). Contradiction. Now that rlvtAF (B) is the minimal unattacked set with respect to B, for all S ∈ {CO, PR, GR, ID} that satisﬁes the directionality criterion, the justiﬁcation status of arguments in B can be independently evaluated with respect to the set of extensions of the induced sub-framework AF ↓ rlvtAF (B) . In this paper, we call ES (AF ↓ rlvtAF (B) ) the partial semantics of the original argumentation framework AF with respect to B. Definition 11. Let AF = A, R be an argumentation framework, B ⊆ A a subset of arguments, and rlvtAF (B) the set of relevant arguments of B. For all S ∈ {CO, PR, GR, ID} that satisﬁes the directionality criterion, the partial semantics of AF with respect to B is deﬁned as ES (AF↓ rlvtAF (B) ). Example 3. Let AF2 = A2 , R2 be an argumentation framework (Figure 2), in which A2 = {α1 , . . . , α6 } and R2 = {(α1 , α2 ), (α2 , α1 ), (α2 , α3 ), (α4 , α3 ), (α5 , α4 ), (α6 , α5 )}. Given B1 = {α2 } and B2 = {α2 , α3 }, we have: rlvtAF2 (B1 ) = {α1 , α2 } rlvtAF2 (B2 ) = A2

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It is obvious that rlvtAF2 (B1 ) is the minimal unattacked set with respect to B1 , while rlvtAF2 (B2 ) is the minimal unattacked set with respect to B2 . Under preferred semantics, the partial semantics of AF2 with respect to B1 is: EPR (AF2↓ rlvtAF2 (B1 ) ) = {E1,1 , E1,2 }, in which E1,1 = {α1 }, and E1,2 = {α2 }; while the partial semantics of AF2 with respect to B2 is: EPR (AF2↓ rlvtAF2 (B2 ) ) = {E1,1 , E1,2 }, in which E1,1 = {α1 , α6 , α4 }, and E1,2 = {α2 , α6 , α4 }.

α1 g

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Fig. 2. Argumentation framework AF2

3.2

Properties

According to previous subsection, given an argumentation framework AF = A, R and a subset B ⊆ A of arguments, we may compute the partial semantics of AF with respect to B independently, without computing the status of other arguments that are irrelevant to B. Furthermore, it is desirable that after the partial semantics of AF with respect to some sets of arguments are obtained, they can be reused in the subsequent computation. This vision is embodied by the following two basic properties of partial semantics of argumentation: monotonicity and combinability. First, given an argumentation framework AF = A, R and two subsets B, B ⊆ A of arguments, the monotonicity of partial semantics of argumentation can be informally expressed as: if B ⊆ B , then the justiﬁcation status of each argument in B evaluated with respect to ES (AF↓ rlvtAF (B) ) is the same as the one evaluated with respect to ES (AF↓ rlvtAF (B ) ). This property indicates that after we get the partial semantics of AF with respect to B (i.e., ES (AF↓ rlvtAF (B ) )), we may evaluate the justiﬁcation status of arguments in B (which is contained in B ) with respect to ES (AF ↓ rlvtAF (B ) ) directly, and do not need to compute the partial semantics of AF with respect to B. Formally, the monotonicity of partial semantics of argumentation is formulated by the following lemma, theorem and corollary: Lemma 1. Given an argumentation framework AF = A, R and two subsets B, B ⊆ A, if B ⊇ B, then rlvtAF (B ) ⊇ rlvtAF (B).

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Proof. Since B ⊇ B, according to Deﬁnition 10, it is obvious that rlvtAF (B ) ⊇ rlvtAF (B). Theorem 1. Given an argumentation framework AF = A, R and two subsets B, B ⊆ A of arguments, if B ⊇ B, then: for all S ∈ {CO, PR, GR, ID} that satisﬁes the directionality criterion, ES (AF ↓ rlvtAF (B) ) = {E ∩ rlvtAF (B) | E ∈ ES (AF↓ rlvtAF (B ) )}. Proof. Since AF ↓ rlvtAF (B) ⊆ AF ↓ rlvtAF (B ) (according to Lemma 1), and both AF↓ rlvtAF (B) and AF↓ rlvtAF (B ) are unattacked sets (according to Proposition 1), according to Deﬁnition 7, it holds that ES (AF↓ rlvtAF (B) ) = {E ∩ rlvtAF (B) | E ∈ ES (AF↓ rlvtAF (B ) )}. According to Theorem 1, we directly have the following Corollary: Corollary 1. Given an argumentation framework AF = A, R and two subsets B, B ⊆ A of arguments, if B ⊇ B, then: for all S ∈ {CO, PR, GR, ID} that satisﬁes the directionality criterion, the justiﬁcation status of each argument of B evaluated with respect to ES (AF ↓ rlvtAF (B) ) is the same as the one evaluated with respect to ES (AF↓ rlvtAF (B ) ). Proof. Obvious. Example 4. According to Example 3, it holds that B1 ⊆ B2 , and rlvtAF2 (B1 ) ⊆ rlvtAF2 (B2 ). It is easy to verify that the justiﬁcation status of argument α2 evaluated with respect to EPR (AF2 ↓ rlvtAF2 (B1 ) ) (i.e., credulously justiﬁed) is the same as the one evaluated with respect to EPR (AF2↓ rlvtAF2 (B2 ) ). Second, given an argumentation framework AF = A, R and two subsets B, B ⊆ A of arguments, the combinability of partial semantics of argumentation can be informally expressed as: if rlvtAF (B) ∩ rlvtAF (B ) = ∅, then the combination of the partial semantics of AF with respect to B and the one with respect to B is equal to the partial semantics of AF with respect to B ∪ B . This property indicates that after we get the partial semantics of AF with respect to some subsets of A respectively, we may get the partial semantics of a larger subset of arguments by means of semantics combination. Let us consider the following example: Example 5. Let AF3 = A3 , R3 be an argumentation framework (Figure 3), in which A3 = {α1 , . . . , α8 } and R3 = {(α1 , α2 ), (α2 , α1 ), (α2 , α3 ), (α3 , α4 ), (α5 , α4 ), (α5 , α6 ), (α6 , α5 ), (α6 , α7 ), (α8 , α7 )}. Let D1 = {α2 , α3 }, D2 = {α5 } and D3 = {α2 , α5 , α8 }. After computing the partial semantics of AF3 with respect to D1 and D2 respectively, we may use them in computing the partial semantics of AF3 with respect to D3 : Let D1 = {α2 }. Since D1 ⊆ D1 , according to Theorem 1, it holds that ES (AF3↓ rlvtAF3 (D1 ) ) = {E ∩ rlvtAF3 (D1 ) | E ∈ ES (AF3↓ rlvtAF3 (D1 ) )}. Let D3 = {α8 }. Since rlvtAF3 (D1 ), rlvtAF3 (D2 ) and rlvtAF3 (D3 ) are pairwise disjoint, we may obtain ES (AF3 ↓ rlvtAF3 (D1 ∪D2 ∪D3 ) ) by combining ES (AF3 ↓ rlvtAF3 (D1 ) ),

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ES (AF3↓ rlvtAF3 (D2 ) ), and ES (AF3↓ rlvtAF3 (D3 ) ). Since D3 = D1 ∪ D2 ∪ D3 , when computing the partial semantics of AF3 with respect to D3 , only the partial semantics of AF3 with respect to D3 is necessary to be computed, while those with respect to D1 and D2 respectively are obtained from the previous results.

α1 f

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α2

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α5 f

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Fig. 3. Argumentation framework AF3

Formally, the combinability of partial semantics of argumentation is formulated by the following theorem: Theorem 2. Given an argumentation framework AF = A, R and two subsets B, B ⊆ A of arguments, if rlvtAF (B) ∩ rlvtAF (B ) = ∅, then: for all S ∈ {CO, PR, GR, ID} that satisﬁes the directionality criterion, it holds that: ES (AF↓ rlvtAF (B∪B ) ) ≡ {E1 ∪ E2 | E1 ∈ ES (AF↓ rlvtAF (B) ), E2 ∈ ES (AF↓ rlvtAF (B ) )}

(3)

Proof. ⇒: For all E ∈ ES (AF↓ rlvtAF (B∪B ) ), according to Deﬁnition 7, it holds that E1 = E ∩ rlvtAF (B) ∈ ES (AF ↓ rlvtAF (B) ) and E2 = E ∩ rlvtAF (B ) ∈ ES (AF↓ rlvtAF (B ) ). ⇐: We prove this part under four diﬀerent argumentation semantics: – First, under complete semantics, for all E1 ∈ ECO (AF ↓ rlvtAF (B) ) and E2 ∈ ECO (AF ↓ rlvtAF (B ) ), according to Deﬁnitions 2 and 3, we may infer that E1 ∪ E2 ∈ ECO (AF↓ rlvtAF (B∪B ) ), in that the following four conditions hold: (1) E1 ∪ E2 ⊆ rlvtAF (B ∪ B ), since: E1 ⊆ rlvtAF (B), E2 ⊆ rlvtAF (B ), and rlvtAF (B) ∪ rlvtAF (B ) ⊆ rlvtAF (B ∪ B ) (according to Lemma 1, rlvtAF (B) ⊆ rlvtAF (B ∪ B ) and rlvtAF (B ) ⊆ rlvtAF (B ∪ B )). (2) E1 ∪E2 is conﬂict-free, in that E1 and E2 are conﬂict-free, and there exist no interactions between E1 and E2 (since rlvtAF (B) ∩ rlvtAF (B ) = ∅). (3) ∀α ∈ E1 ∪ E2 , α is acceptable with respect to E1 ∪ E2 , since: if α ∈ E1 (or α ∈ E2 ), then α is acceptable with respect to E1 (respectively E2 ); since α is not attacked by the arguments outside rlvtAF (B) (respectively, rlvtAF (B )), when we consider α in rlvtAF (B ∪ B ), it is still acceptable with respect to E1 ⊆ E1 ∪ E2 (respectively, E2 ⊆ E1 ∪ E2 ). (4) ∀α ∈ rlvtAF (B ∪ B ), if α is acceptable with respect to E1 ∪ E2 , then α ∈ E1 ∪E2 , in that: since α ∈ rlvtAF (B ∪B ) ⊆ rlvtAF (B)∪rlvtAF (B ) (according to Deﬁnition 9), there are the following two cases: (a) if α ∈ rlvtAF (B), then: since α is acceptable with respect to E1 ∪ E2 and only aﬀected by the arguments in rlvtAF (B) (since rlvtAF (B) ∩ rlvtAF (B ) = ∅), α is acceptable with respect to (E1 ∪E2 )∩rlvtAF (B) = E1 ; since every argument that is acceptable with respect to E1 is in E1 , it holds that α is in E1 ⊆ E1 ∪ E2 ; and

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(b) if α ∈ rlvtAF (B ), then: since α is acceptable with respect to E1 ∪E2 and only aﬀected by the arguments in rlvtAF (B ), α is acceptable with respect to (E1 ∪ E2 ) ∩ rlvtAF (B ) = E2 ; since every argument that is acceptable with respect to E2 is in E2 , it holds that α is in E2 ⊆ E1 ∪ E2 . – Second, under preferred semantics, for all E1 ∈ EPR (AF ↓ rlvtAF (B) ) and E2 ∈ EPR (AF ↓ rlvtAF (B ) ), since a preferred extension is also a complete extension, based on the proof in the case of complete semantics, we only need to prove that E1 ∪ E2 is a maximal (with respect to set-inclusion) complete extension of AF↓ rlvtAF (B∪B ) . Assume that E1 ∪ E2 is not a maximal complete extension. Then, ∃E ⊆ rlvtAF (B ∪ B ), such that E is a preferred extension of AF ↓ rlvtAF (B∪B ) and E ⊇ (E1 ∪ E2 ). Let Φ = E \ (E1 ∪ E2 ). If Φ = ∅, then E = E1 ∪ E2 . In this case, E1 ∪ E2 is a preferred extension. So, we need only to discuss the case when Φ = ∅. Let Φ = Φ ∩ rlvtAF (B) and Φ = Φ ∩ rlvtAF (B ). Since Φ ⊆ rlvtAF (B ∪ B ), it holds that Φ ∪ Φ = Φ = ∅. When Φ = ∅, since E1 ⊆ rlvtAF (B) and E2 ∩ rlvtAF (B) = ∅, it holds that E1 = (E1 ∪ E2 ) ∩ rlvtAF (B). So, E1 ∪ Φ = ((E1 ∪ E2 ) ∩ rlvtAF (B)) ∪ (Φ ∩ rlvtAF (B)) = ((E1 ∪ E2 ) ∪ Φ) ∩ rlvtAF (B) = E ∩ rlvtAF (B). According to Deﬁnition 7, it holds that E ∩ rlvtAF (B) is a preferred extension of AF ↓ rlvtAF (B) . As a result, E1 is not a preferred extension, contradicting E1 ∈ EPR (AF↓ rlvtAF (B) ). Meanwhile, when Φ = ∅, it turns out that E2 is not a preferred extension, contradicting E2 ∈ EPR (AF↓ rlvtAF (B ) ). – Third, under grounded semantics, for all E1 ∈ EGR (AF↓ rlvtAF (B) ) and E2 ∈ EGR (AF↓ rlvtAF (B ) ), since a grounded extension is also a complete extension, based on the proof in the case of complete semantics, we only need to prove that E1 ∪ E2 is a minimal (with respect to set-inclusion) complete extension of AF↓ rlvtAF (B∪B ) . Assume that E1 ∪ E2 is not a minimal complete extension. Then, ∃E ⊆ rlvtAF (B ∪ B ), such that E is a grounded extension of AF ↓ rlvtAF (B∪B ) and E ⊆ E1 ∪ E2 . Let Φ = (E1 ∪ E2 ) \ E . If Φ = ∅, then E = E1 ∪ E2 . In this case, E1 ∪ E2 is a grounded extension. So, we need only to discuss the case when Φ = ∅. Let Φ = Φ ∩ rlvtAF (B) and Φ = Φ ∩ rlvtAF (B ). Since Φ ⊆ rlvtAF (B ∪ B ), it holds that Φ ∪ Φ = Φ = ∅. When Φ = ∅, since E1 ⊆ rlvtAF (B) and E2 ∩ rlvtAF (B) = ∅, it holds that E1 = (E1 ∪ E2 ) ∩ rlvtAF (B). So, E1 \ Φ = ((E1 ∪ E2 ) ∩ rlvtAF (B)) \ (Φ ∩ rlvtAF (B)) = ((E1 ∪ E2 ) \ Φ) ∩ rlvtAF (B) = E ∩ rlvtAF (B). According to Deﬁnition 7, it holds that E ∩ rlvtAF (B) is a grounded extension of AF ↓ rlvtAF (B) . As a result, E1 is not a grounded extension, contradicting E1 ∈ EGR (AF↓ rlvtAF (B) ). Meanwhile, when Φ = ∅, it turns out that E2 is not a grounded extension, contradicting E2 ∈ EGR (AF↓ rlvtAF (B ) ). – Fourth, under ideal semantics, for all E1 ∈ EID (AF ↓ rlvtAF (B) ) and E2 ∈ EID (AF ↓ rlvtAF (B ) ), we need to prove that: (1) E1 ∪ E2 is admissible, (2)

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E1 ∪ E2 is contained in every preferred set of arguments, and (3) E1 ∪ E2 is the maximal (with respect to set-inclusion) ideal set. • According to the proof in the case of complete semantics, it holds that E1 ∪ E2 is also admissible (i.e., it holds that E1 ∪ E2 is conﬂict-free, and every argument in E1 ∪ E2 is acceptable with respect to E1 ∪ E2 ). • Let E1 ∈ EPR (AF ↓ rlvtAF (B) ) and E2 ∈ EPR (AF ↓ rlvtAF (B ) ) be preferred extensions. According to the proof in the case of preferred semantics, it holds that E1 ∪ E2 is a preferred extension of AF ↓ rlvtAF (B∪B ) . Since E1 is contained in E1 and E2 is contained in E2 , it holds that E ⊆ (E1 ∪ E2 ). In other words, E is contained in every preferred set of arguments (E1 ∪ E2 ) ∈ EPR (AF↓ rlvtAF (B∪B ) ). • Assume that E1 ∪E2 is not the maximal ideal set. Then, ∃E ⊆ rlvtAF (B∪ B ), such that E is the ideal extension of AF ↓ rlvtAF (B∪B ) and E ⊇ E1 ∪ E2 . Let Φ = E \ (E1 ∪ E2 ). If Φ = ∅, then E = E1 ∪ E2 . In this case, E1 ∪ E2 is the ideal extension. So, we need only to discuss the case when Φ = ∅. Let Φ = Φ ∩ rlvtAF (B) and Φ = Φ ∩ rlvtAF (B ). Since Φ ⊆ rlvtAF (B ∪ B ), it holds that Φ ∪ Φ = Φ = ∅. When Φ = ∅, since E1 ⊆ rlvtAF (B) and E2 ∩ rlvtAF (B) = ∅, it holds that E1 = (E1 ∪E2 )∩rlvtAF (B). So, E1 ∪Φ = ((E1 ∪E2 )∩rlvtAF (B))∪ (Φ ∩ rlvtAF (B)) = ((E1 ∪ E2 ) ∪ Φ) ∩ rlvtAF (B) = E ∩ rlvtAF (B). According to Deﬁnition 7, it holds that E ∩rlvtAF (B) is an ideal extension of AF↓ rlvtAF (B) . As a result, E1 is not the ideal extension, contradicting E1 ∈ EID (AF↓ rlvtAF (B) ). Meanwhile, when Φ = ∅, it turns out that E2 is not the ideal extension, contradicting E2 ∈ EID (AF↓ rlvtAF (B ) ). In this subsection, we have formulated two properties of the partial semantics of argumentation. The basic ideas can be further explained by the following example. Example 6. Continue Example 5. Under preferred semantics, we ﬁrst compute the partial semantics of AF3 with respect to D1 and D2 respectively: EPR (AF3↓ rlvtAF3 (D1 ) ) = {E1,1 , E1,2 }, where E1,1 = {α1 , α3 }, E1,2 = {α2 }

EPR (AF3↓ rlvtAF3 (D2 ) ) = {E2,1 , E2,2 }, where E2,1 = {α5 }, E2,2 = {α6 }

Second, according to the monotonicity of partial semantics of argumentation (Theorem 1), we get the partial semantics of AF3 with respect to D1 : , E1,2 }, where EPR (AF3↓ rlvtAF3 (D1 ) ) = {E1,1

E1,1 = E1,1 ∩ rlvtAF3 (D1 ) = {α1 }, E1,2 = E1,2 ∩ rlvtAF3 (D1 ) = {α2 }

Third, we compute the partial semantics of AF3 with respect to D3 : }, where E3,1 = {α8 } EPR (AF3↓ rlvtAF3 (D3 ) ) = {E3,1

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Fourth, we get the the partial semantics of AF3 with respect to D3 by means of semantics combination (Theorem 2): EPR (AF3↓ rlvtAF3 (D3 ) ) = {E3,1 , E3,2 , E3,3 , E3,4 }, where E3,1 = E1,1 ∪ E2,1 ∪ E3,1 = {α1 , α5 , α8 }

∪ E2,2 ∪ E3,1 = {α1 , α6 , α8 } E3,2 = E1,1 = {α2 , α5 , α8 } E3,3 = E1,2 ∪ E2,1 ∪ E3,1 E3,4 = E1,2 ∪ E2,2 ∪ E3,1 = {α2 , α6 , α8 }

Example 6 shows that by using the deﬁnition of partial semantics of argumentation and its properties, we not only may evaluate the justiﬁcation status of a subset of arguments independently, but also can make use of the computation results that have been obtained previously.

4

Conclusions and Future Work

In this paper, we have proposed an original approach to evaluate the justiﬁcation status of a part of arguments in an argumentation framework, without evaluating the justiﬁcation status of the whole set of arguments in the argumentation framework. The main contributions of this paper are two-fold: (i) Based on the notion of unattacked sets of an argumentation framework and the directionality criterion of argumentation semantics, the deﬁnitions of partial semantics of argumentation are provided. (ii) Two basic properties of partial semantics (monotonicity and combinability) of argumentation are formulated. Based on the basic theory proposed in this paper, our future work will include the following two parts. On the one hand, we will further study the properties of partial semantics of argumentation. In this paper, when formulating the combinability of partial semantics of argumentation, we only consider the case where the two sets of relevant arguments of two subsets of arguments are disjoint. If we relax this constraint, the combination of the partial semantics of argumentation will be more complex. Meanwhile, in this paper, we only discuss the partial semantics of a static argumentation framework, while leaving that of a dynamic argumentation framework untouched. This part of work will be included in the extended version of this paper. On the other hand, we will develop eﬃcient proof theories and algorithms, on the basis of some existing work [12, 15, 23, 18, 25, 20]. Acknowledgement. We are grateful to the anonymous reviewers of this paper for their valuable comments and suggestions, which helped to improve our work. The research reported in this paper was ﬁnancially supported by the National Natural Science Foundation of China (No.60773177) and Zhejiang Provincial Natural Science Foundation of China (No.Y1100036).

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19. Kakas, A.C., Miller, R., Toni, F.: An argumentation framework for reasoning about actions and change. In: Gelfond, M., Leone, N., Pfeifer, G. (eds.) LPNMR 1999. LNCS (LNAI), vol. 1730, pp. 78–91. Springer, Heidelberg (1999) 20. Kim, E., Ordyniak, S., Szeider, S.: Algorithms and complexity results for persuasive argumentation. Artiﬁcial Intelligence 175(9-10), 1722–1736 (2011) 21. Liao, B., Huang, H.: Angle: An autonomous, normative and guidable agent with changing knowledge. Information Sciences 180(17), 3117–3139 (2010) 22. Liao, B., Jin, L., Koons, R.: Dynamics of argumentation systems: A division-based method. Artiﬁcial Intelligence 175(11), 1790–1814 (2011) 23. Modgil, S., Caminada, M.: Proof theories and algorithms for abstract argumentation frameworks. In: Rahwan, I., Simari, G.R. (eds.) Argumentation in Artiﬁcial Intelligence, pp. 105–129. Springer, Heidelberg (2009) 24. Modgil, S., Luck, M.: Argumentation based resolution of conﬂicts between desires and normative goals. In: Rahwan, I., Moraitis, P. (eds.) ArgMAS 2008. LNCS(LNAI), vol. 5384, pp. 19–36. Springer, Heidelberg (2009) 25. Wakaki, T., Nitta, K.: Computing argumentation semantics in answer set programming. In: Hattori, H., Kawamura, T., Id´e, T., Yokoo, M., Murakami, Y. (eds.) JSAI 2008. LNCS(LNAI), vol. 5447, pp. 254–269. Springer, Heidelberg (2009)

A Dynamic Logic of Knowledge, Graded Beliefs and Graded Goals and Its Application to Emotion Modelling Emiliano Lorini Universit´e de Toulouse, IRIT-CNRS, France [email protected]

Abstract. The paper introduces a logic which allows to represent diﬀerent kinds of mental states of an agent such as knowledge, graded belief, and graded goal, and the notion of epistemic action (as the action of learning that a certain fact ϕ is true.) The logic is applied to the formalization of expectation-based emotions such as hope, fear, disappointment and relief, and of their intensity.

1

Introduction

In this paper, I will present a logic called DL-KGBG (Dynamic Logic of Knowledge, Graded Beliefs and Goals) which allows: (1) to express that a given epistemic action is going to occur; (2) to describe the eﬀects of an epistemic action on an agent’s mental state; (3) to represent diﬀerent kinds of mental attitudes including knowledge, graded beliefs (i.e., believing with a certain strength that a given proposition is true) and graded goals (i.e., wishing with a certain strength a given proposition to be true.) An epistemic action is nothing but the mental action (or process) of learning that a given proposition is true, of changing the agent’s beliefs in the light of a new incoming evidence. I here follow the tradition of Dynamic Epistemic Logic (DEL) [8] in modelling epistemic actions as basic operations of model transformation in the semantics. In the second part of the paper, the logic DL-KGBG will be applied to the formalization of expectation-based emotions such as hope, fear, disappointment and relief. It will be shown that this logic allows to represent the notion of emotion intensity (e.g., how much an agent feels happy or sad, disappointed or relieved, etc.), which is ignored by most of current logical models of emotions [15,1,14].1 Following previous works on the cognitive theory of expectations [6], an expectation-based emotion is here conceived as an emotion that an agent experiences when having an expectation about a certain fact ϕ, that is when: (1) believing that ϕ is true with a certain strength, but envisaging the possibility 1

The only exception is [21]. However, diﬀerent from the approach presented here, Meyer and coll. do not study the cognitive variables determining intensities (e.g., the strength of an agent’s beliefs and the importance of his goals) but instead propose a function describing how the intensity of an emotion decreases over time.

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that ϕ could be false, and (2) either, having the goal that ϕ is true (positive expectation) or, having the goal that ϕ is false (negative expectation). The rest of the paper is organized as follows. Section 2 will be devoted to present the logic DL-KGBG. A complete axiomatization as well as a decidability result for this logic will be given in Section 3. In Section 4, the logic DL-KGBG will be applied to the formalization of expectation-based emotions and of their intensity.

2

The Logical Framework

This section presents the syntax and the semantics of the logic DL-KGBG, as well as a complete axiomatization and a decidability result for this logic. 2.1

Syntax

Assume a countable set of atomic propositions Prop = {p, q, . . .} and a ﬁnite set of natural numbers Num = {x ∈ N : 0 ≤ x ≤ max} with max ∈ N. The language L of DL-KGBG is deﬁned by the following grammar in Backus-Naur Form (BNF): Act : α ::= ∗ϕ Atm : χ ::= p | after|α | exch | desh Fml : ϕ ::= χ | ¬ϕ | ϕ ∧ ϕ | [K]ϕ | [α]ϕ where p ranges over Prop, h ranges over Num, α ranges over Act, and ranges over the set of epistemic action sequences Act ∗ . The other Boolean constructions , ⊥, ∨, → and ↔ are deﬁned from p, ¬ and ∧ in the standard way. I deﬁne the set of objective facts Obj as the set of all Boolean combinations of atomic propositions. An epistemic action ∗ϕ in Act is the mental action (or process) of learning that ϕ is true. As I will show in Section 2.2, technically this amounts to an operation of beliefs’ conditionalization in Spohn’s sense [19]. Act ∗ is the set of epistemic action sequences. An epistemic action sequence is of the form α1 ; . . . ; αn where “;” stands for concatenation. I note nil the empty sequence of epistemic actions. Atm contains special constructions of diﬀerent types which are used to represent both the objective world as well as the mental state of an agent. The formula after|α is a special atom that is read “after the sequence of epistemic actions , the epistemic action α will occur”. The atomic formula afternil |α speciﬁes the actual epistemic action that is going to occur next. Therefore, I deﬁne: def

occα = afternil |α where occα has to be read “the epistemic action α is going to occur”. The special atoms exch are used to identify the degree of plausibility of a given world for the agent. Starting from [10], ranking among possible worlds have been extensively used in belief revision theory. I here use the notion of plausibility ﬁrst introduced by Spohn [19]. Following Spohn’s theory, the worlds that are assigned the smallest numbers are the most plausible, according to the beliefs of the

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individual. That is, the number h assigned to a given world rather captures the degree of exceptionality of this world, where the exceptionality degree of a world is nothing but the opposite of its plausibility degree (i.e., the exceptionality degree of a world decreases when its plausibility degree increases.) Therefore, formula exch can be read alternatively as “the current world has a degree of exceptionality h” or “the current world has a degree of plausibility max−h”. The special atoms desh are used to identify the degree of desirability (or the degree of goodness) of a given world for the agent. Contrary to plausibility, the worlds that are assigned the biggest numbers are the most desirable for the agent. The degree of undesirability (or degree of a badness) of a given world is the opposite of its desirability degree. Therefore, formula desh can be read alternatively as “the current world has a degree of desirability h” or “the current world has a degree of undesirability max − h”. The formula [α]ϕ has to be read “after the occurrence of epistemic action α, ϕ will be true”, while [K]ϕ has to be read “the agent knows that ϕ is true”. This concept of knowledge is the standard S5-notion, partition-based and fully introspective. As I will show in the next Section 2.2, the operator [K] captures a form of ‘absolutely unrevisable belief’, that is, a form of belief which is stable under belief revision with any new evidence. A similar property for the notion of knowledge has been advanced by the so-called defeasibility (or stability) theory of knowledge [11,20,18]. According to this theory, a given piece of information ϕ is part of the agent’s knowledge only if the agent’s justiﬁcation to believe that ϕ is true is suﬃciently strong that it is not capable of being defeated by evidence that the agent does not possess. As pointed out by [4], two diﬀerent interpretations of the term ‘evidence’ have been given in the context of this theory, each giving a diﬀerent interpretation of what knowledge is. The ﬁrst one deﬁnes knowledge as a form of belief which is stable under belief revision with ‘any piece of true information’, while the second one gives a stronger deﬁnition of knowledge as a form of belief which is stable under belief revision with ‘any piece of information’. The concept formalized by the operator [K] captures this latter form of knowledge in a stronger sense. 2.2

Semantics

Definition 1 (Model). DL-KGBG-models are tuples M = W, ∼, κexc , κdes , P, V where: – W is a nonempty set of possible worlds or states; – ∼ is an equivalence relation between worlds in W ; – κexc : W −→ Num and κdes : W −→ Num are functions from the set of possible worlds into the ﬁnite set of natural numbers Num; – P : W × Act ∗ −→ Act is a partial function called protocol, mapping worlds and action sequences to actions; – V : W −→ 2Prop is a valuation function. As usual, p ∈ V(w) means that proposition p is true at world w. The equivalence relation ∼, which is used to interpret the epistemic operator [K], can be viewed

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as a function from W to 2W . Therefore, we can write ∼(w) = {v ∈ W : w ∼ v}. The set ∼ (w) is the agent’s information state at world w: the set of worlds that the agent considers possible at world w or, the set of worlds that the agent cannot distinguish from world w. As ∼ is an equivalence relation, if w ∼ v then the agent has the same information state at w and v (i.e., the agent has the same knowledge at w and v.) The function κexc represents a plausibility grading of the possible worlds and is used to interpret the atomic formulas exch . κexc (w) = h means that, according to the agent the world w has a degree of exceptionality h or, alternatively, according to the agent the world w has a degree of plausibility max−h. (Remember that the degree of plausibility of a world is the opposite of its exceptionality degree.) The function κexc allows to model the notion of belief: among the worlds the agent cannot distinguish from a given world w (i.e., the agent’s information state at w), there are worlds that the agent considers more plausible than others. For example, suppose that ∼ (w) = {w, v, u}, κexc (w) = 2, κexc (u) = 1 and κexc (v) = 0. This means that at world w the agent cannot distinguish the three worlds w, v and u, that is, {w, v, u} is the set of worlds that the agent considers possible at world w. Moreover, according to the agent, the world v is strictly more plausible than the world u and the world u is strictly more plausible than the world w (as max − 0 > max − 1 > max − 2.) The function κdes is used to interpret the atomic formulas desh . κdes (w) = h means that, according to the agent the world w has a degree of goodness (or desirability) h or, alternatively, according to the agent the world w has a degree of badness (or undesirability) max−h. (Remember that the degree of undesirability of a world is the opposite of its desirability degree.) Finally, the protocol function P is used to interpret the atomic formulas after|α . A similar notion of protocol has been studied by [5]. For every world w ∈ W , P(w, ) = α means that, at world w, the epistemic action α will occur after the sequence of epistemic actions . For example, imagine that the agent wakes up in the morning. Then P(w, ∗8pm; ∗rain) = ∗earthquake means that, at world w, after learning in sequence that ‘it is 8 p.m.’ (by looking at the alarm) and that ‘it is raining outside’ (by opening the window), the agent (will switch on the radio) and will learn that ‘during the night an earthquake of magnitude 5 has occurred’. P(w, nil ) = α means that, at world w, the epistemic action α will occur next. P is supposed to be a partial function as I want to allow states in which no action occurs. DL-KGBG-models are supposed to satisfy the following additional normality constraint for the plausibility grading. (NORM κexc ) for every w ∈ W , there is v such that w ∼ v and κexc (v) = 0. Definition 2 (Truth conditions). Given a DL-KGBG-model M , a world w and a formula ϕ, M, w |= ϕ means that ϕ is true at world w in M . The rules deﬁning the truth conditions of formulas are: – M, w |= p iﬀ p ∈ V(w) – M, w |= after |α iﬀ P(w, ) is deﬁned and P(w, ) = α

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M, w |= exch iﬀ κexc (w) = h M, w |= desh iﬀ κdes (w) = h M, w |= ¬ϕ iﬀ not M, w |= ϕ M, w |= ϕ ∧ ψ iﬀ M, w |= ϕ and M, w |= ψ M, w |= [K]ϕ iﬀ M, v |= ϕ for all v with w ∼ v M, w |= [∗ϕ]ψ iﬀ M ∗ϕ , w |= ψ

where the updated model M ∗ϕ is deﬁned according to the Deﬁnition 9 below. Following [19], I extend the exceptionality degree of a possible world to a plausibility/exceptionality degree of a formula viewed as a set of worlds. Definition 3 (Exceptionality degree of a formula). The exceptionality degree of a formula ϕ at world w, noted κw exc (ϕ), is deﬁned as follows: def κw exc (ϕ) = min{κexc (v) : M, v |= ϕ and w ∼ v}. As expected, the plausibility degree of a formula ϕ, noted κw plaus (ϕ), is deﬁned as max − κw exc (ϕ). I do a similar manipulation for the desirability degree of a formula. Definition 4 (Desirability degree of a formula). The desirability degree of a formula ϕ at world w, noted κw des (ϕ), is deﬁned as follows: def κw (ϕ) = min{κ des (v) : M, v |= ϕ and w ∼ v}. des As expected, the undesirability degree of a formula ϕ, noted κw undes (ϕ), is deﬁned as max − κw des (ϕ). Again following [19], I deﬁne the concept of belief as a formula which is true in all worlds that are maximally plausible (or minimally exceptional). Definition 5 (Belief ). At world w the agent believes that ϕ is true if and only if, for every v such that w ∼ v, if κexc (v) = 0 then M, v |= ϕ. The following concept of graded belief is taken from [12]. I say that at world w the agent believes that ϕ with strength at least h if and only if, all possible worlds in which ϕ is false are exceptional at least degree h (or all possible worlds in which ϕ is false are plausible at most degree max − h.) Definition 6 (Graded belief ). At world w the agent believes that ϕ with strength at least h if and only if, κw exc (¬ϕ) ≥ h. I deﬁne the following concept of strong (or certain) belief: an agent has the strong belief that ϕ if and only if either he knows that ϕ is true (i.e., he has an unrevisable belief that ϕ is true) or he believes that ϕ is true with maximal strength max. Definition 7 (Strong belief ). At world w the agent strongly believes that ϕ (or at w the agent is certain that ϕ is true) if and only if, M, v |= ϕ for all v with w ∼ v or κw exc (¬ϕ) = max.

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The following concept of graded goal is the motivational counterpart of the notion of graded belief. This concept has not been studied before in the logical literature. I say that at world w the agent wants (or wishes) ϕ to be true with strength at least h if and only if, all possible worlds in which ϕ is true are desirable at least degree h (or all possible worlds in which ϕ is true are undesirable at most degree max − h.) This implies that, all possible worlds which are desirable at most degree h − 1 satisfy ¬ϕ (or all possible worlds which are undesirable at least degree max − (h − 1) satisfy ¬ϕ.) Definition 8 (Graded goal). At world w the agent wants/wishes ϕ to be true with strength at least h (or the agent has the goal that ϕ with strength at least h) if and only if, κw des (ϕ) ≥ h. The reason why the deﬁnition of graded goal is not symmetric to the deﬁnition of graded belief is that these two concepts satisfy diﬀerent logical properties. As I will show below in Section 2.3, Deﬁnition 6 and Deﬁnition 8 allow to capture interesting diﬀerences between graded belief and graded goal, especially on the way they distribute over conjunction and over disjunction. As the following proposition highlights, the concepts of belief, graded belief, strong belief and graded goal semantically deﬁned in Deﬁnitions 5-8 are all syntactically expressible in the logic DL-KGBG. Proposition 1. For every DL-KGBG-model M and world w: 1. at w the agent believes that ϕ is true if and only if M, w |= [K](exc0 → ϕ) 2. at w the agent believes that ϕ is true with strength at least h if and only if M, w |= l∈Num:l≥h (K (excl ∧ ¬ϕ) ∧ k∈Num:k
Belϕ = [K](exc0 → ϕ) def Bel≥h ϕ = (K (excl ∧ ¬ϕ) ∧ l∈Num:l≥h def

SBelϕ =

def

[K](exck → ϕ))

k∈Num:k
[K](exck → ϕ)

k∈Num:k<max

Goal≥h ϕ =

(K (desl ∧ ϕ) ∧

l∈Num:l≥h

[K](desk → ¬ϕ))

k∈Num:k
where Belϕ, Bel≥h ϕ, SBelϕ and Goal≥h ϕ respectively mean: “the agent believes that ϕ”, “the agent believes that ϕ with strength at least h”,2 “the agent strongly 2

Similar operators for graded belief are studied in [12,3,7].

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believes that ϕ” and “the agent wants/wishes ϕ to be true with strength at least h”. The following two abbreviations deﬁne the concept of “believing that ϕ exactly with strength h” (Belh ϕ) and the concept of “wanting/wishing ϕ to be true exactly with strength h” (Goalh ϕ). For every h < max: Belh ϕ = Bel≥h ϕ ∧ ¬Bel≥h+1 ϕ def

Goalh ϕ = Goal≥h ϕ ∧ ¬Goal≥h+1 ϕ def

In order to have a uniform notation, I use Bel≥max ϕ and Belmax ϕ on the one hand, Goal≥max ϕ and Goalmax ϕ on the other, as interchangeable expressions. Definition 9 (Update). Given a DL-KGBG-model M = W, ∼, κexc , κdes , P, V , M ∗ϕ is the model such that: W ∗ϕ ∼∗ϕ

=W =∼ κexc (w) − κw if M, w |= ϕ exc (ϕ) = for all w, κ∗ϕ exc (w) Cut max (κexc (w) + Δ) if M, w |= ¬ϕ =κ for all w, κ∗ϕ des (w) des (w) α if P(w, ∗ϕ;) = α for all w, P ∗ϕ (w, ) = undeﬁned if P(w, ∗ϕ;) is undeﬁned =V V ∗ϕ where Δ ∈ Num \ {0} and Cut max (x) =

x max

if 0 ≤ x ≤ max if x > max

An epistemic action ∗ϕ makes the protocol P to advance one step forward (see the deﬁnition of P ∗ϕ .) That is, if α will occur after the sequence of events ∗ϕ; then, after the occurrence of the epistemic action ∗ϕ, it is the case that α will occur after the sequence of events . As epistemic actions only aﬀect the agent’s beliefs and do not aﬀect the objective world, the valuation function V is not altered by them (see the deﬁnition of V ∗ϕ .) Moreover, the epistemic action ∗ϕ modiﬁes the plausibility ordering (see the deﬁnition of κ∗ϕ exc ) but does not modify the agent’s information state (see the deﬁnition of ∼∗ϕ ) nor the desirability ordering (see the deﬁnition of κ∗ϕ des .) In particular, the epistemic action of learning that ϕ is true induces a kind of belief conditionalization in Spohn’s sense [19]. The plausibility ranking over possible worlds is updated as follows. – For every world w in which ϕ is true, the degree of exceptionality of w decreases from κexc (w) to κexc (w)−κw exc (ϕ), which is the same thing as saying that, degree of plausibility of w increases from max − κexc (w) to max − w (κexc (w) − κw exc (ϕ)). (Note that, by Deﬁnition 3, we have κexc (w) − κexc (ϕ) ≤ κexc (w).)

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– For every world w in which ϕ is false, the degree of exceptionality of w increases from κexc (w) to Cut max (κexc (w) + Δ), which is the same thing as saying that, degree of plausibility of w decreases from max − κexc (w) to max − Cut max (κexc (w) + Δ). Cut max is a minor technical device, taken from [3], which ensures that the new plausibility assignment ﬁts into the ﬁnite set of natural numbers Num. The parameter Δ is a conservativeness index which captures the agent’s disposition to radically change his beliefs in the light of a new evidence. More precisely, the higher is the index Δ, and the higher is the agent’s disposition to decrease the plausibility degree of those worlds in which the learnt fact ϕ is false. (When Δ = max, the agent is minimally conservative.) I assume that Δ is diﬀerent from 0 in order to ensure that, after learning that p is true, the agent will believe p. That is, from Δ ∈ Num \{0} it follows that [∗p]Belp is valid for every proposition p ∈ Prop. Note that the operation ∗ϕ preserves the constraints on DL-KGBGmodels: if M is a DL-KGBG-model then M ∗ϕ is a DL-KGBG-model too. In the sequel I write |=DL-KGBG ϕ to mean that ϕ is valid in DL-KGBG (ϕ is true in all DL-KGBG-models.) 2.3

Some Properties of Mental Attitudes

The following are some interesting examples of validity. For every h, k ∈ Num we have: |=DL-KGBG Bel≥h ϕ → Bel≥k ϕ if h ≥ k (1) |=DL-KGBG Goal≥h ϕ → Goal≥k ϕ if h ≥ k (2) |=DL-KGBG [K]ϕ → ¬Bel≥h ϕ (3) |=DL-KGBG SBelϕ ↔ ([K]ϕ ∨ Bel≥max ϕ) (4) |=DL-KGBG ¬(Belϕ ∧ Bel¬ϕ) (5) |=DL-KGBG [∗ϕ]Belϕ if ϕ ∈ Obj (6) |=DL-KGBG [K]ϕ → [∗ψ][K]ϕ if ϕ ∈ Obj (7) |=DL-KGBG (Bel≥h ϕ ∧ Bel≥k ψ) → Bel≥min{h,k} (ϕ ∧ ψ) (8) |=DL-KGBG (Goal≥h ϕ ∧ Goal≥k ψ) → (Goal≥max{h,k} (ϕ ∧ ψ) ∨ [K](¬ϕ ∨ ¬ψ)) (9) |=DL-KGBG (Bel≥h ϕ ∧ Bel≥k ψ) → (Bel≥max{h,k} (ϕ ∨ ψ) ∨ [K](ϕ ∨ ψ)) (10) |=DL-KGBG (Goal≥h ϕ ∧ Goal≥k ψ) → Goal≥min{h,k} (ϕ ∨ ψ) (11) According to the validity (1), if the agent believes that ϕ with strength at least h and h ≥ k, then he believes that ϕ is true with strength at least k. Validity (2) is the corresponding property for graded goals. The validity (3) highlights that knowledge and graded belief are distinct concepts. According to the validity (4), the agent has the strong belief that ϕ (i.e., the agent is certain that ϕ) if and only if, either he knows that ϕ is true (i.e., he has an unrevisable belief that ϕ is true) or he believes that ϕ with maximal strength max. According to

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the validity (5) (which follows from the normality constraint NORM κexc given in Section 2.2), an agent cannot have inconsistent beliefs. The validity (6) highlights a basic property of belief revision in the sense of AGM theory [2]: if ϕ is an objective fact then, after learning that ϕ is true, the agent believes that ϕ is true. The validity (7) highlights a basic property of knowledge as unrevisable belief (see Section 2.1 for a discussion): if ϕ is an objective fact and the agent knows that ϕ is true then, after learning a new fact ψ, he will continue to know that ϕ is true. In this sense, knowledge is stable under belief revision. According to the validity (8), if the agent believes that ϕ with strength at least h and believes that ψ with strength at least k, then the strength of the belief that ϕ ∧ ψ is at least min{h, k}. According to the validity (10), if the agent believes that ϕ with strength h and believes that ψ with strength k, then either he believes ϕ ∨ ψ with strength at least max{h, k} or he knows that ϕ ∨ ψ. (Remember that knowledge and graded belief are distinct concepts.) The validities (9) and (11) are corresponding principles for goals. According to the validity (9), if the agent wishes ϕ to be true with strength at least h and wishes ψ to be true with strength at least k, then either he wishes ϕ∧ψ to be true with strength at least max{h, k} or he knows that ϕ and ψ are inconsistent facts. According to the validity (11), if the agent wishes ϕ to be true with strength at least h and wishes ψ to be true with strength at least k, then he wishes ϕ ∨ ψ to be true with strength at least min{h, k}. The interesting aspect is that graded goals distribute over conjunction and over disjunction in the opposite way as graded beliefs. Consider for instance the validity (9) and compare it to the validity (8). The joint occurrence of two events ϕ and ψ is less plausible than the occurrence of a single event. This is the reason why in the right side of the validity (8) we have the min. On the contrary, the joint occurrence of two desired events ϕ and ψ is more desirable than the occurrence of a single event. This is the reason why in the right side of the validity (9) we have the max. For example, suppose Peter wishes to go to the cinema in the evening with strength at least h (i.e., Goal≥h goToCinema) and, at the same time, he wishes to spend the evening with his girlfriend with strength at least k (i.e., Goal≥k stayWithGirlfriend .) Then, according to the validity (9), either Peter wishes to to go the cinema with his girlfriend with strength at least max{h, k} (i.e., Goal≥max{h,k} (goToCinema ∧ stayWithGirlfriend )) or Peter knows that going to the cinema with his girlfriend is impossible (i.e., [K](¬goToCinema ∨ ¬stayWithGirlfriend )), perhaps because he knows that his girlfriend has the ﬂu and cannot go out.

3

Decidability and Axiomatization

Let L-KGBG be the fragment of the logic DL-KGBG without dynamic operators, that is, let L-KGBG be the set of formulas deﬁned by the following BNF: Act : α ::= ∗ϕ Atm : χ ::= p | after|α | exch | desh Fml : ϕ ::= χ | ¬ϕ | ϕ ∧ ϕ | [K]ϕ

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where p ranges over Atm, h ranges over Num, α ranges over Act, and ranges over Act ∗ . Proposition 2. The following formulas are DL-KGBG valid for every h, k ∈ Num, α ∈ Act , ∈ Act ∗ . after|α → ¬after|β if α = β h∈Num exch h∈Num desh K exc0 exch → ¬exck if h = k desh → ¬desk if h = k Proposition 3. The following equivalences are DL-KGBG valid for every p ∈ Atm, h ∈ Num, α ∈ Act , ∈ Act ∗ . ↔p ↔ after∗ϕ;|α ↔ ((ϕ ∧ k,l∈Num:k−l=h (exck ∧ Bell ¬ϕ))∨ (¬ϕ ∧ k∈Num:Cut max (k+Δ)=h exck )) ↔ desh [∗ϕ]desh [∗ϕ]¬ψ ↔ ¬[∗ϕ]ψ [∗ϕ](ψ1 ∧ ψ2 ) ↔ ([∗ϕ]ψ1 ∧ [∗ϕ]ψ2 ) [∗ϕ][K]ψ ↔ [K][∗ϕ]ψ

(R.1) [∗ϕ]p (R.2) [∗ϕ]after|α (R.3) [∗ϕ]exch (R.4) (R.5) (R.6) (R.7)

The equivalences of Proposition 3 provide a complete set of reduction axioms for DL-KGBG. Call red the mapping on DL-KGBG formulas which iteratively applies the above equivalences from the left to the right, starting from one of the innermost modal operators. red pushes the dynamic operators inside the formula, and ﬁnally eliminates them when facing an atomic formula. Proposition 4. Let ϕ be a formula in the language of DL-KGBG. Then 1. red(ϕ) has no dynamic operators [α] 2. red(ϕ) ↔ ϕ is DL-KGBG valid 3. red(ϕ) is DL-KGBG valid iﬀ red(ϕ) is L-KGBG valid. Proof. The ﬁrst item is clear. The second item is proved using Proposition 3 and the rule of replacement of equivalents. The last item follows from the second item and the fact that DL-KGBG is a conservative extension of L-KGBG. Theorem 1. Satisﬁability in DL-KGBG is decidable. Proof. We ﬁrst show that the logic L-KGBG is decidable. Note that the problem of satisﬁability in L-KGBG is reducible to the problem of global logical consequence in S5, where the set of global axioms Γ is the set of all formulas of Proposition 2. That is, we have |=L-KGBG ϕ if and only if Γ |=S5 ϕ. We can show that Γ |=S5 ϕ if and only if Γϕ |=S5 ϕ, where Γϕ = Γ \ { after|α → ¬after|β : after|α ∈ SUB (ϕ) or after|β ∈ SUB(ϕ) }

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and SUB (ϕ) is the set of all subformulas of ϕ. In other words, Γϕ is the subset of Γ containing only formulas of the type after|α → ¬after|β which are ϕ-relevant. Observe that every Γϕ is ﬁnite. It is well-known that the problem of global logical consequence in S5 with a ﬁnite number of global axioms is reducible to the problem of satisﬁability in S5. The problem of satisﬁability checking in S5 is decidable. It follows that the problem of satisﬁability checking in the logic L-KGBG is decidable too. red provides an eﬀective procedure for reducing a DL-KGBG formula ϕ into an equivalent L-KGBG formula red(ϕ). As L-KGBG is decidable, DL-KGBG is decidable too. Theorem 2. The validities of DL-KGBG are completely axiomatized by – – – –

4

some axiomatization of S5 for the epistemic operator [K] the schemas of Proposition 2 the reduction axioms of Proposition 3 ϕ↔ψ the inference rule [α]ϕ↔[α]ψ

Formalization of Expectation-Based Emotions

The modal operators of graded belief and graded goal deﬁned above are used here to provide a logical analysis of expectation-based emotions and their intensities. 4.1

Hope and Fear

According to some psychological models [17,16] and computational models [9] of emotions, the intensity of hope with respect to a given event is a monotonically increasing function of: the degree to which the event is desirable and the likelihood of the event. That is, the higher is the desirability of the event ϕ, and the higher is the intensity of the agent’s hope that this event will occur; the higher is the likelihood of the event ϕ, and the higher is the intensity of the agent’s hope that this event will occur. Analogously, the intensity of fear with respect to a given event is a monotonically increasing function of: the degree to which the event is undesirable and the likelihood of the event. There are several possible merging functions which satisfy these properties. For example, I could deﬁne the merging function merge as an average function, according to which the intensity of hope about a certain event ϕ is the average of the strength of the belief that ϕ will occur and the strength of the goal that ϕ will occur. That is, for every h, k ∈ Num representing respectively the strength of the belief and the strength of the goal, I could deﬁne merge(h,k) as h+k 2 . Another possibility is to deﬁne merge as a product function h × k (also used in [9]), according to which the intensity of hope about a certain event ϕ is the product of the strength of the belief that ϕ will occur and the strength of the goal that ϕ will occur. Here I do not choose a speciﬁc merging function, as this would require a serious experimental validation and would much depend on the domain of application in which the formal model has to be used.

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Let me now deﬁne the notions of hope and fear with their corresponding intensities.3 An agent is experiencing a hope with with intensity i about ϕ if and only if there are h, k ∈ Num such that such that h < max, h is the strength to which the agent believes that ϕ is true, k is the strength to which the agent wishes ϕ to be true and merge(h,k) = i: Hopei ϕ =

def

(Belh ϕ ∧ Goalk ϕ)

h,k∈Num:h<max and merge(h,k)=i

The notion of fear can be deﬁned in a similar way, after assuming that if an agent wishes ϕ to be true, then the situation in which ϕ is false is undesirable for him.4 An agent is experiencing a fear with with intensity i about ϕ if and only if there are h, k ∈ Num such that h < max, h is the strength to which the agent believes that ϕ is true, k is the strength to which the agent wishes ϕ to be false and merge(h,k) = i: Feari ϕ =

def

(Belh ϕ ∧ Goalk ¬ϕ)

h,k∈Num:h<max and merge(h,k)=i

In the preceding deﬁnitions of hope and fear, the strength of the belief is supposed to be less than max in order to distinguish hope and fear, which imply some form of uncertainty, from happiness and sadness (or unhappiness) which are based on certainty. In particular, we have that: |=DL-KGBG Hopei ϕ → ¬SBelϕ i

|=DL-KGBG Fear ϕ → ¬SBelϕ

(12) (13)

This means that if an agent hopes (or fears) ϕ to be true, then he is not certain that ϕ (i.e., the agent does not have the strong belief that ϕ.) For example, if I hope that my paper will be accepted at the next LORI III workshop, then it means that I am not certain that my paper will be accepted. The preceding two validities are consistent with Spinoza’s quote “Fear cannot be without hope nor hope without fear”. Indeed, if an agent hopes that ϕ will be true then, according to the validity (12), he envisages the possibility that ϕ will be false. Therefore, he experiences some fear that ϕ will be false. Conversely, if an agent fears that ϕ will be true then, according to the validity (13), he envisages the possibility that ϕ will be false. Therefore, he experiences some hope that ϕ will be false. On the contrary, to feel happy (or sad) about ϕ, the agent should be certain that ϕ is true. For example, if I am happy that my paper has been accepted at the LORI III workshop, then it means that I am certain that my paper has 3

4

Note that hope and fear, and more generally expectations, are not necessarily about a future state of aﬀairs, but they can also be about a present state of aﬀairs or a past state of aﬀairs. For example, I might say ‘I hope that you feel better now!’ or ‘I fear that you did not enjoy the party yesterday night!’. If an agent wishes ϕ to be true, then in the situation in which ϕ is false he will be frustrated. This is the reason why the latter situation is undesirable for the agent.

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been accepted. More precisely, an agent is experiencing happiness (respectively, sadness) with intensity h about ϕ if and only if, the agent strongly believes (is certain) that ϕ is true and h is the strength to which the agent wishes ϕ to be true (respectively, false). That is: Happinessh ϕ = SBelϕ ∧ Goalh ϕ def

Sadnessh ϕ = SBelϕ ∧ Goalh ¬ϕ def

4.2

Disappointment and Relief

An interesting class of expectation-based emotions are those based on the invalidation of either a positive or a negative expectation. In agreement with Ortony et al.’s psychological model [16], I distinguish two types of these emotions: disappointment and relief. Disappointment is the emotion due to the invalidation of an agent’s hope, while relief is the emotion due to the invalidation of an agent’s fear. On the one hand, the higher is the intensity of the invalidated hope that ϕ is true, and the higher is the intensity of the agent’s disappointment. On the other hand, the higher is the intensity of the invalidated fear that ϕ is true, and the higher is the intensity of the agent’s relief. More precisely, an agent is going to experience a disappointment (respectively, a relief) with intensity i about ϕ if and only if the agent hopes (respectively, fears) ϕ to be true with intensity i and the agent is going to learn that ϕ if false: Disappointmenti ϕ = Hopei ϕ ∧ occ∗¬ϕ def

Relief i ϕ = Feari ϕ ∧ occ∗¬ϕ def

It has to be noted that disappointment and relief entail surprise, where surprise is determined by the invalidation of an agent’s belief (see also [13]). That is: Surpriseh ϕ = Belh ϕ ∧ occ∗¬ϕ def

5

Conclusion

In the preceding section I have only considered the cognitive structure of emotions, without considering the felt aspect of emotions. This is of course a limitation of the model presented in this paper, as the intensity of emotion also depends on the presence of somatic components (e.g., the intensity of fear is ampliﬁed by the fact that, when experiencing this emotion I feel my stomach contracted, my throat dry, etc.) However, an analysis of the cognitive structure of emotions is a necessary step for having an adequate understanding of aﬀective phenomena. An analysis of the relationships between emotion and action (i.e., how an emotion with a given intensity determines the agent’s future reactions), and of the relationships between cognitive structure and somatic aspects of emotions (i.e., how somatic components aﬀect emotion intensity) is postponed to future work.

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References 1. Adam, C., Herzig, A., Longin, D.: A logical formalization of the OCC theory of emotions. Synthese 168(2), 201–248 (2009) 2. Alchourr´ on, C.E., G¨ ardenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. of Symbolic Logic 50(2), 510–530 (1985) 3. Aucher, G.: A combined system for update logic and belief revision. In: Barley, M.W., Kasabov, N. (eds.) PRIMA 2004. LNCS (LNAI), vol. 3371, pp. 1–17. Springer, Heidelberg (2005) 4. Baltag, A., Smets, S.: A qualitative theory of dynamic interactive belief revision. In: Proc. of LOFT 7, Texts in Logic and Games, vol. 3, pp. 13–60. Amsterdam University Press, Amsterdam (2008) 5. van Benthem, J., Gerbrandy, J., Hoshi, T., Pacuit, E.: Merging frameworks for interaction. J. of Philosophical Logic 38(5), 491–526 (2009) 6. Castelfranchi, C., Lorini, E.: Cognitive anatomy and functions of expectations. In: Sun, R. (ed.) Proc. of IJCAI 2003 Workshop on Cognitive Modelling of Agents and Multi-Agent Interactions, pp. 29–36 (2003) 7. van Ditmarsch, H.: Prolegomena to dynamic logic for belief revision. Synthese 147(2), 229–275 (2005) 8. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Synthese Library, vol. 337. Springer, Heidelberg (2007) 9. Gratch, J., Marsella, S.: A domain independent framework for modeling emotion. J. of Cognitive Systems Research 5(4), 269–306 (2004) 10. Grove, A.: Two modellings for theory change. J. of Philosophical Logic 17, 157–170 (1988) 11. Klein, P.D.: A proposed deﬁnition of propositional knowledge. J. of Philosophy 68, 471–482 (1971) 12. Laverny, N., Lang, J.: From knowledge-based programs to graded belief-based programs, part II: oﬀ-line reasoning. In: Proc. of IJCAI 2005, pp. 497–502 (2005) 13. Lorini, E., Castelfranchi, C.: The cognitive structure of surprise: looking for basic principles. Topoi: an International Review of Philosophy 26(1), 133–149 (2007) 14. Lorini, E., Schwarzentruber, F.: A logic for reasoning about counterfactual emotions. Artiﬁcial Intelligence 175(3-4), 814–847 (2011) 15. Meyer, J.J.: Reasoning about emotional agents. International J. of Intelligent Systems 21(6), 601–619 (2006) 16. Ortony, A., Clore, G.L., Collins, A.: The cognitive structure of emotions. Cambridge University Press, Cambridge (1988) 17. Reisenzein, R.: Emotions as metarepresentational states of mind: naturalizing the belief-desire theory of emotion. Cognitive Systems Research 10, 6–20 (2009) 18. Rott, H.: Stability, strength and sensitivity: converting belief into knowledge. Erkenntnis 61, 469–493 (2004) 19. Spohn, W.: Ordinal conditional functions: a dynamic theory of epistemic states. In: Causation in Decision, Belief Change and Statistics, pp. 105–134. Kluwer, Dordrecht (1998) 20. Stalnaker, R.: On logics of knowledge and belief. Philosophical Studies 128, 169–199 (2006) 21. Steunebrink, B., Dastani, M., Meyer, J.J.: A formal model of emotions: integrating qualitative and quantitative aspects. In: Proc. of ECAI 2008, pp. 256–260 (2008)

DEL Planning and Some Tractable Cases Benedikt L¨ owe1,2 , Eric Pacuit3 , and Andreas Witzel4, 1

3

4

Institute for Logic, Language and Computation, Universiteit van Amsterdam, Postbus 94242, 1090 GE Amsterdam, The Netherlands [email protected] 2 Department Mathematik, Universit¨ at Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany Tilburg Center for Logic and Philosophy of Science, Tilburg Universiteit van Tilburg, Postbus 90153, 5000 LE Tilburg, The Netherlands [email protected] Bioinformatics Group, Courant Institute of Mathematical Sciences, New York University, 715 Broadway, New York, NY 10003, USA [email protected]

Abstract. We describe the planning problem within the framework of dynamic epistemic logic (DEL), considering the tree of sequences of events as the underlying structure. In general, the DEL planning problem is computationally diﬃcult to solve. On the other hand, a great deal of fruitful technical advances have led to deep insights into the way DEL works, and these can be exploited in special cases. We present a few properties that will lead to considerable simpliﬁcations of the DEL planning problem and apply them in a toy example.

1

Introduction

Dynamic epistemic logic (DEL) is one of the standard conceptual models for epistemic situations and change. Semantically, it is based on an operation called product update that allows to apply an event model to an epistemic model in order to describe epistemic change. Possible future epistemic states correspond to sequences of event models applied successively to an initial state. DEL is conceptually very clear; this makes it a promising framework for epistemic planning. In this paper, we shall propose a general DEL planning framework. In general, the DEL planning problem is intractable; however, we argue that in some cases,

The third author was supported by a GLoRiClass fellowship funded by the European Commission (Early Stage Research Training Mono-Host Fellowship MEST-CT-2005020841) and an NSF Expeditions in Computing grant. The paper took its ﬁnal form during the LogICCC Meets India meeting in Delhi (January 2011) and a research visit of the ﬁrst author at New York University (March 2011). These trips were funded by the European Science Foundation (LogICCC Networking Activity 280) and the NWO (DN 62-630), respectively. The authors would like to thank Hans van Ditmarsch for discussions about epistemic planning, and the anonymous referees of LORI-III for their comments.

H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 179–192, 2011. c Springer-Verlag Berlin Heidelberg 2011

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our understanding of the DEL semantics will allow us to restrict the attention to a tractable situation. We shall give a number of conditions that will allow us to reduce the complexity of the DEL planning problem drastically, and apply these insights in a toy example. We do not consider DEL planning as a merely theoretical framework, but aim at real applications in the setting of computer games (and ﬁttingly, our toy example is also about a concrete computer game). Making planning into a crucial element of narrative design for computer games has been proposed by Riedl and Young [20, 21]. The ﬁrst and second author have proposed to use formalisms based on epistemic logic for the formalization of narratives [15, 16], and the third author (in collaboration with Kennerly and Zvesper; [29, 13]) has formulated a simple knowledge-based action situation from a computer game. We consider this paper as a ﬁrst step towards combining these logical aspects of computer gaming. Related Work In the logic community, the potential to use dynamic epistemic logic for concrete implementations of reasoning processes permeates the literature; many papers mention concrete applications as motivation for studying dynamic epistemic logic. Renardel de Lavalette and van Ditmarsch [19] discuss updating and maintaining a minimal epistemic model and identifying subclasses of DEL for which that is possible. They provide model minimization for so-called simple actions in order to allow eﬃcient model checking. Our toy application in § 5 has non-simple (though propositional) actions, and uses non-S5 models. Related to our proposal is the work by van der Hoek and Wooldridge [25] on planning with epistemic goals, based on the idea of Giunchilia and Traverso to use model checking as a planning algorithm [11]. Their planning algorithm is based on S5 ATEL using a similar model to the one used in [13]; we argue here that DEL provides a more ﬂexible framework. Also related is the work by ˚ Agotnes and van Ditmarsch on public announcement logic, examining which public announcement to make in a strategic setting with goals (assuming truthfulness) [1]. The aim of [26] is very similar to ours: van Ditmarsch, Herzig and Lima combine DEL and planning; their work is complementary to ours in that the authors start from the situation calculus fragment from [14] and oﬀer a subset of DEL (public announcement and public “questions”) as equivalent to it, while we start from the rich framework of DEL and try to show how planning may be incorporated. The closest cognate of our paper is [5]: Bolander and Andersen consider a setting very similar to the one described in §§ 2 and 3 and prove that the planning problem for the single agent case is decidable [5, Theorem 17] and that the multiagent case is undecidable [5, Theorem 20]. The results of Bolander and Andersen were independently discovered (however, their paper cites a preprint version of this paper in their section on related work). Despite the conceptual agreement

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with Bolander and Andersen, our aim in this paper is quite diﬀerent: while they are interested in decidability and undecidability results, we aim at looking in a “bottom-up” fashion at subclasses for which planning becomes tractable. Outside of the logic literature, there are many approaches to planning under uncertainty and knowledge representation in the large and mature ﬁeld of AI planning. The authors of [18] point out that a syntactic (what they call “knowledge-based” or “knowledge level”) representation can represent uncertainty more succinctly. While we focus on DEL model theory, we do not want to dictate that an implementation has to be model-based and preclude a more proof-theoretic realization. There is a trade-oﬀ of representational complexity versus computational complexity, and these are just two sides of the same coin. Petrick and Bacchus consider a restricted language to stay computationally tractable, while we look at restricted classes of models to stay spatially tractable. In certain cases, one viewpoint may allow more natural or more eﬀective restrictions, and we think that ultimately joining both kinds of restrictions will be necessary. The fact that a semantic approach has its advantages was pointed out by Lakemeyer and Levesque who provide an elegant semantical view on the situation calculus [14]. While these approaches are more expressive than ours in some sense by using (fragments of) ﬁrst- or second-order logic, they deal with only a single agent, and only with S5 knowledge. This simpliﬁes the semantics to just an information set for that player (although Lakemeyer and Levesque in [14] do add some structure in terms of their sensing agreement relation among possible worlds). On the semantic level, the conceptual extension to multiple agents and their (possibly sub-S5) beliefs about each other corresponds to adding more structure in the form of agents’ accessibility relations. This makes the semantic objects and algorithms more complex, so we simplify matters by focusing on propositional modal logic.1 Outline of This Paper In § 2 we shall give a standard introduction to dynamic epistemic logic, following roughly the textbook [27]. In § 3, we deﬁne several versions of DEL planning problem, and argue that we need to ﬁnd eﬃcient techniques for solving it. In § 4, we discuss some examples of techniques to that end, and we apply them in § 5 to the setting of the doxastically enriched computer game Thief from [29, 13]. We close the paper with a few pointers to future work in § 6. 1

Note that we are here using dynamic epistemic logic for a situation in which we describe beliefs. The main diﬀerence between knowledge and belief is that beliefs can be false whereas knowledge—in standard formalizations—cannot. If i discovers that j does know, the product update will produce a model in which i considers no state of the world possible. For a more graceful handling and revising of inconsistent beliefs, we could use a doxastic version of DEL [4]. Since DEL works for the simple examples in this paper, we ignore these issues for the present discussion.

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Dynamic Epistemic Logic

In this section, we give an overview of product updates due to Baltag, Moss and Solecki [3] for the non-expert reader, following closely the textbook [27] (where the reader can ﬁnd more details). Let A be a ﬁnite set of agents and At a set of atomic propositions. An epistemic model M is a tuple W, {Ri }i∈A , V where W =: D(M) is a ﬁnite nonempty set called the domain of M, for each i ∈ A, Ri ⊆ W × W is a binary relation on W (typically an equivalence relation) and V : At → 2W is a valuation function. If w ∈ D(M), we call (M, w) a pointed epistemic model. If w is clear from the context, we may omit it from the notation. The elements of W constitute “states of the world” and the relations Ri are accessibility relations, i.e., for states w, v ∈ W , wRi v means “in state w, agent i would consider state v possible.” The set of multiagent epistemic formulas, denoted LA , is the smallest set of formulas generated by the following grammar: p | ¬ϕ | ϕ ∧ ψ | i ϕ where p ∈ At and i ∈ A. We use the usual abbreviations for the other propositional connectives (∨, →, ↔), and we use LProp to denote the propositional sub-language (i.e., not containing i ). Truth of formulas ϕ ∈ LA is deﬁned as usual in Kripke models. An event (or action) model E is a tuple S, {→i }i∈A , pre, where S is a ﬁnite nonempty set, for each i ∈ A, →i ⊆ S × S is i’s accessibility relation, and pre : S → LA is the pre-condition function. The set S is called the domain of E, denoted D(E). We call E propositional if pre goes into LProp , i.e., all preconditions are propositional. The product update operation updates an epistemic model M = W, {Ri }i∈A , V with an event model E = S, {→i }i∈A , pre and is deﬁned as M ⊗ E = W , {Ri }i∈A , V with (i) W = {(w, e) | w ∈ W, e ∈ S and M, w |= pre(e)}, (ii) (w1 , e1 )Ri (w2 , e2 ) iﬀ w1 Ri w2 in M and e1 →i e2 in E, and (iii) V (p) = {(w, e) ∈ W | w ∈ V (p)}. For pointed models, the point of the product is the pair of the factors’ points. If E = S, {→i }i∈A , pre and E = S , {→i }i∈A , pre are two event models, we can analogously deﬁne the product event model E E = S ∗ , {→∗i }i∈A , pre∗ as follows: (i) S ∗ = S × S , (ii) (e, e ) →∗i (f, f ) if and only if e →i f and e →i f , and

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(iii) pre∗ (e, e ) := pre(e) ∧ pre (e ). It is easy to see that the operation is commutative up to isomorphism (i.e., E E is isomorphic to E E). Note that for non-propositional events, the operations ⊗ and do not necessarily ﬁt together, i.e., it can be that (M⊗E)⊗E is not the same as M ⊗ (E E ).2 However, for propositional event models, we get the following associativity property: Fact 1. If M is an epistemic model, and E and E are propositional event models, then (M ⊗ E) ⊗ E M ⊗ (E E ). The usual notion of equivalence used in modal logic is the weaker notion of bisimulation [27, Deﬁnition 2.14], denoted . Bisimilar pointed models are equivalent in the sense that they satisfy exactly the same formulas. For any model M and events E, E , if E E then M ⊗ E M ⊗ E (note that the converse does not hold, see [28, Observation 13]). For each model M, the union of all bisimulations of M with itself is again a bisimulation. The quotient structure of this bisimulation gives us the most compact model satisfying the same formulas, called the bisimulation contraction. By M , we denote the cardinality of the domain of the bisimulation contraction of M. Now, adding to LA a modal operator E, e for each pointed event model (E, e), we obtain the language LDEL . Truth for these modalities is deﬁned as M, w |= E, eϕ iﬀ M, w |= pre(e) and M ⊗ E, (w, e) |= ϕ. Given a pointed model (M, w) and a formula ϕ ∈ LA , checking M, w |= ϕ can be done in time polynomial in the size of M and ϕ [24, § 3.8]. Some care must be taken with respect to the product update, as at ﬁrst glance it can potentially lead to exponentially growing models (cf. § 4). Planning will happen in the tree of temporal sequences of events: ﬁx a pointed epistemic model (M, w) and a ﬁnite set of (pointed) event models E. For preﬁxes of ﬁnite sequences σ := (E0 , . . . , En ) of models in E, we have a natural notion of immediate successor, viz. extension by one additional model. Let M ⊗ σ := M ⊗ E0 ⊗ . . . ⊗ En ; then the collection of these epistemic models forms a tree structure with successor structure derived from the ﬁnite sequences and M = M∅ at the root. It is this tree structure that we consider to be the natural temporal setting for DEL planning (cf. § 3). Slightly more precisely, for a pointed epistemic model (M, w) and (E, e) ∈ E, we say that (E, e) is possible at (M, w) if M, w |= E, e. We say that a sequence σ is legal if it is empty or its (uniquely determined) immediate 2

Consider an epistemic model M |= p ∧ ¬i p, an event E modelling public announcement of p and another event E modelling public announcement of ¬i p.

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predecessor σ ∗ is legal and (E ∗ , e∗ ) is possible at M ⊗ σ ∗ , with (E ∗ , e∗ ) being the last element of σ. The set of legal sequences, denoted LS, contains exactly those sequences that can be performed in the given order, since the preconditions of each event are met at the appropriate time. They form a subtree of our tree structure. If we want to impose further external restrictions on the possible courses of action, we can consider a subtree T ⊆ LS. To conclude our description of DEL, we should note that we do not consider events that change actual facts (i.e., the valuation function). This is a serious restriction but doesn’t aﬀect the example in § 5. The deﬁnition of the product update can be extended to deal with factual change [24], but for the sake of simplicity, we restricted ourselves to purely epistemic events (cf. § 6).

3

DEL Planning

The classical planning problem consists of a description of the world, the agent’s goal and a description of the possible actions in some appropriate formal language. A planning algorithm consists of sequences of possible actions which when executed will achieve the goal. To attempt an initial deﬁnition of a DEL planning problem, we ﬁx a pointed epistemic model (M, w) and a ﬁnite set E of pointed event models. As mentioned at the end of § 2, we consider the tree LS of legal sequences σ from E as our space of possible plans, and we allow to impose additional constraints on when events can occur in the form of specifying a subtree T ⊆ LS.3 Definition 2 (Absolute DEL planning problem). Given (M, w), E, a subtree T ⊆ LS, and a formula ϕ ∈ LDEL , produce a sequence σ ∈ T such that M ⊗ σ |= ϕ. The absolute DEL planning problem does not talk about agents and whether the plan is realizable by a single agent. In its standard formulation, the DEL formalism does not assign events to particular agents, so we need to supplement the formalism with such an assignment: consider a function power : A → ℘(E) that tells us which events an agent can bring about. Here, if i ∈ A, we interpret E ∈ power(i) as “agent i can perform action E”. If σ is a sequence of actions, we write σ ∈ power(i) if for each E ∈ Set(σ), we have E ∈ power(i). Similarly, for a partial sequence, i.e., a partial function σ : {0, . . . , N } → E (for some N ), we write σ ∈ power(i), if for all n ∈ dom( σ ), we have σ (n) ∈ power(i).4 Definition 3 (Single-agent DEL planning problem). Given (M, w), E, a subtree T ⊆ LS, a function power, an agent i ∈ A, and a formula ϕ ∈ LDEL , produce a sequence σ ∈ T such that M ⊗ σ |= ϕ and σ ∈ power(i). 3

4

Note that this notion is somewhat unusual from an AI point of view, where all constraints concerning an event’s applicability are speciﬁed locally with the event. We still include this “global” protocol-based notion since it is common in the epistemic logic community. These deﬁnitions are easily extended to sets of agents if we want to include actions that can only jointly be performed by a group of agents.

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A special case is the situation where A = {i}. This case is called singleagent epistemic planning in [5]. More interesting and much harder is the multi-agent version that essentially asks for the existence of a winning strategy. Bolander and Andersen called it multi-agent epistemic planning in [5]: Definition 4 (Multi-agent DEL planning problem). Given (M, w), E, a subtree T ⊆ LS, a function power, an agent i ∈ A, and a formula ϕ ∈ LDEL , produce a partial function σ : {0, . . . , N } → E such that 1. σ ∈ power(i), 2. there is a sequence σ ∈ T such that σ ⊆ σ, and 3. for all sequences σ ∈ T with σ ⊆ σ, we have M ⊗ σ |= ϕ. Bolander and Andersen prove that multi-agent epistemic planning in this sense is undecidable [5, Theorem 20].5 Deﬁning the planning problem is only the very ﬁrst step towards solving it. In general, planning problems are intractable (even if they are decidable), and a large part of the work in AI planning is spent on ﬁnding compact representations of large spaces of situations (or plans), and time-eﬃcient ways of traversing them in the search of given goals (cf. [10]). Our DEL planning problems tend to be more complex than classical planning problems: not only can the space of reachable situations explode with the number of events at the planner’s disposal, but even the situations themselves, being product models of previous situations and events, may prima facie grow combinatorially. Consider, for example, the classical STRIPS planning formalism [8], which is by now somewhat dated but still widely used as point of reference. In the most commonly used variant of that formalism, situations are valuations over a certain universe of atomic facts, and the eﬀects of an event (or action) e are speciﬁed by a set e− of facts that it removes (makes false) and a set e+ of facts that it adds (makes true) in any situation that it is applied to.6 While the number of possible situations does grow exponentially with the size of the universe of facts, once that universe is ﬁxed the size of situations is determined and cannot grow out of control. In contrast, which formulas hold after a DEL update depends very much on the situation before the update, so it is not straightforwardly possible to give the eﬀects of an event just by sets e+ and e− (cf. Footnote 6). An embedding of DEL planning into STRIPS thus would necessarily have to encode this information in some complicated way. In our setting, we decided to represent 5

6

Note that this formulation assumes a worst-case scenario. Ultimately, we should like to study a general strategic DEL planning problem where agents take (interfering or cooperative) actions of third agents into account, and may to some extent try to anticipate them. However, while there exists work in the context of DEL on deﬁning the information content of events in strategic settings [9] and modeling goals and preferences [23], things are far less clear in such a case, and we therefore leave it to future work. So, if S is the set of true facts before the event e takes place, then (S\e− ) ∪ e+ is the set of facts that is true after e takes place. Note that e+ and e− are independent of S.

