Modular Ontologies (Frontiers in Artificial Intelligence and Applications, WoMO 2011)

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MODULAR ONTOLOGIES

Frontiers in Artificial Intelligence and Applications FAIA covers all aspects of theoretical and applied artificial intelligence research in the form of monographs, doctoral dissertations, textbooks, handbooks and proceedings volumes. The FAIA series contains several sub-series, including “Information Modelling and Knowledge Bases” and “Knowledge-Based Intelligent Engineering Systems”. It also includes the biennial ECAI, the European Conference on Artificial Intelligence, proceedings volumes, and other ECCAI – the European Coordinating Committee on Artificial Intelligence – sponsored publications. An editorial panel of internationally well-known scholars is appointed to provide a high quality selection. Series Editors: J. Breuker, N. Guarino, J.N. Kok, J. Liu, R. López de Mántaras, R. Mizoguchi, M. Musen, S.K. Pal and N. Zhong

Volume 230 Recently published in this series Vol. 229. P.E. Vermaas and V. Dignum (Eds.), Formal Ontologies Meet Industry – Proceedings of the Fifth International Workshop (FOMI 2011) Vol. 228. G. Bel-Enguix, V. Dahl and M.D. Jiménez-López (Eds.), Biology, Computation and Linguistics – New Interdisciplinary Paradigms Vol. 227. A. Kofod-Petersen, F. Heintz and H. Langseth (Eds.), Eleventh Scandinavian Conference on Artificial Intelligence – SCAI 2011 Vol. 226. B. Apolloni, S. Bassis, A. Esposito and C.F. Morabito (Eds.), Neural Nets WIRN10 – Proceedings of the 20th Italian Workshop on Neural Nets Vol. 225. A. Heimbürger, Y. Kiyoki, T. Tokuda, H. Jaakkola and N. Yoshida (Eds.), Information Modelling and Knowledge Bases XXII Vol. 224. J. Barzdins and M. Kirikova (Eds.), Databases and Information Systems VI – Selected Papers from the Ninth International Baltic Conference, DB&IS 2010 Vol. 223. R.G.F. Winkels (Ed.), Legal Knowledge and Information Systems – JURIX 2010: The Twenty-Third Annual Conference Vol. 222. T. Ågotnes (Ed.), STAIRS 2010 – Proceedings of the Fifth Starting AI Researchers’ Symposium Vol. 221. A.V. Samsonovich, K.R. Jóhannsdóttir, A. Chella and B. Goertzel (Eds.), Biologically Inspired Cognitive Architectures 2010 – Proceedings of the First Annual Meeting of the BICA Society Vol. 220. R. Alquézar, A. Moreno and J. Aguilar (Eds.), Artificial Intelligence Research and Development – Proceedings of the 13th International Conference of the Catalan Association for Artificial Intelligence

ISSN 0922-6389 (print) ISSN 1879-8314 (online)

M Modula ar Onttologiees Proceeedings of the t Fifth In nternational Worksho op (WoMO O 2011)

y Edited by

O Oliver Ku utz Researrch Center on n Spatial Cog gnition (SFB B/TR 8), Universityy of Bremen,, Germany

and

Thom mas Schn neider Deepartment off Computer Science, S Univversity of Breemen, Germaany

Amstterdam • Berrlin • Tokyo • Washington, DC

© 2011 The authors and IOS Press. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without prior written permission from the publisher. ISBN 978-1-60750-798-7 (print) ISBN 978-1-60750-799-4 (online) Library of Congress Control Number: 2011932732 Publisher IOS Press BV Nieuwe Hemweg 6B 1013 BG Amsterdam Netherlands fax: +31 20 687 0019 e-mail: [email protected] Distributor in the USA and Canada IOS Press, Inc. 4502 Rachael Manor Drive Fairfax, VA 22032 USA fax: +1 703 323 3668 e-mail: [email protected]

LEGAL NOTICE The publisher is not responsible for the use which might be made of the following information. PRINTED IN THE NETHERLANDS

Modular Ontologies O. Kutz and T. Schneider (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved.

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Preface Oliver KUTZ and Thomas SCHNEIDER

Modular Ontologies Modularity has been and continues to be one of the central research topics in ontology engineering. The number of ontologies available, as well as their size, is steadily increasing. There is a large variation in subject matter, level of speciﬁcation and detail, intended purpose and application. Ontologies covering different domains are often developed in a distributed manner; contributions from different sources cover different parts of a single domain. Not only is it difﬁcult to determine and deﬁne interrelations between such distributed ontologies, it is also challenging to reconcile ontologies which might be consistent on their own but jointly inconsistent. Further challenges include extracting the relevant parts of an ontology, re-combining independently developed ontologies in order to form new ones, determining the modular structure of an ontology for comprehension, and the use of ontology modules to facilitate incremental reasoning and version control. Still catching up with 40 years of related research in software engineering, ontological modularity is envisaged to allow mechanisms for easy and ﬂexible reuse, generalisation, structuring, maintenance, collaboration, design patterns, and comprehension. Applied to ontology engineering, modularity is central not only to reducing the complexity of understanding ontologies, but also to maintaining, querying and reasoning over modules. Distinctions between modules can be drawn on the basis of structural, semantic, or functional aspects, which can also be applied to compositions of ontologies or to indicate links between ontologies. In particular, reuse and sharing of information and resources across ontologies depend on purpose-speciﬁc, logically versatile criteria. Such purposes include ‘tight’ logical integration of different ontologies (wholly or in part), ‘loose’ association and information exchange, the detection of overlapping parts, traversing through different ontologies, alignment of vocabularies, module extraction possibly respecting privacy concerns and hiding of information, etc. Another important aspect of modularity in ontologies is the problem of evaluating the quality of single modules or of the achieved overall modularisation of an ontology. Again, such evaluations can be based on various (semantic or syntactic) criteria and employ a variety of statistical/heuristic or logical methods. Recent research on ontology modularity has produced substantial results and approaches towards foundations of modularity, techniques of modularisation and modular developments, distributed and incremental reasoning, as well as the use of modules in different application scenarios, providing a foundation for further research and development. Since the beginning of the WoMO workshop series, there has been growing interest in the modularisation of ontologies, modular development of ontologies, and information exchange across different modular ontologies. In real life, however, integration problems are still mostly tackled in an ad-hoc manner, with no clear notion of what to

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expect from the resulting ontological structure. Those methods are not always efﬁcient, and they often lead to unintended consequences, even if the individual ontologies to be integrated are widely tested and understood. Topics covered by WoMO include, but are not limited to: What is Modularity? - Kinds of modules and their properties - Modules vs. contexts - Design patterns - Granularity of representation Logical/Foundational Studies - Conservativity and syntactic approximations for modules - Modular ontology languages - Reconciling inconsistencies across modules - Formal structuring of modules - Heterogeneity Algorithmic Approaches - Distributed reasoning - Modularisation and module extraction - (Selective) sharing and reusing, linking and importing - Hiding and privacy - Evaluation of modularisation approaches - Complexity of reasoning - Reasoners or implemented systems Application Areas - Modularity in the Semantic Web - Life Sciences - Bio-Ontologies - Natural Language Processing - Ontologies of space and time - Ambient intelligence - Collaborative ontology development The WoMO 2011 workshop follows a series of successful events that have been an excellent venue for practitioners and researchers to discuss latest work and current problems. It is intended to consolidate cutting-edge approaches that tackle the problem of ontological modularity and bring together researchers from different disciplines who study the problem of modularity in ontologies at a fundamental level, develop design tools for distributed ontology engineering, and apply modularity in different use cases and application scenarios. Previous editions of WoMO are listed below. The links refer to their homepages and proceedings. WoMO 2006 The 1st workshop on modular ontologies, co-located with ISWC 2006, Athens, Georgia, USA. Invited speakers were Alex Borgida (Rutgers) and Frank Wolter (Liverpool). http://www.cild.iastate.edu/events/womo.html http://sunsite.informatik.rwth-aachen.de/Publications/CEUR-WS/Vol-232

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WoMO 2007 The 2nd workshop, co-located with K-CAP 2007, Whistler BC, Canada. The invited speaker was Ken Barker (Texas at Austin). http://webrum.uni-mannheim.de/math/lski/WoMO07 http://sunsite.informatik.rwth-aachen.de/Publications/CEUR-WS/Vol-315

WoRM 2008 The 3rd workshop in the series, co-located with ESWC 2008, Tenerife, Spain, entitled ‘Ontologies: Reasoning and Modularity’ had a special emphasis on reasoning methods. http://dkm.fbk.eu/worm08 http://sunsite.informatik.rwth-aachen.de/Publications/CEUR-WS/Vol-348

WoMO 2010 The 4th workshop in the series, co-located with FOIS 2010, Toronto, Canada. Invited speakers were Simon Colton (London) and Marco Schorlemmer (Barcelona). http://www.informatik.uni-bremen.de/~okutz/womo4 http://www.booksonline.iospress.nl/Content/View.aspx?piid=16268

Overview of Contributions The invited speakers address modularity in ontologies from three main perspectives: foundational ontologies, bio-medical ontologies and logical approaches. S TEFANO B ORGO, in “Goals of Modularity: A voice from the foundational viewpoint”, discusses the general landscape of modularity from the point of view of foundational ontology. He distinguishes three general ways of understanding and using foundational ontologies. Based on this classiﬁcation, he then analyses different kinds of modules based on their intended usage and with respect to the general understanding of foundational ontology, and ﬁnally sketches the challenges of future module-based ontology engineering. S TEFAN S CHULZ reports about joint work with PABLO L ÓPEZ -G ARCÍA in “Modularity Issues in Biomedical Ontologies”. This talk addresses applications of biomedical ontologies, their requirements concerning modularity, and applicable methods. The most prominent requirement is the extraction of a module that sufﬁciently covers a given subdomain, and preserves entailments. Logic-based and graph traversal based module extraction approaches are evaluated for their suitability. In his talk “Query Inseparability and Module Extraction in OWL 2 QL”, M ICHAEL Z AKHARYASCHEV reports about theoretical and practical results on checking whether two ontologies in the QL proﬁle of the Web Ontology Language OWL give the same answers to conjunctive queries over data with respect to a given signature of interest. This task is an important ingredient of reasoning support for ontology engineering tasks such as composing, re-using, and comparing ontologies, and extracting modules.

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The contributed papers cover a wide range of topics, from empirical studies exploiting modularity, to extracting modules, partitioning ontologies, modularly building new ontologies, multi-perspective and contextualised modelling and reasoning, logical translation, as well as ontology veriﬁcation and ontology repositories. G ÖKHAN C OSKUN, M ARIO ROTHE, K IA T EYMOURIAN, and A DRIAN PASCHKE present methods for extracting thematically related modules from ontologies in “Applying Community Detection Algorithms on Ontologies for Identifying Concept Groups”. Their approach applies methods from social network analysis to the task of ﬁnding semantic groups in ontological structures. These automatically extracted “concept groups” are intended to help users and engineers in getting a better overview of the ontology contents. Different concept grouping algorithms are applied on a set of example ontologies and their results are compared with a gold standard. In “The Modular Structure of an Ontology: Atomic Decomposition and Module Count”, C HIARA D EL V ESCOVO, B IJAN PARSIA, U LI S ATTLER, and T HOMAS S CHNEIDER devise a framework for partitioning an ontology into a linear number of components called atoms. This partition can be obtained via a linear number of module extractions. Equipped with a dependency relation between atoms, the partition can be used to represent the modular structure of an ontology, estimate the number of its modules, and guide the extraction of a single module. J ULIA D MITRIEVA and F ONS J. V ERBEEK, in “Modular Approach for a new Ontology”, present a way to create an integrated ontology from automatically extracted modules of a set of chosen ontologies, based on a set of seed terms of user interest. This contribution elaborates a module extraction algorithm tailored to bio-ontologies, and includes an ontology matching approach based on string distance. M ICHAEL G RÜNINGER, T ORSTEN H AHMANN, and M EGAN K ATSUMI investigate the problem of ontology veriﬁcation in “Exploiting Modularity for Ontology Veriﬁcation”. They understand the problem of ontology veriﬁcation as the task of comparing the intended models of an ontology with the class of all models of a given axiomatised ontology. Proving the ‘equivalence’ of such model classes amounts to verifying an ontology, and a number of ideas are presented that can ease automated theorem proving and showing equivalences of theories by exploiting the modular structure of ontologies. JANNA H ASTINGS, C OLIN BATCHELOR, C HRISTOPH S TEINBECK, and S TEFAN S CHULZ describe several issues concerning the modularisation of the ChEBI ontology, an ontology about the Chemistry domain, in “Modularization requirements in bioontologies: A case study of ChEBI”. Bio-medical ontologies tend to be rather large, and adding further expressivity such as deﬁning equivalent classes can easily impact reasoner performance. The authors discuss the use of modularisation techniques for keeping reasoning over large ontologies manageable and evaluate existing ontology partitioning tools. J OANA H OIS, in “Modeling the Diversity of Spatial Information by Using Modular Ontologies and their Combinations”, distinguishes types of spatial information to specify spatial ontology modules that comply with one of the spatial perspectives. Different combination mechanisms for combining ontology modules are used on the basis of these spatial perspectives. Also, an application scenario is outlined, in which these modularly developed spatial ontologies and their combinations are used.

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M ATHEW J OSEPH and L UCIANO S ERAFINI, in “Simple Reasoning for Contextualized RDF Knowledge”, present a framework for representing and reasoning with contextual knowledge based on the existing RDFS standard. Contextualised knowledge is here meant in the sense that certain statements (RDF triples) are true only at, e.g., certain time intervals, spatial regions, or in certain sub-domains. The authors discuss a corresponding reasoning system and a prototypical implementation. T ILL M OSSAKOWSKI and O LIVER K UTZ, in “The Onto-Logical Translation Graph”, present a common formalisation of most ontology languages in use today based on the framework of institution theory. They distinguish different kinds of logical translation between ontology languages and discuss in detail the various translational relationships that can be obtained. They also show that logical translation interacts well with modularity and how this supports tool reuse and interoperability across different formal ontology languages. DARREN O NG’s and M ICHAEL G RÜNINGER’s contribution “Constructing an Ontology Repository: A Case Study with Theories of Time Intervals” presents a case study of connecting stored ontologies via meta-theoretic properties they share. These relations are obtained using an automated theorem prover. For three speciﬁc theories of time intervals, the relations are presented and discussed.

Contributors to this volume C OLIN BATCHELOR is a Senior Informatics Analyst at the Royal Society of Chemistry in Cambridge, UK. He has been working on natural-language processing of chemical text with members of the University of Cambridge Computer Laboratory as well as on chemical and biomedical ontologies. http://www.rsc.org [email protected]

S TEFANO B ORGO is a researcher in the Laboratory for Applied Ontology (LOA), part of the Institute for Cognitive Sciences and Technologies (ISTC) at the National Research Council (CNR), Italy. He works in the areas of foundational ontology, space representation, object modeling, engineering design and logics for multi-agent systems. He is a co-author of the DOLCE ontology and has been project leader in several national and international projects focusing on the development and application of ontology. http://www.loa-cnr.it/borgo.html [email protected]

G ÖKHAN C OSKUN is a Research Assistant in the Corporate Semantic Web working group at the Freie Universität Berlin (FUB), Germany. His research interests comprise network structure of ontologies and aspect-oriented ontology engineering. He is working on his doctoral thesis investigating a structural analysis and structure-based modularization of ontologies. http://gokhan.coskun.org [email protected]

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C HIARA D EL V ESCOVO is a PhD Student at the University of Manchester, United Kingdom, in the Information Management Group. She graduated in Mathematics at the University Roma 3, Italy. After graduation, she worked in Research & Development department of CM Sistemi, a private informatics company. She is currently working on her PhD project, focused on modularity for ontology comprehension. http://www.cs.man.ac.uk/~delvescc [email protected]

J ULIA D MITRIEVA is a Researcher in Centre de Biophysique Moléculaire Numérique (CBMN) at the University of Liège-Gembloux, Belgium. She did her PhD research at the Leiden Institute of Advanced Computer Science (LIACS) on ontology visualization and integration. Her current research topics are analysis, integration and representation of Protein Protein interaction data. [email protected]

[email protected]

M ICHAEL G RUNINGER is an Associate Professor at the University of Toronto. His current research focuses on the design and formal characterization of ontologies in mathematical logic and their application to problems in manufacturing and enterprise engineering. http://stl.mie.utoronto.ca [email protected]

T ORSTEN H AHMANN is a PhD candidate in the Department of Computer Science at the University of Toronto, advised by Michael Gruninger. As his PhD research he develops a family of multidimensional spatial ontologies for semantic integration of a broad range of spatial theories such as qualitative theories (in particular mereotopologies) and various geometries. He is also active in the veriﬁcation of spatial ontologies and involved in the development of the Common Logic Ontology Repository (COLORE) for semantic integration and modularization of ontologies. http://www.cs.toronto.edu/~torsten [email protected]

JANNA H ASTINGS is a bioinformatician and ontologist in the Chemoinformatics and Metabolism group at the European Bioinformatics Institute, where she works on knowledge representation and reasoning in chemistry in the context of the popular chemical ontology ChEBI. She also works on ontologies for cognition and emotion at the Swiss Center for Affective Sciences. http://www.ebi.ac.uk/~hastings [email protected]

