Lecture Notes in Control and Information Sciences
204
Editor: M. Thoma
¢
Shingo Takahashi and Yasuhiko Takahara
Logical Approach to Systems Theory
Springer London Berlin Heidelberg New York Paris Tokyo Hong Kong Barcelona Budapest
Series AdvisoryBoard A. Bensoussan • M.J. Grimble. P. Kokotovic • H. Kwakernaak J.L. Massey • Y.Z. Tsypkin
Authors Shingo Takahashi, PhD Department of Systems Science, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama 226, Japan Yasuhiko Takahara, PhD Department of Industrial and Systems Engineering, Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo 152, Japan
ISBN 3-54o-19956-X Springer-Verlag Berlin Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 19SS, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. © Springer-Verlag London Limited 1995 Printed in Great Britain The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: Camera ready by authors Printed and bound at the Athenaeum Press Ltd, Gateshead 69/383o-54321o Printed on acid-free paper
PREFACE
This book presents a logical approach to systems theory (LAST). It provides the foundations for "second order" treatment of system models and related concepts from a model theoretic point of view. The logical approach enables us to treat explicitly any types of system models as "objects." It is characterized by the following: (1) Type-free representation of system models. The representation is not only independent of any specific formalisms such as differential equations or finite state automata, but also capable of clarifying each type of system models constructed as a result of multifacetted aspects in modeling. (2) Distinction of system models from their structures. The logical approach provides a language and formal framework that describe the properties of each system model, define its structure and specify the class, determined by the structure of a system model, to which the system model belongs. (3) Hierarchical structure expansion. The relations of inclusion among classes of system models are given as hierarchical relations of the structures of system models. That is, when a class of system models is included in another, the structure of system models of the former class is obtained by expanding the other. The expanded structure inherits the antecedent. LAST places its emphasis mainly on information systems as concrete instances rather than traditional topics in control theory. Even a single information system is so complex that system models of various types coming from diversity of individual objects are included in it. Design of an information system requires to deal with such variety of types of system models and to specify a class of system models of the same type separately from the description of the system models. Thus reusability and extendibility of system models (or components constituting information systems) are expected as key features in information systems. The primary purpose of LAST is to provide an effective framework for using basic concepts in systems theory to meet such requirements of information systems. The use of model theory rather than usual set theory would make this purpose easily
vi
PREFACE
attainable. In this sense, while at first sight this book might give you pedantic impression, LAST aims at a practical device for designing information systems as well as theoretical development on system models and structures. Recent developments in object-orientation are characterized by encapsulation and inheritance, which are also shared by LAST and the requirements for design of information systems. In particular, abstract data type that is one of main components in object-orientation has very similar features to the concept of structure in LAST. These resemblances show a significaalt role of LAST in applying the concepts and methods of systems theory to design or analysis of information systems. Mainly following mathematical general systems theory, especially Abstract Systems Theory (AST) [Mesarovic et a1.1989], this book deals with some core concepts from systems theory. Hence consulting AST would help the reader fully understand this book. However the contents of this book are self-contained, and no advanced knowledge about systems theory is required, since LAST is primarily concerned with providing a rigorous framework for "basic concepts" of systems theory such as system model, structure, morphism or realization. This book is intended as a graduate textbook for an introduction to LAST. Four fundamental concepts in LAST: system model, structure, morphism and universality, are introduced in Chapter 1 through Chapter 5, which are suitable for one semester course. The other chapters might be read as advanced topics. The assumed knowledge about mathematics is standard, i.e., naive set theory. Logic used in this book is the classical first order logic, but its definitions and theorems necessary for LAST are fully given in the chapter. Some notions from category theory are used in formulation of universality, but deep knowledge on it is not necessarily needed. Some necessary basic definitions of category theory are given in appendix. The first version of this book has been used in a graduate systems theory course at Tokyo Institute of Technology. We had a lot of invaluable responses from the students attending the course. We believe that they are reflected in this book. Up to now, we have been deeply indebted to many people. First of all, we would like to thank Professor M.D.Mesaxovic for his encouragement and recommendation for the publication. We are also deeply grateful to Professor B.Nakano, Professor J.Iijima and Professor R.Sato for thoughtful and useful comments and suggestions; and to Professor K.Kijima, Professor T.Takal, Professor T.Asahi and Professor H.Deguchi for various stimulating discussions and encouragement. And We would like to express our special gratitude to Professor B.P.Zeigler for his concern and comments on this book.
PREFACE
vii
We axe thankful to Ms.H.Hayashi, Ms.M.Urata and all the members of our group for their creating good circualstances for the research.
Contents 1
2
INTRODUCTION
1
1.1
System Models: T h e Object of Systems Theory . . . . . . . . . . . .
1
1.1.1
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.2
The Role of System Models in Systems Science . . . . . . . .
2
1.2
W h a t is L A S T ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3
Basic Concepts of LAST . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.4
T h e Chapters of the Book . . . . . . . . . . . . . . . . . . . . . . . .
8
OUTLINE 2.1
3
11
System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.1.1
Systems Viewpoint . . . . . . . . . . . . . . . . . . . . . . . .
11
2.1.2
Representation of System Model
................
14
2.2
Structure
.................................
16
2.3
Morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.4
Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
SYSTEM MODELS AND THEIR STRUCTURES 3.1
4
OF BASIC CONCEPTS
Formulation of System Models
.....................
System Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.1.2
Language for Describing Systems Properties . . . . . . . . . .
30
Description of Systems Behavior
3.3
Structure of a System
3.4
Advantages and Limitations of First Order Logic . . . . . . . . . . .
SIMILARITY
4.2
23
3.1.1
3.2
4.1
23
................
..........................
OF SYSTEM MODELS
....
41 48 53
55
Morphisms for Models of the Same Type . . . . . . . . . . . . . . . .
58
4.1.1
Preservation of Generator - - Homomorphism . . . . . . . . .
58
4.1.2
Preservation of Z - - ~ - h o m o m o r p h i s m . . . . . . . . . . . . .
59
4.1.3
Preservation of
65
Th(M)
--
S-homomorphism . . . . . . . . . .
Morphisms for Models of Different Types
...............
68
CONTENTS
4.2.1
4.3
Preservation of G e n e r a t o r - - F-morphism . . . . . . . . . . .
4.2.2
Preservation of E - - E F - m o r p h i s m . . . . . . . . . . . . . . .
75
4.2.3
Preservation of T h ( . M ) - - SF-morphism . . . . . . . . . . . .
77
Application of F-morphisms . . . . . . . . . . . . . . . . . . . . . . .
83
4.3.1
Equivalence between a Finite A u t o m a t o n and a Petri Net . .
84
4.3.2
Equivalence of Behavior between a Bounded Petri Net and a Finite A u t o m a t o n
4.4 5
6
........................
93
Similarity and Analogy . . . . . . . . . . . . . . . . . . . . . . . . . .
CANONICAL
SYSTEM
MODEL
94
AS REALIZATION
97
5.1
Realization as Universality . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Canonical System Model and its Universality . . . . . . . . . . . . .
102
5.3
Canonical System Model of Stationary Systems . . . . . . . . . . . .
105
5.4
Algebraic Specification . . . . . . . . . . . . . . . . . . . . . . . . . .
113
5.4.1
Programs as System Models . . . . . . . . . . . . . . . . . . .
113
5.4.2
Canonical System Model of Algebraic Specification . . . . . .
115
98
HIERARCHY
119
6.1
Hierarchy and Emergence . . . . . . . . . . . . . . . . . . . . . . . .
119
6.1.1
Levels in a Hierarchy . . . . . . . . . . . . . . . . . . . . . . .
120
6.1.2
Emergent Properties and Hierarchies . . . . . . . . . . . . . .
122
6.2
7
68
Hierarchy in General System Models . . . . . . . . . . . . . . . . . .
126
6.2.1
Associative F-morphism in Structure Expansion
.......
6.2.2
I n p u t - O u t p u t System Models as the Lowest Level Models . .
SYSTEMS PROPERTIES
129
133
7.1
Formulation of Systems Properties
7.2
Characterization of Some Systems Properties
FURTHER
127
...................
TOPICS ON MORPHISM
.............
AND UNIVERSALITY
134 138
147
8.1
Institution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
147
8.2
Canonical System Model as Free Structure . . . . . . . . . . . . . . .
149
SUMMARY
AND FUTURE
9.1
Summary
9.2
Future Problems
PROBLEMS
................................. .............................
155 155 157
Appendix 1
159
Appendix
161
2
CONTENTS
xi
Appendix 3
162
Bibliography
163
Index
171
Chapter INTRODUCTION 1.1
System
1.1.1
Model
Models:
The
Object
of Systems
Theory
Systems Theory is a theory about representations, i.e., models, of systems rather than the systems themselves. It is significant for the theory to reconsider "models" at the first step of the theory. The term "model" is deeply and essentially concerned with human thinking azld activities. We "think" or "recognize" an object existing in the world using a model that is a representation of the object from some specific aspect. The model is used in lots of places even in daily life. Especially in scientific fields it is an indispensable concept for developing theories. A variety of the use of models generates a great diversity of their meanings. Thus it is extremely difficult to answer the question accurately, "What is a model?" without specifying the situation where the model is used. However we believe that there is some common basis for considering models. We therefore observe here the essential factors involved in the concept of models necessary for further consideration on models. Our concern is, however, not in "all" models but in the models utilized in scientific fields. We start with a general definition of a model: "Let A and B be two objects. If B is considered to copy the features of A, B is called a model of A. Then A is a prototype of B." The point is that B not "copy", but "is considered to copy" the features of A. Whether or not B is a model of A is not absolutely determined; i.e., there is no absolute and objective criterion for determining whether one thing is a model of another; a "model" is what a model builder considers as a model according to "his/her" criterion of validity. But the validity determines only whether the model under consideration is "good" or not in reference with some "aims" for the construction and
2
C H A P T E R 1. I N T R O D U C T I O N
use of the model. The aims are closely related to the functions of the model: theory formation, simplitlcation, reduction, extension, adequation, explanation, concretisation, globalisation, action or experimentation.[Appostel 1960] By these functions we can obtain information about an object in accord with our aims of modeling. Considering these, Appostel tried to define a model [Appostel 1960]: "... any subject using a system A that is neither directly nor indirectly interaction with a system B to obtain information about the system B, is using A as a model for B." We should notice that there might be no inevitable relationships between the model A and its prototype B. In that sense A can be a "symbol" of B. We can summarize the factors of a modeling as a relationship of four elements: S: a subject, A: aims, T: a prototype, M: a model, denoted by F(S,A,T,M). hi any discussion or theory concerned with models the "meta"-framework is required which can deal with at least the representation of M itself and the relationship between T and M. The logical framework developed in this book satisfies such requirement.
!
I
I $ F i g u r e 1.1 An Illustration of F(S,A,T,M) For one prototype there can be many models according to the aims. 1.1.2
T h e R o l e o f S y s t e m M o d e l s in S y s t e m s S c i e n c e
It is certain that system models fall into the category of models with the four factors mentioned above, and used as a technical term in systems science. The main doctrine of systems science would be as follows:
1.1. S Y S T E M MODELS: THE O B J E C T OF S Y S T E M S T H E O R Y
3
"A reality itself is so complex that we cannot directly analyze it and obtain effective information from it to improve present situations including problems. To attack the reality it is necessary not only to analyze precisely individual elements, but to 'recognize' the situations in question as a whole entity and 'abstrazt' essential factors to be examined." The above doctrine represents the factors of a modeling and indeed includes the four elements for it: the reality as a prototype and systems scientists or problem solvers as subjects. The a~ms for modeling are reflected in how the reality is recognized and abstracted. The result of the recognition and abstraction provides us with %ystem models". In that sense to understand what system models are is essential for systems science; system models are indispensable for systems science, particularly for logical arguments about systems. Systems theory is therefore not a theory of "systems" themselves that are objects with complexity whose properties we want to examine, but a theory of "system models" for such systems. The significance of system models lies primarily in their roles played in systems science. We can consider the roles of system models, comparing with those of models in natural sciences. There have been various philosophical controversies about the structure of natural sciences as well as models used there.
Here following the aspect of logical
positivism, we consider the structure of theory in natural sciences as shown in Fig. 1.2 [Suppe 1977; Takahara et al.1983].
Interpretation of Theory
Function of Model
Generalization
F i g u r e 1.2 Theory and Model in Natural Sciences
4
C H A P T E R 1. INTROD UCTION
Systems Recognition: Generalization and Abstraction
F i g u r e 1.3 Theory and Model in Systems Science Experience is generalized through natural scientific outlook on the world (it is called Weltanschauung), and formal theory concerning the rules providing our experience is constructed. Then a model is regarded as an interpretation of a formal theory. A model controls a formal theory through many kinds of functions containing those as mentioned in the previous section. Although these imply that there are tight relationships between formal theory and models, it is the leading view that models are inessential concepts, but convenient tools used only in developing formal theory. [Hesse 1968] In natural sciences formal theory is considered more important than models, which are in mere secondary status. In systems science the situation is quite different. As shown in Fig. 1.3, one's experience in a reality specifies a domain of concern through his/her systems recognition that is mainly composed of generalization and abstraction from systems viewpoints. Then a system model of which the domain is its prototype is constructed. System models might be further classified into three levels: proper system models, generalized system models and abstract system models, according as how the ex-
1.2. WHAT IS LAST?
5
tent is of the domain. [Mesarovic et al.1988] Corresponding to these three levels of system models there are three kinds of systems theory with respect to the level of abstraction: individual systemic theory, individual systems theory and general systems theory. Unlike in natural sciences, a systems theory is a theory concerning some system models, not concerning the reality itself. Therefore the starting point of systems theory is to specify some system models of concern. Thus a correct formal framework is required, which provides with the representation of system models and the relationships between system models, system models and systems theory, and system models and the domain of concern.
1.2
W h a t is L A S T ?
Until now we saw how important as central concept system models are. This book attempts to reveal the essentials of system models in a completely formal way based on model theory, which we call LAST(logical approach to systems theory). LAST provides a "meta'-framework to describe and investigate explicitly and deeply both system models and their structures. Roughly speaking, the following relationship holds [Chang et a1.1973]: Model theory = universal algebra + logic Universal algebra is appropriate or proper for describing system models, especially general system models or abstract system models [Mesarovic et al.1989], each of which is expressed as a mathematical structure. Logic generally includes as the main body language, formation rules of formulas, deduction system and satisfaction of formulas in a model. We sometimes use category theoretical formulation as well as universal algebra, which is often effective for the theory of general system models, when we consider a class of all systems satisfying some specific properties, for example, a class of all state space representations. We can characterize LAST by the following features: • Type-free representation • Satisfaction Relation - distinction between model and structure • Hierarchical structure expansion
Type-free representation allows us to deal with "system models" as "objects" to be operated without depending on their individual representations such as differential equation systems or Mealy type automata. This means that in LAST we can
6
C H A P T E R 1. I N T R O D U C T I O N
represent "any" types of system models and clarify the type of each system model obtained from multifacetted aspects in modeling. The greatest advantage of type-free representation is that we can look into "intermodels" or "inter-specifications" that are relationships between system models. This is indispensable for modeling especially of information systems for management, in which various facets are required to build appropriate models. Each facet is strongly connected to a purpose that a model should fulfill to resolve a given problem. Thus we need to develop many kinds of models according to the facets selected. We might have to change one representation or specification applied for expressing models to another.
The framework of type-free representation could provide a "practical"
device to perform these functions as well as a theoretical basis for modeling. We explicitly distinguishes a system model from its structure. They are connected in a model theoretical manner by satisfaction relation.
Language and
formation rules of formulas describe explicitly individual properties of system models such as controllability, stationarity, etc. In LAST we can specify the structure of a system model as an origin that generates the properties of the system model. By specifying the structure, we can formally and generally investigate relationships of structures. The structure of a system model is defined by a language to describe the properties of the system model and a set of properties to serve as axioms. Based on the satisfaction relation, a structure determines the class of system models that satisfy that strucure. Deduction system can represent behavior of system models, which is typically expressed by a sequence of formulas. We can say that LAST is also an approach to a theory about structures of system models. This is a distinctive characteristic of LAST. H i e r a r c h i c a l s t r u c t u r e e x p a n s i o n describes hierarchical relationships between structures of system models. An expanded structure is a specialization that includes an antecedent structure, while the antecedent structure is regarded as a reduction or generalization in the sense that the class of models defined by the generalized structure includes more models than the class defined by the specialized one. Although theoretical development based on the hierarchical structure expansion is a characteristic of Abstract System Theory (AST)[Mesarovic et al.1989], an explicit treatment of the hierarchical structure expansion itself can be done in LAST that provides "language" to describe "structures" and their "hierarchical relationships." These features stated above are closely related to the main characteristics of object-orientation: encapsulation and inheritance.
Type-free representation of a
1.3. BASIC CONCEPTS OF L A S T
7
model and its structure embody modularity and encapsulation. Hierarchical structure expansion shares similar concepts with inheritance in object-orientation. Also from a technical viewpoint, abstract data type or algebraic specification that is a central concept in object-orientation can be described in LAST (see Chapter 5). These resemblances are not incidental.
They give frameworks that enable us to
model a section of the real world of concern as appropriately as possible.
1.3
Basic C o n c e p t s of L A S T
There axe at least four basic concepts to be understood in applying LAST. We will illustrate in the next chapter each of the concepts in detail. (1) S y s t e m m o d e l . As stated previously, system models are objects of study in systems theory. LAST provides a formal framework for representing a s,ystem model to reflect systems recognition of a model builder. The representation should be fully independent of the types of system models, while individual system models employed in individual systems theories have their own specified types. Thus we have to specify the language in such a way that not only system models but their types can be described, that is, what the types of system models is should be clearly defined. LAST gives a natural and suitable way to satisfy such requirement. (2) S t r u c t u r e . Since every system in the reality is recognized only as a system model, the structure of a system is equivalent to that of a system model. If it is allowed to use the term "structure" in defining a system, we could define a system as follows: "A system is a whole entity having its own structure." The concept of structure has been less well-defined than that of a system and rather controversial. However in developing a "meta"-theory of systems we cannot avoid to make clear the concept of structure in a formal way. In our logical approach structure of a system model will be defined by a pair of a language to describe the system model and a set of formulas to specify the behavior of the system model. This definition comprehends essential parts of other definition of structure. (3) M o r p h i s m . Morphism is a conceptual basis for considering similarity between system models. The similarity between two system models is often defined by existing some morphism between them, more precisely, by existing some "homomorphism." However the definition by homomorphism depends on a particular representation or specification such as Mealy type automata being used in defining the similarity. Since morphism is both practically and conceptually significant as such in systems theory, we need to develop some general morphlsm independent of such a particular representation so that it gives similarity between system models not only
8
CHAPTER 1. INTRODUCTION
of the same kind of type, but of different types. The type-freeness of representation of system models in LAST enables us to construct such a general morphism between system models of possibly different types including homomorphism as a truly special case. For example we can construct a general morphism from a finite automaton to a Petri net (see Chapter 4). (4) Universality. Since an aim of LAST is to develop a meta-theory concerning "inter-models," we are interested in universal properties found in a class of system models or their structures rather than in individual models as instances. So far some universal properties significant in systems theory have been examined. This book will concentrate on the realization problem as universality, the problem which deals with how the minimal model in a given class of a structure can be constructed from a given set of input-output pairs. Algebraic specification is one of important examples of the realization as universality.
1.4
The Chapters of the Book
This book is composed of nine chapters including Introduction. The chapters can be divided into two parts: a fundamental part, Chapter 1 through Chapter 5, and an advanced part, Chapter 6 through Chapter 8. The fundamental part is concerned with presenting the four basic concepts in LAST. This part should be considered as a minimum requirement to study LAST. At the same time, it is almost enough as a basis for applying LAST to other problems concerning systems. Chapter 2 describes the four basic concepts in LAST: system model, structure, morphism and universality. This can be served as an illustrative introduction to the basic concepts. Chapter 3, based on the previous discussions on system models, naturally and formally defines a system model as a mathematical structure, and the language to describe system models is introduced as first order logic together with some necessaxy definitions and theorems. The structure of a system is defined as a pair of the language and a set of the axioms specifying the interactions of the elements involved in the system. In Chapter 4, using the morphism concept, we consider structural similarity of system models. This chapter forms a core of this book in the sense that the consideration of similarity of system models is truly fundamental in dealing with all related problems to system models. We discuss the similarity in two cases: similarity o f system models of the same type and of different types. The former is considered as a special case of the latter. Each case is discussed in three subcases with respect
1.4.
THE CHAPTERS OF THE BOOK
9
to preservation of sentences: the preservation of atomic formula, of the axioms and of all sentences satisfied in one system model. Then F-morphism is introduced as a new morphism giving the concept of similarity of system models of different types. And we will show the F-morphism theorem in the case of different types. And we will show the F-morphism theorem in the case of different types corresponding to the homomorphism theorem in the case of the same type. In Chapter 5 we define realization as universality in the sense of category theory and show the universality and minimality of canonical system model. As applications we construct the canonical system models of a stationary system and an algebraic specification. Chapter 6 discusses hierarchy. In Section 6.1 we attempt to formulate hierarchies in general, using emergent properties as a key concept to define levels existing in a hierarchy. In Section 6.2 we introduce associative F-morphisms that play an essential role in structure expansion. Then we show that input-output system models are at the lowest level in the hierarchy of the class of all system models. In Chapter 7 A class of systems properties is defined, and some systems properties are characterized with an equivalence relation defined on them. Chapter 8 is concerned with two further topics on morphism and universality. One is Institution that is a unified framework for relationships between models by category theory. The other is a free ~-structure that employs ~-homomorphism, which is a homomorphism preserving the axioms, in a class of system models for realization. We will give some conditions on which canonical system models are free ~-structures. Chapter 9 gives a summary and future problems as conclusion.
Chapter 2
OUTLINE OF BASIC CONCEPTS This chapter describes four basic concepts of LAST: system model, structure, morphism and universality.
2.1
System
Model
2.1.1
Systems Viewpoint
Systems recognition is based on the aspect that the complexity of an object essentially lies in the interactions among elements in the object. We can specify those interactions through the systems recognition. We should notice that this aspect and the conventional analytical methods are complements each of the other. Then we have a standpoint such that the properties of a complex object can be clarified through the systems recognition to it. We call an approach to an object based on such a standpoint "an approach from a systems viewpoint." Since how one recognizes an object from a systems viewpoint depends on his knowledge and interest, even one object can be studied from more than one systems viewpoint. Specification, through the systems recognition, of interactions involved in an object requires to determine what elements in the object are mutually related and how they interact. Then the object under consideration is called a "system." As a result of the systems recognition, that is, after the determination of "what" and "how," we obtain a "system model" of the object. Thus the system model obtained is characterized by its structure representing "what" and "howl" Consequently, solving complex problems involved in a system or clarifying the properties of a system by an approach from a systems viewpoint implies building a system model by specifying the interactions in the system, and exploring the properties of the system through the system model constructed.
CHAPTER 2. OUTLINE OF BASIC CONCEPTS
12
Since, as stated above, we can take various systems viewpoints even when recognizing only one object, system models of one object cannot be uniquely determined. This fact implies that a system model is not identical with a reality itself. We should adopt a systems viewpoint in accordance with our interest under consideration, aald create a system model from the selected systems viewpoint. Therefore we can see that how we can create a "good" system model in a modeling process is very crucial in systems approaches although this book does not refer to concrete modeling process. Some specific fields such as algebraic specification, control theory, operations research, general equilibrium theory or structurahsm, can be regarded as constituting a part of systems science in the sense that we find that they are developed from some systems viewpoints and the concepts of system models play central roles in developing such fields even though the term of system model is not explicitly employed. (a) Algebraic Specification. Algebraic specification was developed to cope with large software systems that are too complicated to understand easily how the systems work. Due to its abstractness, algebraic specification has a wide applicability covering a range from basic data types to complicated software systems. Even algorithms or programs can be specified using the techniques of algebraic specification. Algebraic specification is of great advantage [Ehrig et al.1985]: it provides a reliable basis for documentation and implementation, and specifies software systems without reference to any particular machine configuration or operating system available. These advantages rise from the formalism of algebraic specification, typically represented in axiomatization of the theory of the data type. In algebraic specification the concept of abstract data type plays a central role. An abstract data type can be considered as a system model that represents the interactions involved in an object through programmers' or users' systems recognition. And algebraic specification represents the structures of system models for software systems as complex objects. As will be clarified as an example in Chapter 5, algebraic specification and abstract data type are formulated, in a natural way, as a structure of system model and a canonical system model respectively. Paradigms object-orientation provides comprehend "modeling the real world as close to a user's perspective as possible [Khonshafian et al.1990]." And Object ori-
ented method that includes abstract data type as a core concept [Graham 1991, Khonshafian et al.1990] shares a lot of features with LAST. Especially two key features of object-orientation, encapsulation and inheritance, are also significant characteristics of LAST. An abstract data type is an abstraction that characterizes a set
2.1. SYSTEM MODEL
13
of objects in terms of an encapsulated data structure and operations on the data structure, and can be represented in a logical form as a system model. (b) Control Theory. The concept of control, especially feedback, has influenced the development of cybernetics advocated by N.Wiener[Wiener 1961], cybernetics that shares common language with systems science. In control theory, a system model of a system is analyzed for the purpose of control of the system. It is usually expressed by some differential or difference equations. Then although the mathematical structure of the equations is quite clear as such, we should notice that it is not clear whether that mathematical structure is the most appropriate as the "structure" of the system, in the sense defined in Section 3.3. For example, we can define deeper and essential structure as a basic linear system [Mesarovic et al. 1988], (c) Operations Research. Operations research has been considered to be one of the most popular methodologies or instrt_Hnents in systems analysis or systems engineering. Here let us consider what the system models in operations research are. Operations research is a scientific method usually arranged as follows, to solve problems in the real world [Makabe et al. 1981]. (Notice that the following represents "traditional" operations research, not a recent new trend such as the soft systems methodology [Checkland 1981].) 1. Definition of the problem. 2. Investigation: Collection of data and analysis of the relationships a~nong factors involved in the problem. 3. Analysis: Construction of a model representing the interactions among factors and comparison of alternatives with the model. 4. Interpretation and evaluation of results. The central matter in operations research is model building at the third step, in which the factors concerned with the problem and their interactions are specified. In that sense the models built in operations research can be called "system models." (d) General Equilibrium Theory. Phenomena economics deals with are also very complex. Therefore we can find that many theories or approaches in economics are developed from systems viewpoints. General equilibrium theory is one of such theories, in which the interactions in market under some conditions that can be also considered as a systems viewpoint, are represented as equation system concerued with demand and supply of goods. This equation system is just a system model in considering general equilibrium; economics in market is dealt with as the problems of general equilibrium of the system model above. Besides general equilibrium theory,
C H A P T E R 2. OUTLINE OF BASIC CONCEPTS
14
structure equation systems in econometrics can be also viewed as a system model [Takahashi 1986].
(e) Structuralism.
Structuralism developed in, say, cultural anthropology by
Levi-Strauss [Levi-Strauss 1958] or psychology by Piaget [Piaget 1968] deals with objects under consideration as unity with interactions involved, and explores "structure," innately contained in the objects in form of deep structure, of system models of the objects.
2.1.2
Representation of System Model
A system model is a whole entity whose essentials lie in the innately complex interactions anlong elements, and which is recognized from a systems viewpoint. An initial step of construction of a system model is to identify elements, attributes the elements should have and relations among the attributes. Let us consider a small but complex object, a "family," as a system. What are elements to be expressed in a system model? This completely depends on the viewpoint we take. We might say that they are a father, a mother, children and sometimes grandparents and the like. However if we consider the family from an economic viewpoint and have concern in its household budget as a subsystem of a whole economic system, then we should select other elements, e.g. income and expense.
Then what about attributes?
If we are interested in the "affections"
involved in the family, we should take an approach from a psychological viewpoint, which lets us choose different attributes from the economic one or others. A "family system" independent of its representation does not exist in the real world in the sense of the systems recognition. From a specific systems viewpoint we recognize a family as a family "system" and represent it as a system model. In general, under a given systems viewpoint, to construct a system model we first choose some elements al, a2,... (not necessarily finite) and attributes V1, V2,..., Vn. Each element has some attributes.
This can be denoted by, for example, a3 E
V2, a4 E V1, V5. Relations R1, R2,..., Rm among the attributes are expressed as corresponding relations on the set of elements A, i.e., a subset of some products of the set of elements. Consequently we have a representation of a system model as follows:
< A;V1,...,Vn, R 1 , . . . , R m > • For example, from a certain viewpoint, suppose father, mother, childl, child2 and grandmother are chosen as elements of a family. Each element has some at-
2.1. S Y S T E M M O D E L
15
tributes, for example, father E FATHER&MALE, mother E M O T H E R & F E M A L E , childl E C H I L D & M A L E , child2 E C H I L D & F E M A L E , grandmother E G R A N D P A R E N T & F E M A L E . This choice of elements and attributes already reflects the selected viewpoint. Next we should specify some relations a~nong the attributes. For example, a parentchild relation, a sister-brother relation and grandparent-grandchild relations. Then, for example, we see that (father, childl ), (father, child2), (mother, childl ), (mother, child2), (grandmother, mother) E P A R E N T
- CHILD;
(child2, childl) E S I S T E R - B R O T H E R ; (grandmother, childl ), (grandmother, child2) e GRANDPARENT
- GRANDCHILD.
Mathematically we have PARENT
- CHILD = ((FATHER U MOTHER) O(GRANDPARENT
SISTER - BROTHER GRANDPARENT
x CHILD)
x (FATHER U MOTHER));
C CHILD x CHILD;
- GRANDCHILD
Let A be the set of the elements:
= GRANDPARENT
x CHILD.
A = ( f a t h e r , mother, grandmother, child1,
child2}. Then each relation given above is a binary relation on A, i.e., a subset of A x A. We should notice that we can specify even a "relation on xrelations" such as I N - L A W on F A T H E R
and G R A N D P A R E N T .
In this case we should consider F A T H E R
and G R A N D P A R E N T
also as ele-
ments. An input-output model, which is a basic system model in systems theory, has two attributes: I N P U T
and O U T P U T .
Each element is recognized as either an
input or output. Then an input-output "system" is specified as a binary relation S on I N P U T
and O U T P U T : S C I N P U T x O U T P U T .
In systems theory we often consider "states" and a "state transition function." Although the state transition function is a relation on inputs and states, it is both semantically and technically convenient to express explicitly functions in a system model. On the other hand since each attribute, V~, stated above can be also expressed as a unary (i.e. 1-ary) relation on the set of elements, we mathematically need no distinction between an attribute and a relation.
16
CHAPTER
2.
OUTLINE
OF BASIC CONCEPTS
Consequently we reach a general representation of a system model: a system model is a collection consisting of the set of elements, relations and functions on it: < A;Rt,...,Rk, fl,...,ft > •
2.2
Structure
A structure of a system model characterizes the system model in the sense that the structure determines to which class of systems the system model pertain. In this sense if a system model is expressed by a collection of some differential equations, we can say that the matrices of the coefficients of the differential equations give a structure of the system model. However, from systems viewpoints, a class of system models should be specified not by the form of differential equations, but by a set of systems properties. Hence the structure of a system model should have at least the following features. First the structure of a system model generates its properties or behavior to be recognized. Second the representation of structure is based on a hierarchical construction. For example, the structure of an input-output linear system model is "hierarchically" constructed from both a linear structure and an input-output structure, in the sense that the input-output linear structure explicitly "inherits" the properties from the linear and input-output structure. Third the structure distinguishes the properties of the class of system models satisfying it from those of an individual system model in the class. One way to fulfill the above requirements is to adopt a "language" that expresses systems properties, and to represent the structure as "axioms." This means that we should abstract basic properties from a class of system models as axioms that are common characteristics of the class. Thus the structure of a "family system" in the previous section is abstracted from concrete family models. For exanlple, we can abstract some axioms such that every father is a male, every mother is a female, father and mother are married, all brothers have the same father and mother and so on. The language such as "father," "male," "every," "is-a," etc., and some "grannnar" to make legal sentences should be chosen before axioms are described. Then the axioms are expressed by some sentences in that language. We should notice that this example of the structure of family does not include all families at all; a family that has brothers whose mothers are different is not included. A language and axioms are chosen from a systems viewpoint that reflects our
2.3.
MORPHISM
17
current interest. In this sense the structure of a system model expresses fundamental interactions we recognize as the system model does. Thus a modeling process contains as its essential part some stages of specifying language and constructing axioms. Consequently the structure of a system model is defined as a pair, (L:;~), of language £ to define the system model and a set of axioms Z to describe the class to which the system model pertain. In LAST every structure is defined in a formal language, and especially in this book first order language is used. However, use of other formal languages than first order is not restricted in LAST. Some advantages and disadvantages of the use of first order language will be mentioned in the next chapter. The formal description of a structure of a system model has some technically outstanding advantages as well as conceptual ones. It enables us to point out what a "systems property" of a given system model is, and to distinguish the system properties from system models that "satisfy" the properties. This relation is provided as satisfaction relation that is one of the main characteristics of LAST as stated in the previous chapter. Thus we can construct and specify a class of system models without depending on the concrete descriptions of individual system models. Finally we notice some relationships between a system model and its structure. For example a system model of integers < Z; + :>, where Z is the set of integers and + addition, belongs to the class of monoids, which is specified by: Axiom 1. For any elements x , y , z , (x + y) + z = x + (y -{- z); Axiom 2. For any element x, • + 0 = 0 + ~c= x. A system model of natural numbers < N ; + >, where N = {0,1,...}, also satisfies the above axioms. Hence < N; + > is also a member of the class of monoids. However if we consider an axiom: Axiom 3. For any x there exists some element y such that x + y = 0, then < N; + > does not satisfy this axiom, while < Z; + > still satisfies it. We should notice that <: N; + > is a subsystem model of < Z; + >, in the sense that a base set N is a subset of Z and + is the restriction of that in Z. This means that hierarchical relations of structures does not accord with those of system models. Conversely, although the structure of input-output system models is a substructure of that of input-output linear system models, an input-output system model is, in a formal sense, not a subsystem model of any input-output linear system model.
2.3
Morphism
Morphisms are concerned with relationshipsbetween system models or structures, while system models and structuresspecify relationshipsamong elements.
CHAPTER 2. OUTLINE OF BASIC CONCEPTS
18
Technically morphisms provide essential "devices" in investigating structural similarity between system models or analogy, for example, in artificial intelligence and cognitive science. (See Section 4.4.) Conceptually morphisms play a role of "bridges" that in modeling process validate system models for objects of concern existing in the world, or in simulation verify system models. A basic idea of morphism is that there is a certain correspondence between two objects, and by the correspondence a "structure" of one object, in a sense, is preserved in the other. In the above the structure to be preserved is not restricted to the structure of the system model stated in the previous section, but rather used here as a general term that indicates some properties of the object. The correspondence that can be a morphism is not necessarily a "function" in the set theoretical sense. Not only "many-one" or "one-one" but "one-many" correspondence are allowed, and even other relations such as inequality can be correspondences. In this sense the image we get by a morphism from an object does not always mean a "simplification" of the original, although a homomorphism that is a typical and the most dominated concept as a morphism provides only a simplified image of the original. Hence it is insufficient to use only homomorphisms as mathematical devices to investigate the problems concerning similarity, although the basic idea of morphism stated above is extracted and developed from the concept of homomorphism. We will define, in this book, a general morphism called F-morphim that includes properly the concept of homomorphism and essentially extends it. The F-morphism provides a general framework for dealing with structural similarity. We start with a homomorphism as a technical and conceptual basis. The essence of homomorphism is often illustrated by a "commutative diagram." For example, a homomorphism from a system model A~I with a base set A to another Ad2 with a base set B is a function h of A to B preserving the structure of A41 in the sense that if for every relation R in AQ, R(a, b) (assuming that R is binary) holds in AQ, then for the corresponding relation R' in A42, R'(h(a), h(b)) also holds in 1v42, and for every function f in A41 and the corresponding function g in Ad2, it follows that
h(f(a)) = g(h(a)) (also assuming that they are unary) for a E A. This relation can be depicted as a commutative diagram. (Fig. 2.1) In homomorphism there are at least two insufficient features to be developed as a general device for investigating structural similarity.
2.3. MORPHISM
19 h A
v
J
B
Jf
A
B h
F i g u r e 2.1 Commutative Diagram First the properties preserved by a homomorphism are rather primitive. We should have other sophisticated classes of properties to be preserved by a morphism. In Chapter 4 two other kinds of classes of properties to be preserved than homomorphism will be specified: the set of axioms as a structure of a system model and the set of all the properties satisfied by a system model. For example, let us consider a system model of a partial order relation <. A homomorphism between two order relations gives an order preserving morphism such that a < b implies h(a) < h(b). Then if we take the axioms of partial ordering (i.e. reflexive, anti-symmetric, transitive) as a structure, the homomorphism preserves that structure. However, if we add the property of the non-existence of the minimum point to the axioms, then the homomorphirm does not transfer this property any more. The following homomorphism h : Z -* N is such an example, which is defined
by h(x) = ~ 0 if x_<0,
[
x
otherwise.
We should notice that a homomorphism that preserves all the properties of a system model is not necessarily an isomorphism, which is a one-to-one homomorphism and also preserves them. The second insufficient feature of the homomorphism is that we cannot consider morphisms between system models of different types. For example, we can only define a homomorphism from a Mealy type automaton to another Mealy type one, but by definition of a homomorphism, cannot construct any homomorphism between a Mealy type automaton and a Moore type one. However we know well that the two types of automata are "essentially" equivalent in the sense that we can simulate one type machine by the other. This kind of equivalence is beyond the homomorphism. We need to develop mathematically a new morphism to deal with such equivalence.
C H A P T E R 2. O U T L I N E OF B A S I C C O N C E P T S
20
One reason of the insufficiency of the homomorphism stems from the "automatic" correspondence of symbols. In saying that R(a, b) implies R~(h(a), h(b)), the relation R t is "automatically" assigned as the corresponding relation of R. This means that two system models of the "same type" are already similar to a great extent.
homomorphism
0/0
I/I
S
b/b
aJb
a / ~ b/a Mealy type X
0/1 Mealy type automaton
F-morphism
y Moore type
F i g u r e 2.2 An Illustration of Homomorphism and F-morphism An F-morphism that can be defined between system models of possibly different types associates a formula with each symbol the language of the system model consists of. If for a relation _< in an original system model there is no directly corresponding relation symbol in a target system model, then we consider, for example, a formula (3x)(a + x = b) - if the target model has the symbol + - as an "interpretation" of the relation a _ b. This formula should be required to satisfy some validation conditions, which will be given in the definition of F-morphism. Furthermore, to deal with classes of properties to be preserved we need a mechanism that "automatically" transforms each formula of a system model to the corresponding "valid" formula in the other. An F-morphism will be actually composed of the "interpretation," Bas, of the symbols and the transformation mechanism, IF, of formulas.
2.4.
21
UNIVERSALITY
In the above exan~ple, B a s ( < ) = ( 3 x ) ( a + x -- b)
and I F ( ( V X ) ( 3 y ) ( x < y)) = ( V x ) ( 3 y ) ( 3 z ) ( x + z = y).
As we can see from the above, an F-morphism does not necessarily give simplification of an original system model, but generally represent similarity itself.
2.4
Universality
Universality has a unique characteristic in LAST. The mathematical definition of
universality comes from category theory rather than logic. However universal property can be found in many areas in mathematical theories. Especially in systems theory minimal realization, which identifies a minimal "structured" system model satisfying a pre-structured data set, is considered to have the universal property. This categorical representation of the universality should be regarded as an abstraction that provides a unified framework to express the universality found everywhere. Suppose given a set of data that might be input-output pairs and a structure ((£; E)) into the models of which the given set of data is embedded. Then we want to construct a "minimal" model among the models satisfying the given structure. The minimal model means that any model in which the given set of data can be embedded preserves the structure of the minimal model in the sense that there is a homomorphism from the minimal model to the given model (Fig.2.3).
embedding ................. = minimalmodel
data e
m
~
homomorphism anymodel Figure2.3
The minimal model will be a actually constructed as a canonical system model. It is generated from the set of data and is composed of only the "terms" that the
22
C H A P T E R 2. OUTLINE OF BASIC CONCEPTS
given structure ((£; ~)) allows to speak about. The caalonical system model as a minimal model satisfying the universal property contains no junk and no terms in an uncontrolled way. For example, suppose we are given only a constant 1 as a datum. Then the problem is to construct a minimal model with a structure (L:; Z) where L: is composed of 1 and addition +, and ~ sentences expressing the commutativity: x + y = y 4- x for all x, y and the associativity: (x 4- y) 4- z = x 4- (y 4- z) for all x, y and z. In this case the base set of the canonical system model consists of all the equivalence classes each of which contains terms with the svane number of l's;e.g. [(1 4- 1) 4- (1 4- 1)]. Any two terms that cannot be implied from ~ are not equal. On the other hand, the system model with the base set {0, 1, 2} of remainders of modulo 3, which is also a model for E, satisfies some properties that cannot be implied from E, for example, 2 4- 2 = 1. Hence this model is not minimal in the sense that it contains "uncontrollable" terms for ~, which Z does not speak about. Actually we can define a homomorphism from the canonical system model to this model. The canonical system model as a minimal realization defined by universality is important not only in modeling but in abstract data type or algebraic specification where the canonical system model gives an initial semantics of a given specification. Although we define realization for a given set of data in terms of category theory, no detailed knowledge on category theory will be assumed and some basic definitions of category theory will be found in Appendix.