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DEL models directly in order to be able to exploit certain structural properties that would not be perspicuous in an embedding into a classical formalism such as STRIPS. Considering planning as a search task in the tree of sequences of event, we should like to highlight and discuss an interesting phenomenon with respect to two standard approaches from the classical planning literature: progression and regression.7 In progression, plan search starts from an initial state and successively expands applicable events until the goal state is reached. In regression, the search starts from the goal state and makes backward steps to ﬁnd situations from which the goal state can be reached, until the initial state is hit. Typically, progression is more straightforward to formalize and implement. This also holds for our model-theoretic view of DEL planning: Given a situation and an event model, it is straightforward to compute the successor situation resulting from application of the event, but it is an open problem how possible predecessor situations can be found, to which the event can be applied yielding the situation in question. However, in a syntactic view of DEL planning, the situation reverses: the central piece of the completeness proof for axiomatizations of DEL is a term rewriting system consisting of axioms that are sometimes called reduction axioms [3, § 4]. These reduction axioms directly yield the required preconditions of an event, given its postconditions, making regression very simple. On the other hand, it is not known whether there is a simply deﬁnable way of associating to each event model E a map FE mapping sets of formulas to sets of formulas such that for all M, if M |= Φ, then M ⊗ E |= FE (Φ). Such a function would correspond to progression in the syntactic view.

4

Some Properties to Ensure Tractability

We ﬁrst introduce some notions. Let E and E be event models. We say that – E is self-absorbing if for all models M we have M ⊗ E ⊗ E M ⊗ E; – E and E commute if for all models M we have M ⊗ E ⊗ E M ⊗ E ⊗ E; and that – E is almost-mutex (“almost-mutually-exclusive”) if there is at most one atomic event e ∈ D(E) with pre(e ) = and e →i e for all i ∈ A, and the formulas pre(e) with e = e are pairwise inconsistent. Lemma 5. Propositional event models commute. Proof. Follows directly from the fact that the product of event models is commutative and Fact 1. 7

We should like to thank Hans van Ditmarsch for comments on the presentation of an earlier version of this paper in Delhi (January 2011) and the e-mail discussion developing from these comments; these formed the basis of the discussion of progression and regression.

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Lemma 6. Almost-mutex event models with transitive accessibility relations are self-absorbing. Proof. Let E = S, {→i }i∈A , pre be an almost-mutex event model with point s ∈ S. We again prove that E E E. Consider the smallest relation ρ ⊆ (D(E) × D(E)) × D(E) such that (e, e)ρe, (e, e )ρe, and (e , e)ρe hold for all e ∈ D(E). We show that this is a bisimulation on the submodels of E E and E generated by (s, s) and s. To see this, note ﬁrst that (s, s)ρs. Next, assume that (e1 , e2 )ρe and (e1 , e2 ) →i (e1 , e2 ). We have to show that there is e with e →i e and (e1 , e2 )ρe . By deﬁnition of ρ, we are in one of three cases: Case 1. e1 = e2 = e. From (e, e) →i (e1 , e2 ) it follows that e →i e1 and e →i e2 . If e1 = e2 then (e1 , e2 )ρe1 by deﬁnition of ρ and we are done. Otherwise e1 = e2 . Since (e1 , e2 ) ∈ D(E E), pre(e1 ) and pre(e2 ) cannot be inconsistent, and with E being almost-mutex it follows that one of the two events is e . If e1 = e then (e1 , e2 )ρe2 by deﬁnition of ρ, and analogously if e2 = e . Case 2. e1 = e and e2 = e . From (e, e ) →i (e1 , e2 ) it follows that e →i e1 and e →i e2 . If e1 = e2 then (e1 , e2 )ρe1 by deﬁnition of ρ. Otherwise e1 = e2 . Since (e1 , e2 ) ∈ D(E E), pre(e1 ) and pre(e2 ) cannot be inconsistent, and with E being almost-mutex it follows that one of the two events is e . If e2 = e then (e1 , e2 )ρe1 by deﬁnition of ρ and we are done since e →i e1 . Otherwise e1 = e , and from e →i e1 = e →i e2 , by transitivity we get e →i e2 . By deﬁnition of ρ, (e , e2 )ρe2 . Case 3. e1 = e and e2 = e. Analogous to the previous case. Finally, assume that (e1 , e2 )ρe and e →i e . We have to show that there is (e1 , e2 ) with (e1 , e2 ) →i (e1 , e2 ) and (e1 , e2 )ρe . This is easy to see with a similar case distinction as above, noting that e →i e by assumption. For a sequence σ = E1 . . . Ek of event models, let Set(σ) = {E1 , . . . , Ek }, and let M ⊗ σ = M ⊗ E1 ⊗ · · · ⊗ Ek for a model M. Proposition 7. For any model M and any sequences σ1 , σ2 of propositional, almost-mutex events with transitive accessibility relations, if Set(σ1 ) = Set(σ2 ) then M ⊗ σ1 M ⊗ σ2 . Proof. Follows immediately from Lemma 5 and 6.

5 5.1

A Toy Application A Formalization of Thief: The Dark Project TM

The video game Thief: The Dark Project TM by Eidos Interactive (1998) is themed as a game of stealth, in which the player (the thief) avoids being detected by computer-simulated guards. The player exploits the guard’s—possibly

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mistaken—beliefs about the thief’s presence. The following is an epistemically enhanced version of a scene from Thief as presented in [13]. We assume that the scene starts with thief and guard present, each uncertain of the other’s presence, and that agents cannot enter or leave. In our formalization, we consider the following kinds of events: – nt , ng : The thief (the guard) makes some noise. – bt , bg : The thief (the guard) sees the other one from behind. – f : Thief and guard see each other face to face. The intuitive epistemic eﬀects of these events are as follows: – nt : The guard learns that a thief is present; the thief learns that, if a guard is present, the guard learns that the thief is present. – bt : The thief learns that a guard is present; the guard believes nothing has happened (he is not paranoid enough to constantly suspect being seen from behind). – f : Thief and guard commonly learn that both are present. The eﬀects of ng and bg are analogous. To model this situation in DEL, we use the set of atomic propositions At = {pt , pg }, with the reading that the thief, respectively the guard, is present. We formalize the initial situation by the pointed model I and the events described above by the set of pointed event models E = {Nt, Ng , Bt , Bg , F }, as depicted in Figure 1. From now on we omit the qualiﬁer “pointed”. Proposition 8. Let σ by any sequence of events from E = {Nt , Ng , Bt , Bg , F } Then I ⊗ σ ≤ 6. Proof (sketch). We are using Proposition 7: First note that F E F for any E ∈ E, so I ⊗ σ = 1 for any σ containing F . Also, Nt Ng F , so the same holds for any sequence containing these two events. Due to symmetry we are left with 6 cases to check: σ ∈ {Nt , Bt , Nt Bt , Nt Bg , Bt Bg , Nt Bt Bg }. Together with the fact that the bisimulation contraction can be computed in linear time [7], this shows that our toy model indeed stays a toy model. As mentioned above, this may not say much about more realistic models, but we expect that in some interesting cases, similar techniques would yield similar results. We can also show the fact stated above, saying that we do not need to consider belief-revision mechanisms in our simple scenario, since the agents never reach inconsistent belief states (although their beliefs may be mistaken). Proposition 9. For any sequence σ of events from E and any agent i ∈ A, I ⊗ σ |= i ⊥. Proof (sketch). Since F E F for any E ∈ E and I ⊗ F |= i ⊥, with Proposition 7 we get that I ⊗ σ |= i ⊥ for any σ containing F . Assume there is some σ with I ⊗ σ |= i ⊥, then there must be no state that i considers possible at the point of I ⊗ σ. By deﬁnition of ⊗, the same would then hold for the point of I ⊗ σ ⊗ F , which is a contradiction.

DEL Planning and Some Tractable Cases

pt ∧ p g

pt , pg g pg

pt ∧ pg g

t

t

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t g

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(c) Ng: guard makes noise

g

t t, g

(d) Bt : thief sees guard from behind

t, g

(e) Bg : guard sees thief from behind

p t ∧ pg (f ) F: face to face

Fig. 1. Models for initial situation and events. Undirected edges represent bidirectional accessibilities. In models without directed edges, reﬂexive accessibilities are omitted.

5.2

Planning for the Toy Application

To sketch how a planning guard may be realized in our simple scenario, we ﬁrst specify which of the events we discussed can be brought about by the guard, and thus should be in his repertoire of actions. For now, we simply assume that the guard can make a noise (ng ), he can let himself be seen from behind (bt ), and he can step out and provoke a face-to-face encounter (f ). These actions are associated with the corresponding DEL events from Figure 1, which also encapsulate the actions’ preconditions. For example, f can only happen if both are present, and correspondingly, the guard only knows that he can make f happen if he knows that both are present. Besides these actions corresponding to our DEL events, we consider the following additional attack actions: – ambush, with precondition pre(ambush) = pt ∧ ¬t pg – trick, with precondition pre(trick) = pt ∧ t pg ∧ ¬t g pt – rush, with precondition pre(rush) = pt . Plans can be assessed as to whether they are legal, believed by the guard to be legal, or believed to be believed to be legal at each intermediate point of the plan (i.e., that plan can be “knowingly” executed). Of course, the guard should only be able to access the last two assessments. For example, two potential plans are P1 = bt, trick and P2 = ng , ambush. In the initial situation I, P1 is legal, but the guard does not know this since

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he is not sure of the thief’s presence. After seeing the thief from behind, the guard does take P1 to be legal, although it is not the case that he would believe it at each point of the plan. Formally, by slight abuse of notation, we have I |= Bt trick but I |= g Bt trick, and I ⊗ Bg |= g Bt trick but I ⊗ Bg |= g Bt g trick. This last thing is due to the fact that we modeled being seen from behind as something that one never assumes to happen. In that sense it is not an action the guard can knowingly perform—it would be more appropriate to formalize such an action as DEL event which does get reﬂected in the guard’s epistemic state, but which looks to the thief like Bt . Due to the modular nature of DEL, such issues can be taken care of simply by modifying or adding the aﬀected event models. With P2 , we have I |= Ng ambush along with I |= g Ng ambush and I |= g Ng g ambush. In fact the same is true in any situation, since by the very act of making noise, the guard destroys the precondition for an ambush. We have implemented a naive planner which considers all plans up to a certain length ending with one of the attack actions, and queries the knowledge module as to their (current) feasibility.

6

Conclusion and Future Work

DEL is about knowledge, but as our toy example showed, the applications we have in mind, are mostly about belief rather than knowledge. So a next step would be to phrase the planning problems of this paper in a doxastic version of DEL [4] instead. There are also a number of simpliﬁcations we made in our set-up that could be removed: We did not consider the possibility of events that change the valuation function (cf. [24]) nor the possibility that an agent’s ability to perform an action may change over time. Generalizing our framework to include these possibilities would be a natural next step. While DEL has a well-developed proof theory, its model theory is particularly appealing due to its clarity and intuitiveness. It will be a topic of future research to determine in what cases it can be directly implemented and in what cases a syntactic representation is better suited (possibly to be used with a generic theorem prover), or a “hybrid” approach such as binary decision diagrams (BDDs). For the model-based approach, two crucial topics for further research are the following: – Examine the long-term expansion of models under iterated updates (related to [22], but we are mostly interested in ﬁnite models) and identify natural and general classes of actions that allow arguments such as our crucial Proposition 8 in § 5. – Find compact representations of models (cf. techniques from model checking [6]) on which the product operation can operate directly. Another interesting question to investigate is whether DEL planning problems can be reduced to a simpler and more established formalism, as in the work by [17] where a certain class of planning problems with uncertainty is translated to

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problems that can be solved using the classical FF planner (“Fast Forward”) by [12]. Classical planning has developed many eﬀective and proven optimization strategies and other techniques, which may be exploited in this way. We envision a use of DEL as a general engine (called “knowledge module” by Kennerly, Witzel, and Zvesper in [29, 13]) which allows for ﬂexible speciﬁcation of situations and events and then maintains the agents’ mental models throughout the progress of the game or scene, much like a physics engine maintains a model of the physical state of the world. Note that, while we focus on maintaining one central epistemic model as part of the simulation engine, such a model can also be distributed and maintained by the individual agents [2].

References ˚gotnes, T., van Ditmarsch, H.: What will they say?—Public announcement 1. A games. Paper presented at Logic, Game Theory and Social Choice 6, Tsukuba, Japan (2009) 2. Aucher, G.: Perspectives on Belief and Change. PhD thesis, Universit´e de Toulouse and University of Otago (2008) 3. Baltag, A., Moss, L., Solecki, S.: The logic of public announcements, common knowledge and private suspicions. In: Gilboa, I. (ed.) Proceedings of the 7th Conference on Theoretical Aspects of Rationality and Knowledge (TARK 1998), pp. 43–56. Morgan Kaufmann, San Francisco (1998) 4. Baltag, A., Smets, S.: A qualitative theory of dynamic interactive belief revision. In: Bonanno, G., van der Hoek, W., Wooldridge, M. (eds.) Logic and the Foundations of Game and Decision Theory (LOFT 7). Texts in Logic and Games, vol. 3, pp. 9–58. Amsterdam University Press, Amsterdam (2008) 5. Bolander, T., Andersen, M.B.: Epistemic planning for single- and multi-agent systems. Journal of Applied Non-Classical Logics 21(1), 9–34 (2011) 6. Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. MIT Press, Cambridge (1999) 7. Dovier, A., Piazza, C., Policriti, A.: A fast bisimulation algorithm. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 79–90. Springer, Heidelberg (2001) 8. Fikes, R., Nilsson, N.: Strips: a new approach to the application of theorem proving to problem solving. Artiﬁcial Intelligence 2, 189–208 (1971) 9. Gerbrandy, J.: Communication strategies in games. Journal of Applied NonClassical Logics 17(2), 197–211 (2007) 10. Ghallab, M., Nau, D., Traverso, P.: Automated Planning: Theory & Practice. Morgan Kaufmann, San Francisco (2004) 11. Giunchiglia, F., Traverso, P.: Planning as model checking. In: Biundo, S., Fox, M. (eds.) ECP 1999. LNCS, vol. 1809, pp. 1–20. Springer, Heidelberg (2000) 12. Hoﬀmann, J., Nebel, B.: The FF planning system: fast plan generation through heuristic search. Journal of Artiﬁcial Intelligence Research 14(1), 253–302 (2001) 13. Kennerly, E., Witzel, A., Zvesper, J.A.: Thief belief. In: L¨ owe, B. (ed.) LSIR-2: Logic and the Simulation of Interaction and Reasoning Workshop at IJCAI-2009, pp. 47–51 (2009)

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14. Lakemeyer, G., Levesque, H.J.: Semantics for a useful fragment of the situation calculus. In: Pack Kaelbling, L., Saﬃotti, A. (eds.) IJCAI-2005, Proceedings of the Nineteenth International Joint Conference on Artiﬁcial Intelligence, Edinburgh, Scotland, UK, July 30-August 5, pp. 490–496. Morgan Kaufmann Publishers Inc., San Francisco (2005) 15. L¨ owe, B., Pacuit, E.: An abstract approach to reasoning about games with mistaken and changing beliefs. Australasian Journal of Logic 6, 162–181 (2008) 16. L¨ owe, B., Pacuit, E., Saraf, S.: Identifying the structure of a narrative via an agentbased logic of preferences and beliefs: Formalizations of episodes from CSI: Crime Scene InvestigationTM . In: Duvigneau, M., Moldt, D. (eds.) Proceedings of the Fifth International Workshop on Modelling of Objects, Components and Agents (MOCA 2009), FBI-HH-B-290/09, pp. 45–63 (2009) 17. Palacios, H., Geﬀner, H.: From conformant into classical planning: Eﬃcient translations that may be complete too. In: Boddy, M., Fox, M., Thi´ebaux, S. (eds.) Proceedings of the Seventeenth International Conference on Automated Planning and Scheduling (ICAPS 2007) (2007) 18. Petrick, R., Bacchus, F.: A Knowledge-Based approach to planning with incomplete information and sensing. In: Ghallab, M., Hertzberg, J., Traverso, P. (eds.) Proceedings of the Sixth International Conference on Artiﬁcial Intelligence Planning and Scheduling (AIPS 2002), pp. 212–221 (2002) 19. Renardel de Lavalette, G.R., van Ditmarsch, H.: Epistemic actions and minimal models, Available on the authors’ websites (2002) 20. Riedl, M.O., Young, R.M.: Character-focused narrative planning (2003) (unpublished manuscript) 21. Riedl, M.O., Young, R.M.: From linear story generation to branching story graphs. IEEE Computer Graphics and Applications 26(3), 23–31 (2006) 22. Sadzik, T.: Exploring the iterated update universe, ILLC Publications PP-2006-26 (2006) 23. van Benthem, J., Liu, F.: Dynamic logic of preference upgrade. Journal of Applied Non-Classical Logics 17(2), 157–182 (2007) 24. van Benthem, J., van Eijck, J., Kooi, B.: Logics of communication and change. Information and Computation 204(11), 1620–1662 (2006) 25. van der Hoek, W., Wooldridge, M.: Tractable multiagent planning for epistemic goals. In: Castelfranchi, C. (ed.) Proceedings of the First International Joint Conference on Autonomous Agents & Multiagent Systems, AAMAS 2002, Bologna, Italy, July 15-19, pp. 1167–1174. ACM, New York (2002) 26. van Ditmarsch, H., Herzig, A., de Lima, T.: From situation calculus to dynamic epistemic logic (2010) (unpublished manuscript) 27. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Synthese Library, vol. 337. Springer, Heidelberg (2008) 28. van Eijck, J., Ruan, J., Sadzik, T.: Action emulation. Draft paper (2008) 29. Witzel, A., Zvesper, J., Kennerly, E.: Explicit knowledge programming for computer games. In: Darken, C., Mateas, M. (eds.) Proceedings of the Fourth Artiﬁcial Intelligence and Interactive Digital Entertainment Conference, Stanford, California, USA, October 22-24, AAAI Press, Menlo Park (2008)

Mathematics of Public Announcements Minghui Ma Institute of Logic and Intelligence, Southwest University Beibei, Chongqing, China, 400715 [email protected]

Abstract. We study some mathematical aspects of public announcement logic (PAL) and its several variants. By a model-theoretic approach, we explore van Benthem’s result that uses recursion axioms to characterize the submodel operation, and show some model-theoretic results on the respecting phenomena. The second approach to understand public announcements is algebraic. Based on a joint work with A. Palmigiano and M. Sadrzadeh, we treat public announcements as devices for getting a new quotient algebra updated by an element in the original one. Then we show the algebraic soundness and completeness result for PAL and generalize this approach to PAL extension of epistemic intuitionistic modal logic. Finally, we give some observations on the PAL extensions of first-order logic as well as epistemic predicate modal logic.

The dynamic turn that has been practiced by logicians in recent years attempts to explore explicit logical theory for informational processes and agency, and thus enriches the range of logic. Among logics on the market, the simplest one for dynamics of information ﬂow and knowledge change by observation or communication is the public announcement logic (PAL)([4], [11] and [15]). It may be supposed that we now possesses a perfectly good understanding of how public announcements work in PAL. However, we lack mathematical knowledge of public announcement logic though it has some applications in practical logics and even in analyzing some interesting mathematical models like games. The basic question is the following: what is the mathematics of dynamic operators? For PAL, how can we understand the semantic role of announcement operators? In this paper, we provide both model-theoretic and algebraic interpretations.

1

Submodel Operation and Respecting Phenomena

The language of PAL consists of epistemic modal language, i.e., a set Φ of proposition letters and the epistemic possibility operator ♦, plus announcement operators. The set of PAL-formulas is given by the following inductive rule: φ ::= p | | ⊥ | ¬φ | φ ∨ ψ | ♦φ | φψ where p ∈ Φ. Deﬁne ϕ := ¬♦¬ϕ and [ϕ]ψ := ¬ϕ¬ψ. The modality is the knowledge operator in epistemic logic (EL), and ϕ is an announcement operator reading as ‘announcing ϕ’. H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 193–205, 2011. c Springer-Verlag Berlin Heidelberg 2011

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PAL-formulas are interpreted in Kripke models. Given a model M = (W, R, V ) and a state w ∈ W , the satisfaction relation M, w |= ϕ is deﬁned as in EL plus the following semantic clause for announcement operators: M, w |= ϕψ iﬀ M, w |= ϕ and M |ϕ, w |= ϕ where M |ϕ is the submodel of M given by the set ϕM of ϕ-states in M . The following completeness theorem under Kripke semantics is well-known. Theorem 1 ([5], Theorem 15.27). The epistemic modal S5 plus the following recursion axioms is a complete axiomatization of PAL: ϕp ↔ ϕ ∧ p, ϕ¬ψ ↔ ϕ ∧ ¬ϕψ, ϕ(ψ ∧ ξ) ↔ ϕψ ∧ ϕξ and ϕ♦ψ ↔ ϕ ∧ ♦ϕψ. The completeness of PAL can be shown via the completeness of the epistemic S5 since PAL reduces every formula with announcements to an equivalent ELformula without announcements. This raises the question to understand the semantic role of announcements: what we get by recursion axioms except that they do reduction? A simple observation is that the submodel operation (.)|ϕ is not a standard algebraic operation within one model, but one from a model to a new model which involves identifying elements of the target with elements in the origin. There are two ways to make the role of these operations explicit. One is an abstract model-theoretic perspective and the other is algebraic. In this section, let us look at the model-theoretic perspective. First note the following van Benthem’s characterization theorem. Theorem 2 ([4]). The eliminative update is the unique model-changing operation satisfying ♥p ↔ p, ♥pEq ↔ p ∧ E♥pq and [♥p]q ↔ (p → (p → [♥p]q)) where E is the existential modality. This theorem characterize the action of public announcement via validity of recursion axioms. However, we may show the following diﬀerent observation on the interaction between recursion axioms and model operations. Fix a ELformula ϕ. For each model M = (W, R, V ), ﬁx a submodel Nφ = (Xϕ , Sϕ , Uϕ ) and announcements can be interpreted as follows: M, w |=∗ ϕψ iﬀ w ∈ Xϕ and Nϕ , w |=∗ ψ. The EL-formulas are interpreted as usual. The following fact is rather clear. Proposition 1. For every EL-formula ξ, M, w |= ξ iﬀ M, w |=∗ ξ. Intuitively, the submodel operation (.)|ϕ maps each model to a submodel. Thus the question is the following: which recursion axioms can be used to characterize the submodel operation? We have the following theorem. Theorem 3. For every EL-formula ϕ, |=∗ ϕ ↔ ϕ iﬀ Nϕ = M |ϕ. Proof. The right-ro-left direction is clear. For the other direction, assume that |=∗ ϕ ↔ ϕ. It suﬃces to show that Xϕ = ϕM . For any state w in M , w ∈ Xϕ iﬀ w ∈ Xϕ and N, w |=∗ iﬀ M, w |=∗ ϕ iﬀ (by assumption) M, w |=∗ ϕ iﬀ (by proposition 1) M, w |= ϕ.

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The above theorem explains the usefulness of recursion axioms and thus refutes the claim that the dynamics in PAL is redundant. There is another modeltheoretic way to understand the behavior of the submodel operation (.)|ϕ. In [4], the following notion is introduced. Definition 1. A model operation O respects bisimulation, if for all bisimular pointed models (M, w) and (N, v), the models (O(M ), w∗ ) and (O(N ), v ∗ ) are bisimular for some states w∗ and v ∗ . Theorem 4 ([4]). The submodel operation (.)|ϕ respects bisimulation for every PAL-formula ϕ. Let us generalize the notion of respecting. For any model operations ρ and τ , we call that ρ respects τ , if for any (pointed) models M and N , τ (M ) ∼ = N implies τ (ρ(M )) ∼ ρ(N ). Now let us consider some familiar model constructions and = related respecting behavior. The disjoint union i∈I Mi of a family {Mi : i ∈ I} of models, point generated submodel, (strong) homomorphism, embedding, isomorphism and bounded morphic image are deﬁned as in [7]. Proposition 2. The submodel operation (.)|ϕ respects disjoint union, embedding, isomorphism and bounded morphic image for every PAL-formula ϕ. Proof. For disjoint union, ( i∈I Mi )|ϕ = i∈I (Mi |ϕ), since i∈I Mi |= ϕ iﬀ Mi |= ϕ for all i ∈ I. For morphisms, take the map restricted to submodels as new morphism between submodels. The argument for bounded morphic image is as follows. Let f be a surjective bounded morphism from M to N . We show that g = f |ϕM is a surjective bounded morphism from M |ϕ to N |ϕ. The atomic and forth conditions are clear. For back condition, assume g(w)Sv . Thus wRw for some w in M and f (w ) = v . By v ∈ ϕN , we have w ∈ ϕM . Let us make more observations on homomorphisms. In general, homomorphism does not respect (.)|ϕ for all ϕ. But for certain formulas, we have the (quasi)respecting result. For instance, think about positive EL-formulas ϕ, i.e., formulas built from proposition letters, and ⊥ by using ∧, ∨, ♦ and . By the preservation result in [16], we may get the following theorem. Theorem 5. Let f be a surjective homomorphism from M to N . For each positive EL-formula ϕ, f |ϕM is a homomorphism from M |ϕ to N |ϕ. Proof. By the fact that all positive EL-formulas are preserved under surjective homomorphism, the homomorphic image under f is still a model consisting of ϕ-states. Then it is easy to show that f |ϕM is the required homomorphism. Now consider only homomorphism. It is known that homomorphism is a special case of simulation ([7, Section 2.7]). A simulation between a model M = (W, R, V ) and N = (X, S, U ) is a non-empty relation Z ⊆ W × X such that for all (w, v) ∈ Z, (i) w and v satisfy the same proposition letters, and (ii) if wRw , then there exists v ∈ X with vSv and vZv . A EL-formula is positive existential, if it is built from proposition letters using only ∧, ∨ and ♦. A PAL-formula is positive existential, if it is built from proposition letters by using only ∧, ∨, ♦ and ϕ. Then we have the following proposition.

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Proposition 3. Every positive existential PAL-formula is equivalent to a positive existential EL-formula and hence preserved under simulations. Proof. By recursion axioms, we get a positive existential EL-formula from a given positive existential PAL-formula. By theorem 2.78 of [7], the result follows. Corollary 1. For every positive existential PAL-formula ϕ, the submodel operation (.)|ϕ respects simulation and hence respects homomorphism. Now let us consider the operation of point generated submodels. If M is point generated from N , then every state in M is reachable from the generator in ﬁnitely many steps. By this fact, we get the following negative result: the submodel operation (.)|p does not respect point generated submodel. Let N = (X, S, U ) and M = (W, R, V ) be models where (i) X = {a, b, c, d, e}, S = {a, c, a, d, b, d, b, e}, U (p) = {a, c, d, e}, and (ii) W = {b, d, e}, R = {b, d, b, e}, V (p) = {d, e}. Then M is generated from N by the state b. However M |p has dead end e such that M |p is not point generated from N |p. Our next model operation is the ultraﬁlter extension. Let M = (W, R, V ) be a model. The ultraﬁlter extension of M is deﬁned as the model M ue = (W ue , Rue , V ue ) where W ue is the set of all ultraﬁlters over W , (ii) uRue v iﬀ ♦X ∈ u for all X ∈ v where ♦X = {w ∈ W : R(w) ∩ X = ∅}, and (iii) V ue (p) = {u : V (p) ∈ u} for each proposition letter p ([7], Section 2.5). By the recursion axioms, we know that the fundamental theorem on ultraﬁlter extension, i.e., M ue , u |= ϕ iﬀ ϕM ∈ u, also holds for all PAL-formulas ϕ. Proposition 4. For any model M and PAL-formula ϕ, (M |ϕ)ue ∼ = (M ue )|ϕ. Proof. Let M = (W, R, V ). Our ﬁrst observation is that any ultraﬁlter u ∈ (ϕM )ue is a proper ﬁlter over W . Thus we may deﬁne a function r : (ϕM )ue → ϕM ue by choosing only one r(u) that extending u in (ϕM )ue . Note that by ϕM ∈ u, we have ϕM ∈ r(u) and hence M ue , r(u) |= ϕ. Now we show that r is an embedding. It is easy to check that r is a bijection. Let r(u1 ) = u and r(u2 ) = u but u1 = u2 . Let X ∈ u1 but −X ∈ u2 . Thus X, −X ∈ u and so X ∩ −X = ∅ ∈ u , a contradiction. Now we show u(R|ϕ)ue v iﬀ r(u)Rue r(v). Assume u(R|ϕ)ue v and X ∈ r(v). If X ∈ v, then ♦X ∈ u and hence ♦X ∈ r(u). Suppose X ∈ v. Consider X ∩ϕM . Suppose X ∩ϕM ∈ v. Then ♦(X ∩ϕM ) ∈ u. By ♦(X ∩ ϕM ) ⊆ ♦X, we have ♦X ∈ u and so ♦X ∈ r(u). Suppose X ∩ ϕM ∈ v. Then −X ∪ −ϕM ∈ v. By ϕM ∈ v, we have −X ∈ v and so −X ∈ r(v), a contradiction. For valuation, we show u ∈ (V |ϕ)ue (p) iﬀ r(u) ∈ V ue (p) ∩ ϕM ue . Assume u ∈ (V |ϕ)ue (p). Then V |ϕ(p) ∈ u and so V |ϕ(p) ∈ r(u). By V (p) ∩ ϕM ⊆ V (p) and V (p) ∩ ϕM ⊆ ϕM , we have V (p) ∈ r(u) and ϕM ∈ r(u). Thus r(u) ∈ V ue (p) ∩ ϕM ue . Conversely, assume r(u) ∈ V ue (p) ∩ ϕM ue . Then V (p) ∈ r(u) and ϕM ∈ r(u). Then V (p) ∩ ϕM ∈ r(u). Thus V |ϕ(p) ∈ r(u). If V |ϕ(p) ∈ u, then −V |ϕ(p) ∈ u and so −V |ϕ(p) ∈ r(u), a contradiction. Hence V |ϕ(p) ∈ u. Conversely, we show that ϕM ue is embedded into (M |ϕ)ue . Deﬁne a map r∗ as follows: for any u ∈ ϕM ue , let Γ = {X ∩ ϕM : X ∈ u }. Then Γ has ﬁnite

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intersection property and so can be extended to an ultraﬁlter u over ϕM . Let r∗ (u ) = u for chosen u. By the choice of ultraﬁlters, r∗ is an embedding. Our last observation is about (.)|ϕ itself. Generally, it is not the case that M |ϕ ∼ = (M |ϕ)|ϕ, i.e., (.)|ϕ does not respect itself. For instance, consider a model M = ({a, b, c}, {a, b, b, c}, V ) where V (p) = {b, c}. Then M |♦p = ({a, b}, {a, b}, V |♦p) but (M |♦p)|♦p = ({a}, ∅, (V |♦p)|♦p). Intuitively, repeating the same announcement about knowledge many times will producing new information. The muddy children example ([4] and [11]) is a good witness. In the axiomatic aspect, in general, ϕϕψ is not equivalent to ϕψ. However, for factual announcement p, (M |p)|p ∼ = M |p. This explains that repeating factual announcement will not produce new information.

2

Algebraic Perspective on Public Announcements

Given a model M = (W, R, V ) and a PAL-formula ϕ, we have an inclusion map i : ϕM → W from M |ϕ to M . Thus from the perspective of algebraic duality, there may be a map from the dual algebra M + to (M |ϕ)+ . By duality again, we may ﬁnd a morphism from ((M |ϕ)+ )+ to (M + )+ which can be used as the algebraic interpretation of (.)|ϕ. In this section, let us give an algebraic semantics and show the algebraic soundness and completeness for PAL. Section 2 and section 3 contains some materials from the joint work in [14]. A map f : W → W is called a forth morphism from a Kripke frame F = (W, R) to F = (W , R ), if wRv implies f (w)R f (v) for all w, v ∈ W . Conversely, a map h : A → A from a Boolean algebra with operator (BAO) A = (A , + , − , 0 , ♦ ) to A = (A, +, −, 0, ♦) is called a continuous morphism, if (i) h is a Boolean homomorphism, i.e., h(a + b ) = h(a )+h(b ), h(− a ) = −h(a ) and h(0 ) = 0; and (ii) ♦h(a ) ≤ h(♦ a ). Recall that the dual of a frame F = (W, R) is a BAO F + = (℘(W ), ∪, −, ∅, ♦) where ♦X = {w ∈ W : R(w) ∩ X = ∅}. Fact 6. If f is a forth morphism from a frame F = (W, R) to F = (W , R ), then the map h : ℘(W ) → ℘(W ) given by h(X ) = f −1 [X ] is a continuous morphism from F + to F + . Conversely, given a BAO A = (A, +, −, 0, ♦), the dual of A is the frame A+ = (U f (A), R) where U f (A) is the set of all ultraﬁlters in A and R is deﬁned by: uRv iﬀ ♦a ∈ u for all a ∈ v. Fact 7. If h : A → A is a continuous morphism from a BAO A to A, then the map f : U f (A) → U f (A ) given by f (u ) = {a ∈ A : h(a) ∈ u } is a forth morphism from A+ to A+ . Now let us show that how we can give an algebraic interpretation for the submodel operation (.)|ϕ. The new information ϕ produces a submodel. Correspondingly, given any BAO A, ϕ denotes one element a in A. Hence the changed algebra will be produced by the meaning a of the formula ϕ in A. The idea is

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rather simple. We just take a quotient algebra modulo an equivalence relation given by a. In Kirpke models, M |ϕ looks like a model which is obtained from M modulo the information given by ϕ. Definition 2. Given a Boolean algebra A = (A, +, −, 0) and a ∈ A, deﬁne a relation ≡a over A by: b ≡a c iﬀ b · a = c · a. Let [b]a = {c ∈ A : b ≡a c} be the equivalent class of b. We may also delete the subscript a if no confusion will arise. The quotient algebra A/ ≡a can also be written as Aa . Fact 8. Let A be a Boolean algebra and a ∈ A. Then ≡a is a congruence in A. Proposition 5. For every b in a Boolean algebra A, there exists an unique c ∈ A with c ∈ [b]a and c ≤ a. Proof. By idempotency, (b · a) · a = b · a ∈ [b]a . If c ∈ [b]a and c ≤ a, then c = c · a = b · a. By uniqueness, we may deﬁne the following maps: (i) Let π : A → Aa be the canonical projection from a Boolean algebra A to Aa , i.e., π(b) = [b]a for each b in A. (ii) Deﬁne a map i : Aa → A by i ([b]a ) = b · a. By proposition 5, the map i is well-deﬁned. Thus each equivalent class in Aa has an unique representative. It is clear that π ◦ i = IdAa , the identity map over Aa . Now let us extend above notions to BAOs. Given any frame F = (W, R) and non-empty X ⊆ W , let F X be the subframe given by X. Recall the inclusion map i : F X → F . We will deﬁne the quotient BAO Aa of a BAO A given by the element a such that (F + )X ∼ = (F X )+ . Then we get the fact that i ([Y ]X ) = i(c) where c is the isomorphic element of [Y ]X in (F X )+ . Definition 3. Let A = (A, ♦) be a BAO where A = (A, +, −, 0) is a boolean algebra. For every a and b ∈ A, deﬁne ♦a [b] := [♦(b · a) · a], which is again equal to [♦(b · a)]. Let Aa = (Aa , ♦a ) be the algebra by adding ♦a to Aa . Proposition 6. For every BAO A = (A, ♦) and a in A, Aa is a BAO. Proof. By fact 8, Aa is a Boolean algebra. It suﬃces to show that ♦a is normal and additive. First, ♦a [⊥] = [♦(⊥ · a)] = [♦⊥] = [⊥]. Second, ♦a ([b] +a [c]) = ♦a ([b + c]) = [♦((b + c) · a)] = [♦(b · a + c · a)] = [♦(b · a) + ♦(c · a)] = [♦(b · a)] +a [♦(c · a)] = ♦a [b] +a ♦a [c]. Proposition 7. Let F = (W, R) be a frame and X is a non-empty subset of W . Then (F + )X ∼ = (F X )+ . Proof. First, (RX )−1 [Y ∩ X] = (R−1 [Y ∩ X]) ∩ X for every Y ⊆ W . Now we show that r : [Y ]X → Y ∩X is an isomorphism from (F + )X to (F X )+ . It is clear that r is a function. The fact that r is a bijection is easy to show. For injection, let Y1 = Y2 . Then Y1 ∩ X = Y2 ∩ X. Thus r([Y1 ]X ) = r([Y2 ]X ). For surjectivity,

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for Y ∈ ℘(X), we have Y = Y ∩ X = r([Y ]X ). Moreover, the following equations show that r is a BAO-homomorphism: (i) r(−X [Y ]) = r([−Y ]) = −Y ∩ X = X − (Y ∩ X) = −r([Y ]), (ii) r([Y1 ] ∪ [Y2 ]) = r([Y1 ∪ Y2 ]) = (Y1 ∪ Y2 ) ∩ X = (Y1 ∩ X) ∪ (Y2 ∩ X) = r([Y1 ]) ∩ r([Y2 ]), (iii) r(♦X [Y ]) = r([♦(Y ∩ X)]) = r([R−1 (Y ∩ X)]) = R−1(Y ∩ X) ∩ X = (RX )−1 (r([Y ])) = ♦r([Y ]). Now let us go to the algebraic semantics for public announcements. Recall that M, w |= ϕψ iﬀ M, w |= ϕ and M |ϕ, w |= ψ. This is equivalent to say that w ∈ ϕψM iﬀ there exists w ∈ ϕM with i(w ) = w ∈ ϕM and w ∈ ψM|ϕ . Since i is the inclusion map, we have w ∈ ϕψM iﬀ w ∈ ϕM ∩i[ψM|ϕ ]. Thus ϕψM = ϕM ∩ i[ψM|ϕ ]. By proposition 7, ϕM ∩ i[ψM|ϕ ] = i ([ψM|ϕ ]). Now let us deﬁne the algebraic semantics for PAL. Let A = (A, ♦) be a BAO. An assignment θ in A is a map from the set Φ of proposition letters to A. An algebraic model is a pair M = (A, θ) of a BAO with an assignment. Given a ∈ A, deﬁne Ma = (Aa , θa ) where Aa = (Aa , ♦a ) and θ a : Φ → A/ ≡a is the assignment such that θa (p) = π(θ(p)) = [θ(p)]a . Definition 4. Given an algebraic model M = (A, θ) over a BAO A = (A, ♦), deﬁne meaning [.]M : P AL → A of PAL-formulas in M recursively by: pM = θ(p), ¬ϕM = −ϕM , ϕ ∧ ψM = ϕM · ψM , ♦ϕM = ♦ϕM and ϕψM = ϕM · i (ψMϕM ) where i : AϕM → A maps each equivalent class to its canonical representative. Under this semantics, we have the algebraic soundness and completeness result. Before proving this, we give the following lemma which is easy to check. Lemma 1. Let Aa = (Aa , ♦a ) be the quotient BAO given by Boolean algebra A and a. Then the following hold: (i) a · i ([−b]) = a · −i ([b]), (ii) i ([b · c]) = i ([b]) · i ([c]), and (iii) a · i ([♦b]) = a · ♦i ([b]). Theorem 9. The PAL axiomatization is sound and complete with respect to the class of all algebraic models. Proof. The completeness is clear since each PAL formula is equivalent to a ELformula. So if P AL ϕ, then BAO ϕ by algebraic completeness of EL. For the soundness, it suﬃces to show the validity of recursion axioms. Given any algebraic model M, we have the following equations: let ϕM = a. (i) ϕpM = a · i (pMa ) = a · i (θa (p)) = a · i ([θ(p)]) = a · (θ(p) · a) = a · pM = ϕ ∧ pM , (ii) ϕ¬ψM = a · i (¬ψMa ) = a · i (−a ψMa ) = a · −i (ψMa ) = a · −(a · i (ψMa )) = a · −ϕψM = ϕ ∧ ¬ϕψM , (iii) ϕ(ψ ∧ ξ)M = a · i (ψM ·a ξM ) = a · i (ψM ) · i (ξM ) = a · ϕξM = ϕψ ∧ ϕξM , and (iv) ϕ♦ψM = a · i (♦ψMa ) = a · i (♦a (ψMa )) = a · ♦i (ψMa ) = a · ♦(a · i (ψMa )) = a · ♦(ϕψM ) = a · ♦ϕψM = ϕ ∧ ♦ϕψM .

3

Intuitionistic Logic and PAL

Recently, van Benthem explored the link between intuitionistic logic and information ﬂow in [3]. One observation is that propositional intuitionistic logic

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(Int) describes stable knowledge at a certain state no matter what the future informational processes will be. The way of doing is to loading the meaning of logical constants. For instance, the intuitionistic negation ¬A says that the current state has no knowledge A and we will never get A in the future. Since we have a static information theory at hand, let us look at its dynamic extension. First we consider only the Int with public announcements, and later we take the intuitionistic epistemic modal logic as the static part. The language of Int consists of a set Φ of proposition letters and connectives ∧, ∨, → and ⊥. The set of Int-formulas is given by the following inductive rule: ϕ ::= p | ⊥ | ϕ → ψ | ϕ ∧ ψ | ϕ ∨ ψ where p ∈ Φ. An Int-frame is a partial ordered set F = (W, R). A subset X ⊆ W is called upward closed, if ∀xy(x ∈ X ∧ xRy → y ∈ X). Let U p(W ) be the set of all upward closed sets. An Int-model M = (W, R, V ) consists of a Int-frame (W, R) and a valuation V : Φ → U p(W ). The satisfaction M, w |= ϕ for any Intmodel M and state w in M and Int-formula ϕ is deﬁned recursively as follows: M, w |= p iﬀ w ∈ V (p) M, w ⊥ M, w |= ϕ → ψ iﬀ ∀v(wRv & M, v |= ϕ ⇒ M, v |= ψ) M, w |= ϕ ∧ ψ iﬀ M, w |= ϕ and M, w |= ψ M, w |= ϕ ∨ ψ iﬀ M, w |= ϕ or M, w |= ψ Deﬁne ¬ϕ := ϕ → ⊥. Then M, w |= ¬ϕ iﬀ ∀v(wRv ⇒ M, v ψ). It is easy to see that the truth set ϕM of each Int-formula ϕ is upward closed. Now let us make the PAL extension of Int. Note that for any Int-model M and formula ϕ, the submodel M |ϕ is still an Int-model. Theorem 10. Let Int be the sound and complete axiomatization. Then Int plus the following recursion axioms is sound and complete Int-PAL with respect the class of all Int-frames: ϕp ↔ ϕ ∧ p, ϕ⊥ ↔ ⊥, ϕ(ψ ∧ ξ) ↔ ϕψ ∧ ϕξ, ϕ(ψ ∨ ξ) ↔ ϕψ ∨ ϕξ and ϕ(ψ → ξ) ↔ ϕ ∧ (ϕψ → ϕξ). Proof. The completeness reduces to that of Int by recursion axioms. For soundness, check only the last one. Assume M, w |= ϕ(ψ → ξ). Then M, w |= ϕ. Suppose wRv and M, v |= ϕψ. Then M, v |= ϕ and M |ϕ, v |= ψ. By assumption, M |ϕ, w |= ψ → ξ. By wRv and w, v ∈ ϕM , we have w(R|ϕ)v. Thus M |ϕ, v |= ξ. Then M, v |= ϕξ. Conversely, assume M, w |= ϕ ∧ (ϕψ → ϕξ), wRv and M, v |= ϕψ. By assumption, M, v |= ϕξ and hence M |ϕ, v |= ξ. It is easy to compute ϕ¬ψ ↔ ϕ∧¬ϕψ in which ¬ is intuitionistic. Now we see that every Int-PAL formula is equivalent to an Int one. By G¨ odel translation T r, Int-PAL is embedded into modal logic S4 ([10]). We may also add the translation clause T r(ϕψ) = T r(ϕ)T r(ψ) and then it is easy to show that Int-PAL is embedded into the PAL extension of S4. A more general question is as follows: assume that a static logic L1 is embedded into L2 . Does the given dynamics

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respect embedding? That is to determine whether the dynamic extension of L1 is embedded into that of L2 or not. It is clear that PAL respects the embedding of Int into S4. Now let us move to our second step: adding knowledge operator into Int-PAL. We consider epistemic intuitionistic modal logic (EIL) as the static base. The set of all EIL-PAL formulas is given by the following inductive rule: ϕ ::= p | ⊥ | ϕ → ψ | ϕ ∧ ψ | ϕ ∨ ψ | ϕ | ♦ϕ | ϕψ | [ϕ]ψ where p ∈ Φ. Note that and ♦, ϕ and [ϕ] are not dual operators now. The minimal intuitionistic modal logic IK introuced in [12] and [9] is sound and complete with respect to the class of all frames of the form F = (X, ≤, R) where X is a non-empty set, ≤ is a quasi-order and R is a binary relation such that (≤−1 ◦R) ⊆ (R◦ ≤−1 ) and (R◦ ≤) ⊆ (≤ ◦R). For algebraic study on intuitionistic modal logic, see [6]. Definition 5. Given a Heyting alegbra (HA) A = (A, ∧, ∨, →, ⊥, ), and a, b ∈ A, deﬁne ♦a and a in the quotient Aa by ♦a [b] := [♦(b ∧ a) ∧ a] = [♦(b ∧ a)] and a [b] := [a → (a → b)] = [(a → b)]. Let Aa = (Aa , ♦a , a ) be the quotient algebra given by a in A. The reason for [a → (a → b)] = [(a → b)] is that the equation x ∧ y = x ∧ (x → y) is valid in Heyting algebras. Fact 11. Both ♦a and a are normal modal operators. Proof. By the following equations: (i) ♦a [⊥] = [♦(⊥ ∧ a)] = [♦⊥] = [⊥], (ii) ♦a [b ∨ c] = [♦((b ∨ c) ∧ a)] = [♦((b ∧ a) ∨ (c ∧ a))] = [♦(b ∧ a) ∨ ♦(c ∧ a)] = [♦(b ∧ a)] ∨a [♦(c ∧ a)] = ♦a [b] ∨a ♦a [b], (iii) a [] = [a → (a → )] = [a → ] (since = a → ) = [a → ] = [], and (iv) a [b ∧ c] = [a → (a → (b ∧ c))] = [a → ((a → b) ∧ (a → c))] = [a → ((a → b) ∧ (a → c))] = [(a → (a → b)) ∧ (a → (a → c))] = [a → (a → b)] ∧a [a → (a → c)]. For (iv), use the valid equation x → (y ∧ z) = (x → y) ∧ (x → z). An algebraic EIL-model is a tuple M = (A, , ♦, θ) where (A, , ♦) is a Heyting algebra with operator (HAO) and θ : Φ → A is an assignment. Deﬁne the quotient model of M given by a ∈ A as Ma = (Aa , a , ♦a , θa ) where θa : Φ → Aa with θa (p) = π(θ(p)) = [θ(p)]a . The meaning of formulas is deﬁned as follows. Definition 6. Given an EIL-algebraic model M = (A, θ) over an EIL-algebra A = (A, , ♦), deﬁne the meaning function [.]M from EIL-PAL formulas to A recursively as in Boolean case plus the following: (i) ♦ϕM = ♦ϕM , (ii) ϕM = ϕM , (iii) ϕψM = ϕM ∧ i (ψMϕM ), and (iv) [ϕ]ψM = ϕM → i (ψMϕM ). Lemma 2. For any quotient algebra Aa = (Aa , a , ♦a ) of EIL-algebra A given by a, (i) i ([b → c]) = a ∧ (i ([b]) → i ([c])) and (ii) i ([b]) = a ∧ (a → i ([b])).

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Proof. For the ﬁrst equation, by the HA-valid equation x ∧ (y → z) = x ∧ ((x ∧ y) → z), we have i ([b → c]) = a ∧ (b → c) = a ∧ ((a ∧ b) → c). Then ((a∧b) → c) = (a∧b) → ((a∧b)∧c) = (a∧b) → ((a∧b)∧(a∧c)) = (a∧b) → (a∧c). Thus i ([b → c]) = a ∧ ((a ∧ b) → (a ∧ c)) = a ∧ (i ([b]) → i ([c])). For the second one, it is argued as follows: i (b) = i (a [b]) = i ([(a → b)]) = a ∧ (a → b) = a ∧ (a → (a ∧ b)) = a ∧ (a → i ([b])). Theorem 12. A sound and complete equational EIL-logic Λ plus the following recursion equations is sound and complete with respect to HAO models: ϕp = ϕ ∧ p ϕ⊥ = ⊥ ϕ(ψ → ξ) = ϕ ∧ (ϕψ → ϕξ) ϕ(ψ ∧ ξ) = ϕψ ∧ ϕξ ϕ(ψ ∨ ξ) = ϕψ ∨ ϕξ ϕψ = ϕ ∧ ϕψ ϕ♦ψ = ϕ ∧ ♦ϕψ

[ϕ]p = ϕ → p [ϕ] = [ϕ](ψ → ξ) = ϕψ → ϕξ [ϕ](ψ ∧ ξ) = [ϕ]ψ ∧ [ϕ]ξ [ϕ](ψ ∨ ξ) = ϕ → (ϕϕ ∨ ϕξ) [ϕ]ψ = ϕ → [ϕ]ψ [ϕ]♦ψ = ϕ → ♦ϕψ.

Proof. The completeness is obtained by recursion equations. We choose only some of them to show their validity. Let ϕM = a. (i) [ϕ](ψ ∨ ξ)M = a → i (ψ ∨ ξM a ) = a → (i (ψM a ) ∨ i (ξM a )) = a → (a ∧ (i (ψM a ∨ i (ξM a )))) = a → ((a ∧ i (ψM a )) ∨ (a ∧ i (ξM a ))) = a → (ϕψM ∨ ϕξM ) = ϕ → ϕψ ∨ ϕξM , (ii) [ϕ](ψ → ξ) = a → i (ψ → ξM a ) = a → (a ∧ (i (ψM a ) → i (ξM a ))) = a → (i (ψM a ) → i (ξM a )) = (a ∧ i (ψM a )) → i (ξM a ) = ϕψM → (ϕψM ∧ a ∧ i (ξM a )) = ϕψM → (ϕψM ∧ ϕξM ) = ϕψM → ϕξM = ϕψ → ϕξM , (iii) ϕψM = a ∧ i (ψM a ) = a ∧ (a → i (ψM a )) = a ∧ [ϕ]ψM = ϕ ∧ [ϕ]ψM , (iv) [ϕ]♦ψM = a → i (♦ψM a ) = a → (a ∧ ♦ϕψM ) = a → ♦ϕψM = ϕ → ♦ϕψM , and (v) [ϕ]ψM = a → i (ψM a ) = a → (a ∧ ([ϕ]ψM )) = a → ([ϕ]ψM ) = ϕ → [ϕ]ψM .

4

First-Order Logic and Public Announcements

In this section, we brieﬂy discuss the PAL extension of ﬁrst-order logic (FO). The factual announcement of FO-formula is rather easy to understand. Take any FO-language and add announcement formulas [ϕ]ψ. It is hard to see what a ﬁrstorder submodel (.)|ϕ will be. But we may interpret [ϕ]ψ as the implication ϕ → ψ in the same model without changing. This is plausible. For instance, the formula [P a]∃xP x says that there exists one individual in the domain satisfying P x after announcing the individual denoted by a satisﬁes P x. For non-factual case, we may think quantiﬁers in the other way round. In [2], a modal interpretation for quantiﬁers is proposed as the foundation of predicate logic. Given a ﬁrst-order model M = (D, I) and an assignment α : V → D where D is the domain, I is the interpretation of non-logical symbols, and V is the set of all individual variables,

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we may interpret existential quantiﬁer as follows: M, α |= ∃xϕ iﬀ there exists β ∈ D V with αRx β and M, β |= ϕ, where αRx β is read as that α agrees with β save possibly for x. Thus ∃xϕ can be taken as a modal formula ♦x ϕ interpreted in the assignment space over M . Definition 7. Let M = (D, I) be a FO-model and V the set of all individual variables. Deﬁne the assignment space over M as S M = (D V , {Rx }x∈V ) where Rx ⊆ D V × DV is deﬁned by: αRx β iﬀ α and β disagree at most at x. Now let us introduce the PAL extension of FO for talking about the information ﬂow in assignment spaces. The formulas of FO-PAL is obtained by adding ϕψ to the inductive rule of FO. Definition 8. Given a FO-model M = (D, I) and FO-PAL formula ϕ, let V S M |ϕ = (Dϕ , {Rxϕ }x∈V ) be the subspace given by the truth set ϕS M = {α ∈ V M V V V = ϕS M and Rxϕ = Rx ∩ (Dϕ × Dϕ ). D : S , α |= ϕ}, i.e., Dϕ The satisfaction relation S M , α |= ϕ is deﬁned obviously. The interpretation of non-logical symbols in S M is the same as that in M . Especially, we have the semantics clause: S M , α |= ϕψ iﬀ S M , α |= ϕ and S M |ϕ, α |= ψ. It is easy to obtain the following theorem. Theorem 13. First-order logic plus the following recursion axioms is sound and complete with respect to all assignment spaces: ϕδ ↔ ϕ ∧ δ for atomic δ, ϕ¬ψ ↔ ϕ ∧ ¬ϕψ, ϕ(ψ ∧ ξ) ↔ ϕψ ∧ ϕξ and ϕ♦x ψ ↔ ϕ ∧ ♦x ϕψ. Now let us move to the ﬁrst-order epistemic modal logic. The quantiﬁed modal logic (QML) has been explored for may years. The following basic syntactic and semantics notions can be found in [13] and [8]. The language of QML is the ﬁrstorder extension of propositional modal language. For semantics, a problem is to treat with the interaction between quantiﬁcation domain and modal accessibility relation. A constant domain model is M = (W, R, D, {Iw }w∈W ) where W is the non-empty set of states, R ⊆ W × W , D is the individual domain and Iw interprets all non-logical symbols in D at the state w. Then we have the following basic semantic deﬁnition. Definition 9. Let M = (W, R, D, {Iw }w∈W ) be a constant domain model and α an assignment. Let α ∼x β mean that α agrees with β on all variables save possibly x. Then deﬁne M, α, w |= ϕ recursively as follows: M, α, w |= P x1 , . . . , xn iﬀ (α(x1 ), . . . , α(xn )) ∈ Iw (P ) M, α, w |= x = y iﬀ α(x) = α(y) M, α, w |= ¬ϕ iﬀ M, α, w ϕ M, α, w |= ϕ ∧ ψ iﬀ M, α, w |= ϕ and M, α, w |= ψ M, α, w |= ϕ iﬀ for all v ∈ W with Rwv, M, α, v |= ϕ M, α, w |= ∀xϕ iﬀ for all assignment β with α ∼x β, M, β, w |= ϕ.

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There are some philosophical problems about Barcan and converse Barcan formulas and related domain conditions in epistemic logic. Here we just look at the constant domain semantics which makes both of them valid. Now let us look at the PAL extension of QML under constant domain semantics. The set of QML-PAL formulas is given by adding [ϕ]ψ to the inductive rule of QML. Given any constant domain model M = (W, R, D, {Iw }w∈W ) and assignment α in M , the truth set of a QML-PAL formula ϕ is deﬁned as ϕM,α := {w ∈ W : M, α, w |= ϕ}. Then we may given the following semantics clause for announcement: M, α, w |= [ϕ]ψ iﬀ M, α, w |= ϕ implies M |ϕ, α, w |= ψ where M |ϕ = (W ϕ , Rϕ , D, {Iw }w ∈ W ϕ ) is the submodel of M given by ϕM,α , i.e., W ϕ = ϕM,α and Rϕ = R ∩ (W ϕ × W ϕ ). Then we may get the following theorem. Let QML be the sound and complete axiomatization of the constant domain semantic validities [8]. Theorem 14. QML plus the following recursion axioms is sound and complete with respect to the class of all constant domain models: [ϕ]δ ↔ (ϕ → δ) f or atomic δ, [ϕ]¬ψ ↔ (ϕ → ¬[ϕ]ψ), [ϕ](ψ ∧ ξ) ↔ ([ϕ]ψ ∧ [ϕ]ξ), [ϕ]∀xψ ↔ (ϕ → ∀xψ) and [ϕ]ψ ↔ (ϕ → (ϕ → [ϕ]ψ)). Proof. For soundness, we show only the recursion axiom for [ϕ]∀xψ. The key fact is that M |ϕ, α, w |= ∀xψ iﬀ M, α, w |= ∀xψ. Suppose M, βxa , w ψ. Since w is in M |ϕ, we have M |ϕ, βxa , w ψ. And vice versa.

5

Conclusions and Open Questions

We have shown some mathematical problems for PAL, including the respecting phenomena and the algebraic semantics for PAL. We also explored two sorts of variant of PAL: the intuitionistic variants and ﬁrst-order ones. However, we have the following open questions that can be explored further. (i) For respecting results, a question is to characterize the syntax of ϕ such that (.)|ϕ respects the given model operation, for instance, simulation and homomorphism. (ii) The current recursion axioms for PAL don’t allow arbitrary substitution for atomic case. Then the natural question is whether all schematic validities of PAL is ﬁnitely axiomatizable. This question is also presented in [4]. (iii) We have introduced the updated algebra as the quotient algebra in section 2. However, we may add announcement operators into BAOs which satisfy, for instance, recursion equations. Then the problem is whether the J´ onsson-Tarski type representation theorem exists. (iv) We may consider the ﬁrst-order extension of the propositional intuitionistic PAL and explore the logical behavior of announcements. Similarly, we may consider more static base logics, for instance, positive or distributive modal logics, guarded fragments of FO, even second-order logic, and so on.

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(v) T. ˚ Agotnes et.al. presented in [1] an extension of PAL with group announcement of the form Gϕ, where G is a group of agents, with the intuitive meaning that G can jointly execute a publicly observable action such that ϕ will be true afterwards. Now the question is how can we extend what we have done in this paper to group announcement logic. Moreover, it is well-known that DEL ([4]) is a natural extension of PAL. Thus the mathematical problems shown above can also be raised for DEL. Acknowledgement. The author is supported by the Key Project of Chinese Ministry of Education (Grant No. 05jzd008) and the Ket Project of Humanities of Chongqing ‘Researches in Modal Model Theory’. The algebraic semantics for PAL and Int-PAL is a part of my joint work with Alessandra Palmigiano (Amsterdam) and Mehrnoosh Sadrzadeh (Oxford). Thanks are given to Prof. J. van Benthem for his unpublished manuscript [4].

References ˚gotnes, T., Balbiani, P., van Ditmarsch, H., Seban, P.: Group announcement 1. A logic. Journal of Applied Logic 8, 62–81 (2010) 2. van Benthem, J.: Exploring Logical Dynamics. CSLI Publications, Stanford (1996) 3. van Benthem, J.: The information in intuitionistic logic. Synthese 167, 251–270 (2009) 4. van Benthem, J.: Logical Dynamics and Information Flow (2010) (manuscript) 5. van Benthem, J.: Modal Logic for Open Minds. CSLI Publications, Stanford (2010) 6. Bezhanishvili, G.: Varieties of monadic heyting algebras part i. Studia Logica 61, 367–402 (1998) 7. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Univeristy Press, Cambridge (2001) 8. Bra¨ uner, T., Ghilardi, S.: First-order Modal Logic. In: Handbook of Modal Logic, pp. 208–219. Elsevier, Amsterdam (2007) 9. Celani, S.: Remarks on intuitionistic modal logics. Divulgaciones Matem´ aticas 9, 137–147 (2001) 10. Chagrov, A., Zakharyaschev, M.: Modal Logic. Clarendon Press, Oxford (1997) 11. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Springer, Heidelberg (2007) 12. Fischer-Servi, G.: Axiomatizations for some intuitionistic modal logics. Rend. Sem. Mat. Polit. de Torino 42, 179–194 (1984) 13. Fitting, M., Mendelsohn, R.: First-order Modal Logic. Kluwer Academic Publishers, Dordrecht (1998) 14. Ma, M., Palmigiano, A., Sadrzadeh, M.: Algebraic semantics and model completeness for intuitionistic public announcement logic. In: van Ditmarsch, H., Lang, J., Ju, S. (eds.) LORI 2011. LNCS(LNAI), vol. 6953, pp. 394–395. Springer, Heidelberg (2011) 15. Plaza, J.: Logics of public communications. In: Proceedings of the 4th International Symposium on Methodologies for Intelligent Systems, pp. 201–216 (1989) 16. de Rijke, M.: Modal model theory. Report CS-R9517, Computer Science/Department of Software Technology, University of Amsterdam (1995)

Logics of Belief over Weighted Structures Minghui Ma and Meiyun Guo Institute of Logic and Intelligence, Southwest University Beibei, Chongqing, China, 400715 mmh.thu,[email protected]

Abstract. We explore logics of belief over weighted structures under the supposition that everything believed by an agent has a weight in the range of agent’s belief. We first define static graded belief logics which are complete with respect to the class of all weighted frames. Furthermore, we discuss their public announcement and dynamic epistemic extensions. We may also define notions of plausible belief by comparing weights of formulas at the current state in a weighted model. This approach is not a new one but we provide new logics and their dynamic extensions which can capture some intuitive notions of belief and their dynamics.

1

Introduction

Knowledge and belief were once taken as proposition attitudes. The eﬀort to give formal semantics for them has been tried by philosophers for a rather long time. One approach to belief is to suppose that every agent owns a collection of beliefs which can be revised. Under this idea, the famous AGM -theory of belief revision developed by C. Alchourr` on, P. G¨ardenfors and D. Makinson in [1] postulates some rational properties of operators performing belief revision. These operators include expansion (addition of a belief without check of consistency), revision (addition of a consistent belief) and contraction (remove of a belief). Another approach begins with J. Hintikka’s idea to interpret belief operator in Kripke semantics [16]. In recent years, belief change modeled by dynamic updates over Kripke models has been explored. Plaza ﬁrst proposed in [23] public announcement logic to model the change of models when a hard information comes. Baltag, Moss and Solecki developed in [3] plausibility models in which belief can be interpreted by plausibility orders between states. Dynamics of plausible belief has been developed recently. For a reference, see [4]. Both approaches to belief mentioned above belongs to the ‘qualitative’ category, while some other approaches are ‘quantitative’. For instance, R. Fagin and J. Halpen [12] proposed a logic of reasoning about knowledge and probability in which we can express that the agent knows that the probability of a proposition is some real number r ∈ [0, 1]. Dynamics in probabilistic epistemic logic has also been developed in [18] and [6]. Moreover, there are some works to capture degrees of knowledge and belief in terms of graded models or probabilistic measures. See, for instance, [2] and [17]. The general idea behind these quantitative approaches is that grading belief is an important measure for distinguishing agent’s degree H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 206–219, 2011. Springer-Verlag Berlin Heidelberg 2011

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of belief. Even the probabilistic approach emphasizes the uncertainty which is a essential part of belief content. We continue to promote the probabilistic approach in this paper to combine quality and quantity together. The idea is the following. We take weighted structures as the ontology for interpreting belief. A weighted frame consists of a set W of states, a binary relation R and a function σ : R → ω + 1 \ {0} where ω + 1 is the set of all ﬁnite cardinals plus ℵ0 . Numbers i ≤ ω are called weights between states. Weights can be used in several ways. First, we can compute the weight of a formula at the current state which can be taken as the degree of the belief. Another use of weights is to interpret the notion of belief. Intuitively, the agent believes φ if she reckons that the weight of φ is at least as that of ¬φ, or equivalently, φ is at least plausible as ¬φ. These interpretations are not new, but we provide interesting and simple logics and their dynamic extensions. The outline of the paper is as follows. Section 2 gives static graded belief logics and their dynamic extensions under the idea that we may form a chain of uncertain beliefs which are interpreted in weighted models. Then we may explore notions of plausible belief deﬁned by comparing weights of formulas in section 3 to section 5. Finally, we give conclusions and open questions in section 6.

2 2.1

Dynamic Graded Belief Logics Epistemic Weighted Frames

Consider the two sentence ‘John believes that it will rain tomorrow (p)’ and ‘John believes with a higher degree that it will snow tomorrow (q)’. In John’s range of belief, he needs to compare the strengths of two future propositions p and q. Then he may explain that the probability of p not to happen is less than 80% and that of q is less than 30%. Thus he believes q with a higher degree than p. We may use B80 p and B30 q to formalize the two propositions. The index number is lower, the degree of belief is higher. These formulas will be interpreted in weighted frames deﬁned as follows. Definition 1. A weighted frame is a triple F = (W, R, σ) where W = ∅ is a set of epistemic worlds (states), R ⊆ W × W is an epistemic accessibility relation and σ : R → ω + 1 \ {0} is a function which assigns a weight in ω + 1 \ {0} to each pair in R. Given a weighted frame F = (W, R) and w ∈ W , deﬁne R(w) = {v ∈ W : wRv}. We may also deﬁne a function σw : R(w) → ω + 1 \ {0} by σw (v) = σ(w, v ). For each subset X ⊆ R(w), deﬁne σw (X) by v∈X σw (v), if X = ∅ σw (X) = 0, otherwise. The expression v∈X σw (v) denotes the sum of all cardinals σw (v). Weighted frames are equivalent to simple structures called multi-graphs in [11]. A multigraph is a pair S = (S, ρ) where S = ∅ is a set of states and ρ : S × S → ω + 1 is a function.

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Given a weighted frame F = (W, R, σ), we deﬁne F • = (W, ρ) by σ(w, v ), if wRv. ρ(w, v) = 0, otherwise. Conversely, given a multi-graph S = (S, ρ), the weighted frame S• = (S, R, σ) can be deﬁned by (i) wRv iﬀ ρ(w, v) > 0, and (ii) σ(w, v ) = ρ(w, v) for each w, v ∈ R. The following fact is easy to show. Fact 1. For any weight frame F and multi-graph S, F ∼ = (F • )• and S ∼ = (S• )• . 2.2

Static Graded Belief Logic

One of the advantages of weighted structures is that we can talk about degrees of belief. This insight has been given by several authors. See, for instance, G. Aucher’s work [2]. In Aucher’s approach, a formula of the form Bn φ is true if φ is true at all accessible states with degree at most n. However, in our setting, we may use ‘weights’ to model the degree of a proposition in a diﬀerent way. A formula Bn φ is true at the current state, if the agent reckons that the sum of weights assigned to each successor ¬φ-states is less than n. Thus the number n is greater, the degree of the belief is lower. When n = 1, the agent completely believes φ. This interpretation suggests that we get a graded modal logic over weighted structures. For graded modal logic, see [13], [9], [24] and [21]. Definition 2. The language for graded belief logic (BLg ) consists of a set Φ of propositional letter and graded belief operators Bn for each natural number n > 0. The set of BLg -formulas is given by the following inductive rule: φ ::= p | ¬φ | φ ∨ ψ | Bn φ where p ∈ Φ and n > 0. Deﬁne Bn φ := ¬Bn ¬φ and Bn !φ := Bn φ ∧ ¬Bn+1 φ. Let B0 !φ := ¬B1 φ. We write B1 φ as Bφ and B1 φ as B φ. A weighted model is a pair M = (F, V ) where F is a weighted frame and V is a valuation which assigns a subset of states to each proposition letter in Φ. Definition 3. Deﬁne truth of a BLg -formula φ at a state w in a weighted model M = (W, R, σ, V ) (notation: M, w |= φ) recursively as follows: M, w |= p iﬀ w ∈ V (p). M, w |= ¬φ iﬀ M, w φ. M, w |= φ ∨ ψ iﬀ M, w |= φ or M, w |= ψ. M, w |= Bn φ iﬀ σw ((W \ φM ) ∩ R(w)) < n. where φM = {w : M, w |= φ}. Other semantic notions are deﬁned as usual.

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Thus for other belief operators, we have M, w |= Bn φ iﬀ σw (φM ∩ R(w)) ≥ n and M, w |= Bn !φ iﬀ σw (φM ∩ R(w)) = n. Intuitively, Bn φ is true, if the agent reckons that the degree of truth of φ is at least n. The formula Bn !φ means that he reckons that the truth of φ is exactly n. Our setting is a little bit diﬀerent from the Kripke semantics for graded modal logic discussed in [13]. However, we use weighted structures to interpret graded operators. We can also show the completeness of the minimal graded modal logic given in [9] can be transferred to our setting. We only give the outline. Details can be found in [21]. Definition 4. The minimal graded belief logic BLg contains the following axiom schemata and inference rules: (1) (2) (3) (4) (5) (6)

All instances of propositional tautologies. Bn+1 φ → Bn φ. B(φ → ψ) ∧ Bn φ → Bn ψ. ¬B1 (φ ∧ ψ) ∧ Bm !φ ∧ Bn !ψ → Bm+n !(φ ∨ ψ). M P : from φ and φ → ψ infer ψ. Gen: from φ infer Bφ.

A normal graded belief logic Λ is a set of BLg -formulas which contains axioms (1)-(4) and closed under M P and Gen. Given a set Σ of formulas, the normal graded belief logic generated by Σ is denoted by Λ = BLg ⊕ Σ. Axiom (2) says that if the agent reckons that the degree of truth of φ is at least n + 1, then he would also reckons that the degree of truth of φ is at least n. Axiom (3) says that if the agent complete believes that φ → ψ and he reckons that the sum of weights assigned to each successor φ-states is at least n, then he would also reckons the sum of weights assigned to each successor ψ-states is at least n. While Axiom (4) says that if the agent reckons that the true states of φ and ψ are disjoint in his current successors, then he reckons that the degree of truth of φ ∨ ψ is exactly the sum of weights of φ and ψ. Let us deﬁne the canonical weighted model for normal graded belief logic by using which we can prove the strong completeness. We assume some knowledge about canonical models for basic modal logic. For details, see [7]. Definition 5. For any normal graded belief logic Λ, deﬁne its canonical weighted model M Λ = (W Λ , RΛ , σΛ , V Λ ) as follows: • W Λ is the set of all maximal Λ-consistent set of formulas. • uRΛ v iﬀ φ ∈ v for all Bφ ∈ u. • σ Λ (u, v ) is deﬁned as follows: for each pair u, v ∈ RΛ , ω, if ∀φ ∈ v∀n > 0(Bn φ ∈ u). σΛ (u, v ) = min{n ∈ ω : Bn !φ ∈ u & φ ∈ v}, otherwise. • V Λ (p) = {u ∈ W Λ : p ∈ u} for each proposition letter p. For the soundness of the deﬁnition of σΛ , it suﬃces to show that the minimal number exists in the second case by the following lemma.

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Lemma 1. Let u be any maximal Λ-consistent set. For every formula φ, either ∀n > 0(Bn φ ∈ u), or ∃n > 0(Bn !φ) ∈ u. The following is the canonical weighted model theorem for any normal belief logic Λ the proof of which can be found in [21]. Theorem 2. For all BLg -formula φ, M Λ , u |= φ iﬀ φ ∈ u. Corollary 1. The minimal graded belief logic BLg is strongly complete with respect to the class of all weighted frames. In the proof of canonical weighted model theorem, we used the following fact which is again useful in proving strong completeness of normal graded belief logics. Here we use capital Latin letters X, Y, Z etc. as variables for non-empty ﬁnite sets of the domain. Fact 3. For any normal graded belief logic Λ, maximal Λ-consistent sets u and v, non-empty ﬁnite set of maximal Λ-consistent sets X, and n > 0, we have (i) σΛ (u, v ) ≥ n iﬀ ∀φ ∈ v(Bn φ ∈ u). Λ X(Bn φ ∈ u). (ii) v∈X σ (u, v ) ≥ n iﬀ ∀φ ∈ Now let us discuss some more interesting graded axioms for belief which are variants of the logic KD45 of belief. The ﬁrst one is Bp → B p, which is equivalent to B , meaning that the agent reckons that p is possible in her range of belief whenever she believes p. This formula corresponds to the condition ∀x∃y(xRy) ∧ ∀x∃y(σ(x, y ) ≥ 1) over weighted frames. However, we may consider the more general form Bn p → Bn p which is not equivalent to Bn since the operator Bn is not distributive over disjunction for n > 1 in our minimal graded belief logic, i.e., the formula Bn (p ∨ q) ↔ Bn p ∨ Bn q is not valid. However, we may consider the formula Bn directly which express some sort of no-dead-end property over weighted frames. Proposition 1. For any weighted frame F = (W, R, σ), F |= Bn iﬀ F |= ∀x∃y(xRy) ∧ ∀x∃Y (σx (Y ) ≥ n). Theorem 4. The logic BLg ⊕Bn is complete with respect the class of weighted frames satisfying the condition ∀x∃y(xRy) ∧ ∀x∃Y (σx (Y ) ≥ n). The next formula is Bp → BBp which corresponds to the condition ∀xyz(xRy ∧ yRz → xRz) ∧ ∀xyz(σ(x, y ) > 0 ∧ σ(y, z ) > 0 → σ(x, z ) > 0). More generally, consider the formula Bn p → BBn p. We obtain the following results. Proposition 2. For any weighted frame F = (W, R, σ), F |= Bn p → BBn p iﬀ F is n-transitive, i.e., F |= ∀xyz(xRy ∧ yRz → xRz) ∧ ∀xY Z(σx (Y ) ≥ n ∧ ∀y ∈ Y (σy (Z) ≥ n) → σx (Z) ≥ n). Theorem 5. The logic BLg ⊕ Bn p → BBn p is complete with respect the class of all n-transitive weighted frames.

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Our last example is Bn p → BBn p. It is a generalization of the Euclidean formula B p → BB p. The Euclidean formula corresponds to ∀xyz(xRy ∧ xRz → yRz)∧∀xyz(σ(x, y ) > 0∧σ(x, z ) > 0 → σ(y, z ) > 0). However, we have the following general correspondence and completeness results over weighted frames. Proposition 3. For any weighted frame F = (W, R, σ), F |= Bn p → BBn p iﬀ F is n-Euclidean, i.e., F |= ∀xyz(xRy ∧ xRz → yRz) ∧ ∀xyZ(σ(x, y ) > 0 ∧ σy (Z) ≥ n → σx (Z) ≥ n). Theorem 6. The logic BLg ⊕ Bn p → BBn p is complete with respect the class of all n-Euclidean weighted frames. Now we get the following generalization of classical belief logic KD45 in [22]: BLg D45n = BLg ⊕ {Bn , Bn p → BBn p, Bn p → BBn p}. This logic is strongly complete. It can be taken as the static logic in following sections to explore dynamics, including public announcement and dynamic epistemic extensions. 2.3

Public Announcement BLg

We may consider the scenario that the agent believes some proposition φ with degree n after the public announcement of a formula α, a piece of hard information. When I heard some people report that it will rain tomorrow, I believe with a large degree n that it will rain tomorrow. Let us ﬁrst give the formal language and semantics for this public announcement logic. Definition 6. The language for public announcement BLg is obtained by adding formulas of the form [!α]φ where α and φ are BLg -formulas. Given a weighted model M = (W, R, σ, V ), deﬁne M !α = (W !α , R!α , σ!α , V !α ) as the submodel of M given by W !α = αM , the truth set of α in M . Then deﬁne truth of public announcement formula [!α]φ by M, w |= [!α]φ iﬀ M, w |= α implies M !α , w |= φ. Then we can get recursion axioms which can be used to understand public announcements over BLg -formulas and prove the completeness theorem by reducing each formula with announcements to an equivalent BLg -formula. This technique is commonly used and can be found in [3], [19], [4] and [5]. Theorem 7. Given a sound and complete graded belief logic Λ, the public announcement logic generated from Λ by adding the following recursion axioms is also sound and complete: [!α]p ↔ (α → p), [!α]¬φ ↔ α → ¬[!α]φ, [!α](φ ∧ ψ) ↔ [!α]φ ∧ [!α]ψ, and [!α]Bn φ ↔ (α → Bn (α → [!α]φ)). Proof. For the validity of the last recursion axiom, note that [!α]Bn φ is true at a state w iﬀ Bn φ is true in the submodel given by αM iﬀ the weight of ¬φ-states in R(w) is less than n. On the other hand, α → Bn (α → [!α]φ) is true at w iﬀ, in αM and within R(w), the set of ¬(α → [!α]φ)-states is equivalent to the set of ¬φ-states.

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Dynamic Epistemic BLg

Now let us consider the generalization of public announcement BLg to dynamic epistemic BLg . The general idea is that after events happen at the current state, the agent may change her beliefs. Let us ﬁrst give our event models and deﬁne the language and updated model. Definition 7. An event model is a structure E = (E, ∼, P re) where E is a nonempty ﬁnite set of events, R is an epistemic relation and P re : E → F ml(BLg ) is a precondition function which assigns a BLg -formula to each event such that |= P re(e) ∧ P re(f ) ↔ ⊥ whenever e = f , i.e., each event has an unique precondition, or preconditions for diﬀerent events are inconsistent. The special condition on the precondition function is assumed. The advantage of this assumption is that it will make the key recursion axiom for the belief formulas valid. Definition 8. Given a weighted model M = (W, R, σ, V ) and event model E = (E, ∼, P re), deﬁne the updated weighted model M ⊗ E = (W ∗ , R∗ , σ∗ , V ∗ ) by • • • •

W ∗ = {(w, e) : M, w |= P re(e)}. (w, e)R(v, f ) iﬀ wRv and e ∼ f . σ ∗ ((w, e), (v, f ) ) = σ(w, v ). V ∗ (p) = {(w, e) : w ∈ V (p)}.