J OANA H OIS is a PhD student at the University of Bremen, Germany. She is a developer of the spatial module for the Generalized Upper Model GUM-Space, and she works on combining formal models of space with spatial language. Her current research activities are focused on modular ontologies of space in different domains and applications as well as on combining ontologies with different kinds of uncertainties. http://www.informatik.uni-bremen.de/~joana [email protected]

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M ATHEW J OSEPH is a second year PhD student in Informatics at University of Trento, Italy. His research is primarily focused on Contextual Knowledge Representation and Reasoning. His PhD supervisor is Luciano Seraﬁni, DKM, FBK-IRST, Trento, Italy. Before this, he worked as a project collaborator in the FBK-IRST, Trento for around one year. He completed his Master degree from Amrita University, India, in 2008. https://dkm.fbk.eu/index.php/Mathew_Joseph [email protected]

M EGAN K ATSUMI is a Master’s of Applied Science student in the Semantic Technologies Lab at the University of Toronto, with a background in Industrial Engineering. Her research currently focuses on development methodologies for expressive, ﬁrst-order logic ontologies, with a speciﬁc focus on the use of automated reasoners to assist the development process. http://stl.mie.utoronto.ca [email protected]

O LIVER K UTZ is a Postdoctoral Research Fellow in the Research Center on Spatial Cognition (SFB/TR 8) at the University of Bremen, Germany. He has published widely in the areas of philosophical, non-classical and spatial logic, ontology engineering, and Artiﬁcial Intelligence. He is a co-designer of the logic SROIQ underlying the web ontology language OWL 2 as well as the E-connections technique that is used in modular ontology design, and is a founding member of the International Association for Ontology and its Applications (IAOA). http://www.informatik.uni-bremen.de/~okutz [email protected]

PABLO L ÓPEZ -G ARCÍA is a PhD student at the University of the Basque Country in San Sebastián, Spain. He has two MSc degrees from the University of Zaragoza/University of the Basque Country for the theses “Avalanche Beacon Rescue Simulator” and “Description Logics Reasoning in Information Systems: An Ontology-driven Menu Recommender System”. He is currently working on ontology customization for home telemonitoring of chronic diseases. http://www.plopez.info [email protected]

T ILL M OSSAKOWSKI is a senior researcher at DFKI GmbH in Bremen and a professor at the University of Bremen. His Habilitation thesis focuses on modular and heterogeneous logical theories and the heterogeneous tool set Hets. He is chairman of the IFIP working group 1.3 “Foundations of Systems Speciﬁcations” and a member of the collaborative research center “Spatial Cognition”. http://www.informatik.uni-bremen.de/~till [email protected]

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DARREN O NG is an active research member in the Semantic Technologies Laboratory, Faculty of Mechanical and Industrial Engineering at the Univerisity of Toronto. He is completing his MASc in the area of Identifying Meta-Theoretic Relationships between and Veriﬁcation of First Order Ontologies of Time under the supervision of Michael Grüninger. His research interests include ontology veriﬁcation, design of modular ontologies, and ontology repositories. http://stl.mie.utoronto.ca [email protected]

B IJAN PARSIA is a Lecturer in the School of Computer Science at the University of Manchester, UK, where he is a member of the Information Management Group (IMG). He has published on many aspects of ontology engineering using description logics including on explanation, modularity, reasoning optimization, visualization, and representing uncertainty. http://www.cs.man.ac.uk/~bparsia [email protected]

A DRIAN PASCHKE is head of Corporate Semantic Web chair (AG-CSW) at the Freie Universität Berlin (FUB). He is research director at the Centre for Information Technology Transfer (CITT) GmbH, director of RuleML Inc., Canada, and vice director of the Semantics Technologies Institute Berlin (STI Berlin). He is steering-committee chair of the RuleML Web Rule Standardization Initiative, co-chair of the Reaction RuleML technical group, founding member of the Event Processing Technology Society (EPTS) and chair of the EPTS Reference Architecture working group, voting member of OMG, and member of several W3C groups such as the W3C Rule Interchange Format (W3C RIF) working group where he is editor of several W3C Semantic Web standard speciﬁcations. http://www.inf.fu-berlin.de/en/groups/ag-csw [email protected]

M ARIO ROTHE is a Student Assistant in the Corporate Semantic Web working group at the Freie Universität Berlin, Germany. His research interests include aspects of ontology versioning and modularization of ontologies. He is currently working on his Master’s thesis. http://www.inf.fu-berlin.de/groups/ag-csw [email protected]

U LI S ATTLER is a professor in the Information Management Group within the School of Computer Science of the University of Manchester. She works in logic-based knowledge representation, investigates standard and novel reasoning problems, and designs algorithms for their usage in applications. Together with Ian Horrocks and others, she has developed the SHIQ family of description logics underlying OWL and OWL 2. Together with various colleagues, she has published several papers on the concept of inferencepreserving modules, their computation, and usage. http://www.cs.man.ac.uk/~sattler [email protected]

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T HOMAS S CHNEIDER is a Postdoctoral Research Associate in the Computer Science Department at the University of Bremen. His PhD was on the computational complexity of hybrid logics, supervised by Martin Mundhenk from the University of Jena. He then worked on the EPSRC funded project "Composing and decomposing ontologies: a logicbased approach" at the University of Manchester, and has published a number of papers on how to use logic-based approaches to modularity to support various tasks in ontology engineering. He has developed the locality-based module extractor that is currently available, e.g., through the OWL API. http://www.informatik.uni-bremen.de/~ts [email protected]

S TEFAN S CHULZ is a physician by training with doctorate in theoretical medicine by the University of Heidelberg, Germany. He has been doing research in biomedical informatics at the University of Freiburg, Germany since 1994, executing projects in the ﬁelds of medical records, eLibraries, language processing, coding and classiﬁcation, ontologies, and information retrieval. He has contributed as author, organiser and advisor to numerous scientiﬁc events and is member of several international committees (e.g. IHTSDO, WHO). Stefan Schulz participated in various European projects and received several national research grants and awards. As a visiting scientist he cooperated with several universities in Brazil. In December 2010 Stefan Schulz was appointed full professor for Medical Informatics at the Medical University of Graz, Austria. http://user.meduni-graz.at/stefan.schulz [email protected]

L UCIANO S ERAFINI is the head of the data and knowledge management research unit at Fondazione Bruno Kessler. His research interests include artiﬁcial intelligence, logic for knowledge representation and multi agent systems, semantic web, ontologies, information integration, and automated reasoning. He has published inﬂuential works in the most important international journals and conferences. He is one of the inventors of a logic of contexts called "Multi Context Systems" which has been applied in the area of formalization of multi-agent systems, information integration, semantic matching and modular ontologies. http://dkm.fbk.eu/seraﬁni seraﬁ[email protected]

C HRISTOPH S TEINBECK is head of the Chemoinformatics and Metabolism group at the European Bioinformatics Institute. He is interested in cheminformatics, bioinformatics and analytical techniques to decipher the metabolic system in living organisms. A profound (but yet largly lacking) understanding of metabolism is the conditio sine qua non for the design of better drugs and the treatment of diseases in general. Towards this goal, his group has worked on algorithms and tools for computer-assisted structure elucidation of biological metabolites and on databases to summarize the current knowledge on small molecules and metabolomics. http://www.ebi.ac.uk/~steinbeck [email protected]

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K IA T EYMOURIAN is a Research Assistant in the group of Networked Information Systems at the Freie Universität Berlin (FUB), Germany. His current research activities are focused on knowledge-based complex event processing as well as large-scale semanticenabled distributed information systems. He is pursuing a PhD focusing on the area of semantic rule-based event processing. http://www.teymourian.de [email protected]

F ONS J. V ERBEEK is a Group Leader of the section Imaging & BioInformatics at the Leiden Institute of Advanced Computer Science (LIACS) at Leiden University, the Netherlands. He obtained his PhD at the Pattern Recognition group of the Delft University of Technology on 3D Image Analysis and Visualization in Microscopy. In the Imaging & BioInformatis group at LIACS a variety of approaches in ontology processing is included in research projects ranging from construction to visualization of ontolgies. A special interest is in applying ontologies in annotation of images, (and other biomedical data) so that these images can be subject to reasoning processes as well as to support the process of image analysis. In the past years results have been published on the subject of ontologies in combination with other research projects. http://bio-imaging.liacs.nl [email protected]

M ICHAEL Z AKHARYASCHEV is Professor at the Department of Computer Science and Information Systems, Birkbeck College London, UK. His research interests include description logic, ontology-based data access, spatial representation and reasoning, and modal logic. http://www.dcs.bbk.ac.uk/~michael [email protected]

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Workshop Organisation Organising Committee Oliver Kutz Thomas Schneider

Research Center on Spatial Cognition (SFB/TR 8), Univ. of Bremen, Germany Department of Computer Science, University of Bremen, Germany

Programme Committee Jie Bao Simon Colton Melanie Courtot Bernardo Cuenca Grau Faezeh Ensan Fred Freitas Silvio Ghilardi Janna Hastings Robert Hoehndorf Joana Hois C. Maria Keet Roman Kontchakov Frank Loebe Till Mossakowski Leo Obrst Bijan Parsia Daniel Pokrywczy´nski Anne Schlicht Marco Schorlemmer Andrei Tamilin Dirk Walther Michael Zakharyaschev

Department of Computer Science, Rensselaer Polytechnic Institute, USA Department of Computing, Imperial College, London, UK Terry Fox laboratory, BC Cancer Care & Research, Vancouver, Canada Computing Laboratory, University of Oxford, UK Faculty of Computer Science, University of New Brunswick, Canada Universidade Federal de Pernambuco, Brazil Department of Computer Science, University of Milan, Italy European Bioinformatics Institute, Cambridge, UK Department of Genetics, University of Cambridge, UK Research Center on Spatial Cognition (SFB/TR 8), Univ. of Bremen, Germany School of Computer Science, Univ. of KwaZulu-Natal, Durban, South Africa Computer Science and Information Systems, Birkbeck College, London, UK Department of Computer Science/IMISE, University of Leipzig, Germany German Research Center for Artiﬁcial Intelligence, Lab Bremen, Germany The MITRE Corporation, McLean, USA School of Computer Science, University of Manchester, UK Department of Computer Science, University of Liverpool, UK KR & KM Research Group, University of Mannheim, Germany Artiﬁcial Intelligence Research Institute, CSIC, Barcelona, Spain Fondazione Bruno Kessler – IRST, Italy Department of Computer Science, Universidad Politecnica de Madrid, Spain Computer Science and Information Systems, Birkbeck College, London, UK

Invited Speakers Stefano Borgo Stefan Schulz Michael Zakharyaschev

Laboratory of Applied Ontology, ISTC-CNR, Trento, Italy Medical University Graz, Austria Computer Science and Information Systems, Birkbeck College, London, UK

Acknowledgements We acknowledge generous ﬁnancial support from the DFG-funded Research Center on Spatial Cognition (SFB/TR 8) situated at the Universities of Bremen & Freiburg, Germany. We would like to thank the PC members and the additional reviewers for their timely reviewing work and our invited speakers—Stefano Borgo, Stefan Schulz and Michael Zakharyaschev—for delivering keynote presentations at the workshop. We would also like to thank the team at IOS Press, in particular Anne Marie de Rover, Carry Koolbergen, and Maarten Fröhlich. Last but not least we would like to thank the ESSLLI summer school for hosting the WoMO workshop. Oliver Kutz Thomas Schneider

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Contents Preface Oliver Kutz and Thomas Schneider

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Goals of Modularity: A Voice from the Foundational Viewpoint Stefano Borgo

1

Modularity Issues in Biomedical Ontologies Stefan Schulz and Pablo López-García

7

Query Inseparability and Module Extraction in OWL 2 QL Michael Zakharyaschev

10

Applying Community Detection Algorithms Gökhan Coskun, Mario Rothe, Kia Teymourian and Adrian Paschke

12

The Modular Structure of an Ontology: Atomic Decomposition and Module Count Chiara del Vescovo, Bijan Parsia, Uli Sattler and Thomas Schneider

25

Modular Approach for a New Ontology Julia Dmitrieva and Fons J. Verbeek

40

Exploiting Modularity for Ontology Verification Michael Grüninger, Torsten Hahmann and Megan Katsumi

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Modularization Requirements in Bio-Ontologies: A Case Study of ChEBI Janna Hastings, Colin Batchelor, Christoph Steinbeck and Stefan Schulz

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Modeling the Diversity of Spatial Information by Using Modular Ontologies and Their Combinations Joana Hois

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Simple Reasoning for Contextualized RDF Knowledge Mathew Joseph and Luciano Serafini

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The Onto-Logical Translation Graph Till Mossakowski and Oliver Kutz

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Constructing an Ontology Repository: A Case Study with Theories of Time Intervals Darren Ong and Michael Grüninger

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Subject Index

127

Author Index

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Modular Ontologies O. Kutz and T. Schneider (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-799-4-1

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Goals of Modularity: A Voice from the Foundational Viewpoint Stefano BORGO LOA-ISTC-CNR, Trento and KRDB FUB, Bolzano

1. One, ten, ... myriad of ontologies If we ask researchers in applied ontology what they think justify the foundational ontological system(s) they develop or use, we get a variety of answers that, for the purposes of this paper, I suggest to split in three general classes: 1. A foundational ontology models reality as it is or, alternatively, as we can best understand it. Most researchers I know in this class work in domains where there is the need to relate and exploit a very large set of notions like in medicine. Several theoretical and practical considerations lead to the search for a single monolithic top-level ontology, generally driven by our scientiﬁc understanding of reality.1 The claim inspired much research in the 1990s, and it is still now quite popular although I see it openly accepted only in limited domains. Note however that this approach is quite tempting in standardization initiatives. Over the years, the idea of a monolithic ontology has been weakened in several ways. Today, the approach is often presented, to my knowledge, as the aim to building a single ontology for each discipline. This change from a unique ontology to a unique ontology per discipline, is substantial since it admits the possible need of a plurality of ontologies, usually called reference or core ontologies, which may differ not only at the level of content but in their structure and assumptions as well. (It will be interesting to see how this view evolves when working within strongly interdisciplinary areas.) As far as I can tell, most people that agree with this view embrace it only partially on the ground of its philosophical attractiveness. The interest arises mainly from practical and engineering considerations, and can be stated quite simply: the complexity of ontology construction and maintenance and the costs of managing multiple ontologies are too high, we can afford only one ontology per discipline if we want to ensure properties like coherence, consistency, robustness, coverage, principled construction and so on. 1 Some

arbitrariness is anyway accepted since not all theoretical questions may be uniquely answered by science. The idea is that these arbitrary choices do not affect the validity of the overall ontology and they are the ﬁrst to be revised if any problem raises in the construction or use of the ontology.

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2. There is a number of ways in which we can make sense of reality. Each foundational ontology models one of these. This attitude is registered in domains with an historical awareness of foundational issues like physics at-large (that is, beyond consolidated areas like classical mechanics), and in areas where there are deep theoretical issues in the integration of empirically developed methodologies like in engineering design. I would argue that this attitude attracts ontologists working in foundational studies for at least two reasons: it explains why people come out with different ontologies even when trying to make sense of the same data or domain; and why two ontologies can be equally well constructed and useful while being mutually inconsistent. Ideally, people in this class claim that there is a ﬁnite (actually, quite small) number of coherent and relevant perspectives and that each information system (database, knowledge base, domain ontology and the like) can be naturally understood as falling under one of these perspectives, or is practically composed of subsystems with this property. In practice, it is expected that agents should be able to understand and use several foundational ontologies in combination depending on the goals and the available sources of information. The overall stand is in part justiﬁed by philosophical results and in part by a mix of considerations from cognitive science, linguistics and commonsense, including direct reference to the state of the art in ontology today. An important challenge for researchers in this group is to fully develop the different top-level ontologies and to ﬁnd inter-ontological relationships to make interoperability a reality. 3. A foundational ontology models basic notions and relations involved in some contextual perspective taken by an agent or knowledge system. This is by far the largest class in interdisciplinary domains like the Semantic Web. It is characterized by a strong form of relativism: one starts from acknowledging that any body of information is somehow biased and that the goal of the ontologist is to identify and capture a coherent way to understand the given information. If another agent wants to use this same knowledge system, the agent should adopt the same contextual perspective otherwise misinterpretations and inconsistencies easily arise. Some researchers here admit that we can ﬁnd fairly stable and widely used constructs (analogous to those in knowledge patterns), which would arise as ontological invariants across different perspectives. The task is then to leverage on these to ﬁnd interrelationships across systems. Nonetheless, most of the people in this class admit that they take an ontology to be nothing more than a consistent logical theory that provides some classiﬁcation mechanism and, possibly, some farther characterization of the categories. The focus indeed is on the logical theory itself, and the underlying assumptions or construction principles are often unanalyzed. The increasing interest in ontological analysis and the recent standardization activities in ontology are having an impact on how researchers in applied ontology understand their work. With the consolidation of the applied ontology research, I expect the ﬁrst class we described to take the lead in most domains where robust technological tools are essential to deliver the required services, there is a normalization process on

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the terminology, and the user is expected to have some technical expertise in the domain, e.g. medical doctors, engineers or accountants. The second class will remain the area in which ontological principles and their combinations are investigated, the goal will likely turn to produce top-levels which balance the need to have a clean and motivated view of reality, while allowing rich capacities to model subtle distinctions. I expect the third class to develop an optimal trade-off between bottom-up techniques and top-down principled constructions. This balance will be however driven by the parameters used to measure the ontologies. Since here there is a tendency to equate ontologies with logical theories, it is clear that mainly formal/logical parameters will be taken into consideration. If this analysis is correct, this line of research will generate optimal tools for constructing, merging, managing and maintaining large logical theories.