Chapter 3
SYSTEM MODELS AND THEIR STRUCTURES In this chapter we discuss what system models and their structure are a~d how they can rigorously represented. To this end we describe system models and their properties in terms of logic. We represent a system model as a mathematical structure, and express the properties of it by first order formulas. Finally, we define the structure of a system.
3.1
Formulation of S y s t e m M o d e l s
In this section we will formulate system models and their description languages in the model theoretical sense. The formulation provides us with some outstanding merits: unambiguity of the meanings of representations and possession of describing power for the succeeding discussion, particularly for the metatreatment of system models. By the metatreatment we can describe an individual system independent of the representation. 3.1.1
System
Model
A direct and natural representation idea that a system model is a whole entity with some interactions among its elements, is that we express it as a mathematical structure that consists of a base set, relations and functions on it. D e f i n i t i o n 3.1.1 ( S y s t e m M o d e l ) A system model A4 is composed of: (1) a base set M; (2) a set of )~(i)-ary relations on M, {P~li E I}, where ~ is a function such that I --* N + (positive integers); (3) a set of #(j)-ary functions on M, {fjlJ E J}, where # is a function such that J --~ N (non-negative integers).
CHAPTER 3. SYSTEM MODELS AND THEIR STRUCTURES
24
Here an n-axy relation or function has n axguments, written as R(al . . . . . a,~), or
f ( a l , . . . , a , , ) . The function A (or #) means that the axity of a relation Ri (or fi) depends on its index i (or j); written as Ri(ai .... , ax(0) or fj(al . . . . , a,(D). Nullaxy functions, which have no arguments, axe called constants.
The pair
< A, # > is called the type of.M. We write AA as follows: A4 = < M; {Ri[i 6 I}, {fj[j 6 J} > We sometimes write [.M[ to indicate the base set M. This definition of a system model has very wide applicability; almost all important systems representations we axe interested in can be reformulated in the above form. We illustrate below some typical and significant examples of system models.
Example
3.1.1 ( I n p u t - O u t p u t S y s t e m Model) A simple but quite important
instance of this definition of a system model is an input-output system model. Although we can have some representations of it, the following one is natural.
.Mi/o = < X tAY; S, X, Y >, where S C X x Y, X is the set of inputs and Y the set of outputs.
X
S
Y
F i g u r e 3.1 An Input-Output System Model .MIIO E x a m p l e 3.1.2 ( T i m e S y s t e m Model) A time system model is an input-output system model whose inputs and outputs axe time functions on input (and output) alphabets. One of natural representations of a time system model is:
.)14T-sys ----< X U Y U A U B U T ; S , X , Y , A , B , % T , <_,+,O, 1 >
25
3.1. F O R M U L A T I O N OF S Y S T E M M O D E L S
where X and Y axe the sets of functions of T to A and B respectively. The set T represents "time," which has linear ordering <, additive properties by + and constants 0 and 1. The symbol o is a binary function that assigns values of the time functions x and y at any time t, i.e., x ~ t stands for x(t).
A
xot(--x(O)
B
M*
an output /~~~y(t)) @
,
~ t (time)
t (time)
F i g u r e 3.2 A Pair of Input and Output in a Time System Model E x a m p l e 3.1.3 ( S t a t i o n a r y S y s t e m M o d e l ) A stationary system model is a time system whose behavior does not depend on the time at which we observe the system. We represent the non-dependency with a shift operator ~. A stationary system model can be defined by: Mst,, = < X U Y U A U B t 3 T ; S ~ X , Y , A , B , o , T ,
<_, +,O, 1, A >
where the meanings of the symbols are the saane as a time system model except A. The symbol A is a binary function that expresses the shift operator by time. For example, A(t, x) o T staalds for x(t + T). The stationary system model must be a system that satisfies the properties of stationarity: S(x, y) --* S(A(t, x), Mr, y)); X ( x ) --* X(A(t, x)) and Y ( y ) -~ Y(A(t, y)). E x a m p l e 8.1.4 ( L i n e a r S y s t e m M o d e l )
A linear system model is a system
model whose input set and output set are vector spaces and whose behavior has the linear property that S(x, y) and S(x', y') implies S ( a x + l~x', ay + fly') for any a and/~ in a field F over which the input and output sets are the vector spaces. The linearity of the input set is represented by the following system model: Mx
=< Ft3X;F,X,+,-,
-1 , • ,OF, 1F, Ox >
26
C H A P T E R 3. S Y S T E M M O D E L S A N D T H E I R S T R U C T U R E S
where X is the set of inputs, F a unary relations that is the set of scalars, 0x the zero vector, 0F the zero scalar, 1F the unit element of F, ÷ a binary function representing both scalar and vector addition, - a unary function representing the additive inverse, a binary function representing scalar multiplication and multiplication of a vector by a scalar and -1 a unary function representing the multiplicative inverse. The linearity of the output set is similarly defined. A linear system model is defined as the union of A / x , M y and the input-output system model with the linear property of behavior, where the union of two system models A,tl and A,t2 is the system model whose base set, functions and relations are respectively the unions of the corresponding sets of the two system models. R e m a r k . In our definition of a system model functions should be defined on the whole domain of the base set. However as in the above examples many functions of system models are not intended to be defined on the whole domain. For example, in a vector space multiplication of two vectors is not defined, while in a linear system model it should be defined. Hence in order to construct a linear system model in the sense of our definition, we need to define artificially multiplication of a pair of vectors, for example, xl • x2 = 0x. This seems inconvenient for developing systems theory. However by using the concept of structure of system defined later, this sort of artificiality will cause no difficulty in LAST. Technically we can avoid such extra definitions if we adopt many-sorted mathematical structures (i.e. structures with an indexed set of base sets each containing objects of a different sorts) as the definition of system model. Since the two methods of representations are logically equivalent, a "one-sorted" system model is selected for the simplicity of representation. E x a m p l e 3.1.5 ( S y s t e m M o d e l o f Stack) A stack is a very common notion in information systems. Here we consider a very simple system model for stacks. A stack stores in a sequence data items from a given alphabet based on the "last in first out" principle. The "push" operator inserts a new element on the top of the stack. The "pop" operator removes the topmost element. The "top" operator looks at the topmost element. If the "top" operator is applied to the empty stack, then an error message, which is expressed by "error" element of the alphabet. Thus a stack system model is defined by: .h48~ac~ =< A (J A* U (error}; A, S T A C K , push,pop, top, error, empty >, where A is an alphabet, A* the set of all strings over A, S T A C K = A*, empty an element of A*, push : A × A* ~ A* defined by p u s h ( a , w ) = aw,
3.1.
FORMULATION
OF SYSTEM
27
MODELS
pop : A* ~ A* with WI
pop(w)=
empty
if W = aw t, otherwise,
top : A* ~ A with
[ a error
top(w)
if w = aw', otherwise.
We can have another aspect to model stacks. For example, if we want to propagate errors as stacks, which is called error propagation, then a distinguished error element should be added to A*. This model is obviously different (i.e., non-isomorphic) from
J~stack'
push
/4pop b
~-
top
c d e f
F i g u r e 3.3 A Stack E x a m p l e 3.1.6 ( D i s c r e t e E v e n t S y s t e m S p e c i f i c a t i o n ( D E V S ) M o d e l ) The DI~VS formalism provides a means of constructing simulation models and a forreal representation of discrete event systems capable of mathematical manipulation just as differential equations serve this role for continuous systems [Zeigler 1990]. A DEVS model that is a basic model in the DEVS formalism consists of a time base, inputs, states, outputs, and functions for determining next states and outputs given current states and inputs:
.h~DEVS = < X U S U Y U R U { ~ } ; X , S , Y , Sint, Sext, A, t a , Q , T > where X is a set of external event types, S a set of sequential state, Y a set of external event types generated as outputs, T the time base, t~ the time advance function from S to the non-negative reals with infinity:ta : S ~ R 0,~, + Q the total state set defined by Q = { ( s , e ) l s E S,O < e < ta(s)}, ~int the internal transition function:Sint : S ~ S, ~ext the external transition function:5~.~ : Q × X ~ S and ), the output function:A : Q -~ Y.
CHAPTER 3. S Y S T E M MODELS AND THEIR STRUCTURES
28
Some essential behavior of a system specified by DEVS, such as the property of the output function that generates an external output just before an internal transition takes place, should be considered to be included implicitly in .MDEVS. As can be seen in the above example, most of system specifications in modeling or simulation can be expressed as system models except that a system model requires to represent explicitly also components that are usually "implicitly" given in specifications, for example the time base or the set of non-negative reals for clock. In the most general sense, a system can be defined as a relation on some attributes V1. . . . . Vn, the relation which is expressed by S C 111 x ... x Vn.[Mesarovic et al. 1989] Starting from this general definition of a system, we develop systems theory by introducing into the attributes some structures such as linearity, stationarity and so on that are suitable to our interests in objects as systems.(cf. Chapter 6) Our definition of a system model realizes this concept of a system in as a formal and general way as possible. Although from the purely formal point of view we can also formulate system models as relational structures, which consist of only relations, or algebraic structures, which consist of only functions, such formulations are inappropriate for the further development of a general theory of system models due to the following reasons: (1) In relational structures, since terms (defined later) are only variables, there is no concept of '~mw" terms generated by functions, which is of great importance in system models defined as in Definition 3.1.1. For exanlple, the concept of canonical system model as will be discussed in Chapter 5 cannot be realized only in relational structures, since it is constructed from the constants contained in it, the constants which are naturally represented as nullary functions. (2) In algebraic structures, a relation can be represented as a truth-value function. Then the base set of a system model must include the set of truth-values. However, this is not suitable to the system model except the case where the system recognition of a model builder for the system model is done also with respect to the truth values. This case is unusual. Independent of individual system models, the truth values should be universally given in a logical fraanework to describe system models. F~arthermore although the base set can be represented as a many sorted one, this is not necessary since each sort can be represented as a unary relation. (cf. Chapter 4) The next definition of subsystem nmdel provides one of typical ways to produce a new system model from an old one, as well as the way of homomorphism introduced in the next chapter.
3.1. FORMULATION OF SYSTEM MODELS
29
D e f i n i t i o n 3.1.2 ( S u b s y s t e m M o d e l ) Let
2Pll =< M1; {R~[i • I}, {f]lj • J} >, ,~42 = < M2; {R21i • I}, {f}lJ • J} >, where A41 and A42 are of the same type. A~i2 is said to be a subsystem model (or
submodel) of ]~41 if the following conditions hold: (1) M2 C M1; (2) R 2 = R 1 N M~ (i) for each i e I; (3) f ] ( a l , . . . , a~,(j)) = fl(al ..... at,(j)) for any a l , . . . , at~(j) • .M2,j • J; Then we write J~42 C ,~41. For exexaple, let J~41 = < {O, 1,2,3};S1,X1,Y1 > andA42 = < {O, 1};S2,X2, Y2 > be input-output system models, where $1 = {(0, 1), (1, 2), (2, 3), (3, 0)}, X1 = Y1 = {0, 1, 2, 3}, $2 = {(0,1)}, X2 = Y2 = {0, 1}. Then ~42 C A41. Next let AA1 = < Z ; + > and A42 = < N ; + >, where Z is the set of integers, N the set of natural numbers and + the usual addition on them. Then A42 C ,~41. We should notice that 2~41, i.e., Z is a group with respect to the addition, but A42, i.e., N is not a subgroup of Z. This implies that the concept of subsystem model does not necessarily accord with that of mathematical subalgebra such as subgroup. This is because only universal sentences, which have only universal quantifiers, can be preserved in a subsystem model in general [Gr£tzer 1979]. In the above example the axioms of group includes the property concerning the existence of inverse elements. However the (first order) sentence expressing it in A41 and J~2 is not "universal." Actually it is usually expressed as (Vx)(3y)(x+y = 0). If each model has a function -1 that assigns an inverse element to each element, then every subsystem model of < Z; +,-1 > as a group is exactly its subgroup since the sentence for the existence of inverse elements is " universal": (Yx) (x + x -a = 0). Another way to produce a new system model from an old one is reduct.
D e f i n i t i o n 3.1.3 ( R e d u c t o f a Model) Let ~'1 and ~2 be types of models and A~I2 a system model of type T2. Let T1 C T2. Then the model, written by AA21rl, obtained from .hA2 by the elimination of the relations and functions not included in T1 is called the reduct of A~t2 to Zl. Notice that the type of A~I21T1 is rl.
30
C H A P T E R 3. S Y S T E M M O D E L S A N D T H E I R S T R U C T U R E S
3.1.2
Language for Describing Systems Properties
The systems properties are the properties possessed by a system model such as linearity, stationarity or causality. To investigate properties of these systems properties, which can be called the meta-treatment of system models, we introduce a formal language to describe systems properties. The use of the formal language characterizes LAST. In this section we give only the formal framework of the language. The fully formal discussions on systems properties will be found in Chapter 7. D e f i n i t i o n 3.1.4 ( L a n g u a g e for a S y s t e m M o d e l ) The language £:(~4) for a system model A~i consists of: (1) A(i)-ary predicate letters Ri for each i E I, where A is a function such that I -+ N + (positive integers); (2) #(j)-ary function symbols fj for each j E J, where # is a function such that J --+ N (non-negative integers). We write Z:(.g4) as follows: L ( M ) = < {aili e I}, {fjlJ e J} > < )~,/~ > is also said to be the type of £(2t4). There is the one-to-one correspondence, denoted by Cot, between the boldface symbols in £:(~4) and the light face symbols for the relations and functions in Az[, i.e., C o r ( R i ) = Ri for each i e I and Cor(fj) = fj for each j e J. Then ~ / i s said to be a realization of the language •(,£4) or model for/:(¢~4) (especially in Chapter 5 where "realization" has another meaning). The languages for two system models of the same type are, up to alphabetic variants, identical. Therefore every system model of the same type as a system model A~t is a realization of the language £(A8). For example, an input-output system model, .~¢Ii/o =< X U Y; S , X , Y
>,
and all the system models of the same type as this system model are realizations of the language, £(A~[//o) = < S, X, Y > . We should notice that for language we customarily use boldface symbols with the same alphabets as a system model, for example, S and S, so as not to confuse language with system models. This usage is only for the sake of convenience. However we should notice that a language, e.g. ~(.MI/O) , is purely syntactic construct. Other models than A4x/o, e.g..h4' --< Z; T, V, W >, can be realizations
3.1. FORMULATION OF SYSTEM MODELS
31
of Z:(,~AI/o) even if they have no property of an input-output system model. The desirable properties that every input-output system model should have are specified not only as a language but as a structure of the system model, as will be mentioned in Section 3.3. To describe systems properties, we need "grammar" that distinguishes "right sentences" from wrong sentences. In our logical approach we assume that the properties of a system can be represented as first order sentences in first order language that plays the role of the "grammar." The first order language consists of the primitive symbols, the formation rules of the terms, the atomic formulas and the well-formed formulas. Definition 3.1.5 ( T h e P r i m i t i v e S y m b o l s o f F i r s t O r d e r L a n g u a g e w i t h E q u a l i t y ) The primitive symbols of the first order language for a system model Z~(.h4) consists of £(.~4) and the following logical symbols: (1) individual variables v 0 , v l , . . . , v n , . . . (for n E N); (2) logical connectives A (and), ~ (not); (3) the universal quantifier V (for all); (4) the binary relation symbol = (identity); (5) parentheses ),(; (6) comma ",'. We should notice that these primitive symbols and the particular combinations of them defined below as terms, formulas and sentences are purely syntactical. The logical connectives mad universal quantifier as primitive symbols have no proper meanings such as "and," "not" and "for all." These intended meanings are realized only when these symbols are interpreted in a specific system model. This realization is called satisfaction. [See Definition 3.1.12] We will usually use other symbols, say x, y, or z, as individual variables. Definition 3.1.6 ( T e r m ) The set of terms of/~(A~), denoted by Term(L(.h4)), is recursively defined as follows: (1) each variable and constant is an element of Term(£(J~4)); (2) for each function symbol fj, j E J, if t l , . . . , t ~ ( / ) E Term(£(A4)), then fj ( t l , . . . , t~(j)) e Term(£(A~)). (3) a string of symbols is a term only if it can be shown to be a term by a finite number of applications of (1) and (2). For exan~ple, let .h4 = < N ; + , 0 > and £(J~4) --< + , 0 >, where + is addition and 0 a constant. Then some instances of terms are:
o, +(o, o), +(x, o), +(x, +(o, y)), +(+(x, y), +(u, w))
. . . . .
CHAPTER 3. SYSTEM MODELS AND THEIR STRUCTURES
32
We sometimes write x + y for a term + ( x , y ) . A term that has no variable is called a closed term. D e f i n i t i o n 3.1.7 ( A t o m i c F o r m u l a ) The set of atomic formulas of £(A4), denoted by Atom(£(A4)), is defined as follows: (1) if tl,t2 E Term(L(A4)), then (tl = t2)E Atom(£(A4)); (2) for each predicate letter Ri, i E I, if t l , . . . , t x ( i )
E Term(£(A4)), then
Ri(tt,...,t:~(i)) e Atom(£(.A4)). Again consider the above examples A4 and £(A4). Some instances of atomic formulas of £(A4) are: 0 = 0,x+y
= 0,(x+y)
+z
= u+w,....
D e f i n i t i o n 3.1.8 ( F o r m u l a ) The set of formulas of £(A4), denoted by
Form(£(A4)), is recursively defined as follows: (1) Atom(L(A4)) c Form(£(A4)); (2) if ¢, ¢ e Form(£(A4)), then (¢ A ¢), -~¢, Vvi¢ e Form(£(A4)), where vi is a vaxiable; (3) a sequence of symbols is a formula only if it can be shown to be a formula by a finite number of applications of (1) and (2). (We sometimes write (Vx)¢ for Vx¢ in (2).) Formulas are basically constructed by some connections of some atomic formulas. some instances of formulas are: (Vx)(x + y = 0),-~(Vx)(x = 0) A (x + y = y + x) . . . . . We will use some abbreviations defined as follows:
(3v)¢ ¢V¢ ¢ --* ¢ ¢ ~--*¢ ¢IAC2A...ACn ¢lV¢2V'"VCn
(VXlX2'"Xn)¢ (::]XIX2"'" Xn)¢
i f f -~(Vv)-~¢ i f f -~((-,¢) A (-~¢)) i f f (-~¢)V ¢ i f f (¢ --* ¢) A (¢ ---*¢) iff (¢IA(C2A"'A¢,)) i f f (¢lV(C2V"'VCn)) i f f (Vx1)(Vx~).'. (VXn)¢ iff (3X1)(3X2)'-'(3Xn)¢
Parentheses may be eliminated unless ambiguity seems likely. Although the abbreviations 3v, ¢ V ¢, and ¢ ~ ¢ actually intend to have the meanings of "for some (or there exist)," "or" and "imply" respectively, as noticed above, these meanings are only realized in a system model.
3.1. FORMULATION OF SYSTEM MODELS
33
The scope of the quantifier Yx i n . . - Vx¢. • • is ¢. A variable x in ¢ E Form(£(Ad)) is bound if either it is immediately after the sign V in ¢ or it is in the scope of Vx in ¢. An occurrence of x that is not bound is free. D e f i n i t i o n 3.1.9 ( S e n t e n c e ) A formula ¢ E Form(£(A4)) is said to be a sentence of £(2¢1) if ¢ has no free variables. We set Sent(L(/vt)) --- {¢1¢ is a sentence of Z(M)}. We illustrate these definitions with examples. Consider a formula: VxR(x, y) A 3yS(x, y). The scope of the quantifier Vx is R(x, y). So the first two occurrences of the variable x are bound while the third occurrence in S(x,y) is free. The first occurrence of the variable y is free, and the second and third occurrences are bound since 3y is the abbreviation of -~(Vy)-~. Formulas VxVy(R(x, y) A S(x, y)) and Vx(R(x) A VyS(x, y)) are sentences. But a formula VxR(x) A VyS(x,y) is not a sentence. A bound variable is bound by the innermost quantifier occurrence within whose scope the variable lies. For example, in the fornmla Vx3yVxR(x, y) the occurrence of x in R(x, y) is bound by the quantifier Vx immediately preceding R aald not by the leftmost quantifier Vx. We define some useful notations below. For any formula ¢ and any unary relation Q, (Vx E Q)¢ stands for (Vx)(Q(x) --* ¢). Similarly (3x E Q)¢ stands for (3x)(Q(x) h ¢). For any formula 0,@(Xl . . . . ,Xn) denotes that the free variables in ¢ are included in the set { x l , . . . ,xn}, and @(tl . . . . ,t,) denotes the formula obtained from O ( x l , . . . ,xn) by replacing free occurrences of X l , . . . ,xn with terms t l , . . . ,tn. In particular, if @ is an atomic formula, ¢(Xl . . . . . xn) also means that the relation symbol in ¢ is n-ary. A formula (I) is said to be an n-ary formula if (I) is free precisely in n variables. For example, a formula O, = (Vx)(R(x, y) -~ S(x, z)), is a 2-ary (or binary) formula written as ¢(y, z). For terms tl and t2, the formula ¢(tl, tz) is
(Vx)(R(x, tl) --* S(x, t2)).
CHAPTER 3. S Y S T E M MODELS AND THEIR STRUCTURES
34
We should take care to substitute an arbitrary term for each free occurrence of a variable in a formula. Vx¢(x) ~
For example, let us consider the following formula:
¢(t), where ~ is a term.
Most formulas of this form are true in all
interpretations (whose precise meaning will be given later). Consider the formula: (Vx3y)(x < y) ~ (3y)(1 < y), where < is a binary relation symbol and i is an individual constant. This is true in all interpretations. In particular, in the domain of natural mlmbers < can be naturally interpreted as "greater than." However if we substitute the variable y for the constant 1, then we get the formula: (Vx3y)(x < y) -* (3y)(y < y) that is not true, for example, in the domain of natural numbers, since ( 3 y ) ( y < y) can be interpreted as saying "There is a number y greater than itself." The following definition is designed to prevent precisely a kind of clash of variables. D e f i n i t i o n 3.1.10 A term t E Term(L) is free for x in ¢ E Form(E.) if either: (1) ¢ is atomic or (2) ¢ = (¢ A ~) and ~ is free for x in ¢ and ~ or (3) ¢ = -~¢ and t is free for x in ¢ or (4) ¢ = Vv¢ and either
v=xor v # y, where y is a free variable of t and t is free for x in ¢. This definition says that a term t is free for x in ¢ if x has no free occurrences in ¢ that lie within the scope of a quantifier Vy where y is a variable occurring ill t. For example, the term f ( x l , x2, c) is free for x2 in the following formulas:
R(xl, x2) A (Vxs)(S(xl, g(x2), xs)), R(X2, C) -"* (VXIX2)(S(xI, h(xl, x2))). In the following formulas the term f(xl, x2, c) is not free for x2:
(Vxl)(R(Xl,X2,Xs)), R(XI, X2) A (VXl)(S(h(Xl, x~), c)). In our logical approach every systems property of an individual system model is expressed by a sentence. For exaanple, an input-output system model .t,4i/o has a basic systems property: "every element of the system is a pair of an input and an output." This property can be expressed by a following sentence: (Vxy)(S(x,y) ~ X(x) A Y(y)).
3.1. FORMULATION OF S Y S T E M MODELS
35
In ordinary mathematical notations, i.e., in set theoretical language, this sentence means that S C X x Y. As another example, an input-output system of functiontype [Mesarovic et al. 1989] can be expressed as: (Vx E X --* (3!y ~ Y ) S ( x , y ) ) , where the notation B!x¢(x) is the abbreviation of the sentence that means "there uniquely exists x such that ¢(x)": (2!x)¢(x) stands for (3x)(¢(x) A (Vy)(¢(y) --* x = y)). We have noticed that a systems property is expressed by a sentence. Conversely a sentence should be interpreted as a systems property in a system model so that the sentence obtains a concrete meaning in the system model. Let us consider a system model M s = < {a, b, c); S, X, Y >, where S --- {(a, b), (a, c), (b, c)} and X = Y = {a, b}. Is a formula S(x, y) --* X(x) A Y ( y ) true in this system model? If we assign a and b to the variables x and y respectively, tile formula is true in that model. However if c to y, then it is not true. To judge the truth of a formula containing some free variables, we need to assign an element of the base set of a system model to each variable. As wiU be stated later, since a sentence has no free variable, we can judge its truth without depending on the assigmnent of variables. For example, a sentence (Vxy)(S(x,y) -* X(x) A Y ( y ) ) is not true in the system model M s , so this model is not an input-output system model. An assignment to each variable is defined by an assignment function. Given a system model .~4 with a base set M, an assignment function p (or briefly assignment) is a function of the set V of variables to M. Notation. We will sometimes use a further notation for an assignment function:
p(y/x) is an abbreviation for the assignment function obtained from p such that p(y/x) (v) = ~ y ( p(v)
if v = x otherwise.
Similarly
p(yl/Xl,...,yn/xn)(V)--
Yi p(v)
if v = x i for each i otherwise.
First we define how to interpret terms of a given language into a system model.
Definition 3.1.11 ( D e n o t a t i o n ) The denotation of a term t in £(A4) with respect
to p, td[p], is defined recursively as follows: (1) i f t is xE V, then td[p] = p(x); (2) if t is f j ( t l , . . . , t~(j)), then td[p] = fj(t~[p],..., td(j)[p]).
CHAPTER 3. S Y S T E M MODELS AND THEIR STRUCTURES
36
Each variable x is replaced by the element p(x) 6 M and each function symbol is replaced by the corresponding function. A denotation with respect to a given assignment can be regarded as a function of terms to the base set of a system model. The concept that a systems property holds in a system model is defined as the satisfaction of formulas. Definition 3.1.12 (Satisfaction) A fol"mula ¢ holds in A4 with an assignment
function p, or p satisfies ¢ in .hA, written A4 ~ ¢[p], is defined recursively: (1) v%4 ~ R i ( t l , . . . ,tA(i)) iff < tall[p],... ,t~(i)[p] >6 Ri; (2) A/[ ~ -~¢[p] iff it is not the case that A4 ~ ¢[p]; (3) A4 ~ ¢1A ¢2[P] iffA4 ~ ¢1[P] and J~4 ~ ¢2[P]; (4) AA ~ Vx¢[p] iff .£4 ~ ¢[p(y/x)] for any y 6 U . This definition provides the intended meanings of the primitive symbols. The logical connectives, -~, A, are given their meanings, "not," "and," which are independent of the context. The universal quantifier, V, is interpreted as "for every element of M ..... " From this definition we can show that the defined connectives as abbreviations, B, V, --% also have intended meanings, "there exists an element of M such that....," "or," "imply." P r o p o s i t i o n 3.1.1 For every ¢ , ¢ 6 Form(f(A4)) and assignment p (1) AA
¢ v ¢[p] i# • b ¢[p] or ~ b ¢[p]; (2) Ad ~ (¢ --* ¢)[p] if] when AA ~ ¢[p] then .h/[ ~ ¢[p]; (3) .hA ~ (¢ ~ ¢)[p] i/~ AA ~ ¢[p] just in case .M ~ ¢[p]; (4) M ~ 3x¢[p] if] Ad ~ ¢[p(y/x)] for some y 6 M. These are easily proved from the definitions of the abbreviations and satisfaction. So the proof is left as an exercise. The above formal definition of satisfaction provides the natural interpretation of formulas in ordinary mathematical representations. Consider an input-output system model .IVlio = < R; Sfg, XR, Y/t > with domain R, the real numbers, XR =
YR = R, and Slg = {(x,y) 6 XR U YR]f (x) <_y <_g ( x ) , f (x) = x 2 and g(x) = x3}. For example, suppose ¢ -- (3y)(S(xl, y) --* S(x2, y)). Let p be an assignment. Then Mio ~ ¢[p]
iff iff
there is a real number r such that if Ad~o ~ S(Xl, y)[p(r/y)], then .h4io ~ S(x2, y)[p(r/y)] if Sfg(xl[p(r/y)],y[p(r/y)]), then S/g(x2[p(r/y)],y[p(r/y)]).
3.1. FORMULATION OF SYSTEM MODELS
37
Now from the definition of denotation (Definition 3.1.11), xl[p(r/y)] = Sl(= p(r/y)(xl)); x2 [p(r/y)] --- s2 (= p(r/y)(x2)); y[p(r/y)] = r. Hence we have: iff
there is a real number r such that if SIg(sl , r), then Sfg(s2, r).
We can see from this example that the truth value of the satisfaction .A4io ~ ¢[p] depends only on the values of sl and s2 corresponding to the free variables Xl and x2 in ¢. This can be shown directly from the Definitions 3.1.11 and 3.1.12. P r o p o s i t i o n 3.1.2 Let X l , . . . , Xk be the free variables of a formula ¢. Then for any
assignment ]unctions p and a, A4 ~ ¢[p] iff,~4 ~ ¢ [ a ( p ( x l ) / X l , . . . , P(Xk)/Xk) ]. Proof: We prove tl~s proposition by induction on the length of ¢, based on the
structural definition of formulas. The length of ¢ is measured by the number of connectives and quantifiers occurring in ¢. The proof consists of four cases. Case 1: ¢ is atomic. Suppose ¢ is of the form Ri(tl . . . . ,t~(i)). Then the free variables X l , . . . , X k of ¢ occur in the terms tl,...,t~(i). From the definition of denotation, for any assignment functions p and a td[p] ----td[a(p(xl)/Xl .... ,p(xk)/Xk) ] for i ----1,...,A(i). Hence by Definition 3.1.12(1) we have .A/[~ Ri($1,... ,CA(1)) iffM ~ Ri(tl .... ,tA(i))[~r(p(Xl)/Xl,...,p(Xk)/Xk) ]. Case 2: ¢ is -~¢. The length of ¢ is less than ~b and the free variables occurring in ¢ are also those of ¢. Hence by the induction hypothesis, for any assignment functions p and a A4 ~ ¢[p] iff A4 ~ ¢ [ a ( p ( x l ) / X l , . . . , P(Xk)/Xk)]. For any formula X either A4 ~ X[P] or A4 ~ "~X[P]. Hence we have A4 ~ ¢[p] iff.hl ~ ¢[~(p(xl)/Xl . . . . , P(Xk)/Xk)]. Case 3: ¢ is ¢1 A¢2. This case can be proven straightforward, using the induction hypothesis for the formulas ¢1, ¢2, and will be omitted.
CHAPTER 3. SYSTEM MODELS AND THEIR STRUCTURES
38
Case 4: ¢ is Vxj¢. Then we should notice that xj is not a free variable of ¢ but may be a free variable of ¢. By definition J~4 ~ ¢[p] iff J~4 ~ ¢[p(y/xj)] for all y E M. From the induction hypothesis we have
M
¢[p(y/xj)]
M
¢[o(y/xj,p(xt)/xt,..., p(Xk)/Xk)].
So ~4 ~ ¢[p] iff for all y E M ~
~ ¢ [ a ( y / x j , p ( x l ) / X l .... ,p(xk)/Xk)].
Hence
J~ ~ ¢[p] iiT J~ ~ ¢[o'(p(x1)/X1,..., p(xk)/Xk)].
[] Having this proposition, we sometimes write ~
~ ¢ ( X l , . . . ,Xn)[al,...~ an] if
p(xi) -- ai for i -- 1 , . . . , n . We have the next corollary for sentences. C o r o l l a r y 3.1.1 For all a E Sent(L(]~t)), we have
either
J~4 ~ a[p] for every assignment p
or
AA ~ -~a[p] for every assignment p.
Proof: Since a sentence a has no free variable, the corollary follows immediately from Proposition 3.1.2.
[]
We can replace each bound occurrence in a formula ¢ of a variable by another variable without changing the meaning of ¢ in any model. L e m m a 3.1.1 Let y be a variable not occurring in a formula ¢ and ¢~ the formula
obrained by replacing each bound occurrence of x in ¢ by y. Then any system model and assignment function p,
Proof: By induction on the length of ¢. Case 1: ¢ is atomic. This is trivial since ¢~ is the same formula as ¢. Case 2: ¢ is ~¢. Then ¢~, is --¢~. So this case is obvious from the induction hypothesis and the definition of the satisfaction. x This case is also straightforward. Case 3: ¢ is ¢1A¢2. Then ¢~ is (¢1)yx A (¢2)y. Case 4: ¢ is Vz¢(z). We have two cases when z is x and when z is not x.
3.1. FORMULATION OF SYSTEM MODELS
39
Subcase 4-1: z is not x. Then ¢~ is Vz¢~(z). By the induction hypothesis, for any a • ]f14],
M ~ ¢(,.)[p(a/,.)] i~ M ~ ¢~(-)[p(a/z)]. Hence
M ~ ¢(z)[p(a/z)] for all a e t.MI iff Ad
I==¢~(z)[p(a/z)] for all a • IMI.
By Definition 3.1.12, we have M
b ¢[P] if[ M b ¢~[P].
Subcase 4-2: z is x. Then ¢ is Vx¢(x) and ¢~ is Vy¢~(y). By the induction hypothesis,
M b ¢(x)[p]
ifr M ~ ¢~(x)[p] iff A4 b ¢~(y)[p(p(x)/y,p(y)/x)].
(3.1)
Since y has no occurrence in ¢, it does not occur also in ¢(x). Then we can easily verify inductively that 3,t ~ ¢(x)[p] iff A4 ~ ¢(y)Lo(p(x)/y,p(y)/x)].
(3.2)
Combining 3.1 and 3.2, we have for all assignment function p,
A4 ~ ¢(y)[p(p(a)/y, p(y)/x)] iff 3,t ~ ¢~(y)[p(p(a)/y, p(y)/x)]. Hence by the definition of satisfaction
b ¢[P] iff M b ¢~[P], which completes the proof.
[]
The next lemma shows a kind of validity of substitution. L e m m a 3.1.2 Let t be a term and free for x in a formula ¢. Then for any assign-
ment function p, M ~ ¢(t)[p] iff M ~ ¢(x)[p(a/x)]
where ta[p] = a. P r o o f : Let X l , . . . , x k be the variables occurring in t. Since t is free for x in ¢, there are no free occurrences of x in ¢ lying within the scope of each quantifier Vxi(i = 1 . . . . , k). First we replace each bound occurrence of x i in ¢(x) by a new
40
C H A P T E R 3. S Y S T E M MODELS AND THEIR STRUCTURES
variable Yi not already occurring in ¢(x) or t. Let ¢*(x) be the formula obtained from ¢(x) as the result of the above replacement. From the hypothesis that t be free for x in ¢, ¢*(t) is the formula when each bound occurrence of xl in ¢(t) is replaced by Yl. From Lemma 3.1.1, we have
~ ¢(x)[p(a/x)] i~ ~ ~ ¢*(x)[p(a/x)] and
~ ¢(t)[p] ifr M ~ ¢* (t)[p]. So it is sufficient to show that M ~ ¢*(t)[p] iff M ~ ¢*(x)[p(a/x)] where td[p] = a. We show this by induction on the length of ¢% Case 1: ¢* is atomic. Then the result is clear from Definition 3.1.12(1). Gaze 2: ¢* is of the form -~¢ or ¢1 A ¢2. We have the result by directly applying the induction hypothesis and the definition of satisfaction. Case 3: ¢* is Vz¢(z). Then z is not one of the variables oft, since ifz is xi, z is bound in ¢*(x) and already replaced by a new variable Yi. The induction hypothesis implies that for all assignment function p, M ~ ¢(z,t)[p] iff M ~ ¢(z,x)[p(a/x)],
where td[p] = a.
Since z is not in t, we have
td[p] = td[p(b/z)] for all b E ]M], where [M[ is the base set of M . Hence for all b E ]M],
AA ~ ¢(z,t)[p(b/z)] iff M ~ ¢(z,x)[p(b/z,a/x)],
where td[p] = a.
So by the definition of satisfaction M ~ vz¢(z,t)Lo] iff M ~ vz¢(z,x)[p(a/x)], where td[p] = a. This completes the proof.
[]
Using the notion of satisfaction, we characterize some formulas of £. Definition 3.1.13 Let M be a system model and ¢ e Form(£(.M)). (1) ¢ is valid in M if M ~ ¢[p] for every assignment p. Then we write M ~ ¢. (2) ¢ is valid if ¢ is valid in every realization of £. Then we write ~ ¢.
3.2. DESCRIPTION OF SYSTEMS BEHAVIOR
41
(3) A model for a set of formulas is a realization of Z: in which each of the formulas is valid. When M is a model for a set of folznulas ~., we write A,t ~ ~. (4) ¢ is satisfiable in J~4 if there is an assignment p that satisfies ¢ in M . (5) ¢ E Form(f.(.M}) is satisfiable if there is a realization of Z: in which ¢ is satisfiable. Note that if a E Sent(~(.M)), then from Corollary 3.1.1 either a or -~a is valid in M . Definition 3.1.14 Let F be a set of formulas o f / : , ¢ a formula of £.
Then I'
logically implies ¢ (or ¢ is a logical consequence of F), written r ~ ¢, if A~ is a realization of £ and p an assignment such that .~4 ~ ¢[p] for every ¢ E P, then
M ~ ¢[p]. Note that a logical consequence of the empty set is a valid formula and in this case we write ~ ¢ as in above definition. We write .M ~ r[p] if A,/ ~ ¢[p] for all
¢ E F C Form(E).
3.2
Description of Systems Behavior
In the previous section we formally defined a system model and provided a logical framework of expressing systems properties of a system model. A systems property of a system model is represented as a first order sentence, for which we can say whether the sentence is satisfied in the system model or not. If it is, the system model is supposed to have the systems property represented by the sentence. A segment of the behavior of a system model - which we call systems behavior -
can be identified by a sequence of systems properties. The whole space of the
behavior consists of all the possible systems properties that would be realized if the system model is analyzed under a possible condition. As will be stated in the next section, a systems property is derived from the structure of a system, which specifies the space of the systems behavior. The reasons why we use logical framework for developing systems theory are mainly based on two assumptions. One is, as described in the previous section, that each systems property can be expressed by a sentence. The other is that a wellorganized deduction system provides the way of description of systems behavior as sequences of systems properties and derivation of systems properties from the structure of a system, the structure of a system which can be also described with the deduction system. These assumptions reflect our recognition that we accept as systems behavior "logically" natural properties.
42
C H A P T E R 3. S Y S T E M MODELS AND THEIR STRUCTURES The basic features of logical deduction system for developing systems theory
are to restrict the space of the behavior description of the system model, and to be consistent.
Furthermore it is desired that the deduction system satisfies "the
completeness" in the sense that the "universal" behavior taken by any system model is all derived from that deduction system. In our logical approach we use the first order logic with equality as such a deduction system. R e m a r k . In the above we consider systems behavior as a systems property or sequences of systems properties that are some derivation from the structure of a system. This seems different from a usual view of behavior of a system that is given as some input-output trajectories. However there is no essential difference in the following senses. (1) We are of more interest in some "properties" of input-output trajectories such as stable or unstable, than trajectories themselves. (2) In specifying an information system, for example, by abstract data type, behavior of a system means "computational" behavior that is calculated from a set of functions given as an initial specification. This accords with our view of systems behavior. (3) Mathematically speaking, behavior as an input-output trajectory can be expressed in a suitable system model as a systems property, i.e., a formula. For exanlple, a trajectory (x(1), y(1)), (x(2), y ( 2 ) ) , . . . , (x(t), y ( t ) ) , . . . , for some pair of x and y is specified by a formula: S ( x , y ) A A ( x o t ) A B ( y <~t) with assignments pt[x/x, y / y , t/t] for * = 1, 2,.... Thus the above formula gives the set of all trajectories. We give below the definitions of some concepts in the first order logic and some basic results that must help further development of systems theory.
Definition 3.2.1 (Axiom Schemata and Rules of Inference) We take as axioms the following schemata and two rules of inference. Let ¢, ¢, X e Form(£(A~t)). A1. ¢ ~ ( ¢ ~ ¢ ) . A2. (¢ ~ (¢ ~ X)) ~ ((¢ ~ ¢) ~ ( ¢ - * X))A3. (-~¢ ~ ¢ ) ~ ((-~¢ --4 -~¢) ~ ¢). A4a. CA ¢ --* ¢. A4b. ¢ A ¢ ~ ¢ .