Definition 9. The language for dynamic epistemic graded belief logic is an extension of BLg by adding formulas of the form [E, e]φ. The truth of [E, e]φ in a weighted model M is deﬁned as: M, w |= [E, e]φ iﬀ M, w |= P re(e) implies M ⊗ E, (w, e) |= φ. Deﬁne E, e φ := ¬[E, e]¬φ. As in the case of public announcement graded belief logic, we also obtain the following completeness theorem for dynamic epistemic graded belief logic. Theorem 8. Given a sound and complete graded belief logic Λ, the dynamic epistemic graded belief logic generated from Λ plus the following recursion axioms is also sound and complete: E, e p ↔ P re(e) ∧ p, E, e ¬φ ↔ P re(e) ∧ ¬E, e φ, E, e (φ ∨ ψ) ↔ E, e φ ∨ E, e ψ, and E, e Bn φ ↔ P re(e) ∧ Bn e∼f E, f φ. Proof. The validity of the last reduction axiom is shown by the unfolding the semantics. The key fact is that by our assumption on preconditions, diﬀerent pairs in the updated model corresponds to diﬀerent states in the original weighted model. The completeness of this logic is reduced to the completeness of Λ. Now let us restrict our ontology to Kirpke models, a special sort of weighted models in which every pair of related states is assigned the number 1. Thus we may interpret formulas of the form Bn φ as follows: given a Kripke model M = (W, R, V ) and w ∈ W , M, w |= Bn φ iﬀ |¬φM ∩ R(w)| < n. Intuitively, Bn φ is true, if the agent reckons that there are less than n states where φ is not true. Moreover, we may deﬁne Bn φ := ¬Bn ¬φ which means that the agent

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reckons that there are least n successor states where φ is true. This semantics is the same as basic graded modal logic in [14] and [9]. Related work has been done in [17]. However, we can get new dynamic logics based on the graded belief logic over Kripke structures. See [20].

3

Logics of Plausible Belief

From this section, we consider belief logic which is interpreted over weighted structures by comparing weights. Intuitively, we may interpret belief as follows. The agent believes a proposition φ (notation: Bφ) at the current state w, if the weight of φ is at least large as that of ¬φ in R(w). The formal language for this belief logic (BL) consists of a set of proposition letters Φ plus Boolean operators and the belief operator B. Formulas are given by the following rule: φ ::= p | ¬φ | φ ∨ ψ | Bφ where p ∈ Φ. Now let us introduce the semantics for BL. Definition 10. Deﬁne truth of a BL-formula φ at a state w in a weighted model M = (W, R, σ, V ) (notation: M, w |= φ) recursively as usual except the following clause for belief: M, w |= Bφ iﬀ σw (φM ∩ R(w)) ≥ σw (¬φM ∩ R(w)). Other semantic notions are deﬁned as usual. Proposition 4. For any BL-formulas φ and ψ, we have • • • • • • •

B(p → q) → (Bp → Bq). Bp ∧ Bq → B(p ∧ q). If |= φ, then |= Bφ. If |= φ → ψ, then |= Bφ → Bψ. If |= φ ↔ ψ, then |= Bφ ↔ Bψ. |= B(φ ∧ ψ) → Bφ ∧ Bψ. |= ¬B¬φ → Bφ.

Proof. Only show the ﬁrst item. For an counter weighted model for the formula, let W = {a, b, c, d}, R = {a, b , a, c , a, d } and σ(a, b ) = 14, σ(a, c ) = 5 and σ(a, d ) = 10. Let V (p) = {b, d} and V (q) = {b}. Then it is clear that a |= B(p → q) ∧ Bp but w Bq. The comparison was once used in probabilistic approach to belief and action ([15]). However, under the weight semantic for belief given above, we see that there may be no minimal normal modal axiomatization for BL. However, we may introduce neighbourhood semantics for BL and give the complete axiomatization. For details of this sort of semantics, see [10]. Definition 11. A neibourhood frame is a pair F = (W, N ) where W = ∅ is a set of states and N : W → ℘(℘(W )) is a function from W to the powerset of its powerset. For each state w ∈ W , N (w) is called the neibourhood of w. A neibourhood model is a tuple M = (W, N, V ) where V is a valuation. Given a neibourhood model M and w ∈ W , interpret Bφ by: M, w |= Bφ iﬀ φM ∈ N (w). Other semantic notions are deﬁned as standard in [10].

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For basic modal logic, we may construct a modally equivalent neighbourhood structure from a given Kripke frame. Now, given a weighted model M = (W, R, σ, V ) and w ∈ W , construct a neibourhood model M = (W, N, V ) by N (w) = {X ⊆ W : σw (X) ≥ σw (−X)}. Then we have the following fact. Fact 9. For all w ∈ W and BL-formula φ, we have M, w |= φ iﬀ M, w |= φ. A neighbourhood frame F = (W, N ) is called monotonic, if for all x ∈ W and X, Y ⊆ W , X ∈ N (w) and X ⊆ Y implies Y ∈ N (w), which is equivalent to that X ∩ Y ∈ N (w) implies X ∈ N (w) and Y ∈ N (w). Then by a copy of a proof from [10], we obtain the following theorems. Theorem 10. The belief logic BL axiomatized by (i) all instances of propositional tautologies, (ii) B(φ ∧ ψ) → Bφ ∧ Bψ, and inference rule (iii) from φ ↔ ψ infer Bφ ↔ Bψ, is complete with respect to the class of all monotonic neighbourhood frames. Theorem 11. The belief logic BL ⊕ B is complete with respect to the class of all monotonic frames F = (W, N ) such that ∀w ∈ W (W ∈ N (w)). Theorem 12. The belief logic BL ⊕ ¬B¬φ → Bφ is complete with respect to the class of all monotonic frames F = (W, N ) such that ∀X ⊆ W ∀w ∈ W (−X ∈ N (w) → X ∈ N (w)). Now we have two semantics at hand for belief. One is the weighted semantics under which we have no complete axiomatization while the other one is the neighbourhood semantics under which we have complete logics. These logics are static. However, we may introduce dynamics under these semantics.

4

Dynamics of Plausible Beliefs

Public Announcement Belief Logic. We ﬁrst introduce the language for public announcement belief logic (P ABL). Formulas of P ABL is given by the following inductive rule: φ ::= p | ¬φ | φ ∨ ψ | Bφ | [!φ]ψ where p is a proposition letter. We have two semantics for this language. Let us start from the weighted semantics. Given a weighted model M = (W, R, σ, V ) and a P ABL-formula φ, deﬁne a new model M !φ = (W !φ , R!φ , σ!φ , V !φ ) as in section 3. Thus we may deﬁne the semantics for public announcements by M, w |= [!φ]ψ iﬀ M, w |= φ implies M !φ , w |= ψ. The dual !φ ψ of [!φ]ψ is deﬁned as ¬[!φ]¬ψ.

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Theorem 13. The following reduction axioms are valid under weighted semantics for P ABL: !φ p ↔ φ ∧ p, !φ ¬ψ ↔ φ ∧ ¬!φ ψ, !φ (ψ1 ∨ ψ2 ) ↔ !φ ψ1 ∨ !φ ψ2 , and !φ Bψ ↔ φ ∧ B!φ ψ. Proof. Only show the validity of the last axiom. Assume that M, w |= !φ Bψ. !φ !φ (ψM !φ ∩R!φ (w)) ≥ σw (¬ψM !φ ∩ Then M, w |= φ and M !φ , w |= Bψ. Hence σw !φ !φ !φ R (w)). It suﬃces to show that σw (ψM !φ ∩ R (w)) = σw (!φ ψM ∩ R(w)), since then σw (¬!φ ψM ∩ (w)) = σw (!φ ¬ψM ∩ R(w)) and so σw (!φ ψM ∩ R(w)) = σw (¬!φ ψM ∩ R(w)). Again it is easy to show that ψM !φ ∩R!φ (w) = !φ ψM ∩ R(w). The other direction is similar. These recursion axioms show that any P ABL-formula is equivalent to a BLformula by reducing announcements to atomic case. We have no complete public announcement belief logic here. However, we may consider public announcement logic under neighbourhood semantics. Definition 12. Given a neighbourhood model M = (W, N, V ) and a P ABLformula φ, deﬁne the updated neighbourhood model given by the announcement of φ as M!φ = (W !φ , N !φ , V !p ) where W !φ = φM , N !φ (w) = {X ∩ φM : X ∈ N (w)} for each w ∈ W !φ , and V !φ (p) = V (p) ∩ W !φ . For each proposition letter p deﬁne the semantics of public announcements by: M, w |= [!φ]ψ iﬀ M, w |= φ implies M!φ , w |= ψ. Other semantics notions are deﬁned as usual. Theorem 14. The following reduction axioms are valid under neibourhood semantics for P ABL: [!φ]p ↔ φ → p, [!φ]¬ψ ↔ (φ → ¬[!φ]ψ), [!φ](ψ1 ∧ ψ2 ) ↔ [!φ]ψ1 ∧ [!φ]ψ2 , and [!φ]Bψ ↔ (φ → B[!φ]ψ). Proof. Only show the validity of the last axiom. Assume that M, w |= [!φ]Bψ and M, w |= φ. Then M!φ , w |= Bψ. Thus ψM!φ ∈ N !φ (w). It is easy to see that ψM!φ = [!φ]ψM ∩ φM . Therefore [!φ]ψM ∈ N (w) and so M, w |= B[!φ]ψ. The other direction is easy. Theorem 15. Any complete belief logic Λ under neibourhood semantics plus recursion axioms in theorem 14 is complete with respect to the class of all neibourhood frames for Λ.

5

Other Notions of Plausible Belief

Let us now introduce more notions of belief under our interpretation of belief via weights. There are also some diﬀerent natural uses of weights. Comparing Weights. We compare the weight of φ and ¬φ in our weighted semantics for Bφ. More generally, we can compare two propositions φ and ψ to decide which one has a higher degree in the agent’s range of belief. We write φ ψ to mean that, at the current state, the agent reckons that the weight of φ is at least large as ψ.

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Definition 13. Given a weighted model M = (W, R, σ, V ) and w ∈ W , deﬁne M, w |= φ ψ iﬀ σw (φM ∩ R(w)) ≥ σw (ψM ∩ R(w)). The dual φ ψ is deﬁned as ¬(¬φ ¬ψ). Thus M, w |= φ ψ iﬀ σw (¬φM ∩ R(w)) < σw (¬ψM ∩ R(w)). Thus Bφ is deﬁned as φ ¬φ. Another insight given by comparison is that we may read φ ψ as that the agent prefers φ to ψ, and so we go into the area of preference logic. Come back to the logic of this binary operator. The ﬁrst observation is that the formula (p1 → q1 ) (p2 → q2 ) → (p1 p2 → q1 q2 ) is not valid. Thus we cannot get a complete minimal normal modal logic. Then what about dynamics over the logic? As in BL, we can show that the following recursion axiom is valid: [!α]φ ψ ↔ α → (α → [!α]φ) (α → [!α]ψ). The proof is similar to the case for [!α]Bφ. Safe Beliefs. The agent may reckons that her belief φ is safe. Under our weighted semantics, this may be expressed by a formula B + φ, and it is true at a state w, if the agent reckons that the weight of φ is ‘always’ at least large as that of ¬φ with respect to w. However, we have the following four ways to deﬁne the notion of ‘being always at least large as’: • • • •

M, w |= B1+ φ M, w |= B2+ φ M, w |= B3+ φ M, w |= B4+ φ

iﬀ iﬀ iﬀ iﬀ

∀v(wRv ∃v(wRv ∀v(wRv ∃v(wRv

& & & &

v v v v

φ → ∃v (σ(w, v ) ≥ σ(w, v ) & v |= φ)). |= φ & ∀v (v φ → σ(w, v ) ≥ σ(w, v ))). |= φ → ∀v (v φ → σ(w, v ) ≥ σ(w, v ))). φ → ∃v (v φ → σ(w, v ) ≥ σ(w, v ))).

We leave the further exploration of logics for these safe belief as an open question. Now let us consider another sort of plausible belief. Conditional Belief. Let B φ ψ formalize the proposition that the agent believes ψ under the condition φ. We have the following natural semantics for this conditional belief in weighted models: M, w |= B φ ψ iﬀ σw (φ ∧ ψM ∩ R(w)) ≥ σw (φ ∧ ¬ψM ∩ R(w)). The weights of ψ and ¬ψ in the range of φ-states over successors are compared. Some logicians observed that conditional beliefs can be read as conditionals in conditional logic ([8] and [25]), and so got a complete logic for conditional belief under the plausibility semantics [4]. Under weighted semantics, we can show the following facts about conditional belief. Proposition 5. Let φ, ψ and ξ be BL-formulas. Then we have • |= B φ φ. • |= ¬B φ ¬ψ → B φ ψ.

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• • • • •

217

If |= B φ ψ, then |= B φ (ψ ∨ ξ). If |= B φ ξ and |= ξ → ψ, then |= B φ∧ψ ξ. The rule {B φ ψ, B φ ξ}/B φ (ψ ∧ ξ) is not sound. The rule {B φ ψ, B ξ ψ}/B φ∨ξ ψ is not sound. The rule {B φ ψ, B φ ξ}/B φ∧ψ ξ is not sound.

Although we haven’t got a complete conditional logic for conditional belief, we may introduce belief revision over weighted models. We are now working in epistemic-doxastic logic (EDL). Definition 14. The language of EDL consists of a set Φ of proposition letters and knowledge operator K and conditional belief. The set of all EDL-formulas is given by the following inductive rule: φ ::= p | ¬φ | φ ∨ ψ | Kφ | B ψ φ where p ∈ Φ. Deﬁne K φ := ¬K¬φ. The knowledge operator is interpreted by the relation R in weighted model as standard, i.e., M, w |= Kφ iﬀ φ is true at all R-successors of w. Let us make some observations about EDL. The belief operator Bφ is deﬁnable as B φ in our language EDL. Thus EDL is more expressive than BL. Again, we have the valid formula Kφ → Bφ which ﬁts our intuition that knowledge is a stronger notion than belief. We may also introduce belief revision into our static logic EDL. However, we just conjecture that two sort of belief revision, radical and conservative revisions, introduce in [4], can be extended to our setting. The idea is to adjust the numbers assigned to relations.

6

Conclusions and Open Questions

We have shown how to interpret notions of belief over weighted structures. Static graded belief logics and their dynamic extensions are explored. We also introduce various plausible beliefs which are interpreted by comparing weights. Now let us list some open questions for further work: • Dynamic logics for these safe beliefs introduced in section 5 are not clear. • We may read Bn φ as ‘φ is true at all successor states with weight at least n’. This is closer to Aucher’s approach to degrees of belief. The problem is to explore the logic under this semantics and its dynamic extensions. • Explore preference logics in weighted structures and preference change. We may also need to compare the quantitative and qualitative approaches in preference logic. Moreover, weights in weighted models allow other readings. For instance, consider regular frames for propositional dynamic logic. The numbers assigned to each program can be considered as the time used for executing the program, and thus we may talk about the fastest path from the current state to states with certain properties.

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Acknowledgments. The ﬁrst author is supported by the Key Project of Chinese Ministry of Education (Grant No. 05jzd008) and the Key Project of Humanities of Chongqing ‘Researches in Modal Model Theory’. The second author is supported by the Foundation for Humanities and Social Sciences by the Ministry of Education of China (08JC72040002), the National Social Science Foundation of China (09CZX033) and the Fundamental Research Funds of Southwest University (SWU0909512).

References 1. Alchourr` on, C., G¨ ardenfors, P., Makinson, D.: On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic 50, 510–530 (1985) 2. Aucher, G.: A Combined System for Update Logic and Belief Revision. Master’s thesis, University of Amsterdam, The Netherlands (2004) 3. Baltag, A., Moss, L., Solecki, S.: The logic of public announcements: Common knowledge and private suspicions. In: TARK, pp. 43–56 (1998) 4. van Benthem, J.: Logical Dynamics and Information Flow (2010) (manuscript) 5. van Benthem, J.: Modal Logic for Open Minds. CSLI Publications, Stanford (2010) 6. van Benthem, J., Gerbrandy, J., Kooi, B.: Dynamic update with probabilities. Studia Logica 93, 67–96 (2009) 7. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Univeristy Press, Cambridge (2001) 8. Burgess, J.: Quick completeness proofs for some logics of conditionals. Notre Dame Journal of Formal Logic 22, 76–84 (1981) 9. de Caro, F.: Graded modalities ii (canonical models). Studia Logica 47, 1–10 (1988) 10. Chellas, B.: Modal Logic: an introduction. Cambridge University Press, Cambridge (1980) 11. D’Agostino, Visser, A.: Finality regained: a coalgebraic study of scott-sets and multisets. Archive for Mathematical Logic 41, 267–298 (2002) 12. Fagin, R., Halpern, J.: Reasoning about knowledge and probability. Journal of the ACM 41, 340–367 (1994) 13. Fattorosi-Barnaba, M., de Caro, F.: Graded modalities i. Studia Logica 44, 197–221 (1985) 14. Fine, K.: In so many possible worlds. Notre Dame Journal of Formal Logic 13, 516–520 (1972) 15. Herzig, A., Longin, D.: On modal probability and belief. In: Nielsen, T.D., Zhang, N.L. (eds.) ECSQARU 2003. LNCS (LNAI), vol. 2711, pp. 62–73. Springer, Heidelberg (2003) 16. Hintikka, J.: Knowledge and Belief. Cornell University Press, Ithica (1962) 17. van der Hoek, W., Meyer, J.J.: Graded modalities in epistemic logic. In: Logical Foundations of Computer Science-Tver 1992, pp. 503–514 (1992) 18. Kooi, B.: Probabilistic dynamic epistemic logic. Journal of Logic, Language and Information 12, 381–408 (2003) 19. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Springer, Heidelberg (2007) 20. Kurzen, L., Vel´ azquez-Quesada, F. (eds.): Logics for dynamics of information and preferences, seminar’s yearbook 2008, pp. 208–219. Institute for Logic, Language and Computation (2008)

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21. Ma, M.: Model Theory for Graded Modal Lnaguages. Ph.D. thesis, Tsinghua University, Beijing (2011) 22. Meyer, J.J., van der Hoek, W.: Epistemic Logic for AI and Computer Science. Cambridge University Press, Cambridge (1995) 23. Plaza, J.: Logics of public communications. In: Proceedings of the 4th International Symposium on Methodologies for Intelligent Systems, pp. 201–216 (1989) 24. de Rijke, M.: A note on graded modal logic. Studia Logica 64, 271–283 (2000) 25. Veltman, F.: Logics for Conditionals. Ph.D. thesis, University of Amsterdam, The Netherlands (2011)

Game Semantics for the Geiger-Paz-Pearl Axioms of Independence Pavel Naumov and Brittany Nicholls Department of Mathematics and Computer Science McDaniel College, Westminster, Maryland 21157, USA {pnaumov,brn002}@mcdaniel.edu Abstract. The paper analyzes interdependencies between strategies of players in a Nash equilibrium using independence relation between two sets of players. A sound and complete axiomatization of this relation is given. It has been shown previously that the same axiomatic system describes independence in probability theory, information flow, and concurrency theory.

1

Introduction

In this paper, we show that the same logical principles describe independence in four diﬀerent settings: probability, information ﬂow, concurrency, and game theory. Independence in Probability Theory. Two events are called independent in probability theory if the probability of their intersection is equal to the product of their probabilities. It is believed [1] that this notion was ﬁrst introduced by de Moivre [2,3]. If A = {a1 , . . . , an } and B = {b1 , . . . , bm } are two disjoint sets of random variables with ﬁnite ranges of values, then these two sets of variables are called independent if forany values v1 , . . . , vn and any values w1 , . . . , wm , events i≤n (ai = vi ) and i≤m (bi = wi ) are independent. We write A B to denote this relation. This deﬁnition can be generalized to independence of sets of variables with inﬁnite ranges through the independence of appropriate σ-algebras. A complete axiomatization of propositional properties of the independence relation between two sets of random variables was given by Geiger, Paz, and Pearl1 [4]: 1. 2. 3. 4.

Empty Set: A ∅, Symmetry: A B → B A, Monotonicity: A B, C → A B, Exchange: A, B C → (A B → A B, C),

where here and everywhere below A, B means the union of sets A and B. Furthermore, Studen´ y [5] showed that conditional probabilistic independence does not have a complete ﬁnite axiomatization. 1

The axiom names shown here are ours.

H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 220–232, 2011. c Springer-Verlag Berlin Heidelberg 2011

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Independence in Information Flow. Sutherland [6] introduced a relation between two pieces of information, which we will call “secrets”, that later became known as the “nondeducibility” relation. Two secrets are in this relation if any possible value of the ﬁrst secret is consistent with any possible value of the second secret. More and Naumov [7] generalized this relation to a relation A B between two sets of secrets and called it independence: sets of secrets A and B are independent if each possible combination of the values of secrets in A is consistent with each possible combination of the values of secrets in B. More and Naumov [7] have shown that the same system of Geiger-Paz-Pearl axioms is sound and complete with respect to deﬁned this way semantics of secrets2 . Cohen [8] presented a related notion called strong dependence. More recently, Halpern and O’Neill [9] introduced f -secrecy to reason about multiparty protocols. In our notation, f -secrecy is a version of the nondeducibility predicate whose left or right side contains a certain function of the secret rather than the secret itself. More, Naumov, and Donders also axiomatized a variation of the independence relation between secrets over graphs [10,11] and hypergraphs [12]. Independence in Concurrency Theory. The third semantics for the GeigerPaz-Pearl axioms of independence was proposed by More, Naumov, and Sapp [13]. Under this semantics, independence is interpreted as “non-interference” between two sets of concurrent processes. A set of processes A interferes with a set of processes B if these two sets can reach a deadlocked state where either set A or set B is not internally deadlocked. For example, if p1 , p2 , p3 , p4 , p5 are ﬁve philosophers seating at a table with ﬁve forks in the Dijkstra’s [14] dining philosopher problem, then neither set {p1 , p2 , p3 } nor set {p4 , p5 } can deadlock by itself (if the other philosophers leave the table). However, the complete set {p1 , p2 , p3 , p4 , p5 } can deadlock. Thus, using our notations we can say that statement {p1 , p2 , p3 } {p4 , p5 } is false. More, Naumov, and Sapp [13] have shown that the same system of axioms 1-4 is sound and complete with respect to this concurrency semantics. Independence in Game Theory. In this paper we consider interdependencies between strategy choices of players in a multi-player game. If no assumptions are made about the players, then each of them can choose any available strategy and, thus, there is no interdependency between these choices. If, however, a player is assumed to be rational, then her choice of strategy might depend on the choices made by the other players. There are diﬀerent possible ways to formally capture the rationality of the player. In this paper we express rationality of all players in the game through the requirement that strategies of all players are in a Nash equilibrium. For example, in the United States, people walk on the right side of a hallway or a sidewalk. In Japan, however, people walk on the left side. There is no law enforcing this in either of the two countries, but walking on the same side as the 2

As long as the same secret can not appear simultaneously on the left and right hand side of the independence symbol. Otherwise, one more axiom should be added to achieve completeness.

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other pedestrians is a Nash equilibrium in the multi-player coordination game played by the pedestrians in both of these countries. By observing a single pedestrian in a hallway, one can predict the side of the hallway the next pedestrian will walk on, without a priori knowledge of the countries in which the observation takes place. This is an example of an interdependency between strategies in Nash equilibria of a strategic game. We say that two players in a multi-player strategy game are independent if knowledge of the ﬁrst player’s strategy in a Nash equilibrium does not reveal anything about the strategy of the second player in the same equilibrium. In other words, for any choice of a strategy for the ﬁrst player that appears in at least one Nash equilibrium and for any choice of a strategy for the second player that appears in at least one Nash equilibrium, there is a Nash equilibrium that uses both of these strategies. Independence can be deﬁned not just between two single players, but also between two sets of players. We say that two disjoint sets of players A and B are independent if for any two Nash equilibria e1 and e2 of the game, there is a Nash equilibrium e such that (i) each player in set A uses the same strategy in equilibria e1 and e and (ii) each player in set B uses the same strategy in equilibria e2 and e. We denote this relation by A B. In this paper we will show that the same axioms 1-4 give a sound and complete axiomatization of properties of independence between sets of players in strategic games. It is easy to see that any strategic game could be viewed as an information ﬂow protocol. Thus, soundness of these axioms in the game setting trivially follows from their soundness in the information ﬂow setting. The main technical contribution of this work is the proof of completeness. More and Naumov [7] have shown that if a formula is not provable from axioms 1-4, then there is an information ﬂow protocol for which this formula is false. In this paper we show that such protocol can be described in terms of a strategic game. The signiﬁcant implication of this result is that the same non-trivial set of axioms captures the properties of independence in four diﬀerent settings: probability, information ﬂow, concurrency, and game theory. In the conclusion we discuss what appears to be a more general independence relation A1 A2 · · · An between several sets of players. We will show, however, that this relation can be expressed through independence between just two sets of players.

2

Semantics

Strategic games are usually deﬁned by specifying for each player either a total preference order on the outcomes or, equivalently, a pay-oﬀ function on the outcomes. We have chosen the second approach since it results in a slightly simpler presentation of our main result. Definition 1. A strategic game is a triple G = (P, {Sp }p∈P , {up }p∈P ), where 1. P is a non-empty ﬁnite set of “players”.

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2. Sp is a non-emptyset of “strategies” of a player p ∈ P . Elements of the cartesian product p∈P Sp are called “strategy proﬁles”. 3. up is a “pay-oﬀ” function from strategy proﬁles into the set of real numbers. For any tuple a = ai i∈I , any i0 ∈ I and any value b, by ai i∈I [i0 → b] we mean the tuple a in which i0 -th component is changed from ai0 to b. In the game theory literature the same modiﬁed tuple is sometimes denoted by (a−i0 , b). Definition 2. Nash equilibrium of a strategic game G = (P, {Sp }p∈P , {up}p∈P ), is a strategy proﬁle sp p∈P such that up(sp p∈P [p0 → s0 ]) ≤ up (sp p∈P )

(1)

for any p0 ∈ P and any s0 ∈ Sp0 . Alternatively, one can deﬁne strict Nash equilibrium by replacing relation ≤ in inequality (1) with strict inequality sign <. The soundness and completeness theorems in this paper are true for both types of equilibria. The set of all Nash equilibria of a game G is denoted by N E(G). Next, we formally deﬁne the set of all formulas that we consider. Definition 3. For any ﬁnite set of players P , the set of formulas Φ(P ) is deﬁned recursively: (i) ⊥ ∈ Φ(P ), (ii) (A B) ∈ Φ(P ), where A and B are two disjoint subsets of P , (iii) φ → ψ ∈ Φ(P ), where φ, ψ ∈ Φ(P ). If x = xi i∈I and y = yi i∈I are two tuples such that xa = ya for any a ∈ A, then we write x ≡A y. We use this notation to deﬁne truth relation G φ between a game G and a formula φ: Definition 4. For any game G = (P, {Sp }p∈P , {up }p∈P ) and any formula φ ∈ Φ(P ), binary relation G φ is deﬁned as follows: 1. G ⊥, 2. G φ → ψ if and only if G φ or G ψ, 3. G A B if and only if for any e1 , e2 ∈ N E(G) there is e ∈ N E(G) such that e1 ≡A e ≡B e2 . The third part of the above deﬁnition is the key deﬁnition of this paper. It formally speciﬁes independence of two sets of players in a strategic game.

3

Axioms

Definition 5. The logic of information ﬂow, in addition to propositional tautologies and the Modus Ponens inference rule, consists of the following axioms: 1. 2. 3. 4.

Empty Set: A ∅, Symmetry: A B → B A, Monotonicity: A B, C → A B, Exchange: A, B C → (A B → A B, C).

Recall from the introduction that these axioms ﬁrst appeared in a work on independence of random variables in probability theory by Geiger, Paz, and Pearl [4].

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Soundness

Theorem 1. For any ﬁnite set of parties P and any φ ∈ Φ(P ), if φ, then G φ for any game G = (P, {Sp }p∈P , {up }p∈P ). Proof. It will be suﬃcient to verify that G φ for each axiom φ of the logic of information ﬂow. Soundness of the Modus Ponens rule is trivial. Empty Set Axiom. Consider any two Nash Equilibria e1 , e2 ∈ N E(G). Let e = e2 . It is easy to see that e ≡∅ e1 and e ≡A e2 . Monotonicity Axiom. Consider any two Nash Equilibria e1 , e2 ∈ N E(G). If e ≡A,B e1 and e ≡C e2 , then e ≡A e1 and e ≡C e2 . Exchange Axiom. Consider any two Nash Equilibria e1 , e2 ∈ N E(G). By the assumption that A B, there is a Nash equilibrium e3 ∈ N E(G) such that e3 ≡A e1 and e3 ≡B e2 . Since D C, there is a Nash equilibrium e4 ∈ N E(G) such that e4 ≡D e2 and e4 ≡C e1 . Finally, by the assumption the A, B C, D, there is a Nash equilibrium e ∈ N E(G) such that e ≡A,B re and e ≡C,D e4 . Thus, e ≡A e3 ≡A e1 , e ≡C e4 ≡C e1 , e ≡B e3 ≡B e2 , and e ≡D e4 ≡D e2 . Therefore, e ≡A,C e1 and e ≡B,D e2 .

5

Completeness

In this section we will prove the completeness of the Geiger-Paz-Pearl axioms with respect to the strategic game semantics. This result is stated in Theorem 2. We start, however, with a sequence of lemmas in which we assume a ﬁxed ﬁnite set of parties P and a ﬁxed maximal consistent set of formulas X ⊆ Φ(P ). 5.1

Critical Sets

The key to understanding axioms 1-4 is the notions of critical pair and critical set. Below is their combined deﬁnition and their basic properties. Later we will deﬁne a separate strategic game for each critical subset of P . Definition 6. A set C ⊆ P is called critical if there is a disjoint partition C1 C2 of C, called a “critical partition”, such that 1. X C1 C2 , 2. X C1 ∩ D C2 ∩ D, for any D C. Lemma 1. Any critical partition is a non-trivial partition. Proof. It will be suﬃcient to prove that for any set A, we have X A ∅ and X ∅ A. The ﬁrst statement is an instance of the Empty Set axiom, the second statement follows from the Empty Set and the Symmetry axioms.

Lemma 2. X A B, for any non-trivial (but not necessarily critical) partition A B of a critical set C.

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225

Proof. Suppose X A B and let C1 C2 be a critical partition of C. By the Monotonicity and Symmetry axioms, X A ∩ C B ∩ C. Thus, X A ∩ C1 , A ∩ C2 B ∩ C1 , B ∩ C2 .

(2)

Since A B is a non-trivial partition of C, sets A and B are both non-empty. Thus, A C and B C. Hence, by the deﬁnition of a critical set, X A ∩ C1 A ∩ C2 and X B ∩ C1 B ∩ C2 . Note that A ∩ C is not empty since A B is a non-trivial partition of C. Thus, either A ∩ C1 or A ∩ C2 is not empty. Without loss of generality, assume that A ∩ C1 = ∅. From (2) and our earlier observation that X A ∩ C1 A ∩ C2 , the Exchange axiom yields X A ∩ C1 A ∩ C2 , B ∩ C1 , B ∩ C2 . By the Symmetry axiom, X A ∩ C2 , B ∩ C1 , B ∩ C2 A ∩ C1 .

(3)

The assumption A ∩ C1 = ∅ implies that (A ∩ C2 ) ∪ (B ∩ C1 ) ∪ (B ∩ C2 ) C. Hence, by the deﬁnition of a critical set, X B ∩ C1 A ∩ C2 , B ∩ C2 . By Symmetry axiom, X A ∩ C2 , B ∩ C2 B ∩ C1 . From (3) and the above statement, using the Exchange axiom, X A ∩ C2 , B ∩ C2 A ∩ C1 , B ∩ C1 . Since A B is a partition of C, we can conclude that X C2 C1 . By the Symmetry axiom, X C1 C2 , which contradicts the assumption that C1 C2 is a critical partition.

Lemma 3. For any two disjoint subsets A, B ⊆ P , if X A B, then there is a critical partition C1 C2 , such that C1 ⊆ A and C2 ⊆ B. Proof. Consider the partial order on set 2A × 2B such that (E1 , E2 ) (F1 , F2 ) if and only if E1 ⊆ F1 and E2 ⊆ F2 . Deﬁne E = {(E1 , E2 ) ∈ 2A × 2B | X E1 E2 }. X A B implies that (A, B) ∈ E. Thus, E is a non-empty ﬁnite set. Take (C1 , C2 ) to be a minimal element of set E with respect to partial order .

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Parity Game

For any subset Q ⊆ P , we deﬁne “parity” game P G(Q). Later we will consider such games only for Q which are critical subsets of P . For now, however, Q is just an arbitrary subset of P . We start with an informal description of the parity game. Players in set Q will be referred to as “active” players, since they will be able to inﬂuence outcome of the game. Players in the set P \ Q are “passive”: they get a pay-oﬀ, but can not inﬂuence its amount. Each active player picks an integer number. If the sum of all picked numbers is even, then pay-oﬀ of each player in set P is zero. If the sum of all picked numbers is odd, then each player in the set P gets one dollar. In the formalization of this game below, we assume that players only pick numbers from the set {0, 1} and that passive players always pick number 0. Definition 7. For any set of parties Q ⊆ P , by parity game P G(Q) we mean game (P, {Sp }p∈P , {up }p∈P ) such that 1. set of strategies of party p ∈ P is {0, 1} if p ∈ Q, Sp = {0} otherwise. 2. pay oﬀ function up is the same for all players p ∈ P . We denote it simply by u. Value of u(sp p∈P ) is either 0 or 1 in such a way that u(sp p∈P ) ≡

sp

(mod 2).

p∈P

Lemma 4.

N E(P G(Q)) = {sp p∈P ∈

p∈P

Sp |

sp ≡ 1

(mod 2)}.

p∈P

Proof. Follows from Deﬁnition 2 and Deﬁnition 7.

Lemma 5. If set Q is not empty, then game P G(Q) has at least one Nash equilibrium. Proof. Let q0 ∈ Q. Consider strategy proﬁle ep p∈P such that ep =

1 if p = q0 , 0 otherwise.

By Lemma 4, ep p∈P ∈ N E(P G(Q)).

Lemma 6. If A and B are two disjoint subsets of Q, then P G(Q) A B if and only if A B is a non-trivial partition of the set Q.

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227

Proof. (⇒) : Suppose that A B is not a non-trivial partition of Q. There are three possible cases to consider: Case I: A is empty. Thus, P G(P ) A B due to soundness of the Empty Set and Symmetry axioms (See Theorem 1). Case II: B is empty. Thus, P G(P ) A B due to soundness of the Empty Set axiom. Case III: there is q0 ∈ Q \ (A ∪ B). Let e , e be any two Nash equilibria of the game P G(Q). We will show that there is e ∈ N E(P G(Q)) such that e ≡A e ≡B e . Indeed, consider strategy proﬁle ep p∈P such that ⎧ if p ∈ A, ep ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ep if p ∈ B, ep ≡ (mod 2) ⎪ 1 + a∈A ea + b∈B eb if p = q0 , ⎪ ⎪ ⎪ ⎪ ⎩ 0 otherwise. Note that ep = eq0 + ea + eb ≡ 1+ ea + eb + ea + eb ≡ 1 p∈P

a∈A

a∈A

b∈B

b∈B

a∈A

(mod 2).

b∈B

Therefore, by Lemma 4, e ∈ N E(P G(Q)). (⇐) : Suppose that P G(P ) A B and A B is a non-trivial partition of Q. Let B B a0 ∈ A and b0 ∈ B. Consider strategy proﬁles eA = eA p p∈P and e = ep p∈P such that 1 if p = a0 , A ep ≡ 0 otherwise

and eB p A

≡

1 if p = b0 , 0 otherwise.

B

By Lemma 4, e , e ∈ N E(P G(Q)). By assumption P G(Q) A B there must be e ∈ N E(P G(Q)) such that eA ≡A e ≡B eB . Since A B is a partition of Q, we have A B ep = ea + eb = eA eB a + b = ea0 + eb0 = 1 + 1 ≡ 0 (mod 2). p∈P

a∈A

b∈B

a∈A

Contradiction with Lemma 4. 5.3

b∈B

Game Composition

Informally, by a composition of several games we mean a game in which each of the composed games is played independently. Pay-oﬀ of any player is deﬁned as the sum of the pay-oﬀs in the individual games.

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Definition 8. Let {Gi }i∈I = {(P, {Spi }p∈P , {uip}p∈P )}i∈I be a ﬁnite family of strategic games between the same set of players P . By product game i Gi we mean game (P, {Sp }p∈P , {up }p∈P ) such that 1. Sp = i Spi , 2. up (sip i∈I p∈P ) = i uip (sip p∈P ).

Lemma 7. NE

i

=

G

i

N E(Gi ).

i

i ∈ NE Proof. First, assume that ep p∈P = i G . We will need to show that eip p∈P ∈ N E(Gi ) for any i ∈ I. Indeed, suppose that for some i0 ∈ I, some p0 ∈ P , and some s0 ∈ Sp0 we have eip i∈I p∈P

(4) uip00 (eip0 p∈P [p0 → s0 ]) > uip00 (eip0 p∈P ). i i Deﬁne strategy proﬁle ˆ ep p∈P = ˆ ep i∈I p∈P of the game i G as follows: s0 if i = i0 and p = p0 , eˆip ≡ eip otherwise. Note that, taking into account inequality (4), up0 (ˆ ep p∈P ) = uip0 (ˆ eip p∈P ) = uip00 (ˆ eip0 p∈P ) + uip0 (ˆ eip p∈P ) = i∈I

=

uip00 (eip0 p∈P [p0

> uip00 (eip0 p∈P ) + =

→ s0 ]) +

i=i0

uip0 (eip p∈P )

>

i=i0

uip0 (eip p∈P ) =

i=i0

uip0 (eip p∈P )

= up0 (ep p∈P ),

i

which is a contradiction with the assumption that ep p∈P is a Nash equilibrium of the game i Gi . Next, assume that {eip p∈P }i∈I is such a set that for any i ∈ I, eip p∈P ∈ N E(Gi ) (5) i i We will prove that ep i∈I p∈P ∈ N E i G . Indeed, consider any p0 and any i i s0 i∈I ∈ i∈I Sp0 . By assumption (5) and Deﬁnition 2, for any i ∈ I uip0 (eip p∈P [p0 → si0 ]) ≤ uip0 (eip p∈P ). Thus, up0 (eip i∈I p∈P [p0 → si0 i∈I ]) =

uip0 (eip p∈P [p0 → si0 ]) ≤

i∈I

≤

i∈I

uip0 (eip p∈P ) = up0 (eip i∈I p∈P ).

Game Semantics for the Geiger-Paz-Pearl Axioms of Independence

Therefore, eip i∈I p∈P ∈ N E

i

229

Gi .

Lemma 8. For any disjoint subsets A and B of the set P , if each of the games {Gi }i∈I has at least one Nash equilibrium, then Gi A B iﬀ ∀i (Gi A B). i

Proof. (⇒) : By the assumption of the theorem, for any i ∈ I there is at least one Nash equilibrium eip p∈P of the game Gi . Suppose that i Gi A B and consider any i0 ∈ I. We will prove that Gi0 A B. Indeed, let f = fp p∈P ∈ N E(Gi0 ) and g = gp p∈P ∈ N E(Gi0 ). We will construct h = hp p∈P ∈ N E(Gi0 ) such that f ≡A h ≡B g. To construct such equilibrium, consider strategy proﬁles fˆ = fˆpi i∈I p∈P and gˆ = ˆ gpi i∈I p∈P for the game i i G such that fp if i = i0 fˆpi = (6) eip otherwise

and gˆpi

=

gp if i = i0 eip otherwise

(7)

By Lemma 7, fˆ, gˆ ∈ N E( i Gi ). Thus, by assumption i Gi A B, there must be ˆh ∈ N E( i Gi ) such that ˆ ≡B gˆ fˆ ≡A h

(8)

Deﬁne strategy proﬁle h for the game Gi0 to be hip0 p∈P . By Lemma 7, h ∈ N E(Gi0 ). From statements (8), (6), and (7), it follows that f ≡A h ≡B g. (⇐) : Assume that ∀i (Gi A B). Let f = fpi i∈I p∈P ∈ N E( i Gi ) and g = gpi i∈I p∈P ∈ N E( i Gi ). We will show that there is e ∈ N E( i Gi ) such that f ≡A e ≡B g. Indeed, by Lemma 7, fpi p∈P ∈ N E(Gi ) and gpi p∈P ∈ N E(Gi ) for any i ∈ I. Thus, by the assumption, for any i ∈ I there is eip p∈P ∈ N E(Gi ) such that gpi p∈P ≡A eip p∈P ≡B gpi p∈P . Thus, fpi i∈I p∈P ≡A eip i∈I p∈P ≡B gpi i∈I p∈P . Pick strategy proﬁle e to be eip i∈I p∈P and notice that, by Lemma 7, e ∈ N E( i Gi ).

5.4

Completeness: The final steps

We are now ready to prove the completeness theorem, which is stated below. Theorem 2. For any set of players P and any φ ∈ Φ(P ), if φ, then there is a game G with set of players P such that G φ.

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Proof. Suppose that φ and let X be a maximal consistent set of formulas containing ¬φ. Let {Ci }i∈I be the ﬁnite set of all critical subsets of P . Let P G(Ci ) be the parity game between set of players P . Pick game G to be i∈I P G(Ci ). Lemma 9. For any disjoint subsets A and B of the set P , GAB

iﬀ

A B ∈ X.

Proof. (⇒) : Assume that A B ∈ / X. Thus, X A B due to maximality of X. Hence, by Lemma 3, there is a critical set C ⊆ P such that (A ∩ C) (B ∩ C) is a critical partition of C. Thus, by Lemma 1, (A ∩ C) (B ∩ C) is a nontrivial partition of the set C. Hence, by Lemma 6, P G(C) A ∩ C B ∩ C. Thus, due to soundness of the Monotonicity and Symmetry axioms (Theorem 1), P G(C) A B. Hence, by Lemma 5 and Lemma 8, i∈I Gi A B. In other words, G A B. (⇐) : Suppose that A B ∈ X. Due to Lemma 5 and Lemma 8, it will be suﬃcient to show that P G(Ci ) A B for any i ∈ I. Assume that P G(C i0 ) A B for some i0 ∈ I. Thus, due to soundness of Symmetry and Monotonicity axioms, P G(Ci0 ) A ∩ Ci0 B ∩ Ci0 . Then, by Lemma 6, A ∩ Ci0 B ∩ Ci0 is a non-trivial partition of Ci0 . Hence, by Lemma 2, X A ∩ Ci0 B ∩ Ci0 . Therefore, by Monotonicity and Symmetry axioms, X A B.

Lemma 10. For any formula ψ in Φ(P ), Gψ

iﬀ

ψ∈X

Proof. Induction on the structural complexity of ψ. Base case is proven in Lemma 9. The induction step follows from the maximality and the consistency of the set X.

To ﬁnish the proof of the completeness theorem, note that ¬φ ∈ X. Thus, φ ∈ /X due to consistency of X. Therefore, by Lemma 10, G φ.

6 6.1

Conclusion An n-ary Independence Relation

In this paper, we have considered the independence relation A B between two sets of players. This binary relation can be naturally generalized to the n-ary relation A1 A2 · · · An between n sets of players by changing part 3 of Deﬁnition 4 to 3. G A1 A2 · · · An if and only if for any e1 , e2 , . . . , en ∈ N E(G) there is e ∈ N E(G) such that e ≡Ai ei for each i ≤ n. It turns out, however, that the n-ary independence relation can be expressed through the binary independence relation studied in this paper. For example, in the case n = 3, the following result holds:

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231

Theorem 3. For any game G = (P, {Sp }p∈P , {up }p∈P ) and any disjoint subsets A, B, and C of set the P , G (A B C) ⇐⇒ (A B, C) ∧ (B C). Proof. (⇒) : Assume G A B C. To prove G A B, C, consider any two equilibria e1 , e2 ∈ N E(G). We will show that there is equilibrium e ∈ N E(G) such that e1 ≡A e ≡B,C e2 . Indeed, by the assumption, there must be equilibrium e ∈ N E(G) such that e ≡A e1 , e ≡B e2 , and e ≡C e2 . To prove G B C, consider any two equilibria e1 , e2 ∈ N E(G). We will show that there is equilibrium e ∈ N E(G) such that e1 ≡B e ≡C e2 . Indeed, by the assumption, there must be equilibrium e ∈ N E(G) such that e ≡A e1 , e ≡B e1 , and e ≡C e2 . (⇐) : Assume G A B, C and G B C. To prove G A B C, consider any three equilibria e1 , e2 , e3 ∈ N E(G). We will show that there is equilibrium e ∈ N E(G) such that e ≡A e1 , e ≡B e2 , and e ≡C e3 . Indeed, by the assumption G B C, there must be equilibrium e4 ∈ N E(G) such that e2 ≡B e4 ≡C e3 . By the assumption G A B, C, there must be equilibrium e ∈ N E(G) such that e1 ≡A e ≡B,C e4 . Therefore, e ≡A e1 , e ≡B e4 ≡B e2 , and e ≡C e4 ≡C e3 .

6.2

Possible Extensions

In this paper we have deﬁned independence A B between two sets of rational players through properties of their strategies in a Nash equilibrium. There are several other ways in which independence between groups of players could be speciﬁed. One can say that set of players A is independent from set of players B if players in set A will not change their strategies if strategies of players in set B are revealed to them. An alternative is to assume that not only players in set A know strategies of players in set B, but they can dictate the choice of these strategies. Yet another way to deﬁne independence is to see if cooperation between coalitions A and B will not increase their pay-oﬀ values. In the future work, we would like to give the precise deﬁnitions of these types of independence and to study their universal properties.

References 1. Moivre, Abraham, de. In: The New Encyclopædia Britannica, 15th edn., vol. 8, p.226. Encyclopædia Britannica (1998) 2. de Moivre, A.: De mensura sortis seu; de probabilitate eventuum in ludis a casu fortuito pendentibus. Philosophical Transactions (1683-1775) 27, 213–264 (1711) 3. de Moivre, A.: Doctrine of Chances (1718) 4. Geiger, D., Paz, A., Pearl, J.: Axioms and algorithms for inferences involving probabilistic independence. Inform. and Comput. 91(1), 128–141 (1991) 5. Studen´ y, M.: Conditional independence relations have no finite complete characterization. In: Transactions of the 11th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, vol. B, pp. 377–396. Kluwer, Dordrecht (1990)

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6. Sutherland, D.: A model of information. In: Proceedings of Ninth National Computer Security Conference, pp. 175–183 (1986) 7. Miner More, S., Naumov, P.: An independence relation for sets of secrets. Studia Logica 94(1), 73–85 (2010) 8. Cohen, E.: Information transmission in computational systems. In: Proceedings of Sixth ACM Symposium on Operating Systems Principles, Association for Computing Machinery, pp. 113–139 (1977) 9. Halpern, J.Y., O’Neill, K.R.: Secrecy in multiagent systems. ACM Trans. Inf. Syst. Secur. 12(1), 1–47 (2008) 10. Miner More, S., Naumov, P.: On interdependence of secrets in collaboration networks. In: Proceedings of 12th Conference on Theoretical Aspects of Rationality and Knowledge, pp. 208–217. Stanford University, Stanford (2009) 11. Donders, M., More, S.M., Naumov, P.: Information flow on directed acyclic graphs. In: Beklemishev, L., de Queiroz, R. (eds.) Proceedings of 18th Workshop on Logic, Language, Information and Computation, Philadelphia, United States, pp. 95–109. Springer, Heidelberg (2011) 12. Miner More, S., Naumov, P.: Hypergraphs of multiparty secrets. In: Dix, J., Leite, J., Governatori, G., Jamroga, W. (eds.) CLIMA XI. LNCS(LNAI), vol. 6245, pp. 15–32. Springer, Heidelberg (2010) 13. Miner More, S., Naumov, P., Sapp, B.: Concurrency semantics for the Geiger-PazPearl axioms of independence. In: 20th Conference on Computer Science Logic (CSL 2011), Bergen, Norway (to appear, September 2011) 14. Dijkstra, E.W.: Hierarchical ordering of sequential processes. Acta Inf. 1, 115–138 (1971)

Algebraic Foundations for Inquisitive Semantics Floris Roelofsen Institute for Logic, Language, and Computation University of Amsterdam [email protected]

1

Introduction

Traditionally, meaning is identiﬁed with informative content. The central aim of inquisitive semantics [1,2,4,5, a.o.] is to develop a notion of semantic meaning that embodies both informative and inquisitive content. To achieve this, the proposition expressed by a sentence ϕ, [ϕ], is not taken to be a set of possible worlds, but rather a set of possibilities, where each possibility in turn is a set of possible worlds. In uttering a sentence ϕ, a speaker provides the information that the actual world is contained in at least one possibility in [ϕ], and at the same time she requests enough information from other participants to establish for at least one possibility α ∈ [ϕ] that the actual world is contained in α. Suppose, for instance, that [ϕ] = {{w1 , w2 }, {w1 , w3 }}, as depicted on the right. Then in uttering ϕ, a speaker prow1 w2 vides the information that the actual world lies in [ϕ] = {w1 , w2 , w3 }, and at the same time requests enough information to establish that the actual world lies in {w1 , w2 } or w3 w4 to establish that it lies in {w1 , w3 }. In this way, [ϕ] captures both the informative and the inquisitive content of ϕ. As soon as the basic notion of meaning is enriched in this way, the question arises how the propositions expressed by complex sentences should be deﬁned in terms of the propositions expressed by their constituents. In particular, if we limit ourselves to a ﬁrst-order language, what is the role of connectives and quantiﬁers in this richer setting? That is, given two propositions that capture the informative and the inquisitive content of ϕ and ψ, how do we construct the propositions that suitably capture the informative and inquisitive content of ¬ϕ, ϕ ∧ ψ, ϕ ∨ ψ, ϕ → ψ, ∃x.ϕ, and ∀x.ϕ? This issue has of course been addressed in previous work [1,2,4,5, a.o.]. However, the clauses for connectives and quantiﬁers that have been formulated so far were motivated based on a limited range of linguistic examples. While such examples can be indicative of whether or not a semantics works the way it is supposed to work, they do not provide a proper foundation for the semantics.

I am very grateful to Ivano Ciardelli, Ga¨elle Fontaine, Jeroen Groenendijk, Bruno Jacinto, Morgan Mameni, Balder ten Cate and Matthijs Westera for discussion of the ideas presented here as well as many closely related topics, and to the anonymous reviewers for useful feedback on an earlier version of this paper.

H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 233–243, 2011. c Springer-Verlag Berlin Heidelberg 2011

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F. Roelofsen

The present paper develops an inquisitive semantics for a ﬁrst-order language that is motivated by general algebraic concerns. As far as connectives are concerned it coincides with, and thus provides an algebraic foundation for the semantics that was developed in previous work. However, its treatment of quantiﬁers will diverge from previous work, and a careful assessment of these diﬀerences will deepen our understanding of the semantics. The paper is structured as follows. In section 2 we brieﬂy review the algebraic foundations of classical logic. In section 3 we develop an algebraically motivated inquisitive semantics, and in section 4 this semantics is compared with previous work. Section 5 concludes.

2

Algebraic Foundations of Classical Logic

To illustrate our approach, let us brieﬂy review the standard algebraic foundations of classical logic. In classical logic, the proposition expressed by a sentence ϕ, which we will denote as [ϕ], is a set of possible worlds that captures the informative content of ϕ: in uttering ϕ, a speaker provides the information that the actual world is contained in [ϕ]. Given this classical way of thinking about propositions, there is a natural order between them: A ≤c B iﬀ A provides at least as much information as B, i.e., iﬀ A ⊆ B. This order in turn gives rise to certain algebraic operations on propositions. For instance, for any two propositions A and B, there is a proposition M (A, B) that is the greatest lower bound of A and B relative to ≤c , which, in algebraic jargon, is called the meet of A and B. Similarly, for every A and B, there is some proposition J(A, B) that is the least upper bound of A and B relative to ≤c , which is called the join of A and B. The existence of meets and joins implies that the set of all classical propositions Πc , together with the order ≤c , forms a lattice. Moreover, this lattice is bounded : it has a bottom element, ∅, and a top element, W (the set of all possible worlds). Finally, for every classical proposition A, there is another proposition C(A) such that (i) the join of A and C(A) coincides with the top element of the lattice, and (ii) the meet of A and C(A) coincides with the bottom element of the lattice. C(A) is called the complement of A, and the fact that every classical proposition has a complement implies that Πc , ≤c forms a Boolean algebra (a speciﬁc kind of bounded lattice). Now, these basic semantic operators, meet, join, and complementation, can be associated with syntactic operators like conjunction, disjunction, and negation, respectively. That is, we can deﬁne: – [¬ϕ] = C([ϕ]) – [ϕ ∧ ψ] = M ([ϕ], [ψ]) – [ϕ ∨ ψ] = J([ϕ], [ψ]) Moreover, it can be shown that C([ϕ]) amounts to W − [ϕ], that M ([ϕ], [ψ]) amounts to [ϕ] ∩ [ψ], and that J([ϕ], [ψ]) amounts to [ϕ] ∪ [ψ]. This yields the familiar clauses for negation, conjunction and disjunction:

Algebraic Foundations for Inquisitive Semantics

235

– [¬ϕ] = W − [ϕ] – [ϕ ∧ ψ] = [ϕ] ∩ [ψ] – [ϕ ∨ ψ] = [ϕ] ∪ [ψ] This is how classical propositional logic is obtained, and the approach can be extended to ﬁrst-order logic [7,8]. Notice that everything starts with a certain notion of propositions, and a natural order on these propositions. This order, then, gives rise to certain operations on propositions—meet, join, and complementation—and classical propositional logic is obtained by associating these semantic operations with the syntactic connectives ∧, ∨, and ¬.

3

Algebraic Inquisitive Semantics

Exactly the same strategy can be applied in the inquisitive setting. Only now we have a richer notion of propositions, and a diﬀerent order on them: propositions are not only ordered in terms of their informative content, but also in terms of their inquisitive content. 3.1

Ordering Propositions

Let us ﬁrst oﬃcially deﬁne what we take propositions to be in the inquisitive setting. In fact, the deﬁnition below is provisional—it will be slightly reﬁned— but it forms the most natural point of departure. Definition 1 (Possibilities and propositions). – A possibility is a set of possible worlds. – A proposition is a non-empty set of possibilities.

[to be reﬁned ]

Propositions embody informative and inquisitive content in the following way. Someone who utters a sentence that expresses a proposition A provides the information that the actual world lies in at least one of the possibilities in A. Thus, she provides the information that the actual world lies in A. We will refer to A as the informative content of A, and denote it as info(A). Definition 2 (Informative content). info(A) = A On the other hand, someone who utters a sentence that expresses a proposition A also requests certain information from other conversational participants. Namely, she requests enough information to establish for at least one possibility α ∈ A, that the actual world is contained in α. Thus, propositions can be ordered in terms of their informative content, but also in terms of their inquisitive content. Just as in the classical setting, one proposition A is at least as informative as another proposition B, A ≤info B, just in case info(A) ⊆ info(B). As for inquisitiveness, we say that one proposition is at least as inquisitive as another proposition B, A ≤inq B, iﬀ A requests at least as much information as B, i.e., iﬀ every response that satisﬁes the request issued

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w1

w2

w3

w4

α

γ

β

w1

w2

w3

w4

α

β (a) A

(b) B

Fig. 1. Two non-identical propositions that are equivalent w.r.t. ≤

by A also satisﬁes the request issued by B. This means that every response that provides enough information to establish some possibility in A must also provide enough information to establish some possibility in B. This holds just in case every possibility in A is contained in some possibility in B. Thus, A ≤inq B if and only if ∀α ∈ A. ∃β ∈ B. α ⊆ β. These two orders can be combined into one overall order: A ≤ B iﬀ both A ≤info B and A ≤inq B. In sum: Definition 3 (Ordering propositions). – A ≤info B iﬀ info(A) ⊆ info(B) – A ≤inq B iﬀ ∀α ∈ A. ∃β ∈ B. α ⊆ β – A≤B iﬀ A ≤info B and A ≤inq B After all, if every Now, notice that A ≤inq B actually implies that A ≤info B. possibility inA is contained in some possibility in B, then A must also be contained in B. This means that A ≤ B if and only if A ≤inq B. Fact 1 (Simplified overall order).

A≤B

iﬀ ∀α ∈ A. ∃β ∈ B. α ⊆ β

Now let us see whether ≤ forms a partial order, i.e., whether it is reﬂexive, transitive, and anti-symmetric. It is easy to see that ≤ is indeed reﬂexive and transitive. However, ≤ is not anti-symmetric. That is, it is possible to ﬁnd two propositions A and B such that A ≤ B and B ≤ A, but A = B. Two such propositions are depicted in ﬁgure 1: the proposition depicted on the left, A, consists of two possibilities, α and β, while the proposition depicted on the right consists of three possibilities, α, β, and γ. Thus, these two propositions are not identical. However, they are equivalent w.r.t. ≤. That is, they are just as informative and inquisitive. To see this, ﬁrst notice that info(A) and info(B), i.e., the union of the possibilities in A and the union of the possibilities in B, clearly coincide. Thus, A and B provide just as much information. To see that A and B also request just as much information, consider a response that satisﬁes the request issued by A. Such a response must either provide the information that the actual world lies in α or it must provide the information that the actual world lies in β. But that means that this response would also satisfy the request issued by B. And

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vice versa, any response that satisﬁes the request issued by B also satisﬁes the request issued by B. Thus, A and B request just as much information. In other words, A and B are just as inquisitive. This shows that, as long as we are just interested in capturing informative and inquisitive content, we don’t really need to deﬁne propositions as arbitrary sets of possibilities. Rather, we could adopt a more restricted notion of propositions, such that any two propositions that are non-identical also really diﬀer in informative and/or inquisitive content. To this end, we will deﬁne propositions as persistent sets of possibilities. Definition 4 (Propositions as persistent sets of possibilities). – A set of possibilities A is persistent if and only if for every α ∈ A and every β ⊆ α, we also have that β ∈ A. – A proposition is a non-empty, persistent set of possibilities. – The set of all propositions is denoted by Π. To see that persistency is a natural constraint on propositions in the present setting, consider the following. We are conceiving of propositions as sets of possibilities, and not just as sets of possible worlds, because in this way it is possible for a proposition to embody a certain issue. An issue can be characterized by the range of responses that resolve it. Thus far, we have been assuming the following relationship between the responses that resolve the issue embodied by a proposition A and the possibilities that A consists of: a response resolves the issue embodied by A iﬀ its informative content is contained in some possibility α ∈ A. But we could also think of this relationship slightly diﬀerently, namely we could say that a response resolves the issue embodied by A iﬀ its informative content coincides with some possibility α ∈ A. And once we assume this relationship between issue-resolving responses and possibilities, we are forced to conceive of propositions as persistent sets of possibilities. After all, suppose that α ∈ A and that R is a response whose informative content coincides with α. Then, given the assumed relationship between possibilities and issue-resolving responses, R must be an issue-resolving response. But then any response that provides even more information than R must also be an issue-resolving response, and this means, again given the assumed relationship between possibilities and issue-resolving responses, that any subset β of α must also be a possibility in A. Given this more restricted notion of propositions as persistent sets of possibilities, the characterization of ≤ can in fact be further simpliﬁed. We said above that A ≤ B iﬀ every possibility in A is contained in some possibility in B. But if propositions are persistent, we could just as well say that A ≤ B iﬀ A ⊆ B. To see this, ﬁrst suppose that A ⊆ B. Then clearly every possibility in A is contained in some possibility in B. Now, for the other direction, suppose that every possibility in A is contained in some possibility in B, and let α be a possibility in A. Then α must be contained in some possibility β in B. But then, since B is persistent, every subset of β, including α, must also be in B. So we must have that A ⊆ B.

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Fact 2 (Further simplification of ≤).

A ≤ B iﬀ A ⊆ B

From this characterization it immediately follows that ≤ forms a partial order over Π. In particular, ≤ is anti-symmetric, which means that every two nonidentical propositions really diﬀer in informative and/or inquisitive content. Fact 3 (Partial order). 3.2

≤ forms a partial order over Π

Algebraic Operations

The next step is to see what kind of algebraic operations ≤ gives rise to. It turns out that, as in the classical setting, any two propositions A and B have a unique greatest lower bound (meet) and a unique least upper bound (join) w.r.t. ≤. Theorem 1 (Meet). For any two propositions A and B, A ∩ B is the meet of A and B w.r.t. ≤. Proof. Clearly, (A ∩ B) ≤ A and (A ∩ B) ≤ B, which means that A ∩ B is a lower bound of A and of B. What remains to be shown is that A ∩ B is the greatest lower bound of A and B. That is, for every C that is a lower bound of A and B, we must show that C ≤ (A ∩ B). To see this let C be a lower bound of A and B, and let γ be a possibility in C. Then, since C ≤ A, γ must be in A, and since C ≤ B, γ must also be in B. But that means that γ must be in A ∩ B. Thus, C ≤ (A ∩ B), which is exactly what we set out to show. So A ∩ B is indeed the greatest lower bound of A and B. 2 Theorem 2 (Join). For any two propositions A and B, A ∪ B is the join of A and B w.r.t. ≤. Proof. Clearly, A ≤ (A ∪ B) and B ≤ (A ∪ B), which means that A ∪ B is an upper bound of A and B. What remains to be shown is that A ∪ B is the least upper bound of A and B. That is, for every C that is an upper bound of A and B, we must show that (A ∪ B) ≤ C. To see this let C be an upper bound of A and B, and let α be a possibility in (A ∪ B). Then α must be in A or in B. Without loss of generality, suppose that α ∈ A. Then, since A ≤ C, α must also be in C. This establishes that (A ∪ B) ≤ C, which is what we set out to show. Thus, A ∪ B is indeed the least upper bound of A and B. 2 As before, we will use M (A, B) and J(A, B) to denote the meet and the join of A and B, respectively. The existence of meets and joins implies that Π, ≤ forms a lattice. And again, this lattice is bounded, i.e., there is a bottom element, {∅}, and a top element, ℘(W ). For every proposition A, we have {∅} ≤ A ≤ ℘(W ). So far, then, everything works out just as in the classical setting. However, unlike in the classical setting, not every proposition has a complement, in the sense that not for every proposition A there is another proposition B such that (i) M (A, B) = {∅} and (ii) J(A, B) = ℘(W ). For instance, suppose that W = {w1 , w2 , w3 , w4 } and let A be the proposition consisting of the possibilities α

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and β depicted in ﬁgure 1(a), and all subsets thereof. Then the only proposition B that satisﬁes (ii) is ℘(W ). But ℘(W ) does not satisfy (i). Thus, given this particular proposition A, it is impossible to ﬁnd a proposition B that satisﬁes both (i) and (ii). The fact that certain propositions do not have a complement in the above sense implies that Π, ≤ does not form a Boolean algebra. However, we will show below that for any two propositions A and B, there is a unique greatest proposition C such that M (A, C) ≤ B. In algebraic jargon, this proposition is called the pseudo-complement of A relative to B, and the existence of relative pseudo-complements implies that Π, ≤ forms a Heyting algebra (the algebra underlying intuitionistic logic). Definition 5. For any two propositions A and B, we deﬁne A/B as follows: A/B = {γ | for every χ ⊆ γ, if χ ∈ A then χ ∈ B} Theorem 3 (Relative pseudo-complementation). For any two propositions A and B, A/B is the pseudo-complement of A relative to B. Proof. First, let us show that for any A and B, M (A, A/B) ≤ B. We know that M (A, A/B) amounts to A ∩ A/B. Now let ξ be a possibility in A ∩ A/B. Then ξ is both in A and in A/B. Since ξ ∈ A/B, it must be the case that if ξ ∈ A then also ξ ∈ B. But we know that ξ ∈ A. So ξ must also be in B. This establishes that M (A, A/B) ≤ B. It remains to be shown that A/B is the greatest proposition C such that M (A, C) ≤ B. In other words, we must show that for any proposition C such that M (A, C) ≤ B, it holds that C ≤ A/B. To see this, let C be a proposition such that M (A, C) ≤ B and let γ be a possibility in C. Towards a contradiction, suppose that γ ∈ A/B. Then there must be some χ ⊆ γ such that χ ∈ A and χ ∈ B. Since C is persistent, χ ∈ C. But that means that χ is in A ∩ C, while χ ∈ B. Thus M (A, C) ≤ B, contrary to what we assumed. So A/B is indeed the pseudo-complement of A relative to B. 2 In terms of relative pseudo-complementation, we can also deﬁne the pseudocomplement of a proposition A simpliciter (not relative to any other proposition B). Definition 6 (Pseudo-complementation). For every proposition A, we deﬁne the pseudo-complement of A, A∗ , as the pseudo-complement of A relative to the bottom element of our algebra: A∗ = A/{∅} Fact 4 (Alternative characterization of A∗ ). A∗ = {

A}.

Thus, starting with a new notion of propositions and an order on these propositions which compares both the informative and the inquisitive content that they embody, we have established an algebraic structure with four operations on propositions, meet, join, and (relative) pseudo-complementation.

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3.3

Connectives

Now suppose that we have a certain language L, whose sentences express the kind of propositions considered here. Then it is natural to assume that this language has certain sentential connectives which semantically behave like meet, join or (relative) pseudo-complementation operators. Below we deﬁne a semantics for the language of propositional logic, LP , that has exactly these characteristics: conjunction behaves semantically like a meet operator, disjunction behaves like a join operator, negation behaves like a pseudo-complementation operator, and implication behaves like a relative pseudo-complementation operator. The semantics assumes a valuation function which assigns a truth-value to every atomic sentence in every world. For any atomic sentence p, the set of worlds where p is true is denoted by |p|. Definition 7 (Algebraically motivated inquisitive semantics for LP ). – – – – –

[p] = {α | α ⊆ |p|} [¬ϕ] = [ϕ]∗ [ϕ ∧ ψ] = [ϕ] ∩ [ψ] [ϕ ∨ ψ] = [ϕ] ∪ [ψ] [ϕ → ψ] = [ϕ] / [ψ]

Natural languages are of course much more intricate than the language of propositional logic. We expect, however, that natural languages will generally also have connectives which behave semantically as meet, join, and (relative) pseudocomplementation operators. For a linguistic case study based on this expectation, see [6]. 3.4

Quantifiers

The approach taken here can straightforwardly be extended to obtain an inquisitive semantics for the language of ﬁrst-order logic, LF O . The universal quantiﬁer can be taken to behave semantically as a generalized meet operator, which does not necessarily operate on just two propositions—like the meet operator considered above—but more generally on a (possibly inﬁnite) set of propositions. Similarly, the existential quantiﬁer can be taken to behave semantically as a generalized join operator.1 Fact 5 (Generalized meet). For any non-empty set of propositions Σ, Σ is the greatest lower bound, i.e., the meet of all propositions in Σ w.r.t. ≤. Fact 6 (Generalized join). For any non-empty set of propositions Σ, Σ is the least upper bound, i.e., the join of all propositions in Σ w.r.t. ≤. As usual, the semantics for LF O assumes a domain of individuals D and a world-dependent interpretation function Iw that maps every individual constant 1

There are several alternative algebraic perspectives on quantiﬁcation as well [7,8]. In future work, we will explore these diﬀerent perspectives in the inquisitive setting.

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c to some individual in D and every n-place predicate symbol R to some ntuple of individuals in D. Formulas are interpreted relative to an assignment function g, which maps every variable x to some individual in D. For every individual constant c, [c]w,g = Iw (c) and for every variable x, [x]w,g = g(x). An atomic sentence Rt1 . . . tn is true in a world w relative to an assignment function g iﬀ [t1 ]w,g , . . . , [tn ]w,g ∈ Iw (R). Given an assignment function g, the set of all worlds w such that Rt1 . . . tn is true in w relative to g is denoted by |Rt1 . . . tn |g . Definition 8 (Algebraically motivated inquisitive semantics for LF O ). – – – – – – –

4

[Rt1 . . . tn ]g = {α | α ⊆ |Rt1 . . . tn |g } [¬ϕ]g = [ϕ]∗g [ϕ ∧ ψ]g = [ϕ]g ∩ [ψ]g [ϕ ∨ ψ]g = [ϕ]g ∪ [ψ]g [ϕ → ψ]g = [ϕ]g / [ψ]g [∀x.ϕ]g = d∈D [ϕ]g[x/d] [∃x.ϕ]g = d∈D [ϕ]g[x/d]

Comparison with Previous Work

In this section we brieﬂy compare the algebraically motivated semantics presented here (i) with the inquisitive semantics for LP speciﬁed in [1,4,5], (ii) with the inquisitive semantics for LF O speciﬁed in [2], and (iii) with the ‘unrestricted’ inquisitive semantics for LP and LF O speciﬁed in [1,3]. The comparison will be brief and not fully self-contained (familiarity with the cited work is necessary to understand and appreciate some of the claims made). First, let us consider the inquisitive semantics for LP speciﬁed in [1,4,5]. Even though this semantics was deﬁned in diﬀerent terms and motivated in a diﬀerent way, it is entirely equivalent with the semantics presented here. Thus, as far as LP is concerned, our algebraic considerations have not led to a new semantics, but rather to a more solid foundation for an existing system. Next, let us consider the inquisitive semantics for LF O speciﬁed in [2]. Interestingly, this semantics does not coincide with the one presented here, and the source of the diﬀerences between the two systems lies in the very basic notion of propositions that is assumed. Here we argued that, as long as we are interested in capturing informative and inquisitive content and nothing more than that, propositions should be deﬁned as persistent sets of possibilities. In this way, we rule out the existence of propositions that are non-identical but equivalent in terms of informative and inquisitive content, such as those depicted in ﬁgure 1. Or in more technical terms, in this way we make sure that ≤ is anti-symmetric. In [2], propositions are sets of possibilities such that no possibility is properly contained in a maximal possibility. That is, a proposition A cannot contain two possibilities α and μ such that μ is a maximal element of A and α ⊂ μ. This rules

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out, for instance, propositions like the one depicted in ﬁgure 1(b). It even assures that no proposition is equivalent but non-identical to the proposition depicted in ﬁgure 1(a). However, it does not generally rule out the existence of non-identical equivalent propositions. In other words, it does not make ≤ anti-symmetric. In fact, several cases are discussed in [2] of sentences that are assigned diﬀerent propositions, even though they are entirely equivalent in terms of informative and inquisitive content. This is presented as a desirable result, which indicates that the system in [2] is really intended to capture more than just informative and inquisitive content. However, it remains to be better understood what kind of content is supposed to be captured by the propositions in this system, besides informative and inquisitive content. Once this is better understood, it should be possible to deﬁne a natural order on those propositions, and to motivate the clauses of the semantics algebraically, as we have done here. For now, we conclude that the ﬁrst-order system developed in the present paper is the most appropriate system as long as we are really only interested in informative and inquisitive content.2 Finally, let us consider the ‘unrestricted’ inquisitive semantics for LP and LF O speciﬁed in [1,3]. These systems are called ‘unrestricted’ because they do not impose any restrictions on the notion of propositions that we started out with—that is, propositions are simply deﬁned as arbitrary non-empty sets of possibilities—and they are very explicitly aimed at capturing more than just informative and inquisitive content. In particular, these systems are based on the idea that, besides informative and inquisitive content, propositions can also be taken to capture attentive content. However, the clauses of these systems have so far not been motivated properly, and indeed, they have certain undesirable features (for instance, conjunction is not idempotent, i.e., it is not generally the case that [ϕ∧ϕ] = [ϕ]). Thus, it would be useful to extend the algebraic approach taken here also to the unrestricted setting. The crucial step in establishing such an extension would be to specify an attentiveness order, which determines when one proposition is more attentive than another. Given such an order, it may again be possible to deﬁne meet, join, and complementation operators, which can then be associated with connectives and quantiﬁers. This direction is currently being pursued, and initial results are reported in [9]. 2

A further connection worth mentioning is the following. The semantics for LP speciﬁed in [1,4,5] is based on the notion of support, a relation between possibilities and formulas. The core of the semantics is a recursive deﬁnition of this support relation. Subsequently, the proposition expressed by a formula ϕ is deﬁned as the set of all maximal possibilities supporting ϕ. In [2] it is argued that this approach can not be extended to the ﬁrst-order setting. Thus, an alternative approach is developed, leading to a direct recursive deﬁnition of the proposition expressed by a formula, bypassing the notion of support. However, if we deﬁne the proposition expressed by a formula ϕ as the set of all possibilities supporting ϕ, rather than the set of maximal possibilities supporting ϕ, we obtain a suitable ﬁrst-order system, in fact one that is equivalent to the system developed in the present paper.

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243

Conclusion

In this paper we have deﬁned an inquisitive semantics for the language of propositional logic and ﬁrst-order predicate logic, motivated by general algebraic concerns. We argued that propositions should be deﬁned as persistent sets of possibilities, and we considered an order on such propositions which determines when one proposition is at least as informative and inquisitive as another. We showed that this order gives rise to a Heyting algebra, with meet, join, and (relative) pseudo-complement operators. Our semantics associates these semantic operators with connectives and quantiﬁers. The semantics for LP presented here is equivalent with the one speciﬁed in earlier work [1,4,5]. Thus, as far as LP is concerned, our algebraic considerations did not lead to a new semantics, but rather to a more solid foundation for an existing system. In the case of LF O , the semantics developed here did diverge from previous work [2] and we argued that, as long as we are only interested in capturing informative and inquisitive content, the present system is most appropriate. In future work, we hope to extend the approach also to the ‘unrestricted’ setting, where propositions do not only embody informative and inquisitive content, but also attentive content.

References 1. Ciardelli, I.: Inquisitive semantics and intermediate logics, Master Thesis, ILLC University of Amsterdam (2009) 2. Ciardelli, I.: A ﬁrst-order inquisitive semantics. In: Aloni, M., Bastiaanse, H., de Jager, T., Schulz, K. (eds.) Logic, Language and Meaning. LNCS, vol. 6042, pp. 234–243. Springer, Heidelberg (2010) 3. Ciardelli, I., Groenendijk, J., Roelofsen, F.: Attention! Might in inquisitive semantics. In: Ito, S., Cormany, E. (eds.) Proceedings of Semantics and Linguistic Theory, SALT XIX (2009) 4. Ciardelli, I., Roelofsen, F.: Inquisitive logic. Journal of Philosophical Logic 40(1), 55–94 (2011) 5. Groenendijk, J., Roelofsen, F.: Inquisitive semantics and pragmatics. Presented at the Workshop on Language, Communication, and Rational Agency at Stanford (2009), www.illc.uva.nl/inquisitive-semantics 6. Groenendijk, J., Roelofsen, F.: Inquisitive algebra and the disjunctive-indeﬁniteinterrogative aﬃnity. ILLC University of Amsterdam (2011) (manuscript) 7. Halmos, P.: Algebraic Logic. Chelsea Publishing Company, New York (1962) 8. Henkin, L., Monk, J., Tarski, A.: Cylindric Algebras–Part I. North-Holland, Amsterdam (1971) 9. Roelofsen, F.: Algebraic foundations for unrestircted inquisitive semantics: the ﬁnite case. ILLC University of Amsterdam (2011) (manuscript)

A Dynamic Analysis of Interactive Rationality Eric Pacuit1 and Olivier Roy2 1

Tilburg Institute for Logic and Philosophy of Science [email protected] 2 Center for Mathematical Philosophy, LMU [email protected]

Abstract. Epistemic game theory has shown the importance of informational contexts in understanding strategic interaction. We propose a general framework to analyze how such contexts may arise. The idea is to view informational contexts as the ﬁxed-points of iterated, “rational responses” to incoming information about the agents’ possible choices. We show general conditions for the stabilization of such sequences of rational responses, in terms of structural properties of both the decision rule and the information update policy.