2. Modularity purposes In the light of the three classes introduce above, I propose to analyze modules by taking as central the following question: modules for what? I argue that ontology research in modularity rightly aims at different module types depending on the above distinctions. I will use the resulting distinctions among modules as an indication of what is needed to move forward in ontology modularization at least from the foundational perspective. 2.1. Modules for a single ontology The attitude of researchers in the ﬁrst class of Section 1, roughly the followers of the monolithic approach, is to introduce modularity mainly to organize and manage domain coverage. An example is given by the OBO Foundry initiative and their policy2 to relate OBO ontologies, roughly modules of the foundational ontology BFO.3 For what concerns us here, the following constraints on OBO are explicative (see also [4]): [...] for each domain there should be convergence upon exactly one Foundry ontology. [...] ontologies should use upper-level categories drawn from Basic Formal Ontology (BFO) [...] together with relations unambiguously deﬁned according to the pattern set forth in the OBO Relation Ontology (RO) [...]. [2, p.1050] The subdivision of the overall ontology in OBO modules is a consequence of the view proposed by the ﬁrst class: once the top-level and the ontology relations are given, the focus is on extending the system to cover the whole domain. Here I ﬁnd interesting the comparison with product design in the manufacturing domain. In manufacturing there is a clear distinction between structural and functional modularity. A product can be divided in modules from the structural perspective and/or from the functional perspective. Although the two aspects of products affect each other, there are methodologies devoted to just one or the other. Roughly, structural considera2 Available 3 Available

at http://www.obofoundry.org/id-policy.shtml at http://www.ifomis.uni-saarland.de/bfo/

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tions aim at optimizing the production, maintenance and disposal phases, both from the engineering and the economic viewpoint. Functional modularity, especially in the form of functional (de)composition, looks at how to divide complex (engineering) functions into simple and easily implementable units. By analogy we can say that the goal of modularity in monolithic approaches is to facilitate construction (more property, extension) and maintenance, thus the focus is towards structural considerations. A crucial feature of this notion of modularity is that a module can be added, dropped or substituted without needs for other changes in the overall system itself. This feature ensures a good overall control on the ontological system and makes it very robust. On the other side, it allows the customer to choose which part of the ontology to use depending on the domain knowledge he needs. 2.2. Modules for several ontologies Proceeding with the comparison, functional considerations are in focus among those that fall in the second class of Section 1, an example of which is given by the WonderWeb approach [3]4 . An inspiring way, in my view, to understand modularity in this approach is by focusing on functionality. Recall the underlying vision: several ontological systems should be developed, each on a par with the others. Beside the actual construction of the different ontologies, research here has been working on techniques and relations to properly build foundational ontologies. These are sometimes called (ontology) building blocks and include theories of essence and of parthood; formalizations of dependence relations; ways to use composition and constitution to relate and deﬁne categories. From the viewpoint of an ontology developer, these ontology blocks are not parts of the ontology in the sense discussed previously since, generally speaking, they do not need to occur in the taxonomy nor to be part of the formal (logical) system. They are better seen as (basic) functionalities that, combined, lead to systems with high expressivity: a theory of parthood in an ontology provides (additional) capacities to describe or relate categories and to express constraints in the ontology. In this sense, the parthood relation may occur only in specialized forms in several parts of the ontology, e.g. on material entities to model a notion of physical part, on events to model a notion of temporal part, on abstracts to relate a notion of conceptual part and so on. Thus, the ontologist uses a precise theory of parthood as a broad guideline in the ontology construction process, and yet the parthood building block itself might not occur anywhere in the ontology. These building blocks are modules developed not to be structural components, they are functional elements and, as such, do not need to be identiﬁable in the ontology system itself, although this might happen. The notion of building block we are discussing isolates an idea of (ontological) module which is disconnected from a speciﬁc ontology or even from speciﬁc ontological views. It can be replicated at different levels since it is not limited to general issues like parthood or dependence. For instance, work in this sense has been done to develop a module for the notion of physical artifact [1] where the notion has been formalized as an historical property of a physical object generated by agent’s intentional acts (speciﬁcally acts of selection and property attribution). Here the module identiﬁes the minimal 4 Note that in [3] the term ‘module’ indicates an ontology in the WonderWeb ontology library. This departs from the use of the term in literature since the library itself is not considered an ontology but a collection of independent ontologies.

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necessary conditions to make sense of the domain notion, and makes only minimal commitments to ontological distinctions: it constrains their existence but not their properties. Indeed, this work on physical artifacts assumes the existence of intentional agents, it uses the constitution relation and the notion of attributed capacity (a property ascribed to an object). The module does not say what an agent is, nor what properties characterize intentionality, it does not model capacities nor constrains constitution: these notions are referred to but left unspeciﬁed. Once the unspeciﬁed notions are connected to notions in an ontological system, the module can be considered a structural module (covering the notion of physical artifacts) tailored for that speciﬁc system. In [1], for example, this work is carried out in the form of an axiomatization based on the DOLCE ontology [3], which could be analogously repeated for other ontologies. In conclusion, if modules in the ﬁst class aim to extend the ontology, modules here aim to help the ontologist in his/her modeling work. 2.3. Modules for... everything When moving to the view of modules in the third class of Section 1, the possible meanings of the term multiply quickly. A module might be the result of: • isolating/developing branches of a taxonomy • collecting categories according to a domain (medicine, engineering, commerce etc.) • isolating (sub)theories to identify a context or local knowledge • isolating primitives and their axiomatizations • isolating patterns (repeating design formats) • isolating (sub)systems by minimizing the number of cross-relationship • dividing/developing the system to improve overall reasoning • separating (sub)systems suitable for different reasoning engines • separating (sub)systems to improve ontology matching • and so on These are orthogonal dimensions along which one can analyze an ontology and identify relevant modules for some purpose. Since the general interest is on formal aspects and techniques to (semi)automatically generate these types of modules, these approaches often lack ontological considerations beyond what we have already seen for the other categories.

3. Final considerations Leaving here aside modularity as understood in the third class since it is of reduced foundational interest, it is evident that modularity based on topic/discipline separation is fairly simple to implement and is being consistently used. Modularity in the sense of building blocks is potentially very powerful but it is hardly adopted: there is no systematic work, nor methodology, to build models in this sense and the use of the few available to, say, expand an ontology still requires a deep understanding of ontological analysis and techniques. The approach, if consistently developed, would improve the quality of ontological systems and the management tech-

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niques but without automatic tools and standardized methodologies, the instantiation of this kind of modules into an ontology remains too complicated. I expect we can move forward by stepping away from a tradition leitmotiv in ontology, namely, the search for systems with a minimal number of strongly pre-characterized primitives. Indeed, we might consider a different construction approach even in foundational ontology. The ﬁrst step is twofold: on the one hand develop ontology frameworks in the form of systems of weakly connected categories at the mesoscopic level and, on the other, develop different building blocks for the corresponding notions. At the second step, build (logically and ontologically) the ontology via the connection of selected modules to the framework. By stating which categories correspond to the pointers of a building block, one enriches the initial framework with specialized axioms (thus augmenting the axiomatization) which, coming from ontological blocks, contribute to characterize the meaning of the categories themselves. Returning to the previous example of the module for physical artifacts, Section 2.2, we can image a pre-ontological system formed by categories like agent, artifact, social organization etc. with a minimal characterization, perhaps just in taxonomic terms. The ontology is then generated by linking these categories to a module for artifact, one for agency, one for social organization and so on. Ontological consistency should be checked (likely beforehand) and so logical consistency (likely afterwards), while the meaning of very top-level notions like that of object and event, if needed, could be obtained indirectly by an analysis of the ﬁnal ontology. If this view is implemented, modularization will be the driving force in ontology construction and new types of relations, dedicated to connecting modules and ontology, will be studied. I would tentatively list among these the relations of analogy, coupling (in the sense of input-output alignment), generalization/specialization and realization. In today’s approaches, these are hardly considered foundational relations but will take a leading role if module structures and module dependences, as presented in 2.2, take the stage.

References Stefano Borgo and Laure Vieu. Artifacts in Formal Ontology. In Anthonie Meijers, editor, Handbook of the Philosophy of the Technological Sciences. Technology and Engineering Sciences, volume 9, pages 273–307. Elsevier, 2009. [2] W. Ceusters and B. Smith. A uniﬁed framework for biomedical terminologies and ontologies. In Proceedings of the 13th World Congress on Medical and Health Informatics (Medinfo 2010), Cape Town, South Africa, 12-15 September 2010, pages 1050–1054, 2010. [3] C. Masolo, S. Borgo, A. Gangemi, N. Guarino, and A. Oltramari. Ontology Library. Deliverable 18, WonderWeb, 2003. [4] B. Smith, M. Ashburner, C. Rosse, J. Bard, W. Bug, W. Ceusters, L.J. Goldberg, K. Eilbeck, A. Ireland, C.J. Mungall, et al. The OBO foundry: coordinated evolution of ontologies to support biomedical data integration. Nature biotechnology, 25(11):1251–1255, 2007.

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Modular Ontologies O. Kutz and T. Schneider (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-799-4-7

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Modularity Issues in Biomedical Ontologies a

Stefan SCHULZ a,1 , Pablo LÓPEZ-GARCÍA b Institute for Medical Informatics, Statistics and Documentation, Medical University of Graz, Austria b University of the Basque Country, San Sebastián, Spain

Abstract. This keynote addresses modularity issues focusing on biomedical ontologies, with a particular consideration of SNOMED CT. Emphasis is put on approaches that carve out high-coverage subsets addressing specialty-speciﬁc documentation needs. Keywords. Biomedical Ontologies, SNOMED CT, Modularization

Background Biology and medicine are highly knowledge-intensive disciplines. This is mirrored by the dynamic evolution of domain-speciﬁc vocabularies and ontologies [2]. Currently, two tendencies deserve special attention: On the one hand, the rapidly growing OBO (Open Biomedical Ontologies) Foundry collection [8], building upon the successes of the Gene Ontology, developed by user communities interested in the annotation of research data. On the other hand, SNOMED CT [1], combining collections of clinical terms with increasingly ontology-based formal descriptions. It contains nearly 300,000 active concepts to represent all aspects of health records. SNOMED CT is centrally developed and maintained by the IHTSDO, an international standards development organization. In OBO ontologies, modularity constitutes one of the basic design principles [8]. Each ontology ideally reﬁnes one ontological top level category (e.g. Material object, Process, Function), commits to a deﬁned granularity level (e.g. cellular components, cells, gross anatomy), and to certain biological taxa (e.g. human, mouse, fruitﬂy). Nearly all classes are primitive and arranged in taxonomic and partonomic hierarchies. Bridging ontologies have recently been semi-automatically built in order to connect classes of OBO modules and to provide full deﬁnitions. One example is the connection between the classes of the Molecular Function branch of the Gene Ontology with chemicals from ChEBI. SNOMED CT, being a monolithic ontology, is nevertheless split into separate subhierarchies like Clinical ﬁnding, Procedure, Body structure, Organism, and Substances, for which well-deﬁned design criteria apply. For instance, the latter two contain only primitive classes and no attributes. Fully deﬁned concepts are frequent in Clinical ﬁnd1 Corresponding Author: Institute for Medical Informatics, Statistics and Documentation, Medical University of Graz, Auenbruggerplatz 2/V, 8036 Graz, Austria; E-mail: [email protected].

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ing and Procedures, linking to Body structure in most of their deﬁning attributes, using description logics axioms not exceeding the expressiveness of EL. Both OBO Foundry and SNOMED CT naturally exhibit a vertical modular structure, with the dissection lines following more (OBO) or less (SNOMED CT) ontologically motivated upper-level categories. However, most use cases from both medical documentation and biological annotation are not served by single modules as they span across several categories. Both usability and computability demands motivate the creation of targeted, use-case speciﬁc modules. Due to the high degree of specialization in health care and life science, very few users require the whole breadth of SNOMED CT. The modularization requirements can be worded as follows: Create a SNOMED CT module M (for a domain of discourse D) which is (i) sufﬁciently compact, and (ii) provides a high coverage of D. Furthermore (iii), the fragment should preserve, as much as possible, the logical entailments that can be derived from the original ontology. It must be mentioned, however, that for the current use cases (iii) is a secondary desideratum, since so far SNOMED CT’s routine use has been restricted to the provision of controlled terms. Given the preliminary and still controversial status of many axioms [6,5], SNOMED CT can not yet be regarded mature enough for supporting clinical decisions by ontology-based reasoning.

Methods and Results We report on an ongoing study in which several modularization approaches are tested with regard to size and coverage, given a set of 20 signatures which represent the domain of discourse to be covered. Each signature corresponds to a patient discharge summary which has been manually annotated with SNOMED CT codes. The texts had been made available by the Hospital de Clínicas de Porto Alegre (Brazil) and describe in-patients of the cardiology department. In contrast to the classical modularization approaches, the task is here not to extract a minimal module [3] but to extract a representative module. The difference is that we consider the input (seed) signatures as typical but not exhaustive for D. For instance, the seed signatures may include several SNOMED CT concepts which represent typical cardiovascular drugs. We expect M to include additional drugs that are likely to be prescribed for cardiovascular disorders, but not, e.g. chemotherapeutic agents used in the treatment of cancer. We applied several variations of a graph-traversal modularization heuristic [7], as well as a technique based on description logics and locality [4]. For every technique, we used 10-fold cross validation over a set of 20 existing fully SNOMED CT-coded patient summaries and measured average coverage and module size. In the best case, an average coverage of 96% was reached with a SNOMED CT subset of about half the size (51%) using graph-traversal heuristics; a medium coverage of 71% was reached with a subset containing 17% of SNOMED CT. The locality-based technique extracted the smallest module (1%), but coverage was strongly affected (55%). These results are, so far, little encouraging. As an alternative strategy, term frequency data from an external source, viz. the biomedical literature database MEDLINE was used. Only SNOMED CT concepts for which a related term appeared at least once in the MEDLINE corpus were considered. Using this method, a medium coverage of 77% was obtained with a SNOMED CT subset of only

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9%. However, using this criterion on the whole SNOMED CT a medium average of 81% could not be hit. The most plausible explanation for this ﬁnding is the content mismatch between clinical texts and scientiﬁc abstracts. Nevertheless, the use of frequency data seems promising, and we will use a clinical corpus in the future. Our case study exposes that common cases in biomedicine, such as the one we have presented, can be particularly challenging for ontology modularization techniques, especially if the terminology, although compact and speciﬁc to a target user group, is spread across a wide range of subhierarchies. This is the case of patient summaries, which include a broad scope of information about reasons for admission, past history, interventions, proposed follow-up, etc. This still ongoing study provides evidence that a combination of graph-traversal strategies and information of data frequency can prune large biomedical ontologies and obtain handy subsets with a still acceptable coverage. A more thorough analysis of the relative gain should be carried out for each subhierarchy, and the frequency data should be obtained from a sufﬁciently large corpus close to the content to be coded. Finally, an open question is whether the modules extracted by our graph-traversal heuristics fulﬁll the safety requirements postulated by [3], i.e. whether they produce exactly the same entailments as the complete SNOMED CT. What may be helpful is that SNOMED CT axioms are rather uniform due to EL expressivity and limited nesting. Further investigation is needed.

References [1] [2]

[3] [4]

[5] [6]

[7]

[8]

SNOMED Clinical Terms. Copenhagen, Denmark: International Health Terminology Standards Development Organisation (IHTSDO), 2010. Fred Freitas, Stefan Schulz, and Eduardo Moraes. Survey of current terminologies and ontologies in biology and medicine. RECIIS - Electronic Journal in Communication, Information and Innovation in Health, 3:1–13, 2009. Bernardo Cuenca Grau, Ian Horrocks, Yevgeny Kazakov, and Ulrike Sattler. Modular reuse of ontologies: Theory and practice. Journal of Artiﬁcial Intelligence Research (JAIR), 31:273–318, 2008. Ernesto Jimenez-Ruiz, Bernardo Cuenca Grau, Thomas Schneider, Ulrike Sattler, and Rafael Berlanga. Safe and economic re-use of ontologies: a logic-based methodology and tool support. In ESWC 2008, Proceedings of the 5th European Semantic Web Conference, Tenerife, Spain, June 1-5, 2008. Springer LNCS, 2008. Stefan Schulz, Ronald Cornet, and Kent Spackman. Consolidating SNOMED CT’s ontological commitment. Applied Ontology, 6:1–11, 2011. Stefan Schulz, Boontawee Suntisrivaraporn, Franz Baader, and Martin Boeker. SNOMED reaching its adolescence: ontologists’ and logicians’ health check. International Journal of Medical Informatics, 78(Suppl 1):S86–94, 2009. Julian Seidenberg and Alan Rector. Web ontology segmentation: analysis, classiﬁcation and use. In Proceedings of the 15th International Conference on World Wide Web, WWW ’06, pages 13–22, New York, NY, USA, 2006. ACM. B. Smith, M. Ashburner, C. Rosse, J. Bard, W. Bug, W. Ceusters, L.J. Goldberg, K. Eilbeck, A. Ireland, C.J. Mungall, OBI Consortium, N. Leontis, P. Rocca-Serra, A. Ruttenberg, S.A. Sansone, R.H. Scheuermann, N. Shah, P.L. Whetzel, and S. Lewis. The OBO Foundry: coordinated evolution of ontologies to support biomedical data integration. Nature Biotechnology, 25(11):1251–1255, 2007.