3.2. DESCRIPTION OF SYSTEMS BEHAVIOR
43
A5. ¢ - ~ ( ¢ ~ ¢ A ¢ ) . A6. Vx¢(x) --* ¢(t) where t is free for x in ¢. AT. Vx(¢ -~ ¢) ~ (¢ ~ Vx¢) where x does not occur free in ¢. AS. Vx(x = x). A9. x = y --* (¢(x,x) ~ ¢(x,y)), where ¢ eAtom(/:). Modus Ponens(MP). From ¢ and ¢ -~ ¢ to infer ¢. Generalization(Gen). From ¢ to infer Vx¢, where x is any variable. (In A6 ¢ may or may not contain free occurrences of x and may contain other free variables.) The axioms from A1 to A5 and Modus Ponens are known as the propositional axioms and rule, while the axioms A6 and A7 and Generalization explicitly involve the universal quantifier. Definition 3.2.2 (Derivation) Let ~ U (¢} C Form(L:(.M)). A derivation of ¢ from ~ is a finite sequence ¢1 .... , Cn of formulas such that Cn is ¢, and for each i one of the following conditions is satisfied: (1) ¢i is an axiom;
(2) ¢~ c ~; (3) ¢i follows by Modus Ponens from preceding members of the sequence; (4) there exists j < i such that ¢i is VxCj, and x is not free in any formula in Z. When there is a derivation of ¢ from ~, we write ~ ~ ¢ and call a theorem of ~. If we can derive a formula ¢ only from the axioms, i.e., ~ is empty, then we write ¢ and call a theorem of the predicate calculus. E x a m p l e 3.2.1 Suppose I- ¢ --~ ¢ and x is not free in ¢. Then ~- ¢ ~ Vx¢. The derivation is as follows: ¢1 : ¢ --* ¢ ¢2 : (Vx)(¢ --~ ¢) ¢3 : (Vx)(¢ --~ ¢) --* (¢ -~ Vx¢) ¢4 : ¢ --~ Vx¢
(given) (from ¢1 by Gen) (instance of A7) (from ¢2 and ¢3 by MP)
R e m a r k . In the definition of derivation (Definition 3.2.2), we restrict the use of Gen: we can apply Gen if the variable to be bound is not free in any formulas in ~. However this restriction can only apply to some minimal, finite ~0 C ~ such that if t- ¢, then there is a derivation of ¢ from ~'0. C Form(£(2vt)) is consistent if there is no formula ¢ e Form(l:(.h4)) such that ~ I- ¢ and E ~- -~¢. E is inconsistent if E is not consistent. E is complete if for any ¢ E Sent(£(./t4)) either ~ I- ¢ or ~ ~- 7¢.
44
C H A P T E R 3. S Y S T E M MODELS AND THEIR S T R U C T U R E S If we axe allowed to use "nullaxy" relations in language, they can be seen as
individual statements that act like sentences in first order language. A well formed formula in a propositional language is built up from such statement letters, called propositional variables, using the propositional connectives &, -~. Tautologies axe the formulas that axe true in any interpretations that assign truth value to each propositional variable in a propositional formula. For example the formulas corresponding to the forms A1 - A5 in the logical axioms (Definition 3.2.1) axe tautologies. It is well known that every tautology is a theorem of the propositional calculus (the completeness theorem). (See reference [Andrews 1986].) D e f i n i t i o n 3.2.3 ¢ E F o r m ( £ ) is a instance of a tautology if there is a tautology ¢ with distinct statement letters P1 . . . . . Pn and ¢1 . . . . . Cn e Form(£) such that ¢ results from ¢ by replacing each occurrence of Pi,i = 1 , . . . , n, by ¢i. P r o p o s i t i o n 3.2.1 /~ ¢ is an instance of a tautology, then t- ¢. Proof: If ¢ is an instance of a tautology, there is a tautology ¢ from which ¢ results by replacing every occurrence of the statement letters in ¢ by a formula ~ . Since ¢ is a theorem of the propositional calculus, there is a proof ¢ 1 , . . . , Cm of ¢ in the propositional calculus. It is easily seen by induction on i that when the formula ¢i is obtained from replacing by a formula ~ each occurrence of the statement letters in the corresponding formula ¢i, then ¢i is a theorem of the predicate calculus for l
Hencet-¢.
[]
T h e o r e m 3.2.1 ( D e d u c t i o n T h e o r e m ) / f ~ U {¢, ¢} C Form(£) and ~ U {¢} I¢, then ~ ~- ¢ ~ ¢. Proof." Let ¢ 1 , . . . , ¢ n be a derivation o r e from ~13 (¢}. We prove that ~ ~¢ ~ ¢i for 1 < i < n by induction on i. Case 1: ¢i is an axiom or in Z. ¢i ~ (¢ ~ ¢i) is an instance of A1. Since ~, ~- ¢i, by MP we have Z t- ¢ --, ¢i. Case 2: ¢~ is ¢. Then ¢ --* ¢i is an instance of a tautology, and hence ]E I- ¢ --. ¢i. Case 3: ¢i is inferred from Cj and Ck = Cj ~ ¢i by MP, where j, k < i. From the induction hypothesis we have E ~- ¢ ~ Cj and ~ ~- ¢ --4 (¢j ~ ¢i). The fornmla (¢ --* (¢j -~ ¢~)) ~ ((¢ --* Cj) --* (¢ ~ ¢~)) is an instance of A2. Two applications of MP yield ]E ~- ¢ ~ ¢i.
3.2. DESCRIPTION OF SYSTEMS BEHAVIOR
45
Case 4: ¢i is inferred from Cj by Gen, where j < i and x is not free in ~3U {¢}. Thus ¢i is VxCj, aald x does not occur free in ~ U {¢}. From the induction hypothesis we have ~ 1- Cj. By Gen (x not free in I3), F- Vx(¢ ~ Cj}. From A7 we have ~ F- ¢ ~ VxCj, which is 12, b ¢ -o ¢i. This completes the proof.
[]
The next theorem is a basic result known as Soundness Theorem. T h e o r e m 3.2.2 ( S o u n d n e s s T h e o r e m ) (1) Every theorem of the predicate cal-
culus is valid. (2) If ~ C Form(£(M)) and M is a model of ~, then every theorem of l] is valid in .44. (3) If {¢} O ~ C Form(f) and ~ F- ¢, then Z ~ ¢. Proof: (1) I f ¢ E Form(f) is an instance of any of Axiom Schemata A1
Ah,
then obviously ~ ¢ since it is also an instance of tautology. Let A4 be a realization of £ and p an arbitrary assignment function. Suppose ¢ is of the form Vx¢(x) --* ¢(t) where t is free for x in ¢. By Proposition
3.1.1(2), ¢k4 ~ ¢[p] iff M ~ Vx¢(x)[p] then M ~ ¢(t)[p]. So if M ~ Vx¢(x)[p], then for all a e IMI M ~ ¢(x)[p(a/x)]. In particular, if a = td[p], M ~ ¢(x)[p(a/x)]. By Lemma 3.1.2 we have f14 ~ ¢(t)[p]. This implies Suppose ¢ is of the form Vx(¢ ~ X) ~ (¢ ~ Vxx) where x does not occur free I f M ~ Vx(¢ ~ X)[P],then for all a E [MI M ~ ¢ ~ X[p(a/x)] • So if M ~ ¢[p(a/x)] then M ~ X[p(a/x)] • By Proposition 3.1.2, since x does not occur
in¢.
free i n ¢ , i f M ~ ¢[p] t h e n M ~ ¢[p(a/x)] for a l l a e IMI. H e n c e M ~ ¢[P] implies vk4 ~ X[p(a/x)] for all a e IM]. Thus we have M ~ ¢ ~ Vxx[P]. This implies ~ ¢. Axioms A8 and A9 can be easily shown. The proof is left as an exercise. Next we show that if ¢ and ¢ --* ¢ are valid, then ¢ is valid. By the definition of satisfaction, vk4 ~ ¢ ~ ¢/[p] iff when ,M ~ ¢[p] then ,M ~ ¢[p]. So if fi,t ~ ¢[p] and ¢~4 ~ ¢ --* ¢[p] then M ~ ¢[p]. Hence ~ ¢ and ~ ¢ ~ ¢ implies ~ 9. Finally we consider Generalization. Suppose A4 ~ ¢[p] for any assignment p. Since p is arbitrary, M ~ ¢[p(a/x)] for all a • lad I. Hence we have ,~4 ~ Vx¢[p]. This shows that ~ ¢ implies ~ Vx¢. (2) From (1) and the defiltition of model of ~, every theorem of the predicate calculus and element of ~ are valid in M . We have to show that every formula
46
CHAPTER 3. S Y S T E M MODELS AND THEIR STRUCTURES
obtained by using MP and Gen is valid in A~i. However this is obvious from a slight modification of the proof of (1). (3) If ~ I- ¢, then there is some finite subset { ¢ 1 , . . . , ¢ n }
of ~ such that
{¢1 . . . . . ¢~} F- ¢. If the subset is empty, the result is trivial by (1). Hence by the Deduction Theorem we have ~- ¢1 -* (¢2 -* "'" --* (¢~ ~ ¢ ) ' " ) .
So by (1)
¢1 ~ (¢2 -* "'" -* (¢~ ~ ¢ ) ' " ")- Since for any model ,~4 of ~ and assignment p, AA ~ ¢~[p], i = 1 , . . . , n, we have ~4 ~ ¢[p]. This concludes E ~ ¢.
[]
Fact 3.2.1 The empty set is consistent.
P r o o f : Suppose there is some ¢ E Form(L) such that }- ¢ and ~- -~¢. Then by Soundness Theorem we have ~ ¢ and ~ -~¢. This is impossible since it contradicts the definition of satisfaction (Definition 3.1.12).
[]
D e f i n i t i o n 3.2.4 ( T h e o r y ) Let T be a set of sentences of L. T is said to be a
theory in L if T is deductively closed, i.e. for each a E Sent(L) T t- a if and only if n E T . P r o p o s i t i o n 3.2.2 ~ C Form(L) is inconsistent if and only 'if for each ¢ E Form(L) ~ t- ¢. P r o o f ; The if part is clear from the definition of inconsistent. Conversely suppose that E is inconsistent. Then for some ¢ E Form(L) ~ ~- ¢ and ~ ~- 9¢. For any
¢ E Form(L) the formula ¢ ~
(-~¢ ~
¢) is an instance of a tantology.
By
proposition 3.2.1, Z }- ¢ --~ (-~¢ --~ ¢). We use MP twice to have Z I- ¢ as required.
[] C o r o l l a r y 3.2.1 A theory T in L is consistent if and only i f T ~ Sent(L). P r o o f : This is easy from the proposition.
[]
The concept of derivability is closely related to the semantic logical consequence. This relationship is explicated by the following completeness theorem. This theorem is both conceptually and technically significant in the development of systems theory. Conceptually, it validates our view that a derivation represents a sequence of behavior of a system model. Technically, it helps that many important properties of system models are explored: the canonical system model in systems theory is a typical exaanple.
3.2. DESCRIPTION OF S Y S T E M S BEHAVIOR
47
T h e o r e m 3.2.3 ( C o m p l e t e n e s s T h e o r e m ) A set ~ C Sen$(L) for countable £:
is consistent if and only if ~ has a model. S k e t c h o f Proof: The if part can be easily proven from Soundness Theorem (3). Indeed if ~ I- a and ~ t- -~a for some a, then no model can satisfy both a and "~O".
A proof of the other implication requires some tough steps including some lemmas, the details of which are beyond the scope of this book. We briefly describe the steps involved in a typical proof of the theorem. (1) A consistent ~ C Sent(£:) is extended to a consistent set E1 C Sent(£:l) as follows. Define ~'
=
{¢ E Form(£:)l¢ has precisely one free variable x and ~ t- 3x¢(x)};
£:1 =
£: U {Col ¢ e ~t}, where C a ¢ £: and C¢ = C o just in case ¢ = ¢;
~1
~, u { ¢ ( C ¢ ) l ¢
=
e
='}.
Then ~l is consistent. (2) We extend ~1 to a complete consistent F1 C Sent(£1). (The countability of £: is used in this extension.) (3) We extend ~ C Sent(£:) to a complete consistent E* C Sent(£*) as follows. Using (1) and (2) alternately and countably often to construct languages Z:0, £:1, £:2," "" and sets of sentences ~.o,~:,[~:,]E2,I'2, -.- such that £:o = £ , ~ o = ~ and ~i C Pi C ~i+] for each i > 1. Then let £:* = U ~ = ] £ .
and ~* = U~=lrn.
Then ~* satisfies the property that if ~* I- 3x¢(x) where ¢ E Form(F.*) and ¢ has precisely one free variable x, then there is a constant C E £:* such that ¢(C) E ~*. (4) A complete consistent ~.* C Sent(Z*) constructed by (3) has a model. (5) The reduct of this model to the language £: is a model for Z.
[]
The above proof is essentially due to Henkin [Henkin 1949]. The model constructed in the step (4) is a canonical system model, which will occupy an important place in LAST. (See Chapter 5.) The condition of countability of language in the theorem is essential. If the language is uncountable, we need the axiom of choice for keeping countability of formulas. Then we have the generalized completeness theorem.
C H A P T E R 3. S Y S T E M MODELS AND THEIR STRUCTURES
48
T h e o r e m 3.2.4 Under the axiom of choice, if ~ C Sen~(£), Z is consistent and
L has cardinality ~, then ~ has a model of cardinality <_ ~;, where ~ is an infinite cardinal number. S k e t c h of P r o o f : The proof is done in almost the same way as in the above countable case except that the assumption of the axiom of choice is explicitly used only in the step (2). The finally constructed ~* and Z:* have each cardinality < ~. and the canonical system model established in the step (4) has the desired cardinal bound.
[]
Several important corollaries of Theorem 3.2.3 are proved. C o r o l l a r y 3.2.2 /] ~ U {¢} C Sen$U:), then ~ ~ ¢ if] ~ ~- ¢. Proof: The if part was proved in Soundness Theorem (3). Conversely suppose it is not the case that Z ~- ¢. Then it is also not the case that E ~- -~-~¢. If ~ U {--¢} is consistent, by Theorem 3.2.3 it has a model. This implies that not ~ ~ ¢. So it is sufficient to show that ~ U {-~¢} is consistent. Since -~-~¢ not derivable from ~, by Proposition 3.2.2 ~ is consistent. Suppose ~ U {-~¢} is inconsistent. By Proposition 3.2.2 ~ t2 {-,¢} ~- ¢. Using Deduction Theorem and the instance of tautology (-~¢ --* ¢) --* ¢ we have Z ~- ¢, which contradicts the hypothesis.
[]
C o r o l l a r y 3.2.3 The theorems of the predicate calculus are precisely the valid for-
mulas. Proof: Let ¢ ( X l , . . . , X n ) E Form(C) be any theorem of the predicate calculus and a the universal closure of ¢ ( X l , . . . , Xn), i.e., a = VXl ... X n ¢ ( X l , . . . , xn). By letting ~ be empty, Corollary 3.2.2 implies that ~ a iff F- a. Since ¢ is valid if al~d only if a is valid ( prove as an exercise) and ¢ is derivable if and only if a is derivable, we have ~ ¢ iff b ¢.
3.3
[]
Structure of a S y s t e m
In the previous sections we formulated system models and the language to describe them. In this section we will formally define structure of a system. The concept of structure of a system is extremely important as one of systems concepts, since system models are characterized by their structures. It, however, has been defined
3.3. S T R U C T U R E OF A S Y S T E M
49
so far depending on the representation of a system or discussed in an informal way. A formal approach in a general framework can make transparent such ambiguity of the concept. Although we cannot well understand system models without considering their structures, it is necessary to discriminate between the concepts of system models and their structures. For example, we can consider the system model of the real number or integer to possess the structure of the natural number system as well as the system model of the natural number. This implies that the real number or integer has also the characteristic of the natural number. Since the structure of a system is obtained through one's systems recognition that reflects his/her aspect to an object what and how elements in the object interact, the structure of a system represents this aspect; it can be defined as follows: D e f i n i t i o n 3.3.1 ( S t r u c t u r e of a S y s t e m ) Let A/[ be a system model, L:(2~4) the language for A4 and ~ a set of sentences of/:(A4) where A/[ ~ ~. Then the structure of a system as a prototype of the system model A/t is defined by (£(A~);~). A given system model necessarily determines £:(¢VI), unique up to alphabetic invariants. We should notice that/:(j~4) is a collection of "symbols," therefore the role of £(A4) in the systems recognition is to point out the nanles and types of the relations that are identified in the system we recognize. On the other hand, 2 provides the rules how elements in a system model interact. Therefore the properties of a system implied by the structure of the system are expressed as the formulas derived from ~; T ( ~ ) = {¢ E Sen~(£(.M))]~ ~- ¢} is the whole of the properties characterized by the structure of the system, (£(A/i);~). If ~ is complete, the properties satisfied by a system model having the structure (£(A4);~) accord with the properties of a system implied by (£(A4);2); that is, let Th(./~4) = {¢ E Sent(Z(.M))]A4 ~ ¢,J~4 ~ ~}, then Th(,~4) = T ( ~ ) . Notice that it follows from the definition that (£(2vi);~) cannot be uniquely determined for one system model J~4 since we can take another ~ as axioms for which A/I is a model. This means that there may be plenty of system models satisfying a given structure (£;~). For example, many models satisfy the Peano's Axioms well known as a structure of the natural number. They are not necessarily isomorphic to the natural numbers [Bridge 1977]. As examples we define below some fundamental structures of system models that play important roles in systems theory.
C H A P T E R 3. S Y S T E M MODELS AND THEIR STRUCTURES
5O
3.3.2 (The Structure
Definition
of Input-Output
structure of input-output system model is defined by
System Model)
The
(f~I/O; •I/O):
~/o = {x,Y,S}, where
X, Y S
: :
unary relation symbols, a binary relation symbol;
1~I/o = { ¢ I / o } ,
where ¢I/O - (Vxy)(S(x, y) --* X(x) A Y(y)). An input-output system model A4Uo - - < X U Y; S, X, Y > (Definition 3.1.1) is a realization of £ U o and model for EUo. Systems theory is developed in the way of "climbing up" the hierarchy of structures of systems from the structure of input-output system models that lies in the b o t t o m of the hierarchy. A well-explored structure in the second layer is that of time system models. In introducing this structure we usually assume the stationary time index on which input and output sets are functions. Definition
3.3.3 (The Structure
of Stationary
Time Index)
The structure
of the stationary time index is defined by (~T; ]~T):
~ r = {T, <,+,o, 1}, where T < + 0,1
: : : :
a unary relation symbol, a binary relation symbol, a binary function symbol, constant symbols;
2T = {¢1 ~ ¢12}, where
¢1 ¢2 ¢3 ¢4 ¢5 ¢6
-----
¢7 = ¢s ¢9 ¢10 ¢11 ¢12
------
(Vt)(T(t) - - t < t) ( V t l t 2 t a ) ( T ( t l ) A T ( t 2 ) A T ( t 3 ) --* (tl _< t2 A t2 _< t3 --* t l _< t3)) ( V t l t 2 ) ( T ( t l ) A T ( t 2 ) --* (tl _< t2 A t 2 ~ t l --" t l : t2)) ( V t l t 2 ) ( T ( t l ) A T ( t 2 ) --* ( t l _< t2 V t 2 _< t l ) ) (Vt)(T(t) ~-* 0_< t) T(0) ^ T(1) ( V t l t 2 t 3 ) ( T ( t l ) A T ( t 2 ) A T ( t 3 ) --* ( t l + t2) -F t3 ---- t l + (t2 + t3)) ( V t ) ( T ( t ) --* t + 0 = t) ( V t l t 2 ) ( T ( t l ) A T ( t 2 ) --* t l + t2 -- t2 + t l ) ( V t s s t ) ( T ( t ) A T ( s ) A T ( s t) --* (t + s = t + s t --* s --- st)) ( V t l t 2 ) ( 3 T ) ( T ( t l ) A T ( t 2 ) --* T(T) A ( t l _< t2 ~ t2 = t l + T ) ) ( V t l t 2 ) ( T ( t l ) A T ( t 2 ) --* T ( t l + t2))
~T implies that T is linearly ordered by <_ and an abelian group with respect to +.
3.3. S T R U C T U R E OF A S Y S T E M
51
The most common indices in systems theory are the set of non-negative real numbers for continuous time system models and the set of non-negative integers for discrete time system models. Remark.
The structure of stationary time index is not an additive group,
because typical exalnples of this structure, as stated above, are the set of nonnegative real numbers and the set of non-negative integers, which are actually not additive group. (See [Mesarovic et a1.1989].) However sometimes in systems theory the set of real numbers is considered as a model of the structure of stationary time index. Then we have some alternatives of the structure of stationary time index: 1) we omit ¢5 from ZT, or 2) we omit ¢5 from ZT mid add the existence of inverse. In case of 1), the set of non-negative real numbers and the set of non-negative integers are also models as well as the set of real numbers; but in case of 2), they are not models at all. Which structure should be selected as the structure of stationary time index depends mainly on how an observer or modeler recognizes a time index as a system and how a subsequent theory would be developed. The structure is not determined a prio~'i at all. Here we follow the definition of stationary time index in AST [Mesarovic et al.1989]. D e f i n i t i o n 3.3.4 ( T h e S t r u c t u r e o f Time System M o d e l )
The structure of
time system model is defined by (£T-,ys; ET-sys): L T - ~ s = £ q O U LT U { A , B , o } , where A, B o
: :
unary relation symbols, a binary function symbol;
~T-sy~ = ~I/O U ~T U {¢t,1 ~ ¢t~4}, where ¢t~l = Cts2 -
Cts3 ~)ts4
(Vxt)(X(x) A T(t) --~ A ( x o t ) ) (Vxx')(3t)((X(x) A X ( x ' ) A (W(t) -~ x o t = x' o r ) ) ~ x = x') (Notice that Cts2 is equivalent to (Vxx') (X(x) A X(x') A (Vt) (W(t) ---* x o t = x-J o t) --~ x = x').) ~-- (Vyt)(V(y) A T ( t ) ---* B ( y o t)) ~--- (Vyy')(3t) ((Y(y) A Y ( y ' ) A (T(t) --* y o t = y'<> t)) ~ y = y') (Notice that Cts4 is equivalent to ( V y y ' ) ( Y ( y ) A Y ( y ' ) A (Vt)(T(t) ~ y o t = y ' o t) --* y = y').)
(¢tsi ~ ¢t~4 represent that both input and output are functions of time scale into alphabets A, B respectively.) A time system model J ~ T - s y s
-~-< X U Y U A U B U T ; S , X , Y , A , B , o , T , < _
, +, 0, 1 > (Definition 3.1.2) is a realization of £:T-~y~ and model for ~T-~y~.
CHAPTER 3. S Y S T E M MODELS AND THEIR STRUCTURES
52
D e f i n i t i o n 3.3.5 ( T h e S t r u c t u r e o f S t a t i o n a r y S y s t e m M o d e l )
The struc-
ture of stationary system model is defined by (£sta; Esta):
[:sta = ~T-sys U {~}, where : a binary function symbol;
where
~)stl Cst2 Cst3 ¢~t4 Cst5
(VxtT)(X(x) A T(t) A TiT ) --~ A(t,x) or = x o it + T)) --= (Vytr)(Y(y) A T(t) A T(~) --* ACt,y) oT = y o (t + T)) ---- (Vxyt)(S(x,y) A T(t) -~ S(A(t,x),A(t,y))) -- (Vxt)(X(x) A T(t) --* X(A(t,x))) ---- (Vyt)(Y(y) A T(t) --* Y(A(t,y)))
(¢8~1 and Cst2 represent that A is a shift operator. ¢,t3 represents AtS C S, that is, any element of S shifted by A belongs to S again. ¢~t4 and ¢,t5 represent that X and Y are closed under A respectively.) A stationary system model A48ta = < X U Y U A U B U T; S, X, Y, A, B, o, T, _< , +, 0,1, ), > (Definition 3.1.3) is a realization of l:sta and model for Est,. The following is an example from abstract data type. Definition 3.3.6 ( T h e S t r u c t u r e of S t a c k S y s t e m Model)
The structure of
stack system model is defined by (£stack; Estate): £,ta~k = {A, S T A C K , P U S H , P O P , T O P , E R R O R , E M P T Y } , where A, S T A C K PUSH POP, TOP ERROR, EMPTY
: : : :
unary relation symbols, a binary function symbol, unary function symbols, constant symbols;
~'~'stack : {¢1 ~ ¢9} where
¢1
A(ERROR) = STACK(EMPTY) ¢2 =(Vxs)(A(x) A S T A C K ( s ) -~ S T A C K ( P U S H ( x , s ) ) ¢3 (Vx)(STACK(x) --* S T A C K ( P O P ( x ) ) ) ¢4 -- (Vx)(STACK(x) --* A ( T O P ( x ) ) ) ¢s - (Vxs)(A(x) A S T A C K ( s ) --* P O P ( P U S H ( x , s)) = s) ¢7 - (Vxs)(A(x) A S T A C K ( s ) --* T O P ( P U S H ( x , s ) ) = x) Cs -= (Vxs)(A(x) A S T A C K ( s ) -~ P O P ( E M P T Y ) = E M P T Y ) ¢9 -= (Vxs)(A(x) A S T A C K ( s ) --* T O P ( E M P T Y ) = E R R O R ) - -
The way of "climbing up" the hierarchy of structures is defined as structure expansion.
3.4. ADVANTAGES AND LIMITATIONS OF FIRST ORDER LOGIC
53
D e f i n i t i o n 3.3.7 ( S t r u c t u r e E x p a n s i o n ) Let A/J1 and Ad2 be system models with structures (£1;~1) and (£2;~,2) respectively. (£2;~2) is said to be a struc-
ture expansion of (£:1;~1), written (£1;~1)C(£2;~2), if £1 C £:2 a~ld ~1 C ~2. A42 is said to be a structure expansive model of,A41 if the base set of A,/1 is a subset of that of A42 and (£1;~q)C (£:2;~2).
For example,
(£I/0; ~'I/O) C (£T; Y]'T) C (£T-sys; ~;]T-sys) C (£sta; Yl]sta). If (£1; E1)C(£2; Es), the structure (£2; E2) inherits the structure
(£1; El). For
example the structure of time system model possesses the structure of input-output system model too. The concept of structure expansion as inheritance plays a crucial role in information systems, especially in object-orientation. It is indispensable for "modularity" that helps reuse software as a component or init.
3.4
A d v a n t a g e s and Limitations o f First Order Logic
We have other logics as alternatives than the first order logic for the description language. Conceptually, the problem which logic should be chosen is equivalent to the problem how we assume the description of properties and behavior of system models. As noticed before, this book assumes that the properties of system models are expressed as "first order" folznulas and the behavior of them as sequences of first order fornmlas. This assumption has both some advantages and limitations in developing systems theory. The primary advantages are: (1) first order logic itself is a well-developed logical system, hence we can enjoy its fruits; (2) expressions we make as systems properties are simple and almost enough to develop the theory of system models; (3) there must be more readers than for other logics. The limitations stem from our assumptions on the representations of systems properties a11d behavior. First there are some systems properties that cannot be expressed as "first order" formula in a given language. It might be true that if we select an appropriate language, "any" systems properties could be expressed as first order fornmlas. However since the language is selected so that it reflects one's systems recognition, the language selected under some particular systems recognition is not necessarily enough to represent "any" systems properties.
CHAPTER 3. SYSTEM MODELS AND THEIR STRUCTURES
54
For example~ if the language does not include time index as a proper relation, then we cannot use "time variables" in expressing properties of a system model that is a realization of that language. This restriction is sometimes crucial for representing dynamical systems; e.g., controllability cannot be represented in a proper sense in control theory. One possible way to get over this limit is to use "higher order" logic or nonclassical logic such as infinitary logic. Though developing a theory using non-classical logic is an interesting problem, our approach in terms of first order logic would occupy a primary position in LAST. Second there axe some situations where other deduction systems as reasoning process than first order logic are appropriate. If we recognize and represent other behavior than the behavior recognized with the axioms, the generalization or Modus Ponens of first order logic, it is natural to use another logic. For example, if we need some modality such as necessity or possibility to represent behavior of system models, modal logic might be appropriate. In particular, if we deal with change of structures of system models, modal logic could be effective especially to represent
self-referential behavior [Smoryflski 1985]. We should notice the difference between the first and the second. In the latter, a formula like a first order fornmla is not denied in the sense that only individual variables occur as variables in the formula.
Chapter 4
SIMILARITY MODELS
OF SYSTEM
One of the most important purposes of systems science is to investigate similarity between system models. Similarity can be divided into two types: structural similarity and behavioral similarity. So far, fixing the type of models, we have studied structural similarity in systems theory using modeling morphisms defined especially between input-output system models. However, as will be made clear in the subsequent discussions, these modeling nmrphisms are defined not in a general way in which we can deal with structural similarity between any system models, but in a specific way based only on homomorphisms. This chapter is devoted to the development of a general theory of structural similarity between system models. The structural similarity is based on the idea: there is a certain correspondence between two objects, such that the properties of one object are preserved in the other by the correspondence. In systems theory or modeling theory the structural similarity that realizes the above idea has been investigated so far solely based on the concept o f a homomorphism [Zeigler 1976,1984][Mesarovic et al.1984]. The approach using the concept of a homomorphism to the structural similarity has the following difficulties: (1) The properties that are preserved by a homomorphism are not explicitly specified. (2) A homomorphism can be defined only between system models of "the same type." The similarity between system models of "different types" is of particular importance. For exanlple, the similarity between an input-output system model and a goal seeking system model, or between an input-output system model and a state space representation, or between a finite automaton and a Petri net, can be considered as "equivalent" representation in the sense that we can transform one system
56
C H A P T E R 4. S I M I L A R I T Y OF S Y S T E M M O D E L S
model into another system model, and vice versa. In that case the main problem is how we realize one system model as an "equivalent" system model to the other one. There are two basic cases for the relationships between system models of different types. (1) Case 1: One system model is generated by forgetting a part of the structure of another one. The relationships of this case are very important in systems theory, and can be often found particularly in modeling [Zeigler 1976,1984]. For example, let S be a general input-output model $1 C X1 x Y1- X1 and Y1 have no structure yet. Each element of them is simply a 'point'. When studying a dynamical system, it is necessary to introduce the time concept into X1 and Y1, that is, we use X~ = { x l x : T ---* A} and Y~ .= {YlY : T ~ B } as input and output sets, respectively, where T is a time index and A and B are input and output alphabets, respectively. Let S~ be a dynamical model S~ fi X~ x Y~, where there is a one-toone correspondence between the elements of S~ and those of $1. Then S1 can be obtained by forgetting T, A, and B of S~. Another example is a relationship between an input-output model representation S C X x Y by the external behavior, and a representation (S, X. Y, C, p) by the internal structure, where C is a state set and p is a function of C x X to Y. It is known [Mesarovic et al.,1975] that the two representations above are equivalent in the sense that given an external representation S fi X x Y, we can always construct an internal structure consistent with S, i.e., such that ( x , y ) E S if and only if
(3c ~ C)(p(c,x) = y). The examples in Case 1 are important instances of associative F-morphisms defined in Chapter 6. (2) Case 2: There is no apparent relationship between the types of two system models. One of the importaalt examples is a relationship between an input-output model S C X x Y and a goal-seeking model D = ( M , X , Y , V , P , G , E ) ,
where X: the set
of inputs, Y: the set of outputs, M: the set of alternative decisions, P: the relation P C M x X x Y, referred to as the process or the internal behavior, G: a function G : M x X × Y --* V that evaluates a behavior of the system, and E: a selection relation E C M x X x Y x V. The two models are obtained based on different viewpoints. The input-output model S represents only the external behavior of a system, but explains no internal structure of it.
On the other hand, the goal-seeking model D does not include
explicitly a structure about an external behavior of a system, but represents an
57 internal structure of it from a goal-seeking point of view. If the two models represented completely different objects, there would be no "isomorphic" relationship between them. However, it is known[Mesarovic et al.,1988] that the two models can be, in a sense, equivalent as general system models. That is, any goal-seeking model can be considered as an input-output model by the following relation: (Vx)(Vy)[(x,y) E S ~ ( 3 m ) [ ( m , x , y , G ( m , x , y ) ) • E & ( m , x , y ) • P]] Conversely, we can derive, at least formally, a goal-seeking model for any inputoutput model to explain its input-output behaviors although the obtained goalseeking model may not always be natural and persuasive [Mesarovic et al.,1988]. Another example is a relationship between a finite automaton and a Petri net. Both are system models to describe a kind of information systems, a~d have quite common features in systems as objects of modeling. It is known [Peterson, 1981] that any finite automaton can be realized by a Petri net. The realized Petri net can be considered to have the ability to describe the properties "equivalent" to those of the original automaton. The concept of "equivalence" as mentioned above is considered only in an intuitive way concerning what is actually equivalent. Furthermore we cannot directly apply the usual concept of isomorphy to the investigations of "equivalence" in the above cases, since the types of two models in question are different. In this chapter we will develop a framework by which we can investigate structural similarity between system models not only of the san~e type, but also of different types. As an example, we will investigate the equivalence between a finite automaton and a Petri net in the framework developed in this chapter. The fraanework includes the usual isomorphy defined using the concept of a homomorphism, as a special case for system models of the same type. In that sense the fraanework can be considered to give a general concept of isomorphy also for system models of different types. As mentioned previously, the concept of structural similarity is based on the idea that for two similar system models there is a morphism between them, and by the morphism some properties of one system model are preserved in the other. Central problems to be considered are therefore how such a morphism between system models regarded as similar can be defined and what properties are preserved by it. In this chapter we will introduce a new morphism, called F-morphism, as a morphism giving structural similarity. An F-morphism is a kind of extension of a homomorphism, in the sense that if system models are of the same type, an Fmorphism between them becomes a homomorphism,
C H A P T E R 4. S I M I L A R I T Y OF S Y S T E M MODELS
58
There are three kinds of properties important as properties to be preserved : a) generator (i.e. the set of atomic formulas); b) a set of axioms ~, which is a part of the structure of a system model; c) all sentences satisfied in a system model. In the sense that both a homomorphism and an F-morphism are defined based on morphisms preserving the generator of the language for a system model, the generator as properties preserved are the most fundamental in considering structural similarity. However, if the properties preserved in one system model are only the generator, the structures of two system models are not "well" similar. The degree of similarity in case b) - - which represents how similar two system models are - - is higher than that in case a), and the degree in case c) is the highest. In that sense case c) is essential for structural similarity. In the subsequent discussion we will use the following notation for c): Th(A/i) = {¢ e Sent(£(AA))IAA ~ ¢}. In Section 4.1 and 4.2 we will study nmrphisms between system models of the same type (Sec.4.1) and of different types (Sec.4.2) with respect to the preservation of properties by the morphisms based on case a),b) and c). In Section 4.3 we will discuss the equivalence between a finite automaton and a Petri net as an example of structural similarity by F-morphisms. 4.1
Morphisms
for Models
of the
Same
Type
In this section we investigate morphisms between system models of the same type, following the three cases mentioned in the previous section. 4.1.1
Preservation
of Generator
--
Homomorphism
A morphism between system models of the same type preserving the generators is usually given by a homomorphism. D e f i n i t i o n 4.1.1 ( t t o m o m o r p h i s m ) Let A~tl = < M1; {R~Ii e I}, {f~lj e J} >, A42 = < M2; {R21i e I}, {f]lJ e S} > . Notice that A/J1 and A/12 are of the same type.) A function h : M1 --* M2 is called a homomorphism of .h41 to .A42 if for any
e I , j e J, a l , . . . , a ~ ( o , a l . . . . ,ap(j),a e M1, ( a l , . . . ,a:~(1)) e R 1 implies ( h ( a l ) , . . . , h(a~(~))) e R 2, h(fJ (al . . . . , a•(j))) = f2(h(al), . . . , h(a,(j)) ).
4.1. M O R P H I S M S F O R MODELS OF THE S A M E T Y P E
59
A bijective (i.e., one-to-one and onto) homomorphism is called an isomorphisn~ From Definition 4,1.1, we can see that a homomorphism preserves only the atomic formulas, which is viewed as the generators of the language for a system model. In systems theory the concept of a homomorphism is defined as a modeling morphism between input-output system models. D e f i n i t i o n 4.1.2 ( M o d e l i n g M o r p h i s m [ M e s a r o v i c et a1.1975,1989]) Let S C X × Y and S ~ C X p x Y ~ be input-output system models. Let hx : X ~ X ~ and h v : Y -~ Y~ be functions, h = (hx, hy) : S ~ S t is called a modeling morphism
o r s to S ~ if for any (x,y) E X x Y , (x,y) E S implies (hx(x),hy(y)) E S'. For example, let us consider.A4 = <
XUY;S,X,Y
> and Ad ~ = <
X ~u
Y~;S~,XP,Y ~ >, where S, Sqbinary relations on X U Y and X , X ~ , Y , Yr: unary relations on X U Y. Suppose that .hal and AJ ~ satisfy S(x, y) ~ X(x) A Y ( y )
S'(x, y) ~ X'(x) A Y'(y) respectively.
Then a homomorphism h of .hal to .A4~ is regarded as a modeling
morphism of S to S J. Notice that from the definition of a homomorphism, h(x) E X ~ for any x G X and h(y) E Y~ for any y E Y. 4.1.2
Preservation
of E --
E-homomorphism
Next we consider homomorphisms preserving axioms H. Recall that the axioms :3 provide the structure of a system. (See Sec.3) Grgtzer defined such homomorphisms as H-homomorphisms [Gr£tzer 1979].
By a H-homomorphism the axioms ~ ave
preserved in a homomorphie image. We formulate a E-homomorphism directly based on this idea. This definition is different from Griitzer's original definition that uses the concept of • - l inverse. Let h be a homomorphism of J~l to J~2. The homomorphic image of h in .A42 is a submodel of.A//2 whose domain is h(M1). We write h(.A//1) as follows to indicate the homomorphic image of h in Ad2. h(Adl) = < h(M1); {Ri2 n h(M:) ~(0 I i e I}, { f ] II h(M:) tL(j) I J e J} > where f ] ]l h(M1) ~'(j) denotes the restriction of f~ to h(M1) •(j). By the property of a homomorphism, f]II h(M:)~(J) is well-defined.
60
CHAPTER 4. SIMILARITY OF SYSTEM MODELS For example, let us consider Adi = < Q; _<> mid A42 = < R; < > , where Q is the
set of rational munbers, R the set of real numbers and _< the usual linear ordering on Q (or R). We define a homomorphisnl h of Q to R by
h(r)=~ for r E Q, where f ineaas the nlaxiinum integer not exceeding r. Then the honlomorphic image of h is h(A4i) = < Z; (_<) A (Z x Z) >, where Z = h(Mi): the set of integers, and (<_) A (Z x Z) ----{(0,0), (0,1),..., (1,1), (1,2),...}. We define a ~-homomorphism as a homomorphism whose homonlorphic image preserves ~. Definition 4.1.3 ( ~ - H o m o m o r p h i s m ) Let Az/i and .A~2 be system models of the
same type, and ~4i ~ Z aa~d ,~42 ~ ~3. A homomorphism h of A4i to A/[2 is called
a Z-homomorphism of M1 to A~t2, if h(M1) ~ ~. For exaalple, we consider again the system models A4i = < Q; _<> and A42 = < R; < > . Suppose that a set of axioms ~ includes the sentences expressing that < is linear ordering and the following sentence: (Yx)(3y)(y < x A --y = x). A homomorphism hi of A~/i to AA2 is defined in the same way; hi(r) -- f for r E Q. Then the homomorphic image of hi, hi(A41), clearly satisfies ~: hl(A4i) ~ Z. So hi is
a
E-honlomorphisnl of M1 to ,~42.
Next consider a holnomorphism h2 of ~41 to .hd2 defined by:
h2(r)={ r 0
if r_>0 otherwise.
Then the homomorphism image of h2 is h2(.h/[1) = < N; (_<) VI (N × N) >, where N is the set of natural mlmbers. Since there is no y in h2(.h41) such that y < 0, h2(M1) does not satisfy the sentence Wx)(3y)(y _< x A --y = x). Hence h2 is not a ~-honlomorphism. Our definition of ~3-homomorphisms is slightly weaker than Gri~tzer's definition using the concept of @ - l inverse. His definition requires that any "inverse" elements should be preserved.
61
4.1. M O R P H I S M S F O R M O D E L S OF T H E S A M E T Y P E
For exmnple, let A/ll and A/12 be system models, where A41 = < M 1 ; R 1 > and .A~2 = < M2; R 2 > with M1 = M2 = {1, 2}, R 1 = {(1, 1), (2, 2)} and R 2 = {(1, 1), (2, 2), (1, 2)}. Let ¢ = (Vx)(3y)(R(x, y)) be an axiom that is satisfied in both A/J1 and A//2. We define a homomorphism h by the identity. Then h(A/[1) = A//2, and h is a ~-homomorphism. On the other hand, it follows that (h(1), 2) E R 2 and there is no element b in A/t1 such that h(b) -- 2 and {1, b) E R 1. This shows that h is not a ~-homomorphism in Gr~tzer's sense. Here we formulate Gr~itzer's concept of ~-homomorphisms as strong I3-homomorphisms.
As clarified later, the strong ~-homomorphisms form a class of the
homomorphisms that preserve E. In order that a homomorphism preserves ~, if a sentence (Vx) (3y) ~ (x, y), for example, holds in a certain system model, there must exist y that is an inverse of x in the system model as the image of the homomorphism. This concept is formulated as (I) - l inverse. For simplicity, but without loss of generality, we set (I) • ~ in the following discussion as: + = (VXl)(3yl)(Vx2)(3y2). • • (VXn_ 1)(3Yn-1)(VXn) ~(xl,Yl,X2,Y2 .... ,Xn-l,Yn_l,xn), where • includes no quantifier. Let e((I)) = n. In a general form of (I), there may be some variables with universal quantifiers between (3yl) and (3yi+l) (or before ( 3 y l ) or after (3Yn-1)). The concept of (I) - l inverse for the general form of (I) can be defined in an essentially same way as the following definitions. D e f i n i t i o n 4.1.4 (@k-relation) Let ]vt ~ ~ and M be the base set of 2¢[. S~ k C M 2k (k =- 1 , 2 , . . . , n - 1) is said to be a Ck-relation, if ( a l , a 2 , . . . , a k , b l , b 2 . . . . ,b~:) • S+k iff
A/[ ~ (VXk+l)(3Yk+l)... (VXn-1)(3Yn-1)(VXn) kOCal,bl,...,ak,blc,Xk+l,Yk+l,...,Xn).