1

Background and Motivation

An increasingly popular1 view is that “the fundamental insight of game theory [is] that a rational player must take into account that the players reason about each other in deciding how to play” [5, pg. 81]. Exactly how the players (should) incorporate the fact that they are interacting with other (actively reasoning) agents into their own decision making process is the subject of much debate. A variety of frameworks explicitly model the reasoning of rational agents in a strategic situation. Key examples include Brian Skyrms’ models of “dynamic deliberation” [34], Ken Binmore’s analysis of “eductive reasoning” [13], and Robin Cubitt and Robert Sugden’s “common modes of reasoning” [19]. Although the details of these frameworks are quite diﬀerent they share a common line of thought: In contrast to classical game theory, solution concepts are no longer the basic object of study. Instead, the “rational solutions” of a game are the result of individual (rational) decisions in speciﬁc informational “contexts”. This perspective on the foundations of game theory is best exempliﬁed by the so-called epistemic program in game theory (cf. [17]). The central thesis here is that the basic mathematical model of a game should include an explicit parameter describing the players’ informational attitudes. However, this broadly decision-theoretic stance does not simply reduce the question of decision-making in interaction to that of rational decision making in the face of uncertainty or ignorance. Crucially, higher-order information (belief about beliefs, etc.) are key 1

But, of course, not uncontroversial. See, for example, [24, pg. 239].

H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 244–257, 2011. c Springer-Verlag Berlin Heidelberg 2011

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components of the informational context of a game2 . Of course, diﬀerent contexts of a game can lead to drastically diﬀerent outcomes, but this means that the informational contexts themselves are open to rational criticism: “It is important to understand that we have two forms of irrationality [...]. For us, a player is rational if he optimizes and also rules nothing out. So irrationality might mean not optimizing. But it can also mean optimizing while not considering everything possible.” [18, pg. 314] Thus, a player can be rationally criticized for not choosing what is best given their information, but also for not reasoning to a “proper” context. Of course, what counts as a “proper” context is debatable. There might be rational pressure for or against making certain substantive assumptions3 about the beliefs of one’s opponents, for instance, always entertaining the possibility that one of the players might not choose optimally. Recently, researchers using methods from dynamic-epistemic logic have taken steps to understanding this idea of reasoning to a “proper” or “rational” context [37,8,9,10]. Building on this literature4 , we provide a general characterization of when players can or cannot rationally reason to an informational context.

2

Belief Dynamics for Strategic Games

Our goal is to understand well-known solution concepts, not in terms of ﬁxed informational contexts—for instance, models (e.g., type spaces or epistemic models) satisfying rationality and common belief of rationality—but rather as a result of a dynamic, interactive process of “information exchanges”. It is important to note that we do not see this work as an attempt to represent some type of “preplay communication” or form of “cheap talk”. Instead, the idea is to represent the process of rational deliberation that takes the players from the ex ante stage to the ex interim stage of decision making. Thus, the “informational exchanges” are the result of the players’ practical reasoning about what they should do, given 2

3 4

That is, strategic behavior depends, in part, on the players’ higher-order beliefs. However, the question of what precisely is being claimed should be treated with some care. The well-known email game of Ariel Rubinstein [32] demonstrates that misspeciﬁcation of arbitrarily high-orders of beliefs can have a great impact on (predicted) strategic behavior. So there are simple examples where (predicted) strategic behavior is too sensitive to the players’ higher-order beliefs. We are not claiming that a rational agent is required to consider all higher-order beliefs, but only that a rational player recognizes that her opponents are actively reasoning, rational agents, which means that a rational player does take into account some of her higher-order beliefs (e.g., what she believes her opponents believe she will do) as she deliberates. Precisely “how much” higher-order information should be taken into account is a very interesting, open question which we set aside in this paper. The notion of substantive assumption is explored in more detail in [31]. The reader not familiar with this area can consult the recent textbook [12] for details.

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their current beliefs. This is in line with the current research program using dynamic epistemic and doxastic logics to analyze well-known solution concepts (cf. [2,8,37] where the “rationality announcements” do not capture any type of communication between the players, but rather internal observations about which outcomes of the game are “rational”). 2.1

Describing an Informational Context

Let G = N, {Si , ui }i∈N be a strategic game (where N is the set of players and for each i ∈ N , Si is the set of actions, or strategies, for player i and ui : Πi Si → R is a utility function).5 The informational context of a game describes the players’ hard and soft information about the possible outcomes of the game. Many diﬀerent formal models have been used to represent an informational context of a game (for a sample of the extensive literature, see [15,37] and references therein). In this paper we employ one such model: a plausibility structure consisting of a set of states and a single plausibility ordering (which is reﬂexive, transitive and connected) w v that says “w is at least as plausible as v.” Originally used as a semantics for conditionals (cf. [26]), these plausibility models have been extensively used by logicians [36,12,9], game theorists [14] and computer scientists [16,25] to represent rational agents’ (all-out) beliefs. We thus take for granted that they provide a natural model of beliefs in games: Definition 1. Let G = N, {Si , ui }i∈N be a strategic form game. An informational context of G is a plausibility model MG = W, , σ where is a connected, reflexive, transitive and well-founded6 relation on W and σ is a strategy function: a function σ : W → Πi Si assigning strategy profiles to each state. To simplify notation, we write σi (w) for (σ(w))i (similarly, write σ−i (w) for the sequence of strategies of all players except i). A few comments about this deﬁnition are in order. First of all, note that there is only one plausibility ordering in the above models, yet we are interested in games with more than one player. There are diﬀerent ways to interpret the fact that there is only one plausibility ordering. One is that the models represent the beliefs of a single player before she has made up her mind about which option to choose in the game. A second interpretation is to think of a model as representing the modeler’s or game theorist’s point of view about which outcomes are more or less plausible given the reasoning of the players. Thus, a model describes a stage of the rational deliberation of all the players starting from an initial model where the players have the same beliefs (i.e., the common prior). The 5

6

We assume the reader is familiar with the basic concepts of game theory. For example, strategic games and various solution concepts, such as iterated removal of strictly (weakly) dominated strategies. Well-foundedness is only needed to ensure that, for any set X, the set of minimal elements in X is nonempty. This is important only when W is inﬁnite – and there are ways around this in current logics. Moreover, the condition of connectedness can also be lifted, but we use it here for convenience.

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private information about which outcomes the players consider possible given their actual choice can then be deﬁned from the conditional beliefs.7 Our second comment on the above deﬁnition is that since we are representing the rational deliberation process, we do not assume that the players have made up their minds about which actions they will choose. Finally, note that the strategy functions need not be onto. Thus, the model represents the player’s(s’) opinions about which outcomes of the game are more or less plausible among the ones that have not been ruled out. Of course, this model can be (and has been: see [9,12]) extended to include beliefs for each of the players, an explicit relation representing the player(s) hard information or by making the plausibility orders state-dependent. In order to keep things simple we focus on models with a single plausibility ordering. We conclude this brief introduction to plausibility models by giving the wellknown deﬁnition of a conditional belief. For X ⊆ W , let M in (X) = {v ∈ X | v w for all w ∈ X } be the set of minimal elements of X according to . Definition 2 (Belief and Conditional Belief ). Let MG = W, , σ be a model of a game G. Let E and F be subsets of W , we say: – E is believed conditional on F in MG provided M in (F ) ⊆ E. Also, we say E is believed in MG if E is believed conditional on W . Thus, E is believed provided M in (W ) ⊆ E 2.2

A Primer on Belief Dynamics

We are not interested in informational contexts per se, but rather how the informational context changes during the process of rational deliberation. The type of change we are interested in is how a model MG of a game G incorporates new information about what the players should do (according to a particular choice rule). As is well known from the belief revision literature, there are many ways to transform a plausibility model given some new information [30]. We do not have the space to survey the entire body of relevant literature here (cf., [12,7]). Instead we sketch some key ideas, assuming the reader is already familiar with this approach to belief revision. The general approach is to deﬁne a way of transforming a plausibility model MG given a proposition ϕ. A transformation τ maps plausibility models and τ (ϕ) propositions to plausibility models (we write MG for τ (MG , ϕ)). Diﬀerent deﬁnitions of τ represent the diﬀerent attitudes an agent can take to the incoming information. The picture below provides three typical examples: 7

The suggestion here is that one can deﬁne a partition model ´ a la Aumann [6] from a plausibility model. Working out the details is left for future work, but we note that such a construction blurs the distinction between so-called belief-based and knowledge-based analyses of solution concepts (cf. the discussion in [17]).

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B

E D

B

E A

ϕ

D

B

E A

ϕ

D

A

ϕ

C

C

C

!(ϕ) : A ≺ B

↑ (ϕ) : A ≺ C ≺ D ≺ B ∪ E

⇑ (ϕ) : A ≺ B ≺ C ≺ D ≺ E

The operation on the left is the well-known public announcement operation [28,21], which assumes that the source of ϕ is infallible, ruling out any possibilities that are inconsistent with ϕ. For the other transformations, while the players do trust the source of ϕ, they do not treat the source as infallible. Perhaps the most ubiquitous policy is conservative upgrade (↑ϕ), which allows the player(s) only tentatively to accept the incoming information ϕ by making the best ϕ-worlds the new minimal set while keeping the old plausibility ordering the same on all other worlds. The operation on the right, radical upgrade (⇑ϕ), is stronger, moving all ϕ worlds before all the ¬ϕ worlds and otherwise keeping the plausibility ordering the same. These dynamic operations satisfy a number of interesting logical principles [12,7], which we do not discuss further here. We are interested in the operations that transform the informational context as the players deliberate about what they should do in a game situation. In each informational context (viewed as describing one stage of the deliberation process), the players determine which options are “rationally permissible” and which options the players ought to avoid (which is guided by some ﬁxed choice rule). This leads to a transformation of the informational context as the players adopt the relevant beliefs about the outcome of their practical reasoning. The diﬀerent types of transformation mentioned above then represent how conﬁdent the player(s) (or modeler) is (are) in the assessment of which outcomes are rational. In this new informational context, the players again think about what they should do, leading to another transformation. The main question is does this process stabilize? The answer to this question will depend on a number of factors. The general picture is τ (D0 )

τ (D1 )

τ (D2 )

τ (Dn )

M0 =⇒ M1 =⇒ M2 =⇒ · · · =⇒ Mn+1 =⇒ · · · where each Di is some proposition and τ is a model transformer. Two questions are important for the analysis of this process. First, what type of transformations are the players using? For example, if τ is a public announcement, then it is not hard to see that, for purely logical reasons, this process must eventually stop at a limit model (see [9] for a discussion and proof). The second question is where do the propositions Di come from? To see why this matters, consider the situation where you iteratively perform a radical upgrade with p and ¬p (i.e.,

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⇑ (p), ⇑ (¬p), . . .). Of course, this sequence of upgrades never stabilizes. However, in the context of reasoning about what to do in a game situation, this situation may not arise thanks to special properties of the choice rule that is being used to describe (or guide) the players’ decisions. 2.3

Deliberating about What To Do

It is not our intention to have the dynamic operations of belief change discussed in the previous section directly represent the players’ (practical) reasoning. Instead, we treat practical reasoning as a “black box” and focus on general choice rules that are intended to describe rational decision making (under ignorance). To make this precise, we need some notation: Definition 3 (Strategies in Play). Let G = N, {Si }i∈N , {ui }i∈N be a strategic game and MG = W, , σ an informational context of G. For each i ∈ N , the strategies in play for i is the set S−i (MG ) = {s−i ∈ Πj=i Sj | there is w ∈ M in (W ) with σ−i (w) = s−i } This set S−i (MG ) is the set of strategies that are believed to be available for player i at some stage of the deliberation process represented by the model MG . Given S−i (MG ), diﬀerent choice rules oﬀer recommendations about which options to choose. There are many choice rules that could be analyzed here (e.g., strict dominance, weak dominance or admissibility, minimax, minmax regret, etc.). For the present purposes we focus primarily on weak dominance (or admissibility), although our main theorem in Seciton 3 applies to all choice rules. Weak Dominance (pure strategies8 ) Let G = N, {Si }i∈N , {ui}i∈N be a strategic game and MG an model of G. For each i and a ∈ Si , put a ∈ Siwd (MG ) provided there is b ∈ Si such that for all s−i ∈ S−i (MG ), ui (s−i , b) ≥ ui (s−i , a) and there is some s−i ∈ S−i (MG ) such that ui (s−i , b) > ui (s−i , a). So an action a is weakly dominated for player i if it is weakly dominated with respect to all of i’s available actions and the (joint) strategies believed to be still in play for i’s opponents. More generally, we assume that given the beliefs about which strategies are in play the players categorize their available options (i.e., the set Si ) into “good” (or “rationally permissible”) strategies and those strategies that are “bad” (or “irrational”). Formally, a categorization for player i is a pair Si (MG ) = (Si+ , Si− ) where Si+ ∪Si− ⊆ Si . (We write Si (MG ) to signal that the categorization depends on current beliefs about which strategies are in play.) Note that, in general, a categorization need not be a partition (i.e., Si+ ∪Si− = Si ) . See [20] for an example of such a categorization algorithm. However, in the remainder of this paper we focus on familiar choice rules where the categorization does form a partition. For example, for weak dominance we let Si− = Siwd (MG ) and Si+ = Si − Si− . 8

This deﬁnition can be modiﬁed to allow for dominance by mixed strategies, but we leave issues about how to incorporate probabilities to another occasion.

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Given a model of a game MG and for each player i a categorization is Si (MG ); the next step is to incorporate this information into MG using some model transformation. We start by introducing a simple propositional language to describe a categorization. Definition 4 (Language for a Game). Let G = N, {Si }i∈N , {ui}i∈N be a strategic game. Without loss of generality, assume that each of the Si is disjoint and let AtG = {Pai | a ∈ Si } be a set of atomic formulas (one for each a ∈ Si ). The propositional language for G, denoted LG , is the smallest set of formulas containing AtG and closed under the Boolean connectives ¬ and ∧. Formulas of LG are intended to describe possible outcomes of the game. Given an informational context of a game MG , the formulas ϕ ∈ LG can be associated with subsets of the set of states in the usual way: Definition 5. Let G be a strategic game, MG = W, , σ an informational context of G and LG a propositional language for G. We define a map [[·]]MG : LG → ℘(W ) by induction as follows: [[Pai ]]MG = {w | σ(w)i = a}, [[¬ϕ]]MG = W − [[ϕ]]MG and [[ϕ ∧ ψ]]MG = [[ϕ]]MG ∩ [[ψ]]MG . Using the above language, for each informational context of a game MG , we can deﬁne Do(MG ), which describes what the players are going to do according to a ﬁxed categorization procedure. To make this precise, suppose that Si (MG ) = (Si+ , Si− ) is a categorization for each i and deﬁne: Doi (MG ) := Pia ∧ ¬Pib a∈Si+

b∈Si−

Then, let Do(MG ) = i Doi (MG ).9 The general project is to understand the interaction between types of categorizations (eg., choice rules) and types of model transformations (representing the rational deliberation process). One key question is: Does a deliberation process stabilize(and if so, under what conditions)? (See [9] for general results here.) In this paper there are two main reasons why an upgrade stream would stabilize. The ﬁrst is from properties of the transformation. The second is because the choice rule satisﬁes a monotonicity property so that, eventually, the categorizations stabilize and no new transformations can change the plausibility ordering. We are now ready to give a formal deﬁnition of a “deliberation sequence”: Definition 6 (Deliberation Sequence). Given a game G and an informational context MG , a deliberation sequence of type τ (which we also call an upgrade sequence), induced by MG is an infinite sequence of plausibility models (Mm )m∈N defined as follows: M0 = MG

Mm+1 = τ (Mm , Do(Mm ))

An upgrade sequence stabilizes if there is an n ≥ 0 such that Mn = Mn+1 . 9

There are other ways to describe a categorization, but we leave this for further research.

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251

Case Study: Iterated Admissibility

A key issue in the epistemic foundations of game theory is the epistemic analysis of iterated removal of weakly dominated strategies. Many authors have pointed out puzzles surrounding such an analysis [4,33,18]. For example, Samuelson [33] showed (among other things) that “common knowledge of admissibility” may be an inconsistent concept (in the sense that there is a game which does not have a model with a state satisfying ‘common knowledge of rationality’ [33, Example 8, pg. 305]).10 This is illustrated by the following game: Bob L R u 1, 1 1, 0 Ann d 1, 0 0, 1 The key issue is that the assumption that players only play admissible strategies conﬂicts with the logic of iteratively removing strategies deemed “irrational”. The general framework introduced above oﬀers a new, dynamic perspective on this issue, and on reasoning with admissibility more generally.11 Dynamically, Samuelson’s non-existence result corresponds to the fact that the players’ rational upgrade streams do not stabilize. That is, the players are not able to deliberate their way to a stable, common belief in admissibility. In order to show this we need the “right” notion of model transformation. Our ﬁrst observation is that the model transformations we discussed in Section 2.2 do not explain Samuelson’s result. Observation 1. Suppose that the categorization method is weak dominance and that Do(M) is deﬁned as above. For each of the model transformations discussed in Section 2.2 (i.e., public announcement, radical upgrade and conservative upgrade), any deliberation sequence for the above game stabilizes. The proof of this Observation is straightforward since the language used to describe the categorization does not contain belief modalities12 . This observation is nice, but it does not explain the phenomena noticed by Samuelson [33]. The problem lies in the way we incorporate information when there is more than one element of Si+ (M) for some agent i. 10

11

12

Compare with strict dominance: it is well known that common knowledge that players do not play weakly dominated strategies implies that the players choose a strategy proﬁle that survives iterated removal of strictly dominated strategies. We do not provide an alternative epistemic characterization of this solution concept. Both [18] and [22] have convincing results here. Our goal is to use this solution concept as an illustration of our general approach. An interesting extension would be to start with a multiagent belief model and allow players not only to incorporate information about which options are “choice-worthy”, but also what beliefs their opponents may have. We leave this extension for future work, focusing here on setting up the basic framework.

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It is well known that, in general, there are no rational principles of decision making (under ignorance or uncertainty) which always recommend a unique choice. In particular, it is not hard to ﬁnd a game and an informational context where there is at least one player without a unique “rational choice”. How should a rational player incorporate the information that more than one action is classiﬁed as “choice-worthy” or “rationally permissible” (according to some choice rule) for her opponent(s)? Making use of a well-known distinction due to Edna Ullmann-Margalit and Sidney Morgenbesser [35], the assumption that all players are rational can help determine which options the player will choose, but rationality alone does not help determine which of the rationally permissible options will be “picked”13. What interests us is how to transform a plausibility model to incorporate the fact that there is a set of choice-worthy options for (some of) the players. We suggest that a generalization of conservative upgrade is the notion we are looking for (see [23] for more on this operation). The idea is to do an upgrade with a set of propositions {ϕ1 , . . . , ϕn } by letting the most plausible worlds be the union of each of the most plausible ϕi worlds:

F C

ϕ1

A

G E

D

ϕ2

B

↑ {ϕ1 , ϕ2 } : A ∪ E ≺ B ≺ C ∪ D ≺ F ∪ G

We do not give the formal deﬁnition here, but it should be clear from the example given above. It is not hard to see that this isnot the same as ↑ϕ1 ∨ · · · ∨ ϕn , since, in general, M in ([[ϕ1 ]] ∪ · · · ∪ [[ϕn ]]) = i M in ([[ϕi ]]). We must modify our deﬁnition of Do(M): for each i ∈ N let: − i Doi (Si (MG )) = {Pai | a ∈ S+ i (MG )} ∪ {¬Pb | b ∈ Si (MG )}

Then deﬁne Do(S(MG )) = Doi (Si (MG )) Do2 (S2 (MG )) · · · Don (Sn (MG )), where if X and Y are two sets of propositions, then let X ∧ Y := {ϕ ∧ ψ | ϕ ∈ X, ψ ∈ Y }. 13

This line of thought led Cubitt and Sugden to impose a “privacy of tie breaking” property which says that players cannot know that her opponent will not pick an option that is classiﬁed as “choice-worthy” [19, pg. 8] (cf. also [4]’s “no extraneous restrictions on beliefs” property). Wlodeck Rabinovich takes this even further and argues that from the principle of indiﬀerence, players must assign equal probability to all choice-worthy options [29].

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Observation 2. Suppose that the categorization method is weak dominance as explained in Section 2.3 and that Do(M) is deﬁned as above. Then, starting with the initial full model of the above game,14 a generalized conservative upgrade stream does not stabilize. The following upgrade stream illustrates this observation:

u, L d, L

u, R

↑D0

d, R

d, L u, L

d, R u, R

↑D1

d, L

d, R

↑D2

u, R u, L

M0

d, R u, R u, L

M1

↑D3

u, L

u, R

d, L

d, R

M2

d, L

M3

M4 = M0

Intuitively, from M0 to M2 the agents have reasons to exclude d and R, leading them to the common belief that u, L is played. At that stage, however, d is admissible for Ann, canceling the reason the agents had to rule out this strategy. The rational response here is thus to suspend judgment on d, leading to M3 . In this new model the agents are similarly led to suspend judgment on not playing R, bringing them back to M0 . This process loops forever: the agents’ reasoning does not stabilize. A corollary of this observation is that common belief in admissibility is not suﬃcient for the stabilization of upgrade streams. Stabilization also requires that all and only those proﬁles that are most plausible are admissible.

4

Stabilization Theorem

In this section we informally state and discuss a number of abstract principles which guarantee that a rational deliberation sequence will stabilize. The principles ensure that the categorizations are “sensitive” to the players’ beliefs and that the players respond to the categorizations in the appropriate way. We start by ﬁxing some notation. Let U be a ﬁxed set of states and G a ﬁxed strategic game. We conﬁne our attention to transformations between models of 14

A full model is one where it is common knowledge that each outcome of the game is equally plausible.

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G whose states come from the universe of states U . Let MG be the set of all such plausibility models. A model transformation is then a function that maps a model of G and a ﬁnite set of formulas of LG to a model in MG : τ : MG × ℘<ω (LG ) → MG where ℘<ω (LG ) is the set of ﬁnite subsets of LG . Of course, not all transformations τ make sense in this context. The ﬁrst set of principles that τ must satisfy ensure that the categorizations and belief transformation τ are connected in the “right way”. One natural property is that the belief transformations treat equivalent formulas the same way. A second property we impose is that receiving exactly the same (ground) information twice does not have any eﬀect on the players’ beliefs. These are general properties of the belief transformation. Certainly, there are other natural properties that one may want to impose (for example, variants of the AGM postulates [1]), but for now we are interested in the minimal principles needed to prove a stabilization result. The next set of properties ensure that the transformations respond “properly” to a categorization. First, we need a property to guarantee that the categorizations depend only on the players’ beliefs. Second, we need to ensure that all upgrade sequences respond to the categorizations in the right way: C2− For any upgrade sequence (Mn )n∈N in τ , if a ∈ Si− (Mn ) then ¬Pia is believed in Mn+1 . C2+ For any upgrade sequence (Mn )n∈N in τ , if a ∈ Si+ (Mn ) then ¬Pia is not believed in Mn+1 Finally, we need to assume that the categorizations are monotonic: Mon− For any upgrade sequence (Mn )n∈N , for all n ≥ 0, for all players i ∈ N , − S− i (Mn ) ⊆ Si (Mn+1 ) − + Mon Either for all models MG , S+ i (MG ) = Si − Si (MG ) or for any upgrade sequence (Mn )n∈N , for all n ≥ 0, for all players i ∈ N , S+ i (Mn ) ⊆ S+ i (Mn+1 ) In particular, Mon− means that once an option for a player is classiﬁed as “not rationally permissible”, it cannot drop this classiﬁcation at a later stage of the deliberation process. Theorem 3. Suppose that G is a finite game and all of the above properties are satisfied. Then every upgrade sequence (Mn )n∈N stabilizes. The proof can be found in the full version of the paper. The role of monotonicity of the choice has been noticed by a number of researchers (see [3] for a discussion). This theorem generalizes van Benthem’s analysis of rational dynamics [37] to soft information, both in terms of attitudes and announcements. It is also closely related to the result in [3] (a complete discussion can be found in the full paper).

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255

Concluding Remarks

In this paper we have proposed a general framework to analyze how “proper” informational contexts my arise. We have provided general conditions for the stabilization of deliberation sequences in terms of structural properties of both the decision rule and the information update policy. We have also applied the framework to admissibility, giving a dynamic analysis of Samuelson’s non-existence result. Throughout the paper we have worked with (logical) models of all out attitudes, leaving aside probabilistic andgraded beliefs, even though the latter are arguably most widely used in the current literature on epistemic foundations of game theory. It is an important but non-trivial task to transpose the dynamic perspective on informational contexts that we advocate here to such probabilistic models. This we leave for future work. Finally, we stress that the dynamic perspective on informational contexts is a natural complement and not an alternative to existing epistemic characterizations of solution concepts [11], which oﬀer rich insights into the consequences of taking seriously the informational contexts of strategic interaction. What we have proposed here is a ﬁrst step towards understanding how or why such contexts might arise. Acknowledgments. We are very grateful to Wlodek Rabinowicz, Denis Bon´ e, Johan van Benthem, and the two anonymous referees for the nay, Paul Egr´ LORI conference for useful comments and suggestions. Olivier Roy would like to gratefully acknowledge the ﬁnancial support of the Alexander von Humboldt Foundation, and of the project “Shared Commitment and Common Knowledge”, hosted by the University of Groningen, and supported by the Nederlandse Organisatie voor Wetenshappelijke Onderzoek (NWO).

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8. Baltag, A., Smets, S., Zvesper, J.: Keep ‘hoping’ for rationality: a solution to the backwards induction paradox. Synthese 169, 301–333 (2009) 9. Baltag, A., Smets, S.: Group belief dynamics under iterated revision: Fixed points and cycles of joint upgrades. In: Proceedings of Theoretical Aspects of Rationality and Knowledge (2009) 10. van Benthem, J., Gheerbrant, A.: Game solution, epistemic dynamics and ﬁxedpoint logics. Fund. Inform. 100, 1–23 (2010) 11. van Benthem, J., Pacuit, E., Roy, O.: Towards a theory of play: A logical perspective on games and interaction. Games 2(1), 52–86 (2011) 12. van Benthem, J.: Logical Dynamics of Information and Interaction. Cambridge University Press, Cambridge (2010) 13. Binmore, K.: Modeling rational players: Part I. Economics and Philosophy 3, 179– 214 (1987) 14. Board, O.: Dynamic interactive epistemology. Games and Economic Behavior 49, 49–80 (2004) 15. Bonanno, G., Battigalli, P.: Recent results on belief, knowledge and the epistemic foundations of game theory. Research in Economics 53(2), 149–225 (1999) 16. Boutilier, C.: Conditional Logics for Default Reasoning and Belief Revision. Ph.D. thesis, University of Toronto (1992) 17. Brandenburger, A.: The power of paradox: some recent developments in interactive epistemology. International Journal of Game Theory 35, 465–492 (2007) 18. Brandenburger, A., Friedenberg, A., Keisler, H.J.: Admissibility in games. Econometrica 76, 307–352 (2008) 19. Cubitt, R., Sugden, R.: Common reasoning in games: A Lewisian analysis of common knowledge of rationality, ceDEx Discussion Paper (2011) 20. Cubitt, R., Sugden, R.: The reasoning-based expected utility procedure. Games and Economic Behavior (2011) (in press) 21. Gerbrandy, J.: Bisimulations on Planet Kripke. Ph.D. thesis, University of Amsterdam (1999) 22. Halpern, J., Pass, R.: A logical characterization of iterated admissibility. In: Heifetz, A. (ed.) Proceedings of the Twelfth Conference on Theoretical Aspects of Rationality and Knoweldge, pp. 146–155 (2009) 23. Holliday, W.: Trust and the dynamics of testimony. In: Logic and Interaction Rationality: Seminar’s Yearbook 2009, pp. 147 – 178. ILLC Technical Reports (2009) 24. Kadane, J.B., Larkey, P.D.: Subjective probability and the theory of games. Management Science 28(2), 113–120 (1982) 25. Lamarre, P., Shoham, Y.: Knowledge, certainty, belief and conditionalisation. In: Proceedings of the International Conference on Knoweldge Representation and Reasoning, pp. 415–424 (1994) 26. Lewis, D.: Counterfactuals. Blackwell Publishers, Oxford (1973) 27. Plaza, J.: Logics of public communications. Synthese: Knowledge, Rationality, and Action 158(2), 165–179 (2007) 28. Plaza, J.: Logics of public communications. In: Emrich, M.L., Pfeifer, M.S., Hadzikadic, M., Ras, Z. (eds.) Proceedings, 4th International Symposium on Methodologies for Intelligent Systems, pp. 201–216 (1989) (republished as [27]) 29. Rabinowicz, W.: Tortous labyrinth: Noncooperative normal-form games between hyperrational players. In: Bicchieri, C., Chiara, M.L.D. (eds.) Knowledge, Belief and Strategic Interaction, pp. 107–125 (1992)

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30. Rott, H.: Shifting priorities: Simple representations for 27 itereated theory change operators. In: Lagerlund, H., Lindstr¨ om, S., Sliwinski, R. (eds.) Modality Matters: Twenty-Five Essays in Honour of Krister Segerberg. Uppsala Philosophical Studies, vol. 53, pp. 359–384 (2006) 31. Roy, O., Pacuit, E.: Substantive assumptions and the existence of universal knowledge structures: A logical perspective (2010) (under submission) 32. Rubinstein, A.: The electronic mail game: A game with almost common knowledge. American Economic Review 79, 385–391 (1989) 33. Samuelson, L.: Dominated strategies and common knowledge. Game and Economic Behavior 4, 284–313 (1992) 34. Skyrms, B.: The Dynamics of Raitonal Deliberation. Harvard University Press, Cambridge (1990) 35. Ullmann-Margalit, E., Morgenbesser, S.: Picking and choosing. Social Research 44, 757–785 (1977) 36. van Benthem, J.: Dynamic logic for belief revision. Journal of Applied Non-Classical Logics 14(2), 129–155 (2004) 37. van Benthem, J.: Rational dynamics and epistemic logic in games. International Game Theory Review 9(1), 13–45 (2007)

Seeing, Knowledge and Common Knowledge Fran¸cois Schwarzentruber IRIT, Universit´e Paul Sabatier 118 Route de Narbonne F-31062 TOULOUSE CEDEX 9, France [email protected]

Abstract. We provide a multi-agent spatially grounded epistemic logical framework to reason about the knowledge of perception (agent a sees agent b) whose potential applications are video games and robotics. Contrary to the classical epistemic modal logic, we prove that in some configurations the logic with the common knowledge operator is as expressive as the logic without the common knowledge operator. We give some complexity results about the model-checking.

Many authors in logic and in Artiﬁcial Intelligence [9] developed epistemic logic and studied mathematical properties of it. Generally speaking many examples of epistemic situation mix time and space [4]. The link between time and knowledge (perfect recall etc.) has been studied and you can ﬁnd a survey in [1]. There exist also some works linking ‘space’ and knowledge: [10], [11] and [5]. In their topological epistemic logic the authors provide an epistemic modal logic based on the accuracy of the observation. For them, the geometrical aspect – topological aspect – is put directly over the set of mental states. However there are few approaches concerning the description of a multiagents in terms of what agents perceive and know about what they perceive whereas applications may be numerous in video games or robotics in which knowledge is related to spatial reasoning. Reasoning about what agents see and know may be useful in order to specify the artiﬁcial behaviors of agents with a high-level descriptive language. Such a descriptive language makes the development/veriﬁcation of software easier. For instance, in a platform game, an enemy e may attack the hero h if e knows that h is not looking at e. This behavior is directly related to perception and knowledge. Common knowledge about perception is important to design artiﬁcial behavior of groups. For instance, the enemy e1 and e2 can commonly decide to attack the hero h if they commonly know that e1 sees h and e2 sees h. Here we based our approach on [12], [2] and [15]: we deal with a spatially grounded epistemic logic. We do not provide modal operators in the language to deal with space but constructions of the form a b (agent a sees agent b) and epistemic modal operators for each agent in the language. The semantics will then directly rely on what agents see. In other worlds we directly describe a situation by the graphical and natural representation of the system or by saying who is seeing who but not with a Kripke structure. The advantage is that the H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 258–271, 2011. c Springer-Verlag Berlin Heidelberg 2011

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model is more concise: as you will see in Section 2, a given conﬁguration of n agents corresponds to a Kripke model of O(2n ) worlds. In that sense, our logic is built-in: the relation Ra is not given in the model but deduced from the data in the possible worlds. This approach is also devoted to teach epistemic modal logic and public announcements using the tool Plaza’s world [3]: this framework oﬀers concrete examples of Kripke structures and then helps the students to understand better the meaning of Kripke semantics. They are also similar works. In [7] agents can perceive propositions but their approaches is not geometrical at all. In [18] the authors provide a logical framework to deal with perception and knowledge and they do not focus on the complexity of the satisﬁability problem. The idea of a built-in logic is not new: for instance in [8] the author develops a logic for rule-based agents where the belief change is built-in. The contributions of this paper are multiple. First we give a uniﬁed framework to speak about what agents see and know for the space of dimension n and in the case where there is no geometrical constraint (Section 1 and 2) inspired by [12], [2] and [15]. In Section 3 we prove that in some cases we can drop the common knowledge operator and keep the same expressivity of the language with a polynomial translation. This leads to a reduction of the satisﬁability problem or model-checking with common knowledge to the satisﬁability problem or model-checking without common knowledge. In Section 4 we give algorithms for model-checking and ﬁgure out the complexity of the model-checking for some fragments. Proofs are given in the technical report [14].

1

Language

Let AGT = {a, b, c . . . } be a countable set of agents. The language of our spatially grounded epistemic modal logic is deﬁned by the following rule: ϕ ::= a b | ⊥ | ¬ϕ | (ϕ ∨ ϕ) | Ka ϕ | CKJ ϕ | [ϕ!]ϕ where a, b ∈ AGT and J is a ﬁnite set of agents. The construction ab is read as “agent a sees agent b”. The construction Ka ϕ stands for “agent a knows that ϕ” is true. CKJ ϕ stands for “there is common knowledge among the group J that ϕ is true. [ϕ!]ψ stands for “after the public announcement ϕ, the formula ψ holds”. We also write a b for ¬ab in order to ease the reading. As usual, (ϕ∧ψ) =def ˆ ¬(¬ϕ ∨ ¬ψ). Ka ψ =def ¬Ka ¬ψ. We follow the standard rules for omission of parentheses. Let agt(ϕ) be the set of all agents occurring in ϕ. The formula Ka b c is read “agent a knows that agent b sees agent c”. We also deﬁne (Ka1 . . . Kan )k ϕ for all k ≥ 0 by induction: – (Ka1 . . . Kan )0 ϕ =def ϕ; – and for all k ≥ 1, (Ka1 . . . Kan )k ϕ =def (Ka1 . . . Kan )k−1 (Ka1 . . . Kan )ϕ.

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Semantics Worlds

A world is a description of the spatial conﬁguration of agents and how they perceive the world. We start with an abstract notion of a world: for all agents a, b ∈ AGT, we say whether a sees b or not. Definition 1 (world). {(a, a) | a ∈ AGT}.

A world w is a subset of AGT2 containing

(a, b) ∈ w means that a sees b in the world w. By convention, we suppose that agent a sees himself. With W all we denote the set of all possible worlds. Given a world w, we note – for all a ∈ AGT, w(a) = {b ∈ AGT | (a, b) ∈ w}. – For all J ⊆ AGT, w(J) = {b ∈ AGT | for all a ∈ J, (a, b) ∈ w}. Informally w(a) denotes all agents that are seen by agent a. w(J) denotes all agents that are commonly seen by agents of J. Agent a sees agent b in w if b ∈ w(a). Agent a sees nobody in w when w(a) = {a}. Now we are interested in worlds that can be instantiated in the euclidean ndimensional space Rn . Its scalar product is noted and Unit is the set of vectors of Rn of euclidean norm equal to 1. A world w ∈ W all is representable in Rn iﬀ there exists an injective function pos : AGT → Rn and a function dir : AGT → Unit such that for all a, b ∈ AGT we have b ∈ w(a) iﬀ dir(a)pos(a)pos(b) ≥ 0. Informally a world w is representable in dimension n iﬀ we can give a position pos(a) and direction dir(a) to each agent a such that a sees b (b ∈ w(a)) when b is in the half-space of dimension n − 1 of direction dir(a). The function pos is injective means that we cannot put two agents at thee same point of Rn . With W n we denote the set of worlds w that are representable in dimension n. 1 W corresponds exactly to the logic of Lineland presented in [12] and a variant of [2]. W 2 corresponds to a variant of the logic of Flatland in [12]. In the logic presented in [12], exact positions of agents are taken into account whereas in this paper only the qualitative relations of seeing are taken in account. We have W n−1 ⊆ W n . The world w1 deﬁned by w1 (a) = {a, b, c}, w1 (b) = {b, c}, w1 (c) = {a, b, c} is representable in dimension 1. (Figure 1).

a

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Fig. 1. The world w1

When card(AGT) ≥ 3, a world w2 deﬁned such that w2 (a) = AGT for all a ∈ AGT (everybody sees everybody) is representable in dimension n ≥ 2 (Figure 2). Nevertheless, the worlds w2 is not representable in dimension 1. Indeed, the representation should look like

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...

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...

a

b

Now the reader may see that if we put agent c on the left we will have c ∈ w2 (a), if we put agent c on the right we will have c ∈ w2 (b) and if we put agent c between a and b, either a ∈ w2 (c) or b ∈ w2 (c). In any case, we have a contradiction. We have proved that it is impossible for three agents to see each other in one dimension. Formally: Proposition 1. Let J be a set of agents. Let w ∈ W 1 . If J ⊆ w(J) then card(J) ≤ 2.

q

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Fig. 2. The world w2

When AGT = {a, b, c, d, e, f, g}, the world w3 deﬁned by w3 (a) = {a, b, c, f }, w(b) = {a, b, c, d}, w3 (c) = {a, b, c, e}, w3 (e) = {e}, w(f ) = {f } and w(g) = {g} is representable in dimension n ≥ 3 Figure 3 gives such a representation (directions of agents d, e, f, g are not reported). Euclidean and linear algebra arguments allows us to show that w3 is not1 in dimension 2. 2.2

Epistemic Kripke Structure over Worlds

Definition 2 (indistinguishable worlds). Two worlds w and u are indistinguishable for agent a, noted wRa u, iﬀ 1. w(a) = u(a); 2. and for all b, c ∈ w(a), (b, c) ∈ w iﬀ (b, c) ∈ u. The condition 1 says that the agent a sees the same agents in both w and u. The condition 2 says that agent a is aware of b sees c or of b does not see c when a sees b and c. The relation Ra is an equivalence relation. Some authors think that requiring the relation to be an equivalence relation is too strong: 1

See http://www.irit.fr/ Francois.Schwarzentruber /documents/inspacelandnotinflatland.pdf

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Fig. 3. The world w3

– Hintinkka rejects the the negative introspection (5) “unless you happen to be as sagacious as Socrates”. [6] He models epistemic reasoning with the logic S4. – Stalnaker [16] studies the combination of the logic of belief KD45 (axiom 5 for beliefs does not yield to a contradiction), the logic of knowledge S4 plus the interactions Ba ϕ → Ka Ba ϕ (positive introspection of beliefs), ¬Ba ϕ → Ka ¬Ba ϕ, Ka ϕ → Ba ϕ (knowledge implies belief), Ba ϕ → Ba Ka ϕ (strong belief) where Ba means “agent a believes that”. In this system, he claims ˆ a Ka ϕ and that the operator Ka veriﬁes the that we can prove Ba ϕ ↔ K principles of the logic S4.2. – Williamson rejects the positive introspection (4) in [17]. Here knowledge of agent a is deﬁned from the perception of agent a. Precisely, the relation Ra is the weak form of equality where we do not take in account what agent a does not perceive. That is why we have an equivalence relation. We also deﬁne for all group J ⊆ AGT the relation RJW as following: – wRJW u iﬀ there exists a ﬁnite sequence w1 , . . . , wn ∈ W and a ﬁnite sequence a1 , . . . an−1 ∈ J of agents such that w1 Ra1 w2 Ra2 . . . wn , w = w1 and u = wn . W In other words, RJ is the reﬂexive and transitive closure of the restriction of j∈J Rj to the worlds that are in W .

Definition 3 (truth conditions). Let W ⊆ W all be a set of worlds. We deﬁne the truth conditions as follows: – – – –

W, w W, w W, w W, w

|= a b iﬀ b ∈ w(a); |= Ka ϕ iﬀ for all u ∈ W , wRa u implies W, u |= ϕ; |= CKJ ϕ iﬀ for all u ∈ W , wRJW u implies W, u |= ϕ; |= [ϕ!]ψ iﬀ w ∈ W [ϕ] implies W [ϕ], w |= ψ where W [ϕ] = {u ∈ W | W, u |= ϕ}.

The truth conditions of Ka ϕ and CKJ ϕ are standard: it is the interpretation of classical epistemic modal logic in the Kripke model made up of set of worlds W , the relations Ra deﬁned by 2 and the ‘valuations’ given directly by the

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particular deﬁnition of a world w (Deﬁnition 1). The truth condition of [ϕ!]ψ is also standard [13]. We note W |= ϕ when for all w ∈ W we have W, w |= ϕ. In that case, we say that ϕ is valid over W . We write w |= a b instead of W, w |= a b because it does not depend on W . Here are some validities over any W ⊆ W all : – – – – – – –

W W W W W W W

|= Ka ϕ → ϕ; |= Ka ϕ → Ka Ka ϕ; |= ¬Ka ϕ → Ka ¬Ka ϕ; |= a b → Ka a b; |= Ka a b ∨ Kaa b; |= a b ∧ b a → CK{a,b} a b; |= a b ∧ a c ∧ b c → Kab c.

Here are some validities over any W n for n ⊆ W all : – W n |= Ka b c → a b. c). – if b = c, W n |= a b ↔ (Ka b c ∨ Ka b Here are some validities over W 1 : – W 1 |= a b ∧ b c → Ka b c; c → Ka b c. – W 1 |= a b ∧ b Note that as W 1 ⊆ W 2 , all valid Boolean formulas over W 2 are valid over W . The following proposition shows the surprising fact that there is no relation in terms of validity between W 1 and W 2 if we allows epistemic modal operators in the formulas. 1

Proposition 2. We have: – Let ϕ1 = (¬a b ∧ a c) → (b a ↔ b c) W 1 |= ϕ1 but W 2 |= ϕ1 . – Let ϕ2 = ¬d a ∧ ¬d b ∧ ¬d c → ¬Kd (¬a b ∧ a c → (b a ↔ b c)) W 2 |= ϕ2 but W 1 |= ϕ2 . Proof. The reader should check that (¬a b ∧ a c → (b a ↔ b c)) is a valid over W 1 but not over W 2 .

3

Rewriting the Common Knowledge Operator

In this section, we are going to prove that in some cases formulas CKJ ϕ are equivalent to a formula without common knowledge operator and whose size is polynomial in ϕ and J. We distinguish two cases: when there exists a, b ∈ J such that a does not see b in the world and when all agents J see them each other.

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When There Exists a, b ∈ J Such That a Does Not See b

First we prove that the property P :=‘there exists a, b ∈ J such that a does not see b’ is an invariant under the relation RJW , that is to say, if P holds in world w and wRJW u then P holds in u. Proposition 3. Let W a set of worlds. Let J a ﬁnite group of agents. Let w ∈ W such that there exists a, b ∈ J such that w |= a b. Let u ∈ W be a world. We have that wRJW u implies there exists c, d ∈ J such that u |= c d. Now we are going to prove that if two worlds w and u satisﬁes the property P then we can reach u from w in a ﬁnite and ﬁxed epistemic transition Ra where a ∈ J. First we introduce a notion enabling this reachability: Definition 4 (imaginable isolation). Isolation is imaginable in W ⊆ W all if for all w ∈ W , for all a ∈ AGT, for all agents b ∈ w(a), there exists a world u ∈ W such that wRa u and u(b) = {b}. Proposition 4. Isolation is imaginable in W all and W n for all n ≥ 1. We give a characterization of accessibility by RJW when isolation is imaginable in W . Theorem 1. Let W be a set of worlds where isolation is imaginable. Let J a ﬁnite group of agents. Let w ∈ W such that there exists a, b ∈ J such that w |= a b. Let u ∈ W be a world. The following are equivalent:

w

u

d

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Ra

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a cI

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6b q b

N Fig. 4. Isolation is imaginable in W n : in w, agent a can imagine a world u in which agent b sees nobody

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1. wRJW u; 2. u is such that there exists c, d ∈ J such that u |= c d; 3. there exists a, b, c, d ∈ J such that wRa u1 Rb u2 Rd u3 Rc u where u1 , u2 , u3 ∈ W. The result of Theorem 1 is optimal for W = W n , for all n ≥ 2 and for W all in the sense that we can ﬁnd two worlds w, u ∈ W such that wRa Rb Rd Rc u and there is no shorter sequence a1 , a2 , a3 ∈ {a, b, c, d} such that wRa1 Ra2 Ra3 u. Candidate worlds w and u are depicted on Figure 5. Let us prove it. Suppose that the shortest sequence of size at most 3: wRa1 w2 Ra2 w3 Ra3 u. a1 ∈ {b, c, d} leads to a contradiction because then Ra1 (w) = {w} makes that we have wRa2 Ra3 u and we get a shorter sequence of size 2. Hence a1 = a. In the same way, a3 = c. Then a2 can not be a because w2 |= a b and w3 |= a b. In the same way, a2 can not be b, c or d. There is a contradiction. Corollary 1. Let W be a set of worlds where isolation is imaginable. For all ﬁnite groups J = {a1 , . . . aj }, for all agents a, b ∈ J we have: 4 – W |= a b → (CKJ ϕ ↔ (K a1 . . . Kaj ) ϕ; – W |= a b → (CKJ ϕ ↔ a,b,c,d∈J Ka Kb Kc Kd ϕ).

Proof. We simply remark that we can build all sequences of length 4 from (Ka1 . . . Kaj )4 .

w

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Fig. 5. How to reach a world u where c does not see d from a world w where a does not see b in W where isolation is imaginable

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The above Corollary covers all sets W n for all dimensions n and also the set of all worlds W all . It is an open question whether we can rewrite the common knowledge operator or not if isolation is not imaginable. The fact that isolation is not imaginable is clearly deﬁnable by the formula χ = a∈J b∈J|a=b a b we can announce the formula χ. Isolation is not imaginable among the agents in J in W [χ]. So rewriting the CKJ operator in the formula [χ!]CKJ ϕ seems to be diﬃcult. 3.2

When for All a, b ∈ J Agent a Sees Agent b

In this subsection, we adopt an similar methodology. The property ‘for all a, b ∈ J , agent a sees agent b’ is an invariant under RJ but it does not characterize RJ . The strong enough invariant Q is the item 2 of the following Theorem 2. Item 3 explains how to reach a world u satisfying the property Q from a world w that also satisﬁes the property Q. Here is the characterization concerning the set of all worlds W all : Theorem 2. Let J be a ﬁnite group of agents. Let w ∈ W all be such that J ⊆ w(J) (i.e. for all a, b ∈ J we have w |= a b). Let u ∈ W all be a world. The following are equivalent: all

1. wRJW u; 2. u is such that w(J) = u(J) and for all c, d ∈ w(J), w |= c d iﬀ u |= c d; 3. w(Ra1 . . . Raj )(Ra1 . . . Raj−1 )u where J = {a1 , . . . aj }. The Item 3 is optimal in the sense that we can ﬁnd two worlds w and u satisfying Item 2 such that Item 3 gives the shortest path from w to u. The Figure 6 shows such worlds w and u: J = {a, b} and in both w and u, both agent a and b see nobody. In 6, directions of agents c and d are not reported. We have wRa Rb Ra u. Let us prove that it is the shortest path by contradiction. Suppose that wRa Rb u. The previous Theorem implies the following rewriting of the common knowledge when all agents of J see themselves. Corollary 2. W all |= → (CKJ ϕ ↔ a,b∈J a b (Ka1 . . . Kaj )(Ka1 . . . Kaj−1 )ϕ) where J = {a1 , . . . aj }. The case of Lineland Concerning worlds that are representable in dimension 1, Proposition recalls us that if all agents of J see themselves, then card(J) ≤ 2. The case card(J) = 1 is trivial. Here is the characterisation for card(J) = 2: Theorem 3. Let a, b be two distinct agents. Let w ∈ W 1 such that w |= a b Ia

... a

The following are equivalent:

Ib

V

and w |= b a. w looks like

...

. . . . Let u ∈ W 1 be a world. b

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1

W 1. wR{a,b} u; Ia

Ib

V

2. u looks like

...

... a

. . . where Ia and Ib are permutations of

b

agents of respectively Ia and Ib ; 3. wRa Rb u. Concerning worlds that are representable in dimension 1, the optimal shortest path is no more of length 3 but 2 (Ra Rb ). Here we have the formal rewriting of the common knowledge in dimension 1 when a sees b and b sees a: Corollary 3. Let a, b be two agents. W 1 |= (ab∧ba) → (CKa,b ϕ ↔ Ka Kb ϕ). The case of W n where n ≥ 2 The rewriting of common knowledge with only classical epistemic operators is an open question for W n where n ≥ 2. The problem in the proof of Theorem 2 is that we are not sure that worlds ui , ui and wF belongs to W n . 3.3

Conclusion

As a conclusion, we can rewrite the common knowledge construction CKJ ϕ exclusively with standard epistemic operators for all worlds W all :

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Corollary 4. Let J = {a1 , . . . aj } be a ﬁnite set of agents. – W all |= CKJ ϕ ↔ (Ka1 . . . Kaj )4 ϕ. – W all |= CKJ ϕ ↔ (Ka1 . . . Kaj )(Ka1 . . . Kaj−1 )ϕ ∧ a,b,c,d∈J Ka Kb Kc Kd ϕ). Proof. By Corollary 1 and Corollary 2. The rewriting of the common knowledge construction CKJ ϕ is easier for worlds that are representable in dimension 1 because Ka Kb ϕ of Corollary 3 is subsumed by a,b,c,d∈J Ka Kb Kc Kd ϕ). Corollary 5. Let J = {a1 , . . . aj } be a ﬁnite set of agents. 4 – W 1 |= (CKJ ϕ ↔ (K a1 . . . Kaj ) ϕ. 1 – W |= (CKJ ϕ ↔ a,b,c,d∈J Ka Kb Kc Kd ϕ).

Proof. By Corollary 1 and Corollary 3. As you can see, both transformations are polynomial. 3.4

Rewriting Common Knowledge Operator as Geometrical Constraints

In this subsection, we will explore the link between common knowledge and positions of agents. We see that the construction CKJ c d is rewritten without epistemic modal operator and only with a b constructions. The following proposition says that for W all or W n for n ≥ 2 the common knowledge among J that agent c sees agent d means that all agents of J sees all agents of J plus c and d and that agent c actually sees agent d (Figure 7).

R >

J

c

)

d

}

Fig. 7. Spatial configuration of agents when there is common knowledge among J that agent c sees d

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Proposition 5. Let W be either W all or W n for n ≥ 2. ⎛ ⎞

ab∧ (a c ∧ a d) ∧ c d⎠ . W |= CKJ c d ↔ ⎝ a,b∈J

a∈J

Note that Proposition 5 and Corollary 2 gives W all |= CKJ c d ↔ (Ka1 . . . Kaj )(Ka1 . . . Kaj−1 )c d. For W 1 , Proposition 3.2 suggests that common knowledge when there is more than three agents is particular. Furthermore, for W 1 , when an agent a sees another c, agent a automatically knows whether a sees d or not for all agents d. We deduce the adapted version of Proposition 5 for Lineland: Proposition 6. 1. If card(J) ≥ 3, then W 1 |= CKJ c d ↔ ⊥; 2. W 1 |= CK{a,b} c d ↔ (a b ∧ b a ∧ a c ∧ b c ∧ c d). Note that Proposition 6 and Corollary 3 gives W 1 |= CK{a,b} cd ↔ Ka Kb c d.

4

Model-Checking Problem

Model-checking for classical epistemic modal logic usually takes a pointed Kripkemodel M, w and a formula ϕ as an input and says whether M, w |= ϕ or not. It is in P [9]. Here the model-checking does not take a whole Kripke model as input but only a ﬁnite description of a world and thus we will see that the modelchecking is here PSPACE-hard. Let W be a subset of worlds. The model-checking in W is the following problem: – input: a formula ϕ and a ﬁnite description of a world w ∈ W (we give a ﬁnite set of pairs (a, b) where a, b ∈ AGT) and a formula ϕ; – output: yes iﬀ W, w |= ϕ. Lemma 1. Let n be an integer. Checking that a ﬁnite description of a world w is such that w ∈ W n can be done with an algorithm running in polynomial space in the size of w. Proposition 7. Even if the formula ϕ does not contain any CKJ -operator or any public announcement operator, the model-checking problems for W = W all and W = W n for all n is PSPACE-hard. Theorem 4. If the formula ϕ does not contain any CKJ -operators, the modelchecking problems for W all and W n for all n is PSPACE-complete. Theorem 5. If there is no public announcements in the formula ϕ, the modelchecking for W all and W 1 is PSPACE-complete.

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Proof. We can apply Corollary 4 or Corollary 5 in order to replace any subformulas CKJ ϕ by an equivalent formula without common knowledge operator. This transformation is polynomial. We then apply the algorithm given by Theorem 4. Now we give a non-surprising result if we authorize both common knowledge and public announcements in the language. Theorem 6. The model-checking for W = W all and W = W n for all n ≥ 2 is in EXPTIME. Note that the satisﬁability problem and the model-checking are ‘inter-reducible’. The satisﬁability problem of a formula ϕ in W is reducible to the model-checking in W : we guess a world w and check if W, w |= ϕ. The model-checking in W of a formula ϕ in a ﬁnite description of a world w is reducible to the satisﬁability problem in W : we check if the formula ϕ ∧ (a,b)∈agt(ϕ)|(a,b)∈w a b ∧ b is satisﬁable. (a,b)∈agt(ϕ)|(a,b)∈w a

5

Conclusion

This paper contributes in providing a semantics for reasoning about what agents see and know about what agents see. We believe that in practical applications, Kripke structures comes from the real situations, so that, in some cases, the reasoning about common knowledge is reducible to reasoning without common knowledge. In those cases, both the model-checking and the satisﬁability problem are PSPACE-complete. This is a good news for practical applications like robotics or video games where epistemic reasoning is related to perception. Concerning the axiomatisation, validities without public announcements over W 1 can be found in [12]. Axiomatization of W all is not really diﬃcult and corresponds to the axiomatization of W 1 in [12], [15] without the Boolean axioms for describing the geometry of the line. Axiomatization of validities for W n for n ≥ 2 is an open problem. There are also other open problems left that are not treated in this article: – In Subsection 2.1, we have given examples of worlds (w1 , w2 and w3 ) in order to prove that W 2 ⊆ W 1 and W 3 ⊆ W 2 . It is an open question whether W n+1 ⊆ W n for n ≥ 3 or not. – Is it possible to remove the common knowledge operator when we are interested about W n for n ≥ 2 or public announcements? To answer this question, we need to adapt the result of Theorem 2. – What is the complexity of the following problem: • Input: a formula ϕ and a set of agents a1 , . . . , an ; • Output: Yes if we can rewrite the common knowledge operator when we restrict the admissible world to W all [ϕ] and we deal with formulas containing agents a1 , . . . , an ? – Is the model-checking for W all or W n for n ≥ 2 EXPTIME-hard? For a real application in robotics or video games, we need to combine this work with others like time and knowledge [4].

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Acknowledgments. I thank Philippe Balbiani, Hans Van Ditmarch, Olivier Gasquet, Andreas Herzig and Emiliano Lorini for their useful comments.

References 1. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. MIT Press, Cambridge (1995), http://library.books24x7.com.libproxy.mit.edu/ toc.asp?site=bbbga&bookid=7008 2. Gasquet, O., Schwarzentruber, F.: Knowledge in Lineland (Extended Abstract) (short paper). In: International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS), Toronto, Canada (2010), doi:10/05/2010-14/05/2010 3. Gasquet, O., Schwarzentruber, F.: Concrete epistemic modal logic: Flatland. In: Blackburn, P., van Ditmarsch, H., Manzano, M., Soler-Toscano, F. (eds.) Tools for Teaching Logic. LNCS, vol. 6680, pp. 70–76. Springer, Heidelberg (2011) 4. Halpern, J., Vardi, M.: The complexity of reasoning about knowledge and time: synchronous systems. In: Proc. 20th ACM Symp. on Theory of Computing, pp. 53–65 (1988) 5. Heinemann, B.: Regarding overlaps in ’topologic’. In: Advances in Modal Logic, pp. 259–277 (2006) 6. Hintikka, J.: Knowledge and belief: an introduction to the logic of the two notions. Cornell University Press, Ithica (1962) 7. van der Hoek, W., Troquard, N., Wooldridge, M.: Knowledge and Control. In: roceedings of the Tenth International Conference on Autonomous Agents and Multiagent Systems, AAMAS 2011 (2011) 8. Jago, M.: Epistemic logic for rule-based agents. Journal of Logic, Language and Information 18(1), 131–158 (2009) 9. Joseph, Y., Halpern, Y.M.: A guide to completeness and complexity for modal logics of knowledge and belief. Artificial Intelligence (1992) 10. Moss, L., Parikh, R.: Topological reasoning and the logic of knowledge. In: Theoretical Aspects of Reasoning about Knowledge (TARK 1992), pp. 95–105 (1992) 11. Parikh, R., Moss, L.S., Steinsvold, C.: Topology and epistemic logic. In: Handbook of Spatial Logics, pp. 299–341. Springer, Heidelberg (2007) 12. Balbiani, P., Olivier Gasquet, F.S.: Epistemic reasoning about what agents see (submitted). Logic Journal of the IGPL (Special issue, FAMAS 2009) (2011) 13. Plaza, J.: Logics of public communications. Synthese 158(2), 165–179 (2007) 14. Schwarzentruber, F.: Seing, knowledge and common knowledge. Tech. rep., IRIT, Toulouse, France (2011) 15. Schwarzentruber, F.: Seeing, knowing, doing: case studies in modal logic. Th`ese de doctorat, Universit´e Paul Sabatier, Toulouse, France (D´ecembre (2010) 16. Stalnaker, R.: On logics of knowledge and belief. Philosophical Studies 128(1), 169–199 (2006) 17. Williamson, T.: Knowledge and its Limits. Oxford University Press, USA (2002) 18. Wooldridge, M., Lomuscio, A.: A logic of visibility, perception, and knowledge: completeness and correspondence results. Journal of the IGPL 9(2) (2001)

Measurement-Theoretic Foundations of Probabilistic Model of JND-Based Vague Predicate Logic Satoru Suzuki Faculty of Arts and Sciences, Komazawa University 1-23-1, Komazawa, Setagaya-ku, Tokyo, 154-8525 Japan [email protected]

Abstract. Vagueness is a ubiquitous feature that we know from many expressions in natural languages. It can invite a serious problem: the Sorites Paradox. The aim of this paper is to propose a new version of complete logic for vague predicates - JND-based vague predicate logic (JVL) which can avoid the Sorites Paradox and give answers to all of the Semantic Question, the Epistemological Question and the Psychological Question given by Graﬀ. To accomplish this aim, we provide JVL with a probabilistic model by means of measurement theory. Keywords: pair comparison probability function, Strong Probabilistic Transitivity, JND, vagueness, Sorites Paradox, intransitivity of similarity, semiorder, measurement theory, representation theorem, bounded rationality.

1

Introduction

Vagueness is a ubiquitous feature that we know from many expressions in natural languages. It can invite a serious problem: the Sorites Paradox. The following argument is an ancient example of this paradox: Example 1 (Sorites Paradox). 1000000 grains of sand make a heap1 . If 1000000 grains of sand make a heap, then 999999 grains of sand do. If 999999 grains of sand make a heap, then 999998 grains do. .. . If 2 grains of sand make a heap, then 1 grain does. 1 grain of sand makes a heap.

You can replace the set of conditional premises with a universally generalised premise. Hyde ([6]) classiﬁed responses to the Sorites Paradox in the following four types: 1

The Sorites Paradox derives its name from the Greek word “σωρ´ oς” for heap.

H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 272–285, 2011. c Springer-Verlag Berlin Heidelberg 2011

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1. 2. 3. 4.

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denying that logic applies to soritical expressions, denying some premises, denying the validity of the argument, accepting the paradox as sound.

In this paper we try to pursue response (2) in which the universally generalised premise is denied. Graﬀ ([[4]: 50]) gives the following three questions to which the theories of vagueness should answer when the universally generalised premise is not true: Problem 1 (Semantic Question). If the universally generalised premise: ∀x∀y(P (x) ∧ R(x, y) → P (y)), where R is a relation that makes this premise seemingly plausible (for example, “is one-grain-of-sand-less than”), is not true, then must its negation, the Sharp Boundaries Claim: ∃x∃y(P (x) ∧ R(x, y) ∧ ¬P (y)), be true? If the Sharp Boundaries Claim is true, how is its truth compatible with the fact that vague predicates have borderline cases? Problem 2 (Epistemological Question). If it is not true, why are we unable to say which one (or more) of its instances is not true? Problem 3 (Psychological Question). If it is not true, why were we so inclined to accept it in the ﬁrst place? Graﬀ ([[4]: 50–54]) goes on to provide a brief description of the most popular theories of vagueness: supervaluationism, epistemicism, and degrees-of-truth theories and context-dependent theories. She considers that the ﬁrst two are incomplete because they cannot give an adequate answer to the Psychological Question, while there is no reason to accept degrees-of-truth theories because they provide no substantial account of what degrees of truth are. She is sympathetic to context-dependent theories which appeal to the context dependence of vague predicates in order to answer the Psychological Question. The aim of this paper is to propose a new version of complete logic for vague predicates - JND-based vague predicate logic (JVL) which can avoid the Sorites Paradox and give answers to all of the Semantic Question, the Epistemological Question and the Psychological Question. In [19] we proposed a logic called SVPL that can avoid the Sorites Paradox and give answers to all of these questions. The model of SVPL is based directly on (1) a measure function f that an experimenter could assign to a subject and (2) a just noticeable diﬀerence (JND) δ. But SVPL has such a defect that we cannot demonstrate how to construct f even though we can know the existence of it. In this paper we eliminate this defect in such a way that in terms of measurement theory, we provide JVL with a model that can give a condition for f and a JND concerning the Strong Probabilistic Transitivity. It must be noted that in this paper we do not aim to show new results in measurement theory, but try to provide inferences about vague predicates with their measurement-theoretic foundations. The structure of this paper is as follows. We deﬁne the language LJVL of JVL, give descriptions of JND, semiorder, pair comparison probability function, Strong Probabilistic Transitivity and weak order, deﬁne a probabilistic model

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M of JVL, provide JVL with a satisfaction deﬁnition and a truth deﬁnition, give an answer both to the Semantic Question and to the Epistemological Question, give an answer to the Psychological Question, provide JVL with a proof system, and touch upon the soundness and completeness of JVL.

2 2.1

JND-Based Vague Predicate Logic JVL Language

We deﬁne the language LJVL of JVL as follows: Definition 1 (Language). – Let V denote a set of individual variables, C a set of individual constants, P a set of one-place vague predicate symbols, DP a dissimilarity relation symbol relative to P ∈ P and S+ P a strong similarity relation symbol relative to P . – The language LJVL of JVL is given by the following rule: t ::= x | a, ϕ ::= P (t) | ti = tj | DP (ti , tj ) | S+ P (ti , tj ) | | ¬ϕ | ϕ ∧ ψ | ∀xϕ, where x ∈ V, a ∈ C, P ∈ P. ⊥, ∨, →, ↔ and ∃ are introduced by the standard deﬁnitions. – DP (ti , tj ) means that ti is dissimilar in P -ness from tj . – S+ P (ti , tj ) means that ti is strongly similar in P -ness to tj . – We deﬁne a weak similarity relation symbol SP relative to P as follows: SP (ti , tj ) := ¬DP (ti , tj ). – The set of all well-formed formulae of LJVL is denoted by ΦLJVL . 2.2

Semantics

JND One possible explanation for the Sorites Paradox is that the intransitivity of similarity results from the fact that we cannot generally discriminate between very close quantities. The psychophysicist Fechner ([3]) explained this inability by the concept of a threshold of discrimination, that is, just noticeable diﬀerence (JND). Given a measure function f that an experimenter could assign to a subject and an object a, its JND δ is the lowest intensity increment such that f (a) + δ is recognised to be higher than f (a) by the subject. Then can we explain all examples of the intransitivity of similarity in terms of this inability? We cannot explain all of them only in terms of it, as the following case illustrates [[1]: 135]:

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Example 2 (Inability in Discrimination and Indiﬀerence as to Discrimination). It is entirely plausible to suppose that an instructor would be indiﬀerent to having the number of students in his seminar be 6 vs. 7, 7 vs. 8, etc., without being indiﬀerent to having it be 6 vs. 15; he might consider 15 students too many for a seminar. But he can certainly discriminate between 6 and 7 students or 7 and 8, etc. We can explain this kind of intransitivity of similarity in terms of the fact that we are indiﬀerent to the results of very ﬁne discriminations. As well as the JNDs that result from the inability in discrimination, we can consider the JNDs that result from the indiﬀerence as to discrimination. In this paper we use the term “JND” in the broad sense that covers both the former and the latter. Semiorder Luce ([8]) introduced the concept of a semiorder 2,3 that can provide a qualitative counterpart of a JND that is quantitative. Scott and Suppes ([[13]: 117]) deﬁned a semiorder as follows: Definition 2 (Semiorder). Let I denote a set of individuals. on I is called a semiorder if, for any w, x, y, z ∈ I, the following conditions are satisﬁed: 1. x x. (Irreﬂexivity), 2. If w x and y z, then w z or y x. 3. If w x and x y, then w z or z y.

(Strong Intervality), (Semitransitivity).

There are two main problems with measurement theory 4 : 1. the representation problem—justifying the assignment of numbers to objects, 2. the uniqueness problem—specifying the transformation up to which this assignment is unique. A solution to the former can be furnished by a representation theorem, which establishes that the speciﬁed conditions on a qualitative relational system are (necessary and) suﬃcient for the assignment of numbers to objects that represents (or preserves) all the relations in the system. Scott and Suppes ([13]) proved a representation theorem for semiorders when I is ﬁnite. The Scott-Suppes theorem was ﬁrst extended by to countable sets by Manders ([10]). Because I of the model M of JVL may be countable, the Manders theorem must be considered. A condition (∼∗ -Connectedness) is necessary for to have a positive threshold even when I is countable. We deﬁne ∼ by as follows: 2

3 4

Van Rooij ([22] and [23]) also argued the relation between the Sorites Paradox and semiorders from a diﬀerent point of view that does not focus on a representation theorem. In [18] and [20] we proposed a new version of complete and decidable preference logic based on a semiorder on a Boolean algebra. [12] gives a comprehensive survey of measurement theory. The mathematical foundation of measurement had not been studied before H¨ older developed his axiomatisation for the measurement of mass ([5]). [7], [15] and [9] are seen as milestones in the history of measurement theory.

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Definition 3 (∼). For any x, y ∈ I, x ∼ y := x y and y x. ∼∗ is deﬁned by ∼ and as follows: Definition 4 (∼∗ ). For any x, y ∈ I, x ∼∗ y := x ∼ y, or, x y and for any z ∈ I, not (x z and z y), or, y x and for any z ∈ I, not (y z and z x). A ∼∗ -chain is deﬁned by ∼∗ as follows: Definition 5 (∼∗ -Chain). Let a0 , . . . , an ∈ I be such that for any k < n, ak ∼∗ ak+1 . Then we call (a0 , . . . , an ) a ∼∗ -chain between a0 and an . ∼∗ -Connectedness is deﬁned by a ∼∗ -chain as follows: Definition 6 (∼∗ -Connectedness). ∼∗ on I is connected if for any x0 , xn ∈ I, there is a ∼∗ -chain between x0 and xn . The Manders theorem can be stated by means of ∼∗ -Connectedness as follows: Theorem 1 (Representation for Semiorders, Manders ([10])). Suppose that is a binary relation on a countable set I and that ∼∗ is deﬁned by Definition 4 and that δ is a positive number. Then is a semiorder and ∼∗ is connected iﬀ there is a function f : I → IR such that for any x, y ∈ I, x y iﬀ f (x) > f (y)+δ.

Pair Comparison Probability Function Although we can know the existence of f , we cannot demonstrate how to construct it. In stead of a model based directly on f and a JND, we provide JVL with a model that can give a condition for f and a JND concerning the Strong Probabilistic Transitivity. Here we deﬁne a pair comparison probability function as follows: Definition 7 (Pair Comparison Probability Function P r). P r : I × I → [0, 1] is called a pair comparison probability function if it satisﬁes the following condition: For any x, y ∈ I such that x = y, P r(x, y) + P r(y, x) = 1. P r(a, b) is interpreted as the relative frequency with which a subject will choose a rather than b when required to make a choice from {a, b}. For example, a subject may at one time say that sound a is louder than sound b and b is not louder than a, and soon thereafter say that b is louder than a and a is not louder than b. If he were totally inconsistent in his judgements, then there would be no hope of developing theories of his behaviour. But in many cases, there is a pattern of consistencies. Although he may not be absolutely consistent, he may be probabilistically consistent. The following is one of the most typical conditions for probabilistic consistency.

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Definition 8 (Strong Probabilistic Transitivity (SPT)). P r is said to satisfy the Strong Probabilistic Transitivity if for any x, y, z ∈ I, If P r(x, y) ≥

1 1 and P r(y, z) ≥ , then P r(x, z) ≥ max {P r(x, y), P r(y, z)}. 2 2

Then we deﬁne a weak order as follows: Definition 9 (Weak Order). on I is called a weak order if, for any w, x, y, z ∈ I, the following conditions are satisﬁed: 1. If x y or y x. (Connectedness), 2. If x y and y z, then x z. (Transitivity).

Cantor ([2]) proved the representation theorem for weak orders. Theorem 2 (Representation for Weak Orders, Cantor ([2])). Suppose I is a countable set and is a binary relation on I. Then is a weak order iﬀ there is a function f : I → IR such that for any x, y ∈ I, x y iﬀ f (x) ≥ f (y).

We deﬁne ≡ on I by as follows: Definition 10 (≡). For any x, y ∈ I, x ≡ y := x y and y x.

The next corollary follows directly from Deﬁnition 10. Corollary 1 (Equivalence Relation). ≡ on I satisﬁes the following conditions: for any x, y, z ∈ I, 1. x ≡ x. (Reﬂexivity), 2. If x ≡ y, then y ≡ x. (Symmetricity), 3. If x ≡ y and y ≡ z, then x ≡ z. (Transitivity). So ≡ is an equivalence relation.

The next corollary follows directly from Theorem 2. Corollary 2 (Representation for Equivalence Relation). Suppose I is a countable set and ≡ is a binary relation on I. Then ≡ is an equivalence relation iﬀ there is a function f : I → IR such that for any x, y ∈ I, x ≡ y iﬀ f (x) = f (y).

Next we deﬁne compatibility, homogeneousness, weak discriminatedness, a binary relation P rλ on I and a binary relation P r− on I as follows: Definition 11 (Compatibility). A semiorder and a weak order are said to be compatible if the following conditions hold: for any z, y, z ∈ I, If x y, then x y, and If x y z and x ∼ z, then x ∼ y and y ∼ z.

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Definition 12 (Homogeneousness). The family of semiorders is called homogeneous if the same weak order is compatible with each member of the family. Definition 13 (Weak Discriminatedness). P r is called weakly discriminated if for any x, y ∈ I if P r(x, y) = 12 , then for any z ∈ I, P r(x, z) = P r(y, z). Definition 14 (Binary Relation P rλ on I). P rλ is a binary relation on I such that for any x, y ∈ I, xP rλ y if P r(x, y) > λ. Definition 15 (Binary Relation P r− on I). P r− is a binary relation on I such that for any x, y ∈ I, xP r− y if P r(x, y) ≥ P r(y, x). We can consider the notion of noticing a JND from a probabilistic point of view. We need to conduct many trials in order to determine the JND. The JND is usually the diﬀerence that a subject notices on 12 or more of trials. Here we consider the JND in terms of the family {P rλ : λ ∈ [ 12 , 1)}. Roberts ([11]) proved the following theorem concerning SPT, P rλ and P r− . Theorem 3 (SPT, Homogeneous Family of Semiorders and Compatible Weak Order, Roberts ([11])). Suppose that P r is a weakly discriminated pair comparison probability function. Then P r satisﬁes SPT iﬀ {P rλ : λ ∈ [ 12 , 1)} is a homogeneous family of semiorders and xP r− y is a weak order compatible with each member of the family. We have the following corollary of Theorem 1, Theorem 2 and Theorem 3. Corollary 3 (Representation for P rλ and P r− ). Suppose that P r is a weakly discriminated pair comparison probability function. Then P r satisﬁes SPT and the relation obtained by Deﬁnition 4 from each member of {P rλ : λ ∈ [ 12 , 1)} is connected iﬀ for any P rλ ∈ {P rλ : λ ∈ [ 12 , 1)}, there is a function f : I → R such that for any x, y ∈ I, xP rλ y iﬀ f (x) > f (y) + δ and xP r− y iﬀ f (x) ≥ f (y).

Model. We now turn to the Sorites Paradox. Suppose that an experimenter observes the relative frequency with which a subject responds that i (1 ≤ i ≤ 1000000) grains of sand look more like a heap than j (1 ≤ j ≤ 1000000) grains of sand. For example, when the relative frequency with which the subject responds that 505000 grains of sand look more like a heap than 500000 grains of sand is 34 and that with which he responds that 500000 grains of sand look more like a heap than 495000 grains of sand is 23 , it is plausible that the relative frequency with which he responds that 505000 grains of sand look more like a heap than 495000 grains of sand should be at least 34 . Then these relative

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frequencies should satisfy SPT. In this paper the Sharp Boundaries Claim (the negation of the universally generalised premise in the Sorites Paradox) is true. We explain this claim in terms of the fact that we cannot generally discriminate between very close quantities or the fact that we are indiﬀerent to the results of very ﬁne discriminations. Moreover, we express it in terms of a binary relation which is deﬁned by the relative frequency with which the subject responds that i grains of sand look more like a heap than j grains of sand. On the basis of SPT and the Sharp Boundaries Claim, we deﬁne a model M of JVL as follows: Definition 16 (Model). M is a sequence (I, aM , bM , . . . , F M , GM , . . . , P rF M , P rGM , . . .), where: I is a nonempty set of individuals, called the universe of M. aM , bM , . . . ∈ I. F M , GM , . . . ⊆ I. P rF M : I × I → [0, 1] is a weakly discriminated pair comparison probability function relative to F M and satisﬁes SPT. . . . . M – P rF M (tM 1 , t2 ) is interpreted as the relative frequency with which the subject responds that t1 is F -er than t2 . . . . . M – There is a P rFλ M ∈ {P rFλ M : λ ∈ [ 12 , 1)} such that there are some tM 1 , t2 ∈ M M M λ M M λ M M I, (t1 ∈ F and (not t1 P rF M t2 and not t2 P rF M t1 ) and t2 ∈ / F M) (Sharp Boundaries Claim). . . . . – The relation obtained by Deﬁnition 4 from P rFλ M is connected. . . . . – – – –

Truth We deﬁne an (extended) assignment function as follows: Definition 17 (Assignment Function). Let V denote a set of individual variables, C a set of individual constants and I a set of individuals. – We call s : V → I an assignment function. – We call s˜ : V ∪ C → I an extended assignment function.