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Query Inseparability and Module Extraction in OWL 2 QL Michael ZAKHARYASCHEV a a

Department of Computer Science and Information Systems, Birkbeck College London, U.K. Abstract. The OWL 2 QL proﬁle of OWL, based on the DL family of description logics, is emerging as a major language for developing new ontologies and approximating existing ones. Its main application is ontology-based data access, where ontologies are used to provide background knowledge for answering queries over data. We give a survey of recent results on the computational complexity of checking query inseparability for OWL 2 QL ontologies and analyse the impact of various OWL 2 QL constructs in the context of query inseparability. We also discuss practical query inseparability checking and minimal module extraction algorithms, as well as experimental results. Keywords. OWL 2 QL, Σ-query inseparability, module extraction

The OWL 2 QL proﬁle of the Web Ontology Language OWL 2 has recently emerged as a major language for ontology-based data access (OBDA). OWL 2 QL was built on a description logic (DL) that was originally introduced under the name DL-LiteR [2,3] and called DL-LiteH core in the more general classiﬁcation of [1]. It can be described as an (almost) maximal sub-logic of SROIQ, underlying OWL 2, which includes most of the features of conceptual models, and for which conjunctive query answering can be done in AC0 for data complexity. One of the consequences of this development is that OWL 2 QL is becoming now an important language for developing ontologies, as well as a target language for translation and approximation of existing ontologies, originally given in more expressive DLs. As a result, efﬁcient reasoning support is required for ontology engineering tasks such as composing, re-using, comparing, and extracting OWL 2 QL ontologies. In the context of OBDA, the fundamental notion underlying various ontology engineering tasks is Σ-query inseparability: given a signature (that is, a ﬁnite set of concept and role names) Σ, we consider two ontologies to be ‘inseparable’ with respect to Σ if they give the same answers to any conjunctive query over any data, both formulated in terms of the signature Σ. In this paper, we give a survey of recent results on the computational complexity of checking Σ-query inseparability for OWL 2 QL ontologies and analyse the impact of various OWL 2 QL constructs in the context of Σ-query inseparability. We also discuss practical Σ-query inseparability checking and minimal module extraction algorithms, as well as experimental results. Technical details can be found in [5,6,7,4].

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References [1] [2] [3]

[4] [5] [6] [7]

A. Artale, D. Calvanese, R. Kontchakov, and M. Zakharyaschev. The DL-Lite family and relations. Journal of Artiﬁcial Intelligence Research (JAIR), 36:1–69, 2009. D. Calvanese, G. De Giacomo, D. Lembo, M. Lenzerini, and R. Rosati. Data complexity of query answering in description logics. In Proc. of KR, pp. 260–270, 2006. D. Calvanese, G. De Giacomo, D. Lembo, M. Lenzerini, and R. Rosati. Tractable reasoning and efﬁcient query answering in description logics: The DL-Lite family. J. of Automated Reasoning, 39(3):385–429, 2007. B. Konev, R. Kontchakov, M. Ludwig, T. Schneider, F. Wolter, and M. Zakharyaschev. Conjunctive query inseparability of OWL 2 QL TBoxes. In Proc. of AAAI, 2011. R. Kontchakov, F. Wolter, and M. Zakharyaschev. Can you tell the difference between DL-Lite ontologies? In Proc. of KR, pages 285–295, 2008. R. Kontchakov, L. Pulina, U. Sattler, T. Schneider, P. Selmer, F. Wolter, and M. Zakharyaschev. Minimal module extraction from DL-Lite ontologies using QBF solvers. In Proc. of IJCAI, pages 836–841, 2009. R. Kontchakov, F. Wolter, and M. Zakharyaschev. Logic-based ontology comparison and module extraction, with an application to DL-Lite. Artif. Intell., 174(15):1093–1141, 2010.

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Applying Community Detection Algorithms on Ontologies for Identifying Concept Groups Gökhan COSKUN a , Mario ROTHE a , Kia TEYMOURIAN a , Adrian PASCHKE a a Freie Universität Berlin, Germany Abstract. Reusing existing Semantic Web ontologies is necessary to avoid heterogeneity as well as redundant modeling efforts, because ontology engineering is a time-consuming and cost-intensive task. In order to decide whether a candidate ontology comprises the right concepts, an analysis process is necessary to understand the conceptual model of the ontology. Driven by the idea that concept grouping simpliﬁes understanding the content of an ontology we investigate the applicability of community algorithms from the ﬁeld of Social Network Analysis on the graph structure of RDF/XML based OWL documents to identify concept groups. In this paper, we present our experiments with different community algorithms on popular ontologies and compare our results with manually created concept groups. Keywords. Concept Grouping, Ontology Partitioning, Ontology Summarization

Introduction Although reusing ontologies is at the core of the Semantic Web and part of various ontology engineering methodologies there are no best practice solutions describing how existing ontologies should be analyzed for their (re)usability. The ﬁrst step towards reuse of existing ontologies is to discover candidate ontologies, which might include required concepts for the application domain of the ontology. For this, online ontology libraries such as Ontolingua1 and OntoSelect2 are available and ontology search engines such as Swoogle3 , Watson4 and Ontosearch5 have been developed. But the second step, which is the major concern of this paper, is the analysis process of the candidate ontologies. In this step the content of each candidate ontology needs to be understood in order to decide whether it covers the domain of interest. Understanding the structural design of the whole ontological model is essential to decide if a candidate ontology is really useful in the application context and whether it needs further customization, e.g. refactoring, narrowing, etc. In particular, in case of larger ontologies with hundreds and thousands of concepts it is nearly impossible for the application engineers to overview and under1 http://www.ksl.stanford.edu/software/ontolingua 2 http://olp.dfki.de/ontoselect 3 http://swoogle.umbc.edu 4 http://watson.kmi.open.ac.uk/WatsonWUI 5 http://www.ontosearch.org

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stand the whole ontology model. Hence, it becomes hard if not impossible for them to decide about the quality of the ontology candidate for the application purpose. Even a smaller ontology such as the Friend of a Friend (FOAF) vocabulary6 uses a grouping of concepts in its speciﬁcation (see Figure 1), in order to provide the reader an easier way to understand the vocabulary.

Figure 1. Concept groups of the FOAF vocabulary in the speciﬁcation (version 0.97)

Aiming at supporting this necessary analysis process we utilize community detection algorithms from the ﬁeld of Social Network Analysis on the graph structure of ontologies in order to identify concept groups which are cognitively easier to understand by ontology engineers. In this work we focus on the conceptual schema part (T-Box model) of the ontology. The rest of this paper is organized as follows. In Section 1, we describe the graph structure of ontologies and discuss the different graph representations for ontologies. Section 2 deﬁnes what we understand by the term "concept grouping" and presents the community algorithms which we have used in our experiments. Section 3 presents the existing ontology module evaluation techniques and contributes with our evaluation technique. In section 4 our experimental results are presented. And ﬁnally, in section 5 we give an overview of related work in the ﬁeld of structure-based analysis and partitioning as well as summarizing ontologies. We conclude the paper and give a brief outlook on our future work in the last section.

1. Graph structure of ontologies In the Semantic Web, ontologies are mostly represented by the Web Ontology Language (OWL) based upon the Resource Description Framework (RDF)7 . RDF allows representing information as triples following the form (Subject, Predicate, Object). The graph syntax of RDF8 maps triples to graphs where the subjects and the objects are nodes and the predicates are directed edges (from subject to object). At this level the inherent semantics of OWL ontologies are not taken into consideration. The nodes and edges have different types, which are reﬂected in their labels (namespace and localname). Since the subjects 6 http://xmlns.com/foaf/spec/ 7 http://www.w3.org/TR/owl-semantics/mapping.html

describes how OWL is mapped to RDF

8 http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/

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as well as the predicates of RDF statements need to be resources and objects might also be resources, it is impossible to organize the edges and nodes into disjoint sets. This is because a resource, which is a subject or an object in one statement might be used as a predicate in another statement. This problem of the RDF graph representation of triples can be avoided if every named entity of the ontology is represented as a node (even the predicate becomes a node, which is connected with the subject and the object). However, since typically the number of properties is much less than the number of resources which are used as subjects and objects, this graph representation would lead to a graph structure in which the properties are central nodes with high degree values. Some predicates such as “hasLabel” or “hasComment” would have a high centrality value. Hence, it is important to ﬁlter and remove such concepts, which have a signiﬁcant impact on the graph structure analysis of an ontology, but which are not necessary in order to understand the content of an ontology. Furthermore, it is important to take different namespaces into consideration. We developed three basic ideas how to represent an ontology as a graph. Firstly, the RDF graph syntax is used as it is, that means the subject and object of each statement are nodes, while the predicate is the connecting edge, directed from the subject to the object (variant V1). Secondly, the predicates are also represented by nodes, where two unlabeled directed edges are created. One edge is directed from the subject to the predicate, while the second is directed from the predicate to the object (V2). Thirdly, a graph is created where only classes are represented as nodes connected by properties as edges, where the direction is from the domain class of the property to the range class of the property (V3). There are also two different extensions of these variants. In the ﬁrst extension (named as VxL) the literals are ﬁltered during the graph creation process. This ﬁlter is enhanced by the second extension (named as VxLX) by excluding concepts from the OWL, RDF, RDFS, and XMLSchema vocabularies like owl#ObjectProperty and rdfschema#Class. Summing up, for our experiments we created nine different graph variants for each ontology. The different variants are shown in Table 1. Table 1. Different graph representations for ontologies Variant name

Description

V1 V1L

Plain RDF graph as V1 without literals

V1LX V2 V2L

as V1L without RDF/RDFS/OWL/XMLSchema nodes Plain RDF graph, but predicates are represented as nodes as V2 without literals

V2LX V3

as V2L without RDF/RDFS/OWL/XMLSchema nodes Class graph

V3L

as V3 without literals

V3LX

as V3L without RDF/RDFS/OWL/XMLSchema nodes

2. Concept Grouping There are different terms in the literature which are used to describe more or less the same process that we call concept grouping. Network partitioning, graph partitioning,

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15

clustering and segmentation are some examples for such terms. We deﬁne the process which we refer as concept grouping as identifying the groups of concepts based on the network structure of the ontology (see Section 1) in such a way, that the concepts within a group are belonging stronger to each other than to the concepts of another group. (We intentionally do not use the term similar because concepts can be externally similar (in worst case even homonyms) but have very different meanings.) The mathematical approach (mostly named as graph partitioning) seeks for subgraphs which are about the same size in such a way that the connections between these subgraphs are minimized. For ontologies this approach does not seem to be suitable because ontologies model various parts (subdomains) of a domain in different levels of detail. E.g. the concept groups of the FOAF ontology shown in Figure 1 have different sizes. We believe that the different subdomains of a domain are reﬂected in an ontology in such a way that the concepts belonging to one subdomain are building a more densely connected graph partition - a community. For that reason a social network analysis approach for detecting communities seems to be more suitable for identifying the concept groups within ontologies. In order to investigate the applicability of different community detection algorithms to the ontologies, we applied the following algorithms on the different structure variants of the ontologies which we introduced in Table 1 . 2.1. Edge Betweenness Community The Edge Betweenness Community (EBC) algorithm introduced in [10] is a divisive hierarchical clustering algorithm which focuses on the edges within a graph. Divisive means that the edges are removed iteratively until different communities are identiﬁed [10]. Its basic idea is that a network comprises densely connected communities which in turn are sparsely connected. By calculating the shortest paths between each node pair the edge with the highest betweenness, which is likely to be connecting two communities, can be identiﬁed and removed. In each step of this algorithm the betweenness of each edge is calculated and only the one with the highest betweenness is removed. 2.2. Walktrap Community Pons and Latapy are proposing an algorithm in [12] which is based on the same community idea as the EBC algorithm. It is stated that “random walks on a graph tend to get “trapped” into densely connected parts corresponding to communities.” For that reason this algorithm is called Walktrap Community (WTC). In contrast to EBC this is not a divisive but an aglomerative hierarchical clustering algorithm, which means that the communities are build step by step by merging vertices into communities. 2.3. Fast Greedy Community The Fast Greedy Community (FGC) algorithm introduced in [3] identiﬁes communities by optimizing a Modularity[10] score, which is a network property and a speciﬁc proposed partitioning of that network into communities. It evaluates the partitioning, in the sense that in a good partitioning there are many edges within communities and only a few between them.

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2.4. Spin Glass Community The Spin Glass Community (SGC) algorithm for community detection was proposed by Reichardt and Bornholdt in [13,10]. It makes use of the model of a spin glass and simulated annealing. The community structure of the network is interpreted as the spin conﬁguration that minimizes the energy of the spin glass with the spin states being the community indices. 2.5. Leading Eigenvector Community Mark Newman proposes in [9] a different approach for the maximization process of the Modularity[10] score, called the Leading Eigenvector Community (LEC) algorithm. It detects communities in network by calculating the leading non-negative eigenvector of the modularity matrix of the graph.

3. Ontology Module Evaluation It is not possible to decide how good an ontology is without knowing the context in which it is intended to be used, because ontologies are build in an application dependent manner. And even if the context is known there are always different ways to create a conceptual model of a domain. The ontology module evaluation techniques which are proposed in literature are based on the structure of the ontology. The common idea behind these techniques is that information provided by different modules, should be as far as possible - independent and disjunct. That means that each module represents a subdomain of the domain which is modeled by the whole ontology. Previous work like [5] and [14] make use of very simple structural information as the number of modules, average module size, size variance, and the connectedness between the modules to evaluate ontology modularization. Calmet et. al. propose in [2] a distance measure for two concepts within an ontology based on the notion of entropy in order to measure the similarity between two modules. This approach is extended in [4] by distinguishing between language level edges and domain level edges, so that two different entropies are calculated, namely the language level entropy and the domain level entropy. By distinguishing between two kinds of edges a ﬁrst step towards a semantic sensitive technique has been made. However, a pure structure-based measure is not adequate to evaluate the structurebased modularization techniques, since the modularization can be always optimized in such a way that the evaluation score is increased. Instead, we use as a "gold standard" (reference model for the evaluation of the modularization technique) the existing concept groupings as they have been designed and described by the ontology engineers. The rational for this evaluation approach is, that the quality of an ontology module usually depends on the application where it is going to be used. Hence, we use the documented groupings which have been introduced by the ontology engineers as test cases for the evaluation of the resulting concept grouping from the applied modularization technique. This means that for evaluating the grouping algorithms we just need to compare the reference model with the concept grouping and calculate the similarity. For that purpose we use two different algorithms. Firstly, we use the Jaccard-Index, which produces a score for the similarity of two sets X and Y.

G. Coskun et al. / Applying Community Detection Algorithms

J(X, Y ) =

|X ∩ Y | |X ∪ Y |

17

(1)

The similarity corresponds to the relation between the number of common elements of a set to the number of the whole elements. In the best case, where two completely identical sets are compared, the Jaccard-Index equals 1. When two sets without any common element are compared, which is the worst-case, the Jaccard-Index equals 0. Because we do not know which of the produced groups refers to which group of the reference model we calculate the Jaccard-Index between each pair. The score value for a produced concept group X is given by Sg (X) = max(J(X, Yj ))

(2)

where Yj are the groups of the reference model. The score value for the algorithm is then the mean value over the scores of each produced concept group divided by the number of groups produced Sg (Xi ) i Sa (A) = (3) i where i is the number of produced groups and Xi is one speciﬁc group. Secondly, we make use of the F-Measure which is a pairs-based approach and described in [15]. It is calculated with F =

2 ∗ precision ∗ recall precision + recall

(4)

where “the precision of a partition is deﬁned as the ratio of intra-pairs in the generated partitioning that are also intra-pairs in the optimal partitioning.” and “the recall of a partition is deﬁned by the ratio of intra-pairs in the optimal partitioning that are also in the generated one.” [15].

4. Experiments For our experiments we implemented a lightweight web application, which uses the R tool9 with the iGraph10 library for the partitioning algorithms. Before calculating the partitions, the ontology documents are loaded with JENA11 and are converted into GraphML ﬁles according to the variants which are shown in Table 1. Before this process is started the ontologies are loaded in two different ways. First we loaded the ontology as a raw model, that means the inference is turned off. In the second way only the inference mechanisms are turned on. As we use hand-made concept grouping to evaluate our partitioning results following the technique presented in Section 3, we made use of three different ways to get the reference partitions for our experiments. First we searched for ontologies, which are divided into concept groups in their documentations. We found the Friend of a Friend 9 http://www.r-project.org 10 http://igraph.sourceforge.net 11 http://jena.sourceforge.net/

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(FOAF) vocabulary12 and the Music Ontology13 . Secondly we decided to merge modularized ontologies into one ontology, and compare the partitioning results with the original modules. We did so with the Semantically-Interlinked Online Communities (SIOC) core ontology and its extensions14 . And ﬁnally, we asked an expert to group the concepts of the Good Relations ontology15 into different groups. We are aware that these ontologies are very small for a complete investigation on the applicability of community algorithms on ontologies. But as ﬁnding existing reference models for our evaluation technique is quite difﬁcult we decided to make use of these small ontologies as a ﬁrst step. Besides, the documentations of the FOAF ontology and the Music Ontology shows that even in small ontologies concept groups are usefull. In the sections 4.1 to 4.4 we illustrate the experimental results with the mentioned ontologies. After brief structural information there are two different tables per ontology. The ﬁrst table shows the results with the Jaccard Index evaluation technique, while the second table shows the results with the F-Measure evaluation technique. In each cell the evaluation result (multiplied by 100) is followed by the number of the created partitions (values in the brackets).