For example, for (I) = (Vx)(3y)(Vu)(3v)~(x, y, u, v), (al, bl) • S+~ iff M ~ (Vu)(3v)~(al, bl, u, v) and (al, a2, bl, b2) • S+~ iff A/[ ~ ~ ( a l , bl, a2, b2). D e f i n i t i o n 4.1.5 ((I)- l I n v e r s e ) We assume the conditions of Definition 4.1.4 andl
< e(~).
Let al . . . . . at,bl • M .
Then blis said to be a @ - I
inverse of
al . . . . . at in ~//, if there exist bl,... ,bl-1 in M such that for every k, 1 < k < l,
( a l , . . . , a k , b l , . . . ,bk) • S¢ k.
CHAPTER 4. SIMILARITY OF SYSTEM MODELS
62
• -inverse will mean (I, - l inverse for some I < e((I)) and ~-inverse will mean (I)-inverse for some (I, E ~. For exanlple, consider the following axiom: (I) _-__(Vx)(3y)(Vu)(Sv)~(x, y, u, v) If for a,b E M, ( V u ) ( S v ) ~ ( a , b , u , v ) holds in M , then b is a (I) - 1 inverse of a. Furthermore if for a, b, c, d E M such t h a t b is a (~ - 1 inverse of a, ~ ( a , b, c, d) holds in M , then d is a (I) - 2 inverse of a, c. A h o m o m o r p h i s m t h a t has the ¢ - 1 inverses is defined as a strong Z-homomorphism.
Definition 4.1.6 (Strong ~ - I - I o m o m o r p h i s m ) Let ~ C Sent(E) be consistent, and M1,A/[2 ~ ]E. Let h:A/[1 --* M 2 be a homomorphism. h is said to be a strong ~-homomorphism of.hA1 to A~2, if the following conditions axe satisfied: For any (I) E ~, any 1 < l < e(~),al,a2,... ,at,b E M1, (1) if b is a (I) - l inverse of a l , . . . ,at in Jk41, then h(b) is a (I) - l inverse of
h(aO,..., Mat) in M 2 ; (2) if b is a ¢ - l inverse h ( a 0 , . . . , h(at) in M 2 , then there exists a b' E M1 such t h a t b = h(b') and bt is a • - l inverse of a l , . . . , at in ]vii. The axioms ]E axe preserved in the homomorphic image on h in M 2 by a strong 2 - h o m o m o r p h i s m h.
Proposition 4.1.1 Let M1 and M2 be system models of the same type. A strong ~-homomorphism is a ~-homomorphism. That is, if h is a strong Z-homomorphism of M1 to M 2 , then the homomo~Thic image of h, h ( M 1 ) , is a model of ~, i.e., h ( M D ~ ~. Proof.' Let h be a strong ~ - h o m o m o r p h i s m of A,/1 to .h~t2. We show Proposition only about formulas in the forms (Vx) (By) • j (x, y) and
(Vx)(3y)(Vu)(~v)%(x, y, u, v), where ~ l ( x , y )
and ~2(x, y, u, v) include no quantifier. The general case can be
proven in induction on the number of quantifiers. In case • -- ( V x ) ( 3 y ) ~ l ( x , y ) :
4.1. MORPHISMS FOR MODELS OF THE SAME TYPE
63
Let h E h(,~41). Then for some a E M1, h(a) = d. Since J~41 ~ ~, there exists b E M1 such that if/1 (a, b). Since h is a strong ~-homomorphism,
•,~2 ~ ~l(h(a),h(b)). Hence h(AJ1) ~ ~l(h,h(b)). T h a t is,
h(M1) ~ ¢. In case ¢ --= (Vx)(3y)(Yu)(3v)k~2(x, y, u, v): Let 5 E h(A41). Then for some a E M1, h(a) = d. Let b E M1 be a ¢ - 1 inverse of a. Then ]~41 ~ (Vu)(3v)~2(a,b,u,v). Since h is a strong Z-homomorphism, h(b) is a • - 1 inverse of h(a). Hence 2~42 ~ (Vu)(3v)~/2(h(a), h(b), u, v). Let ~ E h(A/ll). Then for some c E M1, h(c) = 5. There exists a • - 2 inverse d of a, c such that
.A41 ~ ~12(a,b,c,d). Hence
.A42~2(h(a),h(b),h(c),h(d)) a~ld
h(.hA1) ~ ~t2(~, h(b), ~, h(d)). That is,
h(M1) ~ ¢. This completes the proof.
[]
The converse of Proposition 4.1.1 does not necessarily hold. That is, even if the homomorphic image of a homomorphism satisfies ~, the homomorphism is not necessarily a strong E-homomorphism. A necessary condition for a strong E-homomorphism is that the homomorphic image is the following set.
C H A P T E R 4. S I M I L A R I T Y OF S Y S T E M M O D E L S
64
D e f i n i t i o n 4.1.7 Let A4 be a system model such that AA ~ ~, and 0 ~ H C M. Then the set [H]z is defined by the following.
Ho = H;[In-1 = {a • M l a = t [ a l . . . . . an],t is a term, al . . . . ,a~ • Hn-1}; H~ = *~7~-1U {a • M I there exist b i , . . . ,bt E/[rn_i such that a is a Z-inverse of b l , . . . ,b~ in A4}. Then
[His
=
U(/&li • N).
P r o p o s i t i o n 4.1.2 Let .£4i and JV[2 be system models of the same type, and h a
homomorphism of AA1 to ]~42. Then "if h is a strong Z-homomorphism of J~Ai to JVl2, then h(AAi) = [h(A4i)]s. P r o o f : From the defnition of [h(A41)]s, it is clear that h(AA1) C [h(A4i)]s. Using [h(A4i)]s -- O((h(J~ti))ili • N), we show [h(AAi)]~, C h ( ~ l i ) by induction on i. i = 0. (h(]~l))o = h(M~). Suppose (h(.Adl))i-i C h(AA1). Since h(A41) is closed under functions and every term is constructed from functions and variables, we have (h(.h/[1))i_ 1 C h(fl41). Suppose a E .A~2 and there exist bi . . . . , bt • (h(.Mi))i_l such that a is a ~inverse of bl . . . . , bt in A4s. Then for some a l , . . . , at E Mi, bl -- h ( a l ) , . . . , b t = h(at). From the condition (2) of the definition of a strong ~-homomorphism, there exists an 5 E Mi such that a = h(h) and 5 i s a E-inverse o f a i .... ,at inJ~4i.
Hence
a E h(A41). Therefore for every i E N (h(M1)); C h(M1), so we have [h(.A41)]~. C h(.&/1).
[]
We consider an example of a ~,-homomorphism that is not a strong ~-homomorphism, but whose homomorphic image is equal to [h(A4)]~. Let M i - - < {al,a2, a3};R1 > and M 2 = < {bl,b2,ba};R2 >, where R1 is a binary relation: R1 ----{(al, a2), (a2, a3), (a3, al)}
4.1. M O R P H I S M S F O R M O D E L S OF T H E S A M E T Y P E
65
m:d R2 a binary relation: R2 -- {(bl, bl), (b:, b2), (52, al), (b2, b2), (b3, b3)}. Define h : AR1 ---*.M2 by h(al) -- bl and h(a2) -- h(a3) = b2. Then h(•l)
- - < {b:, b2}; {(b:, bl), (b:, b2), (b2, bl), (b2, b2)} :>.
Let E = {¢}, where ¢ - (Vx)(3y)(R(x,y)). Then clearly h(A,i:) ~ E. So h is a E-honmmorphism. But it is not a strong E-homomorphism, since b: is a & - 1 inverse of h(a:) = b: in ,~42 and there is no a E M: such that b: = h(a) and a is a - 1 inverse of a:. Furthermore in this example we have h(A4i) = [h(A,t:)]~.
Indeed we have
(h(.M1))o = h(.M1) -- {bl,b2} and (h(A41))o -- {bl,b2}.
And if ( h ( M : ) ) , - 1 =
(h(A41))u_l -- {bl, b2}, then (h(A/l:)), -- (b:,
1 inverses of b: (or b2)
b2} since
•-
are bl and b2. Hence we have [h(A4:)]~ ----U ( (h(./vfl) )i]i E N ) = h(AA1). This shows that the converse of Proposition 4.1.2 does not necessarily hold. 4.1.3
Preservation
of Th(M)
-- S-homomorphism
That two system models are isomorphic or of the same structure implies that in a sense the properties of the two system models are equivalent. As seen in Section 4.1.1, a usual honmnmrphism preserves the primitive properties, i.e., the generators. In this section we will define a homomorphism as an S-homomorphism that preserves all sentences satisfied in a system nmdel (Th(A4)). Furthermore we will show that an induced homomorphism in the well known homomorphism theorem in algebra is an S-homomorphism.(Theorem 4.1.1) From this theorem we can see that every morphism for the structural similarity should be an S-homomorphism. Definition 4.1.8 ( S - I - I o m o m o r p h i s m ) Let A41 and Ad2 be system models of the same type, and h : 2~41 ~ Ad2 be a homomorphism of A~I: to ,~42. Then h is called
an S-homomorphism ofA, fi to A~2 if for any sentence ¢ of £ ( M 1 ) J~41 ~ ¢ if and only if h(A41) ~ ¢ From the definition we can immediately see that an S-homomorphism is a Ehomomorphism. We should notice that if every sentence that holds in A41 holds in h ( ) A : ) as well, then h is already an S-homomorphism. Indeed if a sentence ¢ holds in h(A4:)
CHAPTER 4. SIMILARITY OF SYSTEM MODELS
66
and does not hold in A4:, then --¢ holds in 341. Thus, by the above condition, -~¢ holds in h(2~4:), which is a contradiction. If h is an isomorphism of A4: to h(JM:), then h is also an S-homomorphism. In general, isomorphism is stronger than S-homomorphism. However if A4: is finite, i.e., the base set M1 of ¢M: is finite, the:: an S-homomorphism is aa: isomorphism of ¢t4: to h(jM:). To verify this, consider the sentence ~:
l
\:
Then we can easily show that 34: ~ ~ if and only if the base set M: has exactly n elements. Thus in the finite case, if h is an S-homomorphism of 3,t: to 3//2, then the base sets of -'~1 and h(A4:) have the same number of elements. Hence h must be one-to-one correspondence, and an isomorphism of -~l to h(J~l). Let h : 3.t: --+ AA2 be a homomorphism of A/l: onto AA2. Then we define the
quotient system model with respect to h, written by A4:/h: .M1/h = < M1/h; {R~/h I i • I } , { f l / h I J e J} >, where M:=<M:;{R~li•I},{f#lj•J}>, M2 = < M2;{R 2 I i e I}, {f2 I J • J} >;
M:/h is the partitioned set of 3//: by the equivalence relation defined by: a --- b if and only if h(a) = h(b) for any a, b • M1, ([a:],..., [a~(il]) e R}/h if and only if (h(al) . . . . . h(a~(o)) • R'~, =
....
[ai] denotes the equivalence class of ai. The quotient system model, .M1/h, is obviously well-defined. Consider system models ]v4: = < Q; _<> and 2v42 = < Z; <>, where Q is the set of rational numbers, Z the set of integers and < the usual linear ordering. Define h of jr4: onto ]v42 by
h(r) = f for r • Q. Then the quotient system model with respect to h is A~l/h =< Q/h; <_/h >, where Q/h = {[r]lr e Q}, [r] is an equivalence class determined by h(r) = h(r');
4.I. MORPHISMS FOR MODELS OF THE SAME T Y P E
67
Let us consider another example: A41 --< Z; +1 > and A42 = < {0, 1, 2}; +2 :>, where ÷1 is addition on integers and +2 addition of remainders modulo 3. Define h of ~41 onto J~42 by associating its remainder modulo 3 with each integer. Then
M 1 / h =-< Z/h; + l / h >, where Z / h -- {[0], [1], [2]}, [k] = {nln -= 3m + k, m e Z} and
[kl](+l/h)[k'2] = [kl +1 k2]. The well known homomorphism theorem can be considered as a theorem by which an S-homomorphism, h #, is induced. Theorem
4.1.1 ( H o m o m o r p h i s m
Theorem)
Let .hA1 and ]~42 be system mod-
els of the same type, and h a homomorphism o].h41 onto .h42. Then a map h # : A / h / h ~ ~42 is an S-homomorphism, where h # is defined by: h#([a]) = h(a) for any [a] e M1/h.
h # is called the induced homomorphism of h . P r o o f : First we show that h # is an isomorphism. h # is a one-to-one correspondence. Because if In] ¢ [b], then we have h(a) ¢ h(b) from the definition of the equivalence relation. Hence h#([a]) ¢ h#([b]). Furthernmre since h is onto, so is h #. h # is a homomorphism of .h41/h to .tcf2. Since from the definition of .t~41/h the relations of ~41 satisfy the conditions in the definition of a homomorphism, so it suffices to show that the functions f ] / h satisfy them.
h # ( f l / h ( [ a l ] , . . . , lap(j)])
-- h#([fl(al,...,a/4j))] ) =
=
f}(h(al),..., h(a,(s)))
= f2(h# ([al]), • • •, h#([a,,(j)])) Therefore h # is an isomorphism. Since all sentences satisfied in one system model are preserved in another isomorphic system model to it [Gr~itzer 1979], h # is an S-homomorphism.
[]
A difference between the well known homomorphism theorem in the usual form in algebra and Theorem 4.1.1 is that Theorem 4.1.1 points out that the induced isomorphism h # is an S-homomorphism preserving sentences. One of the reasons why the concept of a homomorphism is important is because an S-homomorphism h# can be constructed from a homomorphism h.
CHAPTER 4. SIMILARITY OF SYSTEM MODELS
68 4.2
Morphisms
for Models
of Different
Types
In this section we consider morphisms between system models of different types. Since a homomorphism as seen in the previous section can be defined only for system models of the same type, the concept of a homomorphism is not applicable to the class of system models of different types. We therefore introduce a new morphism, called F-morphism, which is a generalization of homomorphism and can apply to the class of system models not only of the same type but of different types. In this section we also consider the three cases for preservation of properties - - generators,
E and Th(M). We will show F-morphism theorem as a theorem corresponding to the homomorphism theorem. 4.2.1
Preservation
of Generator -- F-morphism
First we define a basic interpretation function and a basic morphism. An F-morphism is defined recursively by using these functions.
Definition 4.2.1 (Basic Interpretation Function) Let AA1 = < M1;{R~]i e / 1 } , { f 1 IJ e J1} >, AA2 = < M2; {R~ I i e I2}, {f2 [ j e J2} > be system models of possibly different types. Then a function Bas of £(A41) to the set of formulas of g(Ad2) is said to be a basic interpretation function of £(]v~1) to £(.~42) if the following conditions are satisfied. (1) For every relation symbol R~ e £(A41), Bas(R 1) is a ()u(i))-ary formula of £(.~42); (2) for every function symbol i~1 E £(M1), Bas(fj1) is a (#l(j) + 1)-ary formula of t:(M.~). A basic interpretation function associates a formula of the second system model with each symbol of the language of the first one. The association is intended to give "interpretation" of the first system model to the second one. The basic interpretation function works as a meaningful interpretation only when a basic morphism with it is defined as follows. D e f i n i t i o n 4.2.2 (Basic M o r p h i s m ) Let .£41 and A42 be as above and Bas be a basic interpretation function of £(2~41) to I:(AA2). A function Io of M1 to M2 is said to be a basic morphism of Adl to .h42 with Bas if the following conditions are satisfied.
4.2. MORPHISMS FOR MODELS OF DIFFERENT TYPES
69
(1) For every relation symbol R 1 E £:(.£41) and every assigmnent p, if Adl ~ R 1 ( X l , . . . , x~l(i})[p], then .h~t2 ~ Bas(R1)(Xl . . . . . X~l(i))[IO o p], where Io o p denotes the composition of Io and p; (2) for every function symbol fjl E £:(A.tl) and every assignment p, if .A~1 ~ ( ~ l ( x l , . . . ,X,l(j)) = X~l(j)+l)[p], then
• 2 ~ Bas(~l)(xl,... ,X#l(j)+l)[Io o p], and satisfies the following condition expressing that Bas(~ 1) is a function:
•AA2 ~ (VX1 ... Xpl(j))(3X.I(J)+I)(Vy#I(J)+I)
(Bas(fjl)(xl, ... , X#l(j),Y#I(J)+I) ~ Xpl(J)+l ----y/~l(J)+l). Bas(R 1) and Bas( fj1) are called basic interpretations o f R 1 and ~1. The identity = is interpreted as the identity of £:(A42),that is, Bas(=£(M~)) --=£(M2).
M1
P
Io
M2
Io°P
L(M2 )
Figure 4.1 An Illustration of a Basic Morphism As illustrated in Figure 4.1, a basic morphism represents, in some sense, a commutativity of models and their languages. Let us consider an example of a basic morphism. Let A41 = < N; <, f > and AA2 = < N; + >, where N is the set of natural numbers, _< the usual linear ordering, + addition, and f a binary function defined by f(a, b) = la - b[ (the absolute value of the difference of a aad b).
C H A P T E R 4. S I M I L A R I T Y OF S Y S T E M MODELS
70
Define a basic interpretation function by
Bas(<)
=
(3z)(x+z=y),
Bas(f)
=
(3u)(x+u=yAz=u)V(3w)(y+w=xAz=w).
Notice that Bas(<_) is actually a binary formula and B a s ( f ) 3-ary formula according to the arities of <__and f respectively. Define a basic morphism Io by h(n) = 2n for n E N. We can verify as follows that Io is actually a basic morphism. Let p be an assignment into M1. Suppose AA1 ~ (x < y)[p]. Then we have p(x) < p(y) in ]vii. Since Io(p(x)) = 2p(x) < 2p(y) = Io(p(y)), there exists a z in N such that Io(p(x)) + z = Io(p(y)). Hence M 2 ~ B a s ( < ) ( x , y ) [ I o o p]. Next suppose A41 ~ (f(x, y) = z)[p]. Then we have ]p(x) - p(y)] = p(z) in A41. If p(x) < p(y) in A/g1, [p(x) - p(y)] = p(y) - p(x) = p(z). Since 2p(y) - 2p(x) = 2p(z), there exists a u in N such that Io(p(x)) + u = Io(p(y)) and Io(p(z)) = u. Hence M 2 ~ (3u)(x + u = y A z = u)[Io o p]. The case p(x) > p(y) can be similarly considered, and we have M 2 ~ (3w)(y + w
o P].
= x A z = w)[Io
Thus we have A42 ~ B a s ( . f ) ( x , y , z ) [ I o o p]. We can easily check that B a s ( f ) satisfies the condition of function. Furthermore from the above explanation, we see that any function of A41 to A42 defined by h(n) = kn, k E N, is a basic morphism of A41 to A42 with this basic interpretation function.
Other examples of basic
morphisms will be given in examples of F-morphisms. For any atomic formula O ( t l , . . . ,tn) of a given language, we define the set T ( ¢ ) of terms as follows. (1) If • is of the form fj ( u l , . . . , u,(j)) = x (or equivalently x = fj (ul . . . . , us(j))), where u l , . . . , u~,(j) are terms and x is a variable, T(q~) = {uk [uk is not a variable, k = 1 . . . . . #(j)}; (2) otherwise, T(O) = {tk ]tk is not a variable, k = 1 . . . . . n}. In case (1) we should notice that f j ( u l , . . . ,u~,(j)) = x corresponds to a formula of the form = (h,t2), where tl = f j ( u l , . . . , u~,(i)) and t2 = x. For example, if (I) = f(g(x, y), z), where x, y, z are variables, then from case (1), T(~) = {g(x,y)}. If (I) = ( f ( g ( x , y ) , z ) = g ( x , y ) ) , x , y , z E V, then from case (2), T ( ¢ ) = {f(g(x, y), z), g(x, y)}. If ¢ = R ( f ( x , y ) , z ) , then from case (2), T ( ¢ ) = {f(x,y)}.
4.2. MORPHISMS FOR MODELS OF DIFFERENT TYPES D e f i n i t i o n 4.2.3 ( F - m o r p h i s m )
71
Let A41 and A42 be as in Definition 4.2.1. An
F-morphisrn~ I : AQ ~ A42, is a pair of functions < Io, IF >, where Io is a basic morphism of A41 to A42 with Bas and IF is a function of the set of formulas of £(A41) to the set of the formulas of £(A42), which is defined as follows. For any formula (I) of/::(A41) (1) if @ is an atomic formula of the form f j ( u l , . . . , us(j)) = x or x = fj(ul . . . . , up(j)), then
IF(t~) = where every
Bas(fj)(ul,... ,u~(j},x),
if T(il)) is the empty set
( 3 X k l . . . X k m ) ( B a s ( f j ) ( X l , . . . , x~(j), x) A(A(IF(xk i = Uk,) [uk, E T(@)))),
if T(@) = { u ~ , . . . , ukm }
Xkp is
a variable not occurring in ~, and x i is ui for ui f~ T(~).
(2) if • is an atomic formula P ( t l , . . . ,tn) other than of the form in (1), then
IF(@) =
Bas(P)(tl,..., tn), if T ( P ) is the empty set (3Xkx... Xkm)(Bas(P)(xl,..., Xn) A(A(IF(Xki = tk,) Irk, E T(P)))), i f T ( P ) = {tk~,...,tk,,}
where every Xkp is a variable not occurring in P, and x i is ui for ui ¢ T(@). (3) otherwise,
IF(~iI~) = -~(IF((I))); 1~(~1 ^ ~2) = (IF(@I)) ^ (Iy(~2));
XF(Vx¢) = (Vx)(X~(~)). Since T(@) eventually becomes empty, Ix is weU-defined.
Form(L 2 )
Form(L 1 ) IF
VAn (h
L
f
f ~ I"~'~
IF (rATe) Bas
---... Bas( / )
F i g u r e 4.2 An Illustration of an F-morphism
CHAPTER 4. S I M I L A R I T Y OF S Y S T E M MODELS
72
For example, if (I) is of the form f(ul, x2) = x and T{~) = {ul }, then
I r ( ~ ) = ( 3 x l ) ( B a s ( f ) ( x l , x2, x) A (IF(Xl = ul))). If ¢ is of the form f(ul, u~) ----g(w) and T ( ¢ ) ~- {f(ul, u2), g(w)}, then IF(C) = (3xlx2)(X1
= x2 A (IF(x1 = f ( u l , us)) A ( I F ( x 2 = g ( w ) ) ) ) .
Let us consider some examples of F-morphisms. E x a m p l e 4.2.1 We can define an F-morphism of (N; _<) to (N; +), where N is the set of natural numbers, < the linear ordering on N and + addition. If we define
Bas by Bas(<_) = ( 3 z ) ( x + z = y) and Io by the identity, then <: IO,IF > with Bas is an F-morphism of (N; _<) to (N; +). We should notice that the function IF is "automatically" defined according to the definition 4.2.3 if Io with Bas is already defined. Furthermore it is clear that if (N; <) ~ ¢, then (N; +) ~
Ip(¢).
For example,
let ¢ be a sentence (Vxyz)(x < y A y <_ z -~ x _< z). Then IF(C)
= IF((Vxyz)(x_< y A y _< z--* x < z)) ----- (Vxyz)(IF(x _< y) A IF(y ~ z) --* Ip(X < z)) = (Vxyz)((3Zl)(X q- zl = Y) A (3z2)(x q- z2 = y) --* (3za)(x + z3 = y)).
So (N; +) ~ IF(C). We should notice that we can define another F-morphism. A trivial example is one which defines Io by the constant mapping to 0. E x a m p l e 4.2.2 Let M ---< G; +, 0 > and M ~ = < G; R > be groups, where if for any a, b, c E G, R(a, b, c) if and only if a + b = c, then we can define an F-morphism of M to A/ff as I = < Io, I f >, where
Io Bas(T) Bas(O) IF(X1 + x2 = X3)
:
the identity,
=
R,
= = =
Co(x) = ( V y ) ( 3 z ) ( R ( y , x , y ) A R ( x , y , y ) A R ( y , z , x ) )
Bas(+)(Xl,X2,X3) R(xl,x2,xa).
For example, let us transform the following axioms of groups by Ip.
(1) (VXlX2Xa)((Xl + x2) + xa = Xl + (x2 + x a ) ) (2) ( V x ) ( x + 0 = x A 0 + x = x )
(-- ¢1) (-- ¢2)
(3) (Vx)(3y)(x -t- y = 0)
(-- ¢3)
4.2. M O R P H I S M S FOR MODELS OF D I F F E R E N T T Y P E S
73
¢1, ¢2, mid ¢3 are transformed as follows. We demonstrate the transformation process only about ¢2. (1') IF(¢1)
(2')/~(¢2)
=
=
(Vxlx2xa)(3x4x~)((x4 = xs) A (3x6) ( t t ( x ~ , x s , x4) A R ( x l , x2, x ~ ) ) A ( 3 x T ) ( R ( x l , x~, x s ) A R ( x 2 , x s , xT))) / F ( ( V x ) ( x + o = x A 0 + x = x))
=
(VX)IF(X+ 0 = x A 0 + x ~ x)
=
( V x ) ( I F ( x + 0 = x) A IF(0 + x = x ) )
=
( V x ) ( ( 3 y ) ( B a s ( + ) ( x , y , x ) A IF(Y = 0)) A (3z)(Bas(+)(z, x, x) A IE(z = 0))) ( ' ¢ x ) ( ( 3 y ) ( R ( x , y , x ) A Bas(O)(y) A (3z)(R(z, x, x) A Bas(O)(z))) ( V x ) ( 3 y ) ( 3 z ) ( R ( x , y , z ) A Bas(O)(z))
= (3') IF(¢3)
=
In the above transformations, ¢1, ¢2 mid Ca are transformed individually. However, if their conjunctive form ¢1 A ¢2 A ¢3 is transformed, the sentence transformed by IF, IF(¢1 A ¢2 A ¢3), is equivalent to the axioms of .h4 ~ as a group. E x a m p l e 4.2.3 A Moore type automaton, M~ --- (A, B, C, Cr, )~r) with A, B, C : finite sets, Cn : C x A ~ C and An : C -* B, is regarded as equivalent to a Mealy type automaton, Me = (A, B, C, Ce, #~) with A, B, C : finite sets, ¢~ : C x A --* C and #e : C × A --* B. Let Moore - - < A U B U C; A, B, C, ¢,., ),,. > and ^
Mealy --< A U B U C; A, B, C, Ce, f*e >, where A, B, C : unaxy relations, Cn, i~, Ce, f*e : arbitrary extensions of Cn, An, Ce, #e respectively, and
for any c, a E A U B U C. Then we can define an F-morphism of Mealy to Moore as follows. Io Bas(~e) Bas(f~e)
: the identity; -~ Cr(X,y,z); = (),r(¢r(x,y)) = z).
B a s ( A ) = A, Bas(B) = B, Bas(C) = C; Obviously these satisfy the conditions of basic morphisms. Then the following sentence, for example, is transformed into a sentence of £(Mealy). Let
74
CHAPTER 4. S I M I L A R I T Y OF S Y S T E M MODELS
P = (Vx e C)(Vy3 e A ) ( 3 y l E A)(qy~. e A)(~e(¢e(X, Yl),Y2) = pe(x~Y3)). IF(P) = (Vx e C)(Vy3 E A ) ( 3 y l e A)(qy2 e A)(3zlz2) ((zl = z2) A I f ( ( z l =/2e(¢e(x, yl),Y2)) A (z2 =/2e(x, Y3)))), because T(P1) -- {/~e(¢e(x, Yl), Y2), fie(X, Y3)}, where
P1 -- (/~e(¢e(x,Yl), Y2) =/2e (x,Y3)). Therefore
IF(P) = (Vx E C)(Vy3 E A)(3yl E A)(3y2 E A)(3zlz2) ((z 1 = z2) A (3w)(/F(Z1 = fie(W, y2) A W = Ce(X, y l ) ) A IF(z~ = ~e(x, y3))), because T(P~) = {¢e(x, yl)}, where P2 = (zl = (/~e(¢e(x, Yl), Y2))). Therefore
IF(P) = (Vx E C)(Vy3 E A)(3yl E A)(3y2 E A)(3ZlZ2) ((z 1 = z2) A (3w)(z I = ~r(~r(W, Y2))
A w = Cr(x, Yl)) A (z 2 = ~r(¢r(X, y3))), which is equivalent to (Vx E C)(Vy3 e A ) ( 3 y l E A)(3y2 E A)(Ar(¢r(¢r(x, Yl),Y2)) = Ar(¢r(x, Y3))). The following corollary shows that a~l F-morphism is an extension of a homomorphism. C o r o l l a r y 4.2.1 Let .A~tl and Jul2 be system models of the same type, and h a homomorphism of.hA1 to .hA2. Then h is a basic morphism of ./~41 to A42, and < h, IF > is an F-morphism o/A41 to A.t2, where IF is a function uniquely determined by h in Definition 4.2.3.
4.2. M O R P H I S M S FOR MODELS OF D I F F E R E N T T Y P E S
75
Proof: It is clear, since for each symbol of Z:(A41) its basic interpretation by h is the corresponding symbol of £(A41).
[]
From the definition of basic morphisms we can see that an F-morphism preserves generators. Notice that it is necessary to give an interpretation IF of the generators in defining an F-morphism, while in the case of a homomorphism IF is trivially defined, and is not explicitly given in usual algebra. The composition is naturally defined as follows. Definition 4.2.4 ( C o m p o s i t i o n ) Let AJ1, A42 and 3/i3 be system models, and /1 : A41 --* A/[2 a n d / 2 : .AzI2--~ it43 F-morphisms. The composition, I : A41 --+ A43, of/1 a n d / 2 is defined as follows: Io : A41 ~ Ad3; Io = (I2)0 o (I1)o, I F : Form(E(.K41)) --* Form(f.(.M3));IF : (I2)r o (I1)F. P r o p o s i t i o n 4.2.1 Let fi41,.K42,A43, I1,I2 and I be as in Definition ~.2.4. For
any symbol S o/L:(A41), i] I is an F-morphism, then Basl(S) = (I2)F(BaSl, (S)). Proof:
Basl(S)
=
/F(S(xl,
=
(I2)F((~l)F(S(xl,
.. • ,x~(i)))
=
(I2)F(Bas~I(S)).
•. •, X~(1)))
[]
The composition of F-morphisms is not necessarily an F-morphism. (See Appendix 2) If the composition of F-morphisms is again an F-morphism, the interpretations by the two F-morphisms can be considered, in a sense, to be "faithful." D e f i n i t i o n 4.2.5 (Faithfulness) Let I1 and /2 be F-morphisms. If the composition of/1 and I2 can be defined in the sense of Definition 4.2.4 and it is an Fmorphism, then we say t h a t / 1 a n d / 2 are faithful or/1 is faithful to/2. The faithfulness of F-morphisms will play an important role in hierarchies. (See Section 6.1) 4.2.2
Preservation of Z --
ZF-morphism
In this section we will define an F-morphism preserving axioms IGas a ~F-morphism. By a ~]r-morphism, Yl. is preserved in an image of an F-morphism. First we define the image of a basic morphis,n.
CHAPTER 4. SIMILARITY OF SYSTEM MODELS
76
Let 341 and 342 be system models as in Definition 4.2.1. Let I = < Io, IF >: 341 --* 342 be an F-morphism. Then the image of a basic morphism, written/(341), is defined by:
1(341) ---< Io(M1); {R 2 N lo(M1) ~a(1) I i • I2},
{f2 1110(M1)P"(J)]f2(al,..., %~.(j)) • Io(M1) for any al .... ,a,2(j ) • Io(M1),j • ]2} ;> Notice that if there exists a j such that f2(al,...,
ap2(j)) ~_ Io(M1)
for a l , . . . ,
a~(j) • 10(M1), then I(341) is of a different type from 342. For example, let us consider 341 = < Q; _<> and 342 = < R; +, - >, where Q is the set of rational numbers, R the set of real numbers, _< the usual linear ordering, + addition and - subtraction. We define a basic morphism Io with Bas of 341 to 342 by to(r) -- f for r • R and Bas(<_) = (Bz)(x + z = y), where ~ means the maximum integer not exceeding r. We can easily verify that Io is a basic morphism. Then the image of Io is
I(341) ---< z; + II z 2, -II z 2 >, where Z = Io(R); the set of integers. But if 341 = < Q+; _<> with Q+ the set of non-negative rational numbers, then
/(341) ----<: N; Jr [I N2 > with N the set of natural numbers. In this case the image of Io,/0(341), does not include the subtraction and is of a different type from 342, since Io(rl) - Io(r2) fL 10(341) if Io(r2) < Io(rl). The primal idea of EF-morphisms is concerned with whether the sentences of the language for 341 transformed by IF hold in I(341). An F-morphism is called an onto F-morphism if its basic morphism is onto. Definition 4.2.6 ( ~ F - M o r p h i s m ) Let A41 and Ad2 be system models and I = <
Io,IF > an F-morphism of 341 to 3,t2. Suppose 3,tl ~ ~. Then I is called a ~F-morphism of 341 to 342 if /(341) ~ / F ( ~ ) , where IF(E) = {/F(~) I ¢ e ~}. Consider again the system models Jvtl ---< Q; <__> and 342 = < R; +, - >. Let P, = {¢}, where ¢ - (Vx)(3y)(y < x A -~y = x).
4.2. MORPHISMS FOR MODELS OF DIFFERENT T Y P E S
77
Define Bas(<) by (3z)(x + z = y). If we define a basic morphism Io of 3`/1 to 3-t2 with Bas(<) by Io(r) = ¢, then IF(C) = (Vx)(3y)((3z)(y + z = x) A --x = y) and I(3`/1) ~ Iv(C). So this is a E-morphism of 3`/1 to 3`/2. However if we define Io by
Io
(r)={ ~ ifr>__O 0
otherwise,
then /(3`/1) =<: N; + II N2 > does not satisfy IF(C) any more. (Note that - is dropped from I (3,/1).) If the composition of Zv-morphisms is again a Nv-morphism, then we say that the two EF-morphisms axe (Er-)faithful. A Er-morphism is a kind of extension of a ZF-homomorphism. The following corollary says that a NF-nmrphism between system models of the same type accords with a E-homomorphism. C o r o l l a r y 4.2.2 Let3`~1 and3`~2 be system models of the same type. Let h : 3`/1 --*
3`/.~ be a homomorphism of A,tl to 3`/2. Suppose 3`/1,3`/2 ~ E. Then h is a Ehomomorphism of 3`/1 to 3`/2 if and only if< h, Ii~ > is a EF-morphism of 3`/1 to 3`/2. Proof: It is clear from the fact that if 3`/1 and 3`/2 are of the same type, then h(.Adl) = I(3`/1) a n d IF(E) = E.
4.2.3
Preservation
[]
of Th(A4) - - SF-morphism
In this section we define a morphism between system models of different types, which preserves Th(.M) of a system model. Furthermore we will show the F-morphism theorem corresponding to the homomorphism theorem in the case of the same type. The F-morphism theorem gives a relationship between an F-morphism and an SFmorphism. Unless mentioned explicitly, in the sequel let A41, Ad2 be system models and I = < Io, IF > an F-morphism of 3,tl to 3,t2. D e f i n i t i o n 4.2.7 ( S F - M o r p h i s m ) An F-morphism I is called an SF-morphism of
A41 to 3,/2 if for any sentence • of £(Adl), Adl ~ @ if aald only if I(A41) ~ IF(~). From the definition an Sv-morphism is a Ev-morphism. The basic morphism of an SF-morphism is not necessarily a one-to-one correspondence. However, in the finite case, by considering again the sentence (I):
l
\l
CHAPTER 4. S I M I L A R I T Y OF S Y S T E M MODELS
78
the base sets of Adi and I(~¢[1) have the same number of elements, hence the basic morphism is "necessarily" a one-to-one correspondence. In general, we can check whether an F-morphism is an SF-morphism by the way based on a structural induction on sentences. But in most cases they are more than routine tasks as the proof of the next theorem shows. For an onto F-morphism I, we define the quotient system model with respect to I by:
M 1 / I =< M1/Io : { R } / I I i e xl}, {f]/x I J e 51}) where M1/Io is the partitioned set of M1 by the equivalence relation ---Io defined
by: a =So b iff Io(a) = Io(b) for any a,b e M1;
R~/I([a,]
.....
[a~l(i)] )
iff Bas(R~)(Io(al) .... ,Io(a~,(O));
fl/I([al] . . . . , [am(j)])
[ f l ( a l , . . . , am(j))];
=
where [a] represents an equivalence class in M1/Io by -=Io. The above definition of f ) / I is well-defined.
Indeed, suppose Io(al) = Io(bl),
.... Io(am(j) ) = Io(bm(j) ) and f ) ( a l , . . . , am(j)) = cl, f ) ( b l , . . . , bt,l(j)) = c2 hold in A41. Then since I is an F-morphism, Bas(~l)(Io(al) .... , Io(%~(j)), Io(cl)) and
Bas(fjl)(Io ( b l ) , . . . , Io(bm(j)), Io(cl)) hold in Ad2. Since Bas(fj 1) is a function from the definition of basic morphism, we have Io (Cl) = Io (c2). For example, consider again the system models A/J1 = < Q; ~ > and A42 = < R;+,-
>. Define Bas(<) by ( 3 z ) ( x + z
= y) a n d I o by Io(r) = ~. Then the
quotient system model with respect to I is
.~gx = < Q/Xo; < / x >, where Q / I o = {[r]lr E Q}, [r] is the equivalence class determined by I0 and
(<_/I) = {([rl], [r2])l there exists a c in N such that ~1 + c = ~2}. The following is one of the main theorems about F-morphisms. T h e o r e m 4.2.1 ( F - M o r p h i s m T h e o r e m ) Let I : .h~ 1 --~ , ~ 2
morphism. Then I~o : M ~ / x -* , ~ 2
is a one-to-one basic moTThism, furthermore I#=<
I#o,I#F >
is an Sf-morphism, where I#o is defined by Io#([a]) = Io(a) for [a] e M1/Io
be an onto F-
4.2. MORPHISMS FOR MODELS OF DIFFERENT TYPES
79
and the basic interpretations are Bas#(R1/I) = Bas(R 1) for i e 11, B a s # ( f l / I ) = Bas(~ 1) for i e J1. I # is called the induced F-morphism of I. Proof: First we show that Io# is a basic morphism. It is sufficient to show that every symbol of ~:(.h~l/I) satisfies Definition 4.2.2. It is clear for the relation symbols from the above definition of A41/I. So we show it only for the function symbols. For any function symbol ~1, we assume that
•A~I/I ~ (~i(x1,.-., X,l(j )) --Xpl(j)+l)[[p] ]. where [p](xi) = [p(xi) ] = [ai],i = 1 , . . . , # l ( j ) + 1. Let f ] ( a b . . . ,a~l(j )) = b in JPil. Since Io is a basic morphism,
fl42 ~ Bas(~l)(xl,..., X~l(j), Xpl(J)+l)[-/o o p'], where b p(x)
p'(x) =
i f x = X~l(j)+ 1 otherwise
By the definition,
fl/I([a1],..., [a/~l(j)]) ~--[fl(al,...,
ap,(D)] = [b].
Hence [apl(j)+l] = [b]. That is
Io(apl(j)+l)=Io(b). So
.M2 ~ Bas(~l)(xl,..., X~l(j), X#l(j)+l)[I O o #]. Since I$([a]) = Io(a),
A42 ~ S a s ( ~ l ) ( x l , . . . , X#l(j), X,l(J)÷l)[/o~ o [p]]. The conditions on the functions are clearly satisfied due to the conditions of Io. Io# is a one-to-one correspondence. Indeed, if [a] ¢ [b], where [a], [b] e A41/I are arbitrary, then =
Io(a) #
Io(b)
=
CHAPTER 4. SIMILARITY OF SYSTEM MODELS
80
For the latter part of theorem, we show the following claim stronger than it.
Claim: For any assignment [p] to M1/Io and any formula ¢ of £(Adl), .M1/I ~ el[p]] if and only if ~M2 ~ IF#(~)[Io# o [p]]. We show the claim by the structural induction on ¢.
Case 1: • is an atomic formula. We divide Case 1 into the following two cases. A) ¢ is of the form f j l ( t l , . . . , t m ( j ) ) = x, where t~1 is a function symbol of
£(JM1/I) and each ti is a term of £:(~41/I). B) • is an atomic formula other than of the form in A). A): We show A) by the induction on I = Func(fl(h,...,tm(j) ) = x), where Func(P) is the number of the function symbols occurring in a formula P. l = 1: Let .M1/I ~ (i~l(Xl,... ,X,l(j)) = X,l(J)+l)[[p] ]. That is
fJ/I([al],...,
[a,~(j)]) = [a,,(~/+l]
Since Io is a basic morphism,
Ad2 ~ Bas(fl)(xl,... ,X,l(j)) = X,l(j)+l)[I O o p]. Therefore J~2 ~ IF~(~I(x1 . . . . ,Xt~l(j) ) = Xttl(J)+l)[Io# o [p]]. Conversely let J~2 ~ ~'F~(~I(x1 . . . . ,X/~l(j) )
Xpl(j)+l)[IO~ o [p]].