We provide JVL with the following satisfaction deﬁnition relative to M: Definition 18 (Satisfaction). What it means for M to satisfy ϕ ∈ ΦLJVL with s, in symbols M |=JVL ϕ[s] is inductively deﬁned as follows: – M |=JVL P (t)[s] iﬀ s˜(t) ∈ P M . – M |=JVL t1 = t2 [s] iﬀ s˜(t1 ) = s˜(t2 ). – M |=JVL DP (t1 , t2 )[s] iﬀ for any P rPλ M ∈ {P rPλ M : λ ∈ [ 12 , 1)}, λ M M λ M tM 1 P rP M t2 or t2 P rP M t1 . + − − M M M – M |=JVL SP (t1 , t2 )[s] iﬀ tM 1 P rP M t2 and t2 P rP M t1 . – M |=JVL . – M |=JVL ¬ϕ[s] iﬀ M |=JVL ϕ[s]. – M |=JVL ϕ ∧ ψ[s] iﬀ M |=JVL ϕ[s] and M |=JVL ψ[s]. – M |=JVL ∀xϕ[s] iﬀ for any d ∈ I, M |=JVL ϕ[s(x|d)], where s(x|d) is the function that is exactly like s except for one thing: for

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the individual variable x, it assigns the individual d. This can be expressed as follows: s(y) if y = x s(x|d)(y) := d if y = x. The next corollary follows directly from Deﬁnition 1 and Deﬁnition 18. Corollary 4 (Satisfaction Condition of SP ). M |=JVL SP (t1 , t2 )[s] iﬀ λ M there is a P rPλ M ∈ {P rPλ M : λ ∈ [ 12 , 1)} such that not tM 1 P rP M t2 and not M λ M t2 P rP M t1 . The next corollary follows from Corollary 3, the satisfaction clause of DP and that of SP . Corollary 5 (Positive Threshold). – M |=JVL DP (t1 , t2 )[s] iﬀ for any P rPλ M ∈ {P rPλ M : λ ∈ [ 12 , 1)}, there is a function f : I → R such that λ M M λ M M M M M tM 1 P rP M t2 or t2 P rP M t1 iﬀ f (t1 ) > f (t2 ) + δ or f (t2 ) > f (t1 ) + δ.

– M |=JVL SP (t1 , t2 )[s] iﬀ there is a P rPλ M ∈ {P rPλ M : λ ∈ [ 12 , 1)} such that there is a function f : I → R, λ M M λ M M M M not tM 1 P rP M t2 and not t2 P rP M t1 iﬀ f (t1 ) − δ ≤ f (t2 ) ≤ f (t1 ) + δ.

The next corollary follows from Corollary 2, Corollary 3 and the satisfaction clause of S+ P. Corollary 6 (Zero Threshold). M |=JVL S+ P (t1 , t2 )[s] f : I → R such that

iﬀ there is a function

− − M M M M M tM 1 P rP M t2 and t2 P rP M t1 iﬀ f (t1 ) = f (t2 ).

We deﬁne the truth in M by means of satisfaction and then deﬁne validity as follows: Definition 19 (Truth and Validity). – If M |=JVL ϕ[s] for all s, we write M |=JVL ϕ and say that ϕ is true in M. – If ϕ is true in all models of JVL, we write |=JVL ϕ and say that ϕ is valid. Our Answer Both to Semantic Question and to Epistemological Question Now we can give an answer to the Semantic Question. Our answer is “Yes”. Because in JVL the bivalence holds, the fact that vague predicates have borderline cases does not imply the truth value gap or so. Both this fact and the truth of the Sharp Boundaries Claim result from the fact that we have only limited ability of discrimination. In this sense JVL has a position of epistemicism. According to epistemicism, vagueness is a type of ignorance. Because vague predicates have extensions with sharp boundaries, bivalence can be retained. We cannot know

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where those boundaries lie. The Stoics were supposed to hold this type of position. It has been recently revived by Williamson ([24] and [25]). Our position in this paper can be considered to be a kind of epistemicism for two reasons: First, the semantic values of vague predicates in the model M of Deﬁnition 16 have sharp boundaries. Second, JNDs reﬂect ignorance. The merit of epistemicism is expressed best by Williamson ([[24]: 162]) when he says: The epistemic view (epistemicism) involves no revision of classical logic and semantics; its rivals do involve such revisions. Classical logic and semantics are vastly superior to the alternatives in simplicity, power, past success, and integration with theories in other domains. In these circumstances it would be sensible to adopt the epistemic view in order to retain classical logic and semantics even if it were subject to philosophical criticisms in which we could locate no fallacy; not every anomaly falsiﬁes a theory. Our answer to the Semantic Question is also our answer to the Epistemological Question. We are unable to say which one (or more) of its instances is not true, because we have only limited ability of discrimination. Sorites Paradox Revisited We now return to the Sorites Paradox. Assume U U that U := (I, aU 1 , . . . , a1000000 , H , P rH U ) is given, where: – I := {a1 , . . . , a1000000 }, where ai denotes i grain(s) of sand, for any i(1 ≤ i ≤ 1000000). – H denotes making a heap. – P rH U is a weakly discriminated pair comparison probability function relative λ to H U and satisﬁes SPT and P rH U satisﬁes the Sharp Boundaries Claim. λ U P r a . – aU 1000000 HU 1 Then we have the following proposition: Proposition 1 (Sharp Boundaries Claim). U |=JVL ((H(a1000000 ) ∧ SH (a1000000 , a999999 )) → H(a999999 ))∧ · · · ∧ ((H(a2 ) ∧ SH (a2 , a1 )) → H(a1 )). From this proposition we obtain the following result: Corollary 7 (Avoidance of Sorites Paradox). |=JVL (H(a1000000 ) ∧ ((H(a1000000 ) ∧ SH (a1000000 , a999999 )) → H(a999999 ))∧ · · · ∧ ((H(a2 ) ∧ SH (a2 , a1 )) → H(a1 ))) → H(a1 ). On the other hand, we have the following proposition: Proposition 2 (Universally Generalised Premise). |=JVL ((H(a1000000 ) ∧ S+ H (a1000000 , a999999 )) → H(a999999 ))∧ (a , a1 )) → H(a1 )). · · · ∧ ((∧H(a2 ) ∧ S+ 2 H

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From this proposition we obtain the following result: Corollary 8 (Invitation of Sorites Paradox). |=JVL (H(a1000000 ) ∧ ((H(a1000000 ) ∧ S+ H (a1000000 , a999999 )) → H(a999999 )) ∧ + · · · ∧ ((H(a2 ) ∧ SH (a2 , a1 )) → H(a1 ))) → H(a1 ). Bounded Rationality and Our Answer to Psychological Question. The standard model of economics is based on global rationality that requires an optimising behaviour. But according to Simon ([14]), cognitive and informationprocessing constraints on the capabilities of agents, together with the complexity of their environment, render an optimising behaviour an unattainable ideal. He dismissed the idea that agents should exhibit global rationality and suggested that they in fact exhibit bounded rationality that allows a satisﬁcing behaviour. Because Corollary 5 implies that the subject has a positive threshold (only limited ability) of discrimination, he is considered to be only boundedly rational. On the other hand, because Corollary 6 implies that the subject has a zero threshold (perfect ability) of discrimination, he is considered to be globally rational. Now we can give an answer to the Psychological Question. Our Answer is that we tend to form the belief that the only boundedly rational subject is globally rational and attribute to him the Universally Generalised Premise (Proposition 2) instead of the Sharp Boundaries Claim (Proposition 1). 2.3

Syntax

Proof System. We extend a proof system of ﬁrst-order logic with an equality symbol in such a way as to add the syntactic counterparts of the Irreﬂexivity of and ≺, the Strong Intervality of and ≺, the Semitransitivity of and ≺, the Sharp Boundaries Claim, ∼∗ -Connectedness, the Reﬂexivity of ≡, the Symmetricity of ≡ and the Transitivity of ≡.5 We deﬁne S∗P that is the syntactic counterpart of ∼∗ as follows: Definition 21 (S∗P ). S∗P (t1 , t2 ) := SP (t1 , t2 ) ∨ (DP (t1 , t2 ) ∧ ∀x¬(DP (t1 , x) ∧ DP (x, t2 ))).

The proof system of JVL consists of the following: 5

Beja and Gilboa proposed another condition necessary for to have a positive threshold when I is countable as follows:

Definition 20 (Positivity). For any x ∈ I and any y1 , y2 , y3 , . . . ∈ I, if y1 y2 , y2 y3 , . . ., then for some n x yn , and if y2 y1 , y3 y2 , . . ., then for some n yn x. In LJVL the syntactic counterpart of Positivity is not expressible, for it contains infinite quantiﬁer sequence and conjunctions. It is possible to propose an incomplete infinitary logic that adopts as an axiom the syntactic counterpart of Positivity.

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Definition 22 (Proof System). 1. all tautologies of ﬁrst-order logic with an equality symbol, 2. ∀x¬DP (x, x) (Syntactic Counterpart of Irreﬂexivity of and ≺), 3. ∀w∀x∀y∀z(((DP (w, x) ∧ DP (y, z)) ∨ (DP (x, w) ∧ DP (z, y))) → ((DP (w, z) ∨ DP (y, x)) ∧ (DP (z, w) ∨ DP (x, y))) (Syntactic Counterpart of Strong Intervality of and ≺), 4. ∀w∀x∀y∀z(((DP (w, x) ∧ DP (x, y)) ∨ (DP (x, w) ∧ DP (y, x))) → (DP (w, z) ∨ DP (z, y)) ∧ (DP (z, w) ∨ DP (y, z))) (Syntactic Counterpart of Semitransitivity of and ≺), 5. ∃x∃y(P (x) ∧ SP (x, y) ∧ ¬P (y)) (Syntactic Counterpart of Sharp Boundaries Claim), 6. ∀x0 ∀xn ∃x1 . . . ∃xn−1 (S∗P (x0 , x1 ) ∧ . . . ∧ S∗P (xn−1 , xn )) (Syntactic Counterpart of ∼∗ -Connectedness), 7. ∀xS+ P (x, x) (Syntactic Counterpart of Reﬂexivity of ≡), + 8. ∀x∀y(S+ P (x, y) → SP (y, x)) (Syntactic Counterpart of Symmetricity of ≡), + + 9. ∀x∀y∀z((SP (x, y) ∧ S+ P (y, z)) → SP (x, z)) (Syntactic Counterpart of Transitivity ≡), 10. Modus Ponens. We deﬁne the provability in JVL as follows: Definition 23 (Provability). – A proof of ϕ ∈ ΦLJVL is a ﬁnite sequence of LJVL -formulae having ϕ as the last formula such that either each formula is an instance of an axiom or it can be obtained from formulae that appear earlier in the sequence by applying an inference rule. – If there is a proof of ϕ, we write JVL ϕ. Then in JVL we can derive the symmetricity of DP and that of SP : Proposition 3 (Symmetricity of DP ). JVL ∀x∀y(DP (x, y) → DP (y, x)).

Proposition 4 (Symmetricity of SP ). JVL ∀x∀y(SP (x, y) → SP (y, x)).

We can also derive the syntactic counterpart of the universally generalised premise: Proposition 5 (Syntactic Counterpart of Universally Generalised Premise). JVL ∀x∀y((S+ P (x, y) ∧ P (x)) → P (y)).

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Metalogic of JVL

We prove the metatheorems of JVL. It is easy to prove the soundness of JVL. Theorem 4 (Soundness). For any ϕ ∈ ΦLJVL , if JVL ϕ, then |=JVL ϕ.

We can also prove the completeness of JVL by Theorem 3, Lindenbaum Lemma and Truth Lemma. Theorem 5 (Completeness). For any ϕ ∈ ΦLJVL , if |=JVL ϕ, then JVL ϕ.

3

Conclusion

In this paper we have proposed a new version of complete logic for vague predicates - JND-based vague predicate logic (JVL) which can avoid the Sorites Paradox and give answers to all of the Semantic Question, the Epistemological Question and the Psychological Question given by Graﬀ. To accomplish this aim, we have provided JVL with a probabilistic model by means of measurement theory. This paper is only a part of our larger measurement-theoretic study. We are now trying to construct such logics as dynamic epistemic preference logic ([17]), dyadic deontic logic ([16]), threshold utility maximiser’s preference logic ([18] and [20]), a logic of interadjective comparison ([21]), and a logic of questions and answers by means of measurement theory. Acknowledgements. This work was supported by Grant-in-Aid for Scientiﬁc Research (C) (23520015). We would like to thank two anonymous reviewers of LORI-III for their helpful comments.

References 1. Ackerman, F.: Roots and consequences of vagueness. Philosophical Perspectives 8, 129–136 (1994) 2. Cantor, G.: Beitr¨ age zur Begr¨ undung der Transﬁniten Mengenlehre I. Mathematische Annalen 46, 481–512 (1895) 3. Fechner, G.T.: Elemente der Psychophysik. Breitkopf und Hartel, Leipzig (1860) 4. Graﬀ, D.: Shifting sands: An interest-relative theory of vagueness. Philosophical Topics 28, 45–81 (2000) 5. H¨ older, O.: Die Axiome der Quantit¨ at und die Lehre von Mass. Berichte u ¨ber die Verhandlungen der K¨ oniglich S¨ achsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physikaliche Classe 53, 1–64 (1901) 6. Hyde, D.: Sorites paradox. Stanford Encyclopedia of Philosophy (2005) 7. Krantz, D.H., et al. : Foundations of Measurement, vol. 1. Academic Press, New York (1971) 8. Luce, D.: Semiorders and a theory of utility discrimination. Econometrica 24, 178– 191 (1956) 9. Luce, R.D., et al.: Foundations of Measurement, vol. 3. Academic Press, San Diego (1990)

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10. Manders, K.L.: On jnd representations of semiorders. Journal of Mathematical Psychology 24, 224–248 (1981) 11. Roberts, F.S.: Homogeneous families of semiorders and the theory of probabilistic consistency. Journal of Mathematical Psychology 8, 248–263 (1971) 12. Roberts, F.S.: Measurement Theory. Addison-Wesley, Reading (1979) 13. Scott, D., Suppes, P.: Foundational aspects of theories of measurement. Journal of Symbolic Logic 3, 113–128 (1958) 14. Simon, H.A.: Models of Bounded Rationality. The MIT Press, Cambridge (1982) 15. Suppes, P., et al.: Foundations of Measurement, vol. 2. Academic Press, San Diego (1989) 16. Suzuki, S.: Measurement-theoretic foundation of preference-based dyadic deontic logic. In: He, X., Horty, J., Pacuit, E. (eds.) LORI 2009. LNCS(LNAI), vol. 5834, pp. 278–291. Springer, Heidelberg (2009) 17. Suzuki, S.: Prolegomena to dynamic epistemic preference logic. In: Hattori, H., Kawamura, T., Id´e, T., Yokoo, M., Murakami, Y. (eds.) JSAI 2008. LNCS(LNAI), vol. 5447, pp. 177–192. Springer, Heidelberg (2009) 18. Suzuki, S.: Prolegomena to threshold utility maximiser’s preference logic. In: Electronic Proceedings of the 9th Conference on Logic and the Foundations of Game and Decision Theory (LOFT 2010), Paper No. 44 (2010) 19. Suzuki, S.: Prolegomena to salient-similarity-based vague predicate logic. In: Onoda, T., et al. (eds.) JSAI-isAI 2010. LNCS(LNAI), vol. 6797, pp. 75–89. Springer, Heidelberg (2011) 20. Suzuki, S.: Measurement-theoretic foundation of threshold utility maximiser’s preference logic. Journal of Applied Ethics and Philosophy 3 (accepted) 21. Suzuki, S.: Measurement-theoretic foundations of interadjective-comparison logic. In: Proceedings of Sinn und Bedeutung, vol. 16 (accepted) 22. Van Rooij, R.: Revealed preference and satisﬁcing behavior. Synthese 179, 1–12 (2011) 23. Van Rooij, R.: Vagueness and linguistics. In: Ronzitti, G. (ed.) Vagueness: A Guide, pp. 123–170. Springer, Heidelberg (2011) 24. Williamson, T.: Vagueness and ignorance. Proceedings of the Aristotelian Society, Supplementary Volumes 66, 145–162 (1992) 25. Williamson, T.: Vagueness. Routledge, London (1994)

An Epistemic Logic with Hypotheses Levan Uridia1 and Dirk Walther2 1 2

Rey Juan Carlos University (URJC), Madrid, Spain [email protected] Technical University of Madrid (UPM), Madrid, Spain [email protected]

Abstract. We introduce a variant of the standard epistemic logic S5 for reasoning about knowledge under hypotheses or background assumptions. The modal operator of necessity expressing what is known is parameterised with a hypothesis. The operator can be described as relative necessity, a notion already used by Chellas to describe conditionality. In fact, the parameterised box resembles a conditional operator and it turns out that our logic is a variant of Chellas’ Conditional Logic. We present an axiomatisation of the logic and show that it bears the same expressivity and computational complexity as S5. Then we consider the extension of our logic with operators for distributed knowledge and show how it can be used to represent knowledge of agents whose epistemic capacity corresponds to any system between S4 and S5. Keywords: epistemic logic, conditional logic, knowledge, distributed knowledge, hypotheses.

1

Introduction

We introduce an epistemic logic for reasoning about knowledge under hypotheses. The resulting logic S5r is an extension of S5 with a modal operator ‘[·]’ that can be parameterised with a hypothesis. The modality [ϕ] represents the knowledge state under the hypothesis ϕ. The formula [ϕ]ψ states that ‘under the hypothesis ϕ, the agent knows ψ’. If ϕ happens to be true at the current world and the agent knows that ϕ implies ψ, then the agent knows ψ; otherwise, i.e., if ϕ is false, the agent knows only what it would know anyway, i.e. without any assumptions. For instance, suppose an agent is interested in whether the street is dry or wet. The agent knows that rain makes the street wet, but it does not know whether or not it is raining outside. The formula [ it-is-raining ] street-is-wet states that the agent knows that the street is wet when adopting the hypothesis that it is raining. We consider two situations: one, where the hypothesis is correct, i.e., it is indeed raining; and another one, where it is false, i.e., it is not raining. Clearly, in the former situation, the street is wet due to the rain and we have that the formula holds true. In case the hypothesis is in fact wrong, the formula H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 286–299, 2011. c Springer-Verlag Berlin Heidelberg 2011

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is not necessarily true. Notice here the diﬀerence to an ordinary implication, which is true whenever the premise is false. Generally, the agent may consider a dry street possible despite adopting the hypothesis that it is raining. But this means that the agent does not know that the street is wet and, thus, the formula is false. The formula is true, however, if the agent does not consider a dry street possible. In this case, the agent already knew that the street is wet before, i.e. without assuming that it is raining. The latter is expressed by the formula: [ ] street-is-wet. The parameterised modal operator can be described as relative necessity, a notion already used by Chellas to describe conditionality [5]. In fact, there is a strong relation with Chellas’ Condition Logic as S5r turns out to be a special case of it. We present an axiomatisation of S5r and we show it is as expressive and complex as S5. In the second part of the paper, we extend S5r with operators for distributed knowledge but use them for combining hypotheses. Distributed knowledge is a standard notion in epistemic logic [9,16]. Generally, distributed knowledge of a group of agents equals what a single agent knows who knows everything what each member of the group knows. Suppose agent a knows p and agent b knows p → q. Then the distributed knowledge between a and b is q, even though neither of them might know q individually. The notion of distributed knowledge is relevant for describing and reasoning about the combined knowledge of agents in a distributed system; see, e.g., [10]. There agents communicate with each other to combine their knowledge. Thus the notion of distributed knowledge is also central to communication protocols and relevant to reasoning about speech acts [11,20]. We may think of distributed hypotheses as the result of combining incoming information from several sources. However, the truthfulness of the incoming information is not assumed. We demonstrate another way to think about distributed hypotheses by using our logic to represent the knowledge of an agent whose knowledge capacity can be characterised by any modal system between S4 and S5. The relative necessity operator of S5r shows up in several places, e.g., in Conditional Logic, Public Announcement Logic and Provability Logic. We now discuss the relationships to these logics in more detail and explain how far the relative necessity is paraconsistent. Sentences in English of the form “If A, then B.” are called conditional sentences. Here, A is called the antecedent (or condition) and B the consequent. Conditional sentences are traditionally put into diﬀerent categories (according to mood or tense) such as indicative/subjunctive or factual/counterfactual. However, there is much disagreement on the logical theory of conditional sentences (in particular that of defeasible conditionals). One logical formalisation is Conditional Logic, which essentially is Propositional Logic extended with a binary operator ‘⇒’ standing for conditionality. Several readings of ‘⇒’ were proposed, among them counterfactual conditional, non-monotonic consequence relation, normality and belief revision. Historically several logical accounts of condition-

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als have been suggested, among them Stalnaker [22], Lewis [14] and Chellas [5]; for an overview on Conditional Logic, see, e.g., [17]. Our logic S5r rejects the common assumption that logics allow one to conclude anything from false premises. To be more precise we borrow the term ‘explosive’ from Paraconsistent Logic, but we refer to conditional operators instead of the logical consequence relation. We say that a conditional operator ‘X’ is explosive if the conditional ϕXψ holds for all conclusions ψ whenever the antecedent ϕ is false. In this sense, implication of Classical Logic and even of Intuitionistic Logic is explosive, so is the conditional operator ‘⇒’ of Conditional Logic [14,22]. On the other hand, the relativised necessity of our logic, which is a special case of Chellas’ conditional operator [5,6], is not explosive. We have that [⊥]ψ is true if, and only if, ψ is universally true. Notice that this does not mean that the consequence relation of our logic paraconsistent; it is not. Epistemic logic traditionally describes the knowledge state of agents at a point in time. To be able to describe the evolution of knowledge over time, we can either combine epistemic logic with a temporal logic, or add dynamic operators for knowledge-changing actions such as communication. The latter approach is followed in the family of Dynamic Epistemic Logics (DELs) [7]. A basic DEL is Public Announcement Logic (PAL) [18] which is the extension of the basic epistemic logic S5 with an operator ‘[·]’ parameterised with a formula expressing the announcement. A formula of the form [ϕ]ψ states that ψ holds after ϕ has been truthfully announced (by someone) to every agent in the system simultaneously. After the announcement, ϕ is incorporated in the knowledge state of every agent, i.e. ϕ becomes common knowledge. This is achieved by employing an update semantics, which cuts oﬀ the model all worlds in which the announcement does not hold. Here we have an important diﬀerence to our logic S5r whose semantics does not change models during the evaluation of a formula. It is well-known that PAL is as expressive as S5 [18,7], but the translation of a PAL-formula to an equivalent formula of S5 causes an exponential blow-up in size. Despite the succinctness of PAL, however, the complexity of the satisﬁability checking problem coincides with that for S5, i.e. NP-complete for the single-agent PAL and PSpace-complete in the presence of multiple-agents [15]. Our relative necessity operator ‘[·]’ bears an interesting relationship to Provability Logic [3]. This is a modal logic, where the modality is considered to capture the metamathematical concept of ‘a proposition being provable in some arithmetical theory’. An important logic in this context is the G¨odel–L¨ob system GL as it characterises provability in Peano Arithmetic. A recent line of research is to modify GL and study the eﬀects of these modiﬁcations wrt. provability [8]. For instance, [8] introduces three variants of the modality in GL and studies their algebraic semantics. Here is where the connection to this paper turns up as one of the modiﬁed modalities corresponds syntactically to the relative necessity operator ‘[·]’ considered in this paper. To be precise, the deﬁnition of the modal operator called ‘Modest Enrichment (Type B)’ in [8] equals Axiom (R) [ϕ]ψ ↔ 2ψ ∨ ϕ ∧ 2(ϕ → ψ), which we introduce below. In this paper, however, we do not investigate further the relationship to Provability Logic.

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The paper is organised as follows. In the next section, the logic S5r is formally introduced as extension of S5. In sections 3 and 4, we show that S5r is a special case of Chellas’ Conditional Logic and we give a sound and complete axiomatisation of S5r , respectively. Then, in Section 5, a polynomial reduction of S5r to S5 is presented together with a discussion on the relation between S5r -modalities and the diﬀerence operator. Section 6 is used to extend S5r with operators for distributed hypotheses that are analog to distributed knowledge. We show how to knowledge of an agent can be represented as distributed hypotheses, where the agents’ knowledge corresponds to any system between S4 and S5. The paper closes with conclusions in Section 7.

2

The Modal Logic

In this section, we introduce the multi-modal logic S5r . Essentially, the language of S5r is the language of Propositional Logic extended with modal operators parameterised with S5r -formulas. Formally, this is done as follows. Definition 1 (Syntax of S5r ). Let Π be a countably inﬁnite set of atomic propositions. Formulas ϕ of the language L are deﬁned inductively over Π by the following grammar: ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | [ϕ]ϕ, where p ranges over atomic propositions in Π.

The logical symbols ‘’ and ‘⊥’, and additional operators such as ‘∧’, ‘→’, ‘↔’ and the dual modality ‘ ·’ are deﬁned as usual, i.e.: := p ∨ ¬p for some atomic proposition p; ⊥ := ¬; ϕ ∧ ψ := ¬(¬ϕ ∨ ¬ψ); ϕ → ψ := ¬ϕ ∨ ψ; ϕ ↔ ψ := (ϕ → ψ) ∧ (ψ → ϕ); and ϕψ := ¬[ϕ]¬ψ. Modal formulas are commonly evaluated in Kripke structures containing a binary relation over the domain, one for each modality in the language. In this case, however, every relation is determined by the valuation of the atomic propositions in the domain. Therefore, it is suﬃcient to consider Kripke structures without relations, which we call basic structures. Formally, a basic structure M is a tuple M = (W, V ), where W is a non-empty set of worlds and V : Π → 2W a valuation function mapping every atomic proposition p to a set of worlds V (p) at which it is true. We will also refer to a basic structure simply as a model. The relations that are required to evaluate the modalities are deﬁned alongside the logical consequence relation. But ﬁrst we introduce an auxiliary notion, a binary operation ‘⊗’ on sets yielding a binary relation. Let X and Y be two sets. Let X ⊗ Y be a binary relation over X ∪ Y such that X ⊗ Y = X 2 ∪ (X × Y ) ∪ Y 2 .

(1)

We illustrate this notion with an example. Example 1. Let X = {x1 , x2 } and Y = {y1 , y2 , y3 } be two sets. Then, according to (1), X ⊗Y is a binary relation over X ∪Y that is composed of the relations X 2 ,

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X × Y and Y 2 by taking their union. We have that X 2 = {(x1 , x2 ), (x2 , x1 )} ∪ id(X), X × Y = {(x1 , y1 ), (x1 , y2 ), (x1 , y3 ), (x2 , y1 ), (x2 , y2 ), (x2 , y3 )} and Y 2 = {(y1 , y2 ), (y2 , y1 ), (y1 , y3 ), (y3 , y1 ), (y2 , y3 ), (y3 , y2 )} ∪ id(Y ). Then the relation X ⊗ Y = X 2 ∪ (X × Y ) ∪ Y 2 contains two fully connected clusters X 2 and Y 2 , and directed edges between every point in X to every point in Y . Figure 1 below gives a graphical representation of X ⊗ Y (leaving out the reﬂexive and symmetric edges). We are now ready to introduce the semantics of L. It diﬀers from the semantics of Public Announcement Logic [18,7] in that the model does not change during the evaluation of formulas. Definition 2 (Semantics of S5r ). Let M = (W, V ) be a basic structure. The logical satisfaction relation ‘|=’ is deﬁned by induction on the structure of Lformulas as follows: For all p ∈ Π and all ϕ, ψ ∈ L, – M, w |= p iﬀ w ∈ V (p); – M, w |= ϕ ∨ ψ iﬀ M, w |= ϕ or M, w |= ψ; – M, w |= [ϕ]ψ iﬀ for all v ∈ W with (w, v) ∈ Rϕ , M, v |= ψ, where Rϕ = (W \ [[ϕ]]M ) ⊗ [[ϕ]]M as deﬁned in (1) and [[ϕ]]M = {x ∈ W | M, w |= ϕ} is the extension of ϕ in M. We say that a S5r -formula ϕ is satisﬁable if there is a model M and a world w in M such that M, w |= ϕ; ϕ is valid in M if M, w |= ϕ for all w in M; and ϕ is valid if ϕ is valid in all models. We will refer to the relation Rϕ as being determined by ϕ and a model. According to the semantics, a formula determines a binary relation in a model. The following proposition states the properties of such relations. Proposition 1. Let ϕ be an S5r -formula and let M = (W, V ) be a basic structure. Then, the relation Rϕ determined by ϕ and M (cf. Deﬁnition 2) is a onestep total preorder, i.e. Rϕ satisﬁes the following conditions: – Rϕ is transitive: ∀x, y, z.(xRϕ y) ∧ (yRϕ z) → (xRϕ z); – Rϕ is total: ∀x, y.(xRϕ y) ∨ (yRϕ x); – Rϕ is one-step: ∀x, y, z.(xRϕ y) ∧ ¬(yRϕ x) ∧ (xRϕ z) → (zRϕ y). Instead of ‘preorder’ also the term ‘quasiorder’ is often used in the literature. Note that totality implies reﬂexivity and that a symmetric total preorder is an equivalence relation. The proposition is readily checked as any relation Rϕ in a model determined by ϕ is deﬁned using the operation ‘⊗’, which always yields a so-called ‘one-step total preorder’. As the domain of a model is non-empty, it contains at least one point and, thus, the smallest relation Rϕ is the edge of a single reﬂexive point. Figure 1 illustrates the relation Rϕ in a model M. The domain of M is partitioned into two clusters, the worlds in each of which are fully connected (reﬂexive and symmetric edges within the clusters are not shown). Between the clusters there are outgoing directed edges from worlds in the cluster on the left to worlds

An Epistemic Logic with Hypotheses

W \ [[ϕ]]M

- u uP * P P @ P PP @ q P : u @ @ @ R u @ u -

291

[[ϕ]]M

Fig. 1. Model M with relation Rϕ

in the cluster on the right-hand side, but not vice versa. Revisit Example 1 to see in detail how Rϕ is computed (where X = W \ [[ϕ]]M and Y = [[ϕ]]M ). Consider the following example, which demonstrates what eﬀect hypotheses can have on an agent’s knowledge. Example 2. Let M = (W, V ) be a basic structure with W = {x, y}, V (ph ) = V (pc ) = {x} and V (pu ) = {x, y}. Intuitively, the three atomic propositions ph , pc and pu stand for hypothesis, conclusion and universal or already established knowledge. Then, [ph ]qu is true at x and y in M. In fact, we have that M, x |= [ϕ]qu for every S5r -formula ϕ, because qu holds everywhere in M. But [ph ]qc holds only at x and not at y, because M, x |= ph and ph implies qc everywhere in M. We conclude this section with a discussion on how S5r could possibly be used to reason about the knowledge of multiple agents; see, e.g., [9,16] for standard references. Syntactically, S5r is a single-agent logic. That is, it does not provide us with syntactic markers to distinguish agents such as a diﬀerent modality for each agent as in S5n . Consequently, there is no way to distinguish diﬀerent agents other than by what they know. In S5r we can represent the individuality of agents in the hypothesis itself. For instance, in order to represent what the agents a and b know, we can use diﬀerent hypotheses pa and pb , which are atomic propositions labelling the states which the agents a and b, respectively, consider possible. Thus [pa ]ϕ states ‘a knows ϕ’ and [pb ]ψ states that ‘b knows ψ’. However, this approach appears to be limited. It is unlikely to be able to encode S5n in this way as this simple complexity-theoretic argument will make clear (i.e. unlikely to the degree the following complexity classes are distinct): The satisﬁability problem of S5n for n > 1 is PSpace-complete [12], whereas it is NP-complete for S5r as it is shown in Section 5 below.

3

Chellas’ Conditional Logic

The language L of S5r coincides with the language of Chellas’ Conditional Logic [5,6,21]. In this section, we make precise the relationship between the two

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logics. It turns out that S5r is a special case of the Conditional Logic considered by Chellas. Chella’s semantics of the Conditional Logic is given in terms of conditional structures. A conditional structure Mc is a tuple Mc = (W, f, V ) which extends basic structures with a condition function f : W × 2W → 2W that assigns worlds w and sets of worlds X to sets f (w, X) of worlds. We also refer to them as conditional models. The set X is also understood as the extension of a proposition. The conditional semantics is deﬁned as follows. Definition 3 (Conditional Semantics). Let Mc = (W, f, V ) be a conditional structure. The logical consequence relation ‘|=c ’ is deﬁned as (we omit the Boolean cases): For all ϕ, ψ ∈ L, – Mc , w |=c [ϕ]ψ iﬀ f (w, [[ϕ]]cMc ) ⊆ [[ψ]]cMc , where [[χ]]cMc is the extension of L-formula χ in Mc .

Now we can make precise the relationship between Chellas’ Conditional Logic and S5r . Let ϕ be a formula of L and M = (W, V ) a model. Rϕ is the relation determined by ϕ and M. Let Mc = (W, f, V ) be a conditional model. We obtain (w, v) ∈ Rϕ

iﬀ

v ∈ f (w, [[ϕ]]cMc ),

whenever the condition function f satisﬁes X if w ∈ X, f (w, X) = W otherwise.

(2)

That is, S5r is Chellas’ Conditional Logic over the class of conditional models whose condition function f satisﬁes (2). Chellas studied an axiomatisation of his logic and he showed that it is complete [5]. It is readily checked that Chellas axioms are sound for S5r as well, whereas completeness remains to be studied.

4

Axiomatisation

In this section, we present a sound and complete axiomatisation of S5r . The axiom system consists of all propositional tautologies and the following axioms: (K) (T) (4) (B) (R)

[ϕ](p → q) → ([ϕ]p → [ϕ]q) []p → p []p → [][]p p → []¬[]p [ϕ]ψ ↔ []ψ ∨ (ϕ ∧ [](ϕ → ψ))

The ﬁrst four axioms are similar to the axioms known from the modal system S5 characterising any modality [ϕ] in our logic S5r as epistemic operator that can be used to represent what is known under the hypothesis ϕ. Notice that, while

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Axiom (T) establishes that what is known must in fact be true, we do not have such an axiom for hypotheses. That is, a hypothesis ϕ can be false and, thus, [ϕ]ψ → ϕ is not a theorem in our logic. The axioms (T), (4), and (B) are for the modality [] only, whereas we need additional instances of the axioms (K) and (R), namely the ones for each modal parameter ϕ (cf. Deﬁnition 1). The reduction axiom (R) states that every modality [ϕ] is deﬁnable in terms of the basic modal operator [], which corresponds to the S5-box or the universal modality. As it was already mentioned in the introduction, Axiom (R) corresponds to the deﬁnition of the modal operator ‘Modest Enrichment (Type B)’ in [8]. Theorem 1. The system S5r is sound and complete wrt. basic structures.

We skip the proof here for reasons of space. The proof follows mainly the standard canonical model construction that can be found in textbooks on modal logic; see, e.g., [2].

5

Reduction

In what follows, we show a polynomial reduction of S5r -formulas into S5. We denote by S5r0 the fragment of S5r containing formulas in which [] is the only modality allowed to occur. It is readily checked that S5r0 is a notational variant of S5. By means of Axiom (R), we obtain a method of rewriting S5r -formulas into formulas of S5r0 . That is, we already know that S5r is no more expressive than S5. At ﬁrst sight this axiom might lead to the impression that S5r may be exponentially more succinct1 than S5 as happens with Public Announcement Logic [18,7]. The following proposition states this is not the case. In fact, we can translate every S5r -formula into a formula of its fragment S5r0 that is equivalent (modulo new symbols) and without exponential blow-up in formula size. The proof Proposition 2 uses a reduction technique similar to the one used in [1], where it is shown that any hybrid language with the diﬀerence operator can polynomially be reduced to its fragment without this operator while preserving satisﬁability. Proposition 2. Let ϕ be an S5r -formula of the form [ψ]θ. Then, there is an S5r0 -formula ϕ of length polynomial in the length of ϕ such that ϕ ≡Σ ϕ , where Σ = sig(ϕ). Recall that ‘≡Σ ’ is logical equivalence relative to a signature Σ (also called semantical Σ-inseparability) deﬁned as: ϕ ≡Σ ϕ if for every model M of ϕ, there is a model M of ϕ such that M|Σ = M |Σ , and vice versa. Proof. Let ϕ be as in the proposition. We describe how to construct ϕ0 . The idea is to replace in a bottom-up fashion all of ϕ’s subformulas that are not 1

A logic L1 is exponentially more succinct than the logic L2 if there is an sequence ϕ1 , ϕ2 , . . . of L1 -formulas such that for every sequence ψ1 , ψ2 , . . . of pairwise equivalent L2 -formulas (i.e. ϕi ≡ ψi for all i ≥ 0), it holds that |ϕi | = O(2|ψi | ).

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in S5r0 by fresh propositional variables, which are then set to be equivalent to the replaced subformula using a formula from S5r0 . We proceed by inductively computing sequences ϕ0 , ϕ1 , . . . and χ1 , χ2 , . . . of S5r -formulas as follows: Set ϕ0 := ϕ. Having computed ϕi , choose a subformula of ϕi of the form [ψi ]θi , where ψi and θi are S5r0 -formulas. If there is no such subformula, ϕi is a formula of S5r0 and we are done. Otherwise, obtain ϕi+1 from ϕi by replacing [ψi ]θi with a fresh propositional variable pi . Let χi = [](pi ↔ [ψi ]θi ). The procedure terminates in n steps, where n ≤ |ϕ|. Whereas ϕn is a formula of S5r0 , the formulas χi are not. So, for every χi with 1 ≤ i ≤ n, we construct an S5r0 -formula χi that is equivalent to χi . First observe that the extension [[[ψ]θ]]M of a S5r -formula of the form [ψ]θ in a model M can be characterised as follows: ⎧ ⎪ if [[θ]]M = W ⎨W [[[ψ]θ]]M = ∅ if [[θ]]M = W and ([[ψ]]M = ∅ implies [[ψ]]M [[θ]]M ) (3) ⎪ ⎩ [[ψ]]M if [[θ]]M = W , [[ψ]]M = ∅ and [[ψ]]M ⊆ [[θ]]M Set χi = χ1i χ2i χ3i χ4i

j=1..4

χji , where:

= []pi ∨ []¬pi ∨ ([](pi ↔ ψi ) ∧ pi ∧ ¬pi ) = []pi → []θi = []¬pi → ¬θi ∧ ( ψi → (ψi ∧ ¬θi )) = [](pi ↔ ψi ) ∧ pi ∧ ¬pi → ¬θi ∧ ψi ∧ [](ψi → θi )

It is readily checked that χi ≡ χi . The three disjuncts of χ1i reﬂect the three possible values of [ψi ]θi ’s extension in a model. The disjuncts are mutually exclusive. The conjuncts χ2i , χ3i and χ4i express the necessary conditions for each of the three possible values of [ψi ]θi ’s extension. Set ϕ = ϕn ∧ i=1..n χi . Clearly, ϕ is a formula of S5r0 . It is readily checked that ϕ ≡Σ ϕ with Σ = sig(ϕ), and that the reduction leads to a blow-up of at most quadratic in the size of the formula. This concludes the proof of Proposition 2. Recall that S5r0 is a notational variant of the (one-agent system) S5. As a corollary from Proposition 2, we obtain that several interesting or desirable properties of S5 carry over to S5r . It follows that: S5r has the polynomially-bounded ﬁnite model property. The computational complexity of the satisﬁability problem for S5r is NP-complete, i.e. no worse than that for propositional logic [13,4]. S5r ’s model checking problem is polynomial in the size of the formula and model. Moreover, every S5r -formula can be translated to an equivalent formula in S5r0 without any nesting of modal operators [16]. We conclude this section by pointing out an interesting similarity between the modalities of our logic S5r and the diﬀerence operator D. A discussion and axiomatisation of the diﬀerence operator can be found in, e.g., [2], its meta theory in [19] and its relation to hybrid temporal logic is investigated in [1]. Recall that the formula Dϕ states that ϕ holds at some point that is diﬀerent to the current one. That is, the diﬀerence operator is rather expressive, e.g., the universal modality can be deﬁned in terms of it as ϕ ∨ Dϕ. This means that

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the modalities [ψ] of S5r can be expressed in terms of the diﬀerence operator (cf. Axiom (R) in Section 4, where [] is the universal modality). The other way around, however, does not hold true, i.e., the diﬀerence operator cannot be expressed in S5r , as a simple complexity-theoretic argument will show: While the satisﬁability problem of S5r is NP-complete, it is PSpace-complete for the logic of the diﬀerence operator [13]. The diﬀerence operator can be deﬁned in terms of the universal modality and nominals [1], but the latter are not available in S5r . Despite the diﬀerence modality being more expressive, an intriguing similarity between the two modalities shows up when comparing the extensions of formulas of the form ψθ and Dϕ. In both cases, the values of their extension fall in solely three categories (cf. the ψ-version of Equation (3) and [1]). That is, the extension is either: (i) the entire domain W of the model; (ii) the empty set; or (iii) W \ [[ψ]] and W \ [[ϕ]], respectively. Note that Case (iii) for Dϕ applies if, and only if, ϕ is a nominal (i.e., [[Dϕ]] = W \ {x} iﬀ [[ϕ]] = {x}, for some point x in W ). Even though in S5r we cannot specify nominals, it seems, intuitively, that we could understand S5r as the logic of the diﬀerence operator “modulo” nominals. The precise relationship between these two logics remains to be studied.

6

Distributed Knowledge

In this section, we extend S5r with modalities for distributed hypotheses that are analogous to modalities for distributed knowledge. Distributed knowledge in modal logic is a well-known notion; standard references include [9,16] and for a more recent discussion, see [11,20]. We show how distributed hypotheses can be used to represent the knowledge of an agent whose epistemic capacity corresponds to any system containing S4. Definition 4 (Syntax of S5Dr ). Let Π be a countably inﬁnite set of atomic propositions. Formulas ϕ of S5Dr are deﬁned inductively over Π by the following grammar: ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | [ϕ]K ϕ | [Φ]D ϕ, where p ranges over atomic propositions in Π, and Φ over ﬁnite sets of S5Dr formulas. To improve readability we index the modalities with ‘K’ and ‘D’ to indicate that they mean knowledge and distributed knowledge, respectively. Formulas of S5Dr are evaluated in basic structures as well. The operators [Φ]D are necessities depending on the formulas in Φ. The semantics of [Φ]D is based on the relations Rϕ , where ϕ ∈ Φ, as follows. Definition 5 (Semantics of S5Dr ). Let M = (W, V ) be a basic structure. The logical consequence relation ‘|=’ and the relations R for formulas of S5Dr are deﬁned as for S5r but extended with the following clauses: For all S5Dr -formulas ψ and all ﬁnite sets Φ of S5Dr -formulas,

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– M, w |= [Φ]D ψ iﬀ for all v ∈ W with (w, v) ∈ RΦ , M, v |= ψ, where RΦ = ϕ∈Φ Rϕ .

The following lemma shows how a preorder (a reﬂexive and transitive relation) can be represented as an intersection of relations that are one-step total preorders. The latter are the type of relations that are determined by a formula in a model (cf. Proposition 1). The proof uses the binary operator ‘⊗’ introduced in Section 2. Before we state the lemma, we introduce an auxiliary notion. Let R be a binary relation over a set W and let w ∈ W . The R-image R(w) at w is deﬁned as R(w) := {v ∈ W | wRv}. Lemma 1. Let W = {w1 , .., wk } be a set. Let R be a preorder overW . For all i ∈ {1, ..., k}, let Ri = (W \ R(wi )) ⊗ R(wi ). Then it holds that R = i=1,..,n Ri . Proof. Let the set W and the relations R, R1 , . . . , Rk be as in the lemma. We show that R = i=1..k Ri holds. First consider ‘⊆’. Suppose (wi , wj ) ∈ R. We need to show that (wi , wj ) ∈ R for all ∈ {1, ..., k}. Suppose not, i.e. (wi , wj ) ∈ / R for some . We obtain wi ∈ R(w ) by deﬁnition of R . That is, wi is an element of the R-image at w . Then we have (w , wi ) ∈ R. But then, together with the assumption (wi , wj ) ∈ R, it follows by transitivity of R that (w , wj ) ∈ R – a contradiction. Consider the other direction ‘⊇’. Suppose (wi , wj ) ∈ =1..k R . That is, (wi , wj ) ∈ R for all ∈ {1, ..., k}. In particular, we have that (wi , wj ) ∈ Ri . We obtain wi ∈ R(wi ) by reﬂexivity of R. But then the construction of Ri yields that wj ∈ R(wi ). That is, wj belongs to the R-image at wi . Thus (wi , wj ) ∈ R. To give an intuition for understanding the lemma, observe that a preorder R induces a partial order (i.e. an antisymmetic preorder) on the set of R-clusters, which are sets of points fully connected by R. In other words, R gives rise to a collection of directed graphs whose nodes are R-clusters. Note that the graph is loopless (and thus antisymmetric). Now, if R is total, all points are connected which gives rise to just one such graph. If additionally R is ‘one-step’, the graph consists of merely two nodes. Intersecting one-step total preorders has the eﬀect of erasing some directed edges from the universal relation. It is not hard to check that the intersection of preorders is again a preorder. Lemma 1 shows that by intersecting a certain selection of one-step total preorders, we can “carve out” the desired preorder. The following example illustrates the scenario. Example 3. Let W = {x, y, z} be a set and R = {(x, y), (x, z)} ∪ id(W ). It is readily checked that R is a reﬂexive and transitive relation. Now let Rw = (W \ R(w)) ⊗ R(w) for all w ∈ W . That is, according to Equation (1), we have Rx = W × W , Ry = {(x, y), (z, y), (x, z), (z, x)} ∪ id(W ) and Rz = {(y, z), (x, z), (x, y), (y, x)} ∪ id(W ). Intersecting these relations we obtain Rx ∩ Ry ∩ Rz = {(x, y), (x, z)} ∪ id(W ), which is equivalent to R. The intersection in Lemma 1 reminds us on the relations RΦ determined by a ﬁnite set Ψ of S5Dr -formulas in a model. In fact, this is the connection we

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seek to establish in order to represent the knowledge of an agent as distributed knowledge. In the following, we state how this is done. Take an arbitrary uni-modal logic L between S4 and S5 (whose satisfaction relation is denoted by |=L ). The necessity operator ‘2’ of L is thought of as representing the knowledge of the agent. Note that the system L contains the axioms (T) and (4), each of which represent important epistemic properties, namely, veridicality and positive introspection, respectively. Of course, L may contain other axioms, in fact, any axiom that can be derived in system S5. For instance, prominent axioms that are considered relevant for epistemics are: (.2) ¬2¬2p → 2¬2¬p, (.3) 2(2p → 2q) ∨ 2(2q → 2p), (.4) p → (¬2¬2p → 2p). We assume that L is determined by a class C of Kripke structures (i.e., the theorems of L are exactly the formulas that are valid on all structures in C). The class C is not required to be ﬁrst-order deﬁnable or deﬁnable in any other formalism. In fact, C may be given by manual selection. Clearly, the structures in C are reﬂexive and transitive. What we require as a precondition is that L has the ﬁnite-model property wrt. C. This means that, if a formula ϕ is not a theorem of L then there is a ﬁnite Kripke structure Mk in C that falsiﬁes ϕ, i.e. Mk , w |= ϕ for some world w in Mk . Before we can state the theorem, we need one more auxiliary notion. Let Mk = (W, R, V ) be a ﬁnite Kripke structure such that the relation R is a preorder. We say that the valuation function V covers R if for every world w ∈ W , there is an atomic proposition pw such that V (pw ) = R(w), i.e. the R-image at w. Theorem 2. Let C be a class of Kripke structures whose relations are preorders. Let Mk = (W, R, V ) be a ﬁnite structure from C such that V covers R. Let M = (W, V ) be a basic structure and let w ∈ W be a world. Let ϕ be a Boolean formula over Π. Then, there is a ﬁnite set Ψ of atomic propositions such that the following are equivalent: (i) Mk , w |=L 2ϕ; (ii) M, w |=S5Dr [Ψ ]D ϕ.

Proof. For every w ∈ W , select an atomic proposition pw such that V (pw ) = R(w). Note that such pw exists since V covers R. Set Ψ = {pw | w ∈ W }. Using Lemma 1 the equivalence of (i) and (ii) can be shown by induction on the structure of ϕ. We remark that the theorem can be generalised since the condition of using ﬁnite models is a bit too strict. Recall the metaphor that views a preorder R as a collection of loopless graphs whose nodes are R-clusters. What is actually required is that the collection of graphs and the graphs themselves are ﬁnite. So, we can still ﬁnd a ﬁnite intersection of relations as desired. The following example illustrates Theorem 2 and discusses the presented notions.

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Example 4. Consider the Kripke model Mk = (W, R, V ), where W and R are as in Example 3, and V (p) = {x, z} and V (q) = {z}. Clearly, Mk is not an S5model as R is not symmetric. Let ϕ.2 , ϕ.3 , ϕ.4 be the instances of the axioms (.2), (.3) and (.4) as shown above. It turns out that only ϕ.3 holds at x, but not ϕ.2 nor ϕ.4 . In fact, ϕ.3 holds at all worlds in Mk . Let us assume that the box (i.e. the epistemic capacity of the agent) is characterised by the system S4.3. Now label the worlds with fresh atomic propositions px , py , pz , i.e., we set V (pw ) = {w} for all w ∈ W . Notice that V covers R. Let Rpx , Rpy , Rpz be the relations determined by the basic structure M = (W, V ) and the fresh propositions (cf. Deﬁnition 2). Notice that Rpw equals Rw from Example 3, for every w ∈ W . Thus Rpx ∩ Rpy ∩ Rpz = R. Now it is immediate that M, w |= [{px , py , pz }]D ϕ iﬀ Mk , w |= 2ϕ, for all w ∈ W and all propositional formulas ϕ without occurrence of any of px , py and pz . In other words, [{px , py , pz }]D simulates the S4.3-box. We can see px , py , pz as hypotheses that another agent has to adopt in order to know what the S4.3-agent knows. In some cases we have an alternative to introducing fresh propositions even though V does not cover R. This means that V covering R is a suﬃcient but not necessary condition for Theorem 2. Here ¬p and q are hypotheses so that [{¬p, q}]D simulates S4.3-box as well. That is, hypotheses do not need to be atomic propositions. Moreover, (parts of) hypotheses may occur in the conclusion as in (W, V ), x |= [{¬p, q}]D ϕ.3 . A candidate for a reﬁned notion of V covering R is the condition that, for every w ∈ W , there is a S5Dr -formula ϕw such that [[ϕw ]] = R(w). In this paper, however, we do not explore this notion further.

7

Conclusion

In this paper, we introduced the logic S5r , which is both an extension of S5 and a special case of Chella’s Conditional Logic. We presented an axiomatisation for this logic and showed that it is as expressive and complex as S5. However, it turns out that unlike Public Announcement Logic S5r is not exponentially more succinct than S5. Nevertheless we argue that S5r is a more intuitive formalism for describing and reasoning about knowledge under hypotheses. In the second part of the paper, we extend the logic with modalities for distributed hypotheses that are analog to modalities for distributed knowledge. We showed how distributed hypotheses can be employed to represent the knowledge of agents whose epistemic capacity corresponds to any system containing S4. Possible directions for future work are to investigate S5Dr in more detail and axiomatise it, generalise Theorem 2 and to investigate an abductive reasoning service, e.g.: Given an agent a with certain epistemic capacity and that a knows ϕ, compute “appropriate” hypotheses such that the combined or distributed hypotheses imply ϕ in S5Dr . Acknowledgements. We would like to thank the anonymous reviewers for their useful comments. This work was supported in part by the Spanish Ministry of Science and Innovation through the Juan de la Cierva programme, the project “Agreement Technologies” (Grant CONSOLIDER CSD2007-0022, INGENIO 2010), and the MICINN projects TIN2006-15455 and TIN2009-14562-CO5.

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References 1. Areces, C., Blackburn, P., Marx, M.: The computational complexity of hybrid temporal logics. Logic Journal of the IGPL 8, 653–679 (2000) 2. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001) 3. Boolos, G.S.: The Logic of Provability. Cambridge University Press, Cambridge (1993) 4. Chagrov, A., Zakharyaschev, M.: Modal Logic, Oxford Logic Guides, vol. 35. Clarendon Press, Oxford (1997) 5. Chellas, B.F.: Basic conditional logic. Journal of Philosophical Logic 4, 133–153 (1975) 6. Chellas, B.F.: Modal Logic: An Introduction. Cambridge University Press, Cambridge (1980) 7. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Synthese Library: Studies in Epistemology, Logic, Methodology, and Philosophy of Science, vol. 337. Springer, Dordrecht (2007) 8. Esakia, L.: Around provability logic. Annals of Pure and Applied Logic 161(2), 174–184 (2009); Festschrift on the occasion of Franco Montagna’s 60th birthday 9. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning about Knowledge. The MIT Press, Cambridge (1995) 10. Fagin, R., Halpern, J.Y., Vardi, M.Y.: What can machines know?: On the properties of knowledge in distributed systems. Journal of the ACM 39, 328–376 (1992) 11. Gerbrandy, J.: Distributed knowledge. In: Twendial 1998: Formal Semantics and Pragmatics of Dialogue. TWLT, vol. 13, pp. 111–124. Universiteit Twente, Enschede (1998) 12. Halpern, J.Y., Moses, Y.: A guide to completeness and complexity for modal logics of knowledge and belief. Artificial Intelligence 54, 319–379 (1992) 13. Ladner, R.E.: The computational complexity of provability in systems of modal propositional logic. SIAM Journal of Computing 6, 467–480 (1977) 14. Lewis, D.: Counterfactuals. Harvard University Press, Cambridge (1973) (revised printing Blackwell) 15. Lutz, C.: Complexity and succinctness of public announcement logic. In: Proceedings of AAMAS 2006, pp. 137–143. ACM, New York (2006) 16. Meyer, J.J.C., van der Hoek, W.: Epistemic Logic for AI and Computer Science. Cambridge Tracts in Theoretical Computer Science, vol. 41. Cambridge University Press, Cambridge (1995) 17. Nute, D.: Conditional logic. In: Handbook of Philosophical Logic, vol. 2, pp. 387– 439. D. Reidel Publishing Company, Dordrecht (1984) 18. Plaza, J.A.: Logics of public communications. In: Proceedings of the 4th International Symposium on Methodologies for Intelligent Systems, pp. 201–216. Oak Ridge National Laboratory, ORNL/DSRD-24 (1989) 19. de Rijke, M.: The modal logic of inequality. The Journal of Symbolic Logic 57(2), 566–584 (1992) 20. Roelofsen, F.: Distributed knowledge. Journal of Applied Non-Classical Logics 17(2), 255–273 (2007) 21. Segerberg, K.: Notes on conditional logic. Studia Logica 48, 157–168 (1989) 22. Stalnaker, R.: A theory of conditionals. Studies in Logical Theory, 98–112 (1968)

Minimal Revision and Classical Kripke Models First Results Jonas De Vuyst CLWF, Vrije Universiteit Brussel [email protected]

Abstract. Dynamic modal logics are modal logics that have statements of the form [π]ψ. The truth value of such statements, when evaluated in a pointed model F, V, w, is determined by the truth value that ψ takes in π the pointed models F , V , w that stand in a relation − → to F, V, w. This paper introduces new dynamic operators that minimally revise finite classical Kripke models to make almost any satisfiable modal for†φ mula φ true. To this end, we define the minimal revision relations −−→ ‡φ †φ ‡φ and −−→, where −−→ revises only the valuation function and −−→ also changes the frame. We show that our language enables us to count the number of accessible worlds and to characterize irreflexive frames. We also demonstrate that any consistent formula can be made true, conditional only on modest seriality and finiteness presumptions. Keywords: dynamic modal logic, revision operators, Kripke semantics.

1

Multi-modal Kripke Semantics

Modal propositional logic extends propositional logic with modalities such as ‘necessarily’, ‘knows that’, ‘believes that’, or ‘it ought to be the case that’. It is commonplace to interpret modal logics in terms of Kripke semantics—to the extent that the terms ‘modal logic’ and ‘Kripke semantics’ are often treated as if they were synonymous. Definition 1. Let the language of multi-modal propositional logic L0 consists of all well-formed formulas (wffs) φ composed as follows: φ ::= p | ¬φ | (φ ∧ φ) | a φ, with p an element of the finite set of atomic propositions Prop and a an element of the set of indices Ind. Read ¬φ as ‘not φ’ and (φ ∧ ψ) as ‘φ and ψ’. a φ is pronounced ‘box-a φ’. By way of example, in epistemic logic this latter formula is interpreted as ‘agent a knows that φ’. Other logical symbols used in this paper are (‘top’, true in every world), ⊥ (‘bottom’, false in every world), ∨ (disjunction), → (implication), and H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 300–313, 2011. c Springer-Verlag Berlin Heidelberg 2011

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↔ (equivalence). It is easily seen that these symbols can be reduced in terms of ¬ and ∧. Similarly, let ♦a (‘diamond’) stand for ¬a ¬. Where in propositional logic you might want to know if a wff holds in a model, in Kripke semantics you would want to know if a wff holds at a ‘world’ in a model. Let us ﬁrst look at how these models are structured. Definition 2. A multi-modal Kripke frame F is a pair (W, R), with W a nonempty set of point-like ‘worlds’ and R : Ind → (W × W ) a function from indices to accessibility relations. The domain W of F is also denoted dom(F). Additionally, a multi-modal pointed frame is a pair (F , w), with (W, R) a multi-modal Kripke frame and w ∈ dom(F ). We call w the current world. Note that a world resembles a point in that only its unique identity is relevant for our purposes. Whenever wR(a)v we will say that v is accessible from w over a. Also, from here on we will use the notation wRa v instead. The intuitive understanding of the accessibility relation depends on the application. For instance, in epistemic logic wRa v means that world w represents agent a as not being able to tell apart v from w. To evaluate wffs we need both a frame and a valuation function. Definition 3. A multi-modal Kripke model is a pair (F , V ), with F = (W, R) a multi-modal Kripke frame and V : Prop → P(W ) a function from atomic propositions to the sets of worlds these propositions hold in. V is called a valuation function for F and a proposition p ∈ Prop is said to be true at (or hold at) w ∈ W if and only if w ∈ V (p). Additionally, a multi-modal pointed model is a tuple F , V, w , with (F , V ) a multi-modal Kripke model and w ∈ dom(F). We can now have a look at the formal deﬁnitions of the formulas in L0 . Definition 4. With p ∈ Prop, a ∈ Ind, and F = (W, R), define the forcing relation between pointed multi-modal Kripke models and formulas of L0 as follows:1 F , V, w p ⇐⇒ w ∈ V (p) F , V, w ¬φ ⇐⇒ not F , V, w φ F , V, w (φ ∧ ψ) ⇐⇒ F , V, w φ and F , V, w ψ F , V, w a φ ⇐⇒ ∀x ∈ Ra [w] : F , V, x φ Notice that we evaluate the modal formula a φ by quantifying over the worlds that are accessible from the current world over a. Speciﬁcally, a φ is deﬁned to be true if and only if φ evaluates to true in all the worlds accessible over a. For example, in the terminology of epistemic logic this means that an agent a knows that φ if and only if φ is true in all the worlds that a cannot distinguish from the current world. Depending on the application the accessibility relations are restricted in various ways. For instance, in epistemic logic reﬂexivity is imposed, which has the 1

For any relation Q : X × Y let Q[x] := {y ∈ Y | xQy}.

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eﬀect of making (a φ → φ) valid. This corresponds to the notion that it is impossible to know false propositions. Of course it is possible to believe false propositions and hence in doxastic logic axiom, in which a φ is read as ‘agent a believes that φ’, the requirement that accessibility relations are reﬂexive is dropped. Instead seriality is substituted for reﬂexivity so that agents cannot believe contradictions. Restrictions on the accessibility relations are known as frame conditions. Definition 5. Given a number of frame conditions σ, we say that F = (W, R) is a σ-frame if and only if Ra meets the conditions of σ (for all a ∈ Ind).2 Similarly, a model that has a σ-frame is called a σ-model. Finally, we deﬁne a notion of equivalence on the level of pointed models. Definition 6. A bisimulation between (the worlds of ) two models (F, V ) and (F , V ) is a relation ρ between W and W such that xρx if and only if the following conditions are met:3 Atom equivalence. x ∈ V (p) ⇐⇒ x ∈ V (p) for all p ∈ Prop. Forth. For every a ∈ Ind and y ∈ Ra [x] there is a y ∈ Ra [x ] such that yρy . Back. For all a ∈ Ind and y ∈ Ra [x ] there is a y ∈ Ra [x] such that yρy . Furthermore, two pointed models F , V, w and F , V , w are bisimilar if and only if there is a bisimulation ρ between them such that wρw .

2

Two Fixed-frame Minimal Revision Operators

In this section we extend the language of multi-modal logic by introducing minimal revision operators that, given a well-formed formula φ to make true, quantify over alternative valuation functions in which φ holds. As we will see, if there is a valuation function that makes φ true at the current world then the minimal revision operators quantify over at least one such function. Moreover, if there is more than one such function then these operators only quantify over the ones that are only minimally diﬀerent from the original valuation function. Let us ﬁrst look at the grammar of our ﬁxed-frame minimal revision logic. Definition 7. With p ∈ Prop and a ∈ Ind, let L† be the set of wffs4 φ of the following form: φ ::= p | ¬φ | (φ ∧ φ) | a φ | [† φ] φ. Read [† φ] ψ as ‘ψ must hold when φ is made true’. Additionally, deﬁne † φ ψ as ¬ [† φ] ¬ψ and read it as ‘ψ could hold after making φ true’. Next, we make the forcing relation accommodate the [† φ] operators. 2 3 4

No particular formal representation of frame conditions is presupposed in this paper. Let F = (W, R) and F = (W , R ). In this section the term wff is used to denote the elements of L† .

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Definition 8. Let the forcing relation be as before, except that L† is the new codomain and except for the addition of the following rule. F, V, w [† φ] ψ ⇐⇒ ∀V : if V is a valuation function for F †φ

and F, V, w −−→ F, V , w then F, V , w ψ. †φ

It should be clear that we only need to deﬁne −−→ between pointed models †φ that share the same frame and current world. Moreover, we want F, V, w −−→ F, V , w to hold if and only if (i) F, V , w φ and (ii) V is only ‘minimally diﬀerent’ from V . To help us with the second requirement we ﬁrst deﬁne a function δ that, given two valuation functions V and V for a single frame, tells us which pairs of worlds and atomic propositions are assigned a diﬀerent truth value by V and V . Definition 9. For any two models (F , V ) and (F , V ), let δ(F , V, V ) := {(w, p) ∈ W × Prop | V (p) ∩ {w} = V (p) ∩ {w}} . In other words, δ(F , V, V ) yields a set of pairs (w, p) such that w ∈ V (p) but w∈ / V (p) or (vice versa) w ∈ / V (p) but w ∈ V (p). †φ

We can now deﬁne −−→. Definition 10. For all φ ∈ L† and pointed models F, V, w and F, V , w , let †φ

F, V, w −−→ F, V , w if and only if 1. F, V , w φ and 2. There is no valuation V for F such that (a) F, V , w φ and (b) δ(F , V, V ) ⊂ δ(F , V, V ). One property that can be read oﬀ the deﬁnition of the ﬁxed-frame minimal †φ revision relation straight away is that −−→ only holds between two pointed models if φ is true in the right hand model. Proposition 1 (Success). [† φ] φ

†φ

For all pointed models F, V, w and F, V , w it is the case that F, V, w −−→ F, V , w implies that F, V , w φ. This success is not a vacuous accomplishment. If there is a way to revise a finite model to make φ true (while keeping the frame and current world constant) then there is a minimal revision that makes φ true. In other words, the minimal revision operators preserve consistency for ﬁnite frames. Proposition 2 (Finite Consistency). (F , w) [† φ] ⊥ =⇒ (F , w) ¬φ For all finite pointed models F, V, w , if a wff φ is satisfiable at (F , w) then †φ

there is a pointed model F, V , w such that F, V, w −−→ F, V , w .

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Table 1. The AGM postulates for belief revision. In this table K is a belief set, Cn closes a set of formulas under entailment, is a belief revision operation, and + is a belief expansion operation. See also [6] Closure Success Inclusion Vacuity Consistency Extensionality Superexpansion Subexpansion

K φ = Cn(K φ) φ∈K φ Kφ⊆K+φ ¬φ ∈ / K =⇒ K φ = K + φ K φ is consistent if φ is consistent (φ ↔ ψ) ∈ Cn(∅) =⇒ K φ = K ψ K (φ ∧ ψ) ⊆ (K φ) + ψ ¬ψ ∈ / Cn(K φ) =⇒ (K φ) + ψ ⊆ K (φ ∧ ψ)

Proof. Consider ﬁrst the following set S. S = {δ(F , V, V ) | V is a valuation function for F and F , V , w φ} We need to show that if S is nonempty then S has minimal elements. To this end we prove that S is ﬁnite. Observe that by deﬁnition of δ it is the case that δ(F , V, V ) ⊆ dom(F )× Prop for all valuation functions V for F . And as by stipulation dom(F) and Prop are ﬁnite sets so is their Cartesian product T = dom(F ) × Prop. And then so is P(T ). It follows that S is ﬁnite since S ⊆ P(T ). A reader familiar with the belief revision literature may recognize the previous two propositions as AGM postulates for belief revision (Table 1). This is also the origin of the following two propositions. The ﬁrst one of which is a trivial result from the fact that we only take semantic elements into account when revising models. Proposition 3 (Extensionality). (φ ↔ ψ) =⇒ ([† φ] χ ↔ [† ψ] χ) If (φ ↔ ψ) then for all pointed models F , V, w and F , V , w it is the case †φ

†ψ

that F , V, w −−→ F , V , w if and only if F , V, w −−→ F, V , w . The minimal revision operators discussed in this paper do not have analogues for all AGM postulates because they revise pointed models and not belief sets. Speciﬁcally, they do not have analogues for the AGM postulates that relate belief revision to belief expansion. Vacuity is one such postulate, although we can record the following proposition about the case where φ is to be made true in a pointed model in which φ is already true (rather than ‘not disbelieved’). Proposition 4 (Special Vacuity). ((φ ∧ ψ) → [† φ] ψ) For all pointed models F , V, w and F , V , w such that F , V, w φ and †φ

F , V, w −−→ F, V , w it is the case that V = V . As is typical for dynamic modal operators, [† φ] is ‘almost’ a normal modal operator. Speciﬁcally, we have the Rule of Necessitation and the K Axiom, but not the Substitution Property.

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Proposition 5 (Rule of Necessitation). ψ =⇒ [† φ] ψ Proof. Whenever [† φ] ψ is evaluated in a pointed model F , V, w , the operator [† φ] quantiﬁes only over models that have the frame F . Hence everything that is valid in F is the case in all revised models. Proposition 6 (K). ([† φ](χ → ξ) → ([† φ] χ → [† φ] ξ)) Proposition 7. ψ =⇒ ψ[p/φ] Proof. It can easily be seen that ((p ∧ q) → [† ¬p] q) but that ((p ∧ p) → [† ¬p] p). †φ

We now take a look at the relational properties of −−→ over the set of all pointed models F , V, w that share the same frame F and current world w ∈ dom(F ). From this vantage point we get two important properties. Proposition 8 (Shift Reflexive). [† φ]([† φ] ψ → ψ)

†φ

For all pointed models F , V, w and F , V , w it is the case that if F , V, w −−→ †φ

F , V , w then F , V , w −−→ F , V , w . †φ

Proof. The premise F , V, w −−→ F, V , w implies that F , V , w φ. Additionally, it is self-evident that δ(F , V , V ) = ∅. Hence there cannot be a pointed model F , V , w such that V = V and such that F , V , w φ and †φ

δ(F, V , V ) ⊂ δ(F , V , V ). It follows that F , V , w −−→ F, V , w . Proposition 9 (Antisymmetric). †φ

For all pointed models F , V, w and F , V , w , if F , V, w −−→ F, V , w and †φ

F , V , w −−→ F, V, w then V = V . †φ

Proof. From F , V, w −−→ F, V , w it follows that F , V , w φ. Further†φ

more, from F , V , w −−→ F , V, w it follows that there is no pointed model F , V , w such that F , V , w φ and such that δ(F, V , V ) ⊂ δ(F, V , V ). Since it is self-evident that δ(F , V , V ) = ∅ and since we already established that F , V , w φ, however, this means that δ(F , V , V ) = δ(F , V , V ) = ∅. It follows that V = V . Propositions 8 and 9 can be summarized in the following property. Proposition 10 (Shift Unique and Shift Reflexive). [† φ]([† φ] ψ ↔ ψ)

†φ

For all pointed models F , V, w and F , V , w it is the case that if F , V, w −−→ †φ

F , V , w then −−→ [ F , V , w ] = { F , V , w }. †φ

It is now also easy to see that −−→ is transitive and dense.5 Moreover, on the basis †φ of their relational properties with respect to −−→ we can partition all pointed models as follows: 5

A relation Q : S × S is dense if and only if ∀x, y ∈ S : xRy xRz and zRy.

=⇒

∃z ∈ S :

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1. Pointed models in which φ does not hold and cannot be made true. These †φ models have no incoming or outgoing −−→-links. 2. Pointed models in which φ does not hold but for which φ can be made true. †φ These models have outgoing −−→-links to pointed models of the third kind †φ and only to such models. They have no incoming −−→-links. 3. Pointed models in which φ holds. These models have reﬂexive links and no other outgoing links. The addition of the ﬁxed-frame minimal revision operators makes our logic more expressive than L0 . For instance, it is possible to count the number of accessible worlds over an index. Theorem 1. Given a pointed model F , V, w it is the case that at least n worlds are accessible from w over a if and only if F , V, w †(♦a φ1 ∧ · · · ∧ ♦a φn ) , 6 where φ1 , . . . , φn are satisfiable conjunctions of literals such that no conjunction (φi ∧ φj ) is satisfiable (with i = j). Proof. Let α := (♦a φ1 ∧ · · · ∧ ♦a φn ). By deﬁnition, F , V, w † α if and †α

only if there is a valuation V for (F , w) such that F , V, w −−→ F, V , w . By Propositions 1–2 such a valuation function exists if and only if α is satisﬁable at (F, w). Finally, since every φi is satisﬁable but no two φj and φk can be true at a single world, α is satisﬁable in (F , w) if and only if there are at least n worlds accessible from w over a. In L† it is also possible to characterize certain types of frames that famously cannot be characterized in L0 . For instance, the following proposition explains how to characterize irreﬂexive frames (when n = 0 and m = 1). Theorem 2. For all pointed models F , V, w and atoms p ∈ Prop it is the case that worlds accessible from w over a in n steps are not accessible over a in m n 7 steps if and only if F , V, w [† na p] [† m a ¬p] a p. Proof. We prove the left to right part of this proposition by contradiction. Suppose that all worlds accessible from w over a in n steps were indeed not accessible over a in m steps. Additionally, suppose that it was not the case that n F , V, w [† na p] [† m a ¬p] a p. † n p

For all valuation functions V for F such that F , V, w −−−a−→ F, V , w it is the case that F , V , w na p by Proposition 1. That is, p holds in all worlds accessible over a in n steps. Of course for all valuation functions V for F such † m ¬p

that F , V , w −−−a−−→ F, V , w it is the case that p does not hold in any of the 6 7

Note that for this to be an actual wff extra parentheses would have to be added. Here n a φ stands for a . . . a φ. n

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worlds accessible over a in m steps. It is a premise of the left to right part of this proof that p also fails to hold in one of the worlds accessible from w in n steps. Call one such world v. Now either v is accessible from w in m steps and then we have arrived at a contradiction (thereby concluding the ﬁrst part of the proof). Suppose that, on the other hand, v is not accessible from w in m steps. Consider then the valuation function V such that δ(F , V , V ) = {(v, p)}. Since v would not be accessible in m steps it would still hold that F, V , w m a ¬p. Note however that δ(F , V , V ) = δ(F , V , V ) \ {(v, p)}. Our previously established † m ¬p

fact that F, V , w −−−a−−→ F , V , w is contradicted by the conclusions that (i) F, V , w φ and (ii) δ(F , V , V ) ⊂ δ(F , V , V ). So again we found a contradiction that concludes the left to right part of this proof. We also prove the right to left part of this proposition by contradiction. Supn pose that it was indeed the case that F, V, w [† na p] [† m a ¬p] a p. Suppose also that there was a world v accessible from w over a in n steps that was also accessible over a in m steps. For all valuation functions V for F such that † n p

F, V, w −−−a−→ F , V , w it would then be the case that F, V , v p. How†φ

ever, for all valuation functions V for F such that F, V , w −−→ F, V , w it would be the case that F, V , v ¬p. But then it would not be the case that F, V , w na p, which contradicts the premise of the right to left part of this proof.

3

Two Generalized Minimal Revision Operators

The ﬁxed-frame operators deﬁned in the previous section allow us to make many formulas true. They are constrained by the available frame, however. Building †φ upon the ﬁxed-frame relation −−→ we will now deﬁne a more general relation ‡φ −−→ between pointed models. But ﬁrst we extend our language with generalized minimal revision operators. Definition 11. With p ∈ Prop and a ∈ Ind, let L‡ be the set of wffs φ of the following form: φ ::= p | ¬φ | (φ ∧ φ) | a φ | [† φ] φ | [‡ φ] φ. Additionally, let ‡ φ be a shorthand for ¬ [‡ ¬φ]. As before we extend the forcing relation. This time around, though, we also add the frame conditions as a parameter. Definition 12. Given frame conditions σ, let the forcing relation σ be as , except that L‡ is the new codomain and except for the addition of the following rule. F, V, w σ [‡ φ] ψ ⇐⇒ ∀ F , V , w : if F , V , w is a pointed σ-model ‡φ

and F, V, w −−→ F , V , w then F , V , w σ ψ.

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The generalized minimal revision relation −−→ is deﬁned as follows. Definition 13. For all formulas φ ∈ L‡ and all pointed models F , V, w and ‡φ

F , V , w , let F , V, w −−→ F , V , w if and only if there is a valuation V for F such that 1. F , V, w ↔ F , V , w and †φ

2. F , V , w −−→ F , V , w . Let us now go through some of the formal properties of these generalizations. First of all, it turns out that the ﬁxed-frame minimal revision relation is a subset of the generalized minimal revision relation. Proposition 11. σ ( † φ ψ → ‡ φ ψ) and σ ([‡ φ] ψ → [† φ] ψ)

†φ

For all pointed models F , V, w and F , V , w it is the case that if F , V, w −−→ ‡φ

F , V ∗ , w then F , V, w −−→ F, V ∗ , w . Note that this property does not hold from right to left, even when keeping the frame and current world constant. Proposition 12. It is not always the case that for pointed models F , V, w and ‡φ

†φ

F , V , w such that F , V, w −−→ F, V , w also F , V, w −−→ F, V , w . Proof. Consider the following frame F . v

a

w

a

u

Consider also two valuation functions V, V for F that diﬀer only in their valuation for p at v and u: Let V (p) = {v} and V (p) = {u}. Since in either case there is one p-world and one ¬p-world accessible from w (and apart from those worlds’ identities there are no diﬀerences), F , V, w ↔ F , V , w . However, †

†

whereas F , V, w −−→ F, V, w it is not so that F , V, w −−→ F, V , w since δ(F, V, V ) = ∅ but δ(F , V, V ) = {(v, p), (u, p)}. †

On the other hand, we do have F , V , w −−→ F, V , w and F , V, w ↔ ‡

F , V , w and this entails that F , V, w −−→ F, V , w . Additionally, because for all pointed models F , V, w and F , V , w such ‡φ

that F , V, w −−→ F , V , w there is a valuation function V for F such that †φ

†φ

‡φ

F , V , w −−→ F , V , w , many properties of −−→ are inherited by −−→. Proposition 13 (Success). σ [‡ φ] φ

‡φ

For all pointed models F , V, w and F , V , w it is the case that F , V, w −−→ F , V , w implies that F , V , w σ φ.