4.1. FOAF Ontology

The FOAF ontology contains 613 statements in its raw form. Aftering running inference on it the model containts 2013 statements. The reference grouping comprises ﬁve groups as shown in Figure 1.

Table 2. Evaluation of the partitioning results with the FOAF ontology (with Jaccard Index) FGC

EBC

SGC

WTC

LEC

21 (9) / 26 (8)

11 (13) / 14 (13)

23 (5) / 29 (5)

18 (8) / 20 (5)

20 (14) / 15 (14)

26 (4) / 32 (3) 23 (11) / 23 (11)

12 (10) / 14 (12) 9 (44) / 10 (38)

26 (5) / 28 (4) —/—

27 (4) / 33 (2) —/—

19 (9) / 23 (7) 26 (8) / 28 (9)

V2L

27 (2) / 27 (2) 30 (3) / 27 (2)

20 (3) / 9 (33) 20 (3) / 9 (33)

21 (4) / 27 (2) —/ 21 (3)

23 (2) / 27 (2) —/ 27 (2)

9 (51) / 24 (3) 13 (19) / 27 (2)

V2LX V3 V3L V3LX

18 (11) / 25 (3) 28 (7) / 23 (2) 28 (7) /— 40 (4) / 36 (3)

9 (32) / 25 (5) 17 (9) / 24 (1) 17 (9) / 24 (1) 20 (4) / 20 (11)

—/— —/— —/— —/—

—/— —/— —/— —/—

15 (9) / 15 (13) —/— —/ 40 (3) 39 (4) / 35 (5)

26.78 / 27.38

15 / 16.56

23.33 / 26.25

22.67 / 26.75

20.14 / 25.88

V1 V1L V1LX V2

AV

12 http://xmlns.com/foaf/spec/20100101.html, we used version 0.97, because the newer version’s documentation does not provide a concept grouping 13 http://musicontology.com/ 14 http://sioc-project.org/ontology 15 http://www.heppnetz.de/ontologies/goodrelations/v1

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Table 3. Evaluation of the partitioning results with the FOAF ontology (with F-Measure) FGC

EBC

SGC

WTC

LEC

26 (9) / 35 (8) 29 (4) / 33 (3) 41 (11) / 40 (11) 31 (2) / 34 (2)

33 (13) / 35 (13) 33 (10) / 35 (12) 4 (44) / 31 (38) 33 (3) / 18 (33)

26 (5) / 38 (5) 29 (5) / 34 (4) —/— 33 (4) / 34 (2)

35 (8) / 31 (5) 33 (4) / 34 (2) —/— 34 (2) / 34 (2)

20 (14) / 19 (14) 21 (9) / 26 (7) 29 (8) / 43 (9) 3 (51) / 33 (3)

33 (3) / 34 (2) 24 (11) / 30 (3) 29 (7) / 22 (2)

33 (3) / 18 (33) 15 (32) / 32 (5) 26 (9) / 24 (1)

—/ 34 (3) —/— —/—

—/ 34 (2) —/— —/—

22 (19) / 34 (2) 23 (9) / 34 (13) —/—

V3LX

29 (7) /— 29 (4) / 26 (3)

26 (9) / 24 (1) 23 (4) / 28 (11)

—/— —/—

—/— —/—

—/ 24 (3) 29 (4) / 30 (5)

AV

30.1 / 31.75

25.1 / 27.22

29.3 / 35

34 / 33.25

21 / 30,38

V1 V1L V1LX V2 V2L V2LX V3 V3L

4.2. SIOC Ontology The SIOC ontology contains 907 statements in its raw form. Aftering running inference on it the model containts 2777 statements. The reference grouping comprises four groups. Table 4. Evaluation of the partitioning results with the SIOC ontology ( with Jaccard Index) FGC

EBC

SGC

WTC

LEC

V1

10 (25) / 8 (24)

5 (61) / 5 (49)

43 (4) / 38 (4)

9 (43) / 11 (12)

5 (53) / 8 (47)

V1L V1LX V2

38 (6) / 42 (4) 12 (21) / 12 (20)

3 (128) / 5 (49) 5 (66) /—

53 (4) / 44 (4) —/—

28 (9) / 51 (3) —/—

21 (13) / 16 (11) —/ 10 (28)

V2L

40 (4) / 67 (2) 53 (3) / 53 (3)

6 (40) / 5 (79) 6 (40) / 4 (108)

41 (4) / 75 (2) —/ 53 (3)

65 (1) / 34 (2) —/ 34 (2)

5 (41) / 5 (40) 6 (44) / 44 (4)

V2LX V3 V3L V3LX

16 (12) / 74 (2) 12 (22) /— 12 (22) /— 22 (10) / 32 (6)

5 (33) / 16 (16) 13 (18) / 5 (43) 13 (18) /— 11 (26) / 7 (48)

—/— —/— —/— —/—

—/— —/— —/— —/—

19 (10) / 43 (4) 9 (28) /— 9 (28) / 10 (26) 21 (6) / 22 (5)

AV

23.89 / 41.14

7.44 / 6.71

45.67 / 52.5

34 / 32.5

11.88 / 19.75

Table 5. Evaluation of the partitioning results with the SIOC ontology ( with F-Measure) FGC

EBC

SGC

WTC

LEC

V1

28 (25) / 31 (24)

49 (61) / 72 (49)

52 (4) / 59 (4)

27 (43) / 63 (12)

16 (53) / 16 (47)

V1L V1LX V2 V2L

45 (6) / 62 (4) 30 (21) / 31 (20) 77 (4) / 69 (2)

2 (128) / 72 (49) 37 (66) /— 66 (40) / 33 (79)

63 (4) / 65 (4) —/— 73 (4) / 77 (2)

64 (9) / 77 (3) —/— 66 (1) / 61 (2)

36 (13) / 28 (11) —/ 24 (28) 32 (41) / 38 (40)

58 (3) / 71 (3)

66 (40) / 11 (108)

—/ 72 (3)

—/ 61 (2)

38 (44) / 59 (4)

V2LX V3 V3L V3LX

59 (12) / 76 (2) 40 (22) /— 40 (22) /— 43 (10) / 36 (6)

51 (33) / 72 (16) 49 (18) / 58 (43) 49 (18) /— 46 (26) / 40 (48)

—/— —/— —/— —/—

—/— —/— —/— —/—

47 (10) / 54 (4) 21 (28) /— 21 (28) / 33 (26) 28 (6) / 31 (5)

46.67 / 53.71

46.11 / 51.14

62.67 / 68.25

52.33 / 65.5

29.88 / 35.38

AV

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4.3. Good Relations Ontology

The good relations ontology contains 1210 statements in its raw form. Aftering running inference on it the model containts 5273 statements. The reference grouping comprises three groups.

Table 6. Evaluation of the partitioning results with the good relations ontology (with Jaccard Index) FGC

EBC

SGC

WTC

LEC

V1

12 (16) / 16 (6)

3 (61) / 4 (26)

29 (3) / 30 (3)

6 (22) / 57 (1)

7 (27) / 14 (12)

V1L V1LX V2 V2L

17 (8) / 25 (3) 18 (9) / 25 (3) 57 (1) / 36 (2)

29 (2) / 4 (26) —/— 6 (16) / 4 (32)

32 (3) / 26 (3) —/— 22 (3) / 24 (3)

10 (9) / 57 (1) —/— 57 (1) / 34 (2)

13 (10) / 13 (13) 7 (24) / 22 (7) 18 (4) / 16 (5)

20 (4) / 23 (3)

6 (16) / 4 (32)

—/ 23 (3)

—/ 25 (3)

4 (49) / 6 (17)

V2LX V3 V3L V3LX

21 (4) / 25 (3) —/— 15 (8) /— —/ 30 (3)

6 (18) / 7 (18) 8 (11) / 9 (22) 8 (11) / 9 (22) —/ 9 (25)

—/— —/— —/— —/ 30 (3)

—/— —/— —/— —/ 18 (5)

12 (9) / 15 (7) —/— 10 (14) /— 18 (8) / 23 (4)

22.86/ 25.71

9.42 / 6.25

27.67 / 26.6

24.33 / 38.2

11.12 / 15.57

AV

Table 7. Evaluation of the partitioning results with the good relations ontology (with F-Measure) FGC

EBC

SGC

WTC

LEC

V1

22 (16) / 49 (6)

29 (61) / 51 (26)

40 (3) / 49 (3)

21 (22) / 60 (1)

12 (27) / 27 (12)

V1L

29 (8) / 49 (3)

59 (2) / 51 (26)

43 (3) / 51 (3)

37 (9) / 60 (1)

30 (10) / 34 (13)

V1LX V2 V2L

32 (9) / 50 (3) 60 (1) / 54 (2) 37 (4) / 51 (3) 37 (4) / 51 (3) —/—

—/— 50 (16) / 42 (32) 50 (16) / 42 (32) 50 (18) / 48 (18) 32 (11) / 26 (22)

—/— 59 (3) / 46 (3) —/ 50 (3) —/— —/—

—/— 60 (1) / 51 (2) —/ 51 (3) —/— —/—

19 (24) / 40 (7) 45 (4) / 50 (5) 18 (49) / 50 (17) 38 (9) / 32 (7) —/—

23 (8) /— —/ 35 (3)

32 (11) / 26 (22) —/ 18 (25)

—/— —/ 35 (3)

—/— —/ 39 (5)

21 (14) /— 26 (8) / 35 (4)

34.29 / 48.43

43.14 / 38

47.33 / 46.2

39.33 / 52.2

26.13 / 38.29

V2LX V3 V3L V3LX AV

4.4. Music Ontology

The music ontology contains 2092 statements in its raw form. Aftering running inference on it the model containts 9281 statements. The reference grouping comprises 23 groups.

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Table 8. Evaluation of the partitioning results with the music ontology (with Jaccard Index) V1 V1L V1LX V2 V2L V2LX V3 V3L V3LX AV

FGC

EBC

SGC

WTC

LEC

31 (19) /— 26 (9) /— 39 (13) / 32 (9) 17 (2) /—

19 (42) /— 9 (1) /— 18 (89) /— 10 (26) /—

29 (17) /— 24 (12) /— —/— 19 (9) /—

16 (30) /— 23 (21) /— —/— 9 (1) /—

28 (32) /— 37 (21) /— 38 (31) / 25 (50) 19 (34) /—

26 (3) /— 30 (5) / 33 (3) 30 (8) / 25 (5)

14 (60) /— 18 (24) /— 17 (38) / 18 (43)

—/— —/— —/—

—/— —/— —/—

16 (78) /— 20 (24) / 29 (8) —/ 23 (20)

30 (8) / 25 (5) —/ 33 (3)

—/— 18 (31) / 18 (47)

—/— —/ 31 (5)

—/— —/ 30 (10)

—/— 28 (8) / 30 (7)

28.63 / 29.6

15.38 / 18

24 / 31

16 / 30

26.57 / 26.75

Table 9. Evaluation of the partitioning results with the music ontology (with F-Measure) V1 V1L V1LX V2 V2L V2LX V3 V3L V3LX

FGC

EBC

SGC

WTC

LEC

29 (19) /— 26 (9) /— 37 (13) / 20 (9) 14 (2) /— 17 (3) /—

11 (42) /— 11 (1) /— 14 (89) /— 9 (26) /— 12 (60) /—

32 (17) /— 27 (12) /— —/— 13 (9) /— —/—

17 (30) /— 25 (21) /— —/— 11 (1) /— —/—

38 (32) /— 36 (21) /— 49 (31) / 32 (50) 22 (34) /— 12 (78) /—

24 (5) / 17 (3) 16 (8) / 8 (5) 16 (8) / 8 (5) —/ 12 (3)

16 (24) /— 6 (38) / 6 (43) —/— 6 (31) / 9 (47)

—/— —/— —/— —/ 13 (5)

—/— —/— —/— —/ 13 (10)

20 (24) / 23 (8) —/ 14 (20) —/— 15 (8) / 19 (7)

22.38 / 13

0.63 / 7.5

24 / 13

17.67 / 13

27.43 / 22

AV

4.5. Conclusion From the experiments we extract the following insights: • The different community algorithms reach very different scores. The scores range from 4 to 77. Values about 20 to 40 are calculated frequently, even though the community algorithms have been applied on the ontologies without any modiﬁcation. We think that some simple modiﬁcations to the algorithms so that the nature of ontologies are respected could lead to better results. The score of the Spin Glass Community algorithm is above the average in each experiment. Being the only one algorithm which was executed with the number of communities it should create it indicates that even such a little modiﬁcation can improve the concept grouping and conﬁrms the consideration that an adaptation of the algorithms might lead to better results. • The different graph variants are not changing the score of the grouping process signiﬁcantly. However, in most cases the results with activated inference are better than the results with deactivated inference. • The worst scores per table are calculated only by two algorithms, namely the edge betweenness centrality algorithm (5 of 8) and the leading eigenvector centrality algorithm (3 of 8). These algorithms are producing much more partitions than the other algorithms.

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• The overall ranking for the algorithms is as follows: 1. SGC (36.42), 2 WTC (33.23), 3. FGC (31.66), 4. LEC (24.26), 5. EBC (21.48) • The overall ranking for the different graph variants is as follows: 1. V2 (39.55), 2. V1L (33.9), 3. V2LX (33.67), 4. V2L (33), 5. V1LX (29.25), 6. V1 (28.15), 7. V3 (26.79), 8. V3L 26.38, 9. V3LX (25.62). The different versions of the variant V2 are mostly better than the other variants while the versions of the variant V3 lead to the worst evaluation values.

5. Related Work In the following we discuss related work in two research ﬁelds. First we analyze techniques for structure-based analysis as well as partitioning of ontologies because our approach to concept grouping is solely based on the graph structure of the ontology. Second, we looked into techniques for ontology summerization, because our main goal is supporting the analysis process in order to simplify the understanding of the content. Driven by the idea that the success of the Semantic Web depends on the existence of ontologies for advanced querying and reasoning services Theoharis et al. state in [16] that there is a need to benchmark repositories and query language implementations. This in turn, requires means to create synthetic ontologies (schemas as well as data). For that reason the structure of 83 selected ontology schemas with more than 100 classes were analyzed in [16] by focusing on power-law degree distribution. They are making a distinction between the property graph and the subsumption graph of the schema and use the Complementary Cumulative Density Function and the Value versus Rank function. The outcome of this analysis is that most schemas which they analyzed approximate a power-law for degree distributions in the property graph and also in the subsumption graph, which indicates the existence of central concepts forming a core. Structural analysis in [6] is motivated by the idea to measure the importance of a node in an RDF graph, without distinguishing between schema and data. For ranking the nodes the closeness centrality values are used. This method is called Node Centrality Ordering (noc-order). The RDF graph is used as an undirected labeled graph. This is a very early work, which needs further investigation. It is planned to extend this method in such a way, that it is able to respect the semantics in OWL. AKTiveRank [1] is a system which is motivated to facilitate reusing existing ontologies. It aims at improving ontology search engines by ranking ontologies based on structural properties of the search terms within the whole ontology. Four different measures are deﬁned, which are calculated separately by ignoring the instances and the resulting values are merged. These are namely Class Match Measure, Density Measure, Semantic Similarity Measure and Betweenness Measure. Except for the ﬁrst one they are all structural measures. The basic idea of the Density Measure is, the more deeply a concept is speciﬁed by other concepts the greater is its density value within the graph structure. That is, the number of relations, subclasses, superclasses and siblings is greater than those values of other concepts. The Semantic Similarity Measure is usable, if more than only one search term is used. The underlying assumption is, that a small distance between them means a close semantic similarity in the conceptual model. For that reason the distance of the terms within the ontology structure is calculated. Finally, the Betweenness Measure calculates the number of shortest paths that pass through each concept. This is

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based on the idea, if a concept has a high betweenness value in an ontology then this class is central to that ontology. In [7] Semantic Network Analysis (SemNA) is introduced to analyze ontologies for the purpose of reuse and re-engineering. Different notions of node centrality are used, namely degree centrality, betweenness centrality and eigenvector centrality. The KAON ontology API was used and the graphs were preprocessed before the centrality values were calculated. Similar to our appraoch all named entities become nodes in the preprocessed graph. In [17] different centrality measures like degree centrality, betweenness centrality and eigenvector centrality are utilized to identify central RDF sentences in the RDF graph. In contrary to our goal, its main goal is not to create concept groups but to extract few RDF sentences which are representative for the content of the ontology. The results are evaluated by human judges who give a score in order to quantify the quality of the outcomes. Work on ontology summarization by identifying key concepts was presented by Peroni et al. in [11]. They used structural criteria along with criteria from cognitive science and lexical statistics. As structural criteria they propose a Density measure which is mainly based on the degree of the nodes. The edges linking subclass relations, properties and instances are weighted with constants, which can be seen as a ﬁrst step towards a semantic-sensitive measure. The evaluation of the results is made by comparing the results of their technique with the choice of human experts. Analyzing the network structure of an ontology as a basis for partitioning the class hierarchy into disjoint and covering set of concepts is presented in [15]. Its main goal is to support distributed maintenance, selective reuse and efﬁcient reasoning. The ontology is preprocessed like in [7], that is each named entity became a node. A reference model is used to evaluate the partitioning result, whereby precision and recall functions are used to calculate the similarity. 6. Future Work In this paper we have experimentally evaluated concept grouping with ﬁve different community identiﬁcation algorithms applied on nine different network representations of ontologies with two different ontology load strategies. In order to evaluate the results we used ontologies whose concepts are manually grouped in the documentation or ontologies which are developed in a modular way or we asked an expert to manually group the concepts of an ontology. We believe that the results of our experiments justify further investigation on the applicability of community algorithms on ontologies for concept grouping. In future work we plan to experiment with algorithms which consider the directed and typed nature of ontologies (e.g. [8]). In case of an appropriate weight function for ontology properties it is also reasonable to investigate the applicability of community algorithms which take edge weights into consideration. In this regard we will investigate how the semantics inherent in ontologies can be taken into account and can be reﬂected as edge weights. The evaluation methods for the concept grouping need also further attention. In [15] precision, recall and a combination of both is used to calculate the similarity between the partitions of a reference model and the partitioning result. Finally, for more signiﬁcant results, experiments with more and larger ontologies are planned.