:
From the definition of Io#, .h~ 2 ~ IF#(fjl(Xl,... ,X#I(j)) -~- Xttl(J)+l)[I O o ill. So
J~2 ~ Bas(fjl)(xl,... ,X~l(j )) : XtLl(j)+l)[Io o fl].
(4.1)
On the other hand, from the definition we have
fJ / I ([al] .... , [apl(j)] ) = [ f l ( a l , . . . ,attl(j))] in ~ l / Z . Hence
J~2 ~ Bas(~l)(xl,... ,X~l(j ))
= Xgl(j)+l)[Io ~ o
p"(x) = ~ f J ( a l , . . . , a m ( j ) ) [ p(x)
if x - - x,l(j)+ 1 otherwise
where
fl"l,
(4.2)
4.2. MORPHISMS FOR MODELS OF DIFFERENT TYPES
81
because Io# is a basic morphism. Since Bas(~ 1) is a function, compaxing (4.1) and (4.2), we have
So (a,1 (j)+l) = I0 (I 1 (al, • • • , up, (j))). Hence [a#l(j)+l] _-- [fl ( a l , . . . ,
a m j))].
So
•A/~I/I ~ ( q ( x l . . . . ,Xttl(j)) --XtLl(J)+l)[[p]].
Induction step: We suppose that for each l _< k, Case B) of the claim holds. We consider the case where Func(f I ( t l , . . . , t ~ o ) ) = x) = k + 1. Let J~i/I ~ (i~l(tl,...,tm(j)) = x)[[p]]. Suppose T(t~l(tl .... ,tin(j) ) = x) = {tk~,...,tk= }. Let bkj be the evaluation of each tk¢ in A~I with respect to p. That is
~ / S I= (% = xki)iiPl(ibk,]lkj) l, where each xkj does not occur in ~ l ( t l , . . . ,tin(j) ) = x, and
[Pl([bkjl/kj)(xi)=
( [bkj] if i = kj [p(xi)] otherwise.
Since Func(tj:~ = xkj) < k, from the induction hypothesis,
•M2 ~ I#F(tkj = Xkj)[Io# o ([pl([b~:,l/kj))].
(4.3)
Since
./V[1/I ~ (~l(xl,... ,Xt~l(j)) = Xttl(J)+l)[[p]([bki]/kj)], •2 ~ Bas(fl)(xl,..., x,~), x,l(>~){Io~ o ([Pl(ibkji/kj))l
(4.4)
Since
i~ ( q (tl,.. •, t,,(~)) = x) =
(3Xkl . . .
Xkm)(Bas(~l)(xl,...,Xpl(j
),X.I(j)+I)
^ (/~(I~(t~, = x h) I tk; e T(~I))), combining (4.3) and (4.4), we have J ~ 2 ~ IF~(~l(tl, ... ,tpl(j )) = X ) [ I o~ o
([D]([bkj]lkj))].
From the definition of [p]([bk,]/kj),
3A2 ~ IF# (~I (t1,..., tin(J)) = x)[Io# o [p]]. Conversely, let
2~42p IF#(~l(tl..... tmu)) = x){Io# o [p]].
(4.5)
CHAPTER 4. SIMILARITY OF S Y S T E M MODELS
82
Then from (4.5) there exist % . . . . . ck,, in 342 such that
3,42 ~ Bas(fl)(xl .... , XtLl(j), Xttl(J)+l)[0 ], and for each j = 1 , . . . , m
.hd2 ~ I#F(tk~ ----Xkj)[0l, where
0 = (Io# o M)(ckl,..., ck~lk,,..., kin). (Notice Io# is onto.) Let Io#([dki]) = ckj and [t$] = [p]([c~j]/kj) for j = 1. . . . . m. Then 0 = Io# o [/$]. By the induction hypothesis, we have
31~/I I= (f~(xl,... ,x,l(j )) = x,l(j)+l)[[~]], and 3411I ~ (tkj = Xkj)[[h]]. Therefore
Z41/S V (~l(tl,..., tin(j)) = x)lN]. B): Let an atomic formula other than in A) be R ( t l , . . . ,tp). Suppose
3111I ~ R(h,... ,tp)[[pl]. Then IF#(R(tI,... ,tp))
=
(3Xkl...Xkm)
(Bas(R(Xl,...,Xp) h(A(IF#(tks = Xkl)
I tkj e T(R)))).
By the induction hypothesis and applying A) to the formulas IF# (tkj = xkj), we have 3t2 ~ IF#(R(tl,... ,tp))[Io# o [pl]. Conversely, let
•M2 ~ IF#(R(tl,... ,tp))[lo# o [PII" By a similar discussion,
31UI ~ R(xl,... ,xp)[[hll and
Mils ~ (% = xkj)[[hl]. Hence
31ili ~
R(I1 ....
,
tp)[[~]].
4.3. APPLICATION OF F-MORPHISMS
83
Case 2: ¢ is of the form ¢1 A ¢2.
~ l / z k ¢1 ^ ¢2[b]] iff A41/I b ¢;[[P]] and A41/I b ¢2[[P]] i~ M., k g ( ¢ , ) [ I o~ o [p]] and ~ k g(¢=)[Io~ o b]]
i~.~ I= (~,~(¢~) ^ z~(¢2)[Zo~ o b]] i~ J~¢[2 k -/'F#(¢1 A 02)[Io~ o [pl]. Case 3: • is of the form -~¢.
M~/I I= ~¢[b]]
iff not A4ilI
b
¢[[P]]
i~ not M~ ~ S~(¢)[g o hi] i~ M2 k ~s~(¢)[So~ o b]] i~ M2 b s~(~¢)[so~ o bl]. Case 4: ~ is of the form Yx¢(x).
MI/± b Vx¢(x)[[p]] iff for any [p] A/fl/I ~ ¢[[p]([a]/x)] iff for any [p] A42 ~ I # ( ¢ ) [ I # o [pl([a]/x)] iff for any ~ • M2 M2 ~ IF#(¢)[(Io# o [p])(~)] where I # ([a]) = h, because I o# is a one-to-one correspondence,
M2 k ±~(vx¢(x))[Io~ o [p]], which completes the proof.
4.3
[]
Application of F-morphisms
A typical way to apply the F-morphism concept to concrete system models would be to construct an F-morphism between them. In this section, as an application of F-morphisms we construct an F-morphism of a given finite automaton structure to a Petri net structure, and show the equivalence between a finite automaton and a Petri net. The equivalence metals here that we can show that there is a Petri net that preserves all the properties of a given finite automaton. It is well known that Petri nets can represent finite automata [Peterson 1981]. But the emphasis in this section is on that the use of the F-morphism concept in considering the equivalence between a finite automaton and a Petri net reveals that each property of the finite automaton precisely (in a formal way) corresponds to some property of the Petri net, and the first order sentences satisfied in the finite automaton are all preserved in the Petri net as corresponding sentences transformed by an F-morphism. Thus the F-morphism concept provides a formal meaning of "equivalence" between system models of different types, while the judgment of the equivalence "without F-morphisms" would depend fully on the intuition of a modeler constructing the correspondence between them.
84
C H A P T E R 4. S I M I L A R I T Y OF S Y S T E M MODELS
4.3.1
Equivalence
between
a Finite Automaton
and a Petri Net
In this section we construct an F-morphism of a given finite automaton structure to a Petri net structure, and show that all the properties holding in the finite automaton also hold in the Petri net.
Definition 4.3.1 (Finite A u t o m a t o n S t r u c t u r e ) A finite automaton structure F A is the following system model. FA =< A u BUC;A,B,C,¢,p > where A, B, C ¢, p
unary relations binary functions such that ¢(a,b) E C, i f a E C and b E A, ¢(a, b) = a, otherwise; and p(a,b) e B , ifaeCandbeA, p( a, b) = a, otherwise.
The conditions on a • C or b ¢~ A for ¢ and p are imposed only to make the functions ¢ and p total, since the first order language we use does not allow partial functions. However, since we will restrict the sentences to the extent as defined later, when we describe the properties of system models, we can regard ¢ and p intrinsically as ¢ : C x A --* C and p : C × A ~ B. D e f i n i t i o n 4.3.2 ( P e t r i N e t S t r u c t u r e ) A Petri net structure P N is the following system model. P N =< P U T U N ; P , T , I , O , N > where
P,T N I,O
: unary relations : the set of natural numbers : the set of constants corresponding to N C PxTxN
P denotes the set of places and T the set of transitions. I(p, t, n) means that there are n arcs from the place p to the transition t. O(p, t, n) means that there axe n arcs from the transition t to the place p. There are some ways to construct P N that is considered to have an equivalent structure to F A [Peterson 1981]. Here following Peterson with some modification, we define P N considered as equivalent to FA. Then our aim is to construct a~l F-nmrphism between F A and P N , and to show that the constructed P N preserves all the properties satisfied in FA.
4.3. A P P L I C A T I O N OF F - M O R P H I S M S
85
D e f i n i t i o n 4.3.3 Given a finite automaton structure FA.
We define the corre-
sponding Petri net structure P N as follows. P N =< P U T O N ; P , T , I , O , N
>
where P T I
= = =
CUADB; {tIIiE(CxA) IiOIo
OAOB};
where
11 0
=
=
~-
{(p, ti, 1)] i = (c,a) E C x A and (p = c or a)} U{(p, ti,1)l i = P E B}, {(p, ti,0)] (p, ti, 1) ¢ I i , p E P, ti E T};
01 U Oo where O1 = { ( p , t , 1)[ i = (c,a) E C × A and (p = ¢(c,a) or p(c,a))} U{(p, ti,1)l i = p E A}, {(p, ti,0) I (p, ti, 1) • 0 1 , p E P, ti e T}. Oo =
0/0
1/1
1/1
0/0 F i g u r e 4.3 A Finite Automaton E x a m p l e 4.3.1 Let F A =< A U B U C; A, B, C, ¢, p >, where A = B = {0,1} and C = {cl,c2}. Fig.4.3 illustrates the state transition of FA. The graph of the
corresponding P N defined in Definition 4.3.3 is depicted in Fig.4.4. Then we can define an F-morphism between F A and the corresponding P N .
C H A P T E R 4. S I M I L A R I T Y OF S Y S T E M MODELS
86
F i g u r e 4.4 A Petri Net D e f i n i t i o n 4.3.4 Let F A and P N be as in definitions 4.3.1 and 4.3.3 respectively. An F-morphism I ----< Io, IF >: F A ~ P N is defined as follows. Io Ir(A(x)) IF(B(x)) IF(C(x))
: = = =
z F ( ¢ ( x , y ) = z)
=
((zF(C(x)) A ZF(A(y))
=
--* (3t E T ) ( I ( x , t , 1) A I ( y , t , 1)AO(z,t, 1) A IF(C(z)))) A(-~IF(C(x)) V -~IF(A(y)) --~ z = x)); ((IF(C(x)) A I F ( A ( y ) ) --* (3t e T ) ( I ( x , t, 1) A I(y, t, 1)AO(z, t, 1) A IF(B(z)))) A(-~IF(C(x)) V -~ir(A(y)) --* z = x)).
I F ( p ( x , y ) = Z)
the inclusion map; (P(x) A (3t E T)((Yp e P ) ( I ( p , t , 0) A O(x, t, 1)))); (P(x) A (3t e W)((Vp e P ) ( O ( p , t , 0) A I(x, t, 1)))); (P(x) A-~IF(A(x))A-~IF(B(x)));
This definition clearly satisfies the condition required for F-morphisms. Also we can see, as the following lemmas show, that the image of the above F-morphism preserves the structure of FA. Lemma 4.3.1
A - - {alPN ~ IF(A(x))[a],a E IPNI}; B = {blPN ~ IF(B(x))[b],b E IPNI}; C : {cIPN ~ IF(C(x))[c],c e IPNI}, where [PN[ is the base set of P N .
4.3. APPLICATION OF F-MORPHISMS
87
P r o o f : We prove A = {aIPN ~ IF(A(x))[a]}. Let a • A. Since Io is the inclusion map, Io (a) = a • P. By the definition of a basic morphism, if F A A(x)[a], then P N ~ IF(A(x))[a]. Indeed from the definition of I and O, it follows that if a • A, then I(p,t~,0) for any p • P and O(a, ta,1). So we have P N IF(A(x))[a]. Conversely let P N ~ Ir(A(x))[a]. Then
P(a) A (3t~ • T)((Vp • P)(I(p, ti,O) h O(a, ti, 1))) holds in PN. P(a) implies a • C U A U B . From (Vp • P)(I(p, ti,O)), we have i • A, hence, a • A by O(a, ti, 1). The rest can be similarly proven.
[]
L e r n m a 4.3.2
{ (al, a2, a3)JFA ~ (¢(x, y) = z)[hi, a2, a3]}
=
!
I
I
I
!
I
l
I
l
(al,a2, a3)lPN ~ Bas(¢)(x,Y,z)[al,as, a3], al, a2, a3 • P};
{(bl, b2, b3)IRA ~ (p(x, y) = z)[bl, b2, 53]}
-
--
Slb' 3 • p}. 1 ~, 1 , b'2 , bt3)iPN ~ Bas(p)(x,y,z)[b~,bS,bt3],b~,bS, b~
P r o o f : Let F A ~ ( ¢ ( x , y ) = z)[al,a2, a3]. Then ¢(al,a2) = a3. Case: al • C and a2 • A. From Lemma 4.3.1, P N ~ I f ( C ( x ) A A(y))[al,a2]. Let i = (al,a2) • C × A. From Definition 4.3.3, (al,ti, 1), (a2, ti, 1) • I and (¢(ai, as),ti, 1) • O. Since
¢(al, a2) = a3 • C from the definition of FA, we have P N ~ Bas(¢)(x, y, z)[al, a2, a3] by Definition 4.3.4. Case: a l ¢ C or as ¢ A. From the definition of FA, al = a3. And from Lemma 4.3.1, P N ~ -~If(C(x) AA(y))[al, a2]. So we have P g ~ Bas(¢)(x,y, Z)[al, a2, a3]. Conversely let P N ~ Bas(¢)(x, y, z)[a~, aS, hi3], where at, a~, a~ • P. Case: a t • C a n d a S • A . Then
P N ~ (3t • T)(I(x, t, 1) A I(y, t, 1) A O(z, t, 1) A IF(C(z)))[a~, a~2,a~3] From the definition of I, ti E T for i = (at,aS) satisfies the above formula. Due to a~ • C and the definition of O, O(a~3,ti, 1) implies ¢(a~, aS) = a~. So we have
F A ~ ( ¢ ( x , y ) - - -.)[a.' as,' a3]. ' Case: a t ¢ C o r a
S~A.
CHAPTER 4. SIMILARITY OF SYSTEM MODELS
88 Then
PN ~ (-~C(x) V-~A(y) --* z = x)[a~, aS, a~]. .~(ar1, a t2J = a~ = a~. Hence So we have a t = a~. From the definition of FA, we have v,~
FA ~ (¢(x, y) = z)[a~, a S, a~]. The rest can be similarly proven.
[]
Our aim is to investigate what properties of FA are preserved by I. To this end, we need to define many-sorted sentences. Definition 4.3.5 ( M a n y - S o r t e d F o r m u l a ) Let L: be a first order language. A
many-sorted formula is a formula of L: and defined recursively. (1) An atomic formula is a many-sorted formula; (2) if ¢ and ¢ are many-sorted formulas, then (¢ A ¢) and -~¢ are many-sorted formulas; (3) if Q is a unary relation symbol of £ and ¢ is many-sorted formula, then (Vx E Q ) ¢ is a many-sorted formula. A many-sorted formula whose variables are all bound is said to be a many-sorted
sentence. Even if we restrict sentences for the description to many-sorted sentences, the capability of the description is not less than that with ordinary first order sentences[Enderton 1972]. The following theorem shows a typical type of equivalence between PN and FA. T h e o r e m 4.3.1 Let I =< Io,IF > be the F-morphism defined in Definition ~.3.~.
Then for any many-sorted sentence • of ~(FA), FA ~ ¢ if] P N ~ IF(¢). P r o o f : First we define the extended models of F A and PN obtained by adjoining to FA and P N a new constant for each element in C U A U B as follows.
< FA, U > = < C U A U B ; A , B , C , ¢ , p , U >, < PN, U > = < PUTUN;P,T,I,O,I~I,U >, where < CUAUB; A, B, C, ¢, p > and < PUTUN; P, T, I, O, N > are the underlying models FA and P N respectively, and U is the set of new constants. Since FA and P N are reducts of < FA, U > and < PN, U > respectively, each many-sorted sentence of FA (or PN) is also a many-sorted sentence of < FA, U > (or < PN, U >). Then for any many-sorted sentence (I) of FA we have
FA~
iff < F A , U > ~ ( ~
4.3. APPLICATION OF F-MORPHISMS
89
and PN~IF(O) iff < P N , U > ~ I ~ ( O ) , where the basic interpretation function
Bas* of I~ is an extension of Bas, that is:
Bas*(P) = { Bas(P)p
i f iPf PEe U£(FA)
So it is sufficient to show that for any many-sorted sentence ¢ of L:(<
FA, U >)
~ ¢ iff ~I~(¢). We show this by induction on the length of 0. (1) q, is an atomic sentence. Let q) be R i ( t l , . . . , t~(0 ), where t l , . . . , t~(i) are closed terms. We should notice that R i is A or B or C or ---, and so A(i) < 2. The proof depends on the munber of the function symbols occurring in Ri. We define Fn (P) for a formula P to indicate the number of the function symbols other than the constants occurring in P. We show the claim in the case of atomic sentences by induction on Fn(¢). The steps for the induction consist of ( a ) F , ( ~ ) = 0 , ( b ) F , ( ¢ ) = 1 and (c)Induction Step. In each step we have two subcases:
Case 1: R i is =; Case 2: R i i s A o r B o r
C.
( a ) F ~ ( ~ ) = o.
Case a-l: If R i is --, by the fact that Bas*(=) is =, the conclusion clearly holds. Case a-2: Ri is A or B or C, and t E U. FA, U > ~ Ri(t). By the definition, (3x)(Bas*(Ri)(x) A I ~ ( x = t)). Since t E U, we have I ~ ( x = t) = (x = t). Let t d be the denotation of t in < FA, U > and < PN, U >. Then t d E P~ C P. Therefore < FA, V > ~ (Ri(x) A (x = t))[td]. Since, by Lemma 4.3.1, Bas*(Ri)(t a) holds in < PN, U >, we have < PN, U >~ (Bas*(Ri)(x)AI~(x = t))[td], and so < PN, V > ~ I~(Ri(t)). Conversely suppose < PN, U > ~ I~(Ri(t)). By the definition of I~, there exists a denotation t d of t in C U A U B such that Bas*(Ri)(t d) holds in < PN, U >. We first show the only if part. Suppose <
I~(Ri(t) ) =
CHAPTER 4. SIMILARITY OF SYSTEM MODELS
90 By L e m m a 4.3.1,
Ri(t d)
holds in < FA, U >.
Hence we have < FA, U > ~
( 3 x ) ( R i ( x ) A (x = t ) ) , which is equivalent to < FA, U > ~ R i ( t ) . (b)F,~(cI,) = 1.
Case b-l: R i is = . We denote ¢ or p by a symbol f . Then we have to show t h a t for any ta, t2, t3 E U
< FA, U > ~ (f(tl,t2)
=
t3)
iff < PN, V > ~
I*F(f(tl,t2 ) =
43). By the definition
of I F and L e m m a 4.3.2, this is clear.
Case b-2: R i is A or B or C. T h e n we can write (I) as l ~ i ( f ( t l , t 2 ) ) , where f is ¢ or p and t l , t 2 E V.
< FA, V > ~ Ri(f(tl,t2))
iff
< FA, U > ~ (Ri(t3) At3 = f(tl,t2)) for some t3 E U < PN, U > ~ (Bas*(Ri)(t3) n I}(t3 = / ( h , t 2 ) ) (from Case 1) < PN, U > ~ (3x)(Bas*(Ri)(x)
iff
A-rb(x = S(t~, t2)) < PN, U > ~ Ib(Ri(f(tl,t2)) ).
iff iff
(c) Induction Step: Suppose the conclusion holds for Fn(¢) < k. We prove it for Fn(O) = k.
Case c-1: ~ is t l = t2, where tl and t2 are any closed terms of £ ( < FA, U >). We first prove the following lemma. Lemma. Let t be a closed term of £ ( < FA, U >). For any a E ]PN] if I ~ ( x = t)[a] holds in < PN, U >, then a E CU A U B.
Proof. By induction on Fn (x = t). The case F n ( x -- t) = 0 is trivial. Let t be f(wl,w2), where f is ¢ or p and Wl and w2 are closed terms. T h e n I ~ ( x = t) = (3yzy2)(Bas(f)(yl, Y2, x) A I ~ ( y l = wl) A I ~ ( y 2 = w2)). If Fn(x = t) = 1, then wl,w2 e U, I ~ ( y l = Wl) = (Yl = Wl) and I ~ ( y 2 = w2) = (Y2 = w2). So Bas(f)(wl,w2,x)[a] holds in < PN, U >. From Definition 4.3.4, we have a E C U A U B. If F n ( x = t) -
k, then F n ( y l = wl) < k and Fn(y2 = w2) < k.
By the
induction hypothesis, there exist bl, b2 E C U A U B such t h a t I ~ ( y l = wl)[bz] and I ~ ( y 2 = w2)[b2] hold in < PN, U >. L e m m a 4.3.1, we have a E C U A U B.
Hence from the definition of Bas(f) and []
4.3. A P P L I C A T I O N OF F-MORPHISMS
91
Now we return to the proof of Case 1-c. It follows that
< FA, U > ~ t l
iff
--t2
there exist some Ul, u2 6 U such that < FA, U > ~ (ul = u2 h u l = tl hu2 = t2).
On the other hand, we have
< P g , U > ~ I*F(tl = t2)
iff
< P N , U > ~ (3XlX2)(xl = x2 A / ~ ( X l --~ tl)
iff
(from the above lemma) there exist ul, u2 6 U such that < P N , U > ~ (Ul = u2 A I~(uz = tl)
AI~(x2 = t2))
AIF(u2 = t~)). So it suffices to show that
< FA, U > ~ (ul = t l ) iff
~I~(ul=tl).
Let tl be fj (sl, s2), where Sl and s2 are closed terms. Then < FA, U > ~ (Ul = tl) iff there exist cl, c2 6 U such that
< FA, U > ~ (ul
= fj(c1, c2) A (c 1 ---- 81)/k (c 2 = 82) ).
iff there exist Cl, c2 6 U such that
< P N , U > ~ (I>(ul = fj(cl, c2)) A I~(cl = sl) A I~(c2 = s2)) (by the induction hypothesis), iff < P N , U > ~
(3XlX2X3)((xa = Ul) A B a s ( f ) ( x l , x 2 , ul) A I ~ ( x 1 ---- Sl) A
I~(x2 =2)) (by the lemlna) iff < P N , U > ~ I}(Ul = tl).
Case c-~. ¢ is R(t), where R is A or B or C, and t is a closed term of £ ( < FA, U >). Then
< FA, U > ~ R(t) iff < FA, U > ~ (R(u) A u = t) for some u 6 U, where u d = t d (i.e. the denotations of u and t axe equivalent). On the other hand,
< P N , U > ~ I~(R(t))
iff iff iff
< P N , U > ~ (3x)(I~(R(x)) A I ~ ( x = t)) < PN, U > ~ I ~ ( R ( u ) A I~(u---- $)) for some u 6 U < P N , U > ~ I ~ ( R ( u ) ) and < P N , U > ~ I~(u = t).
CHAPTER 4. SIMILARITY OF SYSTEM MODELS
92
By the induction hypothesis, we have
< FA, U > ~ R(u) iff < PN, U > ~ I~(R(u)). Since we have already shown that
< FA, U > ~ ( u = t ) iff < P N , U > ~ I ~ ( u = t ) , we have
< FA, U > ~ R(t) iff < PN, U > ~ I;~(R(t)). (2) (I) is ¢1 A ¢2.
< FA, U > ~ ~ iff iff iff iff
< < < <
FA, U > ~ Pg, U >~ PN, U > ~ PN, U > ~
~)1 and < FA, U > ~ ¢2 I,~(¢1) and < PN, U > ~ I~(¢2) I~(¢1 A¢2)
I~(¢).
(3) (I, is -~¢.
< FA, U > ~ ~ iff iff iff iff
(4) • is (Vx
6
not < F A , U > ~ ¢ not < P N , U > ~ I ~ ( ¢ ) < PY, V >~/;~(-~¢) <
Pg, V >~ I~(~).
R)¢(x) where R is a unary relation symbol.
D~
iff iff iff iff iff
for any a E R, < FA, U > ~ (¢(x))[a] for any t 6 U, if the denotation td of ~ is in R, then < FA, U > ~ ¢(t) for any $ 6 U, if t d 6 R, then < PN, U > ~ / ~ ( ¢ ( t ) ) (by the induction hypothesis) for any t d 6 R, < PN, U > ~ I~(¢(x))[t d] for any t d 6 {alPg ~ IF(R(x))[a]},
< PN, U > ~ I~(¢(x))[t d] iff iff which completes the proof.
(by Lemnla 4.3.1) < P g , U > ~ (Vx)(I~(R(x)) ~ I;~(¢(x))) (since I;~(R(x)) = IF(R(x))), < PN, U > ~ Ib(¢),
[]
This theorem implies that the structure of FA is embedded in P N constructed in Definition 4.3.3, and all the properties of FA are preserved there. We should notice that the dynamic behavior of P N by the transition of marking is implied by the relation O of PN, which can also represent the firing of the transitions.
4.3. A P P L I C A T I O N OF F - M O R P H I S M S 4.3.2
Equivalence of Behavior Finite Automaton
between
93 a Bounded
Petri Net and a
P N formulated in Definition 4.3.2 has no rule on firing. In this subsection we formulate a Petri net with marking in the way to specify the transition of marking by firing. Then we discuss the equivalence of behavior between a Petri net with marking and a finite automaton. Through the subsection we deal with only b o u n d e d Petri nets. D e f i n i t i o n 4.3.6 ( P e t r i N e t w i t h M a r k i n g )
A Petri net with marking P N m is
defined as follows.
P N m = < P U T U N U M; P , T , I , O , IV, M , 5 >
where P,T,I,O,N,N M
: as in Definition 4.3.2 : a unary relation, M = { ] l f : P ~ N } : a binary function is an a r b i t r a r y extension of ~ : M x T ~ M
M stands for the whole set of the marking of PNm and ~ stands for the transition of the marking by firing. T h e behavior of P N m is all represented by ~. Let the model < T, M, ~ > be a reduct of P N m , which describes ~, and F A to the state transition system < A, C, ¢ >. These reductions do not lose the generality of the consideration on the equivalence of behavior. Let
PN~ =< TU M;T,M,~ > be a reduct of P N m and
F A ¢~= < A U C ; A , C , ¢ > a reduct of F A . T h e n the following theorem holds. Theorem
4.3.2 If a Petri net with marking P N m is bounded, then there exist F A ¢
and an F-morphism I : PN~m ~ F A ¢ such that
P r o o f : It is clear if we construct F A ¢ by A -- T, C = M and ¢ = ~.
[]
This theorem implies t h a t all the sentences about P N *m are preserved in F A ¢.
94
C H A P T E R 4. S I M I L A R I T Y OF S Y S T E M MODELS
4.4
Similarity and Analogy
Analogy or metaphor is a fundanlental intellectual activity whose basic component is to recognize the "similarity" in the sense that the similarity is found and explicitly used to solve a target problem. System models are represented through one's systems recognition that actually belongs to intellectual activities, and F-morphisms provide similarity between system models. These "facts" suggest a deep relationship between F-morphisms and analogy. In this section, briefly surveying some researches about analogy or similarity mainly in the fields of cognitive science and artificial intelligence, we point out a relationship between F-morphisms and analogy. Some general frameworks for considering analogy have been proposed. For example, Anderson and Thompson [Anderson et al.1989] provided the following steps in which analogical reasoning proceeds. 1. Obtain a goal problem. 2. Find an example similar to the problem. 3. Elaborate the goal. 4. Generate a mapping between the goal and the example. 5. Use the mapping to fill in the goal pattern. 6. Check the validity of the solution. 7. Generalize and form a summarization rule. In the above steps, a goal problem and an example are often called a target and a source (or base) respectively. We should notice that these steps are a conceptual framework for dealing with analogy. In analogical reasoning actually carried out in our brains, the second step of finding a similar example and the forth step of generating a mapping between the goal and the example would be performed at almost the same time. An essential step is to generate a mapping between the source and the target. This mapping should preserve a class of some properties of the source in the target. These features of analogical reasoning show that F-morphisms provide a general formalization for analogy. So far many researchers have formalized the similarity in analogical reasoning.
4.4.
SIMILARITY
AND ANALOGY
95
Tversky [Tversky 1977] represented a source and a target respectively as an object with a set of some features. Let A = { a , b , . . . } be a set of objects, trod {A, B . . . . } sets of some features. Similarity between objects a and b is expressed as a function of common and differential paa'ts of features: s(a, b) = F ( A tJ B , A - B , B - A).
Tversky characterized this similarity s(a, b) as a type of metric functions. However this definition provides no clear relationships between the preservation of properties of objects and the mapping for the similarity. The structure-mapping theory (in short SMT) developed by Gentner [Gentner 1983] includes a mapping between the source and the target as a central concept of analogy. SMT also includes some characterizations for the preservation of properties of the source by some formal constraints of the mapping or representation of the source and the target. It is no doubt that SMT has greatly contributed to the succeeding progress of the research of analogy, especially in cognitive science. However SMT does not deal with the diversity of the representation, on which the formal constraints of the mapping completely depend. In this sense SMT is far from sufficient formal theory of analogy. Indurkhya [Indurkhya 1986] developed a formal theory of metaphor, in which similarity was defined as a partial mapping between domains. His theory is developed in a model theoretical fraanework just like the framework of F-morphisms. However the framework of F-morphisms is more general to consider the preservation of properties of system models. Haraguchi [Haraguchi 1986] formalized analogy as a partial identity in a formal way sharing some features with Indurkhya's theory. Distinctive is that he introduces the concept of maximal analogy and gives a procedure of calculating it. The formalization of similarity by F-morphisms in LAST sufficiently includes the essential parts of the above theories. F-morphisms are more general to the above theories in the sense that F-morphisms can be defined between system models of different types, although the concepts of morphisms involved in the above theories are an extension of homomorphisms. However we should notice that our framework of F-morphisms as well as other theories of analogy, for the present, is concerned with only the step of using the mapping to fill in the goal problem. The most difficult parts of the problems of analogy should be attacked: for example, how to find a source similar to the target problem, how to elaborate the target problem or how to generate a mapping between the target and the source. These problems will occupy part of a central position in developing theory of F-morphisms.
Chapter 5
CANONICAL SYSTEM MODEL AS REALIZATION This chapter is concerned with the problem of realization that is to identify a systems representation of a certain pre-structured set of observational or empirical data. This problem has been one of central issues in systems theory or algebraic specification [Zadeh et al.1969, Mesarovic et al.1989, Ehrig et al.1985]. The realization usually means construction of transformation operators from the input-output pairs of a given system into, for example, a state space representation. The collection of input-output pairs is often regarded as being "pre-structured," while the state space representation is well structured. We will formulate realization by the concept of tmiversality in the category theoretical sense. Category theory can deal with the universality in an explicit and rigorous way so that it describes intuitive feelings for the realization as a precise concept and makes it operational. In systems theory various types of universality have been developed so far [Mesarovic et a1.1989]. An important result the universality provides is the minimality of a system model as a universal element in the sense that any other system model in the class of system models having a given structure (£:; ~) holds the intrinsic properties of the minimal model. In this sense the minimal model is the most general one among the class of system models having the same structure as the minimal model. We will construct this minimal model as the canonical system model, and show that it is universal in the sense of category theory. Besides state space representations in control theory, the way of construction of a structured system nmdel as a universal element can be also found in initial semantics in algebraic specification. It is in the fields of information systems that
the realization as universality would have been getting important rather than in control theory.
CHAPTER 5. CANONICAL SYSTEM MODEL AS REALIZATION
98
We will demonstrate two examples of canonical system models concerning stationary systems and algebraic specification. The canonical system model of stationary systems, in a sense, accords with the stationary systems that is given by stationarization of a causal system by Nerode realization. This implies that the canonical system model as a minimal model of systems plays aal important role in systems theory. The canonical system nmdel of an algebraic specification of an information system occupies an essential place as initial semantics for abstract data type. In this chapter we will show the universality of the canonical system model, the universality which is based on homonmrphism. We can use E-homomorphism to define "free E-structure" that essentially provides a universal element, aa:d show the freeness of the canonical system model. (See Chapter 8.) Through the chapter, constants play a special role. Therefore we explicitly express the constants contained in a system model or a language. Thus a system model A~ is expressed as
< M;{Rili E I } , { f j l j e J},{cklk e K} > where each ck is a constant and no fj, j E J is a constant. The corresponding language £:(2t4) is also expressed as < {Riti e I},{fjlj e J}, {cklk G K} > . Then the type of a system model A~[ or a language £:(A4) is < K, •, # >, where
# : K - - - , N +. We use some notions in category theory: definitions of category, functor and universal map. Appendix supplies the definitions used in this chapter.
5.1
R e a l i z a t i o n as U n i v e r s a l i t y
The realization problem is usually defined as follows: Given a set of input-output pairs as observational data, find the conditions that they must satisfy so that a system model with some specific structure can be constructed and the given set of input-output pairs can be embedded into the input-output set of the model. The above specification involves some aspects on realization. In this section we concentrate on the universal property of realization. The (sub-)problem of realization we deal with in this section is to find transformation as a functor from a
5.1. REALIZATION AS UNIVERSALITY
99
category of observational data to a category of system models with some specific structure, and to construct the most general model in the category of structured system nmdels as a universal solution. First, we define a category of observational data {c k [k • K} as follows.
Definition 5.1.1 (Category STA" o f Observational Data) Let L be a language having only constants with a type K, that is, L = {{ck[k • K}}. The category
STa- is constructed as follows: 1. the class of objects of STK, Ob(STI~.), consists of all models for £; 2. a set of morphisms of STK, Mor(54,54t), consists of all mappings of 54 to A4 1 that maps the constants to the corresponding ones (Notice that these mappings are homomorphisms from 54 to 541, although there is no function symbol and predicate symbol in L.); 3. composition is a usual one of functions. While an object of category STI,- is a system model having a structure with only the constants, we can expand it into a larger structure including E. Then the constants {Ck} are considered as observed source data. In this stage the constants as data are not structured even as input-output pairs. A central concern in realization is to interpret them in an expanded model for L. D e f i n i t i o n 5.1.2 ( E x p a n s i o n ) Let A4 be a model for L, and A4 ~ ~, where
P, C Sent(L). Let L C L*, ~ C I]* C Sent(E*), and A4* be a model for L* with the same base set M as AA interpreting new symbols of L*. If 54* ~ ~*, then we call 54* an expansion of 54 and A4 a reduct of A4*. (See Definition 3.1.3.)
Definition 5.1.3 ( E x p a n s i b l e Class) Let Mod(~) = {A41A4 is a model for L and A4 ~ ~}. The expansible class, EX(£, )2,,£*, ~*), of Mod(~) for L C L* and C ~* C Sent(L*) is defined by
EX(L,r.,L*,~*) =
{./t,f • Mod(~)]there is an A4* such that A4* ~ ~* and jr4* is an expansion of A4}.
When we assume that L = {ck[k e K}, then E X ( £ , ¢ , L ' , Z ) ,
where ¢ is the
empty set and ~ is an arbitrary consistent set of sentences of L t D L, is considered as a class of models that can interpret the observational data {ck]k E K} by the structure (L ~, ~). When we let (L ~, ~), for example, be an I / O structure, EX(L, ¢, L t, ~)
CHAPTER 5. CANONICAL SYSTEM MODEL AS REALIZATION
100
is the class of models interpreting the observational data as I/O data, where I / O structure is (£I/O; ~I/O), a pair of £ I / o (composed of an input relation symbol X, an output relation symbol Y and a system relation symbol S between X and Y) and a sentence ¢I/o - (k/xy)(S(x,y) --* X ( x ) A Y(y)) representing the property that a system model consists of an input element and an output one. It is quite natural and important that we consider the nfinimal model in a given expansible class. Definition 5.1.4 ( C a t e g o r y STrE of S t r u c t u r e d models) Let £ be a language with a type 7-, and P, a consistent set of Sent(£). The category STrE is constructed as follows: 1. the class of objects of ST~P., Ob(STr~), consists of all models for £ satisfying
2. a set of morphisms of STr~, Mor(A4,A4'), consists of all honmlnorphisms from A/I to A4'; 3. composition is a usual one of functions.
STIr- and STr~ are obviously categories due to the definitions. Definition 5.1.5 ( F t m c t o r G : STrP, ---*STK) Let r = < K, A, # > be a type, and
C Sent(E) consistent. The functor G : S T ~ ~ STK is defined by: 1. for each A4 = < M; {Rill 6 I}, {fjlJ 6 J}, {cklk • K} >• Ob(STr~), G(A4) = < M, {cklk • K} >60b(STK); 2. for each f • Mor(STK),
G(f) = f • Mor(STr~). This functor is called a forgetful functor. There is a following relationship between the expansible class and functor G.
Proposition 5.1.1 E X (~:, ¢, ~', P.) = G(ST.P,),
where £ = {cklk 6 K} and f ' is a language of a type 7- including the constants {cklk 6 g } . P r o o f : Let A4 = < M; {ck} > 6 EX(£, ¢, £', I~). By the definition of EX(£, ¢, £',X~), there exists a system model A4* = < M;{Rili 6 I},{fjlj 6 J},{ck[k 6 K} >6 Mod(~) such that A/f* is an expansion of A4.
5.1. REALIZATION AS UNIVERSALITY
101
So
a(~4*) =< M; {cklk E K} > = A~t. Therefore
M ~ a(STWO. Conversely, let ¢~4 E G(STr~). Then there exists a system model ,~4" = < M; {RiIi E I}, {fjlJ E J}, {cklk E K} > such that AA = G(AA*). Since A~t* e Mod(~), we have A4 E EX(£, ¢, fJ, ~).
[] From the aspect of universal property the realization problem can be formulated as a problem to find a universal element in G(STr~). Definition 5.1.6 (Realization) Let G : STr~ -* STK be a functor as in Definition 5.1.5. If there is a functor F : STK ~ STr~ such that (u,F(C~)) for any
Cc, = < {ck]k E K}; {ckik E K} >E Ob(STK) is a universal map for Cc~ with respect to G, where u : C~k --* GF(C~) is an insertion, then the system model F(C~) is a realization for Cck. From the definition of a universal map, if F(Cck) is a realization for Cck, for every system model .~t ~in STr~ and every homomorphism f of Cck to G(A4 r) there exists a unique homomorphism f of F(Cck) to A~C such that f = G(f) o u. R e m a r k . If G(2~#) has a fixed interpretation of the constants Cck, then there is exactly one homomorphism of Cck to G(A~t~), and accordingly only one homomorphism of F(C~k) to .M ~ exists.
F
G
F i g u r e 5.1 Realization as Universality
CHAPTER 5. CANONICAL SYSTEM MODEL AS REALIZATION
102
For the consideration of the minimaiity, we introduce an order on the class of objects of STrE. D e f i n i t i o n 5.1.7 For any A4, A4 t • Ob(STrE), "< is defined by: A4 _ A4 t A4~A41
iff iff
there is an f such that f : A4 --* A4 t is a homomorphism fl4--
It is easy to show that this order is a partial order. The minimality of F(Cck) with respect to this order is given by the next corollary. C o r o l l a r y 5.1.1 F(Cck) is the minimal in Ob(STrE) with respect to the order -<
of Definition 5. i. 7. P r o o f : It is clear due to Definition 5.1.6.
[]
F(Cck) is the minimal in EX(f~, ¢, L:t, E) under the santo condition as in Proposition 5.1.1 with respect to a similar order to _.
5.2
Canonical System Model and its Universality
A canonical system model is a model originally used in the Henkin's style proof of the completeness theorem of the first order logic [Bridge 1977]. We will show in this section that the canonical system model is a realization for any object of category
STI~., and is the minimal model in the sense that it is the most general structure. D e f i n i t i o n 5.2.1 ( C a n o n i c a l S y s t e m M o d e l CAN(E)) If ~ C Sent(E) is consistent a n d / : contains at least one constant, the canonical system model determined by E is:
CAN(E) =< CT; {Rdi E I}, {fj[j E J}, {ckl k E K} >, where CT = {It]It is a closed term }; tl,~t2 < [tl],...,[t~(0 ] >E Ri
if and only if if and only if
E~tl--t2; ~ b Ri(h,...,t~(i));
fj([tx],..., [t•(j)]) = [fj(tl,... ,t,(j))],j • J; ck = [Ck], k • K. We should notice that it is known [Bridge 1977] that if E is complete, CAN(E) satisfies Z, but it does not necessarily satisfy E. We illustrate a simple example.