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Like its ﬁxed-frame variant, the generalized minimal revision relation is consistent and extensional. Proposition 14 (Finite Consistency). For all pointed models F , V, w it is the case that if there are finite pointed models F , V , w and F , V , w such that F , V, w ↔ F , V , w and F , V , w φ, then there is a pointed model ‡φ

F , V , w such that F , V, w −−→ F , V , w . Theorem 3 (Finite D-Consistency). σ [‡ φ] ⊥ =⇒ σ ¬φ If σ includes seriality and if φ ∈ L‡ is satisfiable in a finite σ-frame then for all finite pointed σ-models F , V, w there is a pointed model F , V , w such that ‡φ

F , V, w −−→ F , V , w . Proof. As F , V, w is a ﬁnite σ-model and as φ is satisﬁable in a ﬁnite σ-frame, a ﬁnite pointed σ-model F ◦ , V ◦ , w◦ can be constructed such that F , V, w ↔ F ◦ , V ◦ , w◦ and such that φ is satisﬁable at (F ◦ , w◦ ). But ﬁrst, let (F , w ) be a ﬁnite frame such that φ is satisﬁable at (F , w ). Where F = (W, R), F = (W , R ), and F ◦ = (W ◦ , R◦ ), construct F ◦ , V ◦ , w◦ as follows: – – – –

W◦ = W × W Ra◦ : W ◦ × W ◦ such that (x, x )Ra◦ (y, y ) ⇐⇒ xRa y and x Ra y V ◦ : Prop → P(W ◦ ) such that V ◦ (p) := V (p) × W w◦ = (w, w )

Notice that the frame is computed as in action model logic. The valuation function, on the other hand, ignores W and is deﬁned solely in terms of W . We show that one bisimulation ρ between F , V, w and F ◦ , V ◦ , w◦ is as follows: ρ : W × W ◦ such that xρ(y, y ) ⇐⇒ x = y For consider the following: – Atom equivalence. For all x ∈ W and x ∈ W such that xρ(x, x ), by deﬁnition x ∈ V (p) ⇐⇒ (x, x ) ∈ V ◦ (p) for all p ∈ Prop. – Forth. We need to prove that for all x ∈ W, x ∈ W , if xρ(x, x ) then for every a ∈ Ind and y ∈ Ra [x] there is a (y, y ) ∈ Ra◦ [(x, x )] such that yρ(y, y ). By deﬁnition of Ra◦ , however, (y, y ) ∈ Ra◦ [(x, x )] if and only if (i) y ∈ Ra [x] and (ii) y ∈ Ra [x ]. Now (i) is the antecedent of the implication we are interested in so we can assume it holds (since if it doesn’t then we’re in the clear also). As for (ii), there is at least one y ∈ Ra [x ] because we presuppose that F is a serial frame. – Back. We need to prove that for all x ∈ W, x ∈ W , if xρ(x, x ) then for every a ∈ Ind and (y, y ) ∈ Ra◦ [(x, x )] there is a y ∈ Ra [x] such that yρ(y, y ). By deﬁnition of Ra◦ , however, (y, y ) ∈ Ra◦ [(x, x )] if and only if (i) y ∈ Ra [x] and (ii) y ∈ Ra [x ]. Thus the antecedent of the implication we are interested in—viz. (y, y ) ∈ Ra◦ [(x, x )]—entails its consequent y ∈ Ra [x].

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Additionally, it is obvious that wρ(w, w ). An entirely analogous argument shows that φ is satisﬁable in (F ◦ , w◦ ). We omit this argument here. Moreover, observe that since W and W are ﬁnite, so is its Cartesian product ◦ W = W × W . It is also easily seen that since R and R are serial relations so is R◦ . Finally, by Proposition 2 there is a pointed model F ◦ , V , w◦ such that †φ

F ◦ , V ◦ , w◦ −−→ F ◦ , V , w◦ . Since F , V, w ↔ F ◦ , V ◦ , w◦ this means that ‡φ

F , V, w −−→ F ◦ , V , w◦ . It should be obvious that the generalized minimal revision operators too are extensional. Proposition 15 (Extensionality). σ (φ ↔ ψ) =⇒ σ ([† φ] χ ↔ [† ψ] χ) Whereas special vacuity itself does not hold for the generalized minimal revision relation, the following property comes close. Proposition 16 (Special Quasi-Vacuity). For all pointed models F , V, w

‡φ

and F , V , w such that F , V, w σ φ and F , V, w −−→ F , V , w it is the case that F , V, w ↔ F , V , w . Note, however, that whereas ((φ∧ψ) → [† φ] ψ) corresponds to Proposition 4, the formula ((φ ∧ ψ) → [‡ φ] ψ) is not valid for all ψ ∈ L‡ . Proposition 17. σ ((φ ∧ ψ) → [‡ φ] ψ) Proof. We show that (( ∧ †(♦a p ∧ ♦a ¬p) ) ∧ ¬ [‡ ] †(♦a p ∧ ♦a ¬p) ) is satisﬁable. First, consider the following frame F: v

a

w

a

a

w

u

Now compare F to the frame F ’: v

Finally, let V and V be valuation functions for F and F ’ such that the extension for every proposition is the empty set. It is obvious that F , V, w ↔ F , V , w . †

Since, moreover, it is trivially established that F , V , w −−→ F , V , w we ‡

also have F , V, w −−→ F , V , w . However, F , V, w σ †(♦a p ∧ ♦a ¬p) but not F , V , w σ †(♦a p ∧ ♦a ¬p) . And again we are dealing with ‘almost’ normal modal operators.

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Proposition 18 (Rule of Necessitation). σ ψ =⇒ σ [‡ φ] ψ Proof. Trivial result as we restrict quantiﬁcation for [‡ φ] and ‡ φ to σ-models. Proposition 19 (K). σ ([‡ φ](χ → ξ) → ([‡ φ] χ → [‡ φ] ξ)) Proposition 20. ψ =⇒ ψ[p/φ] Shift Reﬂexivity is another property that is preserved. Proposition 21 (Shift Reflexive). σ [‡ φ]([‡ φ] ψ → ψ)

‡φ

For all pointed models F , V, w and F , V , w , if F , V, w −−→ F , V , w

‡φ

then F , V , w −−→ F , V , w . Antisymmetry and shift uniqueness are almost preserved. That is, the derived properties that result from substituting bismilarity for equality hold. Proposition 22 (Quasi-Antisymmetric). For all pointed models F , V, w

‡φ

‡φ

and F , V , w , if F , V, w −−→ F , V , w and F , V , w −−→ F, V, w then F , V, w ↔ F , V , w . ‡φ

Proof. From F , V, w −−→ F , V , w it follows that there is a valuation func†φ

tion V ◦ for F such that F , V, w ↔ F , V ◦ , w and such that F , V ◦ , w −−→ ‡φ

F , V , w . Similarly, from F , V , w −−→ F, V, w it follows that there is a valuation function V for F such that F , V , w ↔ F , V , w and such that †φ

F , V , w −−→ F , V, w . ‡φ

From F , V, w −−→ F , V , w it follows that F , V , w σ φ and from ‡φ

F , V , w −−→ F, V, w it follows that F , V, w σ φ. But then it also follows that F , V ◦ , w σ φ and that F , V , w σ φ since F , V ◦ , w ↔ F , V, w

and F , V , w ↔ F , V , w . By Proposition 4 it follows that V = V ◦ and V = V . This, in turn, entails that F , V, w ↔ F , V , w . Proposition 23 (Shift Quasi-Unique and Shift Reflexive). σ [‡ φ](ψ → [‡ φ] ψ) For all finite pointed models F , V, w and F , V , w it is the case that if ‡φ

‡φ

F , V, w −−→ F , V , w then also { F , V , w } ⊆ −−→ [ F , V , w ] ⊆ ↔[ F , V , w ]. ‡φ

Proof. First. By Proposition 21 it follows from F , V, w −−→ F , V , w that ‡φ

‡φ

F , V , w −−→ F , V , w . This proves that the set −−→ [ F , V , w ] contains the element F , V , w . Second. Using Proposition 13 we can deduce that F , V , w σ φ. By Proposition 16 it then follows that for all pointed models F , V , w such that ‡φ

F , V , w −−→ F , V , w it is the case that F , V , w ↔ F , V , w . ‡φ

This proves that the set −−→ [ F , V , w ] contains only pointed models that are bisimilar to F , V , w .

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Related Work

Our approach is diﬀerent from most other dynamic logics in several ways. First, many dynamic modal logics have operators that directly refer to abstract semantic entities. For instance, in action modal logic—as described in [5]— the dynamic operators refer to ‘action models’. New epistemic models are computed by an operation on the current epistemic model and the speciﬁed action model. Our dynamic operators instead refer to propositions. Second, some dynamic modal logics where the dynamic operators refer to propositions only allow a subset of non-modal propositions to be used for this purpose. One example is the logic of ‘introspective forgetting’ described in [4], which can only express forgetting of atoms. Our operators accept all formulas of the object language. Third, dynamic modal logics that do permit all well-formed formulas for parameters of dynamic operators typically add extra semantic structure to their models. For instance, in [3] (and many other papers) plausibility orderings are used to sensibly change doxastic models. We know of two logics that are not diﬀerent from ours on the above criteria. First, the dynamic context logic of [2] has dynamic operators for adding and removing φ-worlds to and from doxastic models. However, the authors do not offer a criterion for prudently choosing a set of φ-worlds to add. Instead they opted to make this a nondeterministic choice. By contrast, in this paper revisions are restricted to minimal valuation changes—although we grant that other notions of minimal change exist and that in the absence of applications it is impossible to decide whether our notion of minimality is ultimately a desirable one. Second, [1] explains how to do AGM belief revision on ‘internal belief models’ that are sets of pointed multi-modal Kripke models. This is done using centered ordering relations on pointed Kripke models. These orderings are based on a notion of n-bisimilarity, though, and as such only take into account whether two worlds have diﬀerent valuations and ignore the degree by which they are diﬀerent. From a technical point of view it is interesting that action modal logic and our minimal revision logic have almost the same constraints when it comes to realizing more or less arbitrary formulas. To wit, [5] contains a result that if φ is satisﬁable in a pointed frame (F , w) such that its current world is part of a serial subframe (F , w) then there is an update that realizes φ (see Deﬁnition 3.1 and Corollary 3.4). This is reminiscent of Theorem 3 in this paper. As a matter of fact, in this respect action modal logic updates are slightly more powerful than minimal revisions as for a minimal revision to be successful it must have a serial subframe (F , w) that can satisfy a state bisimilar to the original pointed model. Spelling out this property would yield a statement halfway between Proposition 14 and Theorem 3.

5

Future Work

The search for theorems has only just begun and it is not yet clear if the minimal revision operators are axiomatizable. Moreover, many questions remain

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regarding the expressivity of L† , L‡ , and the language that results when the formulas containing ﬁxed-frame operators are subtracted from L‡ . So far we do not have applications for minimal revision. We do, however, intend to investigate whether minimal revision has applications in doxastic and epistemic logic. For instance, we are eager to ﬁnd out if minimal revision can be used to extend the forgetting operator of [4] so that the forgetting of arbitrary modal formulas can be modeled. It might also be worthwhile to search for ‘expansion’ operators that validate counterparts of the remaining AGM postulates—i.e. those postulates that relate belief revision to belief expansion. Finally, we want to explore what happens when the generalized revision relation is decoupled from the ﬁxed-frame relation. For instance, the latter could be made a parameter of the generalized relation. Alternatively, an operator could be introduced that quantiﬁes over the pointed models bisimilar to the current one. Acknowledgements. I thank Patrick Allo for our regular meetings during the genesis of this paper. I also thank Benedikt L¨ owe as well as Davide Grossi and Peter van Ormondt for previous discussions on closely related topics. Lastly, this paper beneﬁted from the comments by the anonymous reviewers.

References 1. Aucher, G.: Generalizing agm to a multi-agent setting. Logic Journal of IGPL (2010) 2. Aucher, G., Grossi, D., Herzig, A., Lorini, E.: Dynamic context logic. In: He, X., Horty, J.F., Pacuit, E. (eds.) LORI 2009. LNCS, vol. 5834, pp. 15–26. Springer, Heidelberg (2009) 3. Baltag, A., Smets, S.: A qualitative theory of dynamic interactive belief revision. In: Logic and the Foundations of Game and Decision Theory (LOFT 7). Texts in Logic and Games, vol. 3, pp. 13–60. Amsterdam University Press, Amsterdam (2008) 4. van Ditmarsch, H., Herzig, A., Lang, J., Marquis, P.: Introspective forgetting. Synthese 169, 405–423 (2009) 5. van Ditmarsch, H., Kooi, B.: Semantic results for ontic and epistemic change. In: Logic and the Foundations of Game and Decision Theory (LOFT 7). Texts in Logic and Games, vol. 3, pp. 87–117. Amsterdam University Press, Amsterdam (2008) 6. Hansson, S.O.: Logic of belief revision. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Spring 2009 edn.) (2009)

On Axiomatizations of PAL Yanjing Wang Department of Philosophy, Peking University [email protected]

Abstract. In the literature, diﬀerent axiomatizations of Public Announcement Logic (PAL) were proposed. Most of these axiomatizations share a ‘core set’ of the so-called reduction axioms. In particular, there is a composition axiom which stipulates how two consecutive announcements are composed into one. In this paper, by designing non-standard Kripke semantics for the language of PAL, we show that without the composition axiom the core set does not completely axiomatize PAL. In fact, most of the intuitive ‘axioms’ and rules we took for granted could not be derived from the core set. The non-standard semantics we proposed is of its own interest in modelling realistic agents. We show that with the help of diﬀerent composition axioms we may axiomatize PAL w.r.t. such non-standard semantics.

1

Introduction

The language of Public Announcement Logic (PAL) [10,7] is usually presented as follows: φ ::= | p | ¬φ | φ1 ∧ φ2 | 2i φ | [ψ]φ As usual, we deﬁne ⊥, φ ∨ ψ, φ → ψ and ψφ as the abbreviations of ¬, ¬(¬φ ∧ ¬ψ), ¬φ ∨ ψ and ¬[ψ]¬φ respectively 1 . In this paper we call the [φ]-free fragment of PAL language the language of Epistemic Logic (EL). Given a Kripke model over a non-empty set of basic propositions P, and a non-empty set of agents I: M = (S, {→i | i ∈ I}, V ), the truth value of a PAL formula φ at a state s in M is deﬁned as follows: M, s ⇔ always M, s p ⇔ p ∈ V (s) M, s ¬φ ⇔ M, s φ M, s φ ∧ ψ ⇔ M, s φ and M, s ψ M, s 2i ψ ⇔ ∀t i s : M, t ψ M, s [ψ]φ ⇔ M, s ψ implies M|ψ , s φ where (∀t i s : . . . ) denotes ‘for all t : s →i t implies . . . ’, and M|ψ = (S , {→i | i ∈ I}, V ) such that: S = {s | M, s ψ}, →i =→i |S ×S and V (p) = V (p) ∩S . Note that in this paper we do not restrict ourselves to S5 model unless speciﬁed. 1

To simplify discussion, we do not consider common knowledge in this paper.

H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 314–327, 2011. c Springer-Verlag Berlin Heidelberg 2011

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315

In the literature, diﬀerent axiomatizations of PAL were proposed(cf. e.g.,[10], [1],[2],[5]). Most of these axiomatizations are based on the following proof system PA: Axiom Schemas TAUT all the instances of tautologies DISTK 2i (φ → ψ) → (2i φ → 2i ψ) !ATOM [ψ]p ↔ (ψ → p) !NEG [ψ]¬φ ↔ (ψ → ¬[ψ]φ) !CON [ψ](φ ∧ χ) ↔ ([ψ]φ ∧ [ψ]χ) !K [ψ]2i φ ↔ (ψ → 2i (ψ → [ψ]φ)) Rules φ NECK 2i φ φ, φ → ψ MP ψ

where φ, ψ, χ denote arbitrary formulas, p denotes an arbitrary propositional letter in P or , and i denotes an arbitrary index in I 2 . However, it is not clear whether the above system is complete. Moreover, based on the above system, there are diﬀerent proposals for a complete system with extra axioms or rules. It is not clear either that, among those additional axioms and rules that are valid, which are necessary in the proof system of PAL, and which can be derived from PA3 . In this paper, we will try to answer such questions. We ﬁrst list the additional axiom schemas and rules that we will discuss in this paper as follows: Axiom Schemas DIST! [ψ](φ → χ) → ([ψ]φ → [ψ]χ) !COM [ψ][χ]φ ↔ [ψ ∧ [ψ]χ]φ WDIST! [ψ](φ → χ) ↔ ([ψ]φ → (ψ → [ψ]χ)) SDIST! [ψ](φ → χ) ↔ ([ψ]φ → [ψ]χ) !K [ψ]2i φ ↔ (ψ → 2i [ψ]φ) EA! (ψ → [ψ]φ) → [ψ]φ DIA! (ψ ∧ [ψ]φ) ↔ ψφ Rules φ NEC! [ψ]φ φ↔χ RE ψ ↔ ψ[χ/φ] φ↔χ RE¬ ¬φ ↔ ¬χ φ↔χ RE∧ (ψ ∧ φ) ↔ (ψ ∧ χ) φ↔χ RE2 2i φ ↔ 2i χ φ↔χ RE! [ψ]φ ↔ [ψ]χ 2 3

Note that PA does not include the rule of uniform substitution (US). A discussion on the decidability of the US-closed fragment of PAL can be found in [8]. φ A rule ψ is derivable from a system S if all the instances of this rule can be derived by using φ, axioms and inference rules in S.

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The selection of these axioms and rules is not arbitrary: RE is the rule of replacement of equivalents where ψ[χ/φ] denotes any formula obtained by replacing one or more occurrences of φ in ψ with χ (cf. e.g., the axiomatization in [10]). !COM is the composition axiom featured in many expositions of PAL proof system (cf. e.g., [5]). Other usual suspects in the above table include !K , NEC! and DIST!. !K is often used in the literature as an “equivalent” of !K (cf., e.g., [2]), while NEC! and DIST! are [φ]-versions of the well-known necessitation rule and distribution axiom in basic modal logic, which sometimes appear too in the axiomatizations of PAL (cf. e.g. [1]). WDIST! and SDIST! are (weaker/stronger) variations of DIST! that we will use. Finally EA! and DIA! are usually taken for granted but EA! does play an important role in our later discussions. In the literature, it is shown that the following systems are sound and complete w.r.t. the standard semantics of PAL: PA + !COM (cf. e.g., [5]), PA + RE (cf. [10]). The main technical contributions of this paper can be summarized as follows: – WDIST!, DIA!, RE¬, RE∧, and RE2 can be derived in PA. – None of !COM, NEC!, RE!, RE, DIST!, SDIST!, EA!, and !K can be derived in PA. – PA, PA + !K + EA! + DIST!, PA + NEC! are not complete w.r.t. the standard PAL semantics while PA + DIST! + NEC! is complete. – To prove the above results, we introduce two non-standard semantics of PAL such that both validate PA. Moreover, we show PA plus a variation of the composition axiom [ψ][χ]φ ↔ [ψ ∧ χ]φ)(!COM∧) is sound and complete w.r.t one of the non-standard semantics. Under this semantics PAL is still equivalent to EL qua expressive power. The points we would like to make are as follows: – Axiomatizating PAL and other dynamic epistemic logics (DEL) is more subtle than it may look, which invites careful investigations. – There are two general ways to conduct the reductions from DEL to the base logic EL: ‘inside-out’ (by using RE) and ‘outside-in’ (by using composition axioms). Composition axioms give explicit information about the dynamics of the system. Various forms of composition axioms can be used in axiomatizing DEL under non-standard semantics when RE rule is not be valid. – The development of DEL is so far in principal semantics-driven, however, the syntactic perspective may bring new insights. Other useful variations of DEL can be designed for good reasons e.g., for modelling richer phenomenons regarding interactions between updates. The paper is organized as follows: In Section 2, we review some known results about the axiomatizations of PA and make some useful observations linking many axioms and rules. In Section 3, we show that !COM, NEC!, RE!, RE, DIST!, EA!, !K are not derivable from PA by giving two non-standard semantics which validate PA but invalidate (all or some of) the above mentioned axioms and rules, thus showing that PA is not complete w.r.t. the standard PAL semantics.

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Moreover, we prove that by adding the right composition axiom we can obtain a complete system w.r.t. the ﬁrst non-standard semantics which intuitively models a ‘slow’ agent. Finally we conclude with discussions and future work in Section 4.

2

Preliminaries

Proposition 1. PA is sound w.r.t to the standard PAL semantics. Proof: Cf. e.g., [5].

Moreover, it is an easy exercise to show that all the other axioms and rules mentioned in the introduction are valid w.r.t. to the standard PAL semantics: Proposition 2. Axiom schemas DIST!, !COM, WDIST!, SDIST!, !K , EA!, DIA!, and inference rules NEC!, RE, RE¬, RE∧, RE2, RE! are all valid w.r.t. the standard PAL semantics. A natural question to ask is: are they derivable in PA? We list a few positive answers here. Proposition 3. RE¬, RE∧, RE2 can be derived in PA. Proof: RE¬, RE∧ are trivial by using TAUT. Here we only show the (standard) reasoning behind RE2. 1 PA 2 PA 3 PA 4 PA 5 PA 6 PA

φ↔χ φ→χ TAUT 2i (φ → χ) NECK 2i (φ → χ) → (2i φ → 2i χ) DISTK 2i φ → 2i χ MP(3, 4) 2i φ ↔ 2i χ repeat 2-5 for χ → φ, TAUT

Note that the above proof uses NECK and DISTK, however, as we will see in Section 3 the [φ] versions of them (DIST! and NEC!) can not be derived in PA. Based on the above proposition, we know that the following restricted version of RE holds. Proposition 4. The following rule REr is valid: Given PA φ ↔ χ, we have PA ψ ↔ ψ where ψ is obtained by replacing some occurrences of φ in ψ with χ, provided that these occurrences of φ are “good”, i.e., they do not appear in the scope of any announcement operator. Proof: Suppose PA φ ↔ χ and ψ is obtained from ψ by replacing some “good” occurrences of φ by χ. It is not hard to see that we can construct ψ and ψ from φ, χ and other formulas by using the equivalence preserving operations: RE¬,RE∧,RE2. For example, from PA φ ↔ χ we can show that PA ([ψ]φ → 2i φ) ↔ ([ψ]φ → 2i χ) by taking [ψ]φ as one of the atomic build blocks.

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Proposition 5. DIA! is a theorem schema of PA. Proof:

1 PA 2 PA 3 PA 4 PA

[ψ]¬φ ↔ (ψ → ¬[ψ]φ) !NEG ¬[ψ]¬φ ↔ ¬(ψ → ¬[ψ]φ) RE¬(1) ¬(ψ → ¬[ψ]φ) ↔ (ψ ∧ [ψ]φ) TAUT ¬[ψ]¬φ ↔ (ψ ∧ [ψ]φ) REr (2, 3)

Proposition 6. WDIST! is a theorem schema of PA. Proof: Note that φ → χ is the abbreviation of ¬(φ ∧ ¬χ). Thus [ψ](φ → χ) is the abbreviation of [ψ]¬(φ ∧ ¬χ). 1 PA 2 PA 3 PA 4 PA 5 PA 6 PA 7 PA 8 PA 9 PA 10 PA

[ψ]¬(φ ∧ ¬χ) ↔ (ψ → ¬[ψ](φ ∧ ¬χ)) !NEG [ψ](φ ∧ ¬χ) ↔ ([ψ]φ ∧ [ψ]¬χ) !CON ¬[ψ](φ ∧ ¬χ) ↔ ¬([ψ]φ ∧ [ψ]¬χ) RE¬(2) [ψ]¬χ ↔ (ψ → ¬[ψ]χ) !NEG ¬[ψ](φ ∧ ¬χ) ↔ ¬([ψ]φ ∧ (ψ → ¬[ψ]χ) REr (3, 4) [ψ](φ → χ) ↔ (ψ → ¬([ψ]φ ∧ (ψ → ¬[ψ]χ)) REr (5, 1) [ψ](φ → χ) ↔ (ψ → ([ψ]φ → (ψ ∧ [ψ]χ))) TAUT [ψ](φ → χ) ↔ ((ψ ∧ [ψ]φ) → (ψ ∧ [ψ]χ)) TAUT [ψ](φ → χ) ↔ ((ψ ∧ [ψ]φ) → [ψ]χ) TAUT [ψ](φ → χ) ↔ ([ψ]φ → (ψ → [ψ]χ)) TAUT

Note that PA [ψ]χ → (ψ → [ψ]χ), thus if PA EA! then PA [ψ]χ ↔ (ψ → [ψ]χ). Based on this observation and REr , it is clear that if PA EA! then SDIST! (and DIST!) can be proved in PA. However, PA EA! as we will see in Section 3. If we extend PA with EA! and NEC!, then RE is derivable. Proposition 7. RE! and RE are derivable in PA+EA!+NEC! and PA+DIST!+NEC!. Proof: PA+EA! DIST!, together with NEC! we can derive RE! (cf. the proof of Proposition 3). Then based on the proof of Proposition 4, RE can be derived. Proposition 8. EA! is a theorem schema of PA+!COM. Proof: By induction on the structure of φ (cf. [5, pp.251]).

By using the reduction axioms in PA and the above restricted substitution rule we can translate most of PAL formulas to equivalent EL formulas by iteratively replacing the inner part of the formula with an equivalent announcement-free formula. However formulas in the shape of [ψ][χ]φ may be problematic since RE! is missing in PA. Here we mention a few completeness results by using such reductions.

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Theorem 1. PA+!COM is sound and (weakly) complete w.r.t. the standard semantics of PAL. Proof: We only sketch the proof in [5]4 . We ﬁrst deﬁne a translation t : PAL → EL: t() = t([ψ]) = t(ψ → ) t(p) =p t([ψ]p) = t(ψ → p) t(¬φ) = ¬t(φ) t([ψ]¬φ) = t(ψ → ¬[ψ]φ) t(φ1 ∧ φ2 ) = t(φ1 ) ∧ t(φ2 ) t([ψ](φ1 ∧ φ2 )) = t([ψ]φ1 ∧ [ψ]φ2 ) t(2i φ) = 2i t(φ) t([ψ]2i φ) = t(ψ → 2i (ψ → [ψ]φ)) t([ψ][χ]φ) = t([ψ ∧ [ψ]χ]φ) Based on a suitable deﬁnition of the complexity of formulas (cf. [5]) we can show that the translation/rewriting always reduce the complexity thus it will terminate at some point. Note that in the process of the rewriting, t(ψ) never falls in the scope of any announcement operator. Based on this observation, by induction on the complexity of the formulas we can show that PA+!COM φ ↔ t(φ) (using reduction axioms and RE∧, RE¬, and RE2). By soundness of PA+!COM, φ ↔ t(φ). Now suppose φ then t(φ). Thus by the completeness of the basic modal logic K, K t(φ). Therefore PA+!COM t(φ). Since PA+!COM φ ↔ t(φ), we have PA+!COM φ by MP. Theorem 2 ([10]). PA+RE is sound and weakly complete w.r.t. the standard semantics of PAL. Proof: Similar to the above proof, we only need to revise the last item of the translation function t as follows: t([ψ][χ]φ) = t([ψ]t([χ]φ)) Note that now we do need the full power of RE since t does fall in the scope of announcement operators. As a straight forward corollary, we have: Corollary 1. PA+DIST!+NEC! and PA+EA!+NEC! are sound and complete w.r.t. the standard semantics of PAL. Proof: From Proposition 7 and the above theorem.

Note that the translation of [ψ][χ]φ formulas deﬁned in Theorem 2 is in the fashion of ‘inside-out’ while the translation in Theorem 1 is ‘outside-in’.

3

PA is Not Complete

In this section we give two alternative semantics for the language of PAL which validate PA but make many intuitive axioms and rules invalid. 4

We need to adapt the proof just a little bit to ﬁt !K in the proof instead of !K used in [5].

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A Context-Dependent Semantics

Inspired by the semantics developed in [6,11,3], we deﬁne the satisfaction relation w.r.t. a context ρ (notation: ρ ), which is used to record the information from previous announcements. Given a Kripke model over P, I: M = (S, {→i | i ∈ I}, V ), the truth value of a PAL formula φ at a state s in M is recursively deﬁned as based on ρ where ρ is a formula in the language of PAL: M, s φ ⇔ M, s φ M, s ρ ⇔ always M, s ρ p ⇔ p ∈ V (s) M, s ρ ¬φ ⇔ M, s ρ φ M, s ρ φ ∧ ψ ⇔ M, s ρ φ and M, s ρ ψ M, s ρ 2i φ ⇔ ∀t i s : M, t ρ implies M, t ρ φ M, s ρ [ψ]φ ⇔ M, s ψ implies M, s ρ∧ψ φ Note that instead of updating the model we somehow remember the announcements and recall them as the context only in evaluating 2i formulas. Remembering the context is an alternative way of doing model relativization. We say that φ is valid w.r.t. this non-standard semantics if φ (i.e. φ). Example 1. Consider the following (S5) model M with two worlds s, v: i

s:po

i

i

/ v : ¬p

M, s ¬2i p ⇐⇒ M, s 2i p ⇐⇒ (∃t i s : M, t and M, t p) i Since p ∈ V (v) and s → v, M, s ¬2i p. M, s p 2i p ⇐⇒ (∀t i s : M, t p implies M, t p p). Clearly, M, s p 2i p. Similarly M, s ∧p 2i p. M, s [p]2i p ⇐⇒ (M, s p implies M, s ∧p 2i p) ⇐⇒ M, s ∧p 2i p. Thus, M, s [p]2i p (based on the above example). M, s [p ∧ ¬2i p]2i p ⇐⇒ (M, s p ∧ ¬2i p implies M, s ∧p∧¬2i p 2i p) ⇐⇒ (M, s p and M, s ¬2i p) implies M, s ∧p∧¬2i p 2i p ⇐⇒ M, s ∧p∧¬2i p 2i p (from the above examples) ⇐⇒ ∀t i s : M, t ∧ p ∧ ¬2i p implies M, t ∧p∧¬2i p p. It is easy to see that M, s [p ∧ ¬2i p]2i p. M, s [p][¬2i p]⊥ ⇐⇒ (M, s p implies M, s ∧p [¬2i p]⊥) ⇐⇒ M, s ¬2i p implies M, s ∧p∧¬2i p ⊥. Thus M, s [p][¬2i p]⊥. On the other hand, it is easy to verify that M, s [p][¬2i p]⊥ (recall that denotes the standard semantics). ♣ In the example, it seems that coincides with except for the formulas with consecutive announcements. We will show that it is not a coincidence.

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Proposition 9. coincides with on EL formulas. Proof: Note that without [ψ] operators, ρ can never be changed to any nontrivial formula during the evaluation of a formula. Since M, s is always true, it is easy to see that the deﬁnition of coincides with for EL formulas. Before going further we ﬁrst prove two useful propositions. Let !COM∧ be the axiom schema [ψ][χ]φ ↔ [ψ ∧ χ]φ which diﬀers from !COM. Proposition 10. !COM∧ is valid w.r.t. . Proof: M, s [ψ][χ]φ ⇐⇒ M, s ψ implies M, s ∧ψ [χ]φ ⇐⇒ M, s ψ implies (M, s χ implies M, s ∧ψ∧χ φ) ⇐⇒ (M, s ψ and M, s χ) implies M, s ∧ψ∧χ φ ⇐⇒ (M, s ψ ∧ χ) implies M, s ∧ψ∧χ φ ⇐⇒ M, s [ψ ∧ χ]φ

Proposition 11. For any PAL formulas χ, ψ, and φ : if χ ↔ ψ then for all pointed model M, s: M, s χ φ ⇐⇒ M, s ψ φ. As a consequence, the following rule is valid: φ↔χ !RE : [φ]ψ ↔ [χ]ψ Proof: By induction on the structure of φ. The Boolean cases are trivial. Now let φ = 2i φ . Note that M, s ρ 2i φ ⇐⇒ ∀t i s : M, t ρ implies M, t ρ φ . Since χ ↔ ψ, for all M, t: M, t χ ⇐⇒ M, t ψ. Therefore, based on the induction hypothesis that M, t χ φ ⇐⇒ M, t ψ φ , M, s χ 2i φ ⇐⇒ M, s ψ 2i φ. Now consider φ = [φ ]φ . According to the semantics of conjunctions, it is not hard to see that if ψ ↔ χ then for any φ we have (ψ ∧ φ ) ↔ (χ ∧ φ ). Now according to the truth condition of [φ ]φ and induction hypothesis, M, s χ [φ ]φ ⇐⇒ M, s ψ [φ ]φ . Based on the these observations, it is easy to show that ψ ↔ χ implies [χ]φ ↔ [ψ]φ. Remark 1. The admissible rule !RE is itself interesting in axiomatizing PAL. We conjecture that it is not derivable from PA but leave it for future work. In the following we show that PA is sound w.r.t. . Actually many other rules and axiom schemas are also valid under as we will see soon. Lemma 1. TAUT, MP, NECK, and DISTK are valid w.r.t. . Proof: For TAUT and MP: Trivial (check the truth conditions for Boolean cases). For NECK: Suppose φ then for all models M, s φ. Suppose towards a contradiction that there is a model M, s ¬2i φ. According to the semantics there exists t i s M, t and M, t ∧ ¬φ, contradiction. For DISTK: Suppose M, s 2i (φ → ψ) then for all t i s M, t φ → ψ. Now suppose M, s 2i φ then for all t i s: M, t implies M, t φ. It is clear that for all t i s: M, t implies M, t ψ. Thus M, s 2i ψ. Therefore M, s 2i (φ → ψ) → (2i φ → 2i ψ).

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Lemma 2. !ATOM, !NEG, !CON, !K, !K , and EA! are valid w.r.t. . Proof: For !ATOM: M, s [ψ]p ⇐⇒ (M, s ψ implies M, s ∧ψ p) ⇐⇒ (M, s ψ implies M, s p) ⇐⇒ M, s ψ → p. For !NEG: M, s [ψ]¬φ ⇐⇒ (M, s ψ implies M, s ∧ψ ¬φ) while M, s ψ → ¬[ψ]φ ⇐⇒ (M, s ψ implies M, s ¬[ψ]φ) ⇐⇒ M, s ψ implies (M, s ψ and M, s ∧ψ ¬φ) ⇐⇒ M, s ψ implies M, s ∧ψ ¬φ. Thus M, s [ψ]¬φ ↔ (ψ → ¬[ψ]φ). For !CON: M, s [ψ](φ ∧ χ) ⇐⇒ (M, s ψ implies M, s ∧ψ φ ∧ χ) ⇐⇒ (M, s ψ implies M, s ∧ψ φ) and (M, s ψ implies M, s ∧ψ χ) ⇐⇒ M, s [ψ]φ ∧ [ψ]χ. For !K: M, s [ψ]2i φ ⇐⇒ M, s ψ implies M, s ∧ψ 2i φ while M, s ψ → 2i (ψ → [ψ]φ) ⇐⇒ M, s ψ implies M, s 2i (ψ → [ψ]φ) ⇐⇒ M, s ψ implies (∀t s : M, t implies (M, t ψ implies M, t [ψ]φ)) ⇐⇒ M, s ψ implies (∀t s : M, t ψ implies M, t [ψ]φ) ⇐⇒ M, s ψ implies (∀t s : M, t ψ implies (M, t ψ implies M, t ∧ψ φ)) ⇐⇒ M, s ψ implies (∀t s : M, t ψ implies M, t ∧ψ φ) ⇐⇒ M, s ψ implies (∀t s : M, t ∧ ψ implies M, t ∧ψ φ) ⇐⇒ M, s ψ implies M, s ∧ψ 2i φ

Thus M, s [ψ]2i φ ↔ (ψ → 2i (ψ → [ψ]φ)) Similarly, we can verify that !K is valid w.r.t. . For EA!: immediate from the implication form of the truth condition of [ψ]φ.

Based on the lemmata 2 and 1, we can prove the soundness of PA (and some of its extensions) w.r.t. . Theorem 3. For all PAL formulas φ: PA+EA!+!K φ implies φ. Now we prove that many axioms and rules we mentioned in the introduction are not derivable from PA, by showing that they are not valid w.r.t . Lemma 3. None of !COM,NEC!,RE!,RE is valid under . Proof: For !COM: We consider [p][2i p]⊥ and [p∧[p]2i p]⊥. From Proposition 10, [p][2i p]⊥ ↔ [p ∧ 2i p]⊥. Note that [p]2i p is valid w.r.t. thus [p]2i p ↔ . From Proposition 11, [p ∧ [p]2i p]⊥ ↔ [p ∧ ]⊥ ↔ [p]⊥. However, [p ∧ 2i p]⊥ ↔ [p]⊥ is not valid e.g., on the following (S5) model: i

s:po

i

i

/ v : ¬p

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For NEC!: It is not hard to verify that [¬2i p ∨ ¬p](¬2i p ∨ ¬p) is valid. From Proposition 10, ([p][¬2i p ∨ ¬p](¬2i p ∨ ¬p)) ↔ ([p ∧ (¬2i p ∨ ¬p)](¬2i p ∨ ¬p)). From Proposition 11, ([p][¬2i p ∨ p](¬2i p ∨ ¬p)) ↔ ([p ∧ ¬2i p](¬2i p ∨ ¬p)). However, [p ∧ ¬2i p](¬2i p ∨ ¬p) is clearly not valid in the above (S5) model. For RE! and RE: From the proof of the above case of NEC!, we have a valid equivalence: ([¬2i p ∨ ¬p](¬2i p ∨ ¬p)) ↔ . However, although [p] is still valid, [p][¬2i p ∨ ¬p](¬2i p ∨ ¬p) is not valid, as we have shown. Therefore RE! is not valid w.r.t. , thus RE is not valid too. From Lemma 3, Theorem 3 we have: Theorem 4. None of !COM, NEC!, RE, RE! can be derived from PA + EA! + !K . Proof: From Theorem 3, for all φ: φ implies PA+EA!+!K φ. Moreover, since the rules in PA + EA! + !K preserve validity, we can show that if a rule is not valid w.r.t. , then it is not derivable in PA + EA! + !K . However, Lemma 3 says none of !COM, NEC!, RE, RE! is valid w.r.t. . Since DIST! is derivable from PA + EA! then the following corollary is immediate: Corollary 2. PA + !K + EA! + DIST! and its subsystems are not complete w.r.t. . We know that PA + !COM is sound and complete w.r.t. standard semantics. Now we give a complete axiomatization of PAL under . Recall that !COM∧ is the axiom schema [ψ][χ]φ ↔ [ψ ∧ χ]φ. We can show the completeness of PA+!COM∧ w.r.t. our new semantics. Theorem 5. PA + !COM∧ is sound and weakly complete w.r.t. . Proof: Soundness follows from Theorem 3 and Proposition 10. For completeness, clearly we can use the reduction axioms in PA+!COM∧ to translate a PAL formula in to an equivalent EL formula w.r.t. (cf. the proof of Theorem 1). From proposition 9 and the completeness of K w.r.t. , the desired completeness can be obtained. Despite the technical motivation behind PA + !COM∧, it also stipulates a particular kind of updates which may be reasonable in modelling real agents. What !COM∧ says is that the agents are not ‘instant updaters’ in the sense that they postpone the update until they hear all the consecutive announcements and collect them all together as a conjunction. Here are two realistic scenarios which may exemplify this rationale: 1. two announcements are made right after each other, and in the ﬂash of time between the two agent may not manage to update their information according to the ﬁrst announcement. Therefore they may take the two announcements as a conjunction; 2. Agents may intentionally postpone the updates according to the announcements: it makes sense if we are considering announcements from diﬀerent (reliable/unreliable) sources which may contradict each other.

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!COM∧ stipulates a special case of announcement composition diﬀerent from the standard one. We may well assume that agents have limited memory in remembering the previous announcements and employ diﬀerent forgetting mechanisms and so on. By stipulating diﬀerent composition axioms and designing non-standard semantics accordingly, we can model diﬀerent types of agents/ updates. A systematic study of such non-standard PAL is left for future work. 3.2

Another Non-standard Semantics

The rest of this paper is devoted to the axioms DIST!, SDIST!, !K , EA! and rule NEC!. First note that EA! is valid w.r.t. the above semantics thus DIST!, SDIST!, !K are also valid (by soundness of PA + EA!). We do not know yet whether these axioms are derivable from PA, and moreover it is unclear whether PA + NEC! is complete. To show that DIST!, SDIST!, !K , and EA! are not derivable in PA+NEC!, we now deﬁne another semantics () which diﬀers from in the clause of [ψ]φ. In the sequel, we say that a formulaφ is special if, modulo associativity and commutativity of ∧, φ = 1≤i≤n φi ∧ 1≤j≤m φj where n ≥ 1, m ≥ 0, and φi are in the shape of [χ]χ but none of φj is in such a shape. If φ is special then we write φ = φ[] ∧ φ−[] where φ[] and φ−[] are the corresponding conjunctions of announcement formulas and non-announcement formulas respectively. Given a Kripke model over P, I: M = (S, {→i | i ∈ I}, V ), the new truth conditions are as follows: M, s φ[] if M, s ψ and φ is special M, s [ψ]φ ⇔ M, s ψ implies M|ψ , s φ otherwise Intuitively, the new semantics for [ψ]φ depends on the form of φ thus RE! is expected to be not valid under this semantics. In the case that ψ is false and φ involves announcement formulas, we simply skip the false announcement of ψ (an agent does not go mad if hearing a false announcement followed by other announcements: they can just skip the ﬁrst one.) According to this semantics, we can show that DIST!, SDIST!, !K and EA! are not valid. Lemma 4. EA!, DIST!, SDIST!, and !K are not valid w.r.t. . Proof: For EA!: Consider (p → [p][q]¬q) → [p][q]¬q and the following (S5) model M: s : ¬p, q It is clear that M, s p → [p][q]¬q. However, M, s [p][q]¬q ⇐⇒ M, s [q]¬q ⇐⇒ (M, s q implies M|q , s ¬q). Thus M, s [p][q]¬q. For DIST! and SDIST!: Consider the above model again, it is easy to verify that [p](p → [q]¬q) → ([p]p → [p][q]¬q) is not valid. For !K : consider [p]2i [q]⊥ ↔ (p → 2i [p][q]⊥) and the following (S5) model: i

s : p, ¬q o

i

i

/ t : ¬p, q

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M, s [p]2i [q]⊥ ⇐⇒ M, s p implies (M|p , s 2i [q]⊥), and M, s p → 2i [p][q]⊥ ⇐⇒ (M, s p implies M, s 2i [p][q]⊥). Note that M, t [p][q]⊥ ⇐⇒ M, t [q]⊥ ⇐⇒ M|q , t ⊥. Therefore M, s 2i [p][q]⊥. Thus M, s [p]2i [q]⊥ but M, s p → 2i [p][q]⊥. Now we prove that PA is sound w.r.t. this semantics. Compared to , since we do not change the semantics for Boolean formulas and 2i φ formulas, the proof of Lemma 1 also works here w.r.t. : Lemma 5. TAUT, MP, NECK and DISTK are valid w.r.t. . Lemma 6. !ATOM, !NEG, !CON, and !K are valid w.r.t. . Proof: The case for !ATOM is trivial. !CON is a tricky one and we will see how the complicated case-divided semantics of [ψ]φ pays back. For !CON: First note that φ ∧ χ is not special iﬀ φ and ψ are both not special. Now we consider two cases: – If φ∧χ is not special, then neither φ nor ψ is special. M, s [ψ](φ∧χ) ⇐⇒ (M, s ψ implies M|ψ , s φ ∧ χ) ⇐⇒ M, s ψ implies (M|ψ , s φ and M|ψ , s χ) ⇐⇒ (M, s ψ implies M|ψ , s φ) and (M, s ψ implies M|ψ , s χ) ⇐⇒ (M, s [ψ]φ and M, s [ψ]χ) – If φ ∧ χ is special then at least one of φ and χ is special. Suppose w.l.o.g. that χ is not special and φ is special thus φ = φ[] ∧ φ−[] . Here are again two cases to be considered: • suppose M, s ψ then the new semantics coincides with the standard one thus M, s [ψ](φ ∧ χ) ↔ [ψ]φ ∧ [ψ]χ). • suppose M, s ψ then M, s [ψ](φ ∧ χ) ⇐⇒ M, s φ[] ⇐⇒ M, s φ[] and (M, s ψ =⇒ M|ψ , s χ) ⇐⇒ M, s [ψ]φ and M, s [ψ]χ The proofs for !NEG and !K are almost as before under the standard semantics . We only need to handle the extra special cases. Now suppose φ is special. Clearly ¬φ and 2i φ are not special. For !NEG: We only need to consider the case when M, s ψ since otherwise the proof for the standard semantics suﬃces. Then it is clear that M, s ψ → ¬[ψ]φ and M, s [ψ]¬φ since ¬φ is not special. Thus M, s [ψ]¬φ ↔ (ψ → ¬[ψ]φ). For !K: It is clear that if M, s ψ then M, s [ψ]2i φ ↔ (ψ → (2i (ψ → [ψ]φ))). However, it does not suﬃce since even ψ is true at M, s it is still possible that ψ → (2i (ψ → [ψ]φ)) diﬀers from the standard semantics due to the appearance of [ψ]φ in the scope of 2i . Now suppose M, s ψ. M, s [ψ]2i φ ⇐⇒ (M, s ψ implies M|ψ , s 2i φ) ⇐⇒ M|ψ , s 2i φ. On the other hand, M, s ψ → 2i (ψ → [ψ]φ) ⇐⇒ (M, s ψ implies M, s 2i (ψ → [ψ]φ)) ⇐⇒ M, s 2i (ψ → [ψ]φ) ⇐⇒ (∀t s : M, t ψ implies (M, t [ψ]φ). Note that the new semantics only diﬀers from the standard one if ψ is false. Thus (∀t s : M, t ψ implies (M, t [ψ]φ) ⇐⇒

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(∀t s : M, t ψ implies (M, t ψ implies M|ψ φ) ⇐⇒ (∀t s : M, t ψ implies M|ψ , t φ) ⇐⇒ (∀t s : t exists in M|ψ implies M|ψ , t φ) ⇐⇒ M|ψ , s 2i φ. Moreover, we can show that NEC! is valid w.r.t. . Lemma 7. NEC! is valid under . Proof: Suppose φ. Now consider [ψ]φ. There are two cases: – φ is not special: Trivial. – φ is special: It is in the shape of φ[] ∧ φ−[] . To verify M, s [ψ]φ there are again two cases. Suppose M, s ψ, then M, s [ψ]φ ⇐⇒ M|ψ , s φ which is true since φ. Now suppose M, s ψ, then M, s [ψ]φ ⇐⇒ M, s φ[] . Since φ, φ[] ∧ φ−[] thus φ[] . Therefore, M, s φ[] . This concludes the proof. Lemmata 5, 6, and 7 showed that PA + NEC! is sound w.r.t. . Together with 4 we have: Theorem 6. None of DIST!, SDIST! !K , EA! can be derived from PA + NEC!. As an immediate corollary: Corollary 3. PA + NEC! is not complete w.r.t. standard semantics .

4

Discussion and Future Work

We have shown that PA and many natural extensions of it are not complete w.r.t. the standard semantics on arbitrary Kripke models. It is also natural to ask whether PA plus the usual S5 axioms: T (φ → 3φ), 4:(2φ → 22φ), B: (φ → 23φ) is complete under standard semantics on S5 models. We conjecture that all of our incompleteness results still hold if we replace PA by PA+T+4+B. For now, to be more conﬁdent about this conjecture, observe that we only use S5 counter models to show the invalidity of axioms and rules. Similar discussions can be carried out in the context of PAL with common knowledge and dynamic epistemic logic with action models (cf. e.g., [1,2]). We hope that this paper has demonstrated the subtleness of the axiomatizations of dynamic epistemic logics and the importance and the use of the composition axiom in various forms. As we mentioned, the composition axioms are not just technical patches to make the system complete, actually they are essential components stating how the updates are interacting with each other, thus deserving independent study. To accompany diﬀerent composition axioms we may design diﬀerent nonstandard semantics and keep the expressive power the same. One possible application is to use reasonable composition axioms (e.g., !COM∧) to control the

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iteration of updates which is responsible for the undecidability in the standard setting (cf. [9]). Diﬀerent composition axioms may also be useful in the setting of iterated belief revision (cf. e.g., [4]). The syntactic driven approaches to dynamic epistemic logic will bring new insights and raise new questions to this ﬂourishing ﬁeld of research. Acknowledgement. This work is supported by SSFC grant 11CZX054. The author would like to thank Meiyun Guo, Fenrong Liu, Hans van Ditmarsch, Johan van Benthem and two anonymous reviewers of LORI-3 for their useful comments on earlier versions of this paper.

References 1. Baltag, A., Moss, L.: Logics for epistemic programs. Synthese 139(2), 165–224 (2004) 2. van Benthem, J., van Eijck, J., Kooi, B.: Logics of communication and change. Information and Computation 204(11), 1620–1662 (2006) ´ e, P.: Inexact knowledge with introspection. Journal of Philosoph3. Bonnay, D., Egr´ ical Logic 38(2), 179–227 (2009) 4. Delgrande, J., Jin, Y.: Parallel belief revision. In: Proceedings of AAAI 2008, pp. 430–435 (2008) 5. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic, 1st edn. Synthese Library. Springer, Heidelberg (2007) 6. Gabbay, D.M.: A theory of hypermodal logics: Mode shifting in modal logic. Journal of Philosophical Logic 31(3), 211–243 (2002) 7. Gerbrandy, J., Groeneveld, W.: Reasoning about information change. Journal of Logic, Language and Information 6(2), 147–169 (1997) 8. Holliday, W., Hoshi, T., Icard III, T.F.: Schematic Validity in Dynamic Epistemic Logic: Decidability. In: van Ditmarsch, H., Lang, J., Ju, S. (eds.) LORI 2011. LNCS(LNAI), vol. 6953, pp. 87–96. Springer, Heidelberg (2011) 9. Miller, J., Moss, L.: The undecidability of iterated modal relativization. Studia Logica 79(3), 373–407 (2005) 10. Plaza, J.A.: Logics of public communications. In: Emrich, M.L., Pfeifer, M.S., Hadzikadic, M., Ras, Z.W. (eds.) Proceedings of the 4th International Symposium on Methodologies for Intelligent Systems, pp. 201–216 (1989) 11. Wang, Y.: Indexed Semantics and Its Application in Modelling Interactive Unawareness. Master’s thesis, University of Amsterdam (2006)

Public Announcement Logic with Distributed Knowledge Y`ı N. W´ ang1 and Thomas ˚ Agotnes1,2 1

2

Department of Computer Engineering, Bergen University College, Norway [email protected] Department of Information Science and Media Studies, University of Bergen, Norway [email protected]

Abstract. While dynamic epistemic logics with common knowledge have been extensively studied, dynamic epistemic logics with distributed knowledge have so far received far less attention. In this paper we study extensions of public announcement logic (PAL) with distributed knowledge, in particular their expressivity and axiomatisations. PAL extended only with distributed knowledge is not more expressive than standard epistemic logic with distributed knowledge. Our focus is therefore on PACD, the result of adding both common and distributed knowledge to PAL, which is more expressive than each of its component logics. Our main result is a completeness result for PACD. The axiomatisation is not surprising: it is the combination of well-known axioms. The completeness proof, however, is not trivial, and requires novel combinations and extensions of techniques for dealing with S5 knowledge, distributed knowledge, common knowledge and public announcements at the same time.

1

Introduction

Dynamic epistemic logics such as public announcement logic [12] extend standard epistemic logic with dynamic operators. Dynamic epistemic logics with common knowledge have been extensively studied, one reason being the intuitively close relationship between events like public announcements and common knowledge. Dynamic epistemic logics with distributed knowledge on the other hand have so far received far less attention. In standard multi-agent epistemic logic [5,11], a fact is distributed knowledge among a group of agents if, informally speaking, it follows from the combined knowledge of the agents in the group. Although distributed knowledge is sometimes mentioned in dynamic epistemic logic settings, we are not aware of any existing meta-logical results, like completeness or characterisations of expressive power, for dynamic epistemic logics with distributed knowledge. In this paper we study variants of public announcement logic [12] with distributed knowledge, assuming the S5 properties of knowledge. In particular, we H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 328–341, 2011. c Springer-Verlag Berlin Heidelberg 2011

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study their expressive power and complete axiomatisations. It is easy to see that public announcement logic (PAL) extended with only distributed knowledge (PAD) is not more expressive than standard epistemic logic with distributed knowledge, and an axiomatisation is obtained by adding a reduction axiom. Public announcement logic with both common and distributed knowledge (PACD) does not have a similar property; it is strictly more expressive than standard epistemic logic with common and distributed knowledge, as well as both PAC and PAD, and therefore more interesting and the main focus of this paper. A main technical result is a completeness result for PACD. Completeness and expressivity results and tools have been developed for different variants of public announcement logic (without distributed knowledge) including PAC; in particular see [4]. When it comes to distributed knowledge, axiomatisations of diﬀerent variants of standard epistemic logic have been studied by several authors [6,9,8,10]. When it comes to expressivity, it is well known that distributed knowledge is not invariant under standard bisimulation. Roelofsen [13] develops a notion of collective bisimulation for distributed knowledge, as well as model comparison games which completely characterises the expressive power of epistemic logic with distributed knowledge. Given these existing results there are few surprises when we combine the two types of logics in this paper. For example, PACD is completely axiomatised by extending an axiomatisation of PAC with standard axioms for distributed knowledge. But even if this result is unsurprising, it is valuable to have it formally stated and proved, even more so since its proof is not straightforward. Although we make extensive use of existing techniques, their combinations and extensions is not trivial. In particular, the completeness proof for PACD requires combinations and extensions of techniques for dealing with S5 knowledge, distributed knowledge, common knowledge and public announcements at the same time. In the next section we give a brief overview of epistemic and public announcement logics. We study the expressivity of PAD and PACD in Section 3, and their complete axiomatisations in Section 4. We conclude in Section 5.

2

Background

Standard multi-agent epistemic logic [5,11] is a propositional modal logic with a modality Ka for each agent a, where Ka ϕ is intended to mean that “agent a knows ϕ”. Knowledge modalities for groups are often also considered: CA ϕ, DA ϕ and EA ϕ denotes “group A has common knowledge of ϕ”, “A has distributed knowledge of ϕ”, and “everybody in A knows ϕ”, respectively. Public announcement logic (PAL) [12] extends epistemic logic with a modality [ϕ] for each formula ϕ. The intended meaning of [ϕ]ψ is “if ϕ is true then after it is announced ψ becomes true”. Formally, take a non-empty set prop of propositions and a ﬁnite non-empty set ag of agents. Let gr be the set of all ﬁnite non-empty groups of agents.

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Definition 1 (Languages). The following grammar rules deﬁne languages for epistemic logics and public announcement logics, with p ∈ prop, a ∈ ag, A ∈ gr. | ϕ ∧ ϕ | Ka ϕ | ϕ ∧ ϕ | Ka ϕ | CA ϕ | ϕ ∧ ϕ | K a ϕ | DA ϕ | ϕ ∧ ϕ | Ka ϕ | CA ϕ | DA ϕ | ϕ ∧ ϕ | Ka ϕ | [ϕ]ϕ | ϕ ∧ ϕ | Ka ϕ | [ϕ]ϕ | CA ϕ | ϕ ∧ ϕ | Ka ϕ | [ϕ]ϕ | DA ϕ | ϕ ∧ ϕ | Ka ϕ | [ϕ]ϕ | CA ϕ | DA ϕ. We write EA ϕ as a shortcut for a∈A Ka ϕ. When A is a singleton set {a}, we simply write Ca ϕ and Da ϕ instead of C{a} ϕ and D{a} ϕ. By the unqualiﬁed word “formula”, we usually mean a PACD-formula. (S5) ϕ ::= p (S5C) ϕ ::= p (S5D) ϕ ::= p (S5CD) ϕ ::= p (PAL) ϕ ::= p (PAC) ϕ ::= p (PAD) ϕ ::= p (PACD) ϕ ::= p

| | | | | | | |

¬ϕ ¬ϕ ¬ϕ ¬ϕ ¬ϕ ¬ϕ ¬ϕ ¬ϕ

Epistemic models are used to model agents’ knowledge and ignorance. For technical reasons we also deﬁne two broader classes of structures, pseudo models and pre-models. Definition 2 (Epistemic models, pre-models and pseudo models). An epistemic model ( model for short) is a tuple M = (M, ∼ag , V ) where: – M is a non-empty set of states; – ∼ag maps every agent a to an equivalence relation, ∼a , on M ; – V : prop → ℘(M ) is an evaluation function.

Given a group A∈ gr, let ∼CA be the reﬂexive transitive closure of a∈A ∼a , and let ∼DA be a∈A ∼a . A pre-model is a tuple M = (M, ∼ag , ∼gr , V ) where ∼gr maps every A ∈ gr to an equivalence relation, ∼DA , on M , and the other components are as in a model. ∼CA is deﬁned as for a model, while ∼DA is a primitive notation. A pseudo model is a pre-model M = (M, ∼ag , ∼gr , V ) where: – For any a ∈ ag, ∼Da =∼a ; – For any A, B ∈ gr, A ⊆ B implies ∼DB ⊆∼DA . A pointed model (pointed pre-model, pointed pseudo model) is a tuple (M, m) consisted of a model (pre-model, pseudo model) M and a state m in M. Definition 3 (Satisfaction and validity). Let M = (M, ∼ag , ∼gr , V ) be a pre-model, and m a state in M . Satisfaction at (M, m) is deﬁned as follows: M, m |= p M, m |= ¬ϕ M, m |= ϕ ∧ ψ M, m |= Ka ϕ M, m |= CA ϕ M, m |= DA ϕ M, m |= [ϕ]ψ

⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒

m ∈ V (p) M, m |= ϕ M, m |= ϕ & M, m |= ψ (∀n ∈ M)(m ∼a n ⇒ M, n |= ϕ) (∀n ∈ M)(m ∼CA n ⇒ M, n |= ϕ) (∀n ∈ M)(m ∼DA n ⇒ M, n |= ϕ) M, m |= ϕ ⇒ M|ϕ, m |= ψ,

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where M|ϕ = (N, ≈ag , ≈gr , ν) is a pre-model such that N = {m ∈ M | M, m |= ϕ}, ν(p) = V (p) ∩ N for p ∈ prop,

≈a = ∼a ∩(N × N ) for a ∈ ag, ≈DA = ∼DA ∩(N × N ) for A ∈ gr.

Satisfaction in a pre-model M (denoted by M |= ϕ) is deﬁned as usual. Semantics with respect to a model is deﬁned similarly to the above, except that every ∼DA is not a primitive relation in that case. We use |= ϕ to denote validity with respect to models, i.e. M, m |= ϕ for any pointed model (M, m). Note that we are ultimately only interested in the notion of satisfaction in (“proper”) models, but the notion of satisfaction in pre- and pseudo models will be useful later. Satisfaction is well deﬁned for pseudo models by the following. Proposition 1. If M is a pseudo model, and ϕ is a formula such that M, m |= ϕ for some m ∈ M, then M|ϕ is also a pseudo model. Let L and L be two of the languages introduced above. L is at least as expressive as L iﬀ for every L -formula there is an L-formula equivalent to it. This is denoted as L L. L and L are equally expressive iﬀ L L and L L. This is denoted as L ≡ L . L is more expressive than L iﬀ L L but L ≡ L. This is denoted as L ≺ L. We also say that L is less expressive than L in this case. L and L are not comparable iﬀ neither L L nor L L. Definition 4 (Modal degree). The modal degree of an S5CD-formula is given by the function d : S5CD → N, with p ∈ prop, a ∈ ag and A ∈ gr: d(p) = 0 d(¬ϕ) = d(ϕ) d(ϕ ∧ ψ) = max(d(ϕ), d(ψ)) d(Ka ϕ) = d(CA ϕ) = d(DA ϕ) = d(ϕ) + 1. Definition 5 (Bisimulations). Given two models M = (M, ∼ag , V ) and N = (N, ≈ag , ν), a non-empty relation Z ⊆ M × N is called a bisimulation between M and N, if for all a ∈ ag, m ∈ M and n ∈ N with mZn, it holds that: ( at) for all p ∈ prop, m ∈ V (p) iﬀ n ∈ ν(p); (zig) for every m ∈ M , if m ∼a m , there is an n ∈ N such that n ≈a n and m Zn ; and (zag) for every n ∈ N , if n ≈a n , there is an m ∈ M such that m ∼a m and m Zn . We say pointed models (M, m) and (N, n) are bisimilar, if there is a bisimulation Z between M and N linking m and n. It is well known that satisfaction of S5, S5C, PAL, and PAC-formulae are invariant under bisimulation.

3 3.1

Expressivity PAD

PAL is not more expressive than S5 as shown already in [12]. This is most easily shown by reduction axioms that can be used to “get rid of” announcement

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operators in any formula without changing the meaning of the formula. In particular, the following is a valid relationship between knowledge and announcement: [ϕ]Ka ψ ↔ (ϕ → Ka [ϕ]ψ). It is easy to see that we have a similar relationship between distributed knowledge and announcement: Proposition 2. |= [ϕ]DA ψ ↔ (ϕ → DA [ϕ]ψ) From this it is easy to show that adding public announcements to S5D does not increase the expressive power of the language. Of course, adding distributed knowledge to PAL increases the expressive power, like it does for S5. Proposition 3. PAD ≡ S5D and PAL ≺ PAD. Theorem 1. PAD and S5C are not comparable. Proof. We make use of the fact that PAD ≡ S5D. The following two models: M:

l ¬p

b

mp

a

n ¬p

N:

mp

ab

n ¬p

can be used to show that there is a PAD-formula, Dab p, that discerns between (M, m) and (N, m) but is not equivalent to any S5C-formula, since the two pointed models are bisimilar, leading to S5C-equivalence. For the other direction, we can show the S5C-formula, Cab ¬p, discerns between ﬁnite and inﬁnite spine models [4, p.226], and we can use model comparison games for S5D [13, Deﬁnition 17] together with [13, Proposition 18] to show that no S5D-formula can discern between the spine models. 3.2

PACD

The situation for PACD is diﬀerent: no (suﬃcient) reduction axioms exist, as we now formally show. Since PACD includes both PAC and S5CD, the interesting question (albeit with a rather obvious answer) is whether or not PAC and S5CD are comparable: if they are not, PACD must be strictly more expressive than both. Of course it should come as no surprise that they are not comparable, but we now show this formally. The following is a straightforward extension of Roelofsen’s model comparison games for S5D [13, Deﬁnition 17] to S5CD. Definition 6 (S5CDP games). Fix a set P of propositions. For any two pointed models (M, m) = (M, ∼ag , V, m) and (N, n) = (N, ≈ag , ν, n), the r-round S5CDP game between spoiler and duplicator on (M, m) and (N, n) is the following. If r = 0, spoiler wins iﬀ V (p) = ν(p) for some p ∈ P. If r = 0, spoiler can initiate one of the following scenarios: – C-forth Spoiler chooses a group A and a state m ∈ M such that m ∼CA m . Duplicator responds by choosing a state n ∈ N such that n ≈CA n . – C-back Spoiler chooses a group A and a state n ∈ N such that n ≈CA n . Duplicator responds by choosing a state m ∈ M such that m ∼CA m .

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– D-forth Spoiler chooses a group A and a state m ∈ M such that m ∼DA m . Duplicator responds by choosing a state n ∈ N such that n ≈DA n . – D-back Spoiler chooses a group A and a state n ∈ N such that n ≈DA n . Duplicator responds by choosing a state m ∈ M such that m ∼DA m . For any of the above scenarios, the rest of the game is the (r − 1)-round S5CDP game on (M, m ) and (N, n ). If either player cannot perform an action prescribed above, that player loses. Theorem 2. Let (M, m) and (N, n) be two pointed models and P be a ﬁnite set of propositional variables. Let S5CDP be the sublanguage of S5CD which uses only propositional variables in P. Then the following are equivalent for all r ∈ N: 1. Duplicator has a winning strategy for the r-round S5CDP game on (M, m) and (N, n). 2. (M, m) and (N, n) satisfy exactly the same S5CDP -formulae of modal degree at most r. Theorem 3. PAC and S5CD are not comparable. Proof. Similarly to the proof of Theorem 1, the two models can still be used to show the S5CD-formula, Dab p, is not equivalent to any PAC-formula. The converse can be shown analogously by using the hairpin models given in [4, pp. 231–232] together with the S5CDP game (Deﬁnition 6) and Theorem 2. We immediately get the following. Corollary 1. PACD is more expressive than PAC, S5CD, and PAD. Proof. Obviously, PAC PACD and S5CD PACD. Since PAC and S5CD are not comparable, there is a formula in PAC which is not expressible in S5CD. But since PAC PACD, that formula is expressible in PACD, and thus S5CD ≺ PACD. By similar reasoning, PAC ≺ PACD. That PAD ≺ PACD follows from the fact that PAD ≡ S5D ≺ S5CD, which again follows from Theorem 1. What about bisimulations? While it is well known that distributed knowledge is not invariant under standard bisimulation (see the proof of Theorem 1), Roelofsen [13] shows that S5D satisfaction is invariant under a notion of collective bisimulation. We now show that collective bisimulation “works” also for PACD. For technical reasons we state a slightly more general result, namely for pre-models rather than just models (we will use the more general property in the completeness proof in the next section). Definition 7 (Collective bisimulations between pre-models [13]). Let M = (M, ∼ag , ∼gr , V ) and N = (N, ≈ag , ≈gr , ν) be two pre-models. A nonempty binary relation Z ⊆ M × N is called a collective bisimulation between M and N, if the following hold for all m ∈ M and n ∈ N with mZn, and for all τ of type a or DA :

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( at) m ∈ V (p) iﬀ n ∈ ν(p) for all p ∈ prop, (zig) ∀m ∈ M. [m ∼τ m ⇒ ∃n ∈ N. (n ≈τ n & m Zn )], (zag) ∀n ∈ N. [n ≈τ n ⇒ ∃m ∈ M. (m ∼τ m & m Zn )]. We write Z : (M, m) (N, n) if Z is a collective bisimulation between M and N linking m and n, and we say pointed pre-models (M, m) and (N, n) are collectively bisimilar, denoted by (M, m) (N, n), if such a Z exists. Theorem 4. For all pointed pre-models (M, m) and (N, n), (M, m) (N, n) 3.3

implies

M, m |= ϕ ⇔ N, n |= ϕ for any ϕ ∈ PACD.

Expressivity Relations: Summary

The following ﬁgure summarises expressivity results for public announcement logics with distributed knowledge and related logics. An arrow goes from one logic to another means that the ﬁrst is strictly less expressive (the relationships not explicitly discussed above are either well known or follow easily from known results). / S5CD S5D (PAD) II 7 q8 o q II o qq oo II q o o q II o q o q o II q oo qq $ / / / S5 (PAL) S5C PAC PACD

4 4.1

Axiomatisation PAD

Figure 1 shows the Hilbert-style proof system PAD. It extends the axiomatisation of PAL in [4] with the reduction axiom for distributed knowledge (Proposition 2). We refer to a sketch of completeness proof of S5D in [7, pp. 64-65], and showing the following is an easy exercise. Let P AD denote derivability in PAD. Theorem 5. PAD is sound and complete: for any ϕ ∈ PAD, |= ϕ iﬀ P AD ϕ. Note that completeness in the above theorem can be easily generalised to a strong version, i.e., for any Φ and ϕ, Φ |= ϕ implies Φ P AD ϕ. 4.2

PACD

Figure 2 shows the Hilbert-style proof system PACD. In the remainder of this section we show that it is a sound and complete axiomatisation of PACD with respect to the class of all models. PACD is a straightforward extension of the proof system for PAC given in [4] with standard axioms for distributed knowledge. In the following we use ϕ to denote that the PACD-formula ϕ is derivable in PACD (ϕ is a PACD-theorem). A set Γ of formulae is consistent if Γ ⊥, and inconsistent otherwise. Theorem 6 (Soundness and pseudo soundness). Every PACD-theorem is valid in any model and in any pseudo model.

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Axioms and Rules PC KK 4K KD 4D DK1 Red[]p Red[]∧ Red[]D MP

All instances of propositional tautologies Ka (ϕ → ψ) → Ka ϕ → Ka ψ TK Ka ϕ → ϕ Ka ϕ → Ka Ka ϕ 5K ¬Ka ϕ → Ka ¬Ka ϕ DA (ϕ → ψ) → DA ϕ → DA ψ TD DA ϕ → ϕ DA ϕ → DA DA ϕ 5D ¬DA ϕ → DA ¬DA ϕ Ka ϕ ↔ Da ϕ, if a ∈ ag DK2 DA ϕ → DB ϕ, if A ⊆ B [ϕ]p ↔ (ϕ → p) Red[]¬ [ϕ]¬ψ ↔ (ϕ → ¬[ϕ]ψ) [ϕ](ψ ∧ χ) ↔ ([ϕ]ψ ∧ [ϕ]χ) Red[]K [ϕ]Ka ψ ↔ (ϕ → Ka [ϕ]ψ) [ϕ]DA ψ ↔ (ϕ → DA [ϕ]ψ) Red[][] [ϕ][ψ]χ ↔ [ϕ ∧ [ϕ]ψ]χ ϕ& ϕ→ψ⇒ψ NK ϕ ⇒ Ka ϕ Fig. 1. The axioms and rules of the proof system PAD

Completeness Proof. The completeness proof is based on the completeness proof for PAC given in [2,4], and several of the constructions and intermediate results in the following are adaptions or extensions of constructions and intermediate results in [4]. These proofs deal with common knowledge using a ﬁnitary canonical model technique, similarly to the technique used to deal with iteration in PDL. In our proof we must extend the technique to deal with distributed knowledge as well. There are several completeness results and proofs for logics with distributed knowledge [6,9,8,10] in the literature. None of them have public announcements as well and most of them consider only distributed knowledge for the grand coalition (only a single D operator without subscript). Two steps are typically used: ﬁrst, construct (some form of) canonical model, which will not have the intersection property required by distributed knowledge. Second, transform the canonical model into a proper model without changing its theory. We use the same general strategy for dealing with distributed knowledge here. Our approach is most similar to that for distributed knowledge in [6], which uses an unravelling technique to transform the canonical model. So do we, but the particular unravelling technique we use is diﬀerent. We start with deﬁning the closure of a formula, which is then used to deﬁne a ﬁnite canonical pseudo model. Recall that a pseudo model is like a model except that it lacks one direction of the intersection property. The proof is then completed by ﬁrst showing a truth lemma for the canonical pseudo model (Lemma 3), and then transforming the canonical pseudo model into a model by ﬁrst unravelling it to a tree-like pre-model (Deﬁnition 13, Theorem 8) which is then ﬁnally used to construct an equivalent model (Lemma 6). Definition 8 (Closure). Given a formula ϕ, the closure of ϕ is given by the function cl : PACD → ℘(PACD) which is deﬁned as follows: 1. 2. 3. 4.

ϕ ∈ cl(ϕ), and if ψ ∈ cl(ϕ), so are all of its subformulae; if ϕ is not a negation, then ϕ ∈ cl(ϕ) ⇒ ¬ϕ ∈ cl(ϕ); Ka ψ ∈ cl(ϕ) iﬀ Da ψ ∈ cl(ϕ); CA ψ ∈ cl(ϕ) ⇒ {KaCA ψ | a ∈ A} ⊆ cl(ϕ);

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Y.N. W´ ang and T. ˚ Agotnes Axioms and Rules PC TK 5K CK1 KD 4D DK1 Red[]p Red[]∧ Red[]D MP NC Red[]C

All instances of tautologies KK Ka (ϕ → ψ) → Ka ϕ → Ka ψ Ka ϕ → ϕ 4K Ka ϕ → Ka Ka ϕ ¬Ka ϕ → Ka ¬Ka ϕ KC CA (ϕ → ψ) → CA ϕ → CA ψ CA ϕ → ϕ ∧ EA CA ϕ CK2 CA (ϕ → EA ϕ) → ϕ → CA ϕ DA (ϕ → ψ) → DA ϕ → DA ψ TD DA ϕ → ϕ DA ϕ → DA DA ϕ 5D ¬DA ϕ → DA ¬DA ϕ Ka ϕ ↔ Da ϕ, if a ∈ ag DK2 DA ϕ → DB ϕ, if A ⊆ B [ϕ]p ↔ (ϕ → p) Red[]¬ [ϕ]¬ψ ↔ (ϕ → ¬[ϕ]ψ) [ϕ](ψ ∧ χ) ↔ ([ϕ]ψ ∧ [ϕ]χ) Red[]K [ϕ]Ka ψ ↔ (ϕ → Ka [ϕ]ψ) [ϕ]DA ψ ↔ (ϕ → DA [ϕ]ψ) Red[][] [ϕ][ψ]χ ↔ [ϕ ∧ [ϕ]ψ]χ ϕ& ϕ→ψ⇒ψ NK ϕ ⇒ Ka ϕ ϕ ⇒ CA ϕ N[] ϕ ⇒ [ψ]ϕ χ → [ϕ]ψ & χ ∧ ϕ → EA χ ⇒ χ → [ϕ]CA ψ

Theorems and Derived Rules 4C 5C C2 D1 ND

CA ϕ → CA CA ϕ ¬CA ϕ → CA ¬CA ϕ CA ϕ → CB ϕ, if B ⊆ A Ka ϕ → DA ϕ, if a ∈ A ϕ ⇒ DA ϕ

TC C1 C3 K+ []

CA ϕ → ϕ CA ϕ → E A ϕ Ka ϕ → Ca ϕ [ϕ](ψ → χ) ↔ ([ϕ]ψ → [ϕ]χ)

Fig. 2. The axioms and rules of the proof system PACD. Also shown are some derived theorems and admissible rules; these are well known and/or easy to prove.

5. [ψ]CA χ ∈ cl(ϕ) ⇒ (([ψ]χ) ∈ cl(ϕ) & {Ka [ψ]CA χ | a ∈ A} ⊆ cl(ϕ)); 6. for all of the six reduction axioms, such as Red[]p and Red[]¬ , if the left-hand side of the equivalence is in cl(ϕ), so is the right-hand side. That is, for each reduction axiom of the form α ↔ β: α ∈ cl(ϕ) ⇒ β ∈ cl(ϕ). It is easy to observe that cl(ϕ) is ﬁnite. For any formula α, we say a set Γ of formulae is maximal consistent in cl(α) if: Γ ⊆ cl(α); Γ is consistent; and Γ is maximal in cl(α), i.e., there is no Δ ⊆ cl(α) such that Γ Δ and Δ is consistent. A ﬁnitary version of Lindenbaum’s lemma can now be shown: for any α, any consistent subset of cl(α) is a subset of a maximal consistent set in cl(α). ˆ a as the dual operator of Ka , i.e. K ˆ a ϕ ↔ ¬Ka ¬ϕ; we write Ka Γ We use K ˆ for {K a ϕ | Ka ϕ ∈ Γ }; similarly used are DA and DA Γ ; we use Γ as shorthand for Γ when Γ is a ﬁnite set of formulae. Definition 9 (Canonical pseudo model). Let α be a formula. The canonical pseudo model Mc = (Mc , (∼ag )c , (∼gr )c , V c ) for cl(α) is deﬁned as follows: – – – –

M c = {Γ | Γ is maximal consistent in cl(α)}; Γ ∼ca Δ iﬀ Ka Γ = Ka Δ; Γ ∼cDA Δ iﬀ DB Γ = DB Δ whenever B ⊆ A; V c (p) = {Γ ∈ M c | p ∈ Γ }.

∼cCA is deﬁned analogously to be the reﬂexive transitive closure of

a∈A

∼ca .

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Proposition 4. The canonical pseudo model for any cl(α) is a pseudo model. Before proving a truth lemma for the canonical pseudo model, we need some properties (following the technique for PAC in [4]). Lemma 1. Let α be a formula and S = {Γ | Γis maximal consistent in cl(α)}. Then it holds that Γ ∈S Γ and that ϕ ↔ ϕ∈Γ ∈S Γ for any ϕ ∈ cl(α). Lemma 2. If Γ and Δ are maximal consistent in cl(α), then 1. 2. 3. 4. 5. 6. 7. 8.

Γ is deductively closed in cl(α), i.e., for any ϕ ∈ cl(α), Γ ϕ ⇔ ϕ ∈ Γ ; if ¬ϕ ∈ cl(α), then ϕ ∈ Γ ⇔ ¬ϕ ∈ / Γ; if ϕ ∧ ψ ∈ cl(α), then ϕ ∧ ψ ∈ Γ ⇔ ϕ ∈ Γ & ψ ∈ Γ ; ˆ a Δ is consistent, Γ ∼ca Δ; and if Γ ∧ D ˆ A Δ is consistent, Γ ∼c Δ; if Γ ∧ K DA if Ka ψ ∈ cl(α), then Ka Γ ψ ⇔ Ka Γ Ka ψ; if DA ψ ∈ cl(α), then DA Γ ψ ⇔ DA Γ DA ψ; if CA ϕ ∈ cl(α), then CA ϕ ∈ Γ ⇔ ∀Δ.(Γ ∼cCA Δ ⇒ ϕ ∈ Δ); if [ϕ]CA ψ ∈ cl(α), then [ϕ]CA ψ ∈ Γ ⇔ (∀n ∈ N)∀Γn .(Γ0 ∼ca0 · · · ∼can−1 Γn & ϕ ∈ ni=0 Γi ⇒ [ϕ]ψ ∈ ni=0 Γi ), where a0 , . . . , an−1 ∈ A, Γ0 = Γ , and each Γi (0 ≤ i ≤ n) is maximal consistent in cl(α).