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Acknowledgements This work has been partially supported by the “InnoProﬁle-Corporate Semantic Web" project funded by the German Federal Ministry of Education and Research (BMBF) and the BMBF Innovation Initiative for the New German Länder - Entrepreneurial Regions.

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[2] [3] [4] [5] [6]

[7]

[8] [9] [10] [11]

[12] [13] [14]

[15] [16] [17]

Harith Alani and Christopher Brewster. Metrics for ranking ontologies. In Denny Vrandeˇci´c, Mari del Carmen Suárez-Figueroa, Aldo Gangemi, and York Sure, editors, Proceedings of the 4th International Workshop on Evaluation of Ontologies for the Web (EON2006) at the 15th International World Wide Web Conference (WWW 2006), pages 24–30, Edinburgh, Scotland, May 2006. Jacques Calmet and Anusch Daemi. From entropy to ontology. In Proc. of From Agent Theory to Agent Implementation (AT2AI-4), 2004. A. Clauset, M. E. J. Newman, and C. Moore. Finding community structure in very large networks. Physical Review E, 70:066111, 2004. Paul Doran, Valentina A. M. Tamma, Terry R. Payne, and Ignazio Palmisano. An entropy inspired measure for evaluating ontology modularization. In K-CAP, pages 73–80, 2009. Bernardo Cuenca Grau, Ian Horrocks, Yevgeny Kazakov, and Ulrike Sattler. Modular reuse of ontologies: Theory and practice. J. of Artiﬁcial Intelligence Research (JAIR), 31:273–318, 2008. Alvaro Graves, Sibel Adali, and Jim Hendler. A method to rank nodes in an rdf graph. In Christian Bizer and Anupam Joshi, editors, International Semantic Web Conference (Posters & Demos), volume 401 of CEUR Workshop Proceedings. CEUR-WS.org, 2008. Bettina Hoser, Andreas Hotho, Robert Jäschke, Christoph Schmitz, and Gerd Stumme. Semantic network analysis of ontologies. In Proceedings of the 3rd European Semantic Web Conference, volume 4011 of LNCS, pages 514–529, Budva, Montenegro, June 2006. Springer. E. A. Leicht and M. E. J. Newman. Community structure in directed networks. Phys. Rev. Lett., 100(11):118703, 2008. M. E. J. Newman. Finding community structure in networks using the eigenvectors of matrices. Physical Review E, 74(3):036104, 2006. M. E. J. Newman and M. Girvan. Finding and evaluating community structure in networks. Physical Review E, 69:026113, 2004. S. Peroni, E. Motta, and M. d’Aquin. Identifying key concepts in an ontology through the integration of cognitive principles with statistical and topological measures. In Third Asian Semantic Web Conference, Bangkok, Thailand, 2008. Pascal Pons and Matthieu Latapy. Computing communities in large networks using random walks. J. of Graph Alg. and App. bf, 10:284–293, 2004. J. Reichardt and S. Bornholdt. Statistical mechanics of community detection. Arxiv preprint condmat/0603718, 2006. Anne Schlicht and Heiner Stuckenschmidt. A ﬂexible partitioning tool for large ontologies. Web Intelligence and Intelligent Agent Technology, IEEE/WIC/ACM International Conference on, 1:482–488, 2008. Heiner Struckenschmidt. Network analysis as a basis for partitioning class hierarchies. In Workshop on Semantic Network Analysis, ISWC, 2006. Yannis Theoharis, Yannis Tzitzikas, Dimitris Kotzinos, and Vassilis Christophides. On graph features of semantic web schemas. IEEE Transactions on Knowledge and Data Engineering, 20:692–702, 2007. Xiang Zhang, Gong Cheng, and Yuzhong Qu. Ontology summarization based on rdf sentence graph. In Carey L. Williamson, Mary Ellen Zurko, Peter F. Patel-Schneider, and Prashant J. Shenoy, editors, WWW, pages 707–716. ACM, 2007.

Modular Ontologies O. Kutz and T. Schneider (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-799-4-25

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The Modular Structure of an Ontology: Atomic Decomposition and Module Count Chiara DEL VESCOVO a Bijan PARSIA a Uli SATTLER a and Thomas SCHNEIDER b a University

of Manchester, UK {delvescc,bparsia,sattler}@cs.man.ac.uk Bremen, Germany [email protected]

b Universität

Abstract Extracting a subset of a given ontology that captures all the ontology’s knowledge about a speciﬁed set of terms is a well-understood task. This task can be based, for instance, on locality-based modules. However, a single module does not allow us to understand neither topicality, connectedness, structure, or superﬂuous parts of an ontology, nor agreement between actual and intended modeling. The strong logical properties of locality-based modules suggest that the family of all such modules of an ontology can support comprehension of the ontology as a whole. However, extracting that family is not feasible, since the number of locality-based modules of an ontology can be exponential w.r.t. its size. In this paper we report on a new approach that enables us to efﬁciently extract a polynomial representation of the family of all locality-based modules of an ontology. We also describe the fundamental algorithm to pursue this task, and report on experiments carried out and results obtained. Keywords. locality-based modules, decomposition, ontology comprehension

1. Introduction Why modularize an ontology? Modern ontologies can get quite large as well as complex, which poses challenges to tools and users when it comes to processing, editing, analyzing them, or reusing their parts. This suggests that exploiting modularity of ontologies might be fruitful, and research into this topic has been an active area for ontology engineering. Much recent effort has gone into developing logically sensible modules, that is, parts of an ontology which offer strong logical guarantees for intuitive modular properties. One such guarantee is called coverage. It means that a module captures all the ontology’s knowledge about a given set of terms (signature)—a kind of dependency isolation. A module in this sense is a subset of an ontology’s axioms that provides coverage for a signature, and each possible signature determines such a module. Coverage is provided by modules based on conservative extensions, but also by efﬁciently computable approximations, such as modules based on syntactic locality [5]. We call the task of extracting one module given a signature GetOne; it is well understood and starting to be deployed in standard ontology development environments, such

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as Protégé 4,1 and online.2 Locality-based modules have already been effectively used for ontology reuse [14] and as a subservice for incremental reasoning [4]. Despite its usefulness, the service GetOne does not provide information about the ontology as a whole. It cannot help us to exploit an ontology as a one-piece of software, and understand its topicality, connectedness, structure, superﬂuous parts, or agreement between actual and intended modeling. To gain that understanding, we aim at revealing an ontology’s modular structure, a task that we call GetStruct. That structure is determined by the set of all modules and their inter-relations, or at least a suitable subset thereof. From a naïve point of view, a necessary step to achieve GetStruct is GetAll, the task of extracting all modules. This is the case as long as we have not speciﬁed what a “suitable subset of all modules” is, or do not know how to obtain such a subset efﬁciently. It might well be that GetAll is feasible and yields a small enough structure, in which case it would solve GetStruct. While GetOne is well-understood and often computationally cheap, GetAll has hardly been examined for module notions with strong logical guarantees, with the works described in [7, 8] being promising exceptions. GetOne also requires the user to know in advance the right signature to input to the extractor: we call this a seed signature for the module and note that each module can have several such seed signatures. Since there are non-obvious relations between the ﬁnal signature of a module and its seed signature, users are often unsure how to generate a proper request and confused by the results. If they had access to the overall modular structure of the ontology determined by GetStruct, they could use it to guide their extraction choices. While GetAll seems to be a necessary step to perform GetStruct, we note that in the worst case, the number of all modules of an ontology is exponential in the number of terms or axioms in the ontology, in fact in the minimum of these numbers. In [20], we have shown cases of (artiﬁcial) ontologies with exponentially many modules w.r.t. their sizes, and obtained empirical results conﬁrming that in general ontologies have too many modules to extract all of them, even with an optimized extraction methodology. Then, some other form of analysis would have to be designed. In this paper, we report on new insights regarding the modular structure of ontologies which leads to a new, polynomial algorithm for GetStruct (provided that module extraction is polynomial) that generates a linear (in the size of the ontology), partially ordered set of modules and atoms which succinctly represent all (potentially exponentially many) modules of an ontology. We use this decomposition to give an estimate of the number of modules of an ontology, and compare these numbers with the real number of modules (when possible), obtained following the same approach as in [20]. For full proofs and more details, the reader is referred to [9]. Related work. One solution to GetStruct is described in [7, 6] via partitions related to E-connections. When this technique succeeds, it divides an ontology into three kinds of disjoint modules: (A) those which import vocabulary from others, (B) those whose vocabulary is imported, and (C) isolated parts. In experiments and user experience, the numbers of parts extracted were quite low and often corresponded usefully to user understanding. For instance, the tutorial ontology Koala, consisting of 42 logical axioms, is partitioned into one A-module about animals and three B-modules about genders, de1 http://www.co-ode.org/downloads/protege-x 2 http://owl.cs.manchester.ac.uk/modularity

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grees and habitats. It has also been shown in [7] that certain combinations of these parts provide coverage. Partitions can be computed efﬁciently. Ontology partitions based on E-connections require rather strong conditions to ensure modular separation. However, it has been observed that ontologies with fairly elaborate modular structure have impoverished E-connections based structures. For the ontology Periodic,3 for example, such a combination is still the whole ontology, even though Periodic seems well structured. Furthermore, the robustness properties of the parts (e.g., under vocabulary extension [17]) are not as well-understood as those of locality-based modules. Finally, there is only a preliminary implementation of the partition algorithm. Among the other approaches to GetStruct we ﬁnd the tool ModOnto [2], which aims at providing support for working with ontology modules, that borrows intuitions from software modules. This approach is logic-based and a-posteriori but, to the best of our knowledge, it has not been examined whether such modules provide coverage. Another procedure to partition an ontology is described in [22]. However, this method only takes the concept hierarchy into account, therefore it cannot guarantee to provide coverage. In [15], it was shown how to decompose the signature of an ontology to obtain the dependencies between its terms. In contrast to the previous ones, this approach is syntax-independent. While gaining information about term dependencies is one goal of our approach, we are also interested in the modules of the ontology. Among the a-posteriori approaches to GetOne, only some provide logical guarantees. Those are usually restricted to “small” DLs where deciding conservative extensions— which underly coverage—is tractable. Examples are the module extraction feature of CEL [25] and the system MEX [16]. However, we want to cover DLs up to OWL 2. There are several logic-based approaches to modularity that function a-priori, i.e., the modules of an ontology have to be speciﬁed in advance using features that are added to the underlying (description) logic and whose semantics is well-deﬁned. These approaches often support distributed reasoning; they include C-OWL [24], E-connections [19], Distributed Description Logics [3], and Package-Based Description Logics [1]. Even in these cases, however, we may want to understand the modular structure of the syntactically delineated parts (modules), because decisions about modular structure have to be taken early in the modeling which may enshrine misunderstandings. Currently there is no requirement that these modules provide coverage, so GetStruct can be useful to verify the imposed structure throughout the development process. Examples were reported in [7], where user attempts to capture the modular structure of their ontology by separating the axioms into separate ﬁles were totally at odds with the analyzed structure. 2. Preliminaries Underlying description logics. We assume the reader to be familiar with OWL and the underlying description logics (DLs) [12, 11]. We consider an ontology to be a ﬁnite set of axioms, which are of the form C D or C ≡ D, where C, D are (possibly complex) concepts, or R S, where R, S are (possibly inverse) roles. Since we are interested in the logical part of an ontology, we disregard non-logical axioms such as annotation and declaration axioms. However, it is easy to add those in retrospect once the logical part of a module has been extracted. This is included in the publicly available implementation of locality-based module extraction in the OWL API.4 3 http://www.cs.man.ac.uk/~stevensr/ontology/periodic.zip 4 http://owlapi.sourceforge.net

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Let NC be a set of concept names, and NR a set of role names. A signature Σ is a set of terms, i.e., Σ ⊆ NC ∪ NR . We can think of a signature as specifying a topic of interest. Axioms using only terms from Σ are “on-topic”. For instance, if Σ = {Animal, Duck, Grass, eats}, then Duck ∃eats.Grass is on-topic, while Duck Bird is (α ). off-topic. Given an ontology O (axiom α), its signature is denoted with O Conservative extensions and locality. Conservative extensions (CEs) capture the above described encapsulation of knowledge. They are deﬁned in [5] as follows. Deﬁnition 2.1. Let L be a DL, M ⊆ O be L-ontologies, and Σ be a signature. 1. O is a deductive Σ-conservative extension (Σ-dCE) of M w.r.t. L if for all axioms ⊆ Σ, it holds that M |= α if and only if O |= α. α over L with α 2. M is a dCE-based module for Σ of O if O is a Σ-dCE of M w.r.t. L. Unfortunately, CEs are hard or even impossible to decide for many DLs, see [10, 17]. Therefore, approximations have been devised. We focus on syntactic locality [21] (here for short: locality). Locality-based modules can be efﬁciently computed and provide coverage, that is, they capture all the relevant entailments, but not necessarily only those [5, 13]. Although locality is deﬁned for the DL SHIQ, an extension to SHOIQ(D) is straightforward [5, 13] and has been implemented in the OWL API. For the sake of completeness, we deﬁne locality and locality-based modules below. However, the atomic decomposition introduced later does not rely on locality because it will work for almost every notion of a “module for a signature”. Deﬁnition 2.2. An axiom α is called syntactically ⊥-local ( -local) w.r.t. signature Σ if it is of the form C⊥ C, C C , R⊥ R (R R ), or Trans(R⊥ ) (Trans(R )), where C / Σ (R ∈ / Σ), and C⊥ and C

is an arbitrary concept, R is an arbitrary role name, R⊥ ∈ are from Bot(Σ) and Top(Σ) as deﬁned in Table (a) (Table (b)) below. (a) ⊥-Locality

∈ Top(Σ), n¯ ∈ N \ {0} Let A⊥ , R⊥ ∈ / Σ, C⊥ ∈ Bot(Σ), C(i)

Bot(Σ) ::= A⊥ | ⊥ | ¬C | C C⊥ | C⊥ C | ∃R.C⊥ | n¯ R.C⊥ | ∃R⊥ .C | n¯ R⊥ .C Top(Σ) ::= | ¬C⊥ | C1 C2 | 0 R.C (b) -Locality

∈ Top(Σ), n¯ ∈ N \ {0} Let A , R ∈ / Σ, C⊥ ∈ Bot(Σ), C(i)

Bot(Σ) ::= ⊥ | ¬C | C C⊥ | C⊥ C | ∃R.C⊥ | n¯ R.C⊥ Top(Σ) ::= A | | ¬C⊥ | C1 C2 | ∃R .C | n¯ R .C | 0 R.C

It has been shown in [5] that M ⊆ O and all axioms in O \ M being ⊥-local (or all is sufﬁcient for O to be a Σ-dCE of M. The converse axioms being -local) w.r.t. Σ ∪ M does not hold: e.g., the axiom A ≡ B is neither ⊥- nor -local w.r.t. {A}, but the ontology {A ≡ B} is an {A}-dCE of the empty ontology. A locality-based module is computed as follows [5]: given an ontology O, a seed and an empty set M, each axiom α ∈ O is tested whether it is local signature Σ ⊆ O with respect to Σ; if not, α is added to M, the signature Σ is extended with all terms in , and the test is re-run against the extended signature until M is stable. M is denoted α as -mod(Σ, O) or ⊥-mod(Σ, O), respectively. Sometimes the resulting modules are quite large; for example, given the ontology O = {Ci D | 1 ≤ i ≤ n}, the module -mod({D}, O) contains the whole ontology. In or-