5.2. CANONICAL SYSTEM MODEL AND ITS UNIVERSALITY
103
Example 5.2.1 Let £ = < +, S, 1 >, where + is a binary function symbol, S a binary relation symbol, and 1 a constant. Let E = {¢1, ¢2, ¢3}, where ¢1 ¢2
= -
+3 -
Vxy(x+y=y+x), Vxyz((x+y)+z---x+(y+z)),
Vx(S(x,x+l)).
Every term is a combination of the constant 1, + and parentheses. By ¢1 and ¢2, any two terms with the same number of 1 axe derived to be equivalent even if the occurrences of parentheses are different: that is, for example,
E I'- ( 1 + 1) -I- ( 1 + 1) = 1-1- ((1-t- 1) + 1). So we denote by n any closed term including n of l's: 1 + 1 + ... + 1. The base set CT of CAN(E) is the set of equivalence classes of n for every
n 6 N: CT = {[n]ln 6 Y}. The function + is defined by [n] + [m] = [n + m]. The relation S is defined by < [n],[n + 1] > 6 S iff E ~- S ( n , n +
1): S = {<
[n], [n + 11 > in e N}. Thus the canonical system model CAN(E) is < CT; +, S, [1] >. We have the next theorem that shows the canonical system model as a realization. T h e o r e m 5.2.1 Let C A N ( E ) is the canonical system model determined by E with
constants Cck. I~ C A N ( E ) 6 0 b ( S T r E ) , then CAN(E) is a realization for Cck, where Cc~ = < {ck[k 6 g } ; {c~lk 6 K} > 6 0 b ( S T K ) .
P r o o f : We define a functor F : STK ~ ST~E by F(A,~c) = CAN(E) for any
J~4c 6 0 b ( S T K ) and F ( f ) = idcAy(~) for any f 6 Mor(STrE). Then from the definition of a realization, we have to show that (u, CAN(E)) is a universal map for Cc~ with respect to G, where and u : Cc~ ~ G(CAN(E)) is an insertion. For any A / / 6 0 b ( S T ~ E ) and any f : Cck --* G(¢~4) we construct f~ : C A N ( E ) --~ j~4, where ~A = < M~{S~Ii 6 I},{9jl j E J},{dklk 6 K} >, as follows: (1) if t = Ok, then f~(t) = f(ck); (2) if t = fj ( t b . . . , t,(j)), then f~(t) = gj(fc(tl),..., fc(t,(j))). Since the commutativity of the following diagram is clear, it is sufficient to show that f¢ is a unique homomorphism.
CHAPTER 5. CANONICAL SYSTEM MODEL AS REALIZATION
104
ill Cck
P, G(CAN(Z ))
CAN( Z )
G(r¢)
f¢
G(M)
M
D i a g r a m 5.1 First we show that f~ is a homomorphism. < t:,...,t~(i) >E Ri iff ~ t- R i ( t l , . . . ,t~(i)) (by the definition of CAN(Z)) iff A4 ~ 1~i(~1,... ,t~(i) ) (by the completeness theorem) iff <(fc(tl) . . . . . fc(gA(i)) > e Si (by the definitions of the denotation a~ld ~); if fj(t: . . . . , t~(j)) ----t, then fc(t) = gj(fc(tl),..., fc(t~(j))) by the definition of f~; if c k = t, then f~(t) = f(ck). Therefore fc is a homomorphism, that is, f¢E Mor(CAN(~), )v4). Next we show the uniqueness of re. Suppose that f = G(f~) o u for some f~ : CAN(~,) ~ N[. Since
/'(¢k) = G(f')(¢k) = (G(f:)
o
,,)(Ok) = / ( O K ) =/~(¢k),
so for any Ck,k E K, f~ takes the same value as ft.
If fc(~j) = f~(tj) for any
= f j ( t : , . . . ,tz(j)) E CT, then fc(t)
=
g j ( A ( t l ) . . . . ,fc(t.(;)))
=
gj(f~(t:),...,f~(t.(~)))
=
f~(t)
Therefore
f~=:'~. This completes the proof.
[]
Theorem 5.2.1 directly yields the following. C o r o l l a r y 5.2.1 If CAN(E) ~ E, then CAN(E) is the minimal in Ob(STrE) with
respect to the order -< of Definition 5.1.7.
5.3. CANONICAL S Y S T E M MODEL OF STATIONARY SYSTEMS
105
Thus G ( C A N ( Z ) ) is the minimal in E X ( £ , ¢, £~, ~). As explained in Chapter 2, the essential meaning of minimality is that the minimal model contains neither junk nor terms not implied from I3. For example, consider Example 5.2.1. Let M be a system model with the base set {0,1, 2} of remainders modulo 3 and S = {(0, 1), (1, 2), (2, 0)}. Then A4 is a model for Z, i.e., A4 ~ Z. While A4 satisfies a property 2 + 2 = 1, this kind of properties cannot be derived from ~. (For example (qx)(x + x = 1)~)
5.3
Canonical System Model of Stationary Systems
We consider stationary systems as an example of a canonical system model. To construct the canonical system model of stationary systems, first recall the stationary structure (£sta; ~sta). (Definition 3.3.5) Our aim is that given input-output data we construct the canonical system model for (£sta; ~sta). In Reference[Takahara et al.1983], the functor that stationarizes a causal system not necessary to be stationary, is considered. The stationarized system of an original system S is given as S = U(AtS]t E T). The Nerode realization is one of concrete examples of this functor. The system S obtained is a "free" construction of the original system S. We will see that this result accords with the catmnical system model of stationary systems, CAN(Ze-,t~), that we will construct. We describe below the stationary structure again. We call this structure "basic" since our target structure for the construction of canonical system model should have explicitly some constants as input-output data, which structure will be called "extended" stationary structure. D e f i n i t i o n 5.3.1 ((Basic) S t a t i o n a r y S t r u c t u r e )
The (basic) stationary struc-
ture is defined by (£sta; Zsta):
;-.,ta = Z:T-~, u {A}, where A : a binary function symbol;
~sta ~" ~T--sys U {¢stl ~' Cst5}, where
~stl =
(Vxtr)(X(x) A T(t) A T(T) ~ A(t,x) o r -- x o (t + r))
~st2 -q)st3 ~gst4 -~gst5 =
(Vytr)(Y(y) A T(t) A T(T) ~ A(t, y) o T -- y o (t + r)) (Vxyt)(S(x,y) A T(t) -, S(A(t,x), A(t,y)))
(Vxt)(X(x) A T(t) -~ X(A(t,x))) (Vyt)(Y(y) A T(t) - , Y(A(t,y)))
CHAPTER 5. CANONICAL SYSTEM MODEL AS REALIZATION
106
(¢8tl and ¢a2 represent that A is a shift operator. Cst3 represents AtS C S, that is, any element of S shifted by I belongs to S again. ¢~t4 and ¢,t5 represent that X and Y are closed under ~ respectively.) A stationary system actually observed is considered as a model in which all sentences of ~8,~ are satisfied. We assume that we have input-output data Gx and Gy as constants. The canonical system model we want to construct is a stationary system model that realized these constants. The information that Gx and Gy are inputs and outputs respectively, that is, the input-output structure of Gx and Gy should be included in the stationary structure. Hence we define "extended" stationary structure with input-output data. D e f i n i t i o n 5 . 3 . 2 ( E x t e n d e d S t a t i o n a r y S t r u c t u r e w i t h Gx a n d Gy) The
extended stationary structure with input-output data Gx and Gy, (£e-aa; Ee-,ta), is the minimum structure obtained by extending (£sta; Eaa) such that: 1. Gx and Gy are sets of finite nmnber of constants respectively, where Gx = { d l 0 , d l l , . . . , d l r - 1 } , Gy = {d20,d21 .... ,d2s-1}, and G x M G y = {0, 1}MGx = {0,1} N Gy = ¢; 2. Gx, Gy C ~e--sta; 3. (Vdli E Gx)(Vd2j E G y ) ( X ( d l i ) , Y ( d 2 j ) , S ( d l i , d2j ) E ~e-aa). The constants Gx and Gy can be considered as generators. We construct the canonical system model for (£e--aa;~e--,ta). For the sake of the subsequent discussion we use the following notations:
A n N
= {~O+...+~k_tl~i=O,l, k e N } (N is the set of natural numbers); = 1 + 1 + . . . + l ( n times); =
{n[neN}.
We define a realization of f~e-sta in which we can interpret the symbols of £:~-sta. Definition 5.3.3 A realization A//~-,ta of l:,-,ta is defined by
A4e-sta = { C T ; T , A , B , X , Y , S , o , <,+,O, 1,A,G~,Gy, E} where
CT is the set of all closed terms of £e-aa, T=A, X--- Gx U {A(t0, )~(tl,..., A(tn-1, dl))"" ")ldl E G x , n E N, t i E A},
5.3. CANONICAL S Y S T E M MODEL OF S T A T I O N A R Y S Y S T E M S Y = G y U {A(to, A ( t l , . . . , A ( t n _ l , d 2 ) ) . . . ) l d 2
107
• G y , n E N , t i • A},
A = { x o t l x e X , t • A}, B = {YotlY e Y,t e A}, S = {(Sl,S2)ldl • G x , d 2 E G y , n • N , t i • A,
~ ( t l , . . . , ) ~ ( t n - l , d l ) ) " ,), s2 : )~(to, )~(tl,.-., ~(tn-1, d2))"" ")}, _< = { ( t l , t 2 ) ] f ( t l ) _< f ( t 2 ) , t l , t 2 • A} s 1 = )~(t0,
where f : A --- N; f(0) = 0, f(1) = 1, f ( t l + t2) = f ( t l ) + f ( t 2 ) (the right-hand addition of f is the usual one in natural numbers),
tl O~2 ----t l ot2, tl + t2 = t l + t2,
/~(tl, ~2) ----~(tl, t2), Gx=Gx, Gy=Gy,
0=0,1=l,
E = ETUE X UEyUEAUE
B
where ET = { ( t l , t z ) [ f ( t l ) = f ( t 2 ) , t l , t 2 • A}
Z x = { ( t l , t 2 ) i g ( t l ) = g ( t 2 ) , t l , t 2 • X} h : Gx --* NN; h(dli)(k ) = rk + i, d l i E G x (Notice that the cardinality of Gx is r.)
g : X ---+NN;g(dli) = h ( d l i ) , d l i • Gx, g(~(t, x))(n) = g(x)(n + / ( t ) ) , E y is similarly defined to E x ,
EA = { ( t a , t 2 ) I p ( t l ) = p ( t 2 ) , t x , t 2 • A} p: A --+ N ; p ( x o t ) = g ( x ) ( / ( t ) ) , EB is similarly defined to EA. Lemma
5.3.1
•]~e-sta ~ Ee-sta.
Proof." Although it is sufficient to show indeed that every sentence of ~e-st~ is satisfied in A4e-sta, it is clear by the definition of A4e-~ta except ¢t~2, ¢ts4, ¢~tl, ¢~t2 and
Cst3. ¢t82: let x , x ' E X.
Suppose that ( x o t , x ' o t )
E EA for a n y t
E A. Then
p ( x o t ) -- p ( x ' o t ) . Hence g ( x ) ( f ( t ) ) = g ( x ' ) ( / ( t ) ) , which implies g(x)(n) = g(x')(n) for n • N. So we have g(x) = g(x') and (x, x') • E x C E. The satisfiability of ¢t~4 is similar to that of ¢t~2. ¢~t1: let x • X and t , t ' • A.
CHAPTER 5. CANONICAL SYSTEM MODEL AS REALIZATION
108
Then p(A(t,x) oT)
= =
g(A(t,x))(f(r)) g ( x ) ( / ( r ) + Y(t))
=
+t))
=
p ( x o (t + r)).
Hence (A(t,x) o r , x o (t + ~)) E EA C E. The satisfiability of Cst2 is similar to that of e r a . Cst3: let (x,y) E S a n d t E A. By the definition of S, we h a v e x e X and y E Y. By the definition of X , Y , we have A(t,x) E X and A(t,y) E Y. Hence (A(t,x),A(t,y)) • S. [] L e m m a 5.3.2
Ee-sta I- (Vtlt2X)(T(tl) A T(t2) A X(X) A(tl, A(t2, X)) = A(tl + t2, x)). Proof: We prove that (Vr)(T(T) -* A(tl, A(t2, x)) o r = A(t I + t2, X) O r).
By e r a , )~(tl,A(t2,x))oT
: =
)~(t2,x)o(t l + ' r ) xo(t l+t 2+T).
)~(tl,)~(t2,x))<>T
= =
)~(t2,x)o(t I + T ) xo(t l+t 2+r).
By Cstl,
Similarly, ),(tl + t 2 , x ) OT = X¢ (tl + t2 +'r). Hence A(tl, )~(t2, X)) O T = )~(t 1 + t2, X) O ~-).
[] L e m m a 5.3.3 r~_s~a ~- (Vx)(X(x) -~ A(0, X) = x). Proof." A(0,x) o r = x o ( 0 + r )
=xor.
Hence A(0, x) = x.
[]
The model .Me-st, itself is not a canonical system model. The following theorem shows that .Me-st,,~ ,.,, each of whose relations is the quotient set of the corresponding relation of .hde-sta, is a canonical system model determined by Ee-sta.
5.3. CANONICAL S Y S T E M MODEL OF STATIONARY SYSTEMS Theorem
109
5.3.1
C A N ( ~ - , t a ) = .A4e-~u,I "" =
{ C T / , . , ; T , A , JD,2,Y,D,o,~_,-J-,O, 1, A,G,,G~,$}
where
A
=
A/~
=
=
AI,.,
=
D
=
BI~
=
2 ?
= =
Xl~ y/,..
= =
D k
= =
Sl~ E/~
= =
_~
=
< / ~
=
{[n]]n • N}, {[dl o n ] ] d l • G x , n E N}, {[d2 o n]]d2 E Gy, n • N}, {[A(n, dl)]]dl • Gx, n • N}, {[A(n, d2)]]d2 • Gy, n • N}, {([A(n, dl)], [A(n, d2)])]d 1 • Gx, d2 • Gy, n • N}, {([t], [t])It • E}, {([nl], [n2])]jr(nl) _< f(n2), nl, n2 • N ) .
P r o o f : First We prove 2b = A/..~. By the definition of the canonical system model, it is sufficient to show that t • A if and only if Ee-~ta t- T(t). Only if part: suppose t • A. Then there exists a k • N such that t ~ so + ' "
+ ~ k - l , ~ i = 0,1.
We use the induction on k. If k = 0, then t = 0 or t = 1. By ¢T06 and the logical axiom A4, we have E t- T ( t ) . Since it holds that
Z~--sta I'- T(a0 + " " + ak-2 + 0) and E~-~ta i- T(a0 + ' - . + ak-2 + 1) by Ee-sta f- T ( 0 ) A T(1), ¢r6, Modus Ponens and the induction hypothesis, we have
~e-sta t- W(t). If part: suppose ~ - s t a I- T(t). By the completeness theorem, Ee-sta ~ T ( t ) . Since A4~-sta ~ ~e-sta by Lemma 5.3.1, t E A.
A / ~ = {[nil[n] e N}. Since for any t E A there exists an n E N such that jr(t) = jr(n), so t ~ n. Furthermore, for any n, n' e N, if n # n', then f ( n ) # jr(n), so [n] # [n']. R = X~ ~:
It is sufficient to show that t E X if and only if ~e-sta F- X(t). Only if part: suppose t E X. Then there exist t l , . . . , t n - 1 E A, x E G X such that t ----A(to, A(tl . . . . , A ( t n _ l , X ) ) . . . ) ) or t -= x.
C H A P T E R 5. CANONICAL S Y S T E M MODEL AS REALIZATION
110
By Cst4, we have E~-,ta F- X ( A ( t n _ l , X ) ) .
Again applying
Cst4,
we
have Ee-sta F-
X ( A ( t n - 2 , A ( t n - l , X ) ) ) . By repeating this way, we have
~e-sta [- X ( A ( t 0 , A ( t l , . . . , A ( t n - l , X ) ) ' " '))). Hence Ee-,ta F- X ( t ) . The case t _= x is trivial. The if p a r t is similar to t h a t of T = A / . - , .
X~ ~ = {[A(n, d l ) ] l d l e G x , n e N}: For any t E X there exist a t E h and an n E N such t h a t if t --- A ( t o , A ( t l , . . . , A ( t n _ l , d l ) ) . . . ) )
and d l e G X ,
then t ~ A(T, dl) ~ A(n, dl) w h e r e T --= t I -{- • • • + t n _ 1 a n d T ~ n. If
t = dl E G x ,
then t ~ A(T, d l ) where ~ -- 0 E N. The proof of Y is similar to t h a t of )~. ~i = A / ~ : We prove t h a t t E A if and only if Ee-sta F- A ( t ) . Only if part: suppose t E A. Then there exist a n x
E X and a u
E A such
t h a t t - x o u. Since t E X if and only if Ee-,ta F- X ( t ) a~d t E A if and only if
Ee-sta I-- T ( t ) , so by Ctsl, Ee-sta I'- A ( t ) . The if p a r t is similar to t h a t of T = A/,,~.
A~ ~ = {[dl o n ] l d l E G x , n E N}: First we prove t h a t for any a E A there exist a d 1 E G x and an n E N such t h a t a ~ d l o n. Now let a =- x o t ,
(t E A), x - = A ( t o , A ( t l . . . . , A ( t n - l , d l ) ) ' " )
and d l E G x .
By using L e m m a 5.3.2 repeatedly,
Ee-sta [- A(to, A ( t l , . . . , / ~ ( t n - 1 , d l ) ) " ' ") =
A(tl
+'"
+
t n - 1,
dl).
5.3. CANONICAL S Y S T E M MODEL OF S T A T I O N A R Y S Y S T E M S By Cstl,
[Je-sta [- X(tl
+ "'" + tn-l,dl) O T = d I o (t I + ... + tn_ 1 + T).
Hence a ~ dl o n. Next we claim that if dlon~d~on where dl, d~ • G X
l
and n, n' E N, then d I = d~ and n = n ~.
Indeed by the definition of the equivalence relation,
~e-sta ~" d l
o n = d~ o n'.
Hence •A4e-sta ~ dl o n -- d~ o n'. So
g(dl)(f(n))
= g(d~.)(f(n')).
By the definitions of g and f, d l = d~ and n = n'. The proof of 1} is similar to that of A. = Sl ~:
We prove that ( t l , t 2 ) e S if and only if Ze-~ta ~- S ( t l , t 2 ) . Only if part: suppose ( t l , t 2 ) E S. Since t l ~ d l E GX or t l ~ A(to, A ( t l , . . . , A ( t n _ l , d l ) ) ' " ) where t i E A, and t2 ~ d2 E G y or t2 -= A ( t o , A ( t l , . . . , A ( t n - l , d 2 ) ) " "') where t i • A, so by S ( d l , d 2 ) • ~e-sta and ¢,t3, we have ~e-sto t- S ( t l , t 2 ) . The if part is similar to that of/b = A / ~ . S / N = {([A(n, dl)], [A(n, d2)])ldl • G x , d 2 e G y , n • IN}: This is clear due to Lemma 5.3.1, 5.3.2 and/b = A / ~ . = E/~:
We prove that (tl,t~) • E if and only if ~e-~ta ~- t l = t2. Only if part: suppose ( t l , t 2 ) • E.
111
112
CHAPTER 5. CANONICAL SYSTEM MODEL AS REALIZATION The case ( t l , t 2 ) E ET. Then since f ( t l ) = f(t2), by ¢~1,¢r2 a~ld ¢~3,
Ee_stal-tl . . . . .
1+...+1=1+'"+1 ](tl)times
.....
t2.
S(t2)times
The case ( t l , t 2 ) E Ex. Let t l -= ~(T0, ~(~1,... , ~ ( r , - 1 , x ) ) ' "
"),
t2 -- A(*o, A(T1,..., A(~'m-1, x))"" "). Since g ( t l ) = g(t2), so x = x r and fiT0 + ' "
+ ~n -- 1) = S ( ~ + '
I
' +*m-l)"
By ¢~,I and ¢~,2, we have ~,-~t~ t- t l = t2. The case ( t l , t 2 ) E EA. Let t l =-- A(vo, A(*I . . . . , A(vn-1, x ) ) " ") o t, t2 =- A(T0, M * I , . . . , A ( r ~ - I , x ) ) " " ") oV. Since p ( t l ) = p(t2), so x = x' and f ( t + ~o + .... + ~ _ ~ ) = f ( t ' + ~ + . . . + ~'_~). By ¢~1, there exists a t E N sudl that ~e-**~ I- t 1 = x o t and ~e--sta ~- t 2
=
xot.
Hence 5"]~e--sta t- t l = t2. Other cases are similarly proven. The if part is similar to that of T = A/-~. J~ = {([t], [t])lt e E}: ([tl], [t21) e El
iff iff iff
~e-sta }'- t l = t2 t l " t2 [tl] = [t2].
We prove that ( t l , t 2 ) E< if and only if ~ , - , t a t- t l _< t2. Only if part: suppose ( t l , t 2 ) ~<_. Since f ( t l ) _< f ( t 2 ) , f ( t 2 ) = f ( t l + n) for some n E N. So ( t 2 , t 1 + n ) E E. Hence ~e-s*a t- t2 = t l + n and by ¢r5, we have ~ - s t a t- t l < t2. The if part is similar to that of 5b = A / ~ . T h e o r e m 5.3.2
CAN(~e-s~a) ~ ~3e-sta.
[]
5.4. ALGEBRAIC SPECIFICATION
113
Proof: It is clear due to the definition of the canonical system model.
[]
Theorem 5.3.3 CAN(~e-sta) is the minimum in the class of objects o/ STrZ, where T is the type o/ £e-sta. Since ~e-sta may not be complete, we cannot apply Corollary 8.2.1 to C A N ( The input-output pairs of the stationary system itself is represented by the relation S in CAN(~e-~t,). This S can be essentially interpreted as S = U(£tStlt e T) where S t -- Gx x Gy, Gx C X and Gy C Y. This result accords with the stationarized system S = U()~tSIt E T) by Nerode realization. This implies that the framework for realization by universality in this chapter is an essential and sufficient generalization of the traditional concept of realization.
5.4
Algebraic Specification
In this section, we give another example of the canonical system model that is known as initial semantics of algebraic specification. This section also includes the reformulation of the algebraic specification within our framework. Basically a signature in algebraic specification corresponds to a language, an algebraic specification to a structure, a spec-model to a system model for ~ and initial semantics to canonical system model. We start our discussion with clarifying the relationships between system models and software systems, especially programs.
5.4.1
P r o g r a m s as S y s t e m M o d e l s
Computers are indispensable tools for solving complex problems existing in the real world. Since a computer works through a computer algorithm that is realized as a program written in a certain "suitable" programming language, solving a problem with a computer requires to represent the problem as a process in which the real world objects connected with the problem are operated. For the representation of the problem, a programmer or a system engineer who is engaged in solving the problem either explicitly or implicitly describes a "process model" concerning the problem. This model is obtained through his/her systems recognition for the real world objects. Then a program can be considered to "realize" this process model in a programming language. Therefore a program can be regarded as a kind of a system model of a process in the real world from the point of view of computers through a programmer. (Fig. 5.2)
114
C H A P T E R 5. CANONICAL S Y S T E M MODEL AS REALIZATION
From this aspect we can see that algebraic specification serves as a powerful formalism by which system models obtained in programming can be represented not depending on particular programming languages. Algebraic specification, to be exact, represents the structure of a system model, and algebras satisfying an algebraic specification provide system models. In that sense, algebraic specification can be considered as general representation of system models in accord with our framework, for which some constraints with respect to programming or computer should be imposed on representation. These constraints concretely reflect the "algebraic" aspect of algebraic specification; it allows only operations as languages and equations as axioms. We should notice that many sorted representation is not essential for the constraints; indeed we will try in the next section to formulate algebraic specification in terms of "one" sorted logic of our framework. Real World A Process In Real World
The Programmer's Systems Recognition for the Problem
Interpretation of Results
1
Computer Algorllhm
System Model
Figure 5.2 An Illustration of the Relationships between Programs and Problems
5.4. A L G E B R A I C SPECIFICATION
115
Abstract data type can be specified by means of algebraic specification, which specification means that data type is considered as a system model obtained through progranmlers' or users' systems recognition. Then an abstract data type represents the "structure" of a data type as a system model. There are some applications of algebraic specification to specification of systems other than software; for example, the specification of production systems by algebraic specification [Ko et al.1988] may be. available in the sense that algebraic specification represents the structure of the system. Such applications are, however, not always suitable for modeling of system models, since, as mentioned before, algebraic specification is concerned with system models on which some constraints from the point of view of the programming are imposed. In the previous example of the production systems, boolean, which is introduced due to formal motive, is necessarily included in the obtained system model for a factory; it is not clear what is meant by boolean in the system model for the "factory." 5.4.2
Canonical System Model of Algebraic Specification
Initial semantics is well known as semantics of algebraic specification. We formulate algebraic specification in this section within our framework, and show that the well known initial semantics accords with the canonical system model determined by the structure of algebraic specification. Definition
5.4.1 ( S i g n a t u r e ) A signature is a language defined by: E~ =< S, OP, {CslIts E S} >,
where S C {RsIRs is a unary relation symbol}; O P = U(OP~sIRs ~ S), where
OP~
C {f~s : a --- R s [ a = R 1 x . . . x R s n , R i
E
S,i
= 1.... ,sn};
Cs is a constant symbol for each Rs E S. A signature is a laalguage allowing only unary relation symbols, operations and constant symbols. Each element is called a "sort," collection of which forms a domain of, say, data type. Notice that each function in O P is not "partial," therefore, f~s is applicable to elements other than a within our framework. This matter is slightly different from the usual definition of a signature [Ehrig et a1.1985].
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116
D e f i n i t i o n 5.4.2 ( A l g e b r a i c S p e c i f i c a t i o n ) An algebraic specification is defined
by: < £8; ~s >, where £8 is a signature; where ZE is a set of universal atomic sentences that permit only the equality as relation symbols, ~Iunc consists of the following sentences: for all f~s E OP~ C O P ,
(VXI"'" Xsn)(Rl(x1) A ' " A l~s n(xs n) --~ l~(fas(X1,...,Xsn))), and for all Cs E {Cs]Rs E S},
R~(c.). A spec-model of < £8; Zs > is a realization of £s that is a model for Z8 and has a base set A with A = U(RsIRs E S). Notice that algebraic specification and spec-models defined axe not many sorted because we formulate them with "first" order logic. The canonical system model determined by a given algebraic specification < £~; Zs > is as follows: T h e o r e m 5.4.1 Given an algebraic specification < £s; Zs >, the canonical system
model determined by ~s is:
CAN(~8) =< CT;{/?/siltsE S}, {/,,sif~s EOPas},{CsIRs e S} >, where CT = {[f~s(tl,... ,ts~)]if~s e OP, ti : closed term, i = 1 . . . . . sn} U {[Cs]}; R^ , = / [ ~ fs ( ~ , . . . . ' ~ " ) ] I ~s arbitrary such that f~s EOP, }~ U{[Cs]}; ..... a: }- Ri(ti),ti : closed term, i = 1,... ,sn j k I / a s ( [ t l ] , . . . , [t,,]) = [fas(tl,... ,t,,)], for any f~s e OF; Cs -- [Cs], for any Cs. P r o o f : Clear from the definition.
[]
There is no difference between the above canonical system model and the well known initial semantics of an algebraic specification defined with a many sorted algebra.
5.4. ALGEBRAIC SPECIFICATION
117
It is crucial for the universality of the canonical system model that we have the following theorem that does not in general hold in canonical structures. This matter arises from that the canonical system model determined by an algebraic specification essentially allows only equations as substantial axioms, namely ~E. T h e o r e m 5.4.2 Let < £8;~s > be an algebraic specification and CAN(~8) the
canonical system model determined by ~8. Then CAN(Es) ~ Y1,8. Proof: Case. Let ¢ E ZE. From the logical axiom A6, any sentence, written Ct/x, obtained from ¢ by substituting any closed terms for the variables occurring in ¢ is derivable from ~E, that is,
~E ~- Ct/x" From the definition of ~E, ¢,/x is of the form: A(tl . . . . . to) = S(to+l . . . . . to+m), where A ( x l , . . . , Xn) is a term with free variables Xl . . . . . Xn and A ( t l , . . . , to) is the closed term obtained from A(Xl .... , Xn) by substituting t l , . . . , to for X l , . . . , Xn; B ( t o + l , . . . , t~+m) is similarly obtained. By the definition of the equivalence relation,
[A(tl,..., to)] = [B(to+l,..., t~+m)], which implies
A([tl],..., [tn]) = B([to+l],..., [t,+m]), since A and B are either variables or functions. Since [tl],..., [to+m] are arbitrary elements of CT, and ¢ is universal, we have
CAN(~8) ~ ¢. Case. Let ¢ E ~fun~, that is, ~b -= (Vxl • • •Xsn) (R1 (Xl) A.'- A R s n (Xsn) ~ P ~ (f~s (Xl, • • •, Xsn ))),
for some a, s. For any [tl] E/~1,..., [ts~] E/~sn, we have
118
CHAPTER 5. CANONICAL SYSTEM MODEL AS REALIZATION
Hence which implies
[f~s(tl,... ,t,o)] • R~. So
f~s([tl],..., [t,n]) • k,, and CAN(E~) ~ ¢. This completes the proof.
[]
Chapter 6
HIERARCHY Simply put, systemic properties, which come mainly from systems concepts such as emergence, hierarchy, wholeness, openness, communication and the like, are the properties of systems properties. This requires a second order treatment for the investigations of systemic properties.
In our framework with the concept of F-
morphisms, it can be naturally and effectively realized since we are at a stage to be able to formulate from a higher order point of view system models, systems properties and their relationships using the F-morphisms. In this chapter, paying our attention especially to hierarchy as the most fundamental systemic property, we will formulate hierarchical systems and emergent properties, and consider the structure expansion in general system models in detail, especially the role of inputoutput system models as the lowest level models.
6,1
Hierarchy
and
Emergence
It is no exaggeration to say that every complex system necessarily possesses a hierarchical structure. It is our or systems scientists' conviction that truly complex objects cannot be understood with knowledge described only at one level. To understand the complexity we have to take a hierarchical view in which knowledge is described in the languages at some different levels. The hierarchical view results in a hierarchical structure of a system. It might be recognized as a structure formed by a system model. In the hierarchical view a system consists of subsystems that mutually operate, and each of which is at a level for which a subsystem at a higher level has the emergent properties that are meaningless in the language appropriate to a lower level. In this section we will formulate the hierarchy in reference to emergence. Note that our purpose is not to construct a "system model" of the hierarchy, but to formulate the "concept" of hierarchy, i.e., to clarify what is the distinction between
CHAPTER 6. H I E R A R C H Y
120 a higher level and a lower one. 6.1.1
Levels in a Hierarchy
When recognizing a system as having a hierarchical structure, we have to select the levels of description so that each subsystem constituting the hierarchy is described in the language at the selected level, which is called "stratum." [Mesarovic et al. 1970] That selection is done in a course of a modeling process, and hence is the first step for recognizing the hierarchy. Besides strata can we take other types of levels from functional viewpoints such that every subsystem operates as a decision-making unit and the like; these levels are called layers or echelons. Here we pay our attention solely to strata type of levels, since the functional levels are concerned with actually determining the concrete structure of subsystems and the interactions among them, which determination is a part of a modeling process and ascribes constructing a system model of an overall system. Let K be a class of system models each of which is obtained at a level of description through the systems recognition of an object (as a system). We can consider the class K as the class of subsystems constituting the hierarchy. How the system models in K interact is one of the central problems of hierarchy theory [Checkland 1981], but it is beyond the scope of this book. Every system model in K has the language and the set of axioms (Recall these are the structure of the system model.) appropriate to the level of description. Different levels of description require to change the languages and the axioms in such an essential way that every sentence on the lower level system model can be interpreted in the higher level system model, however, the converse does not hold. Definition 6.1.1 (Closed u n d e r Faithfulness) Let A be a class of ~F-morphisms. If every composition of any UF-morphisms in A is also a ~F-morphism in A, then A is said to be closed under ~f-faithfulness.
Definition 6.1.2 (Hierarchical Class) Let K be a class of system models obtained through the (hierarchical) systems recognition of an object. Let A be a class of ZF-morphisms between system models in K. If A is closed under ZF-faithfulness and for any two system models in K there exists some ZF-morphism in A of one to the other, then (K, A) is called a hierarchical class of system models. The whole of K forms one object to be examined through a hierarchical view. A represents the relations between the systems recognition for system models in K. In that sense A implies part of the systems recognition for K as well as the structure
6.1. H I E R A R C H Y A N D E M E R G E N C E
121
of a system model. Thus we obtain the following definition of levels in a hierarchy but without consideration of emergence. Definition 6.1.3 (Level in a H i e r a r c h y ) Let (K, A) be a hierarchical class of system models. For any system nmdels, ]~4i, Adj in K, A,Ij is said to be at a higher
level than A4i if there exists some ~f-morphism in A of .~4i to .A~j and there exists no ~F-morphism in A of 2¢Ij to A,ti; 2~4j is said to be at the same level as J~ti, if there exists some ~F-morphisms in A of ~Ai to A,Ij and of ]~j to ./~i.
P r o p o s i t i o n 6.1.1 Let (K, A) be a hierarchical class of system models. If the num-
ber of the system models in K is finite, then K has a maximal element with respect to the following ordering. For A4i, A4j in K , .Mi < J~4j iff .hdj is at a higher level than A~i. P r o o f : By induction on the number of the system models in K. ]K[ = 1: This case is trivial. We assume that when ]K] = n, K has a maximal element ~4h-. Consider the case ]K] -- n + 1. If we remove a system model Me from K and the ZF-morphisms concerning A4e from A, then we obtain another hierarchical class (K,/~) with ]/(] = n. By the induction hypothesis R has a maximal element 2~4/~. Now we divide the situation into two cases. Case 1: There exists some ~F-morphism in A of M R to )~4e. Consider the ordering relation between A4e and another system model AAi other than A4 R. Subcase (a) of Case 1: J~4£. is at the same level as J~4i. If there exists some ZF-morphism in A of Ade to A4i, then .~te is at the same level as A4~ and .h4/i. due to the faithfulness of the ~F-morphism. If there exists some ~F-morphism in A of A4i to Me, then A4e is at a higher level than .~4 R and ~4i. Subcase (b) of Case 1: J~4~. is at a higher level than AAi. AAe is at a higher level than Adi because if there exists some ~f-morphism in A of A4e to A4i, then there exists some ~r-morphism in A of .h4 R to A/li, which existence contradicts the assumption. Therefore in Case 1 A4e is a maximal element in K. Case 2: There exists some ~F-morphism in A of Me to AA~.. Subcase (a) of Case 2: ~ R is at the same level as A4i.
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122
If there exists some ~F-morphism in A of A4i to A4e, then A4e is at the sanle level as A4i and A4 R due to the faithfulness of the ]BF-morphism. If there exists some ~F-morphism in A of A4e to A4i, then is at a higher level than A4~. Subcase (b) of Case 2: A/R is at a higher level than .44;. A4 R is at a higher level thai1 A4~ regardless of the direction of the ~f-morphism between A4e and A4i. Therefore in Case 2 A4 R is a maximal element in K. This completes the proof.
[]
We can elaborate on Definition 6.1.3 by clarifying how emergent properties ave concerned with the hierarchy. To this end definitions of emergent properties are necessary.
6.1.2
Emergent Properties and H i e r a r c h i e s
Each structure, (L:;E~), possessed by each system model although the structure might not be uniquely determined, operates on the system model as constraints, in the sense that the system nmdel 'must' satisfy that structure. Thus the emergent properties of the system model result from the constraints, i.e., (£;~,,). Therefore we have at least two kinds of the emergent properties: one associated with £, and the other associated with E~. The emergent properties associated with £ come from the idea that the emergent properties at a higher level cannot be expressed in the language appropriate to a lower level; they cannot be defined and interpreted in the lower language. The interpretation is provided by the ZF-morphisms in the hierarchical class under consideration. As for the definability we define it as follows: D e f i n i t i o n 6.1.4 (Definable) Let A4 be a system model, and (L:;E~) the structure of A4. Let S be a symbol in/2. Then S is said to be definable in .h4 if the following conditions are satisfied: If S is a relational symbol Ri, then there exists a formula CR i of/: such that J ~ ~ (VXl".xA(i))(Ri(xl . . . . ,xA(i)) ~
CRi(xl,...,xA(i)))
where CR i has A(~) free variables and does not contain Ri; if S is a functional symbol fj, then there exists a formula Cfj of £ such that
.A~ ~ (VX1-. "X#(j))(fj(X 1 .....X/~(j))= Xp(j)+l ~ ~bfj(Xl,... ,x/~(j)+I))
6.1. H I E R A R C H Y AND EMERGENCE
123
and Cfj is a function of X l , . . . , x~(j), i.e.,
J~ ~ (VX1 ""'X~(j))(~X~(j)+l)(¢fj(X1,.-., X~(j)+l)) A ( V X l ' " x # ( j ) y ) ( ¢ ~ ( X l , . . . ,x~(j)+l) ACfj(xx . . . . ,x/~(j),y) --4 x~(j)+l ----y), where Cfj has/~(j) + 1 free variables and does not contain fj. Then we can define an emergent property associated w i t h / : as follows: D e f i n i t i o n 6.1.5 ( E m e r g e n t P r o p e r t y w i t h r e s p e c t t o / : )
Let (K,A) be a hi-
erarchical class, and .M1 and A42 be in K where A42 is at a higher level than M1 and the structure of A41(or A42) is (/:1;~1) (or (/:2;~2)). Then a sentence of/:2 is said to be an emergent property in ( K , A ) with respect to/:1 if the sentence contains a symbol that is not definable in A42 and is not contained in/:1. And a sentence of /:2 is said to be a complete emergent property in K with respect to/:1 if for any A concerning the system models in K the sentence is an emergent property in ( K , A ) with respect t o / : 1 . D e f i n i t i o n 6.1.6 ( R e d u c i b l e ) Let (K,A), .hA1 and A42 be as in Definition 6.1.5. Then a sentence of/:2 is reducible in ( K , A ) to .h41 if and only if it is not an emergent property with respect to/:1. E x a m p l e 6.1.1 Let K = {A~I,M2}, where A41 = <
N ; S , 0 > and M 2 = <
N ; S , O , W >; S is a unary function, 0 is a constant, W is a unary relation and N is the set of natural numbers. Let ~1 and Z2 be as follows: r~l = ~,2 = {¢i,¢2},
where ¢1
=
(Vx)(-~S(x)
¢2
--
(Vxy)(S(x)=S(y)
= o),
~
x--y).
We define the following set for each formula ¢ , that contains only x as a fi'ee variable: Wo = { n t ~ 2 ~ ~,[n]}, where a -- [¢x] (the Ghdel number of Cx, see [Andrews 1986]). W is defined by:
W --
n
n is the Ghdel number of a formula that contains only x as a free variable, and n q[ Wn
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124
Furthermore the member of A is the embedding from AzI1 to Az[.2. Then every sentence that contains W is an emergent property in (K,A) with respect to £(/%A1). To prove this it is sufficient to show that W is not definable in/%42. Suppose W is definable in AJ2. For some ¢(x) E Form(£(Ag2)), from the definition,
M2 ~ (Vx)(W(x) ~ ¢(x)). Let [¢(x)] - - p . Then
pEW
iff iff iff
M2 ~ ¢(x)[p] p E Wp
peW,
(from the definition of definable) (from the definition of W~) (from the definition of W),
which is contradiction. The second type of emergent properties is associated with the axioms ~, which idea is based on that an emergent property cannot be derived from the axioms at a lower level. D e f i n i t i o n 6.1.7 ( E m e r g e n t P r o p e r t y w i t h r e s p e c t to Z)
Let (K,A), AA1
and .hA2 be as in Definition 6.1.5. A sentence ¢ of £2 is said to be a (weak) emergent
property in ( K , A ) with respect to E1 if either there is no F-morphism of/%A2 to Adl or there is an F-morphism I of .A/J2 to J ~ l such that E2 ~- ¢ holds and it is not the case that E1 t- IF(C) or E1 ~ -YF('~¢); a strong emergent property in ( K , A ) with
respect to E1 if for any F-morphism I 0 Az[2 to AA1, E2 F- ¢ holds and it is not the case that E1 F- If(C) or E1 I- IF(-~¢). It follows immediately from the definition that if E1 is complete, no sentence of
£2 is an emergent property with respect to El. The following proposition gives a typical example concerning the existence of an emergent property. P r o p o s i t i o n 6.1.2 Let (K,A ), .hall and A42 be as in Definition 6.1.5, where £1 = £2, E2 = ~]IU {¢}, ¢ E Sent(£1) and A42 is a submodel of Agl. f l i t is not the case
that Az[1 ~ ¢, then ¢ is a weak emergent property with respect to El. Proof: Let Io : A2 ---+A1 be the embedding of :%A2 to :~41. The embedding is obviously an F-morphism of A42 to AA1. Then for any ¢ E Sent(£2), IF(C) = ¢. From the condition, it does not follow that
Therefore it is not the case that E 1 }- ~.