Definition 10 (Complexity of formulae). The complexity of a formula is deﬁned by the function c : PACD → N: c(p) = 1, c(¬ϕ) = c(Ka ϕ) = c(CA ϕ) = c(DA ϕ) = 1 + c(ϕ), c([ϕ]ψ) = (4 + c(ϕ)) · c(ψ), c(ϕ ∧ ψ) = 1 + max(c(ϕ), c(ψ)). Proposition 5. For any formulae ϕ, ψ and χ, 1. 2. 3. 4.

c(ϕ) ≥ c(ψ) if ψ is a subformula of ϕ 5. c([ϕ]Ka ψ) > c(ϕ → Ka [ϕ]ψ) c([ϕ]p) > c(ϕ → p) 6. c([ϕ]CA ψ) > c([ϕ]ψ) c([ϕ]¬ψ) > c(ϕ → ¬[ϕ]ψ) 7. c([ϕ]DA ψ) > c(ϕ → DA [ϕ]ψ) 8. c([ϕ][ψ]χ) > c([ϕ ∧ [ϕ]ψ]χ). c([ϕ](ψ ∧ χ)) > c([ϕ]ψ ∧ [ϕ]χ)

Lemma 3 (Pseudo Truth lemma). Let Mc = (M c , (∼ag )c , (∼gr )c , V c ) be the canonical pseudo model for cl(α), given a formula α. For all Γ ∈ M c , and all ϕ ∈ cl(α): it holds that ϕ ∈ Γ iﬀ Mc , Γ |= ϕ. Proof. We show this lemma by induction on c(ϕ). Despite the slight diﬀerence that we are using pseudo models here, the cases for boolean operators, individual knowledge operators, common knowledge operators, and public announcement operators are like those cases in the proof of the Truth Lemma for PAC [4, p.193] (the only missing subcase is [ψ]DA χ which can be easily shown). Thus, we only discuss the case when ϕ is of the form DA ψ. From left to right. Suppose DA ψ ∈ Γ . Take an arbitrary Δ ∈ M c . Suppose that Γ ∼cDA Δ. So DA ψ ∈ Δ by the deﬁnition of ∼cDA . Since DA ψ → ψ (Axiom TD ), and Δ is deductively closed, it must be the case that ψ ∈ Δ. This is equivalent to Mc , Δ |= ψ by induction hypothesis. Since all the Δs “∼cDA reachable” from Γ forces ψ, therefore Mc , Γ |= DA ψ by the semantics.

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From right to left. Suppose Mc , Γ |= DA ψ. Therefore Mc , Δ |= ψ for any Δ such that Γ ∼cDA Δ. We must show DA ψ ∈ Γ . Suppose not, then ¬DA ψ ∈ Γ , ˆ A ¬ψ ∈ Γ . Hence Γ ∧ D ˆ A ¬ψ is consistent, and then Γ ∧ D ˆA i.e. D ¬ψ∈Θ∈M c Θ is consistent by Lemma 1. This is equivalent to the situation that ¬ψ∈Θ∈M c (Γ ∧ ˆ A Θ) is consistent. It entails that there is a Ψ ∈ M c such that ¬ψ ∈ Ψ and D ˆ A Ψ is consistent. From ¬ψ ∈ Ψ , we have ψ ∈ Γ ∧D / Ψ , and therefore Mc , Ψ |= ψ c ˆ A Ψ is by induction hypothesis. Hence M , Ψ |= ¬ψ. From the fact that Γ ∧ D c c ˆ consistent, we have Γ ∼DA Ψ by Lemma 2(4). Therefore M , Γ |= DA ¬ψ, which contradicts with the assumption that Mc , Γ |= DA ψ. We now move on to transforming the canonical pseudo model into a model. Some auxiliary deﬁnitions and results are needed. Definition 11 (Tree-like pre-models). Given an arbitrary pre-model M = (M, ∼ag , ∼gr , V ), a path of M from a state m0 to a state mn is a ﬁnite nonempty sequence of the following form: m0 ∼τ0 · · · ∼τn−1 mn , where each mi (0 ≤ i ≤ n) is a state of M, each ∼τj (0 ≤ j < n) is a relation of M such that mj ∼τj mj+1 . The reduction of a path is obtained by recursively running the following procedure: – replace every segment of the type x ∼τ y ∼τ z by x ∼τ z; – replace every segment of the type x ∼τ x by x. The reduction of a path is still a path, because all relations are transitive. A reduced path is a path which is precisely the same as its reduction. A pre-model M is tree-like, if for any two states m, n ∈ M there is at most one reduced path from m to n. For convenience, a path of M ended with m is denoted by m. Definition 12 (τ -extensions and τ -grafts). Let M = (M, ∼ag , ∼gr , V ) be a pre-model, and τ be of the type a or DA . Let m = m0 ∼τ0 · · · ∼τi−1 mi and n = n0 ∼θ0 · · · ∼θj−1 nj be two paths of M. m is called a τ -extension of n iﬀ m = n0 ∼θ0 · · · ∼θj−1 nj ∼τ mi . m is called a τ -graft of n iﬀ m and n are diﬀerent τ -extensions of the same path. Definition 13 (Unravelling). Given a pseudo model M = (M, ∼ag , ∼gr , V ), its unravelled structure TM = (T, Rag , Rgr , VT ) is deﬁned as follows: – T is the set of all reduced paths of M. – Let τ be of any type a or DA . Rτ is a relation on T , and for any m, n ∈ T , mRτ n holds if and only if, i) m = n, or ii) m is a τ -graft of n, or iii) m is a τ -extension of n, or iv) n is a τ -extension of m. – VT : prop → ℘(T ) is such that m ∈ VT (p) iﬀ m ∈ V (p). Theorem 7. Unravelling of a pseudo model generates a pre-model.

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Theorem 7 shows that an unravelled pseudo model is a pre-model. Nevertheless, it is not necessarily a pseudo model, since neither the equivalence of ∼a and ∼Da nor the superset relation between ∼DA and ∼DB (A ⊆ B) is preserved. Lemma 4. Let X = x0 , Rτ0 , . . . , Rτn−1 , xn be a reduced path of an unravelled pre-model TM . The following hold for X: 1. (Once growing, always growing): If xi (1 ≤ i ≤ n) is a τi−1 -extension of xi−1 , then every xj (i ≤ j ≤ n) is a τj−1 -extension of xj−1 . 2. (Once grafted, always growing diﬀerently): If xi (1 ≤ i ≤ n) is a τi−1 -graft of xi−1 , then every xj (i < j ≤ n) is a τj−1 -extension of xj−1 and does not contain xi−1 as an initial segment. 3. (Once pruned, never growing back): If xi−1 (0 < i ≤ n) is a τi−1 -extension of xi , then every xj (i ≤ j ≤ n) does not contain xi−1 or any τi−1 -graft of xi−1 as an initial segment. Theorem 8. Given a pseudo model M, its unravelled pre-model TM is tree-like. Proof. Let X = x0 , Rτ0 , . . . , Rτs−1 , xs and Y = y0 , Rθ0 , . . . , Rθt−1 , yt be two reduced paths of TM such that x0 = y0 and xs = yt . We must show that X and Y are identical. Here is a sketch of the proof. We can show that X and Y are identical, if the states at the same position of the paths match. The if-condition can furthermore be shown as follows: First, show xn = yn for 0 ≤ n ≤ min(s, t), which is done case-by-case with the help of Lemma 4. This leads to the fact that X and Y are identical if s = t. Second, suppose s = t. A contradiction can be reached. Lemma 5 (Invariance of unravelling). Let (M, m) be any pointed pseudo model. Then for every m ∈ TM , (M, m) and (TM , m) are equivalent. That is, M, m |= ϕ

iﬀ.

TM , m |= ϕ,

for any formula ϕ.

Proof. We can show that (M, m) and (TM , m) are collectively bisimilar, and the lemma follows by Theorem 4. Lemma 6. If a formula ϕ is satisﬁed at a pointed pseudo model, then ϕ is also satisﬁed at some pointed model. Proof. Let (M, m) be a pointed pseudo model. The unravelled tree-like premodel TM = (T, Rag , Rgr , VT ) of M can be translated into a model (TM )∗ = (T, Qag , VT ), where every Qa is the transitive closure of Ra ∪ RDA0 ∪ · · · ∪ RDAn such that A0 · · · An is the list of all groups containing a. We show by induction on the construction of ϕ that ∀ϕ.∀(TM , m).(TM , m |= ϕ ⇔ (TM )∗ , m |= ϕ), and then the lemma follows by Lemma 5. Here we only show the cases for distributed knowledge operators and public announcement operators. First, assume that ϕ is of the form DA ψ. Assume that TM , m |= DA ψ. Thus, there is an n ∈ T such that mRDA n and TM , n |= ψ. By the induction hypothesis (IH), (TM )∗ , n |= ψ. Since RDA ⊆ QDA , it follows that (TM )∗ , m |= DA ψ.

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Assume that (TM )∗ , m |= DA ψ. Thus, there is an n ∈ T such that mQx n for all x ∈ A and (TM )∗ , n |= ψ. By IH, TM , n |= ψ, and M, n |= ψ by Lemma 5. Suppose A = {a}. It follows that mQa n. Then there must be a reduced path m, R0 , . . . , Rj , n such that every Ri (0 ≤ i ≤ j) is either Ra or some RDX such that a ∈ X. Thus, there is a path m ∼0 · · · ∼k n of M, where each ∼i (0 ≤ i ≤ k) is either ∼a or some ∼DX such that a ∈ X. Since M is a pseudo model, we have ∼a =∼Da and ∼DX ⊆∼Da . It holds that M, m |= (Da )k+1 ψ, which entails M, m |= Da ψ since Da ψ → (Da )k+1 ψ. Therefore TM , m |= Da ψ. Otherwise |A| ≥ 2. Let a0 , . . . , ak be a sequence of all A-members. It follows that mQai n holds for each ai (0 ≤ i ≤ k). For any agent ai , there must be a reduced path Pai = m, R0i , . . . , Rxi i , n such that every Rji (0 ≤ j ≤ xi ) is either Rai or some RDX such that ai ∈ X. However, because TM is tree-like, all the reduced paths from m to n must be the same. Therefore, x0 = · · · = xn and every Rji ∈ Pai is in fact a constant. It follows that Rji cannot be Rai , and has to be a certain RDX such that ai ∈ X for each ai ∈ A. Therefore, the reduced path from m to n is of the form m, RDA0 , . . . , RDAk , n where A ⊆ Aj ⊆ ag for 0 ≤ j ≤ k. It follows that there is a path m ∼DA0 · · · ∼DAl n of M where every ∼DAj (0 ≤ j ≤ l) is such that A ⊆ Aj . Since M is a pseudo model, ∼DAj ⊆∼DA holds for 0 ≤ j ≤ l. Thus, M, m |= (DA )l+1 ψ, which entails M, m |= DA ψ as DA ψ → (DA )l+1 ψ is a theorem. Therefore TM , m |= DA ψ, as was to be shown. Second, assume that ϕ is of the form [χ]ψ. We have that TM , m |= [χ]ψ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒

M, m |= [χ]ψ M, m |= χ ⇒ M|χ, m |= ψ TM , m |= χ ⇒ TM|χ , m |= ψ (TM )∗ , m |= χ ⇒ (TM|χ )∗ , m |= ψ (TM )∗ , m |= χ ⇒ (TM )∗ |χ, m |= ψ (TM )∗ , m |= [χ]ψ,

(Lemma 5) (Lemma 5) (IH) (†)

where m is any state of TM|χ which ends with m ∈ M, and (†) follows from ((TM|χ )∗ , m ) ((TM )∗ |χ, m). To show this, we can deﬁne a relation Z = {(m , m) | m ∈ (TM|χ )∗ and m ∈ (TM )∗ |χ} and then verify the conditions for it being a collective bisimulation. We omit the details, but just point out that the corresponding pre-models, TM|χ and TM |χ, being tree-like makes it suﬃcient to consider only a unique reduced path from, say, m to another state. A weak completeness result now follows immediately from Lemmas 3 and 6. Theorem 9 (Completeness). For every formula ϕ, if |= ϕ then ϕ.

5

Discussion

In this paper we have studied the expressivity and axiomatisation of public announcement logic with distributed knowledge. We argued in the introduction that distributed knowledge has received less attention in dynamic epistemic logics than common knowledge, and in one sense it is indeed less interesting: PAD is

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reducible to S5D. PACD is more interesting; it is more expressive than its parts. We adapted model comparison games to S5CD and showed that collective bisimulation works for PACD. When it comes to axiomatisation, it should come as no surprise that PACD is completely axiomatised by adding standard axioms for distributed knowledge to an axiomatisation of PAC. However, proving completeness is not trivial. We made extensive use of existing techniques, but combined them in new ways. We believe the technique we use to deal with distributed knowledge is simpler than in many proofs involving distributed knowledge in the literature, which also often do not give all details and usually only consider a single distributed knowledge operator (for the grand coalition), and could be of more general interest than for public announcement logic. As for future work, we believe that most of the results in this paper can be extended to action model logic [1]. Van Benthem [3] pointed out that there is an intuitive relationship between distributed knowledge and public announcements in the sense that public announcements can be used by a group to share their knowledge; in further work we are looking more closely at that relationship. Also interesting for future work is to relax the S5 assumption. Acknowledgments. We thank the anonymous LORI reviewers, P˚ al Grøn˚ as Drange, and Patrick Girard for helpful remarks, suggestions, and correcting typos.

References 1. Baltag, A., Moss, L.: Logics for epistemic programs. Synthese 139, 165–224 (2004) 2. Baltag, A., Moss, L., Solecki, S.: The logic of public announcements, common knowledge, and private suspicions. In: Proc. of TARK VII, pp. 43–56 (1998) 3. van Benthem, J.: Open problems in logical dynamics. In: Math. Problems from Applied Logic I: Logics for the XXIst Century. Int. Math. Series, vol. 4, pp. 137– 192. Springer, Heidelberg (2006) 4. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Springer, Heidelberg (2007) 5. Fagin, R., Halpern, J., Moses, Y., Vardi, M.: Reasoning about Knowledge. MIT Press, Cambridge (1995) 6. Fagin, R., Halpern, J., Vardi, M.: What can machines know? On the properties of knowledge in distributed systems. Journal of the ACM 39(2), 328–376 (1992) 7. Gerbrandy, J.: Bisimulations on Planet Kripke. Ph.D. thesis, ILLC (1999) 8. Halpern, J., Moses, Y.: A guide to completeness and complexity for modal logics of knowledge and belief. Artif. Intell. 54(3), 319–379 (1992) 9. van der Hoek, W., Meyer, J.J.: Making some issues of implicit knowledge explicit. International Journal of Foundations of Computer Science 3(2), 193–224 (1992) 10. van der Hoek, W., Meyer, J.J.: A complete epistemic logic for multiple agents: Combining distributed and common knowledge. In: Bacharach, M., Gerard-Varet, L., Mongin, P., Shin, H. (eds.) Epistemic Logic and the Theory of Games and Decisions, pp. 35–68. Kluwer, Dordrecht (1997) 11. Meyer, J.J., van der Hoek, W.: Epistemic Logic for AI and Computer Science, vol. 41. Cambridge University Press, Cambridge (1995) 12. Plaza, J.: Logics of public communications. In: Proc. of ISMIS, pp. 201–216 (1989) 13. Roelofsen, F.: Distributed knowledge. J. of Applied Non-Classical Logics 16 (2006)

An Alternative Logic for Knowability Xuefeng Wen, Hu Liu, and Fan Huang Institute of Logic and Cognition and Department of Philosophy, Sun Yat-sen University, Guangzhou, 510275, China [email protected], [email protected]

Abstract. We give an alternative logic for knowability to arbitrary public announcement logic. In the new semantics, ‘knowable’ means ‘known after an information update’ rather than the more speciﬁc interpretation of ‘known after an announcement’. An update is modelled by the reﬁning of accessibility relations. We compare our logic to arbitrary announcement logic and subset space logic and thus establish a link between the latter two. Keywords: knowability, arbitrary announcement logic, dynamic epistemic logic, subset space logic.

1

Introduction

Knowability is usually analysed as the the combination of knowing and possibility. When using standard epistemic logic and modal logic1 to represent them, however, it leads to a surprising result: if all truths are knowable, then all truths are actually known. This is the so-called Fitch’s paradox, originally formulated by Fitch in [10].2 Traditionally, the possibility for analysing knowability is treated as a normal modal operator. It is interpreted in standard Kripke semantics using accessibility relation between possible worlds. In [4], van Benthem suggested a dynamic interpretation for possibility. In the dynamic view, ‘knowable’ means ‘known after an announcement’. Following this interpretation, Balbiani et al. [1,2] proposed an extension of public announcement logic, in which another modal operator 3 is augmented to represent the modality of knowability. The logic is called arbitrary public announcement logic (AP AL, henceforth). Though it may not solve Fitch’s paradox, it gives a new understanding of knowability and connects it to some important notions in dynamic epistemic logic, such as successful formulas.3 In AP AL, 3ϕ means that there is some truthful announcement ψ after which ϕ is true. But public announcement is only one possible way of gaining new knowledge. There are certainly other possibilities. One motivation of our paper is to give a more general semantics for 3, keeping the dynamic interpretation 1 2 3

Here modal logic is in narrow sense. The modal operators in it are interpreted as necessity and possibility. See [6] for a detailed survey on Fitch’s paradox. See [8] for an excellent introduction to dynamic epistemic logic.

H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 342–355, 2011. c Springer-Verlag Berlin Heidelberg 2011

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of knowability meanwhile. We call our alternative logic LK and use SLK to denote the single-agent version of LK. The idea of LK is to focus on the result of information update rather than on its procedure. When new information is obtained, by whatever means, the agents will be able to distinguish more possible states. This is represented by ﬁner accessibility relations in the updated epistemic model. All epistemic models with ﬁner accessibility relations are taken to be possible updates. In this interpretation, 3ϕ means that there is a possible update after which ϕ is true. Since the update does not involve speciﬁc formulas, LK does not need the announcements to represent knowability any more. Interestingly, the alternative semantics yields the same logic as AP AL in the single-agent situation. But when it comes to multi-agent systems, the two logics are diﬀerent. AP AL preserves the invariance property of bisimulation, whereas LK does not. On the other hand, unlike AP AL, the truth value of a formula ϕ in LK does not depend on the valuations of propositional variables outside of ϕ. The technical idea of AP AL is to interpret modality by the relation between models rather than worlds in a single model. LK follows the same approach. This technique, however, is not totally new. A much more earlier logic combining knowledge and possibility is called subset space logic (SSL, henceforth). It was proposed by Moss and Parikh in [16,7] with a rather diﬀerent motivation.4 They intended to use SSL for topological reasoning. In SSL, unlike standard possible world semantics, a formula is not evaluated at a point, but at a pair of a point and a set containing that point. Every set can be regarded as a Kripke frame with a universal relation. The modal operator for knowledge is interpreted by the universal relation, while the operator for possibility is interpreted by the subset relation between sets. Another motivation of our paper is to establish a link between AP AL and SSL, by giving a semantics similar to both of them. The semantics of LK keeps the intuitiveness of AP AL but is also simple enough to be easily translated into SSL. In [2], the authors gave a very brief comparison between AP AL and SSL, indicating that the update in the former is temporal whereas in the latter it is spatial. The diﬀerence, however, may not be so great. We give a detailed comparison between SLK and SSL. We prove that SLK is equivalent to an extension of SSL. Since SLK is equivalent to a fragment of the single-agent AP AL, our comparison establishes a link between AP AL and SSL. As the paper involves several logical systems, we list them here for reference: ALK AP AL EL LK SALK SEL SLK SSL 4

the fragment of AP AL without announcements arbitrary public announcement logic standard multi-agent epistemic logic (without common knowledge) our alternative logic of knowability single-agent version of ALK standard single-agent epistemic logic single-agent version of LK subset space logic

For recent developments of SSL, see for example [17,13,14].

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2 2.1

Syntax and Semantics of LK

We ﬁx a set P V of propositional variables and a set n = {1, . . . , n} of agents throughout the paper. We write EL and SEL for standard multi-agent epistemic logic (without common knowledge) and for single-agent epistemic logic, respectively. We denote by Lpl and by Lel the sets of well-formed formulas of P L and EL, respectively. The language Llk of LK is Lel augmented with a unary operator 3, formally given by the following BNF: ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | Ki ϕ | 3ϕ where p ∈ P V and i ∈ n. The other Boolean connectives are deﬁned as usual. The intended interpretation of 3ϕ is that ϕ is informatively possible, i.e. there is some information update after which ϕ is true. We write Li and 2 for the duals of Ki and 3, respectively. Models for LK are the same as those for EL, deﬁned as follows. Definition 1. A (multi-agent) epistemic model is a tuple M = (S, ∼, V ), which consists of a non-empty set S of states, an accessibility function ∼: n → ℘(S ×S) assigning to each agent i an equivalence relation (called accessibility relation) ∼i on S, and a valuation V : P V → ℘(S) assigning to each propositional variable a set of states. When more information is acquired, the agents can distinguish more states. This is represented by ﬁner accessibility relations in the epistemic model. Compared to AP AL, we concentrate on the result of the update rather than on how the update is achieved. Definition 2. Let M = (S, ∼, V ) and M = (S , ∼ , V ) be two epistemic models. We say that M is an update of M , denoted M ≤ M (or M ≥ M ), if S = S, V = V and ∼i ⊆ ∼i for all i ∈ n. For brevity, we often write ∼ ⊆ ∼ if ∼i ⊆ ∼i for all i ∈ n. It is easily seen that the update relation ≤ is reﬂexive and transitive. Definition 3. Given an epistemic model M = (S, ∼, V ) and a state s ∈ S, we deﬁne the satisfaction relation |=lk inductively as follows: – – – – –

M, s |=lk M, s |=lk M, s |=lk M, s |=lk M, s |=lk

p iﬀ s ∈ V (p), ¬ϕ iﬀ M, s |=lk ϕ, ϕ ∧ ψ iﬀ M, s |=lk ϕ and M, s |=lk ψ, Ki ϕ iﬀ M, t |=lk ϕ for all t such that s ∼i t, 3ϕ iﬀ M , s |=lk ϕ for some M ≥ M ,

Dually, M, s |=lk 2ϕ iﬀ M , s |=lk ϕ for all M ≥ M . The semantic entailment lk 5 is deﬁned as usual. We often omit the subscript lk in |=lk and lk if the context makes them clear. 5

Note the minor diﬀerence between the symbols |= and .

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345

Some Semantic Results

Clearly, all standard axioms and rules for epistemic logic are still valid in LK. As for modality, besides that all axioms and rules of S4 hold in LK, we have: Proposition 1. The following hold in LK: 1. 2. 3. 4.

23ϕ ↔ 32ϕ (p → 2p) ∧ (¬p → 2¬p), for p ∈ P V Ki 2ϕ → 2Ki ϕ 2Ki ϕ → Ki 2ϕ

Proof. Let M = (S, ∼, V ) be an arbitrary epistemic model. 1. For the direction from left to right, suppose M, s |= 23ϕ. Deﬁne M ∗ = (S, ∼∗ , V ) such that ∼∗i = {(s, s) | s ∈ S} for all i ∈ n. Then M ≤ M ∗ . Thus M ∗ , s |= 3ϕ. Note that the unique update of M ∗ is M ∗ itself. Hence M ∗ , s |= ϕ and M ∗ , s |= 2ϕ. By M ≤ M ∗ again, M, s |= 32ϕ. For the other direction, suppose M, s |= 32ϕ. Then there is M = (S, ∼ , V ) such that ∼ ⊆ ∼ and M , s |= 2ϕ. Let M = (S, ∼ , V ) be an arbitrary update of M , i.e. ∼ ⊆ ∼. Deﬁne N = (S, ∼N , V ) such that ∼N i = ∼i ∩ ∼i for all i ∈ n. Then M ≤ N . From M , s |= 2ϕ, it follows that N, s |= ϕ. Since M ≤ N , we have M , s |= 3ϕ. By the arbitrariness of M , M, s |= 23ϕ. 2. This is obvious, since the valuation does not change in updated models. By an easy induction, it can be generalised to arbitrary formulas in Lpl , stated by Lemma 1. 3. Suppose M, s |= Ki 2ϕ. Then M, t |= 2ϕ for all t such that s ∼i t. Let M = (S, ∼ , V ) be an arbitrary update of M and s ∼i t . Since ∼i ⊆ ∼i , we have s ∼i t . From M, s |= Ki 2ϕ, it follows that M, t |= 2ϕ, which implies that M , t |= ϕ, since M ≤ M . By the arbitrariness of t , M , s |= Ki ϕ. By the arbitrariness of M , M, s |= 2Ki ϕ. 4. Consider the model M = (S, ∼, V ) such that S = {s, t}, ∼i = S × S and V (p) = {t}. Then M, s |= 2Ki ¬Ki p. But M, s |= Ki 2¬Ki p. Lemma 1. If M ≤ M then for all s ∈ S and π ∈ Lpl , M, s |= π iﬀ M , s |= π. 2.3

Various Representations of the Knowability Thesis in LK

The knowability thesis says that all truths are knowable. The standard representation of it in modal logic is (KT1) ϕ → 3Ki ϕ, namely, given any truth, it is possible to know it. Considering the scope of the modality in the knowability thesis, it is possible to give another reading of it: for any proposition p, it is possible that if p is true then p is known. This is represented in LK by (KT2): (KT2) 3(ϕ → Ki ϕ)

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Note that it is diﬀerent from 3∀p(p → Kp), which actually says that omniscience is possible. Intuitively, even if every truth may come to be known, human beings are not omniscient at any time. In LK, (KT2) is semantically equivalent to: (KT3) 2ϕ → 3Ki ϕ, which follows from (KT1), given that 2ϕ → ϕ is valid in LK. Hence, (KT1) is stronger than (KT2) and (KT3). Actually, unlike (KT1), (KT2) and (KT3) are valid in LK. The equivalence between (KT2) and (KT3) is shown by the following equivalents: 3(ϕ → Ki ϕ) ↔ 3(¬ϕ ∨ Ki ϕ) ↔ (3¬ϕ ∨ 3Ki ϕ) ↔ (¬2ϕ ∨ 3Ki ϕ) ↔ (2ϕ → 3Ki ϕ) We propose another representation of the knowability thesis: (KT4) 32ϕ → 3Ki ϕ It does not require ϕ to be true from now on for its knowability, as in (KT3). It only requires that ϕ will always be true from a future moment. We suggest (KT4) to be an acceptable knowability thesis. It is also valid in LK and implies (KT3), given the axiom ϕ → 3ϕ.

LK and Arbitrary Public Announcement Logic

3

The language of arbitrary public announcement logic is Llk augmented with public announcement formulas [ϕ]ψ. For comparison, we only consider the fragment of AP AL without announcements. We call it arbitrary announcement logic of knowability and denote it by ALK. Its single-agent version is denoted by SALK. The semantic diﬀerence between ALK and LK lies only in the interpretation of 3ϕ. Given an epistemic model M = (S, ∼, V ) and s ∈ M , 3ϕ is interpreted in ALK as follows: – M, s |=alk 3ϕ iﬀ there is ψ ∈ Lel s.t. M, s |=alk ψ and M |ψ , s |=alk ϕ, where M |ψ = (S , ∼ , V ) is deﬁned as follows: S := {s ∈ S | M, s |=alk ϕ} ∼ := ∼ ∩(S × S ) V (p) := V (p) ∩ S 3.1

Single-Agent Systems

First we consider LK and ALK for their single-agent versions: SLK and SALK. Recall that we use SEL for single-agent epistemic logic. It has been proved in [2] that SALK can be reduced to SEL. One of the pivots to prove the reduction is the following lemma. We prove that it also holds for LK. Lemma 2. Let π, π0 , π1 , . . . , πn ∈ Lpl and ϕ, ψ ∈ Llk . Then6 6

This is not the original form of the lemma in [2]. We combine two clauses in the original lemma and omit the other clauses, which are redundant for the reduction.

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1. 3(ϕ ∨ ψ) ↔ 3ϕ ∨ 3ψ 2. 3(π ∧ Ki π0 ∧ Li π1 ∧ · · · ∧ Li πn ) ↔ π ∧ π0 ∧ Li (π0 ∧ π1 ) ∧ · · · ∧ Li (π0 ∧ πn ) Proof. Clause 1 is easy. We prove clause 2 for the case of n = 1. The general case is proved similarly. Let M = (S, ∼, V ). Suppose M, s |= 3(π ∧Ki π0 ∧ Li π1 ). Then there is M ≥ M such that M , s |= π ∧ Ki π0 ∧ Li π1 . Thus M , t |= π0 ∧ π1 for some t accessible from s in M . By Lemma 1, M, t |= π0 ∧ π1 . Since M ≤ M , t is also accessible from s in M . Hence M, s |= Li (π0 ∧ π1 ). On the other hand, since M , s |= π ∧Ki π0 , we have M , s |= π ∧π0 . Thus M, s |= π ∧π0 by Lemma 1 again. Therefore M, s |= π ∧ π0 ∧ Li (π ∧ π1 ). For the other direction, suppose M, s |= π∧π0 ∧Li (π0 ∧π1 ). Then M, s π∧π0 and M, t |= π0 ∧π1 for some t such that s ∼i t. Construct a model M = (S, ∼ , V ) such that ∼ is the same as ∼ except that ∼i = {(x, x) | x ∈ S} ∪ {(s, t), (t, s)}. Then M ≤ M . By Lemma 1, we have M , s |= π ∧ π0 and M , t |= π0 ∧ π1 . It follows that M , s |= π ∧ Ki π0 ∧ Li π1 . Therefore M, s |= 3(π ∧ Ki π0 ∧ Li π1 ). Another pivot for the reduction of SLK to SEL is the fact that any SEL formula is equivalent to a formula that contains at most one layer of knowledge operators. More precisely, any SEL formula is equivalent to a disjunction of conjunctions of the form π ∧ Kπ0 ∧ Lπ1 ∧ · · · ∧ Lπn , where π and all πi are purely propositional formulas.7 Using this fact and the above lemma by an induction on the number of 3, it should be easy to see that all formulas in SLK can be equivalently reduced to those in SEL. The reduction also gives the axiomatisation and decidability of SLK. Let’s denote by (DD) and (RDn) the axiom schemes of 3(ϕ ∨ ψ) ↔ 3ϕ ∨ 3ψ and 3(π ∧ Kπ0 ∧ Lπ1 ∧ · · · ∧ Lπn ) ↔ π ∧ π0 ∧ L(π0 ∧ π1 ) ∧ · · · ∧ L(π0 ∧ πn ), respectively. Then all the axioms and rules for SEL plus (DD) and all (RDn) give a complete axiomatic system for SLK. Since the reduction axioms for SLK and SALK are the same, namely, (DD) and (RDn), we have the following theorem. Theorem 1. For any (single-agent) epistemic model M , any s ∈ M and ϕ ∈ Lslk , M, s |=slk ϕ iﬀ M, s |=salk ϕ. Hence, Σ slk ϕ iﬀ Σ salk ϕ. The theorem above implies that in the single-agent situation any information update (in our sense) can be realized by a public announcement. This is not the case when it comes to multi-agent systems. 7

See [15] for the proof.

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Multi-agent systems

Unlike the relation between SLK and SALK, LK is not equivalent to ALK. Since the update in ALK is by public announcement, the agents reﬁne their accessibility relations simultaneously. As a result, under the semantics of ALK, if two agents have the same set of knowledge at ﬁrst, then it is impossible for one of them to know more than the other in future. This is not the case in LK, since the agents reﬁne their accessibility relations independently. The most important diﬀerence between LK and ALK concerns expressivity. Unlike ALK, LK does not preserve the invariance property of bisimulation8 . The following is a counter-example. Example 1. Let M and N be two epistemic models depicted as follows. b : ¬p

2

b : ¬p

1

a:p

1 2

a :p

M

¯b : ¬p 1

a ¯:p N

It can be veriﬁed that E = {(a, a ¯), (a , a ¯), (b, ¯b), (b , ¯b)} is a bisimulation between (M, a) and (N, a¯). But we have M, a |=lk 3(K1 p ∧ ¬K2 K1 p) and N, a ¯ |=lk 3(K1 p ∧ ¬K2 K1 p). That LK is not invariant under bisimulation may be undesirable, compared to ALK. There is, however, an advantage of LK over ALK. In LK, the truth value of a formula ϕ only depends on the valuations on the propositional variables in ϕ. Formally, we have the following proposition, which can be proved by an easy induction. Proposition 2. Let M = (S, ∼, V ) and M = (S, ∼, V ) be two epistemic models such that V (p) = V (p) for all p ∈ P V (ϕ), where P V (ϕ) is the set of all propositional variables in ϕ. Then for any s ∈ S, M, s |=lk ϕ iﬀ M , s |=lk ϕ This property does not hold for ALK, as illustrated by the following example. Example 2. Let M and M be two epistemic models depicted as follows. Then M and M are the same model except that they give diﬀerent valuations on q. But M, a |=alk 3(K1 p ∧ ¬K2 p) and M , a |=alk 3(K1 p ∧ ¬K2 p). 8

We refer the reader to [5] for the deﬁnition of bisimulation and will not repeat it here.

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b : ¬p 2

a : p 2

a:p

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b : ¬p

M

b : ¬p, q

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a : p, q

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a : p, ¬q 2

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LK and Subset Space Logic

4 4.1

Single-Agent Systems

The main result of this subsection is that SLK is sound and complete with respect to the class of downwards closed subset frames. First, we introduce the semantics of SSL.9 Definition 4. A subset frame is a pair F = (X, O), where – X = ∅, consisting of points, – O ⊆ ℘(X) is a set of subsets of X, called opens or observations. A subset model is a triple M = (X, O, V ), where (X, O) is a subset frame and V : P V → ℘(X) is a valuation, assigning to each propositional variable a subset of X. If M = (X, O, V ), we say that M is based on (X, O) and call (X, O) the underlying subframe of M . The subset space logic is characterized by the class of all subset frames and the following truth conditions. Definition 5. Given a subset model M = (X, O, V ), the satisfaction of ϕ at (s, u) in M for s ∈ u ∈ O , denoted M, s, u |=ssl ϕ, is inductively deﬁned as follows: 1. 2. 3. 4. 5.

M, s, u |=ssl M, s, u |=ssl M, s, u |=ssl M, s, u |=ssl M, s, u |=ssl

p iﬀ p ∈ V (s), for p ∈ P V ¬ϕ iﬀ M, s, u |=ssl ϕ ϕ ∧ ψ iﬀ M, s, u |=ssl ϕ and M, s, u |=ssl ψ Kϕ iﬀ M, t, u |=ssl ϕ for all t ∈ u 3ϕ iﬀ M, s, v |=ssl ϕ for some v ∈ O such that s ∈ v ⊆ u

Note that |=ssl is not deﬁned on all pairs (s, u) ∈ X × O, but only on pairs (s, u) with s ∈ u ∈ O. The semantic entailment ssl is deﬁned by the truth preservation on all such pairs. We will omit the subscript ssl in |=ssl and ssl if the context makes them clear. We say that ϕ is valid in the subset frame F , denoted F |= ϕ, if ϕ is true in all subset models based on F . A subset frame F = (S, O) is (ﬁnitely) downwards closed if for any u ∈ O, all (ﬁnite) non-empty subsets of u are also in O. A subset model is (ﬁnitely) downwards closed if its underlying subset frame is (ﬁnitely) downwards closed. 9

See [7] for details.

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Observe that if a subset frame is downwards closed, then it is also closed under arbitrary intersection. The axiomatic system of SSL is the combination of S5 for K and S4 for 3 plus the following axioms:10 – (p → 2p) ∧ (¬p → 2¬p), for p ∈ P V – K2ϕ → 2Kϕ It has been shown in Proposition 1 that the above axioms are valid in SLK. Since 2ϕ → 3Kϕ is valid in SLK but not in SSL, SSL is a proper sub-logic of SLK. A more interesting result is that one direction of the reduction axioms (RDn) characterizes the ﬁnite downward closure condition for subset frames. Precisely, we have the following proposition. Proposition 3. For any subset frame F , F is ﬁnitely downwards closed iﬀ for all n ∈ N, F |= p ∧ p0 ∧ L(p0 ∧ p1 ) ∧ · · · ∧ L(p0 ∧ pn ) → 3(p ∧ Kp0 ∧ Lp1 ∧ · · · ∧ Lpn ), where p and all pi are diﬀerent propositional variables. Proof. The direction from left to right can be easily veriﬁed. For the other direction, suppose F = (X, O) is not ﬁnitely downwards closed. Then there is u ∈ O and ∅ = v ⊆ u such that v is ﬁnite and v ∈ / O. Let v = {0, 1, . . . , n} for some n ≥ 0. Deﬁne V : P V → ℘(W ) such that v(p) = {0}, v(p0 ) = v and v(pi ) = {i} for 1 ≤ i ≤ n. Let M = (X, O, V ). Then M, 0, u |= p ∧ p0 ∧ L(p0 ∧ p1 ) ∧ · · · ∧ L(p0 ∧ pn ). Let w ∈ O be an arbitrary subset of u containing 0. Since v ∈ / O, w = v. Thus w either contains an element outside of v or does not contain all the elements of v. In the former case, we have M, 0, w |= Kp0 . Hence M, 0, w |= p ∧ Kp0 ∧ Lp1 ∧ · · · ∧ Lpn . In the latter case, suppose j ∈ v is not in w. Then M, 0, w |= Lpj . Hence M, 0, w |= p ∧ Kp0 ∧ Lp1 ∧ · · · ∧ Lpn . By the arbitrariness of w, we have M, 0, u |= 3(p ∧ Kp0 ∧ Lp1 ∧ · · · ∧ Lpn ). Therefore F |= p ∧ p0 ∧ L(p0 ∧ p1 ) ∧ · · · ∧ L(p0 ∧ pn ) → 3(p ∧ Kp0 ∧ Lp1 ∧ · · · ∧ Lpn ), as required. It has been proved in [7] that SSL is more expressive than SEL. The above result shows that ﬁnite downward closure is a suﬃcient frame condition for the reduction of SSL to SEL. Whether the condition is necessary is open to us. It can be easily veriﬁed that all axioms and rules of SLK are valid in ﬁnitely downwards closed subset frames. It follows from Proposition 3 that the axiomatic system of SLK is sound and complete with respect to the class of all ﬁnitely downwards closed subset frames. Since there are ﬁnitely downwards closed subset frames that are not downwards closed, the formulas in Proposition 3 can not characterize the downward closure condition. But it turns out that SLK is also complete with respect to the class of downwards closed subset frames, which will be proved in the rest of this subsection. 10

The proof of the completeness is non-trivial. See [7] for details.

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First, we generalize the notions of submodels and generated submodels for subset models. Given two subset models M = (X, O, V ) and M = (X , O , V ), we say that M is a submodel of M if X ⊆ X, O ⊆ O, and V (p) = V (p)∩X for all p ∈ P V . We say that M is a generated submodel of M if M is a submodel of M and for any u ∈ O if v ⊆ u and v ∈ O then v ∈ O . Like standard generated submodels, generated submodels for subset models also preserve truth values, as shown by the following lemma. Lemma 3. Let M = (X , O , V ) be a generated submodel of M = (X, O, V ) and s ∈ u ∈ O . Then for all ϕ ∈ Lslk , M, s, u |= ϕ iﬀ M , s, u |= ϕ Now we prove that any epistemic model is equivalent to a downwards closed subset model. Definition 6. Given an epistemic model M = (S, ∼, V ), we deﬁne the derived subset model M s = (S, O, V ) from M by O := {[x]∼ | x ∈ S and ∼ ⊆ ∼ is an equivalence relation}, where [x]∼ is the equivalence class of x with respect to ∼ . Lemma 4. Let M = (S, ∼, V ) be an epistemic model and M = (S, ∼ , V ) an update of M . Let M s = (S, O, V ) and M s = (S, O , V ) be the derived subset models from M and M , respectively. 1. For any u = [x]∼ ∈ O, x ∈ v ⊆ u iﬀ v = [x]∼ for some equivalence relation ∼ ⊆ ∼ . 2. M s and M s are downwards closed. 3. M s is a generated submodel of M s . Proof. For Clause 1, consider ∼ = ∼ ∩ (v × v). Clause 3 follows from 2, which follows from 1 directly. Theorem 2. Let M = (S, ∼, V ) be an epistemic model and s ∈ M . Then for all ϕ ∈ Lslk , M, s |=slk ϕ iﬀ M s , s, [s]∼ |=ssl ϕ Proof. By induction on ϕ. The only interesting cases are ϕ = Kψ and ϕ = 3ψ. For the ﬁrst case, we have M, s |=slk ϕ iﬀ M, t |=slk ψ for all t ∈ [s]∼ iﬀ M s , t, [t]∼ |=ssl ψ for all t ∈ [s]∼ iﬀ M s , t, [t]∼ |=ssl Kψ, noting that [s]∼ = [t]∼ . The second ‘iﬀ’ is by induction hypothesis. For the second case, we have M, s |=slk ϕ iﬀ M , s |=slk ψ for some M = (S, ∼ , V ) such that ∼ ⊆ ∼ iﬀ M s , s, [s]∼ |=ssl ψ for some M = (S, ∼ , V ) such that ∼ ⊆ ∼ iﬀ M s , s, [s]∼ |=ssl ψ for some equivalence relation ∼ ⊆ ∼ iﬀ M s , s, v |=ssl ϕ for some v ⊆ [s]∼ such that s ∈ v ∈ O iﬀ M s , s, [s]∼ |=ssl 3ψ. The second ‘iﬀ’ is by induction hypothesis. The third and fourth ‘iﬀ’ are by Lemma 3 and Lemma 4.

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Next we prove that any downwards subset model is equivalent to an epistemic model. Definition 7. Given a subset model M = (S, O, V ) and u ∈ O, we deﬁne the derived epistemic model (M, u)e = (S, ∼, V ) from the pair (M, u) by ∼ := {(x, y) | x, y ∈ u or x = y} It is easily seen that ∼ deﬁned above is an equivalence relation. Hence (M, u)e is indeed an epistemic model. For any epistemic model M and s ∈ M , we write M |s for the generated submodel of M by s. Note that here the notion of generated submodel is standard. Lemma 5. Let M = (S, O, V ) be a downwards closed subset model and (M, u)e the derived epistemic model from (M, u). 1. If v ∈ O and v ⊆ u, then (M, u)e ≤ (M, v)e . 2. If (M, u)e ≤ M , then for any s ∈ u, there is v ∈ O such that s ∈ v ⊆ u and (M, v)e |s = M |s . Proof. 1. Suppose (M, v)e = (S, ∼ , V ) and (M, u)e = (S, ∼, V ). By Deﬁnition 7, from v ⊆ u it follows that ∼ ⊆ ∼. Hence (M, u)e ≤ (M, v)e . 2. Let (M, u)e = (S, ∼, V ) be the derived epistemic model from (M, u) and M = (S, ∼ , V ) an update of (M, u)e . For any s ∈ S, let v = [s]∼ . Then s ∈ v ⊆ u. Since M is downwards closed, v ∈ O. Now it should be easy to verify that (M, v)e |s = (v, v × v, V |v ) = M |s , where V |v is the valuation such that V |v (p) = V (p) ∩ v for p ∈ P V . Theorem 3. Let M = (S, O, V ) be a downwards closed subset model and s ∈ u ∈ O. Then for all ϕ ∈ Lslk , M, s, u |=ssl ϕ iﬀ (M, u)e , s |=slk ϕ Proof. By induction on ϕ. Suppose (M, u)e = (S, ∼, V ). The only interesting cases are ϕ = Kψ and ϕ = 3ψ. For the ﬁrst case, we have M, s, u |=ssl ϕ iﬀ M, t, u |=ssl ψ for all t ∈ u iﬀ (M, u)e , t |=slk ψ for all t ∈ u iﬀ (M, u)e , t |=slk ψ for all t such that s ∼ t iﬀ (M, u)e , s |=slk Kψ, where the second ‘iﬀ’ is by induction hypothesis and the third ‘iﬀ’ is by the deﬁnition of ∼. For the second case, suppose M, s, u |=ssl ϕ. Then M, s, v |=ssl ψ for some v ∈ O such that s ∈ v ⊆ u. By induction hypothesis, we have (M, v)e , s |=slk ψ. By Lemma 5.1, we have (M, u)e ≤ (M, v)e . It follows that (M, u)e , s |=slk 3ψ. For the other direction, suppose (M, u)e , s |=slk ϕ. Then M , s |=slk ψ for some M ≥ (M, e)e . Thus M |s , s |=slk ψ. By Lemma 5.2, there is v ∈ O such that s ∈ v ⊆ u and (M, v)e |s = M |s . Hence (M, v)e , s |=slk ψ. By induction hypothesis, we have M, s, v |=ssl ψ. Therefore, M, s, u |=ssl 3ψ.

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Let’s denote by DSSL the logic characterized by the class of downwards closed subset frames. Then by Theorem 2 and 3, we have the following theorem. Theorem 4. SLK is equivalent to DSSL, i.e. Σ slk ϕ iﬀ Σ dssl ϕ. Since SLK can be reduced to EL and EL has the ﬁnite model property, it follows from Theorem 2 and 3 that DSSL also has the ﬁnite model property, which is absent from SSL.11 4.2

Multi-Agent Systems

SSL is a single-agent logic and there is no standard formalisation of multi-agent SSL yet. In [3], several multi-agent SSL semantics were proposed. But the resulted semantics for Ki is quite diﬀerent from standard multi-agent epistemic logic. For example, K1 K2 ϕ ↔ K2 K1 ϕ is valid in the semantics. Another drawback is that the 3 operator also splits into n operators. In [12], Heinemann gave another multi-agent SSL. In addition to the K operator, there is an operator i for each agent i. Heinemann suggested to use i Kϕ to express Ki ϕ. According to his semantics, however, Ki does not satisfy the axiom of negative introspection any more. Moreover, the remaining K operator does not have an intuitive reading. A most possible reading of Kϕ is that some agent knows ϕ. But K(ϕ → ψ) → (Kϕ → Kψ) is valid in the semantics, which is counter-intuitive if Kϕ means that someone knows that ϕ. Here we give a new semantics for multi-agent SSL. It generalizes SSL naturally. One merit of the semantics is that the operators Ki behave in the same way as those in standard multi-agent epistemic logic. Moreover, the relation between LK and SSL in the single-agent situation is preserved in the multi-agent version. First, we deﬁne the models for multi-agent SSL. Definition 8. A multi-agent subset frame is a pair F = (S, O), where – S = ∅, consisting points, – O : n → ℘(℘(S)), assigning to each agent i a set Oi of subsets of S. The multi-agent subset model based on F is deﬁned as usual. A multi-agent subset frame F = (S, O) is downward closed if each (S, Oi ) is downward closed. For brevity, we write u for (u1 , . . . , un ). We write s ∈ u if s ∈ ui for all i ∈ N . Similarly, we write v ⊆ u if vi ⊆ ui for all i ∈ N . By slightly abusing the notation, we also use O for Oi . Definition 9. Given a multi-agent subset space model M = (S, O, V ), the satisfaction of ϕ at (s, u) in M for s ∈ u ∈ O , denoted M, s, u |=mssl ϕ, is inductively deﬁned as follows: – M, s, u |=mssl p iﬀ s ∈ V (p), for p ∈ P V – M, s, u |=mssl ¬ϕ iﬀ M, s, u |=mssl ϕ 11

See [7] for a counter-example.

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– M, s, u |=mssl ϕ ∧ ψ iﬀ M, s, u |=mssl ϕ and M, s, u |=mssl ψ – M, s, u |=mssl Ki ϕ iﬀ M, t, u |=mssl ϕ for all t ∈ ui – M, s, u |=mssl 3ϕ iﬀ M, s, v |=mssl ϕ for some v ∈ O such that s ∈ v ⊆ u All the other deﬁnitions and theorems in Section 4.1 can be obtained translationally for multi-agent SSL. For example, we can deﬁne the multi-agent derived epistemic model from a multi-agent subset model as follows. Definition 10. Given a multi-agent subset model M = (S, O, V ) and u ∈ O, we deﬁne the derived (multi-agent) epistemic model (M, u)e = (S, ∼, V ) from the pair (M, u) by ∼i := {(x, y) | x, y ∈ ui or x = y} for all i ∈ n. In particular, we have the following theorem. Theorem 5. LK is equivalent to DM SSL, i.e. Σ lk ϕ iﬀ Σ dmssl ϕ, where DM SSL is the logic characterized by the class of downward closed multi-agent subset frames.

5

Conclusion

We give an alternative logic LK for knowability to arbitrary public announcement logic. By comparing LK to arbitrary public announcement logic and subset space logic, we establish a link between the latter two logics. It turns out that the single-agent arbitrary public announcement logic without announcements can be characterized by the class of downwards closed subset frames. Though we call LK a logic, it is mainly a semantics in this paper. The axiomatisation of LK is still open to us. We hope its solution may also help to solve the axiomatisation of ALK, which is an open problem pointed out in [2]. Observe that though the information update we give in this paper is general enough, it is still a special one. In our modelling, when an agent gets new information, the other agents know that she has got new information, though she may not know what exactly the information is. But in reality, an agent can certainly updates her information secretly. That’s why the action model (a.k.a. event model) is required to model a more general information update and the notion of knowability based on it, which has already been suggested in [2]. Further research includes the comparison between LK and the simulation logic proposed in [9] and the partition logic proposed in [11]. This paper is only a ﬁrst step for the interaction between dynamic epistemic logic and topological reasoning. The connection between these two areas may help to borrow both philosophical ideas and proof techniques from each other. It may also help to connect dynamic epistemic logic and temporal logic, especially the logic of branching time, since the latter has a very similar semantics to subset space logic.

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Acknowledgements. The paper was partially supported by National Social Science Foundation of China (Grant No. 09CZX031). We would like to thank the anonymous referees for their helpful comments.

References 1. Balbiani, P., Baltag, A., van Ditmarsch, H., Herzig, A., Hoshi, T.: What can we achieve by arbitrary announcements?: A dynamic take on Fitch’s knowability. In: Proceedings of TARK, pp. 42–51 (2007) 2. Balbiani, P., Baltag, A., van Ditmarsch, H., Herzig, A., Hoshi, T., de Lima, T.: ‘Knowable’ as ‘known after an announcement’. Review of Symbolic Logic 1, 305– 334 (2008) 3. Baskent, C.: Topics in Subset Space Logic. Master Thesis, University of Amsterdam (2007) 4. van Benthem, J.: What one may come to know. Analysis 64(282), 95–105 (2004) 5. Blackburn, P., de Rijke, M., Venema, Y.: Modal logic. Cambridge University Press, Cambridge (2001) 6. Brogaard, B.: Fitch’s paradox of knowability. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (2009), http://plato.stanford.edu/entries/fitch-paradox/ 7. Dabrowski, A., Moss, L., Parikh, R.: Topological reasoning and the logic of knowledge. Annals of Pure and Applied Logic 78, 73–110 (1996) 8. van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Springer, Heidelberg (2007) 9. van Ditmarsch, H., French, T.: Simulation and Information: Quantifying over Epistemic Events. In: Meyer, J.J., Broersen, J. (eds.) KRAMAS 2008. LNCS, vol. 5605, pp. 51–65. Springer, Heidelberg (2009) 10. Fitch, F.: A logical analysis of some value concepts. Journal of Symbolic Logic 28(2), 135–142 (1963) 11. French, T.N.: Bisimulation Quantiﬁers for Modal Logics. Ph.D. thesis, The University of Western Australia (2006) 12. Heinemann, B.: Topology and Knowledge of Multiple Agents. In: Geﬀner, H., Prada, R., Machado Alexandre, I., David, N. (eds.) IBERAMIA 2008. LNCS (LNAI), vol. 5290, pp. 1–10. Springer, Heidelberg (2008) 13. Heinemann, B.: Reﬁning the Notion of Eﬀort. In: Proceedings of ECAI, pp. 1071– 1072 (2010) 14. Heinemann, B.: Using hybrid logic for coping with functions in subset spaces. Studia Logica 94(1), 23–45 (2010) 15. Meyer, J.J., van Der Hoek, W.: Epistemic Logic for AI and Computer Science. Cambridge University Press, Cambridge (1995) 16. Moss, L., Parikh, R.: Topological reasoning and the logic of knowledge: preliminary report. In: Proceedings of TARK, pp. 95–105 (1992) 17. Moss, L., Parikh, R., Steinsvold, C.: Topology and Epistemic Logic. In: Aiello, M., Pratt-Hartmann, I., van Benthem, J. (eds.) Handbook of Spatial Logic. Springer, Heidelberg (2007)

Conditional Ought, a Game Theoretical Perspective Xin Sun Department of Philosophy, Tsinghua University, Beijing, China [email protected]

Abstract. This paper presents a new consequentialist deontic logic in which the relation of preference over sets of possible worlds and the relation of conditional dominance are both transitive. This logic validate the principle that absolute ought can be derived from conditional ought whenever the conditional statement is the agent’s absolute ought. Ought about conditionals is not implied by conditional ought in this logic. Keywords: consequentialist deontic logic, transitivity, conditional dominance.

1

Introduction

Lucy is playing the matching pennies game with Lily. They choose, simultaneously, whether to show the head or the tail of a coin. If they show the same side, Lucy receives one dollar; if they show diﬀerent sides, Lucy receive 0 dollar. This situation can be depicted in Figure 1:(The letters ϕ and ψ are not referred in current discussion, but are involved in the proof of proposition 21.) Lily head tail ϕ ϕ 0 head 1 w1 ψ w2 Lucy γ tail w 0 ψ w 1 3 4 Fig. 1.

It is obvious that under the condition of Lily showing head, Lucy ought to see to it that she shows head. Our question is, does Lucy ought to see to it that if Lily shows head, then she shows head? The answer seems to be positive at ﬁrst sight. But here we claim the answer is negative. The conclusion can be achieved by following reasoning: Denote the situation in which Lucy shows tail and Lily shows head as situation γ. First note that to Lucy, both showing head and showing tail are optimal, which implies Lucy is permitted to show tail. Since Lucy showing tail may lead to situation γ, Lucy is permitted to lead to situation H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 356–369, 2011. c Springer-Verlag Berlin Heidelberg 2011

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γ. Next, assume the answer is positive, that is, Lucy ought to see to it that if Lily shows head, then Lucy shows head. This means Lucy ought to prohibit the the following outcome of the game: Lily shows head but Lucy shows tail. Hence Lucy ought to prohibit situation γ. Contradiction. We refer sentences of the form “under the condition of ϕ, agents ought to see to it that ψ” as conditional ought; and we use ought about conditionals to name sentences of the form “agents ought to see to it that if ϕ then ψ”. [1] suggests that our desired theory of conditional obligation should include the principle that conditional ought implies ought about conditionals. In [2], this principle is involved in the strongly normal system G. But our matching pennies example indicates that this principle is doubtable. In this paper, we are going to present a consequentianlist deontic logic in which ought about conditionals is not implied by conditional ought. Our consequentianlist deontic logic is based on the logic of [5], [7], [6] and [11]. [5] is a notable book, which represents a major advance in the ﬁeld of stit-based deontic logic. However, there are some technical mistakes which are derived from an inappropriate deﬁnition of preference over sets of possible worlds and causes several problems when discussing the properties of conditional ought. [7], inspired by [5], developed a consequentianlist deontic logic which can be used to analysis moral conﬂicts between diﬀerent groups of agents with diﬀerent moral codes. [6] adds conditional ought to consequentianlist deontic logic. Although the semantic of conditional ought in [6] is slightly diﬀerent from that of [5], it potentially bears similar problems as [5]. In our new consequentianlist deontic logic, most mis-proved theorems in [5] can be proved and the potential problems of conditional ought in [6] are avoided. Furthermore, as we have already mentioned, this new logic can distinguish conditional ought and ought about conditionals. Consequentianlist deontic logic is in some sense a kind of preference based deontic logic, related work in this ﬁeld include [12] and [3]. The structure of this paper is as follows: Section 2 is an introduction to our new consequentianlist deontic logic, including the language and semantics. In Section 3 we analyze some principles of conditional ought using our new logic. Section 4 is conclusion and future work. The proves for propositions are listed in the Appendix.

2 2.1

A New Consequentianlist Deontic Logic Language

The language of consequentianlist deontic logic is built from a ﬁnite set A of agents and a countable set P of atomic propositions. We use p and q as variables for atomic propositions in P , use F and G, where F , G ⊆ A, as groups of agents. The consequentianlist deontic language L is given by the following Backus-Naur Form: F F ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | ♦ϕ | [G]ϕ | G ϕ | G (ϕ/ϕ)

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Intuitively, ♦ϕ can be read as “ It is possible that ϕ”. [G]ϕ can be read as F “Group G sees to it that ϕ”. G ϕ can be read as “In the interest of group F , F group G ought to see to it that ϕ”. G (ϕ/ψ) can be read as “In the interest of group F , group G ought to see it that ϕ under the condition of ψ”. We use to F PF ϕ as an abbreviation of ¬ ¬ϕ, which can be read as ‘In the interest of G G group F , group G is permitted to lead to a situation in which ϕ is true.” 2.2

Consequentialist Frame

The semantics of consequentianlist deontic logic is based on consequentianlist frames. Similar to [6], our deﬁnition of consequentianlist frame is as follows: Definition 1 (consequentianlist frame). A consequentianlist f rame F is a quadruple W ,A,Choice, {V alueF }F ⊆A , where W is a nonempty set of possible worlds, A is a ﬁnite set of agents, Choice is a choice function, and V alueF , represents the preference of some group of agents F ⊆ A, is a function from W to the set of real numbers R. Formally, V alueF : W → R. The choice function Choice is a function from the power set of A to the power set of the power set of W , i.e. Choice : ℘(A) → ℘(℘(W )). Choice is built from the individual Choice function IndChoice: A → ℘(℘(W )). IndChoice must satisfy the following three conditions: (1) for each agent i∈A it holds that IndChoice(i) is a partition of W ; (2) for each selection function s that assigning toeach agent i ∈ A a set of possible worlds s(i) ∈ IndChoice(i), it holds that i∈A s(i) in nonempty; (3) for each i ∈ A, the set IndChoice(i) is ﬁnite. Let Select be the set of all selection functions, then Choice(G) = { i∈G s(i) : s ∈ Select} if G is nonempty. Otherwise, Choice(G) = {W }. For any two world w and w , if there exist a K ∈ Choice(G) such that w ∈ K and w ∈ K, we denote it as w ∼G w . Intuitively, w ∼G w means the choice of group G cannot sperate w and w . Take the Prisoner’s Dilemma in [9] as an example: player β quiet ﬁnk quiet player α

ﬁnk

w1

3, 3

w3 4, 0

w2

0, 4

w4 1, 1

Fig. 2.

In this example, A={α β},W = {w1 , w2 , w3 , w4 }, IndChoice (α)={{w1 , w2 },{w3 ,w4 }}, IndChoice(β) ={{w1 , w3 },{w2 ,w4 }}. Apparently both

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IndChoice(α) and IndChoice(β) are partitions of W . And there are four selection functions, Select ={s1 , s2 , s3 , s4 }, where: s1 (α) = {w1 , w2 }, s1 (β) = {w1 , w3 } s2 (α) = {w1 , w2 }, s2 (β) = {w2 , w4 } s3 (α) = {w3 , w4 }, s3 (β) = {w1 , w3 } s4 (α) = {w3 , w4 }, s4 (β) = {w2 , w4 } So we have for each s ∈ Select, i∈A s(i) is not empty. Therefore the two conditions of individual choice are both satisﬁed. Then we have Choice(A) = { i∈A s(i) : s ∈ Select}= {{w1 }, {w2 }, {w3 }, {w4 }}. Having deﬁned consequentianlist frames, we are able to deﬁne preferences over sets of worlds. In [5], the deﬁnition is like following: Definition 2 . Let X ⊆ W , Y ⊆ W be two sets of worlds, F a group of agents from a consequentialist frame. Then X ≤F Y if and only if for each w ∈ X, for each w ∈ Y , V alueF (w) ≤ V alueF (w ). As we have mentioned, it is this deﬁnition that causes mistakes. According to this deﬁnition, for any set X ⊆ W , X ≤F ∅ is true and for any set Y ⊆ W , ∅ ≤F Y is also true. Hence this preference relation can’t be transitive. But intuitively preference relations should be transitive and in fact lots of theorems in [5] are based on the transitivity of this preference relation. So we must modify the deﬁnition to make it transitive. Our attempt is following: Definition 2 (preferences over sets of worlds; ≤F ,
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Proposition 5. Let X and Y be sets of worlds, F a group of agents from a consequentialist frame. Then: 1. If X ≤F Y and Y ≤F Z, then X ≤F Z. 2. If X ≤F Y and Y
Proposition 7. Let F , G be groups of agents from a consequentialist frame, and let K, K ∈ Choice(G). Then K
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tent with condition X. We can further deﬁne conditional dominance relation over agent’s choice. The intuition is, to compare whether the agent’s choice K is dominated by K under the condition X, we only need to concern other agents’ choices which are consistent with the condition X and one of K and K . Definition 10 (conditional dominance; ≤F G/X ). Let F , G be groups of agents from a consequentialist frame, X a set of worlds in the frame. Let K, K ∈ Choice(G/X). Then K ≤F G/X K iﬀ for all S ∈ Choice((A − G)/(X ∩ (K ∪ K ))), K ∩ X ∩ S ≤F K ∩ X ∩ S. K ≤F G/X K can be read as “in the interest of group F , K weakly dominates K F under the condition of X”. And we use K
Proposition 11. Let F , G be groups of agents from a consequentialist frame, X a set of worlds in the frame. Let K, K ∈ Choice(G/X). Then K
(2)

w2

(0)

K2 w (3) 3

X w4

(1)

K3 w5 (3)

w6

(1)

w1

X

X

Fig. 3.

Here W = {w1 , ..., w6 }, A = {α, β} Choice({α}) = {K1 , K2 , K3 }, Choice({β}) = {S1 , S2 }, K1 = {w1 , w2 }, K2 = {w3 , w4 }, K3 = {w5 , w6 }, S1 = {w1 , w3 , w5 },

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S2 = {w2 , w4 , w6 }, X = {w2 , w3 , w6 }, and the number in the brackets represents the value of the world in the interest of group A. According to Horty’s deﬁnition, A A we have K3 ≤A {α}/X K2 and K2 ≤{α}/X K1 but we don’t have K3 ≤{α}/X K1 . F Therefore in [5] the ≤G/X relation is not transitive and Proposition 5.4 is not totally true. In [6], the deﬁnition of restricted choice set and conditional dominance is diﬀerent from ours. See the following: Definition 9 Let F , G be groups of agents from a consequentialist frame, X a set of worlds in the frame. Then Choice(G/X) = {K ∩ X : K ∈ Choice(G) and K ∩ X = ∅} Definition 10 Let F , G be groups of agents from a consequentialist frame, X a set of worlds in the frame. Let k, k ∈ Choice(G/X). Then k ≤F G/X k iﬀ for all S ∈ Choice(A − G) and for all w and w ∈ W it holds that if w ∈ k ∩ S and w ∈ k ∩ S, then V alueF (w) ≤ V alueF (w )

It’s easy to verify that according to above deﬁnition, Choice(G/X) = {{w2 }, A A {w3 }, {w6 }} and {w6 } ≤F G/X {w3 }, {w3 } ≤{α}/X {w2 }, but {w6 } ≤{α}/X {w2 } does not hold. Hence this version of conditional dominance is not transitive. 2.3

Semantics

As in traditional modal logic, a model is a frame plus the valuation function. Definition 13 (consequentialist model) A consequentialist model M is an ordered pair F, V where F is a consequentialist frame and V a valuation function that assigns to each atomic proposition p ∈ P a set of worlds V (p) ⊆ W . In our semantics, we use the optimal choice and condition optimal choice to interpret our deontic operator. The deﬁnition of optimal (Deﬁnition 14) and conditional optimal (Deﬁnition 16) is rather simple. F Definition 14 (OptimalG ) Let F , G be groups of agents from a consequentialist frame, F OptimalG = {K ∈ Choice(G) : there’s no K ∈ Choice(G) such that K
Proposition 15 Let F , G be groups of agents from a consequentialist frame, F F , there exist K ∈ OptimalG such that then for each K ∈ Choice(G) − OptimalG F K
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F OptimalG/X = {K ∈ Choice(G/X) : there’s no K ∈ Choice(G/X) such that K
Proposition 17. Let F , G be groups of agents from a consequentialist frame, F F for each K ∈ Choice(G/X) − OptimalG/X , there exist K ∈ OptimalG/X such F that K
w1 (3)

K2 w 4 K3 w7 (2)

X w2 (0)

X w3

(1)

X w6

w5 X w8

(4)

X

w9 (5)

X

Fig. 4.

In above case, W = {w1 , ..., w9 }, A = {α, β}, Choice({α}) = {K1 , K2 , K3 }, Choice({β}) = {S1 , S2 , S3 }, K1 = {w1 , w2 , w3 }, K2 = {w4 , w5 , w6 }, K3 = {w7 , w8 , w9 }, S1 = {w1 , w4 , w7 }, S2 = {w2 , w5 , w8 }, S3 = {w3 , w6 , w9 } X = {w1 , w2 , w5 , w6 , w7 , w9 }, the numbers in the brackets represent the value of the world in the interest of A. If Deﬁnition 2 is replaced by Deﬁnition 2 , A A we would have K1
M, w M, w M, w M, w M, w M, w M, w

|= p iﬀ w ∈V(p); |= ¬ϕ iﬀ it is not that M, w |= ϕ; |= ϕ ∧ ψ iﬀ M, w |= ϕ and M, w |= ψ; |= ♦ϕ iﬀ there is a w such that M, w |= ϕ; |= [G]ϕ iﬀ for all w with w ∼G w it holds that M, w |= ϕ; F F |= G ϕ iﬀ K ⊆ ||ϕ|| for each K ∈ OptimalG ; F F |= G (ϕ/ψ) iﬀ K ⊆ ||ϕ|| for each K ∈ OptimalG/ψ .

F F Here ϕ = {w ∈ W : M, w |= ϕ}. OptimalG/ψ is shorthand for OptimalG/||ψ|| . We say ϕ is true in the world w of a consequentialist model M if M, w |= ϕ. Just like the standard modal logic in [4], we introduce the concept of validity as following: a formula ϕ is valid in a world w of a consequentialist frame F ( notation: F, w |= ϕ) if ϕ is true at w in every model F, V based on F; ϕ is valid in a consequentialist frame F (notation: F |= ϕ) if it is valid at every world of F; ϕ is valid (notation: |= ϕ) if it is valid in the class of all consequentialist frames.

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Revisit Principles of Conditional Ought

[1] suggests that the logic of commitment or conditional ought should satisfy six principles. In language, are as follows: our they F F (1) (ψ ∧ G (ϕ/ψ)) → G ϕ. F F F (2) ( G ψ ∧ G (ϕ/ψ)) → G ϕ. (3) (PF Gψ ∧

F

G (ϕ/ψ))

→ PF G ϕ.

F F F (4) ( G (ϕ/ψ) ∧ G (χ/ϕ)) → G (χ/ψ) (5)

F

(6)

F

G (ϕ/ψ)

→

G (ϕ/¬ϕ)

F

→

G (ψ

F

G

→ ϕ)

ϕ

The strongly normal system G in [2] excludes principle (1) and (4) and involves the others. In our logic, however, only principle (2) and (6) are valid. The invalidity of principle (1) and (4) and the validity of principle (6) are easy to prove, here we skip it. For the rest three principles, we have following propositions: F F F Proposition 19. The statement ( G ψ ∧ G (ϕ/ψ)) → G ϕ is valid. Proposition 20. The statement (PF Gψ ∧ Proposition 21. The statement

F

G (ϕ/ψ))

F

G (ϕ/ψ)

→

F

→ PF G ϕ is not valid.

G (ψ

→ ϕ) is not valid.

The invalidity of principle (3) can be illustrated by a variation of the matching pennies game. In this new game, Lucy has three choices, showing head, showing tail and refraining from showing. If Lucy refrains from showing, then no matter how Lily acts, Lucy will receive 50 dollars. The detailed payoﬀ of Lucy is indicated by numbers in brackets in Figure 5. Head

Tail

Head

w1 (100)

w2 (0)

Tail

ϕ (20) w3

ϕ (30) w4 ψ

Refrain w (50) 5

w6 (50)

Fig. 5.

ψ

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Denote the situation in which Lucy shows one side of her penny and Lily shows tail as situation ψ. Lucy is permitted to lead to situation ψ, since showing head is one of Lucy’s optimal action and this action could lead to situation ψ. Apparently under situation ψ, Lucy ought to see to it that she shows tail. But Lucy is not permitted to show tail because this action is dominated by refraining.

4

Conclusion and Future Work

The main point of this paper is to introduce a new consequentianlist deontic logic in which the relation of preference over sets of worlds and the conditional dominance relation are both transitive. With transitivity a bunch of good properties could be proved. Our logic support the principle that absolute ought can be derived from conditional ought whenever the conditional statement is the agent’s absolute ought. According to our semantics, conditional ought does not imply ought about conditionals. One line of our future work is to create an axiomatic system for consequentianlist deontic logic and prove its soundness and completeness. Another line is to add epistemic modality to our system. For some related work in this area, see [10] and [8]. Acknowledgments. The author would like to thank Fenrong Liu for her encouragement and suggestion in the whole process of writing this paper. And special thanks to Johan van Benthem, John Horty, Leon van der Torre, Allard Tamminga, Xavier Parent and three anonymous referees for their comments.

References 1. Anderson, A.: On the logic of commitment. Philosophical Studies 10(1959), 23–27 (1959) 2. Aqvist, L.: Deontic logic. Handbook of Philosophical Logic 2(1994), 147–264 (1994) 3. van Benthem, J., Grossi, D., Fenrong, L.: On the two faces of deontics: Semantic betterness and syntactic priority (2011) (manuscript) 4. Blackburn, P., Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001) 5. Horty, J.: Agency and Deontic Logic. Oxford University Press, Oxford (2001) 6. Kooi, B., Tamminga, A.: Conditioanl obligations in strategic situations. In: Boella, G., Pigozzi, G., Singh, M., Verhagen, H. (eds.) Proceedings of the 3rd International Workshop on Normative Multiagent Systems (2008) 7. Kooi, B., Tamminga, A.: Moral conflicts between groups of agents. Journal of Philosophical Logic 37(2008), 1–21 (2008) 8. Loohuis, L.: Obligations in a responsible world. In: He, X., Horty, J., Pacuit, E. (eds.) LORI 2009. LNCS, vol. 5834, pp. 251–262. Springer, Heidelberg (2009) 9. Osborne, M., Rubinstein, A.: A Course in Game Theory. The MIT Press, Cambridge (1994) 10. Pacuit, E., Parikh, R., Cogan, E.: The logic of knowledge based obligation. Synthese 149(2006), 311–341 (2006) 11. Tamminga, A.: Deontic logic for strategic games (2011) (manuscript) 12. van der Torre, L.: Reasoning about Obligations: Defeasibility in Preference-based Deontic Logic. Phd thesis, Erasmus University Rotterdam (1997)

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Appendix Proof of Lemma 3. Straightforward. Proof of Proposition 4. Left to right. Assume X V alueF (w). But we already have X = ∅ and Y = ∅. Hence for some w ∈ Y , for some w ∈ X V alueF (w ) > V alueF (w), then (2) is true. Right to left. By (2) we know X = ∅ and Y = ∅. This plus (1) implies X ≤F Y . So it’s suﬃcient to prove it is not the case that Y ≤F X. Suppose Y ≤F X, then for each w ∈ Y and for each w ∈ X, V alueF (w ) ≤ V alueF (w). But according to (2), for some w ∈ Y and for some w ∈ X, V alueF (w ) > V alueF (w). Contradiction. Proof of Proposition 5. Here we just prove Clause 1. Assume X ≤F Y and Y ≤F Z, then X = ∅, Y = ∅ and Z = ∅. Let w be arbitrary history in X, w be arbitrary history in Z. By Y = ∅ we know there exist some w ∈ Y . By X ≤F Y and Y ≤F Z we know V alueF (w) ≤ V alueF (w ), V alueF (w ) ≤ V alueF (w ), hence V alueF (w) ≤ V alueF (w ). Therefore X ≤F Z. F Proof of Proposition 7. Left to right. Assuming K
Proof of Proposition 8. See Proposition 4.7 of [5]. Proof of Proposition 11. Similar to the proof of Proposition 7. To prove Proposition 12, we need following Lemmas: Lemma A. Let F , G be groups of agents from a consequentialist frame, X a set of worlds in the frame. Let K, K ∈ Choice(G/X). If K ≤F G/X K , then Choice((A − G)/(X ∩ K))=Choice((A − G)/(X ∩ K )). Proof of Lemma A. We are going to prove Choice((A − G)/(X ∩ K)) ⊆ Choice((A−G)/(X ∩K )) and Choice((A−G)/(X ∩K)) ⊇ Choice((A−G)/(X ∩ K )). For Choice((A − G)/(X ∩ K)) ⊆ Choice((A − G)/(X ∩ K )), assume there exist some S ∈ Choice(A − G) such that S ∈ Choice((A − G)/(X ∩ K)) but

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S ∈ Choice((A − G)/(X ∩ K )). Then S ∩ X ∩ K = ∅ and S ∩ X ∩ K = ∅. Therefore S ∩ X ∩ (K ∪ K ) = ∅ and S ∈ Choice((A − G)/(X ∩ (K ∪ K ))). Now by K ≤F G/X K we have K ∩ X ∩ S ≤F K ∩ X ∩ S. This plus Lemma 3 implies K ∩ X ∩ S = ∅. Contradiction. Hence Choice((A − G)/(X ∩ K)) ⊆ Choice((A − G)/(X ∩ K )). The case for Choice((A−G)/(X ∩K)) ⊇ Choice((A−G)/(X ∩K )) is similar. Lemma B. Let G be a group of agents, X and Y be sets of worlds from a consequentialist frame. If Choice((A − G)/X) = Choice((A − G)/Y ), then Choice((A − G)/X) = Choice((A − G)/(X ∪ Y )). Proof of Lemma B. For Choice((A − G)/X) ⊆ Choice((A − G)/(X ∪ Y )). If S ∈ Choice((A − G)/X), then S ∈ Choice((A − G)/Y ) and S ∩ X = ∅, S ∩ Y = ∅. Hence (S ∩ X) ∪ (S ∩ Y ) = ∅, S ∩ (X ∪ Y ) = ∅. So we have S ∈ Choice((A − G)/(X ∪ Y )). For Choice((A − G)/X) ⊇ Choice((A − G)/(X ∪ Y )). If S ∈ Choice((A − G)/(X ∪ Y )), then S ∩ (X ∪ Y ) = ∅, (S ∩ X) ∪ (S ∩ Y ) = ∅. Now assume S∈

Choice((A − G)/X), then S ∈ Choice((A − G)/Y ). Hence S ∩ X = ∅ and S ∩ Y = ∅. So we have (S ∩ X) ∪ (S ∩ Y ) = ∅. Contradiction. Proof of Proposition 12. Here we just prove clause 1. Other clauses are similar. F Assume K ≤F G/X K and K ≤G/X K . By Lemma A, Choice((A − G)/(X ∩ K))=Choice((A − G)/(X ∩ K ))=Choice((A − G)/(X ∩ K )). By Lemma B, we now have Choice((A−G)/(X∩K))=Choice((A−G)/((X∩K)∪(X∩K )))=Choice((A− G)/(X ∩ (K ∪ K ))), Choice((A−G)/(X∩K))=Choice((A−G)/((X∩K)∪(X∩K )))=Choice((A− G)/(X ∩ (K ∪ K ))), Choice((A−G)/(X∩K ))=Choice((A−G)/((X∩K )∪(X∩K )))=Choice((A− G)/(X ∩ (K ∪ K ))). Hence for each S ∈ Choice((A − G)/(X ∩ (K ∪ K ))), we have S ∈ Choice((A − G)/(X ∩ (K ∪ K ))) and S ∈ Choice((A − G)/(X ∩ (K ∪ K ))). Therefore F by K ≤F we G/X K we have K ∩ X ∩ S ≤F K ∩ X ∩ S, by K ≤G/X K have K ∩ X ∩ S ≤F K ∩ X ∩ S. Since the relation ≤F is transitive, we have K ∩ X ∩ S ≤F K ∩ X ∩ S. Therefore K ≤F G/X K . Proof of Proposition 15. Similar to Proposition 4.11 of [5]. Proof of Proposition 17. Similar to Proposition 5.7 of [5]. Proof of Proposition 19. Assume this formula is not valid, then there is a consequentialist frame F and a world w in F such that for some model M

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F F F based on F, M, w |= G ψ ∧ G (ϕ/ψ) but M, w G ϕ. Hence there must be F some K ∈ OptimalG such that K ||ϕ||, K ⊆ ||ψ||. As K ⊆ ||ψ||, for arbitrary S ∈ Choice(A − G), for arbitrary K ∈ Choice(G), we must have S ∩ (||ψ|| ∩ (K ∪ K )) = ∅ because S ∩ K ⊆ S ∩ (||ψ|| ∩ (K ∪ K )) and S ∩ K = ∅ by the deﬁnition of choice function. Hence S ∈ Choice((A − G)/(||ψ|| ∩ (K ∪ K ))) and Choice(A − G) = Choice((A − G)/(||ψ|| ∩ (K ∪ K ))). F F F Obviously either K ∈ OptimalG/ψ or K ∈ / OptimalG/ψ . If K ∈ OptimalG/ψ , F F / OptimalG/ψ . then by M, w |= G (ϕ/ψ), K ⊆ ||ϕ||. Contradiction. Therefore K ∈ Then by Proposition 17 there exist some K ∈ Choice(G/ψ) with K
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F have M, w1 |= PF G ψ and M, w1 |= G (ϕ/ψ). But as {w1 , w2 } ∩ ||ϕ|| = ∅ and {w3 , w4 } ∩ ||ϕ|| = ∅, we have M, w1 PF G ϕ. Proof of Proposition 21. It’s suﬃcient to constructa model M such that F F for some world w in M , M, w |= G (ϕ/ψ) but M, w G (ψ → ϕ). Revisit Figure 1. Let M = W, A, Choice, {V alueF }F ⊆A , V , W = {w1 , ..., w4 }, A = {α, β}, Choice ({α}) = {{w1 , w2 }, {w3 , w4 }}, Choice({β}) = {{w1 , w3 }, {w2 , w4 }}, V alue{α}(w1 ) = 1, V alue{α} (w2 ) = 0, V alue{α}(w3 ) = 0, V alue{α} (w4 ) = 1. Let F = {α}, G = {α}, ||ϕ|| = {w1 , w2 }, ||ψ|| = {w1 , w3 }. F F In this situation, OptimalG/ψ = {{w1 , w2 }}, hence M, w1 |= G (ϕ/ψ). As F OptimalG = {{w1 , w2 }, {w3 , w4 }} and {w3 , w4 } ||ψ → ϕ||, we have M, w1 F G (ψ → ϕ).