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der to make modules smaller, we will nest alternatively ⊥- and -module extraction. The resulting sets are again dCE-based modules, denoted ⊥ -mod(Σ, O) or ⊥-mod(Σ, O), depending on the type of the ﬁrst extraction [21]. We can keep nesting the extraction until a ﬁxpoint is reached. The number of steps needed to reach it can be at most as big as the number of axioms in O [21]. The ﬁxpoint, denoted as ⊥∗ -mod(Σ, O), does not depend on the type of the ﬁrst extraction [9]. In contrast, - and ⊥-modules do not have to be equal—in fact, the former are usually larger than the latter. Through the nesting, ⊥∗ -mod(Σ, O) is always contained in -mod(Σ, O) and ⊥-mod(Σ, O). From now on, we will denote by x-mod(Σ, O) the x-module M extracted from an ontology O by using the notion of x-locality w.r.t. Σ, where x ∈ { , ⊥, ⊥ , ⊥, . . . , ⊥∗ }, including any alternate nesting of these symbols. Finally, we want to point out that, for nor M ⊆ Σ needs to hold. M = x-mod(Σ, O), neither Σ ⊆ M Properties of locality-based modules. We list in this paragraph the properties of locality-based modules of interest for this paper. Proofs can be found in the papers cited. Proposition 2.3. Let O be an ontology, Σ be a signature, x ∈ {⊥, , ⊥∗ }; let M = Then x-mod(Σ , O) = M. x-mod(Σ, O) and Σ be a signature with Σ ⊆ Σ ⊆ Σ ∪ M. (For x ∈ {⊥, }, see [5]; the transfer to nested modules is straightforward). Locality is anti-monotonic: a growing seed signature makes no more axioms local. Corollary 2.4. Let Σ1 and Σ2 be two sets of terms, and let x ∈ { , ⊥}. Then, Σ1 ⊆ Σ2 implies x-local(Σ2 ) ⊆ x-local(Σ1 ) (see [5]). Remark 2.5. Some obvious tautologies are always local axioms, for any choice of a seed signature Σ. Hence, they will not appear in locality-based modules. Anyway, they do not add any knowledge to an ontology O. Proposition 2.6. In general, the following are not modules (see [9]): the union, intersection or complement of modules. a signature. Deﬁnition 2.7. Let O be an ontology, M ⊆ O a module, and Σ ⊆ O M is called self-contained if O is a (Σ ∪ M)-dCE of M. M is called depleting if O \ M is a (Σ ∪ M)-dCE of the empty ontology. Proposition 2.8. If S is an inseparability relation that is robust under replacement, then every depleting SΣ -module is a self-contained SΣ -module (see [18]). Theorem 2.9. Let S be a monotonic inseparability relation that is robust under replacement, T a TBox, and Σ a signature. Then there is a unique minimal depleting SΣ -module of T (see [18]). Remark 2.10. From now on, we use the notion of ⊥∗ -locality from [21]. However, the results we obtain can be generalized to every notion of module that guarantees the existence of a unique and depleting module for each signature Σ. In particular, the same conditions guarantee also that such notions of modules satisfy self-containedness.

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Fields of sets and atoms. We want to describe the relationships between an ontology O and a family F(O) of subsets thereof by means of a well-understood structure. To this end, we introduce in what follows some notions of algebra. Deﬁnition 2.11. A ﬁeld of sets is a pair (O, F), where O is a set and F is an algebra over O i.e., set of subsets of O that is closed under intersection, union and complement. Elements of O are called points, while those of F are called complexes. Given a ﬁnite set O and a family F of subsets of O, we can build the set B(O, F) = (O, F ), where F is the closure of F under union, intersection and complement. Then B(O, F) is clearly a ﬁeld of sets, as well as a partial order w.r.t. the inclusion relation “⊆”, because ⊆ is reﬂexive, transitive and antisymmetric. We focus on the minimal elements of B(O, F), i.e., elements a ∈ B(O, F) such that there exists no non-empty element b of B(O, F) \ {a} with b ⊂ a. Deﬁnition 2.12. The minimal elements of B(O, F) \ {0} / with respect to “⊆” are called atoms.5 The principal ideal of an element a ∈ B(O, F) is the set (a] := {x ∈ B(O, F) | x ⊆ a}. 3. The Atomic Decomposition Modules and atoms. In what follows, we are using the notion of ⊥∗ -locality from [21]. However, the approach we present can be applied to any notion of a module that is monotonic, self-contained, and depleting. These properties have a deep impact on the modules generated, as described in Proposition 3.1. See [18] for more details. Proposition 3.1. Any module notion that satisﬁes monotonicity, self-containedness, and depletingness is such that any given signature generates a unique module. We are going to deﬁne a correspondence among ontologies with relative families of modules and ﬁelds of sets as deﬁned in Deﬁnition 2.11. Axioms correspond to points. Let then F(O) denote the family of ⊥∗ -modules of O (or let Fx (O) be such family for each corresponding notion x of module if not univocally speciﬁed). Then F(O) is not, in general, closed under union, intersection and complement: given two modules, neither their union nor their intersection nor the complement of a module is, in general, a module; hence, only some complexes correspond to modules. Next, we introduce the (induced) ﬁeld of modules, that is the ﬁeld of sets over F(O). This enables us to use properties of ﬁelds of sets also for ontologies. Deﬁnition 3.2. Given an ontology O and the family F(O) of ⊥∗ -modules of O, we deﬁne the (induced) ﬁeld of modules B(F(O)) as the closure of the set F(O) under union, intersection and complement. Deﬁnition 3.3. A syntactic tautology is an axiom that does not occur in any module and O). A global axiom is an axiom that occurs in each hence belongs to O \ ⊥∗ -mod(O, ∗ / O). module, in particular in ⊥ -mod(0, 5 Slightly

abusing notation, we use B(O, F) here for the set of complexes in B(O, F).

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Remark 3.4. To make the presentation easier, we assume that O contains no syntactic tautologies or global axioms. This is no real restriction: we can always remove those unwanted axioms that occur in either all or no module, and consider them separately. An (induced) ﬁeld of modules is, by construction, a ﬁeld of sets. It is partially ordered by ⊆ and, due to the ﬁniteness of O, can thus be represented via its Hasse diagram. Next, we deﬁne atoms of our ﬁeld of modules as building blocks of modules of an ontology; recall that these are the ⊆-minimal complexes of B(F(O)) \ {0}. / Deﬁnition 3.5. The family of atoms from B(F(O)) is denoted by A(F(O)) and is called atomic decomposition. An atom is a set of axioms such that, for any module, it either contains all axioms in the atom or none of them. Moreover, every module is the union of atoms. Next, we show how atoms can provide a succinct representation of the family of modules. Before proceeding further, we summarize in the following table the four structures introduced so far and, for each, its elements, source, maximal size, and mathematical structure. Structure Elements Source Maximal size Mathem. object

O

F(O)

B(F(O))

A(F(O))

axioms α ontology engineers baseline set

modules M module extractor exponential family of sets

complexes closure of F(O) exponential complete lattice

atoms a, b, . . . atoms of B(F(O)) linear poset

Atoms and their structure. The family A(F(O)) of atoms of an ontology, as in Deﬁnition 3.5, has many properties of interest for us. Lemma 3.6. The family A(F(O)) of atoms of an ontology O is a partition of O, and thus #A(F(O)) ≤ #O. Hence the atomic decomposition is succinct; we will see next whether its computation is tractable and whether it is indeed a representation of F(O). The following deﬁnition aims at deﬁning a notion of “logical dependence” between axioms: the idea is that an axiom α depends on another axiom β if, whenever α occurs in a module M then β also belongs to M. A slight extension of this argument allows us to generalize this idea because, by deﬁnition of atoms, whenever α occurs in a module, all axioms belonging to α’s atom a occur. Hence, we can formalize this idea by deﬁning a relation between atoms. Deﬁnition 3.7. (Relations between atoms) Let a = b be atoms of an ontology O. Then: – a is dependent on b (written a b ) if, for every module M ∈ F(O) such that a ⊆ M, we have b ⊆ M. – a and b are independent if there exist two disjoint modules M1 , M2 ∈ F(O) such that a ⊆ M1 and b ⊆ M2 . – a and b are weakly dependent if they are neither independent nor dependent; in this case, there exists an atom c ∈ A(F(O)) which both a and b are dependent on. We also deﬁne the relation “” to be the inverse of “”, i.e., b a ⇔ a b. Proposition 3.8. For every pair of distinct atoms exactly one of the relations in Deﬁnition 3.7 applies.

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The logical dependence between atoms can, in general, be incomplete: for example, consider the following (hypothetical) family of modules: F(O) = {M1 , M2 , M3 , M4 } where M1 = a ∪ b, M2 = a ∪ c, M3 = a ∪ b ∪ d and M4 = a ∪ c ∪ d. Following Deﬁnition 3.7, the atoms b, c and d depend on a. However, we want our structure to reﬂect that b and c act as “intermediates” in the dependency of d on a, i.e., that d depends via “c or b” on a. Since in Def. 3.7 we do not capture disjunctions of occurrences of atoms, we call the pairs (d, b) and (d, c) problematic. Fortunately, problematic atom pairs do not exist in an atomic decomposition obtained via locality-based modules, as Lemma 3.9 shows. Its consequences on the dependency relation on atoms are captured by Proposition 3.12. Lemma 3.9. Since the ⊥∗ notion of module is monotonic, self-contained, and depleting, there are no problematic pairs in the set A(F(O)) of atoms over O. The key to proving Lemma 3.9 is the following result. , O) is the smallest containing α. Proposition 3.10. The module ⊥∗ -mod(α Proof. We recall ⊥∗ -mod satisﬁes the properties as in Prop. 2.3. Then: (i) Mα is not empty since it contains α (recall that O does not contain syntactic tautologies) (ii) Mα is the unique and thus smallest module for the seed signature α results in a superset of Mα (iii) by monotonicity, enlarging the seed signature α , O) = ⊥∗ -mod(M ∪ α , O) ⊇ ⊥∗ -mod(α , O) by self(iv) M = ⊥∗ -mod(M containedness and monotonicity, thus any module M that contains α needs to contain also Mα . Corollary 3.11. Given an atom a, for every axiom α ∈ a we have that Mα =

⊥∗ -mod( a, O). Moreover, a is dependent on all atoms belonging to Mα \ a. Proposition 3.12. The binary relations “ ” and “ ” are partial orders over the set A(F(O)) of atoms of an ontology O. Deﬁnition 3.7 and Proposition 3.12 allow us to draw a Hasse diagram also for the atomic decomposition A(F(O)), where independent atoms belong to different chains, see Figure 1 for the Hasse diagram of Koala. The edges in this diagram denote dependency: an edge from node a to b means that b a, i.e., atom a depends on b. Some atoms depend on more than one atom. Their nodes have more than one outgoing edge. Atoms as a module base. As an immediate consequence of our observations so far, a module is a disjoint ﬁnite union of atoms. Conversely, it is not true that arbitrary unions of atoms are modules. However, the atomic decomposition satisﬁes another interesting property: from each atom, it is straightforward to identify the smallest module containing it. Deﬁnition 3.13. The principal ideal of an atom a is the set (a] = {α ∈ b | b a} ⊆ O. Proposition 3.14. For every atom a, (a] is a module. Deﬁnition 3.15. A module is called compact if there exists an atom a in A(F(O)) such that M = (a].

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Given the (possibly exponential w.r.t. the ontology size) family F(O) of all modules of an ontology O, there is a well-deﬁned injection that maps every module M to the set its image of atoms in ℘(A(F(O))) whose union is M: given the module signature M, is the set of all atoms that are relevant w.r.t. M’s terminology, deﬁned in the following. Hence, A(F(O)) is indeed a succinct representation of all modules. Deﬁnition 3.16. We say that an atom a is relevant w.r.t. its terminology for a module M if its signature a is contained in the module’s signature M. The well-deﬁnedness of Def. 3.16 follows from the properties of depletingness and self-containedness that locality-based modules satisfy. We can however restrict our attention to just some relevant (w.r.t. its terminology) atoms to identify our module within the atomic decomposition. Deﬁnition 3.17. Let (P, ≥) be a poset, and (P, ≤) its dual. Then, an antichain is a set of pairwise incomparable elements A ⊆ P, i.e. such that for each a, b ∈ A, neither a ≥ b nor b ≥ a (dually, neither a ≤ b nor b ≤ a). Proposition 3.18. Let M ⊆ O be a module. Then, there exists an antichain of atoms a1 , . . . , aκ such that M = κi=1 (ai ]. In particular, the set of compact modules is a base for the set F(O) of all modules. 4. Computing the atomic decomposition As we have seen, the atomic decomposition is a succinct representation of all modules of an ontology: its linearly many atoms represent all its worst case exponentially many modules. Next, we will show how we can compute the atomic decomposition in polynomial time, i.e., without computing all modules, provided that module extraction is polynomial (which is the case, e.g., for syntactic locality-based modules). Our approach relies on modules “generated” by a single axioms, which can be used to generate all others. Deﬁnition 4.1. Given an ontology O and decomposition A(F(O)), we call module M: , O). 1) α-module if there is an axiom α ∈ O such that M = ⊥∗ -mod(α 2) fake if there exist two incomparable (w.r.t. set inclusion) modules M1 = M2 with M1 ∪ M2 = M; a module is called genuine if it is not fake. Please note that our notion of genuinity is different from the one in [20], where the incomparable “building” modules were also required to be disjoint. The following lemma provides the basis for the computation in polynomial time of the atomic decomposition since it allows us to construct A(F(O)) via α-modules only. Lemma 4.2. The notions of compact (as in Def. 3.15), α and genuine modules coincide. Algorithm 1 gives our procedure for computing atomic decompositions that runs in time polynomial in the size of O (provided that module extraction is polynomial), and calls a module extractor as many times as there are axioms in O. It considers, in ToDoAx, all axioms that are neither tautologies nor global, see Remark 3.4, and computes all genuine modules, all atoms with their dependency relation and the cardinalities of all modules and atoms. For each axiom α “generating” a module, that module is stored in Mod(α)

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Algorithm 1 Atomic decomposition Input: An ontology O. Output: The set G of genuine ⊥∗ -modules; the poset of atoms (A(F(O)), ); the set of generating axioms GenAx; for α ∈ GenAx, the cardinality CardAt(α) of its atom. O) \ ⊥∗ -mod(0, / O) ToDoAx ← ⊥∗ -mod(O, GenAx ← 0/ for each α ∈ ToDoAx do , O) {= 0} / Mod(α) ← ⊥∗ -mod(α new ← true for each β ∈ GenAx do if Mod(α) = Mod(β ) then At(β ) ← At(β ) ∪ {α} CardAt(β ) ← CardAt(β ) + 1 new ← f alse end if end for if new = true then At(α) ← {α} CardAt(α) ← 1 GenAx ← GenAx ∪ {α} end if end for for each α ∈ GenAx do for each β ∈ GenAx do if β ∈ Mod(α) then At(β ) At(α) end if end for end for A(F(O)) ← {At(α) | α ∈ GenAx} G ← {Mod(α) | α ∈ GenAx} return [(A(F(O)), ), G, GenAx, CardAt(·)]

Name

#logical axioms

DL

#Gen. #Con. #max. #max. mods comp. mod. atom

Koala 42 ALCON(D) 23 Mereology 44 SHIN 17 University 52 SOIN(D) 31 People 108 ALCHOIN 26 miniTambis 173 ALCN(D) 129 OWL-S 277 SHOIN(D) 114 Tambis 595 ALCN(D) 369 Galen 4, 528 ALEHF+ 3, 340

5 2

18 11

7 4

11

20

11

1

77

77

85

16

8

1

57

38

119

236

61

807

458

29

Table 1. Experiments summary

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12

23

19

2

14

18

11

7

21

13

15

22

8

3

20

9

16

5

Figure 1. The atomic decomposition of Koala

and the corresponding atom is constructed in At(α); those functions are undeﬁned for axioms outside GenAx. We prove the correctness of Algorithm 1 in [9]. 5. Empirical evaluation We ran Algorithm 1 on a selection of ontologies6 , including those used in [8, 20], and indeed managed to compute the atomic decomposition in all cases, even for ontologies where a complete modularization was previously impossible. Table 1 summarizes ontology data: size, expressivity, number of genuine modules, number of connected components, size of largest module and of largest atom. Our tests were obtained on a 2.16 GHz Intel Core 2 Duo Macbook with 2 GB of memory running Mac OS X 10.5.8; each atomic decomposition was computed within a couple of seconds, (resp. 3 minutes for Galen). 6 Ontologies

and their decompositions can be found at http://bit.ly/i4olY0 .