~ r t h e r m o r e if ~1 F- -~¢, then since
A42 ~ El, we have AA2 ~ -~¢, which is contradiction. Hence it is not the case that E 1 ~- ~(~. Since E2 ~- ¢, ¢ is a weak emergent property with respect to El.
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6.1. H I E R A R C H Y AND EMERGENCE
125
P r o p o s i t i o n 6.1.3 Let (K,A), ]~41 and .M2 be as in Proposition 6.1.2. I/for any
F-morphism of .A42 to .£41, A,t2 ~ ¢ implies .h42 ~ IF(C), and it ,is not the case that A~: ~ IF(C), then ¢ is a strong emergent property 'in (K,A) with respect to
Proof: Let I be an arbitrary F-morphism of.AA2 to .A,~I satisfying the conditions. Suppose ~: ~- IF(C). Since AA: ~ ~:, we have AA: ~ IF(a), which contradicts the condition. Next suppose ~: }- IF(-~¢). Since A,12 ~ ~:, we have ,~42 ~ IF(-~d), which contradicts the condition that A~2 ~ ¢. It is clear that ~2 F- ¢, hence this completes the proof.
[]
E x a m p l e 6.1.2 Let K = {M1,M2}, where M I = < N; x > and M2 = < {0, 1}; x >; x is the multiplication. E1 consists of the following sentences: ¢1
~
¢3
-
(VXlX2X3)((X 1 X X2) X X 3 = X 1 X (X 2 X X3)), (3z)(Vx)(x x z = x).
Z2 = :E: U {¢}, where = (3x:x2)(Vxs)(-~xl
= x2 ^ ( x s = x : v x3 = x~)).
A consists of all ~F-morphisms of M1 to M2. Then d is a strong emergent property in (K,A) with respect to ~:. Indeed, since the embedding I of M2 to A,t: is an F-morphism of M2 to 2~4: and IF(C) ----¢, so we have M2 ~ IF(C) and not M1 ~ IF(C). E x a m p l e 6.1.3 Let K = { . A ~ l , ~ 2 , . J ~ 3 } , where J~41 = < N ; + >, ~42 < N; x > and .;~43 =<: N; +, x >; + is the addition and x is the multiplication. E1 (or E2) is a set of axioms expressing that ~ 1 (or ~42) is a monoid with + (or x). E3=E: U E2 U {¢}, where d - (Vxlx2xs)(xi
× (x2 + x s )
= (xl × x2) + (x: × xs)).
Let /1 : AA1 --* ~43 and /2 : ,~42 -* A~3 be the embeddings of A,t: to A~3 and of ~42 to A~t3, respectively; (I1)0 and (/2)o are the identities, Bas(+) = ÷ and
Bas( x ) = x. A = {/1,/2 and all homonmrphisms of 2~4: to A~t2 and of A~2 to A,tl}. Then d is a weak emergent property in (K,A) with respect to ~.1 and ~2.
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126
We prove it only with respect to Z1. Let I : A43 --* 341 be an F-morphism defined by: Io(N) = {0}, Bas(+) = Bas(x) = +. Obviously IF(C) is equivalent to the following sentence: (Vxlx2xa)(xl + (x2 + x3) = (xl + x2) + (xl + x3)). It does not follow that E1 t- IF(C), since A41 ~ 1F(¢) does not hold. Since the model Ado = < {0};+ > satisfies ¢, E1 b- If(-'¢) does not hold. E2 b- ¢ is trivial. [ ]
In a hierarchical structure a system model at a higher level, in some sense, possesses emergent properties that cannot be expressed at a lower level. Therefore the definition of a hierarchy is required to provide a relationship between higher and lower levels, concerning emergence. Considering this, we try below to define hierarchies from the viewpoint of emergence. Definition 6.1.8 ( H i e r a r c h i c a l S y s t e m ) Let (K,A) be a hierarchical class. K is said to be a hierarchical system provided that for any two system models 3,t I and A42 in K irA42 is at a higher level than A41, then there exists a sentence of £2 that is an emergent property in (K,A) with respect to/:1 or ~1. We should notice that a hierarchical system above is not a "formal" system model as defined in the fraanework developed in this book. As mentioned before, to construct the system model of a hierarchical system is a central problem of the hierarchy theory. 6.2
Hierarchy
in General
System
Models
What is important in the mathematical theory of general systems [Mesarovic et al.1975,1989] is that the basic systems concepts are introduced by means of forrealization. Starting from the concept of an input-output system that is the most fundamental structure in any systems theory, the mathematical theory of general systems is further developed by adding more mathematical structure as needed for the investigation of various systems according to our interests. The usual way to expand the structure is as follows: the starting point is, as mentioned above, an input-output system model. Then we have two directions of the expansion by adding more mathematical structure: to the sets of inputs and outputs, or to the elements itself of the input and output sets. The typical structures for the former are linearity and state space representation; linearity in which the vector space is introduced, and state representation in which the concept of states is introduced to functionalize the system.
6.2. HIERARCHY IN GENERAL SYSTEM MODELS
127
For the latter case the structure of time is essential for the dynamics of the system. After the introduction of a time system, some concepts related to it a~-e defined, for example, the concepts such as causality or stationarity. Combining the structure of time with some related concepts and the state space representation, we obtain dynamical system representation. Furthermore by adding linearity to it, we obtain basic linear systems. (see Fig. 6.1) The way of the expansion of the structures as shown above forms a hierarchy in an intuitive sense. In this section we will show the following two points: (1) The expansion results in a hierarchical structure in the sense of the definition in Section 5.2; (2) Input-output system models are, in a sense, at the lowest level than any other system models. 6.2.1
Associative F-morphism
in Structure E x p a n s i o n
In considering the expansion of structures the concept of associative F-morphism is more useful than "general" F-morphism. In this section we define an associative F-morphism and give a relationship of associative F-morphisms and structure expansions.
Definition 6.2.1 (Associative F - M o r p h i s m ) Let A,tl and A,t2 be system models, and ~'1 and r2 the types of A~I and Ad2 respectively. Let ~'1 C T2. Then an F-morphism of 2~41 to M2, I =< Io, IF >: Adl --* A42, is called an associative F-
morphism of A~l to A42. Furthermore the basic morphism Io is called an associative basic morphisr~ Let Ad2]Ti be the reduct to Ti (see Definition 3.1.3).
Proposition 6.2.1 Let ./~41, A42, rl and T2 be as in Definition 6.2.1. Let I = < IO,IF >: .£41 ~ A,t2 be an F-morphism of.M1 to ,M2. Suppose Io is onto. Then . M i / I is isomorphic to .~2[T1.
Proof: By F-morphism Theorem, Io is a one-to-one basic morphism. Since the type of .M1/I and ~ 2 ] r l is T1, Io is an isomorphism. Hence A41/I is isomorphic
to M21~1.
[]
We recall the definition of structure expansion.
Definition 6.2.2 (Structure Expansion (Definition 3.3.7)) Let A~I and .A42 be system models with structures (£1;~1) and
(/:2;~2)
respectively. (£2;~2) is said
C H A P T E R 6. H I E R A R C H Y
128
to be a structure expansion of (£1;E1), written (£1;E1)C(£2;E2), if £1 C £2 mid E1 C E2. Ad2 is said to be a structure expansive model of eL41 if the base set of 2Vfl is a subset of that of AA2 and (£1;E1)C (£2;E2). From the definition we can see that if ¢L42 is a structure expansive model of 2L41, then M2 ~ E1 and
d~f2]T1~
E 1.
AS the next proposition shows, structure expansion is very close to the associative F-morphism. This implies that the associative F-morphism plays a central role in specification of systems, since effective specification of systems is done based on the notion of structure expansion.
Proposition 6.2.2 Let .h,tl and .hal2 be system models. If Ad2 'is a structure expansive model of A4i, then there exists an associative Er-morphism of Adl to J~2 that is faithful to any F-morphism of 2L42 to any system model. Proof." Let I : Adl --* Ad2 be the embedding such that for any symbol S of £1,
Bas(S) = S. Then for ¢ E El, IF(C) = ¢. Since AA2 ~ E2, we have
Hence I is an associative Er-morphism. We show that I is the desired ~r-lnorphism. Let Ad3 be an arbitrary system model and IR : AA2 ~ Ad3 an arbitrary F-nmrphism. We show that IR o I is an F-morphism. Let S be a relation symbol in £1. If .Adl ~ S(x I . . . . . xA(i)))[al . . . . . aA(1)] where al . . . . . a~(1) E M1, then .hd2 ~ I r ( S ( x l . . . . . x:~(i)))[Io(al) . . . . , Io(a:~(1))]. By the definition of I, .A42 ~ S ( x l , . . . , xA(i))[al,..., aA(i)]. Since IR is an F-morphism, •A43 ~ IR(S(Xl . . . . , x)~(i))[(IR)O(al),..., (IR)o(a)~(i))]. Hence M3 ~ IR o I ( S ( x x , . . . , x~(i))[(IR o I)o(al),..., (In o I)o(ax(i))]. The cases of the function symbols and the constant symbols can be similarly proven.
[]
6.2. HIERARCHY IN GENERAL SYSTEM MODELS 6.2.2
Input-Output
129
S y s t e m M o d e l s as t h e L o w e s t L e v e l M o d e l s
One of the reasons why input-output system models are located in the starting points of the development of the mathematical theory of general systems is that we can naturally observe every real system as an input-output system, and such an observation provides us with invaluable benefits. This observation implies that we construct a EF-morphism of an input-outpu t system model to the real system. (The term "the real system" means here a system model obtained from a "real" world through the systems recognition.) From the aspect of the hierarchy in our framework we see, in a sense, that input-output system models are at the lowest level in the class of all system models. The following theorem shows this matter. Theorem 6.2.1 For any system model.h,{ there exists an input-output system model A4I/0 and a Ef-molThism I of A4Uo to A4 such that for any system model M R
and F-morphism IR of .td to M R , I is faithful to IR, where Jt4l/O is assumed to have the structure of input-output system model (~I/O ;EI/O ). (See Section 3.3) Proof: Let A4 be a system model. We divide the proof into four cases according to the symbols involved in f14. (1) M has a unary relation R. A41/o = < U; X, 1I, S > is defined as follows: U= X= Y-s =
M, where M is the universe of M , M, R, {(x,v)lx ~ x , y e y}.
Io is defined as the identity. The basic interpretations are defined by:
X(x) Y(y) S(x,y)
--,
R(x) v -,R(x),
~ R(y), -~ (R(x) V ~R(x)) A R(y).
It is clear that the F-morphism defined from the above basic interpretationsand Io is a :EF-morphism of A 4 U o to A4. Next we show the faithfulnessof the F-morphism I defined as above. Let IR be an arbitrary F-morphism of A4 to any system model M R . W e show that IR o I is an F-morphism. Let A4//O ~ X(x)[a], where a E M. Then
M ~ rF(X(x))[Io(a)].
CHAPTER 6. HIERARCHY
130 Hence
M ~ (R(x) V ~R(x))[a]. Since M R ~ (IR)F(R(x))[(IR)o(a)] or M R ~ -~(IR)F(R(x))[(IR)o(a)],
A4R ~ ((IR)r o IF)(X(x))[((IR)o o Io)(a)]. Let A J U o ~ Y(y)[a], where a e M. Then
A4 ~ R(y)[Io(a)]. By the definition of IR,
A4 R ~ (IR) F(R(Y ) )[( (IR)o o Io )(a)]. Hence
.A4R ~ ((IR)F o IE)(Y(y))[((IR)o o Io)(a)]. Let A4x/o ~ S(x,y)[a, b], where a,b E M. Since .MI/o ~ ~I/O, we have
Mx/o ~ X(x)[~] and
•A41/o ~ Y(y)[b]. Hence
MR ~ ((IR)r o XF)(X(x))[((irR)Oo Io)(a)] and
M R ~ ((IR)F o IF)(Y(y))[( (IR)o o Io)(b)]. Hence
M R ~ (IR)F o IF(S(x,y))[((IR)o o Io)(a), ((IR)o o Io)(b)]. (2) M has an n-cry relation R" , where n > 2. M U o = < U; X, Y, S > is defined as follows:
U= X = Y = S=
M, {x]x e M , ( 3 y l ' " y n - t ) R n ( x , yl "''yn-1)}, {y[y e M , ( S x l " " X n - 1 ) R n ( X l "' "Xn--l,y)}, ((x,y)]x E X, y E Y}.
Io is defined as the identity. The basic interpretations are defined by: X(x) Y(y) S(x,y)
--, (Syl •. • Y n _ l ) R n ( x , y l • • • Yn_l), --* ( 3 x l . • • X n _ l ) R n ( x l • •. Xn_l, y), --* ( 2 y l . . . Y n _ l ) R n ( x , y l ' - . Y n _ l ) A ( 3 X 1 • • •X n _ 1 ) R n (x1 •• •Xn_l, y).
It is clear that the above F - m o r p h i s m
is a ~ F - m o r p h i s m
of A 4 U o
to A{.
6.2. HIERARCHY IN GENERAL SYSTEM MODELS
131
I B.~=u.. sy,~., I
I by=
,
F i g u r e 6.1 A Hierarchy of General System Models The relationships between levels of the system models are based on structure expansion. Faithfulness: Let IR be an arbitrary F-morphism of A{ to A4R, where M R is an arbitrary system model. Let .h,tl/O ~ X(x)[a], where a e M. Then
.M ~ Ir(x(~))[ro(a)], that is,
.h/[~ (3yl... Yn_l)Rn(x, Yl"'" Yn-1)[a] • Hence there exists some bl,...,bn-1 E M such that
A4 ~ Rn(x, Y l " ' " Yn-1)[a, bl . . . . . bn-1]
CHAPTER 6. HIERARCHY
132 Hence
]~4R ~ (IR)F(Rn(x, Y l " " Yn-1))[(IR)o(a), (IR)O(51),..., (IR)o(b,~-l)]. Therefore A4R ~ ( 3 y l " '" Yn-1)(IR)F(Rn(X, y l "'" Yn-1))[((IR)o o Io)(a)]. The proofs about Y and S are similar to that of X. (3) A4 has an n-ary function fn, where n >_ 1. A Q / o = < U; X, Y, S > is defined as follows: g ~--
M~
X= Y = s =
M, {y[y e M n f ( M n ) } , {(x,y)lx e X,y e Y}.
Io is defined as the identity. The basic interpretations are defined by:
x(x) Y(Y)
S (x, y)
(3Xl""" Xn-lY)(f(x, xl,..., Xn-1) = Y), -' (~x1""x,)(f(xl,...,x,) = y), "-* (3Xl""Xn_lY)(f(x, xl .... ,Xn-1) -- y)
A ( 3 x l ' ' ' X n ) ( f ( x l , . . . , Xn) = y). The rest can be similarly proven to (2). (4) A4 has a constant co.
.A4i/o = < U; X, Y, S > is defined as follows: U ~
M,
X= Y= s =
M, {co},
{(z,y)lx e x , y e Y}.
Io is defined as the identity. The basic interpretations are defined by: X(x) Y(y) S(x, y)
~ x=coVx=co, --* y - - c o , ~ (x = co V -~x = co) ^ y = co.
The rest can be similarly proven to (1).
[]
As seen in the proof, we can regard any system model quite naturally as an input-output system model.
Chapter 7
SYSTEMS
PROPERTIES
Since we obtain a system model as a result of a modeling process, which fully depends on how a model builder recognizes an object as a system, the obtained system model and the language to describe it reflect the systems recognition of the model builder. The multifacetted systems recognition, which can be done at various levels of recognition to an object, results in various types of system models. Thus we get a class of system models according to our interests. Systems properties are the properties possessed by system models obtained through modeling and so represented by sentences of the languages for the models. A systems property of a system model is expressed as a set of sentences of the language for the system model rather than as only a sentence of it. Furthermore when an object is recognized as a class of system models, a systems property of that object should be represented as a kind of conjunction of sentences of the languages for the system models, the sentences that represent how the systems property is recognized in each system model. We will define a class of systems properties for a given class of system models as a set of cartesian products of sentences. Each component of the products represents how a system property is recognized in a system model. We will also define an equivalence relation on a given class of systems properties, using the concept of an F-morphism, which can clarify the relationships of the systems recognition used for getting the system models in the given class, since an F-morphism gives an interpretation between the system models. By this equivalence relation we can characterize a class of systems properties with respect to the interests used for the system models whose languages express the systems properties. As a concrete example of characterization, we will consider two types of state transition models: $1
=
S2 =
,
,
CHAPTER 7. SYSTEMS PROPERTIES
134
where C: the state set, 6t: the state transition at t,i.e., 6t : C --* C, and/~: the state transition, 6 : C × T ~ C. These two models are usually considered to be able to represent the same class of systems properties. However, the systems recognition of the two is different with respect to time. From the point of view of the role of time we will characterize some systems properties.
7.1
Formulation of Systems Properties
First we extend the concept of satisfaction to the system models of possibly different types to compare formulas that are represented by different type languages, although in the usual first order logic the satisfaction of a formula is defined only for a realization of the language that can express that formula. D e f i n i t i o n 7.1.1 ( S a t i s f a c t i o n in a Class K of Models) Let K be a class of models of possibly different types. Let A~i, 2vlj E K. For any P E Form(E.(Mi)), an assignment p F-satisfies P in
J~j
if and only if there exists an F-morphism I of
]vll to Jk4j such that Io(p) satisfies IF(P) in J~j. Then we write
]V4j ~1~"P[p]. If P is a sentence, we omit p and write M j ~ K P. In Example 4.2.2, IF(¢I),IF(¢2) and IF(¢3) are satisfied in ~4 r. Therefore ¢1,¢2 and ¢3 are F-satisfied in ]vt r, where K = {J~4,J~}, although the usual satisfaction concept is not applicable to them in .M ~. If two languages £(.A~41) and £:(M2) are not disjoint, we can make them disjoint by renaming the symbols belonging to the intersections of £(Jk41) and Z:(A~2). In the following discussions, K denotes a class of models whose types are not necessarily the same, and without loss of generality we assume that the languages of the models in K are disjoint unless explicitly mentioned otherwise. D e f i n i t i o n 7.1.2 (Class o f M o d e l s d e t e r m i n e d b y P in K ) Given K, let A/[i E K. For any P E Form(E~(Jvti)) and any assignment p the class of models determined by P in K ,MLp(P, K), is defined by:
MLp(P,K) = {MjlAAj e K, M j ~K P[P]}. C o r o l l a r y 7.1.1 Let M E K. If P E Sent(£(M)), then for any assignments p, pr
MLp(P, K) = ML#(P, K).
7.1. FORMULATION OF SYSTEMS PROPERTIES
135
Proof: We assume that 2vii ~ MLp(P, K). If A4~ = ~4, then since P is a sentence, 2~4i ~ P[p] implies A4i ~ P[pq. Hence
Mi e ML# (P, K). If M i ¢ ]vi, then
that is, there exists aal F-morphism such that
M~ ~ XF(P)[Zo(p)]. Since from the definition of F-morphism, the number of the free vaxi~bies in ~ formula is not increased by IF, IF(P) is also a sentence of £(.M~). Therefore
.Mi ~ IF(P)[Io(p')], that is,
Hence 2vii E ML#(P,K). So we have MLp(P, K) C ML#(P, K). The converse inclusion, ML# (P, K) C MLp(P, K), can be similarly proven. [ ] When P is a sentence, we write ML(P, K) for MLp(P, K) simply. We define the equivalence relation on the class of formulas that may be described by more than one type of languages, using F-satisfaction.
Definition 7.1.3 ( K - E q u i v a l e n c e on F o r m u l a s ) Set
Form(K) = U(Form(£(.Mi))lA~i E K). Let P,Q E Form(K), where there exist ,~4i,AAj E K such that P E Form(f~(.Mi)) and Q ~ Form(£(A4j)). P and Q axe said to be K-equivalentwith respect to p and p~, denoted by
P =--l,-(p,#)Q, if MLp(P, K) = MLp! (Q, K). C o r o l l a r y 7.1.2 Let P, Q E Form(K). If there exist A4i,.Mj E K such that P E
Sent(g(AA~)) and Q E Sent(£(2vIi)), then for any assignment p and p~, P =--K(p,#)Q if and only if ML(P, K) -- ML(Q, K).
CHAPTER 7. SYSTEMS PROPERTrES
136 P r o o f : Let P =g(p,¢) Q. By the definition of K-equivalent,
MLp(P, K) = ML¢ (Q, K). Since P and Q are sentences, from Corollary 7.1.1 we have
ML(P, K) = MLp(P, K) and ML(Q, K) = ML¢(Q, K). Hence ML(P, K) = ML(Q, K). Conversely, let ML(P, K) = ML(Q, K). By Corollary 7.1.1, for any p and pr we have
ML(P, K) = i i p ( P , K) and ML(Q, K) = ML¢(Q, K). Hence for any p and pl,
MLp(P, K) = ML¢(Q, K).
[] Based on Corollary 7.1.2, we define the equivalence relation on the class of sentences as follows. Definition 7.1.4 ( K - E q u i v a l e n c e on Sentences) Set Sent(K) =
e K).
Let P,Q E Sent(K), where there exist ¢~4i,,~4j E K such that P E Sent(£(AAi)) and Q E Sent(£(.Mj)). Then P and Q are said to be K-equivalen~ denoted by
P-Q, if ML(P, K) = ML(Q, K). There may be a systems property that cannot be represented by a given language. Since we want to compare with systems properties represented by more than one type of languages, we introduce the empty sentence that denotes that there is no sentence representing a given systems property.
Definition 7.1.5 (Empty Sentence) Given K, let ~:(K) = {£(.~i)lJ~i E K } . Let 7r be a symbol that is not included in any £ E £(K). For any £ E £(K) we set Sent*(£) =
Sent(L:) u {Tr},
where Sent(E) N {~r} = 0. We call r the empty sentence. To extend the notation ML(P, K), we define ML(r, K) = ~r.
7.1. FORMULATION OF SYSTEMS PROPERTIES
137
Definition 7.1.6 Given E C Sent*(~.), we set
ML(E, K) = ~ f-I(ML(¢, K ) I ¢ E E),
l
7r,
if 7r is not in E; ifTr E E.
Definition 7.1.7 ( K - E q u i v a l e n c e o n Classes o f Sentences) Let E, r C Sent*
(K) = Sent(K) U {Tr}, where there exist 2k4i, J~j • K such that E C Sent*(f~(Adi)) and F C Sent*(f.(A/tj)). Then E aald F are said to be K-equivalent, denoted by E ---=KF, if ML(E, K) = ML(I', Z). Systems properties are represented by each language in £.(K).
If we cannot
represent a systems property by a given language, we assign 7r to the property. Definition 7.1.8 (Class of S y s t e m s P r o p e r t i e s ) Given K,
SPK C II(~(Sent*(Z))iZ • £(K)) is said to be a class of systems properties on K, where II denotes the usual cartesian product of sets, and p(X) denotes the power set of X. A systems property on K is represented as an element of SPK. It is necessary to specify SPK according to a problem we want to examine. A systems property belonging to SPK is different from such systems properties as emergence, complexity, hierarchy, and so on although these can be surely said to be a kind of systems properties of systems theory in the usual sense. Since we can consider SPt( as a subset of the following set, { f i r : Z ( K ) ~ U(p(Sent*(£.))I£ • £ ( g ) ) , f ( £ . ) • p(Sent*(£))} so we can define a projection as follows:
PR£ : SPI~ --* p(Sent*(Z)), P R £ ( f ) = f(£.). We define K-equivalence on SPK using this projection. Definition 7.1.9 ( K - E q u i v a l e n c e o n SPK) Given K and SPK, let P, Q •SPtt'. P and Q are said to be K-equivalent on SPI~., denoted by
P - s p ~ Q, if for any Z • Z:(K), PR£(P) =-x~ PR£(Q).
CHAPTER 7. SYSTEMS PROPERTIES
138
The meaning of P -sP~,- Q can be considered as follows. Let K = {AA1,A42. . . . . A/In}, P and Q be E1 x E2 x ... x En and F1 x F2 x • .. x Fn, respectively, where for each i, Ei, Fi C Sent*(f.(A4i)). Each model in K can be considered to be obtained according to one's interest. That is, the languages £:(A41), £(¢~A2),..., and £(A~tn) reflect one's interest. Since a language is specified by its type, the type of a language represents a part of the systems recognition represented by a model (another part of the systems recognition is represented by the set of axioms specifying the structure of the model together with the language) in the sense that we can identify of what relations the model consists by specifying the type. That a property P is represented by E1 x E2 x ... x En means that the ith aspect of P is represented by the structure .hAi. An F-morphism between models interprets the language of one model by the formulas of another one, and transforms sentences of the former. If the transformed sentence is satisfied in the latter model, the sentence and the transformed one express the same meaning in the two models with respect to the systems recognition represented by them. In that sense the class of models ML(Ei, K) is a set of the models in which P expresses the same meaning with respect to the systems recognition represented by A4i. Therefore
PR£(A4i)(P) =--g PR£(A4i)(Q) means that P and Q are equivalent with respect to the systems recognition represented by AAi. That is, P -sPu Q means that the representations of P and Q are equivalent with respect to the systems recognition for each model in K. We should notice that P =-sPK Q neither mean that P is the "same property" as Q nor P is derived from Q and vice versa.
7.2
Characterization of Some Systems Properties
In this section, using the equivalence relations defined in the previous section, we will characterize some systems properties. To make the discussion clear, we restrict our attention to four systems properties of a state transition system: reachability, existence of a cyclic state and a transition of states, and finiteness of the state set. Suppose that a discrete time state transition system has a state set C and a transition function 6. Using this system and the four systems properties we will investigate the role of the concept of time in our 'systems recognition'. Let K be a class of two models, A4a and .t~42 defined as follows: .A41 = A42 =
< CUT;C1,T, 6t >, ,
where C U T and C are the universes of A41 and A42, respectively, C1, C2 and T
7.2. CHARACTERIZATION OF SOME SYSTEMS PROPERTIES
139
are unary relations, 61 : C x T --* C is a binary function of A41, and b2 : C --* C is a unary function of ,£42. Both A41 and A//2 represent models of state transition systems with no inputs. We should notice that while in J~41 the time index is included, in A42 it is assumed that the time index is given a priori. This difference between A41 and A42 is reflected in the state transition functions, 61 and 62, that is, 61 ca~ take the elements of T as an argument, but 62 cannot do so. (1) SPECIFICATION OF SPK We specify a class of systems properties and languages for A41 and A/t2. Here we consider the four systems properties: reachability(P1), existence of cyclic state with n steps(P2), existence of transition of states(P3) and finiteness of the state set(P4). Since the finiteness of the domain cannot be represented by the first order sentences[Bridge,1977], we consider the state set with n elements. First we define the languages of .~41 and A42:
~(J~l)
and £:(A42).
Z(A41)
should include the structure of the time index, because A41 has a unary relation T that means the time index. Recall the structure of the time (see Definition 3.3.3). Then Z:(A//1) and £(,~42) are: £(A41) L(•2)
= {C1,T,6,} UT, = {c~,62},
where
C1,C2,T 61
62
: unary relation symbols, : binary function symbol, : unary function symbol.
Since 61 and 62 denote the state transition functions of A41 and A42 respectively, they must satisfy the following semigroup properties of 61 and 62 as transition functions. For 61: r,~ 1 = { ¢ }
where ¢ =---( V t l t 2 c ) ( T ( t l ) A T ( t 2 ) A C(e) --* ~ l ( c , t 1 q- t2) = ~l(~l(c, t l ) , t 2 ) ) ; For ~2: The semigroup property of 62 can be considered as t times of compositions of 62, denoted by 62. This can be explicitly defined by: r,~2 = {¢tlt ~ N }
C H A P T E R 7. S Y S T E M S PROPERTIES
140
where for all $ • N ¢t - (Vc)(C2(c) -~ 6~+1(¢) = 6~(62(~))).
This is the usual composition of functions. Since we consider a 'discrete' time state transition system at the present, we can assume that the time index T in A42 is the set of the natural numbers N. We can define a state transition function of A42 as a class of functions with the index t such as {6tit E N}. However, the semigroup property of it is equivalent to that of 62 as mentioned above. Hence to characterize properties concerned with the time index it is sufficient to include not a class of functions but 62 alone in £(A42) without the index t. The four properties P1,P2,P3 and P4 are represented as follows, where (c~1) and (a2) are sentences represented by L:(A//1) and £(A42), respectively. Notice that since the properties include the transition functions or elements of the time index, we have to take ~ ,
YI,~2and ~T into consideration as a part of the properties. However
since ~ 2 is always satisfied in A42 as the usual composition of 62, we need not add ~ 2 to systems properties explicitly. PI: Reachability (~1) (Vcl, c2 • C1)(3t • T)(61(Cl, t) = c2) and ~T U ~'61, (e2) (VCl,C2 • C 2 ) ( v ( 6 t ( c l ) = c2[~ • g ) ) . P2: Existence of cyclic state with n steps (c~1) (3c • C1)(Vt • T)(61(c,t + n) = 61(c,t)) and ~T U ~$1,
(c~2)(3c • C2)(h(6t+n(c) = ~t(c)l~C N)). P3: Existence of transition of states (c~1) (3c E C1)(-~61(c, 1) = c) and ~3T U ~ 1 , (~2) ( ~ c • c 2 ) ( ~ 6 2 ( c )
= c).
P4: The state set with n elements
(e~) (3"!c)(C~(c)), (~) (~"!c)(C2(c)), where for any unary (1-ary) formula ¢, (~!¢)(¢(c))
= ( ~ c ) ( ¢ ( c ) ^ ( V x ) ( ¢ ( c ) -~ x =
¢))
(3~!c)(¢(c)) =- (3c)(¢(c) ^ (~"-4x)(¢(x) A -~x = c)).
7.2. C H A R A C T E R I Z A T I O N OF SOME S Y S T E M S PROPERTIES
141
Strictly speaking, Pl(a2) is not a first order sentence. We cannot represent it without using an infinitary logic. The property P2(a2) is not also a first order sentence. However, P2(a2) can be represented by the first order logic by considering P2(a2) as a countable set of the sentences for each t E N: P2(~2) --= {P2(a2)t[P2(a2)t = (3c E C2)(~t+n(c) = 5t(c)),t E N}, We take P1,P2,P3 and P4 as the systems properties under consideration, that is, SPK = {Pl, P2, P3, P4} C v(Sent*(£(A41))) × ~)(Sent*(£(.t~42))), where K = {AA1, 2A2}, P1 -----P l ( a l ) X {~¢}, P2 - P 2 ( a l ) x P2(a2), P3 -- P 3 ( a l ) x P3(a2), P4 --- P4(al) x P4(a2). Since we can take various sets as the state set C, K is not uniquely determined. This is crucial for the construction of F-morphisms as considered in the next subsection.
(2) CONSTRUCTION OF F-MORPHISM We assume that .M1 ~ P l ( a l ) A P2(al) A P 3 ( a l ) A P4(al) and •£42 ~ P2(a2) A P3(a2) A P4(a2). First we have to define a basic morphisn~ between A41 and 2¢t2. Since Adl includes the time index, but Ad2 does not, it is difficult to define a basic morphism of 2~41 to A42 unless the structure of the state set C in A42 includes that of the time index. On the other hand, we can define naturally a basic morphism and basic interpretations of A42 to .~41 in the following way: A basic morphism, Io : C ~ C U T, is defined as an embedding: The state set C2 of.h~2 Call be interpreted as the state set C1 of A~I. Hence a basic interpretation
of C2 is defined by C1 (x). The state transition function 52 of AA2 is equivalent to the state transition function 51 of 2dl at the time 1. So we get quite naturally a basic interpretation of 52 as 51(x, 1) = y, that is,
IF(52(X) =
y) = (51(x, 1) = y).
A function It- of F-morphism I = < Io, IF > is determined by the above basic interpretations. Using IF let us transform the four systems properties, Pl(a2), P2(a2), P3(a2) and P4(a2), and evaluate the classes of models determined by them in K.
CHAPTER 7. SYSTEMS PROPERTIES
142
Since P l ( a 2 ) = {~}, we have ML(PI(a2), K) = ~r by Definition 7.1.6. For any t,
Ir(P2(o~2)t) ---- (3c)(3XlX2)(3yl ... Y t + n _ l ) ( 3 z l . . . Zt_l)(Cl(c ) A (xl ----x2) A(51(yl,
1)
= x l ) A (51(y2, 1) = Yl) A (51(Y3,
1)
= Y2)
A . . . A (51(c, 1) = Yt+n-1) A(51(zl, 1) = x2) A (51(z2, 1) = Zl) A (~1(Z3, 1) ----Z2)
A"" A ((~1(c, 1) ---- Zt_l) which is equivalent to (qc 6 C1)(51(c t + n) = 51(c,t)). Therefore for any t,
.A41 ~ IF(P2(a2)t), and so
.'~1 ~ IF(P2(o~2)). Hence we have
ML(P2(a2), K) = {J£41, A42} Similarly,
IF(P3(a3) = (3¢)(C1(c) A -~51(¢, 1) ----c) which is equivalent to (3c E C1)(~l(C, 1) -~ c). Hence
AA1 ~ IF(P3(a2)), so we have
ML(P3(c~2), K) = {A41, A42} Furthermore, obviously ML(P4(a2), K) ----{A41,A42}. As mentioned above, it is difficult to generally define a basic morphism of A41 to A42, because it depends on whether or not the state set in fl42 has the structure of the time index.
Therefore for the evaluation of the classes of models for K
determined by P l ( a l ) , P 2 ( a l ) , P 3 ( a l ) and P 4 ( a l ) we have the following two cases: Case A: In A42 there is an interpretation of the time index of AA1; Case B: A~2 has no interpretation of the time index of M1. We can easily find examples satisfying the above cases as follows.
7.2. CHARACTERIZATION OF SOME S Y S T E M S PROPERTIES
143
Example of Case A: K = {J~1,./~42},
.A/J1 ----- and A42 =< C;C,~2 > where
C = {cl,c2}, Cl # c2, T is the set of the n a t u r a l numbers, and ~1 and 6a are defined by: for any t • T, ~fl(cl,t) = c2 and 6l(Ca, t) = ca; ~f2(cl) = c2 and ~fa(c2) = ca. I n this case a basic morphism, Io : C U T --, C, can be defined by: Io(x) = cl, if x = Cl • C or 0 • T, and Io(x) = c2, otherwise. Hence basic interpretations of the language for A41 can be defined as follows:
Cl(X)
-,
C2(x)
T(x)
-,
C2(x)
~fl(x,t) -- y t l_t 2 x+y=z
--,
x = 0 x = 1
~f2(x) = y A t = t
---*
t 1 =t 2A~2(tl)=t
2
--* ( - ~ 6 2 ( x ) = y ~ z - - - - - x ) A ( - ' b 2 ( Y ) - - x - - ' z = Y ) A(6a(x) = y A ~f2(y) = x ~ z = x) ~ -~62(x) = x --* ~f2(x) = X
Each interpretation is one of the possible ones. We can define other basic interpretations. The systems properties transformed from P1(~1), P 2 ( a l ) , P 3 ( ~ l ) and P 4 ( ~ l ) by F - m o r p h i s m with the above basic interpretations are trivially satisfied in .A42.
Example of Case B: K = {A41,A42}, J~tl = < C U T ; C , T , ~ b < , + , O , 1 > and .A42 =< C;C,~2 >, where
C , T and 61 axe the same as those of Case A, and ~f2 is defined by: ~f2(Cl) = c2 and ~2(C2) = Cl.
In this example even if we can define a basic morphism of A41 to Ad2, we cannot find a formula q)(x, y ) • Form(£(.A42)) with free variables x and y such t h a t (~(x, y ) satisfies the axioms of the linear ordering < on A41 as shown below.
Lemma 7.2.1 Let q'(x) be a formula of £(M2) that is free precisely in x. Then for cl, c2 • C M2 ~ ¢(x)[cl]
i/and only il M2 ~
O(x)[c2].
Proof." Let (I)(x) be equivalent to the following prenex normal form: • (x) = ( Q l X l ) ( Q 2 x 2 ) ' ' " ( Q n x n ) V ( A ~is( x, x l , . . . ,Xn)) i j where each Qi(i = 1,... ,n) is V or 3 and ~ i j ( x , x l . . . . . Xn) is a literal. We assume t h a t M 2 ~ ~(x)[cl].
CHAPTER 7. SYSTEMS PROPERTIES
144
Let d l , . . . , dn (where di = Cl or c2) be assignments of X l , . . . , Xn holding that ,a~t2 ~ V ( A q~ij(x, X l , . . . , Xn))[cl, d l , . . . , d,l. i j Since each @q(X, Xl . . . . . Xn) is a literal, it is one of the following forms: C(8~(x)),-~C(5~(y)), 5~(x) = 5~n(y),-~8~(x) = 5~n(y), where 6~ denotes n times compositions of 62. For i = 1 , . . . , n , we set d ~ = { c2, cl,
ifdi=cl ifdi=c2
Then the truth value of q~ij(X, Xl .... , Xn)[cl, d l , . . . , dn] in ~t2 is the same as that of
~ij (x, X l , . . . , Xn)[c2, d~,.. •, d~] in AA2. Hence we have
M2 The if part can be similarly proven.
[]
L e m m a 7.2.2 Let ¢ ( x , y ) be a formula of £(.&42) that is free precisely in xand y.
Then for cl, c2 E C M2 ~ ¢(x, y)[cl, c2] if and only if.A42 ~ q~(X,Y)[C2, Cl].
[]
Proof: Similar to the proof of Lemma 7.2.1.
The above two lemmas show that in ¢~42 any interpretation of T and < of A~I cannot distinguish between cl and c2. Therefore we have the following proposition. P r o p o s i t i o n 7.2.1 There is no formula O(x,y) of £(,£42) interpreting the linear
ordering < of ./t~1 such that O(x,y) satisfies ET, where O(x,y) is free precisely in x and y. Proof: We assume that ¢(x, y) is a basic interpretation of < satisfying ET. Then ¢(x, y) necessarily satisfies the following sentences belonging to ET: (Vxy) (¢T(X) A ¢T(Y) ~ q~(x, y) V O(y, x))
(7.1)
(corresponding to ¢4) (Vxy)(OT(X) A ¢T(Y) --* (¢(x,y) A ¢ ( y , x ) ~ x = y))
(7.2)
7.2. CHARACTERIZATION OF SOME S Y S T E M S PROPERTIES
145
(corresponding to ¢3) i8 a basic interpretation of T. By the definition of F-morphism, ¢T(X)[Cl] or CT(X)[C2] holds in M2. Hence by Lemma 7.2.1, CT(X)[Cl] and ~T(X)[C2] hold in M2. From 7.1 ~(x,y)[cl,c2] or • (x, y)[c2, cl] holds in M2. By Lemma 7.2.2, ¢(x,y)[cl, c2] and ¢(x, y)[c2, Cl] hold in M2. Therefore from 7.1 we get cl = c2, which contradicts cl # c2. [] w h e r e (I)T
Proposition 7.2.1 means that no formula ¢(x,y) of £(~42) satisfies I3T. Since P l ( a l ) , P2(Crl) and P3(c~1) include I3T, they cannot be F-satisfied in M2. On the other hand, P4(al) does not include T or 61. Hence P4(a]) is F-satisfied in M2. Thus we can surmnarize the results about the characterization of SPIc as follows: T h e o r e m 7.2.1 (,4) If there is an interpretation of the time index in M2, we have
the following result. (Table 7.1) P1 I P2 M L ( P i ( a l ) , K ) {M1,M2} [ {M1,M2} ML(Pi(a2),K) rc {M1,M2}
P3 {M1,.)~2"}' {M1,M2}
P4 {M1,M2} {M1,M2}
Table 7.1 Therefore SPK is partitioned into the two classes: S P K / -=--K= {{P1}, {P2, P3, P4}}. (B) Otherwise, we have the following.(Table 7.2) P4 ML(Pi(al),K) ML(Pi(ot2),g)
{
1}
{.A't1} {.A/t1} {,~41, A/f2} {M1,A42}
{Aal,2a2} {M1,~2}
Table 7.2 Hence SPIt'/----K = {{P1}, {P2, P3}, {P4}}. Proof." It is clear from the above discussion.
[]
.A42 does not include the structure of the time as a part of the language of A//2. This shows that when building the model .&42, we recognize the structure of the time at the different level from, say, the state set. If there is an interpretation of the time index in ,£42 by an F-morphism, we can bridge the gap between the two kinds of recognition; that is, the properties (P2 and P3) represented by the sentences including the time index are equivalent with respect to the systems recognition for A/t1 and ,~A2 to the property (P4) represented by the sentences not including the
146
C H A P T E R 7. S Y S T E M S PROPERTIES
time index.(Theorem 7.2.1(A)) However without interpretations of the time index, the Sentences including the time index (P2(al) and P3(~1)) do not express the same meaning as P4 with respect to the systems recognition for ¢~42. (Theorem 7.2.1(B)) On the other hand, there is a property (P1) that cannot be represented by sentences unless the language of a model includes the structure of the time. Thus we get at least the following three categories about systems properties: (1) Properties that essentially depend on the structure of the time in the sense that we cannot represent them unless the languages for describing them include the structure of the time.(Say P1.) (2) Properties for whose description models have to include the structure of the time, but that are equivalent to properties not including the structure of the time with respect to the systems recognition for the models in K, if there is an interpretation of the structure of the time in the models.(Say P2 and P3.) (3) Properties that axe independent of the structure of the time.(Say P4.)