The Categorial Logic of Vacuous Components in Natural Language Chongli Zou1, Kesheng Li2, and Lu Zhang1 1

Institute of Philosopy, the Chinese Academy of Social Sciences No. 5, Jianguomennei Avenue, Beijing, 100732, China [email protected], [email protected] 2 Department of Foreign Languages, Hefei Normal University, No. 1689, Lianhua Road, Hefei, Anhui, 230601, China [email protected]

Abstract. This paper develops LMP --- a categorial type logic that is meant to formulate vacuous components in natural language. The vacuous components are those expressions which have no semantic referents, but syntactically function as grammatical links between expressions on its left and right hand side. In order to legitimatize the derivations of sentences containing these components in categorial type logic, the present paper proposes system LMP, which is based on a ternary composite category [A{B}C]. LMP’s axiomatic presentation and Gentzen presentation are thereafter presented, and LMP’s soundness, completeness and decidability are established. Keywords: Vacuous Components, Composite Category, Categorial Type Logic.

1 Introduction In natural language, there are some expressions which can be called vacuous components. Semantically, these components have no referents, but syntactically, they function as a link between the expressions which do have referents as their semantic interpretations. More exactly, vacuous components are those syntactic components which make no semantic contributions to the whole expression. For instance, some of structural auxiliaries in Chinese, such as “de”, and some of prepositions in English, such as “to”, can be seen as vacuous components. In Mandarin Chinese, “de” in “da de mayi” (big + de + ant) is a semantically redundant element which makes no semantic contribution to the expression of “da de mayi”. That is, syntactically, “de” serves as a link between adjective “da” (big) and common noun “mayi” (ant), but semantically, no difference can be found between the meaning of “da de mayi” and that of “da mayi” (big ant). In the Compound Predicate Logic proposed by [1], “da de mayi” is semantically analyzed as the concatenation between the predicate symbol corresponding to “da” and the predicate symbol corresponding to “mayi”. There is no logical term corresponding to the structural auxiliary “de”. H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 370–381, 2011. © Springer-Verlag Berlin Heidelberg 2011

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A similar treatment can be found in Wang’s doctoral dissertation. Wang argues that “de” is both semantically and syntactically redundant. She illustrated her idea with the comparison between “zhongshi jiaju” (Chinese-style furniture) and “zhongshi de jiaju” (Chinese-style + de + furniture), both of which mean “Chinese-style furniture” in English. The derivations are given below [2]:

a. zhongshi jiaju n/n n n

b. zhongshi de jiaju

n/n

n n

Fig. 1.

As illustrated in Figure 1, both “zhongshi” (Chinese-style) and “zhongshi de” (Chinese-style+ de) are of the same category n/n while “de” makes no difference to their categories. Therefore, “de” is redundant both syntactically and semantically. There is another “de” in Mandarin Chinese, which, unlike the one given above, is used in verb phrases to link adverb and the head-verb it modifies. “de” in this usage is also viewed as a vacuous component. For instance, the common practice to formulate the semantics of the verb phrase such as “feikuai de paobu” (rapidly+ de+ run) is: let α be the logical form of “feikuai” (rapidly) and β be the logical form of “paobu” (run), then the logical form of “feikuai de paobu” is α(β). Obviously, in English, “de” has no semantic counterpart referents either in noun phrases or in verb phrases. For instance, “da de mayi” and “feikuai de paobu” correspond to “big ant” and “run rapidly” in English respectively. In these English counterparts, there is no lexical unit corresponding to “de” in Chinese. Thus, “de” is a vacuous component. Similar vacuous components can also be found in English. For instance, the copular “be” in “John is running” and complementizer “that” in “Brooke believes that Francis schemed” from a type-logical perspective. In [3], some English prepositions, say “to” in “Ann introduces Maria to Jacob”, are proposed to be treated as vacuous components. The authors provide the following ad hoc rule to cope with these vacuous components.

PP If Į has the form of P ȕˈthen ɊĮɊ=ɊȕɊ Fig. 2.

The rule given in Figure 2 reads as: if α is a prepositional phrase, then ║α║=║β║, which means semantically the preposition is a vacuous component and the meaning of α is only determined by β. In the current literature on Type Logical Grammar, it is common to treat these vacuous components in lexicalism approach, that is, to assign the vacuous components some specific categories in the lexicon. For instance, in order to analyze “John is running” under type-logical framework, Carpenter gives “is” the following lexical entry [4]:

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John j : np

is OV.V : (np\s)/(np\s) run : np\s run(j): s

running run : np\s

Fig. 3.

In Figure 3, it seems that “is” has “λV.V” as its semantic representation. However, λV.V is an identity function which indicates a process of function application. Thus, the result of the derivation is shown as follows : John is running  run(j): s

(2)

Finally, “is” has no semantic representation in “run(j): s”. Similar treatment has been applied to various vacuous components under Categorial tradition. In [4], lexical entry “that  λx.x: sc/s” is added to the lexicon so that sentence “Brooke believes that Francis schemed” can be derived in type-logical grammar. And also in [5], particle “to” is explained as “to  λg.g: PP/NP”. Lexical entries assigned to both “that” and “to” indicate: (1) these vacuous components serve as the indicator of function application; (2) there is no semantic position for these words in the final result during the derivation. Similar treatment can also be found in [6, 7, 8, 9]. Unlike the lexicalism approach, our proposal is to adopt a specific rule. This rule is able to eliminate these vacuous components from the very beginning of the derivation. The specific rule EV we introduced in Figure 4 aims at proposing a general operation on eliminating vacuous components. Figure 4 is following:

John j : np

is OV.V : (np\s)/(np\s) j : np

running run : np\s

EV

run : np\s run(j): s Fig. 4.

The derivation in Figure 4 is obviously shorter than that of Figure 3. As a result, what we need at the moment is a new type-logical rule EV to legitimatize the derivation from the first step to the second one. The present paper proposes a system of LMP (Lambek Calculus with the Limited Monotonicity and Permutation), aiming at capturing the linguistic intuition on vacuous components in natural language.1 An overview of this paper is as follows. In section 2, we present an axiomatic presentation for LMP, in which we provide a ternary composite category [A{B}C], and prove the system is sound and complete. In the third section, we present Gentzen-

1

The omission of the expressions which have no semantic referents is the subject of contextsensitive grammar, and we are not concerned with this problem in this paper.

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style sequent presentation of LMP, in which we establish its decidability, and prove the equivalence between the two presentations. The last section is a summary to this paper, where some further thinking is proposed.

2 LMP — A Categorial Type Logic First of all, we propose a ternary composite category [A{B}C]. It can be employed to formulate the semantics of vacuous components and function as a link from the first step and the second one in the derivation given in figure 4. In this derivation, category B is “(np\s)/(np\s)”, which indicates the vacuous component. As usual, we will give the ternary composite category first, and then present the categorial type logic LMP, which concerns the logic properties of the ternary composite category. The ternary composite category is semantically defined as2: ║ [A{B}C]║ = {x | ∃yz[Sxyg(B)z & y ∈║A║ & z ∈ ║C║]}

(3)

In this definition, the semantic representations of A and C are y and z respectively, while B has no definite semantic representation. Sxyg(B)z means x is the result obtained from the concatenation of y, g(B) and z. g(B) is the object in ║B║. It functions vacuously in semantics but as a bridge linking A and C together syntactically. Then we have the following meaning postulate: MP 0: ∀B∀x[x ∈║B║  x ~ g(B)]

(4)

“x ~ g(B)” can be understood as: x is g(B), an object which functions vacuously. MP0 means the members in ║B║ are all objects which functions vacuously. In LMP, there are two axioms derivating EV given in Figure 4: Ax 1: (A • B) • C → [A{B}C] Ax 2: [A{B}C] → (A • C) Axioms related to product “•” include: Ax 3: D • [A{B}C] → [(D • A){B}C] Ax 4: [A{B}C] • D → [(A • D){B}C] Ax 5: D • [A{B}C] → [A{B}(D • C)] Ax 6: [A{B}C] • D → [(A{B}(C • D)] Ax 7: [A{B}C] • [D{B}E] → [(A • D){B}(C • E)] Besides, there are two rules in LMP:

2

The following definitions based on the frame semantics is given in the similar way in [10]. However, a new meaning has been given to g(B) and 4-tuple relation S is added.

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Rule 1: Rule 2:

A→B [A{D}C] → [B{D}C] A→B [D{C}A] → [D{C}B]

Lastly, the Identity Axiom and five rules in L proposed by Lambek3 remain valid in LMP. Application for those axioms about the operation between the ternary composite category and product “•” can be found in natural language derivation. For instance, from “congming” (clever) and “meili de guniang” (beautiful + de + girl), “congming meili de guniang” (clever and beautiful + de + girl) can be generated by means of Ax 3 4 . “feikuai de paobu bing tanhua” (run rapidly and talk) can be generated from “feikuai de paobu” (run + de + rapidly) and “tanhua” (talk) by means of Ax 6. Following standard practice, we give LMP’s frame semantics, which is constituted of a ternary relation and a quaternary relation. The semantics of the Lambek calculus L5 remains. The meaning postulates corresponding to Ax 1 — 7 are as follows. MP 1:∀xyzuvw[Rxyz & Ryuv & v ~ w  Sxuwz] MP 2:∀xyzu[Sxyzu  Rxyu] MP 3:∀xyzuvw[Rxyz & Szuwv  ∃t[Sxtwv & Rtyu]] MP 4:∀xyzuvw[Rxyz & Syuvw  ∃t[Sxtvw & Rtuz]] MP 5;∀xyzuvw[Rxyz & Szuvw  ∃t[Sxuvt & Rtyw]] MP 6:∀xyzuvw[Rxyz & Syuvw  ∃t[Sxuvt & Rtwz]] MP 7 :∀xyzuvwst[Rxyz & Syuvw & Szsvt  ∃ab[Sxavb & Raus & Rbwt]] In the following, we will prove the soundness of LMP: Proof: Ax 1: Suppose x ∈ ║(A • B) • C║, then we have: ∃yz[Rxyz & y ∈║A • B║& z ∈║C║]. Furthermore, it follows ∃yz[Rxyz & ∃uv[Ryuv & u∈║A║& v ∈║B║]& z ∈║C║]. Take out v ∈║B║, hence v ~ g(B) based on MP0. Take out Rxyz, Ryuv and v ~ g(B), due to MP 1 and Modus Ponens, we get : Sxug(B). Take out u∈║A║ and z∈║C║, and introduce existential quantifier twice, therefore x ∈║[A{B}C]║. Ax 2: Similar to the proof of Ax 1. Ax 3: Suppose x∈║D • [A{B}C]║, then we have: ∃yz[Rxyz & y∈║D║ & z∈║[A{B}C]║]. Furthermore, it follows ∃yz[Rxyz & y∈║D║ & ∃uv[Szug(B)v & u ∈║A║& v ∈║C║]]. Take out Rxyz and Szug(B)v, due to MP3 and Modus Ponens, we get: ∃t[Sxtg(B)v & Rtyu]. Take out Rtyu, y∈║D║ and u∈║A║, hence t ∈║D • A║. Take out Sxtg(B)v, t∈║D • A║ and v∈║C║, introduce existential quantifier twice, and thus it follows: x ∈║[(D • A){B}C]║. Ax 4 — Ax7: The proofs are similar to that of Ax 3.

3 4 5

For further reference, see [11] or [12]. “he” and “bing” in the next example both mean “and” in English. For further reference, see [10] or [11].

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The validity concerning the rules is proved as follows. Rule 1: Assume that if y∈║A║, then y∈║B║; Suppose x ∈║[A{D}C]║, then we have: ∃yz[Sxyg(D)z & y∈║A║ & z∈║C║]. Take out y∈║A║; due to above assumption and Modus Ponens, hence y ∈║B║; Take out Sxyg(D)z and z ∈║C║, together with y ∈║B║, it follows: ∃yz[Sxyg(D)z & y∈║B║ & z∈║C║] after introducing existential quantifier twice. Therefore, x ∈║[B{D}C]║. Rule 2: Similar to that of Rule 1. Axioms and rules inherited from Lambek calculus L remain valid in LMP.

■

Next, we will prove the completeness of LMP. We start with the construction of a canonical model under the framework’s specific properties provided by MP1 - MP7, according to which we have: f(p) = { x | ⊢ x → p} Rxyz iff ⊢ x → y • z Sxyzu iff ⊢ x → [y{z}u] x ~ y iff ⊢ x → y g(B)＝B We will prove that all MPs are satisfied by this canonical model: Proof: MP1: Suppose Rxyz & Ryuv & v ~ w. Thus, in the canonical model, we get: x→y•z

y→u•v

v→w

y • z → (u • v) • z (u • v) • z → (u • w) • z Ax 1: (u • w)• z → [u{w}z] x → [u{w}z] (applying transitive rule repeatedly) Due to the definition of [ ] in the canonical model, hence Sxuwz. MP 3: Suppose Rxyz & Szuwv. Thus, in the canonical model, we have: x→y•z

z → [u{w}v] y • z → y • [u{w}v]

Ax 3: y • [u{w}v] → [(y • u){w}v]

x → [(y • u){w}v] (applying transitive rule repeatedly) Due to the definition of [ ] in canonical model, it follows Sx(y • u)wv. Hence, there is a t such that Sxtwv and t = (y • u), and thus Rtyu. MP 2 and MP 4 — 7: They can be proved in a similar way. MP 0: MP 0 is satisfied by the canonical model due to a truth lemma: ∀D∀x [ x ∈║D║ iff ⊢ x → D ]

(5)

Let’s prove it via induction on D = [A{B}C]. Suppose x ∈║[A{B}C]║, then we have: ∃yz[Sxyg(B)z & y ∈║A║ & z ∈║C║]. Due to the definition of S and induction hypothesis, therefore:

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x → [y{B}z]

y→A

z→C

Rule 1 [y{B}z] → [A{B}z] Rule 2 [A{B}z] → [A{B}C] x → [A{B}C] (applying transitive rule repeatedly) Suppose x → [A{B}C], by the definition of S in the canonical model, it follows: SxABC. Then, by the definition of g, there is SxAg(B)C. Since A→A and C→C are hold in LMP, there are A∈║A║ and C∈║C║ by induction hypothesis. Introduce existential quantifier twice, we get x ∈║[A{B}C]║. Other cases can be treated in the same way6.

3 A Gentzen-Style Presentation of LMP In order to discuss LMP’s decidability, we need to construct LMP’s Gentzen-style presentation. Recently, various hybrid logics have been developed in categorial type logics, permitting various structural controls. It means making use of structural rules such as associativity, commutativity or permutation. Our studies indicate that in LMP’s Gentzen-style presentation, structural rules, including Permutation and limited Monotonicity, are needed during the proof of Cut Elimination theorem7. As to the proof of the equivalence between LMP’s Gentzen presentation and its axiomatic presentation, commutativity is needed to be added into the latter as Ax00: A • B ↔ B • A, together with which a meaning postulate needs to be added: ∀xyz[Rxyz → Rxzy]. Thus, the right and left rules for the ternary composite category ([ ]R and [ ]L) are given as follows: Γ′, A1, Y1, …, An, Yn  A

Γ, C1, Z1, …, Cn, Zn  C [ ]R Γ′, Γ, [A1{B}C1], Y1, Z1, …, [An{B}Cn], Yn, Zn  [A{B}C] X, A,Y, C, Z  D X, [A{B}C], Y, Z  D

[ ]L

Before we start to prove the equivalence of two presentations, we need to prove a lemma first. Lemma 1: σ(Γ′, Γ, [A1{B}C1], Y1, Z1, …, [An{B}Cn], Yn, Zn) → [σ(Γ′, A1, Y1, …, An, Yn ){B} σ(Γ, C1, Z1, …, Cn, Zn )] holds in the axiomatic presentation. Proof: The lemma can be proved via induction over n. Suppose n = 1, σ( Γ′, Γ, [A1{B}C1], Y1, Z1) → [σ(Γ′, A1, Y1){B}σ(Γ, C1, Z1)] holds in axiomatic presentation by repeated application of Ax 3--6 and the definition of σ.

6 7

For further reference, see [10]. That is: A  D A, B  D.

┣

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By induction hypothesis, the lemma holds when n = k, we need to prove that the lemma holds when n = k + 1. By induction hypothesis, σ(Γ′, Γ, [A1{B}C1], Y1, Z1, …, [Ak{B}Ck], Yk, Zk) → [σ(Γ′, A1, Y1, …, Ak, Yk ) {B}σ(Γ, C1, Z1, …, Ck, Zk )] holds in the axiomatic presentation. Obviously, by Ax 4 and 6, it follows: σ([Ak+1{B}Ck+1], Yk+1, Zk+1) → [σ(Ak+1, Yk+1){B}σ(Ck+1, Zk+1)]. Due to monotonicity of “•”, we get: σ(Γ′, Γ, [A1{B}C1], Y1, Z1, …, [Ak{B}Ck], Yk, Zk) • σ([Ak+1{B}Ck+1], Yk+1, Zk+1) →[σ(Γ′, A1, Y1, …, Ak, Yk ){B}σ(Γ, C1, Z1, …, Ck, Zk )] • [σ(Ak+1, Yk+1){B}σ(Ck+1, Zk+1)] Apply the definition of “σ” to the antecedent, and Ax 7 and the definition of “σ” to the succedent, we have: σ(Γ′, Γ, [A1{B}C1], Y1, Z1, …, [Ak{B}Ck], Yk, Zk, [Ak+1{B}Ck+1], Yk+1, Zk+1) →[σ(Γ′, A1, Y1, …, Ak, Yk, Ak+1, Yk+1){B}σ(Γ, C1, Z1, …, Ck, Zk, Ck+1, Zk+1)] ■ The lemma is proved. Equivalence Theorem: ⊢X A in the Gentzen sequent presentation iff ⊢ σ(X) → A in the axiomatic presentation. Proof: Only-if direction: to prove that both [ ]R and [ ]L are derivable in the axiomatic presentation. Firstly, we prove that the theorem holds for rule [ ]R. By induction hypothesis, [ ]R’s two premises σ(Γ′, A1, Y1, …, An, Yn) → Α and σ(Γ, C1, Z1, …, Cn, Zn) → C are hold in the axiomatic presentation. From σ(Γ′, A1, Y1, …, An, Yn) → Α, the following is hold in the axiomatic presentation by rule 1: [σ(Γ′, A1, Y1, …, An, Yn){B}σ(Γ, C1, Z1, …, Cn, Zn)] → [A{B}σ(Γ, C1, Z1, …, Cn, Zn)] From σ(Γ, C1, Z1, …, Cn, Zn) → C, we have: [A{B}σ(Γ, C1, Z1, …, Cn, Zn)] → [A{B}C] in the axiomatic presentation by rule 2. Repeatedly applying transitive rule, finally we get: [σ(Γ′, A1, Y1, …, An, Yn){B}σ(Γ, C1, Z1, …, Cn, Zn)] → [A{B}C] in the axiomatic presentation. Via the lemma 1: σ(Γ′, Γ, [A1{B}C1], Y1, Z1, …, [An{B}Cn], Yn, Zn) → [σ(X, A1, Y1, …, An, Yn ){B}σ (Γ, C1, Z1, …, Cn, Zn )] holds in the axiomatic presentation. Repeatedly applying transitive rule, then the following result: σ(Γ′, Γ, [A1{B}C1], Y1, Z1, …, [An{B}Cn], Yn, Zn) → [A{B}C] holds in the axiomatic presentation. That means the theorem holds for [ ]R.

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Secondly, the proof for the theorem holds for rule [ ]L is a direct consequence of the limited monotonicity and permutation in the Gentzen presentation. The situation of the other rules remains as usual. If-direction direction: to prove that Ax00---As7 and the rule 1--2 are derivable in Gentzen presentation. Obviously, the Gentzen presentations corresponding to the σ-schemes of axioms concerning the ternary composite category are all results based on rule [ ]R. The σ-scheme for Ax 00: σ(A, B) → B • A (obviously) The σ-scheme for Ax 1: σ((A, B), C) → [A{B}C]. Its proof is as follows: For any n, suppose An = Yn = Cn = Zn = ∅. Then [An{B}Cn] = ∅. Suppose further that Γ′= A, Γ = C. Then the left premise of rule [ ]R turns to A  A, and the right premise turns to C  C. Both of them are identity axiom. Therefore, applying [ ]R, we get: A, C  [A{B}C]. Via limited Monotonicity, and Cut rule, we get: (A, B), C  [A{B}C] is derivable in Gentzen presentation. The σ-scheme for Ax 2: σ([A{B}C]) → A • C. Via [ ]L, [A{B}C]  A • C is obviously derivable. The σ-scheme for Ax 3: σ(D, [A{B}C]) → [(D • A){B}C]. We have: Let A = D • A, Γ′ = D, A1 = A, C1 = C, while the others are empty. Then the left premise of rule [ ]R becomes D, A  D • A, which is derivable in Gentzen presentation. And the right premise of rule [ ]R becomes C  C, which is the identity axiom. Therefore, via rule [ ]R: D, [A{B}C]  [(D • A){B}C] is derivable in Gentzen presentation. The proof for Ax 4 — Ax6 are of similar case. The σ-scheme for Ax 7: σ ([A{B}C], [D{B}E]) → [(A • D){B}(C • E)]. Suppose A = A • D, Γ′ = A, A1 = D, C = C • E, Γ = C, C1 = E, while the others are empty. Then the right and left premises are both derivable from the Gentzen presentation. via [ ]R, we get: A, C, [D{B}E]  [(A • D){B}(C • E)] Via rule [ ]L, [A{B}C], [D{B}E]  [(A • D){B}(C • E)] is derivable in Gentzen presentation. Next, we need to prove that the equivalence theorem holds for rule 1. That is, in the axiomatic presentation: if σ(A) → B holds, then σ([A{D}C]) → [B{D}C] holds. That means, we need to prove that in the Gentzen presentation, if A  B holds, then [A{D}C]  [B{D}C] holds. Suppose A  B holds, meanwhile we have the axiom: C  C. Let B = D, A = B, A1 = A, C1 = C, while the others are empty, via rule [ ]R, [A{D}C]  [B{D}C] is derivable in Gentzen presentation. ■ We establish the Equivalence Theorem. Cut Elimination Theorem: If LMP ⊢X  A, then there is a Cut-free sequent proof of X  A. Proof: There are three cases in the proof. The first case is the same as usual; and in the second case, the followings are cases related to the ternary composite category:

The Categorial Logic of Vacuous Components in Natural Language *c, A1, Y1, }, An, Yn A *, C1, Z1, }, Cn, ZnC

379

X, A,Y, C, ZD

[]R *c, *, [A1{B}C1], Y1, Z1, }, [An{B}Cn], Yn, Zn[A{B}C]

[]L X, [A{B}C], Y, ZD Cut X, *c, *, [A1{B}C1], Y1, Z1, }, [An{B}Cn], Yn, Zn, Y, ZD *c, A1, Y1, }, An, Yn A X, A,Y, C, ZD Cut X, *c, A1, Y1, }, An, Yn, Y, C, ZD *, C1, Z1, }, Cn, ZnC Cut X, *c, A1, Y1, }, An, Yn, Y, *, C1, Z1, }, Cn, Zn, ZD []L X, *c, [A1{B}C1], Y1, }, An, Yn, Y, *, Z1, }, Cn, Zn, Z D

ӭ

use []L n times X, *c, [A1{B}C1], Y1, , }, [An{B}Cn], Yn, Y, *, Z1, }, Zn, Z D use Permutation repeatedly X, *c, *, [A1{B}C1], Y1, Z1, }, [An{B}Cn], Yn, Zn, Y, ZD

In the third case (in which at least one of the premises is not active formulae), the followings are cases related to the ternary composite category: X, A,Y, C, ZD

[]L

X, [A{B}C], Y, ZD

D, A1, *, C1, *c E D, [A1{B}C1], *, *c E

X, [A{B}C], Y, Z, [A1{B}C1], *, *c E X, A,Y, C, ZD D, A1, *, C1, *c E X, A,Y, C, Z, A1, *, C1, *c E

[]L Cut

Cut []L

X, [A{B}C], Y, Z, A1, *, C1, *c E X, [A{B}C], Y, Z, [A1{B}C1], *, *c E

[]L

and˖ D, A, *, C, *c E XD

D, [A{B}C], *, *c E

Cut

X, [A{B}C], *, *c E XD D, A, *, C, *c E Cut

X, A, *, C, *c E X, [A{B}C], *, *c E

and˖

[]L

[]L

380

C. Zou, K. Li, and L. Zhang D, A1, Y1, }, An, Yn A XD

*, C1, Z1, }, Cn, ZnC

D, *, [A1{B}C1], Y1, Z1, }, [An{B}Cn], Yn, Zn[A{B}C]

X, *, [A1{B}C1], Y1, Z1, }, [An{B}Cn], Yn, Zn[A{B}C] XD D, A1, Y1, }, An, Yn A Cut X, A1, Y1, }, An, Yn A

*, C1, Z1, }, Cn, ZnC

X, *, [A1{B}C1], Y1, Z1, }, [An{B}Cn], Yn, Zn[A{B}C]

[]R Cut

[]R

In any derivation given above, the formulae after reduction are of smaller complex degree than those before reduction. If we keep doing this, all uses of Cut rule will be eliminated. ■ Decidability Theorem: Derivability in LMP is decidable.

■

4 Further Thinking (1)The primary objective of LMP is to device a new categorical type logic mechanism, in order to eliminate the vacuous components existing in natural languages. As a result, it seems necessary to reformulate the Principle of Compositionality in a more strict way. Not all components of composite expressions in natural language functions semantically. To put it in other words, some components make no semantic contribution to the whole expression. Therefore, the Principle of Compositionality should be restated more accurately as: (a) the meaning of the composite expression which involves no vacuous components is the function of the meanings of its all components; (b) the meaning of the composite expression which involves vacuous components is the function of the meanings of its nonvacuous components. That is, there are some composite expressions whose meaning is the function of their non-vacuous components rather than all of their components. For instance, the meaning of “introduce John to Mary” is determined by the meaning of it non-vacuous components “introduce”, “John” and “Mary”, but not “to”. (2) We propose the 3-tuple composite category [A{B}C] to capture such a linguistic intuition which we call “Vacuous Slot”. In a word, the category of [A{B}C] is a tool whose function is to reveal and then eliminate vacuous components. When Ax 1 is applied, [A{B}C] can be derived from the sequence of categories “(A • B) • C”. With the help of a certain logical tool such as the function g, Ax 1 reveals B to be a vacuous component. Then application of Ax 2 will eliminate this vacuous component. Thus without Ax 1 and Ax 2, it would be impossible to employ the axiom of (A • B) • C → A • C to capture the intuition about vacuous components. Ax 3 — Ax 7 reveal how product “•” and vacuous slot may compose with each other, which are necessary for the categorical type logic analysis over sentences in natural languages. For instance, according to Ax 3: D • [A{B}C] → [(D • A){B}C], we can

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obtain “congming meili de guniang” (clever and beautiful + de + girl) from “congming” (clever) and “meili de guniang” (beautiful + de + girl) in Mandarin Chinese.

References 1. Liu, Z.H.: A Logical System of Compounded Predicate. Supplement to Studies in Dialectics of Nature (2000) 2. Wang, X.: Type-Logical Grammar and The Chinese shi and de, Dissertation, Beijing Language and Culture University (2009) 3. Heim, I., Kratzer, A.: Semantics in Generative Grammar. Basil Blackwell, Oxford (1998) 4. Carpenter, B.: Type-Logical Semantics. MIT Cambridge, Mass. (1997) 5. Steedman, M.: Surface Structure and Interpretation. MIT Press, Cambridge (1996) 6. Steedman, M.: Dependency and coordination in the grammar of Dutch and English. Language 61, 523–568 (1985) 7. Steedman, M.: Combinatory grammars and parasitic gaps. Natural Language and Linguistic Theory 5, 403–439 (1987) 8. Steedman, M.: Combinators and grammars. In: Oehrle, R., Bach, E., Wheeler, D. (eds.) Categorial Grammars and Natural Language Structures, pp. 417–442. Reidel, Dordrecht (1988) 9. Steedman, M.: The Syntactic Process. MIT Press, Cambridge (2000) 10. Jäger, G.: Anaphora and Type Logical Grammar. Springer, Heidelberg (2005) 11. Lambek, J.: The Mathematics of Sentences Structure. The American Mathematical Monthly 39, 154–170 (1958) 12. Moortgat, M.: Categorial Type Logics. In: van Benthem, J., ter Meulen, A. (eds.) Handbook of Logic and Language, pp. 93–178. Elsevier, MIT Press (1997)

A Logic for Strategy Updates Can Ba¸skent Department of Computer Science, Graduate Center, City University of New York [email protected]

1

Introduction

While people play games, they observe, learn, recollect and update their strategies during the game as well as adopting deontological strategies and goals before the game. In this paper, we focus on what we call move updates where some moves become unavailable during the game, and present a formal framework for extending strategy logic which was introduced by Ramanujam and Simon [2].

2

Restricted Strategy Logic

Restricted strategy logic (RSL) is based on the strategy logic (SL) [1,2]. The focus is on games played between two players. The most basic constructions in SL are strategy speciﬁcations. The speciﬁcation [ϕ → a]i at player i’s position stands for “play a whenever ϕ holds”. The speciﬁcation σ1 + σ2 means that the strategy of the player conforms to the speciﬁcation σ1 or σ2 and σ1 · σ2 means that the strategy of the player conforms to the speciﬁcations σ1 and σ2 . In RSL, we add an additional speciﬁcation [σ!a]i that stands for the updated speciﬁcation where the player is not allowed to make an a move. Then, [σ!a] conforms with the model where the outgoing move is not a. Now, based on the strategy speciﬁcations, the syntax of the strategy logic SL is given as follows. p | ¬ϕ | ϕ1 ∧ ϕ2 | aϕ | (σ)i : a | σ i ψ We read (σ)i : a as “at the current state the strategy speciﬁcation σ for player i suggests that the move a can be played”, and σ i ψ as “following strategy σ player i can ensure ψ”. The syntax of RSL, then, is the same as SL as we have incorporated the update modality [σ!a] at the speciﬁcation level. The truth deﬁnition for the strategy formulas are as follows: a

M, s |= aϕ iﬀ ∃s such that s ⇒ s and M, s |= ϕ M, s |= (σ)i : a iﬀ a ∈ σ(s) M, s |= σ i ψ iﬀ ∀s such that s ⇒∗σ s in Ts |σ, we have M, s |= ψ ∧ (turni → enabledσ ) where σ(s) is the set of enabled moves at s for strategy σ, and ⇒∗σ denotes the reﬂexive transitive closure of ⇒σ . Furthermore, Ts is the tree that consists of the unique path from the root (s0 ) to s and the subtree rooted at s, and Ts |σ is the least subtree of Ts that contains a unique path from s0 to s and from s H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 382–383, 2011. c Springer-Verlag Berlin Heidelberg 2011

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onwards, for each player i node, all the moves enabled by σ, and for each node of the opponent player, all possible moves . The proposition turni denotes that it is i’s turn to play. Finally, deﬁne enabledσ = a∈Σ (a ∧ (σ)i : a). The axioms of RSL is given as follows based on SL [2]. – – – – – – – – – –

All the substitutional instances of the tautologies of propositional calculus [a](ϕ → ψ) → ([a]ϕ → [a]ψ) aϕ → [a]ϕ a → ([ψ → a]i : a)i for all a ∈ Σ [turni ∧ ψ ∧ ([ψ → a]i )i : a)] → a

turni ∧ ([ψ → a]i )i : c ↔ ¬ψ for all a = c (σ + σ )i : a ↔ (σ : a)i ∨ (σ : a)i (σ · σ )i : a ↔ (σ : a)i ∧ (σ : a)i σ i ψ → [ψ ∧ invσi (a, ψ) ∧ invσ−i (ψ) ∧ enabledσ ] (σ!a)i : c ↔ turni ∧ ¬((σ)i : a) ∧ (σ)i : c

Here, invσi (a, ψ) = (turni ∧ (σ)i : a) → [a](σ i ψ) which expresses the fact that after an a move by i which conforms to σ, the statement σ i ψ continues to hold, and invσ−i (ψ) = turni → (σ i ψ) states that after any move of −i, σ i ψ continues to hold. Moreover, ϕ ≡ a∈Σ aϕ and ϕ ≡ ¬ ¬ϕ. The inference rules that SL employs is as follows: modus ponens and generalization for [a] for each a ∈ Σ. The induction rule is a bit more complex: From the formulas ϕ ∧ (turni ∧ (σ)i : a) → [a]ϕ, ϕ ∧ turn−i → ϕ, and ϕ → ψ ∧ enabledσ derive ϕ → σ i ψ. Note that the additional axiom we need for RSL is the last one, and the previous ones are the axioms for SL. The ﬁnal axiom explains how updates behave in RSL. Note that SL is sound and complete with respect to the given semantics [2]. Theorem 1. RSL is complete with respect to the given semantics. Theorem 2. The model checking problem for SL and RSL are in PSPACE. The proof of the completeness of RSL is by reduction. We can show that any RSL formula can be reduced to an SL formula. For the complexity of model checking, we can use either the standard arguments for the model checking for basic modal logic, or we can give a polynomial translation between CTL* and RSL. Acknowledgements. We especially acknowledge the help and encouragement of Sujata Ghosh and Rineke Verbrugge. This paper is the product of author’s visit to the Department of Artiﬁcial Intelligence of the University of Groningen. We also thank Sergei Artemov and Rohit Parikh for their feedback.

References 1. Ghosh, S., Ramanujam, R., Simon, S.: On strategy composition and game composition (2010) (unpublished manuscript) 2. Ramanujam, R., Simon, S.: A logical structure for strategies. In: Logic and the Foundations of Game and Decision Theory (LOFT 7). Texts in Logic and Games, vol. 3, pp. 183–208. Amsterdam University Press, Amsterdam (2008)

Eﬃcient Action Extraction with Many-to-Many Relationship between Actions and Features Jianfeng Du1 , Yong Hu1 , Charles X. Ling2 , Ming Fan3 , and Mei Liu4 1

3

1

Guangdong University of Foreign Studies, Guangzhou 510006, China [email protected] 2 The University of Western Ontario, London, Ontario N6A 5B7, Canada Foster School of Business, University of Washington, Seattle, WA 980195, USA 4 Vanderbilt University, Nashville, TN 37232, USA

Motivation

Real-world problems often call for eﬃcient methods to discovery actionable knowledge on which business can directly act [3]. Some works for discovering actionable knowledge [3,5] view actions as behaviors which render a state of an instance into a preferred state, where a state is represented by feature values of the instance and whether a state is preferred is determined by a classiﬁer. Actions usually have many-to-many relations with features of an instance. That is, an action may aﬀect multiple features of an instance, and vise versa, a feature may be inﬂuenced by multiple actions. This type of many-to-many relationships between actions and features is prevalent in real-world applications. However, most existing works [3,5] only deal with one-to-one relationship and ignore manyto-many relationship between actions and features. In these works, an action is treated as a behavior with a ﬁxed execution cost. Restricting to a one-to-one relationship between actions and features may not yield an action set (i.e. a set of actions) with the minimal total execution cost. Moreover, one-to-one relationship is simply a special case of many-to-many relationship, and hence the latter will be applicable to more real-world problems. Therefore we aim to extract action sets from a classiﬁer for which the total execution cost is minimal based on many-to-many relationship between actions and features. More precisely, the problem to be addressed is to compute an action set (called a cost-minimal action set) from a classiﬁer such that the total execution cost is minimal among all preferred action sets w.r.t. a given instance, where a preferred action set w.r.t. an instance is a set of actions that render the state of the instance into a preferred state, namely a state with which the instance is classiﬁed to a preferred class by the given classiﬁer.

2

Method and Experimental Evaluation

Obviously, the problem can be solved by a traditional generate-and-test method, in which the following step is repeated until all preferred action sets are found, then a cost-minimal action set is selected from the preferred action sets. The step H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 384–385, 2011. c Springer-Verlag Berlin Heidelberg 2011

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is to select an action set from the search space and test whether it is preferred by applying the action execution process and the classification process in turn. The action execution process transforms the original state of the given instance to a new state by applying all actions in the action set. The classiﬁcation process determines whether the new state is preferred by applying a given classiﬁer. However, this method has drawbacks in eﬃciency because the given classiﬁer is applied as a black-box and the classiﬁcation process cannot be anatomized to facilitate the search of action sets. Hence, we consider encoding both the classiﬁcation process and the action execution process as rules. These rules can be treated as the input of a propositional satisﬁability (SAT) problem and anatomized by existing SAT techniques to direct the search of action sets and prune the search space. Based on this idea, we propose an eﬃcient method to extract a cost-minimal action set from a classiﬁer that can be encoded as rules. We present the method with a well-known ensemble classiﬁer, namely random forest [2], in order to show that the method can also be applied to an ensemble classiﬁer as long as its base classiﬁers can be encoded as rules. Random forest uses decision trees as base classiﬁers and outputs the class that is the mode of the classes output by individual decision trees. The method is based on a reduction from the original problem to an extended SAT problem, called Linear Pseudo-Boolean Optimization problem [4], which consists of a minimization object function nj=1 cj xj and m pseudo-Boolean constraints of the form nj=1 aij xj ≥ bi , where i ∈ {1, ..., m}, xj ∈ {0, 1}, bi is an integer, cj is a nonnegative integer cost associated with the variable xj and a1j , ..., amj are integer coeﬃcients of xj . The Linear Pseudo-Boolean Optimization problem can be solved by existing pseudo-Boolean solvers [4]. We applied the proposed method to a real-world application, namely software project risk management planning [1], to extract cost-minimal action sets from a random forest for risk mitigation. Experimental results show that the proposed method is able to extract a cost-minimal action set in a few seconds no matter how many decision trees are assembled in the given random forest. It is by orders of magnitude faster and much more scalable than a traditional generate-and-test method which uses the branch-and-bound strategy to prune the search space.

References 1. Boehm, B.W.: Software risk management: Principles and practices. IEEE Software 8(1), 32–41 (1991) 2. Breiman, L.: Random forests. Machine Learning 45(1), 5–32 (2001) 3. Cao, L., Zhang, C., Yang, Q., Bell, D., Vlachos, M., Taneri, B., Keogh, E.J., Yu, P.S., Zhong, N., Ashraﬁ, M.Z., Taniar, D., Dubossarsky, E., Graco, W.: Domain-driven, actionable knowledge discovery. IEEE Intelligent Systems 22(4), 78–88 (2007) 4. Manquinho, V.M., Roussel, O.: The ﬁrst evaluation of pseudo-boolean solvers (pb’05). Journal on Satisﬁability, Boolean Modeling and Computation 2(1-4), 103– 143 (2006) 5. Yang, Q., Yin, J., Ling, C.X., Pan, R.: Extracting actionable knowledge from decision trees. IEEE Transactions on Knowledge and Data Engineering 19(1), 43–56 (2007)

Reflections on Vote Manipulation Jan van Eijck, Floor Sietsma, and Sunil Simon CWI, Amsterdam {jve,f.sietsma,s.e.simon}@cwi.nl

The notion of non-manipulability (or: strategy-proofness) used in the famous Gibbard-Satterthwaite theorem is too strong to make useful distinctions between voting rules. We explore alternative deﬁnitions and suggest how these can be used to classify voting rules. If A is a ﬁnite set of alternatives, with |A| > 2, then an A-ballot is a linear ordering of A. Let {1, . . . , n} be a set of voters. An (A, n)-profile is an n-tuple of A-ballots. If P is an (A, n)-proﬁle, then P can be written as (1 , . . . , n). i , the i-th component of proﬁle (1 , . . . , n ), is the ballot of voter i. i expresses “what voter i wants.” If P(A) is the set of all (A, n)-proﬁles, for given n ∈ N, then a function V : P(A) → A is a resolute voting rule for A, and a function V : P(A) → P + (A) is a voting rule for A. Let P ∼i P express that P and P diﬀer only in the ballot of voter i. A resolute voting rule V is non-manipulable (NM) (or: strategy-proof) if P ∼i P implies V (P) i V (P ), where i denotes the i-preference in P. A voting rule V is non-imposed (NI) if any candidate can be a winner: ∀a ∈ A∃P : a ∈ V (P). The famous Gibbard/Satterthwaite theorem (GS, [2,3]) says that any resolute voting rule that is NM and that is NI is a dictatorship. We take the fact that GS has an easy proof as a sign that the notion of manipulability (forbiding any adjustment of the vote) is too strong, and we investigate what happens if we distinguish between reasonable adjustments and perverse adjustments. Following [1], we deﬁne the knights and knaves of a voter i, given proﬁle P and voting rule V : The knights of a voter i, given proﬁle P and resolute voting rule V , are the alternatives that are above V (P) on the i-ballot. The knaves of a voter i, given proﬁle P and resolute voting rule V , are the alternatives that are below V (P) on the i-ballot. Thus, if i has ballot a b c d in P, and the outcome of the vote is c, then a and b are knights of i in P, and d is a knave of i in P. Using this, we can distinguish between benign adjustments and perversions by means of a check whether the manipulation involves knight demotion or knave promotion. A resolute voting rule V is demotion pervertible (DP) if there exists an iminimal pair of proﬁles P, P such that (i) V (P) ≺i V (P ), and (ii) ∃x : V (P) ≺i x ≺i V (P). A resolute voting rule V is NDP (non-demotion-pervertible) if V is not DP. Note that the demotion of knight x from above V (P) to a new position below V (P) is the perversion. For example, suppose i has ballot abcd in P, and the outcome of the vote is c, and P ∼i P where i has ballot bcad in P , and the H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 386–387, 2011. c Springer-Verlag Berlin Heidelberg 2011

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outcome of the vote in P is b. Then V is demotion pervertible, for we have that V (P) = c ≺i b = V (P ), and V (P) = c ≺i a ≺i c = V (P), that is to say, a was demoted from a knight to a knave position (from the perspective of P). In a similar way, we can deﬁne (non)-promotion pervertibility. An i-minimal pair of proﬁles P, P invites decency towards knights if the following hold: (i) V (P) ≺i V (P ) implies ∀x : V (P) ≺i x ⇒ V (P) i x, and vice versa: (ii) V (P) i V (P ) implies ∀x : V (P ) ≺i x ⇒ V (P ) i x. This says: “If the shift from i to i is an improvement, then no knight was demoted”, and similarly in the other direction. In a similar fashion, we can deﬁne decency towards knaves (“don’t promote them to where they do not ﬁt”). The above deﬁnition of NDP is not suitable yet to classify voting rules, for what matters is not whether the vote can be perverted by demotion, but whether such perversion is essential. The following deﬁnition takes this into account. A voting rule V is not single-winner demotion pervertible if for any two proﬁles P ∼i P , where V (P) ≺i V (P ) and there is an x i V (P) i x, there exists another proﬁle Q such that P ∼i Q and V (Q) = V (P ). What this says is that perversions of the vote may be possible, but they are never essential to achieve the goal: for every vote perversion there is a non-perverted alternative that works just as well. Theorem 1. The plurality rule is not single-winner promotion or demotion pervertible. Proof. Suppose P ∼i P and V (P) ≺i V (P ). Since V (P) = V (P ), either i’s top candidate in P is V (P) or i’s top candidate in P is V (P ). In the ﬁrst case, V (P) i V (P ), which contradicts our assumption. In the second case, let Q be the proﬁle with i’s ballot identical to that in P, with the only diﬀerence being that V (P ) got moved to the top. Since V (P ) i V (P), this cannot involve promotion or demotion and since in the plurality rule only the top element of the ballot counts, V (Q) = V (P ).

The following theorem can be proved by example: Theorem 2. The weak Condorcet rule is single-winner promotion pervertible. What these examples show is that our reﬁned notions of manipulability can be used to make useful distinctions. In future work we intend to further classify positional voting rules with respect to the new notions of manipulability.

References 1. van Eijck, J.: A geometric look at manipulation. In: Leite, J., et al. (eds.) CLIMA XII 2011. LNCS(LNAI), vol. 6814, pp. 92–104. Springer, Heidelberg (2011) 2. Gibbard, A.: Manipulation of voting schemes: A general result. Econometrica 41, 587–601 (1973) 3. Satterthwaite, M.: Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory 10, 187–217 (1975)

Playing Extensive Form Negotiation Games: A Tool-Based Analysis (Abstract) Sujata Ghosh1 , Sumit Sourabh2 , and Rineke Verbrugge1 1

2

Department of Artificial Intelligence University of Groningen PO Box 407, 9700 AK Groningen The Netherlands sujata,[email protected] Department of Mathematics and Statistics Indian Institute of Technology Kanpur 208 016, India [email protected]

Introduction. This paper reports the development of a simple tool, NEGEXT, written in the platform-independent Java language. NEGEXT has been constructed to aid real people doing actual negotiations, when the ways to negotiate are simply too many to be computed by a normal human mind. This toolkit will also help in planning one’s strategic moves in negotiation situations when the opponents’ possible moves can be approximated. Even though some visualization tools for extensive form game trees already exist, we believe we are the ﬁrst to make a tree-based negotiation toolkit that incorporates the possibility of representing learning from game to game, by sequential and parallel composition (cf. [1]). Moreover, the toolkit has a model-checking component which computes whether and how an individual or a speciﬁc coalition can achieve a given objective. NEGEXT toolkit. We have developed the NEGEXT toolkit1 to represent and aid in the strategic interactions that happen during negotiations. Many negotiation situations can be aptly represented as extensive form games and their sequential and parallel combinations. The software is represented by a Game tree simulator (Figure 1), written in Java version 1.6.0 22. It uses an applet for displaying the graphical user interface and can run on any system having a Java Runtime Environment (JRE) installed or Java-enabled web browser. The users are provided with an option to draw game trees using diﬀerent input modes. The trees thereby generated are rendered using Graphviz for the ﬁnal output. NEGEXT provides an option to combine the trees sequentially and in parallel. The main feature of this software is to check for the existence of a strategy for achieving a formula for an individual or a set of players, as well as displaying that strategy (or all such strategies), in case one exists (Figure 1). 1

Available at http://www.ai.rug.nl/~ sujata/negext.html

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Fig. 1. Strategy checking in the graphical user interface

Discussions. This toolkit can be used to represent negotiations which span over a ﬁnite time and consider the actions of the negotiators one after another in response to each other. The NEGEXT toolkit can help users to learn how to respond in order to achieve what they want, also in situations where it is not easy to compute the best response. We can consider simple game trees, build up complex game structures from those trees by sequential and parallel combination, and then compute group and individual strategies. The current version of NEGEXT is restricted to perfect information situations. However, in many real-life negotiations, the information dilemma looms large: Which aspects to make common knowledge and which aspects to keep secret, or to divulge only to a select subset of co-players? It would be fascinating to extend NEGEXT so that also such aspects of incomplete, imperfect and asymmetric information can be incorporated in its tree representations and its strategic advice, but that is still in the future. Acknowledgements. The ﬁrst author acknowledges the Netherlands Organisation of Scientiﬁc Research grant 600.065.120.08N201, and the second and third authors acknowledge the Netherlands Organisation of Scientiﬁc Research grant 227-80-001.

Reference 1. Ghosh, S., Ramanujam, R., Simon, S.E.: Playing extensive form games in parallel. In: Dix, J., Leite, J., Governatori, G., Jamroga, W. (eds.) CLIMA XI. LNCS, vol. 6245, pp. 153–170. Springer, Heidelberg (2010)

The Dynamics of Peer Pressure Zhen Liang and Jeremy Seligman 1

Institute of Logic and Intelligence Southwest University Chongqing, China [email protected] 2 Department of Philosophy University of Auckland, New Zealand [email protected]

Abstract. Following the general programme of [2], we investigate the eﬀect of social relationships on the dynamics of preference change within a community. Speciﬁcally, we are interested in the phenomenon of ‘peer pressure’, according to which a person’s preferences are changed in response to the preferences of a ‘peer group’. This involves both aggregation of preferences, to determine the group’s preferences and preference change. We propose a simple model of peer pressure which is still suﬃciently non-trivial to display some interesting dynamics, and show how the stable conﬁgurations can be expressed logically. Keywords: preference logic, logic in the community, aggregation, social dynamics, hybrid logic.

A community of teenage girls is divided between two current fashions. Some strictly prefer A, others strictly prefer B, others are I indiﬀerent and yet others are conﬂicted O. They are inﬂuenced by their friends’ choices in the following way. If all of their friends strictly prefer one of the styles then they will too. We’ll call this strong suggestion. Even if some of their friends are indiﬀerent, if the rest strictly prefer one style, they will still be inﬂuenced, but not as strongly. We’ll call this weak suggestion. In this case, they will change their preference so that they regard the style preferred by their non-indiﬀerent friends so that it is at least as good as the other one. This means that if they are conﬂicted they will resolve the conﬂict in favour of their friend’s choice, and prefer the other style, they will become indiﬀerent. The resulting dynamics has some interesting features. Representing the friendship relation as a graph we see the following looping patterns: A

B

A B A

B

A

A

B B

A

B

B

A A

B

A

A

B B

··· ···

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If one of the friends is indiﬀerent, however, the pattern stabilizes: A

B

A

A

B

B

B

I

A conﬁguration

A

A

A I

B

I

B

A

B

I

I

A

B

B

B

I

I

A

B

I

B B

B

B

B

A

I

B B

with an indiﬀerent mutual friend is stable and even when

B

the indiﬀerent friend is out on a wing of the group, the pattern stabilizes quickly to that of the mutual friend: A

B

B I

B

I

Indiﬀerence is not so easily shaken, as

B

I

B B

is stable. But those who are con-

I A

ﬂicted are slightly more easy to inﬂuence: I

I

O A

A

A A

A A

Two friends of the same type will stand in solidarity, no matter what their environment: B B

I

I

O

O

A A

When a couple of agents have opposite preferences, they usually keep them. However, if an agent who has an indiﬀerent opinion joins in, a comprise results: A B I

I I I

In the paper [1], we develop a logic of preference and preference change as an extension of ‘facebook logic’ [2] and use it to represent the process of ‘suggestion’. This is an idealised model of peer pressure, which we use as an example to show how the dynamic properties of change within a community can be represented logically. In particular, we study the conditions under which a community has a distribution of preferences that is stable with respect to our chosen dynamics. The main theorem states a formula that characterises precisely when a distribution of preferences will eventually stabilize. This uses novel proof techniques and so sets an agenda for similar logic studies of community dynamics.

References 1. Liang, Z., Seligman, J.: The dynamics of peer pressure (2011), http://auckland.academia.edu/JeremySeligman/Papers/672669/ 2. Seligman, J., Liu, F., Girard, P.: Logic in the community. In: Banerjee, M., Seth, A. (eds.) Logic and Its Applications. LNCS, vol. 6521, pp. 178–188. Springer, Heidelberg (2011)

On Logic of Belief-Disagreement among Agents Tian-Qun Pan Institute of Modern Logic and Its Application, Nanjing University, China [email protected]

1

Three Kinds of Belief-Disagreement

Epistemic disagreement and agreement between two agents have been considered by game theorists and logicians[1][2]. But they are discussed only from the view of knowledge. A belief-disagreement on a certain proposition p in a group with n-agent (n ≥ 2) is defined as the following: Definition 1. (Belief − Disagreement : Weak, Moderate, and Strong) (1)(Weak Belief-Disagreement): There exists a weak belief-disagreement on p among agents iﬀ one agent does not believe p and another agent does not believe ¬p. (2)(Moderate Belief-Disagreement): There exists a moderate beliefdisagreement on p among agents iﬀ one agent believes p and another one does not believe p. (3)(Strong Belief-Disagreement): There exists a strong belief-disagreement on p among agents iﬀ one agent believes p and another one believes ¬p. We use D W p, D M p and DS p to denote weak belief-disagreement, moderate disagreement, and strong belief-disagreement on p. Thus the definition 1 becomes: D S p ↔ def Bi p ∧ Bj ¬p DM p ↔ def Bi p ∧ ¬Bj p DW p ↔ def ¬Bi p ∧ ¬Bj ¬p

i = j i = j i = j

By the D axiom of belief logic Bφ → ¬B¬φ, a relation among the three kinds of belief-disagreement is (DS p → DM p) ∧ (DM p → D W p) The above formula shows that, among the three operators, DS is the strongest, and DW is the weakest.

2

A Belief Logic Containing Belief-Disagreement Operators

We construct a belief-disagreement logic system BD(n) by adding DW p, D M p and D S p in definition 1 to the multi-agent belief system KD4(n) which contains the axioms of propositional tautologies, Bi (φ → ψ) → (Bi φ → Bi ψ), H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 392–393, 2011. c Springer-Verlag Berlin Heidelberg 2011

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Bi φ → ¬Bi ¬φ and Bi φ → Bi Bi φ (for 1 ≤ i ≤ n), and the inference rules of N (Necessitation), MP, Sub. Suppose BD(n) φ. By N, BD(n) Bi φ, for 1 ≤ i ≤ n. Then BD(n) ¬D W φ by definition 1. Thus, we have a derived rule of BD(n) : RD BD(n) φ ⇒BD(n) ¬Dw φ RD and (DS p → DM p)∧(DM p → DW p) show that it is impossible for agents to have any kinds of belief-disagreement on any logical law. The formulas of BD(n) can be interpreted by possible worlds semantics. Since BD(n) contains the same axioms and the inference rules exactly as KD4(n) does, we get: Theorem 1. (Soundness and Completeness). Given the system BD(n) , for any formula φ, BD(n) φ ⇔|= φ with respect to any Kripke models in which each accessible relation Ri is serial transitive.

3

Some Theorems in BD(n)

Theorem 2. (1) D S φ ↔ DS ¬φ (2) DS (φ ∧ ψ) → (DS φ ∨ D M ψ) ∧ (DM φ ∨ D S ψ) (3) D S (φ ∨ ψ) → (DS φ ∨ D M ¬ψ) ∧ (DM ¬φ ∨ D S ψ) (4) D S (φ → ψ) → (D S ψ ∨ DM ¬ψ) ∧ (DM φ ∨ D S ψ) Theorem 3. (1) D M ¬φ → DW φ (2) D M (φ ∧ φ) → (D M φ ∨ DM ψ) (3) D M (φ ∨ ψ) → (DM φ ∨ D W ψ) ∧ (DW φ ∨ D W ψ) (4) DM (φ → ψ) → (D M ψ ∨ DW φ) ∧ (DM ¬φ ∨ D W ψ) Theorem 4. (1) D W φ ↔ DW ¬φ (2) DW (φ ∧ ψ) → (DW φ ∨ D W ψ) (3) DW (φ ∨ ψ) → (DW φ ∨ D W ψ) (4) DW (φ → ψ) → (DW φ ∨ D W ψ) It is easy to prove Theorem 2-4 by the definition D W p, DM p and DS p and some theorems of KD4(n) .

References 1. Aumann, R.J.: Agreeing to disagree. The Annals of Statistics 6(1236-1239) (1976) 2. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. The MIT Press, Cambridge (2006)

Algebraic Semantics and Model Completeness for Intuitionistic Public Announcement Logic Mehrnoosh Sadrzadeh1 , Alessandra Palmigiano2, and Minghui Ma3 1

Oxford University Computing Laboratory, Ofxord, UK [email protected] 2 Institute for Logic, Language and Information, University of Amsterdam [email protected] 3 Institute of Logic and Intelligence, Southwest University, Chongqing, China [email protected]

Abstract. In this paper, we start studying epistemic updates using the standard toolkit of duality theory. We focus on public announcements. We give the dual characterization of the corresponding submodelinjection map, as a certain pseudo-quotient map between the complex algebras respectively associated with the given model and with its relativized submodel. The dual characterization we provide naturally generalizes to much wider classes of algebras, which include, but are not limited to, arbitrary BAOs and arbitrary modal expansions of Heyting algebras (HAOs). As an application, we axiomatize the intuitionistic analogue of PAL, which we refer to as IPAL, and prove soundness and completeness of IPAL w.r.t. both algebraic and relational models.

We start a line of investigation which aims at deﬁning and studying dynamic epistemic logics based on weaker-than-classical propositional logic. The conceptual relevance of this investigation is providing a point-free, non-classical account on the phenomenon of epistemic actions, which naturally generalizes the classical account. The approach we adopt is based on the dual characterization of the map which represents the update induced by an epistemic action. This dual characterization provides an equivalent description of the epistemic update, in an algebraic environment. The beneﬁt brought in by this dual characterization is that it immediately generalizes to much more general setting, in a modular way. In the present paper, we have treated the simplest epistemic actions, namely public announcements. We axiomatically deﬁned IPAL, a logic of public announcements based on intuitionistic propositional logic. The static fragment of IPAL is the intuitionistic modal logic MIPC, which is widely considered the intuitionistic counterpart of S5. We have shown that IPAL is sound and complete w.r.t. MIPC-models, with an entirely analogous proof to the soundness and completeness proof for the classical PAL. Basic facts for intuitionistic modal logics used in our work refer to [2], [3] and [6]. The language is an extension of intuitionistic logic by adding modal operators and ♦ which are not duals of each other since the negation is intuitionistic now. The logic IK is obtained from intuitionistc logic by adding modal axioms as in H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 394–395, 2011. c Springer-Verlag Berlin Heidelberg 2011

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[4] and MIPC is from [6]. Take these intuitionistic logics as the static part which allow intuitionistic epistemic interpretations. Basically, they are logics about our intuitionistic notion of knowledge. Let us consider their public announcement extensions. For public announcement logic, see [5] and [1]. The idea we take to treat with public announcement operators in algebra is to form pseudo-quotients by an element which is the meaning of the announced formula. Given a ∧-semilattice A and a in A, deﬁne an equivalence relation ≡a by: b ≡a c iﬀ b ∧ a = c ∧ a. Thus we get a quotient set A/ ≡a . We may show that ≡a is a congruence if A is a Boolean algebra or Heyting algebra. Given a Heyting algebra with operator (A, ♦), we may deﬁne operators ♦a in A/ ≡a by: ♦a [b] = [♦(b ∧a)∧a]. For the box, we may deﬁne a [b] = [a → (a → b)]. Thus for every modal Heyting algebra (A, ♦, ) and a in A, we may check that (Aa , ♦a , a ) is a modal Heyting algebra. For an algebraic semantics for IPAL, we need go from the quotient back to the original algebra. Fortunately, every equivalent class [b] in the quotient algebra Aa has an unique representative b ∧ a. Thus we may deﬁne the meaning of announcements in an algebraic model M = (A, V ) where A is a modal Heyting algebra and V is an assignment from proposition letters to the carrier set of A as follows: αφM = αM ∧ i (φM a ) and [α]φM = αM → i (φM a ) where a = αM , M a is the quotient model based on the quotient algebra and i maps each equivalent class to its representative. By using this semantics, we get the following main result: Theorem 1. The intuitionistic logic MIPC plus the following recursion axioms is a algebraicly sound and complete axiomatization for IPAL: (i) α⊥ ↔ ⊥, α ↔ α, [α] ↔ and [α]⊥ ↔ ¬α; (ii) αp ↔ α ∧ p and [α]p ↔ (α → p); (iii) α(φ ∨ ψ) ↔ αφ ∨ αψ and [α](φ ∨ ψ) ↔ (α → αφ ∨ αψ); (iv) α(φ ∧ ψ) ↔ αφ ∧ αψ and [α](φ ∧ ψ) ↔ [α]φ ∧ [α]ψ; (v) α(φ → ψ) ↔ α ∧ (αφ → αψ) and [α](φ → ψ) ↔ (αφ → αψ); (vi) α♦φ → α ∧ ♦αφ and [α]♦φ ↔ (α → ♦αφ); (vii) αφ ↔ α∧[α]φ and [α]φ ↔ (α → [α]φ). We also give a relational semantics for IPAL and show that the above logic is sound and complete with respect to MIPC-models. Our method is a general algebraic approach to epistemic actions. It may be extended to other settings beside the PAL extension of intuitionistic modal logic.

References 1. van Benthem, J.: Logical Dynamics and Information Flow (2010) (manuscript) 2. Bezhanishvili, G.: Varieties of monadic heyting algebras part i. Studia Logica 61, 367–402 (1998) 3. Bezhanishvili, G.: Varieties of monadic heyting algebras part ii. Studia Logica 62, 21–48 (1999) 4. Fischer-Servi, G.: Axiomatizations for some intuitionistic modal logics. Rend. Sem. Mat. Polit. de Torino 42, 179–194 (1984) 5. Plaza, J.: Logics of public communications. In: Proceedings of the 4th International Symposium on Methodologies for Intelligent Systems, pp. 201–216 (1989) 6. Simpson, A.: The Proof Theory and Semantics of Intuitionistic Modal Logic. Ph.D. thesis, University of Edingurgh, UK (1994)

Bayesianism, Elimination Induction and Logical Reliability Renjie Yang1 and Min Tang2 1

University of South Carolina, Department of Philosophy, Columbia, SC, USA [email protected] 2 Renmin University of China, Department of Philosophy, Beijing, China [email protected]

Abstract. The logic of scientific justification is a central problem in the philosophy of science. Bayesianism is usually taken as the leading theory in this area. After a brief review of Bayesian account of scientific justification and learning theorists’ objection against Bayesianism, this paper proposes an argument defending Bayesianism. It is shown that Bayesian conditionalization has the necessary equipment to capture the idea of elimination induction, which functions as an indispensible component in a satisfactory account of scientific justification. Bayesianism has two central ideas: the degrees of scientists’ belief about scientific theories should be quantified in a way that satisfies the principles of probability calculus; and learning from experience is through conditionalization which makes essential use of the Bayes’ theorem. Learning theorists criticized that there are certain crucial characterizations of scientific justification that can be captured by their account of logical reliability or mean-ends analysis, but not by Bayesian strategy of the explication of the relationship of confirmation. (Kelly and Glymour , 2004) The characterizations they listed are the intrinsic difficulty and efficiency of Bayesian conditionalization in the process of approaching the truth. They concluded that Bayesian updating is just one method or strategy among many that may or may not be justified depending on how efficiently it answers the question at hand. My response to this line of objection is that Bayesian conditionalization could also answer certain crucial problem about the justification of scientific theories which could not be answered by the formal learning theory. 1, the degree to which a given piece of evidence support a theory can be measured by how many possible models in which the theory is right under our knowledge before the discovery of evidence E could be eliminated by E. The idea of explicating evidential support in terms of the elimination of the failing circumstances of the theory can be called ‘elimination induction’. Bayesianism has its own criteria in determining the degree of elimination: if the proportion of the eliminated models of the theory in the eliminated models under our knowledge is smaller than the prior probability of the theory, then the piece of evidence confirms the theory; and vice versa. Formal learning theory could not provide the degree of elimination in cases where the hypothesis should not be changed under a new piece of evidence. 2, people sometimes care more about the evidential support of one piece of evidence on a given theory. This is especially true in scientific disciplines in which evidence is hard to gain. If a H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 396–397, 2011. © Springer-Verlag Berlin Heidelberg 2011

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theory survives the elimination of evidence E, it is legitimate to place more confidence on the theory, because the theory turns out to be not wrong in certain circumstances where it could have been wrong. The evidential support of E on the theory does not necessarily mean that the theory contains more truth or has more true contents. In cases where the truth lies in a theory with little support by a specific piece of evidence E, the theory itself is subject to significant modification because it also contains many falsehoods that could be eliminated by Ehe abstract should summarize the contents of the paper and should contain at least 70 and at most 150 words. It should be set in 9-point font size and should be inset 1.0 cm from the right and left margins. There should be two blank (10-point) lines before and after the abstract. This document is in the required format. Keywords: Probability, Logical Reliability, Bayesianism.

References 1. Earman, J.: Bayes or Bust. Bradford-MIT, Cambridge, MA (1992) 2. Kelly, K., Glymour, C.: Why Probability Does not Capture the Logic of Justification. In: Hitchcock, C. (ed.) Contemporary Debates in Philosophy of Science, pp. 94–114. Blackwell, Malden (2004)

A Logic of Questions for Rational Choice Zuojun Xiong and Jeremy Seligman 1

Institute of Logic and Intelligence Southwest University Chongqing, China [email protected] 2 Department of Philosophy University of Auckland, New Zealand [email protected]

Abstract. When making a choice between competing alternatives, we are primarily guided by our preferences. But the process is typically aided by asking questions. The questions serve to expand the set of possibilities we consider. Nonetheless a reasonable condition we might impose on this process is that the order in which questions are asked is ultimately irrelevant. Someone for whom this is not the case can be manipulated into making unfortunate choices by a careful choice of questions. We develop a logic for reasoning about such processes, use this to provide an independent justification for the rationality of having transitive preferences, and explain what goes wrong in situations where preferences are not transitive, such as Condorcet’s voting paradox. Keywords: rational choice, preference, closed and open questions, hybrid logic, transitivity of preferences, Condorcet, dynamic logic.

1

How Questions Guide Choices Alice is considering moving house. She is unhappy with the fact that her house is far from the bus stop. She searches the listings for a house that is better located and sees several that she likes better. She goes to visit the one of them with Betty, her good friend. When Betty sees the house, she says ‘what about a garden?’ This is not a question that Alice had considered before. Her own house doesn’t have one, but she is inﬂuenced by Betty to go back to the listings and check out houses with gardens. Eventually, she ﬁnds a house and moves. It has a nice big garden. But a few months later, she visits Chandra, a friend of Betty’s who lives in a concrete house. Alice ﬁnds in quite charming. Her new house is timberframed, like her old house and every house she has ever lives in. That night, she goes back to the listings. . .

The story illustrates how the process of practical decision making is guided by the questions one asks. Alice may well have asked very direct questions, such H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 398–399, 2011. c Springer-Verlag Berlin Heidelberg 2011

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as ‘does it have a roof?’ and ‘can I aﬀord it?’ but often it is the more open ended questions such as ‘what about a garden?’ that helped her to enlarge the options available to her, and it is these questions that we will focus on. We will develop a language and logic for reasoning in this way in the following sections. For now, consider a further development in the story: Alice is so impressed with the houses she looks at. Why had she never thought of houses being made out of anything but wood? Next month, she has moved into her new plaster house, which looks very modern and stylish. But after a while she happens to walk past her old house - the ﬁrst one. Taken by it’s quaint charm and worn woodwork, she realises that she prefers it to her new house.

2

Summary of the Paper

In [4], we introduce an operator D to model the act of making a decision to get a simple (hybrid modal) logic of rational choice. We then transform this into a ceteris paribus logic, following the work of [3] and [2], in order to interpret ‘open’ questions as opening up further states for investigation. Even though ‘other things’ may be equal, we can discover more by considering those things that are not. These questions are modelled as dynamic operators [?Q] that transform the model. We also consider ‘closed’ questions [!Q], which close oﬀ options by determining the facts. Using this idea, we propose a model of actual decision as involving the sequence [?Q]D[!Q], in which questions are opened, a decision is made and those questions are then closed. This is use to analyse the above example, and others concerning non-transitive preferences, such as Condorcet’s voting paradox.

References 1. van Benthem, J., Minic˘ a, S ¸ .: Toward a dynamic logic of questions. In: He, X., Horty, J., Pacuit, E. (eds.) LORI 2009. LNCS, vol. 5834, pp. 27–41. Springer, Heidelberg (2009) 2. Seligman, J., Girard, P.: Being flexible about ceteris paribus (2011), http://auckland.academia.edu/JeremySeligman/Papers/672679 3. van Benthem, J., Girard, P., Roy, O.: Everything else being equal: a modal logic for ceteris paribus preferences. Journal of Philosophical Logic (August 2008), http://www.springerlink.com/content/p756008882505667/ 4. Xiong, Z., Seligman, J.: A logic of questions for rational choice (2011), http://auckland.academia.edu/JeremySeligman/Papers/672670

Capturing Lewis’s “Elusive Knowledge” Zhaoqing Xu Department of Philosophy, Peking University Beijing, China, 100871 [email protected]

Abstract. David Lewis developed one of the most prominent versions of epistemic contextualism (EC) in Elusive Knowledge [2], which is also known as a version of relevant alternatives theory (RAT). In this work, I propose a simple formalization of Lewis account, which results in our basic Lewisian epistemic logic (LEL). Trivial as it may seem, my formalization focuses directly on Lewis’s theory, and thus provides a ﬁrm basis to discuss his theses on anti-skepticism, infallible knowledge, rules of relevance, and especially “knowledge without belief”. Moreover, my formalization not only avoids problems of two former existed formalizations [1],[3], but also partly answers Holliday’s question how a relevant alternatives theorist should handle higher-order knowledge [1]. Keywords: Epistemic contextualism, relevant alternatives, elusive knowledge, Lewisian epistemic logic.

1

Introduction

In daily life, it is clear whether we know something or not. However, a problem arises as soon as we engage in philosophical contemplation. Consider the following example: Example 1. You and your friend went to the Zoo. You saw there was a horse-like pinto animal in the pen, and you knew that it was a zebra. But what could you respond if your friend asked: “How do you know that it is not a cleverly disguised mule which just looks like a zebra?” For people who have no idea about the morphic distinctions between zebra and mule, such a question is really tough. It seems that we do know that the animal is a zebra, and it also seems that we do not know it is not a cleverly disguised mule. But how is that possible? Since if we know something is a zebra, then we also know that it is not a mule, let alone a cleverly disguised mule. This gives us the notorious skeptical problem, or skeptical paradox: (a) We know that it is a zebra (Kp); (b) We do not know that it is not a cleverly disguised mule (¬Kq); (c) We know that if something is a zebra, we also know that it is not a cleverly disguised mule (Kp → Kq). H. van Ditmarsch, J. Lang, and S. Ju (Eds.): LORI 2011, LNAI 6953, pp. 400–401, 2011. c Springer-Verlag Berlin Heidelberg 2011

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Clearly, (a), (b) and (c) are incompatible, so one of them must go. Mooreans do modus ponens from (a) and (c), and thus reject (b); skeptics do modus tollens from (b) and (c), and thus reject (a); tracking theorists accepts both (a) and (b) but rejects (c). All of the above accounts have their drawbacks, which results in a forth solution: epistemic contextualism (EC). David Lewis developed one of the most prominent versions of EC in Elusive Knowledge [2], which is also known as a version of relevant alternatives theory (RAT). In this work, I propose a simple formalization of Lewis account, which results in our basic Lewisian epistemic logic (LEL). Trivial as it may seem, my formalization focuses directly on Lewis’s theory, and thus provides a firm basis to discuss his theses on anti-skepticism, infallible knowledge, rules of relevance, and especially “knowledge without belief”. Moreover, my formalization not only avoids problems of two former existed formalizations [1], [3], but also partly answers Holliday’s question how a relevant alternatives theorist should handle higher-order knowledge [1]. The rest of the paper is organized as follows. In section 2, I briefly review the history of RAT and EC, as well as two former formalizations of RAT [1] and EC [3], respectively. In section 3, I summarize Lewis’s theory of elusive knowledge. Section 4 includes my formalization of Lewis’s account with static context (LEL), and short comparisons with two former formalizations. Section 5 is my formalization of the dynamic mechanism of context change. In section 6, I revisit Lewis’s anti-skepticism and infallible knowledge in my formal framework. In section 7, I investigate Lewis’s thesis that “knowledge does not imply belief” by extending LEL with two belief operators, namely, belief and “ought to” belief. Section 8 is the conclusion. All proofs are collected in the Appendix. Acknowledgments. Thanks to Can Baskent and two anonymous referee of LORI-3 conference for their comments on earlier drafts. Special thanks to Prof. Rohit Parikh for inviting me to CUNY and many other helps. The work is granted by China Scholarship Council (CSC), No.2010601219.

References 1. Holliday, W.: Epistemic logic and relevant alternatives. In: Slavkovik, M. (ed.) Proceedings of the 15th Student Session of the European Summer School in Logic, Language, and Information, pp. 4–16 (2010) 2. Lewis, D.: Elusive knowledge. Australasian Journal of Philosophy 74(4), 549–567 (1996) 3. Rebuschi, M., Lihoreau, F.: Contextual epistemic logic. In: Degrmont, C., Keiﬀ, L., Rckert, H. (eds.) Dialogues, Logics and Other Strange Things – Essays in Honour of Shahid Rahman, pp. 305–335 (2008)

Author Index

˚ Agotnes, Thomas

328

Bach, Christian W. 51 Ba¸skent, Can 382 Benthem, Johan van 1 Cabessa, J´er´emie 51 Chow, Ka Fat 61 Du, Jianfeng

Fan, Ming

386

384

Ghosh, Sujata 388 Grossi, Davide 74 Guo, Meiyun 206 Hoek, Wiebe van der 97 Holliday, Wesley H. 87 Hoshi, Tomohiro 87, 111 Hu, Yong 384 Huang, Fan 342 Huang, Huaxin 151 Icard III, Thomas F. Iliev, Petar 97 Isaac, Alistair 111 Ju, Fengkui

87

125

Kaneko, Mamoru

139

Li, Kesheng 370 Liang, Zhen 390 Liao, Beishui 151 Ling, Charles X. 384 Liu, Fenrong 125 Liu, Hu 342 Liu, Mei 384 Lorini, Emiliano 165 L¨ owe, Benedikt 179 Ma, Minghui

Pacuit, Eric 1, 179, 244 Palmigiano, Alessandra 394 Pan, Tian-Qun 392 Paul, Soumya 28 Ramanujam, R. 28 Roelofsen, Floris 233 Roy, Olivier 244

384

Eijck, Jan van

Naumov, Pavel 220 Nicholls, Brittany 220

193, 206, 394

Sadrzadeh, Mehrnoosh 394 Schwarzentruber, Fran¸cois 258 Seligman, Jeremy 390, 398 Sietsma, Floor 386 Simon, Sunil 386 Sourabh, Sumit 388 Sun, Xin 356 Suzuki, Nobu-Yuki 139 Suzuki, Satoru 272 Tang, Min

396

Uridia, Levan

286

Verbrugge, Rineke 388 Vuyst, Jonas De 300 Walther, Dirk 286 Wang, Yanjing 314 W´ ang, Y`ı N. 328 Wen, Xuefeng 342 Witzel, Andreas 179 Wooldridge, Michael 97 Xiong, Zuojun Xu, Zhaoqing Yang, Renjie

398 400 396

Zhang, Lu 370 Zou, Chongli 370

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