Chapter 8
FURTHER TOPICS MORPHISM AND UNIVERSALITY 8.1
ON
Institution
In this section, we will present a unified framework for morphisms using the concepts of category theory. The categorical framework defined in this section is called "institution," introduced by Goguen [Goguen et al. 1985]. Since the language in an institution for description of the properties of a system model is not specified by a particular language such as the first order language employed in this book, Institution can be considered to give a general framework for the relationships mnong system models. We, however, should notice that the institution developed by Goguen is a general framework not for the F-morphisms, but for the associative F-morphisms. The following theorem [Goguen et al. 1985], called "Satisfaction Condition," will play a main role in institution. T h e o r e m 8.1.1 (Satisfaction Condition) Let M1 and J~2 be system models, and T1 and 7-2 the types of M1 and M 2 respectively. Sent(£(M1)),
Let T1 C T2. For any ¢ E
M2 ~ IF(C) if and only if M2{T1 ~ ¢. Proof: Let ¢ e Sent(£(M1)). By F-morphism Theorem,
M2 V xF(¢) if and only if AA1/I ~ ¢. By Proposition 6.2.1, A,t l / I ~ ¢ if and only if A'/2]T1 ~ ¢.
148 C H A P T E R 8. FURTHER TOPICS ON MORPHISM AND UNIVERSALITY Therefore ]v42 ~ IF(C) if and only if AA2[n ~ ¢.
[] We define an institution for the associative F-morphisms. Definition 8.1.1 An institution for the associative F-morphisms A F is defined as the following quadruplet: A F =< AssSig, AssSen, AssMod, ~ > where AssSig: a category of languages defined by: Ob(AssSig) = {£1£ is a first order language }, Mor(Ll,£2) -- {(¢1,¢2)1£:1 C £:2}, ¢1 ; {l~i]i E /'1} "~ {l~ili e 12}; 1:~| ~ 1:~|, ¢2: {fjlJ e J1} "* {filJ e J2}; fj ~ fj, AssSen: a functor giving sets of sentences, defined by: AssSen : AssSig ---* Set AssSen(£:) = Sen~(£:), AssSen(¢) : AssSen(£:l) ~ AssSen(£:2), where ¢ e Mor(£:l, £:2). Note that since there trivially exists a function IF : Form(£:1) ~ Form(£:2) such that for any symbol S in £:1, Bas(S) = S, AssSen(¢) can be defined as the restriction of that IF to Sent(£:l). AssMod: a functor giving models, defined by: AssMod:AssSig --* Cat °p AssMod(£:) is the category of the models for £: (with homomorphisms as morphisms). AssMod(¢) : AssMod(£:2) ~ AssMod(£:l), Assiod(¢)(.hA) = A~tt£:l, AssMos(¢)(f) = ft£:1, where fi£:1 is a homomorphism in models for £:t that is the same as f for the symbols in £:1. is the relation of satisfaction between models and sentences in the first order language such that for any ¢ E Mor(£:l, £:2) in AssSig, .M ~ AssSen(¢)(¢) if and only if AssMod(¢)(.A4) ~ ¢
149
8.2. CANONICAL SYSTEM MODEL AS FREE STRUCTURE holds for any .M in Ob(AssMod{~2)) and any ¢ in AssSen(~l).
AF is essentially the same as the institution of a many-sorted first order logic defined by Goguen [Goguen et al. 1985]. An institution is defined as a generalization of AF. D e f i n i t i o n 8.1.2 ( I n s t i t u t i o n ) An institution Z is a quadruple 2" = < Sign, Sen, Mod, ~£:>, where
Sign:a category of languages, Sen: a functor giving sets of sentences, i.e., Sen : Sign ~ Set, Mod:a functor giving models for £:, i.e. , Mod : Sign ~ Cat °p, ~£C Ob(Mod(£) × Sen(£)) is a satisfaction relation such that for any ¢ e Mor(£, ~) in Sign, ~4 ~d Sen(¢)(¢) if a~ld only if Mod(¢)(2~4) ~
¢
holds for any ¢Q in Ob(Mod(~)) and any ¢ in Sen(£). Notice that languages consisting of the category Sign are not necessarily first order languages. In fact Goguen defined Sign as a category of "siguatures" without specifying the type of language [Goguen et al. 1985]. The institution defined above is not only concerned with the associative Fmorphisms such as in AF, but intended to give a general framework for t h e relationships of models. However, the condition on satisfaction relation shows that the institution is essentially concerned with the morphisms between models possessing the same kind of the "associative" properties that can be found in the associative F-morphisms. If we use F-morphisms as morphisms in the category of models, the condition on SF-morphisms could be considered to correspond to that on satisfaction relations.
8.2
C a n o n i c a l S y s t e m M o d e l as Free S t r u c t u r e
In characterizing the realization by the universality, the existence of a homomorphism between system models plays a main role. Even if a system model satisfies a set ~ of sentences, however, a homomorphism does not necessarily preserve ~.
150 C H A P T E R 8. FURTHER TOPICS ON MORPHISM AND U N I V E R S A L I T Y Hence it is natural to consider the realization that uses homomorphisms preserving E in a class of system models; that is, strong E-homomorphisms. This section introduces the concept of free E-structure that is essentially a universal element in terms of E-homomorphisms, and shows that the canonical system model is a free E-structure and thus is a minimal model with respect to the order determined by strong E-homomorphisms. We define free E-structure, following Gr£tzer [Gr£tzer 1979]. Recall the following definition. D e f i n i t i o n 4.1.7 (repeated) Let J~4 be a system model such that AA ~ E, and 0 ¢ H C M. Then the set [H]~ is defined by the following. Ho = H;f-In-1 = {a e Mla = t[al,... ,an],t is a term, a l , . . . ,a~ E Hn-1}; Hn = ~rn-1 l.J {a e M[ there exist bl,... ,bt e I-In-1
such that a is a E-inverse of b l , . . . , bt in AA). Then [H]s = U(Hili E N). D e f i n i t i o n 8.2.1 Let M be a system model such that M ~ E, and O ~ H C M. If ~4' is a subsystem model of M and M' = [H]~., where M' is the base set of M ' , then we say that H E-generates .hAr and H is a E-generating set of ]~4'. We can see easily that M ' E-generated by H is a smallest subsystem model of A4 among subsystem models that satisfy E and preserve E-inverses. D e f i n i t i o n 8.2.2 (Free E - S t r u c t u r e on M) Let a be an ordinal, and M C Mod( E). F z ( a ) is a free E-structure on M with a E-generators if the following conditions are satisfied:
1. F~(a) ~ E; 2. FE(~) is E-generated by elements xo,... ,x~,... (~/< ~); 3. if~.4 E M, ao,... ,a~,... E M(T < a), then any mapping f : x~ ~ av,'y < a, can be extended to a strong E-homomorphism of Fv~(a) into A4. Every C A N ( E ) is not necessarily a free E-structure. We need the following conditions to show that C A N ( E ) is free: (A) E is complete; (B) for any • E E, any I < e(O), • - l inverse is uniquely determined;
8.2. CANONICAL SYSTEM MODEL AS FREE STRUCTURE
151
(C) an extended strong p,-homomorphism of f of the third condition in Definition 8.2.2 is an onto-map. We impose the condition (C) on M in Definition 8.2.2. We assume t h a t a consistent E and CAN(P,) are given. For Mod(E) a mapping ~/of CAN(P,) to 3,4 is defined as follows: 'l([¢k]) = ck (where ck is an interpretation of the constant c k in f14); ,/([fj (tl . . . . . tv(j))]) = fj (U(~I), • • •, ~(tvij)). This ~/is obviously well-defined. By the definition, ~/is a homomorphism. We define the following class on which the condition (C) is imposed:
K~ = {.A4]A4 E Mod(p,) and ~7 is onto.}. The following theorem on the minimality of CAN(~) with respect to _E holds. As mentioned before, if ~ is complete, we have CAN(~) ~ 2. So we can omit the condition t h a t CAN(P,) ~ 2. T h e o r e m 8.2.1 If both (.4) and (B) hold, then CAN(P,)is a free E-structure on K2 with the set of constants as a p,-generating set. Proof: 1. By the assumption, CAN(P,) I==P,. 2. It is clear t h a t CAN(P,) is p,-generated by the set of constants. 3. It is sufficient to show t h a t for A4 E K~, ~ : CAN(P,) ~ .A4, as defined in the above, is a strong P,-homomorphism. Since ~? is a homomorphism, we show t h a t preserves ¢ - 1 inverses. For simplicity as usual, we show it about a formula ¢ =_ ( V x ) ( ~ y ) ( V u ) ( 3 v ) ( V w ) V ( x , y, u, v, w ) E P,, where ~I' has no quantifiers. In general we can prove theorem by induction on e(~). Since e ( ~ ) -- 3, it is sufficient to show it about ~ - 1 and ~ - 2 inverses. We assume t h a t b E CAN(YJ) is a ¢ - 1 inverse of a E CAN(P,) in CAN(P,). We want to show t h a t y(a) is a • - 1 inverse of y ( b ) in f14. Let c and e E M be arbitrary. Since ~ is onto, there exist c and e E CAN(P) such t h a t ~?(c) = c and ~?(e) = e. Since b is a • - I inverse of a, we have for some d
CAN(~) ~ ~ ( a , b, e, d, e).
152 CHAPTER 8. FURTHER TOPICS ON MORPHISM AND UNIVERSALITY Let ~ be equivalent to the following disjunctive form:
~-= VhRij, i j
where Rij is a literal(i.e, either atomic or its negation), and
Vi and
tions and conjunctions, respectively. If i = j = 1, then we have A4 ~ ~ ( a , b , c , d , e ) . Because in case • -----R (atomic),
CAN(Z) ~ R ( a , b , c , d , e ) iff
Z F R(a, b, c, d, e) (by the definition of CAN(Z))
which implies 3,t ~ R(a, b, c, d, e)
because A4 ~ Z,
and in case • - -~R,
CAN(Z) ~-~R(a,b,c,d,e) iff
not ~ F R(a, b, c, d, e)
iff
E F -~R(a,b,c, d, e) (~. is complete)
which implies A4 ~ -~R(a, b, c, d, e). If it is not the case that i = j = 1,
CAN(Z) ~
V
i j
iff
V(A(cAN(Z)~ i
Rti)).
j
Since CAN(~) ~ tt|j implies Ad ~ R U for each i and j, and
V(A(M kRu))iffA4 I= VAau, i
j
i j
We have A4 ~ ~(a, b, c, d, e). Since ~} gives the denotation of • into A4, we have A4 ~ ~(~/(a),~}(b),~}(c),z}(d),~/(e)).
Aj are disjunc-
8.2. CANONICAL SYSTEM MODEL AS FREE STRUCTURE
153
Since ~(c) E M is arbitrary, we have A4 ~ (Vu)(3v)(Vw)fft(Tl(a), y(b), u, v, w). Hence ~7(a) is a q2 - 1 inverse of 71(b) in A4. Next we assume t h a t d E
CAN(E) is a • - 2 inverse of a,cE CAN(E). We want
to show t h a t ~/(d) is a (I) - 2 inverse of ~/(a),y(c) in A4. It is sufficient to show t h a t there exists a b E M such t h a t
(71(a), ~) ~ S~, and (~](a), b, ~/(c), ~/(d)) E S¢ 2. By the assumption, it holds that there exists a b in
CAN(E) such t h a t
CAN (E) ~ (Vu)(3v)(Vw)O(a, b, u, v, w) and CAN(E) ~ qz(a, b, c, d, e). By a similar argument above, we have A4 ~ @(~/(a), ~/(b), y(c), ~/(d), y(e)).
CAN(F~), then y ( b ) is a (I) - 1 inverse of ~(a) in A/l, so by p u t t i n g y ( b ) to/~, y ( d ) is a (I) - 2 inverse of ~(bfa), ~l(bfc) in A4. Since if b is a (I) - 1 inverse in
Conversely, we assume that b E M is a (I) - 1 inverse of y(a) in A4, where
a E CAN(~). Then we will show t h a t there exists a b E CAN(E) such t h a t ~(b) = b and b is a (I) - 1 inverse of a in CAN(E). By(I) E ~, b- (3y)(Vu)(Sv)(Vw)k~(a, y, u, v, w). So there exists a b such t h a t
CAN(E) ~ (Vu)(3v)(Vw)il~(a, b, u, v, w). T h a t is, b is a • - 1 inverse of a in
(8.1)
CAN(E).
By the previous argument, 7](b) is a • - 1 inverse of y(a) in Ad. Since by the assumption (B) (I) - 1 inverse is unique, we get y ( b ) --- b. Next let d E M be a ~ - 2 inverse of y(a), ~/(c) E M in 3,4. Then we will show t h a t there exists a d E
CAN(E) such t h a t y ( d ) = d and d is a g2 - 2 inverse of a,¢ in
CAN(~). 8.1 also holds about c, and CAN(~) ~ ( 3 v ) ( V w ) ~ ( a , b, c, v, w).
154 CHAPTER 8. FURTHER TOPICS ON MORPHISM AND UNIVERSALITY So there exists a d E CAN(P`) such that
CAN(P,) ~ ~ ( a , b , c , d , e ) . Therefore d is a ¢D- 2 inverse of a,c in CAN(P`). By the previous argument, ~?(d) is a ¢ - 2 inverse of ~(a),~](c) in ,M. Since by the assumption (B) ~ - 2 inverse is unique, we get ~](d) = d, which completes the proof.
[]
The condition (A) seems too strong. However, the theorem caamot hold without this condition. (See Appendix 3) For showing the minimality of CAN(P`) we introduce a partial order on the class of objects of STr~.
D e f i n i t i o n 8.2.3 For any J~4, d~4' E Ob(STrp`), J~t _ AA'
iff
A,~ "~ M '
iff
there is an f such that f : JM --* A,ff is a strong p`-homomorphism, .M U ,~4' and ~d' _C .M.
C o r o l l a r y 8.2.1 Under the conditions Theorem 8.2.1, CAN(P,) is the minimal in K~ with respect to the order U_.
P r o o f : It is clear due to Theorem 8.2.1.
[]
Chapter 9
SUMMARY PROBLEMS 9.1
AND
FUTURE
Summary
We believe that LAST is quite effective to obtain "meta"-knowledge about system models, that is, knowledge about a class of system models rather than about a specific system model. We summarize below the main results reported in this book. Any system model can be represented as a mathematical structure. If the language to describe system models and their properties is specified, the structure of a system is given as a pair of the description language and the axioms providing the interaction involved in the system. Since the interaction is specified through one's systems viewpoint and reflected in the language and the axioms, the structure of a system represents how he/she recognizes the object as a system, which recognition is termed the systems recognition. It is essential for the structural similarity between system models to construct a morphism which gives a correspondence between them. In Chapter 4 we classified the morphisms into six categories according to the two bases: 1. whether the types of two models under consideration are the same or not, and 2. what properties of one system model are preserved in another. Table 9.1 illustrates the classification. The F-morphism theorem is a generalized version of the homomorphims theorem into the case of different types; it establishes the connection between F-morphisms and Sr-morphisms. Besides the general concept of an F-morphism, associative F-morphisms were specified, which played a main role in developing systems theory. Institution originally introduced by Goguen could be considered as a general framework for associative F-morphisms not depending on language used.
CHAPTER 9. SUMMARY AND FUTURE PROBLEMS
156
.......... SAME TYPE
GENERATOR AXIOMS TH(.M)
Homomorphism ~-homomorphism S-homomorphism Homomorphism Theorem
DIFFERENT TYPE F-morphism ~F-morphism SF-morphism F-morphism Theorem
T a b l e 9,1 Classification of the Morphisms
Since the concept of an F-morphism gives a fully formal way to interpret one system model in terms of another, we can make a formal discussion of equivaience between two system models of different types, though such discussions have been done so far only in informal way. The equivalence between a finite automaton and a Petri net considered in Chapter 4 serves as an example of such discussions. In Chapter 5, we showed that the canonical system model was a realization of a given pre-structured system model in the sense that it has the universality. Algebraic specification is one of abstract and general models for software system. Within our framework algebraic specification was formulated, and its initial semantics was established as a canonical system model. Chapter 6 and 7 investigated properties of system models constituting part of central problems in systems theory. The properties of system models were divided into two categories according to the scope with which the properties were concerned: systems properties and systemic properties. As for systems properties, a class of systems properties was defined. As an example, specifying some specific systems properties, we obtained a characterization of the time index in a dynamical system. As for systemic properties, we tried some formal definitions of emergent properties and hierarchical structure. These formal definitions would provide a basis for developing formal hierarchy theory. In this book we investigated system models without specifying their types. The advantage of this is invaluable for a meta-approach to system models. (One of outstanding results is F-morphism.) On the other hand, mathematical general systems theory is also regarded as a meta-approach to system models, though input-output system models that are specific type models are employed as fully general ones obtained from systemic generalization of real life [Mesarovic et al.1988]. It should be answered in what sense general system models are general. In Chapter 6, we gave an answer to it. As further topics, in Chapter 8 we presented Institution for unified framework
9.2. F U T U R E P R O B L E M S
157
of similarity and E-freeness of the canonical system model. Since the structure of a system is specified by not only the language but the axioms E, i~ would be expected that the importance of E-homomorphisms should increase in systems theory rather than homonmrphisms.
9.2
Future Problems
This book is just a beginning of the development of LAST. We can find many problems to be solved within the framework developed here, for exanlple, the necessary and sufficient condition for the existence of an F-morphism, construction of the canonical system model of other systems and so on. However, we focus here on more general problems concerned with the framework itself. (1) Problems of the selection of language. In this book first order language was used as description language of system models. But Foo [Foo 1979] pointed out that there are essentially higher order properties like continuity. In selecting language the problems are how other languages such as type theory [Andrews 1986] are applied to the franlework for investigating system models, and what essential distinction or advantage is when using other languages. (2) Axiomatization and categorization of F-morphism. The essence of F-morphism lies in interpretation between system models or systems theories. If this matter is abstracted and axiomatized, the concept of an F-morphism would be effectively applied to the development of systems theory, say, hierarchy theory, without being restricted to "first" order language. Categorization is another way of abstraction of F-morphism. It may be possible to define an F-morphism as a functor between two suitably defined categories. Institution partly realized this possibility. (3) Development of hierarchy theory. It is significant to develop hierarchy theory based on the definitions given in Chapter 5. In developing it abstraction of F-morphism might be necessary, which problem should be considered in association with (2). (4) Validity and approximation. As mentioned in Introduction the problem how to construct a valid system model is crucial for the activities in systems science. In an actual validation process there is no formal manner [Flood et al.1986,Flood 1987]. However, homomorphisms play a basic role as a criterion of validity [Zeigler 1976,1984]. We therefore expect that F-morphisms essentially serve as a general criterion of validity. The point is how to deal with a domain of concern.
158
C H A P T E R 9. S U M M A R Y AND FUTURE PROBLEMS (5) Problems of the preservation of systems properties. One of the great merits of model theoretic approach is to enable us to consider
that properties are preserved under a transformation between system models such as a homomorphism [Foo 1979,Iijima 1983,1985]. For example, a well-known theorem by Lyndon [Lyndon 1959] says that "a sentence of the predicate calculus is preserved under homomorphism if and only if it is equivalent to a positive sentence." Other than homomorphism some theorems about the preservation are known. [Gr£tzer 1979] It is significant and interesting for systems theory to establish a unified theory of the preservation of systems properties under formation of system models such as homomorphism, parallel and serial connection a~ld feedback. The latter two formations require further consideration on the preservation theorem like Lyndon's theorem. (6) Preference of F-morphisms and application to general modeling process For given two system models, we can define, in general, many "natural" Fmorphisms between them. In the situation where we have several "natural" Fmorphisms, we often have to select the "best," or at least "not bad" one. However we have no objective criterion to decide which F-morphism is the best. If we consider an F-morphism as a modeling morphism, the above problem is closely related to a modeling process in which selecting a good modeling morphism is indispensable element for good modeling. In a modeling process a "good" modeling morphism depends on how a model builder "sees" an object and "selects" elements from the object. A suggestive approach to this problem is to use the concept of purpose. We first identify some purposes related to a model being built in a modeling process. Then we consider "fitness" of modeling morphisms to the identified purposes. The main (and hard) problems are 1) how we should represent such purposes and 2) how we should construct a measure for the "fitness." The above problem of selecting a good modeling morphism can be included a more general problem, called representation change [Korf 1980;Benjamin et al.1990], that seaxch a structure representing a system model. Also in this general problem, the selection and preference of modeling morphisms are a central issue, and the approach stated above would be useful for a basic consideration of the problem.
APPENDIX Appendix 1 We give below some definitions in category theory. Definition 9.2.1 ( C a t e g o r y ) A categoryC is composed of a class of objects Oh(C) and a class of morphisms Mor(d). For any pair (A, B) of objects, there is a set
Mot(A, B) of morphisms such that the following conditions are satisfied. (1) There is a composition operation o for morphisms such that for any f E Mot(A, B) and g e Mor(B, C), g o f E Mor(a, c). (2) The composition operation satisfies the associativity: For any f e Mor(A,B),g e Mor(B,C),h E Mor(C,D)
ho(go f ) = ( h o g ) o f. (3) For any object A there is the identity morphism idA E Mor(A, A) such that for any object B, f E Mor(A,B) and g E Mor(B,A)
f OidA = f and ida o g = g. For exanlple, the category of sets Set whose class of objects is the class of all sets; and for A, B E Ob(Set), Mot(A, B) is the set of all functions from A to B. The category of topological spaces Top
whose class of objects is the class of all
topological spaces; a~d for topological spaces A and B, Mot(A, B) is the set of all continuous functions from A to B. Definition 9.2.2 ( F u n c t o r ) Let d and P be categories. A functor F from d to Z) is composed of two functions: a function F from Ob(C) to Ob(Z)) and a function (also denoted "by abuse of notation" by) F from Mor(C) to Mot(D) satisfying the following conditions. (1) F preserves identities; for any A E Ob(C) F(idA) = idF(m). (2) F preserves composition; for any f,g e Mot(C)
F ( f o g) -= F(f) o F(g) whenever f o g is defined.
160
APPENDIX For example, there is a functor F : T o p --* Set that assigns to any object A
of T o p , the underlying set A a~d to any morphism of T o p , the corresponding function on the underlying sets. This functor is called a forgetful functor (which "forgets" topological structure). D e f i n i t i o n 9.2.3 ( U n i v e r s a l M a p ) Let G : A ~ B be a functor and let B E Ob(B). A pair (u, A) with A • Ob(A) and u : B ~ G(A) is called a universal map for B with respect to G if for every A' • Ob(A) and every f : B -~ G ( X ) there exists a unique morphism ] : A ~ A' such that f = G(]) o u holds;i.e., the triangle U
B
D G(A)
G(A') commutes.
A
A'
APPENDIX 2
161
Appendix 2 The composition of F-morphisms are not necessarily an F-morphism. The following is such an example. Let A~I = < N; R1 >, .&42 = < N; R2 > mid A43 = < Re; R3 > be system models, where R1 = N: a unary relation, R2 = N × N and R3 = N x N: binary relations, respectively. N and Re axe the sets of natural numbers and real numbers, respectively. Let /1 : 2¢tl -~ 2vi2 and I2 : Iv/2 ~ A43 be F-morphisms, where (I1)0 is the identity a~d (/2)0 is the inclusion. (I1)F a~d (Ig_)F are defined by:
Basl (R1)
=
Bas2(R2)
= R3.
(Vx)R2 (x, y), i.e., (I1)F(R1 (x)) = (Vv) (1t.2 (v, x)),
Then for any a, b, c E N, M1 ~ Rl(x)[a]
Ms ~ R2(x,y)[b,e]
implies implies
.A42 ~ (Vx}I:t2(x,y)[(I1)o(a)] ,
.&43~ R3(x,y)[(I2)o(b), (I~)o(c)].
However for any a E N,
M, ~ r h (x)[a] and
not M3 ~ ((Z2)F o (Zl)F)(RI(x))[((/.2)o o (I,)o)(a)], where ((I2)F o (I1)F)(Rl(x)) ----(Vx)R3(x, y). This shows that not all compositions of F-morphisms are F-morphisms. We should notice that if the basic interpretation of R1, Basl(R1), is (3x)R2(x,y), then the composition of/1 and/_9 is again an F-morphism.
APPENDIX
162
Appendix 3 Let us consider the following counter example when the condition (A)(Section 8.2) does not hold. Let (L:;~) be a structure, where
E -~ {R, 1,2,3} with R: a binary relation symbol, 1, 2, 3: constaa~tsymbols;
= {¢1, ¢2} with ¢i
~
¢2 ~
(Vx)(3!y)-~R(x,y),
R(1, I) A R(2, 2) h R(3, 3) A R(1, 2) A R(2, 1) A R(3, I).
Then
CAN(E) =< {1, 2, 3}; R, 1, 2, 3 >, where
R : {(1,1), (2,2), (3,3),(1,2), (2,1),(3,1)}. And we define a model A4 of ~ by: M : < {1,2,3};RM,1,2,3 >, where R • = {(i, 1), (2, 2), (3, 3), (1, 2), (2, 1), (3, 1), (1, 3)}. Obviously CAN(E) and A4 satisfy E. Since ¢1 requires that ¢1 - 1 inverse of x is unique, the condition (B) is satisfied. But E is not complete, because it is not the case that E ~- -,R(1, 3). Indeed we have
CAN(E) ~ -~R(1, 3)
(9.1)
not 34 ~ -~R(I, 3).
(9.2)
and
(9.1) implies that 3 is a ¢I - i inverse of I in CAN(E), and (9.2) implies that r/(3) is not a ¢i - I inverse of ~}(1) in A4. This shows r/isnot a strong E-homomorphism.
Hence CAN(E) is not a free E-structure on K~.
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Index K-equivaience
commutative diagram, 18 137
--- on classes of sentences, 137
completeness theorem, 4 7 , 102 computer algorithm, 113
- - on formulas, 135
consistent, 43
--
-
-
on SPK,
on sentences, 136
- l inverse, 61 • k-relation, 61 •-inverse, 62 ~-homomorphism, 59, 60 strong --, 62 E-inverse, 62 ~F-morphism, 75 abstract data type, 12, 115 abstraction, 4 algebraic specification, 12, 113, 116 analogy, 94 anthropology, 14
control theory, 13 cybernetics, 13 deduction system, 5, 42, 54 Deduction Theorem, 44 denotation; 35 derivation, 43 discrete event system specification, 27 emergence, 119 emergent property, 122 expansible class, 99 expansion, 99 F-morphism, 18, 68, 71
assignment function, 35
associative--, 127
automaton, 73
composition o f - - , 75
axiom schemata, 42 base set, 23 basic interpretation function, 68 basic linear system, 13 basic morphism, 68 canonical system model, 47, 102 category -
-
of observations data, 99
of structured models, 100 category theory, 97, 159 causality, 30 -
-
F-morphism Theorem, 78 faithfulness, 75 finite automaton, 83 formula, 32 atomic --, 32 many-sorted--, 88 free ~.-structure, 150 free for x, 34 function, 30 function symbol, 30 functor, 100, 159 forgetful--, 100
172
INDEX
general equilibrium theory, 13 Generalization, 43
model, 1 - - for £(A~), 30
generalization, 4 generating set, 150
type o f - - , 24 modeling
goal-seeking model, 57 hierarchical class, 120 hierarchy, 119 level in a - - , 121
-
-
-
-
process, 133 process, 17
factors o f - - , 2 modeling morphism, 59 Modus Ponens, 43
hierarchy theory, 126 homomorphic image, 59 homomorphism, 58 Homomorphism Theorem, 67
natural science structure o f - - , 3 object oriented method, 12
inference
object-orientation, 12
rules o f - - , 42 initial semantics, 97, 113
operations research, 13
institution, 147 isomorphism, 59 language, 30 for a system model, 30 first order - - , 31
-
-
formal--, 30 programming--, 113 linearity, 25 logic, 5
Petri net, 83 predicate calculus theorem of the --, 43 predicate letter, 30 properties of a --, 31 prototype, 1 quantifier universal--, 31 quotient system model -
-
-
-
first order - - , 53 higher order - - , 54 infinitary--, 54 logical connectives, 31 logical consequence, 41 logical framework, 41 logically imply, 41 mathematical structure
with respect to I, 78 with respect to h, 66
teachability, 140 realization, 97, 98, 101 of the language, 30 -
-
Nerode --, 105 reducible, 123 reduct - - of a model, 29
many-sorted--, 26 metaphor, 95
relation, 30
minimal model, 97
Sv-morphism, 77
173
INDEX S-homomorphism, 65 satisfaction, 5, 31, 36
state transition - - , 138 system model, 2, 13, 23
- - in a class of models, 134
- -
of stack, 26
abstract - - , 4
scope
behavior o f - - , 6, 41
- - of quantifier, 33 self-referential behavior, 54
general--, 126
sentence, 33
generalized--, 4
first order - - , 31
i n p u t - o u t p u t - - , 24, 30, 129
many-sorted - - , 88
l i n e a r - - , 25
signature, 115
proper - - , 4
similarity, 55
s t a t i o n a r y - - , 25
behavioral--, 55
structure o f - - , 6
s t r u c t u r a l - - , 18, 55
t i m e - - , 24
Soundness Theorem, 45
system models
spec-model, 113
- -
stationary structure, 105 e x t e n d e d - - , 106
of different types, 68
- - of the same type, 58 systemic theory, 5
stationary system, 105
systems analysis, 13
structuralism, 14
systems behavior, 41
structure, 7
systems property, 34, 133, 134, 137
- -
of a system, 48, 49
systems recognition, 4, 11, 49, 133 multifacetted--, 133
- - of input-output system, 50 - - of stack system model, 52
systems science, 4
- - of stationary system model, 52
systems theory, 3
- - of stationary time index, 50
general - - , 5
- - of time system model, 51
individual--, 5
algebraic - - , 28
logical approach to - - , 5
finite automaton - - , 84 f r e e - - , 149 Petri net - - , 84 relational--, 28
systems viewpoint, 4, 11 tautology, 44 instance of a - - , 44 term, 31
structure expansion, 52 structure-mapping theory, 95 subsystem model, 29
closed--, 32 theory, 46 type
system, 11 - -
- -
o f
£.(M),
30
of function-type, 35
hierarchical--, 126
universal algebra, 5
INDEX
174 universal map, 160
f r e e - - , 33
universality, 21, 97
individual--, 31
variable bound - - , 33
Welt anschauung, 4
172
INDEX
general equilibrium theory, 13 Generalization, 43 generalization, 4
model, 1
generating set, 150
modeling
- - for £(A~), 30 type o f - - , 24
goal-seeking model, 57 hierarchical class, 120 hierarchy, 119 level in a - - , 121
-
-
-
-
process, 133 process, 17
factors o f - - , 2 modeling morphism, 59 Modus Ponens, 43
hierarchy theory, 126 homomorphic image, 59 homomorphism, 58 Homomorphism Theorem, 67 inference rules o f - - , 42 initial semantics, 97, 113 institution, 147 isomorphism, 59 language, 30 for a system model, 30 first order - - , 31 -
-
formal--, 30 programming--, 113 linearity, 25 logic, 5
natural science structure o f - - , 3 object oriented method, 12 object-orientation, 12 operations research, 13 Petri net, 83 predicate calculus theorem of the --, 43 predicate letter, 30 properties of a --, 31 prototype, 1 quantifier universal--, 31 quotient system model -
-
-
-
first order - - , 53 higher order - - , 54 infinitary--, 54 logical connectives, 31 logical consequence, 41 logical framework, 41 logically imply, 41 mathematical structure
with respect to I, 78 with respect to h, 66
teachability, 140 realization, 97, 98, 101 of the language, 30 Nerode --, 105 -
-
reducible, 123 reduct - - of a model, 29
many-sorted--, 26 metaphor, 95
relation, 30
minimal model, 97
Sv-morphism, 77
173
INDEX S-homomorphism, 65 satisfaction, 5, 31, 36
state transition - - , 138 system model, 2, 13, 23
- - in a class of models, 134
- -
of stack, 26
abstract - - , 4
scope
behavior o f - - , 6, 41
- - of quantifier, 33 self-referential behavior, 54
general--, 126
sentence, 33
generalized--, 4 i n p u t - o u t p u t - - , 24, 30, 129
first order - - , 31
l i n e a r - - , 25
many-sorted - - , 88 signature, 115
proper - - , 4
similarity, 55
s t a t i o n a r y - - , 25 structure o f - - , 6
behavioral--, 55
t i m e - - , 24
s t r u c t u r a l - - , 18, 55 Soundness Theorem, 45
system models
spec-model, 113
- -
- - of the same type, 58
stationary structure, 105 e x t e n d e d - - , 106 stationary system, 105
of different types, 68
systemic theory, 5 systems analysis, 13
structuralism, 14
systems behavior, 41
structure, 7
systems property, 34, 133, 134, 137
- -
of a system, 48, 49
systems recognition, 4, 11, 49, 133 multifacetted--, 133
- - of input-output system, 50 - - of stack system model, 52
systems science, 4
- - of stationary system model, 52
systems theory, 3
- - of stationary time index, 50
general - - , 5
- - of time system model, 51
individual--, 5
algebraic - - , 28
logical approach to - - , 5
finite automaton - - , 84 f r e e - - , 149 Petri net - - , 84 relational--, 28
systems viewpoint, 4, 11 tautology, 44 instance of a - - , 44 term, 31
structure expansion, 52 structure-mapping theory, 95 subsystem model, 29
closed--, 32 theory, 46 type
system, 11 - -
- -
o f
£.(M),
30
of function-type, 35
hierarchical--, 126
universal algebra, 5
INDEX
t universal map, 160
f r e e - - , 33
universality, 21, 97
individual--, 31
variable bound - - , 33
Welt anschauung, 4
Lecture Notes in Control and Information Sciences Edited by M. Thoma 1992-1995 Published Titles: Vol. 167: Rao, Ming
Vol. 176: Rozovskii, B.L.; Sowers, R.B.
integrated System for Intelligent Control. 133 pp. 1992 [3-540-54913-7]
(Eds) Stochastic Partial Differential Equations and their Applications. Proceedings of IFIP WG 7.1 International Conference, June 6-8, 1991, University of North Carolina at Charlotte, USA. 251 pp. 1992 [3-540-55292-8]
Vol. 168: Dorato, Peter; Fortuna, Luigi; Muscato, Giovanni Robust Control for Unstructured Perturbations: An Introduction. 118 pp. 1992 [3-540-54920-X] Vol. 169: Kuntzevich, Vsevolod M.; Lychak, Michael Guaranteed Estimates, Adaptation and Robustness in Control Systems. 209 pp. 1992 [3-540-54925-0] Vol. 170: Skowronski, Janislaw M.; Flashner, Henryk; Guttalu, Ramesh S. (Eds) Mechanics and Control. Proceedings of the 4th Workshop on Control Mechanics, January 21-23, 1991, University of Southern California, USA. 302 pp. 1992 [3-540-54954-4] Vol. 171: Stefanidis, P.; Paplinski, A.P.;
Gibbard, M.J. Numerical Operations with Polynomial Matrices: Application to Multi-Variable Dynamic Compensator Design. 206 pp. 1992 [3-540-54992-7] Vol. 172: Tolle, H.; Ers~J, E. Neurocontroh Learning Control Systems Inspired by Neuronal Architectures and Human Problem Solving Strategies. 220 pp. 1992 [3-540-55057-7] Vol. 173: Krabs, W. On Moment Theory and Controllability of Non-Dimensional Vibrating Systems and Heating Processes. 174 pp. 1992 [3-540-55102-6] Vol. 174: Beulens, A.J. (Ed.}
Optimization-Based Computer-Aided Modelling and Design. Proceedings of the First Working Conference of the New IFIP TC 7.6 Working Group, The Hague, The Netherlands, 1991. 268 pp. 1992 [3-540-55135-2] Vol. 175: Rogers, E.T.A.; Owens, D.H. Stability Analysis for Linear Repetitive Processes. 197 pp. 1992 [3-540-55264-2]
Vol. 177: Karatzas, I.; Ocone, D. (Eds) Applied Stochastic Analysis. Proceedings of a US-French Workshop, Rutgers University, New Brunswick, N.J., April 29-May 2, 1991. 317 pp. 1992 [3-540-55296-0] Vol. 178: ZolSsio, J.P. (Ed.)
Boundary Control and Boundary Variation. Proceedings of IFIP WG 7.2 Conference, Sophia-Antipolis, France, October 15-17, 1990. 392 pp. 1992 [3-540-55351-7] Vol. 179: Jiang, Z.H.; Schaufelberger, W.
Block Pulse Functions and Their Applications in Control Systems. 237 pp. 1992 [3-540-55369-X] Vol. 180: Karl, P. (Ed.) System Modelling and Optimization. Proceedings of the 15th IFIP Conference, Zurich, Switzerland, September 2-6, 1991. 969 pp. 1992 [3-540-55577-3] Voi. 181: Drane, C.R. Positioning Systems - A Unified Approach. 168 pp. 1992 [3-540-55850-0] Vol. 182: Hagenauer, J. (Ed.) Advanced Methods for Satellite and Deep Space Communications. Proceedings of an International Seminar Organized by Deutsche Forschungsanstalt fQr Luft-und Raumfahrt (DLR), Bonn, Germany, September 1992. 196 pp. 1992 [3-540-55851-9] Vol. 183: Hosoe, S. (Ed.)
Robust Control. Proceesings of a Workshop held in Tokyo, Japan, June 23-24, 1991. 225 pp. 1992 [3-540-55961-2]
Vol. 184: Duncan, T.E.; Pasik-Duncan, B. Vol. 193: Zinober, A.S.I. (Ed.) Variable Structure and Lyapunov Control (Eds] Stochastic Theory and Adaptive Control. 428 pp. 1993 [3-540-19869-5] Proceedings of a Workshop held in Vol. 194: Cao, Xi-Ren Lawrence, Kansas, September 26-28, Realization Probabilities: The Dynamics of 1991. Queuing Systems 500 pages. 1992 [3-540-55962-0] 336 pp. 1993 [3-540-19872-5] Vol. 185: Curtain, R.F. (Ed.); Bensoussan, A.; Lions, J.L. (Honorary Eds) Vol. 195: Liu, D.; Michel, A.N. Dynamical Systems with Saturation Analysis and Optimization of Systems: Nonlinearities: Analysis and Design State and Frequency Domain Approaches 212 pp. 1994 [3-540-19888-1] for Infinite-Dimensiona~ Systems. Proceedings of the lOth international Conference, Sophia-Antipolis, France, June Vol. 196: Battilotti, S. Noninteracting Control with Stability for 9-12, 1992. Nonlinear Systems 648 pp. 1993 [3-540-56155-2] 196 pp. 1994 [3-540-19891-1] Vol. 186: Sreenath, N. Systems Representation of Global Climate Yol. 197: Henry, J.; Yvon, J.P. (Eds) Change Models. Foundation for a Systems System Modelling and Optimization 975 pp approx. 1994 [3-540-19893-8] Science Approach. 288 pp. 1993 [3-540-19824-5] Vol. 198: Winter, H.; N~J~er, H.-G. (Eds) Advanced Technologies for Air Traffic Flow Vol. 187: Morecki, A.; Bianchi, G.; Management Jaworeck, K. (Eds) 225 pp approx. 1994 [3-540-19895-4] RoManSy 9: Proceedings of the Ninth CISM-IFToMM Symposium on Theory and Vol. 199: Cohen, G.; Quadrat, J.-P. (Eds) Practice of Robots and Manipulators. 1 l t h International Conference on 476 pp. 1993 [3-540-19834-2] Analysis and Optimization of Systems Discrete Event Systems: SophiaVol. 188: Naidu, D. Subbaram Antipolis, June 15-16-17, 1994 Aeroassisted Orbital Transfer: Guidance 648 pp. 1994 [3-540-19896-2] and Control Strategies. 192 pp. 1993 [3-540-19819-9] Vol. 200: Yoshikawa, T.; Miyazaki, F. (Eds) Experimental Robotics II1: The 3rd Vol. 189: Ilchmann, A. International Symposium, Kyoto, Japan, Non-Identifier-Based High-Gain Adaptive October 28-30, 1993 Control. 624 pp. 1994 [3-540-19905-5] 220 pp. 1993 [3-540-19845-8] Vol. 190: Chatila, R.; Hirzinger, G. (Eds) Experimental Robotics I1: The 2rid International Symposium, Toulouse, France, June 25-27 1991. 580 pp. 1993 [3-540-19851-2] Vol. 191: Blondel, V.
Simultaneous Stabilization of Linear Systems. 212 pp. 1993 [3-540-19862-8]
Vol. 201: Kogan, J.
Robust Stability and Convexity 192 pp. 1994 [3-540-19919-5] Vol. 202: Francis, B.A.; Tannenbaum,
A.R. (Eds) Feedback Control, Nonlinear Systems, and Complexity 288 pp. 1995 [3-540-19943-8]
Vol. 203: Popkov, Y.S. Macrosystems Theory and its Applications: Vol. 192: Smith, R.S.; Dahleh, M. (Eds) The Modeling of Uncertainty in Control Equilibrium Models 344 pp. 1995 [3-540-19955-1] Systems. 412 pp. 1993 [3-540-19870-9]