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We have also generated a graphical representation using GraphViz7 . Our atomic decompositions show atom size as node size, see e.g. Fig. 1. It shows four isolated atoms, e.g., Atom 22, consisting of the axiom DryEucalyptForest Forest. This means that, although other modules may use 22’s terms, they do not “need” 22’s axioms for any entailment. Hence, removing (the axioms in) isolated atoms from the ontology would not result in the loss of any entailments regarding other modules or terms. Of course, for entailments involving both DryEucalyptForest and Forest and possibly other terms, axioms in isolated atoms may be needed. A similar structure is observable in all ontologies considered, see the graphs at http://bit.ly/i4olY0 . 6. Labelling The atomic decomposition partitions the ontology into highly coherent fragments. However, we still need to understand their structure and access their content. To this aim, it can be useful to label an atom with the terms that we ﬁnd relevant. An obvious candidate is simply the signature of the corresponding genuine module. However, genuine modules, and hence their signatures, can be too numerous, as well as unstructured. Another candidate is suggested by Proposition 3.18: we could label an atom a with the set of all its minimal seed signatures for which a is relevant. As before, a genuine module can have in principle a large number of such signatures, even more numerous than the number of axioms it contains. So, we suggest here different candidates for a labelling and discuss them; but we leave applying them for future work. Deﬁnition 6.1. Given: an ontology O; the atomic decomposition of the ontology A(F(O)) = {a1 , a2 , . . . , an }; the set of genuine modules G = {Mi | Mi = (ai ], 1 ≤ i ≤ n}. We deﬁne the following labelling functions Lab j (.) from A(F(O)) to O: Lab1 (ai ) := ai Lab3 (ai ) := Σ∈mssig(Mi ) Σ Lab2 (ai ) := ai \ b≺ai Lab2 (b) Lab4 (ai ) := Σ∈mssig(Mi ) Σ \ b≺ai Lab4 ((b]) Lab1 is deﬁned to label each atom with the vocabulary used in its axioms. However, an atom a can be large and reuse terms already introduced in the atoms that a is dependent on. To better represent the “logical dependency” between terms, we recursively deﬁne Lab2 to label an atom only with the “new terms” introduced. We want to note that such label can be empty, as in the following example: let us consider the ontology O = {A B, C D, A C BD}. This ontology generates 3 atoms, one for each axiom, such that the atom a3 = A C B D is dependent on both the other 2, which are independent of each other. Clearly, Lab2 (a3 ) is empty, because (a3 ] reuses terms from the other atoms. Moreover, let us consider the axiom A B (C ¬C). Then, all the labelling deﬁned so far will include the term C in the label for the atom containing this axiom, even if this axiom does not say anything about it. This behaviour does not occur for labellings Lab3 and Lab4 , because C is not necessary in any of the minimal seed signatures for (a3 ]. Moreover, these labellings are also useful to discover “hidden relations” between an atom and terms that do not occur in it. For example, let us consider the ontology O = {A ≡ B, B C, B D C E, D E, E ≡ F}. Then, each axiom identiﬁes an atom, and O equals the principal ideal of the atom a3 containing the axiom B D C E. Although the signature of a3 contains neither A nor F, the set Σ = {A, F} is indeed a minimal seed signature of the genuine module (a3 ]. The need of this axiom for the signature Σ is not evident at ﬁrst sight. However, the set of all minimal seed signatures of a module M is in principle exponential in the size of M. 7 http://www.graphviz.org/About.php

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7. Module number estimation via atomic decomposition

Ratio 2a : #modules

In order to test the hypothesis that the number of modules does not grow exponentially with the size of the ontology, in [8] we tried to compute a full modularization for the ontologies of different size listed in Table 1 but managed to compute all modules for two ontologies only, namely Koala and Mereology. Then, we sampled subontologies of these ontologies, and extracted all of their modules. The results we obtained made us tend towards rejecting the hypothesis, but they were not strong enough for a clear rejection. One plausible application of the atomic decomposition is an estimate of the number of modules of an ontology: Proposition 3.18 implies that a module is the union of principal ideals of the atoms over an antichain. In general, the converse does not hold, but prima facie this seems to be a reasonable approximation, and can help us in understanding whether or not the number of modules is exponential w.r.t. the size of the ontology: as a matter of fact, if all antichains of an atomic decomposition generate distinct modules, then an efﬁcient way to ﬁnd a lower bound of the number of antichains of a poset is simply extracting the size a of the maximal antichain and compute 2a . Unfortunately, the measure 2a is not always a lower bound of the actual number of modules. For example, consider the ontology O = {Ai Ai+1 | i = 0, . . . , n − 1}, which consists of a single subsumption path p. The atomic decomposition of O consists of n independent atoms: ⊥∗ -mod({Ai , Ai+1 }, O) = {Ai Ai+1 }, for every i = 0, . . . , n − 1. Hence, the maximal antichain is of size n, and we would estimate that O has 2n modules. However, the modules of O are all subpaths of p: for seed signatures Σ of size < 2, ⊥∗ -mod(Σ, O) = 0; / for all other Σ, ⊥∗ -mod(Σ, O) is the smallest subpath of p containing all concepts in Σ. The actual module number is therefore only n(n−1) 2 . The explanation for the difference lies in the fact that atoms are not really independent, since they share parts of the minimal seed signatures of their induced modules. Based on the module numbers from that previous experiment, we have now performed an atomic decomposition of all the subontologies, computed the length a of the maximal antichain as well as the ratio between 2a and the number of modules for the respective ontology. If that ratio is greater (less) than 1, then the value 2a overestimates (underestimates) the module number. The picture below contains plots of the measured ratios against the subontology size for 3 ontologies. The y-axis is scaled logarithmically, ensuring that ratios r and 1/r have the same vertical distance from the value 1. 10,0

10,00

10,00

1,00

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0,01 70

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To interpret the plots for every ontology O and its collections of subsets, the following observations are of interest. How much does the maximal, minimal, or average ratio differ from 1? If it tends to differ much in one direction, the estimate needs to be scaled. If it differs erratically, then the estimate will not be useful.

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Does the maximal (minimal) ratio grow (shrink) when the size of O grows? If it does, the the growth (shrinkage) function needs to be qualiﬁed for the estimate to be useful. It is problematic to predict the function if it differs between ontologies. Are the differences to the “ideal” ratio 1 the same for the ratios >1 and those <1? If they are not and if such an imbalance only occurs for some ontologies, then we should ask the question what property of the ontology is responsible for it. The degree of imbalance could then serve as gauge for that property. How much do the maximal and the minimal ratio differ? Their quotient represents a margin for the estimate. E.g., if the maximal and minimal ratio are 3.0 and 0.5, then we can conclude from the measured value x = 2a that O has between 0.333x and 2x modules. The quotient is 6; therefore we can estimate the module number up to one order of magnitude. Quotients > 10 decrease precision to more orders of magnitude. We made the following observations for the ontologies we examined. Koala.

The ratio ranges from 0.36 to 2.61. For example, if we measure a maximal antichain of length 10 for any subontology of Koala, then we can estimate that the mod210 210 ≈ 392 and 0.36 ≈ 2, 844. The plot shows an even balule number is between 2.61 ance between “> 1” and “< 1” ratios. The minimal ratio seems to be constant with growing subontology size, but the maximal ratio seems to grow slightly. The quotient between max and min is 7.25. Mereology. The observations are similar, with a slight imbalance towards ratios < 1. The min and max ratio are 0.40 and 1.42, yielding a quotient of only 3.55. People. The ratio is almost always < 1; it ranges from 0.09 to 1.14. This yields a quotient of 12.67, i.e., the prediction of the module number is only up to two orders of magnitude. For example, for a maximal antichain of length 10, the number of modules can now be between 898 and 11,378. Furthermore, the underestimation appears to grow with the ontology size. University. The ratio is evenly distributed and ranges from 0.25 to 5.35. The quotient of 21.4 is even larger than for People. Galen. There is almost always a ratio < 1, and the underestimation appears to grow with the subontology size. For the ﬁrst 28 subontologies of very small size (up to 26 out of Galen’s 4,528 axioms), we already obtain a quotient of 1.14/0.04 = 28.5. In summary, the ratio behaves quite differently for these ﬁve ontologies, and this restricts its use as an estimate of the module number. For some ontologies, the measured value 2a tends to underestimate the module numbers, for others, there is no tendency. For some ontologies, the margin for the estimate obtained from 2a is simply too large. 8. Conclusion and outlook We have presented the atomic decomposition of an ontology, and shown how it is a succinct, tractable representation of the modular structure of an ontology: it is of polynomial size and can be computed in polynomial time in the size of the ontology (provided module extraction is polynomial), whereas the number of modules of an ontology is exponential in the worst case and prohibitely large in cases so far investigated. Moreover, it can be used to assemble all other modules without touching the whole ontology and without invoking a direct module extractor.

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Future work is three-fold: ﬁrst, we will try to compute, from the atomic decomposition, more precise upper and lower bounds for the number of all modules to answer an open question from [20]. Second, we will continue to investigate suitable labels for atoms, e.g., suitable representation of seed and module signatures, and how to employ the atomic decomposition for ontology engineering, e.g., to compare the modular structure with engineers’ intuitive understanding of the domain and thus detect modelling errors, and to identify suitable modules for reuse. Third, we will investigate when module extraction using the atomic decomposition is faster than using a module extractor. References [1] J. Bao, G. Voutsadakis, G. Slutzki, and V. Honavar. Package-based description logics. In [23], pp. 349–371. [2] C. Bezerra, F. Freitas, A. Zimmermann, and J. Euzenat. ModOnto: A tool for modularizing ontologies. In Proc. WONTO-08, vol. 427 of ceur-ws.org, 2008. [3] A. Borgida and L. Seraﬁni. Distributed description logics: Assimilating information from peer sources. J. Data Semantics, 1:153–184, 2003. [4] B. Cuenca Grau, C. Halaschek-Wiener, and Y. Kazakov. History matters: Incremental ontology reasoning using modules. In Proc. ISWC-07, vol. 4825 of LNCS, pp. 183–196, 2007. [5] B. Cuenca Grau, I. Horrocks, Y. Kazakov, and U. Sattler. Modular reuse of ontologies: Theory and practice. J. of Artif. Intell. Research, 31:273–318, 2008. [6] B. Cuenca Grau, B. Parsia, and E. Sirin. Combining OWL ontologies using E-connections. J. of Web Sem., 4(1):40–59, 2006. [7] B. Cuenca Grau, B. Parsia, E. Sirin, and A. Kalyanpur. Modularity and web ontologies. In Proc. of KR-06, pp. 198–209. AAAI Press, 2006. [8] C. Del Vescovo, B. Parsia, U. Sattler, and T. Schneider. The modular structure of an ontology: an empirical study. In Proc. of WoMO-10, vol. 211 of FAIA, pp. 11–24. IOS Press, 2010. [9] C. Del Vescovo, B. Parsia, U. Sattler, and T. Schneider. The modular structure of an ontology: atomic decomposition. Technical report, University of Manchester, 2011. Available at http://bit.ly/i4olY0. [10] S. Ghilardi, C. Lutz, and F. Wolter. Did I damage my ontology? A case for conservative extensions in description logics. In Proc. of KR-06, pp. 187–197, 2006. [11] I. Horrocks, O. Kutz, and U. Sattler. The even more irresistible SROIQ. In Proc. of KR-06, pp. 57–67, 2006. [12] I. Horrocks, P. F. Patel-Schneider, and F. van Harmelen. From SHIQ and RDF to OWL: The making of a web ontology language. J. of Web Sem., 1(1):7–26, 2003. [13] E. Jiménez-Ruiz, B. Cuenca Grau, U. Sattler, T. Schneider, and R. Berlanga Llavori. Safe and economic re-use of ontologies: A logic-based methodology and tool support. In Proc. of ESWC-08, vol. 5021 of LNCS, pp. 185–199, 2008. [14] A. Jimeno, E. Jiménez-Ruiz, R. Berlanga, and D. Rebholz-Schuhmann. Use of shared lexical resources for efﬁcient ontological engineering. In SWAT4LS-08, ceur-ws.org, 2008. [15] B. Konev, C. Lutz, D. Ponomaryov, and F. Wolter. Decomposing description logic ontologies. In Proc. of KR-10, pp. 236–246. AAAI Press, 2010. [16] B. Konev, C. Lutz, D. Walther, and F. Wolter. Logical difference and module extraction with CEX and MEX. In Proc. of DL 2008, vol. 353 of ceur-ws.org, 2008. [17] B. Konev, C. Lutz, D. Walther, and F. Wolter. Formal properties of modularization. In [23], pp. 25–66. [18] R. Kontchakov, L. Pulina, U. Sattler, T. Schneider, P. Selmer, F. Wolter, and M. Zakharyaschev. Minimal module extraction from DL-Lite ontologies using QBF solvers. In Proc. of IJCAI-09, pp. 836–841, 2009.

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[19] O. Kutz, C. Lutz, F. Wolter, and M. Zakharyaschev. E-connections of abstract description systems. Artiﬁcial Intelligence, 156(1):1–73, 2004. [20] B. Parsia and T. Schneider. The modular structure of an ontology: an empirical study. In Proc. of KR-10, pp. 584–586. AAAI Press, 2010. [21] U. Sattler, T. Schneider, and M. Zakharyaschev. Which kind of module should I extract? In DL 2009, vol. 477 of ceur-ws.org, 2009. [22] H. Stuckenschmidt and M. Klein. Structure-based partitioning of large concept hierarchies. In Proc. of ISWC-04, vol. 3298 of LNCS, pp. 289–303. Springer-Verlag, 2004. [23] H. Stuckenschmidt, C. Parent, and S. Spaccapietra, eds. Modular Ontologies: Concepts, Theories and Techniques for Knowledge Modularization, vol. 5445 of LNCS. Springer, 2009. [24] H. Stuckenschmidt, F. van Harmelen, P. Bouquet, F. Giunchiglia, and L. Seraﬁni. Using COWL for the alignment and merging of medical ontologies. In Proc. KR-MED, ceur-ws. org, pp. 88–101, 2004. [25] B. Suntisrivaraporn. Module extraction and incremental classiﬁcation: A pragmatic approach for EL+ ontologies. In Proc. of ESWC-08, vol. 5021 of LNCS, pp. 230–244, 2008.

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Modular Ontologies O. Kutz and T. Schneider (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-799-4-40

Modular Approach for a New Ontology Julia DMITRIEVA a , Fons J. VERBEEK b a [email protected] b [email protected] Abstract. In the life sciences researchers are working with large amount of data from different domains that frequently overlap. Overlapping information can be utilized at the moment the domains are integrated. A typical case is the drug discovery process in which the information from different domains, e.g. diseases, proteins, pathways, drugs, etc. need to be integrated in order to connect a disease with genes, pathways and ﬁnd a potential chemical compound that can be active as a drug. However, information from different domains is often available in different ontologies. In order to combine these data an ontology integration approach is required. In this paper we demonstrate an approach in which a new integrated ontology is created from modules that are extracted from different ontologies. Module extraction is based on well deﬁned notions of modularity, locality and conservative extensions. The signature of the modules is based on symbols of the user interest. Subsequently, the mappings between the similar concepts are generated. Finally, on the basis of these mappings we integrate modules in one ontology. Keywords. module extraction, ontology reuse, ontology integration, ontology mapping

Introduction In life sciences, ontologies, in particular available in the OBO F OUNDRY [16] and B IO P ORTAL [15] repositories contain information about species, proteins, chemicals, genomes, pathways, diseases, etc. Information in these ontologies might overlap, and it is possible that a certain concept is deﬁned in different ontologies from a different point of view and at different level of granularity. Therefore, the combination of information from different ontologies is useful for the creation of a new ontology. Case Study The integration will be illustrated with a case study on Toll-like receptors. As it is known, Toll-like receptors are important in immune response, they recognize molecules speciﬁc for pathogens and activate immune system. In order to create an ontology about Toll-like receptors it is important to use ontologies which can provide information about immune response, cell, cell membrane, proteins, biological pathway, biological process, diseases, etc. Therefore, we have chosen for ontologies in the biomedical domain provided by OBO F OUNDRY [16]. If we want to investigate what kind of information about Toll-like receptors is available in the M OLECULE ROLE O NTOLOGY (MoleculeRoleOntlogy) [16], then we will learn that Toll-like receptors are deﬁned as pattern recognition receptors (see Figure 1). In the B IOLOGICAL P ROCESS O NTOLOGY (part of GO) [16] the Toll-like receptors are

J. Dmitrieva and F.J. Verbeek / Modular Approach for a New Ontology

(1)

41

(2)

Figure 1. Visualization of (1) the concept TLR from M OLECULE ROLE O NTOLOGY and (2) the concept Toll-like receptor from P ROTEIN ONTOLOGY.

described in the context of a signaling pathway and are subsumed by the pattern recognition receptor signaling pathway concept. In P ROTEIN ontology [16], a Toll-like receptor is just a protein (see Figure 1). In NCI_T HESAURUS ontology [3], Toll-like receptors are deﬁned as Cell Surface Receptors. From these examples follows that multiple ontologies model different aspects of the same concept and the combination of the available information provides more knowledge about concepts that are of potential interest to ontology developer. A number of methods have been developed for ontology integration. Some of these methods are dedicated to reuse parts of ontology, such as modular approach [8] and MIREOT approach [5], other are devoted to connect concepts from different ontologies with each other, such as Distributed Description Logics (DDL) [4] and E−connections [12]. We are interested in the creation of a new ontology about a topic of user interest. Therefore, we introduce an approach for generating a new ontology from different ontologies obtained from the OBO F OUNDRY repository/library. In this method ﬁrst, we extract modules from these ontologies, on the basis of the well deﬁned modularity approach [8]. As a signature for the modules we are using the symbols that match the terms of interest. In our case study we create an ontology about Toll-like receptors, therefore we use two seed terms (Toll, TLR). Subsequently, we create mappings between concepts in the modules. It has already been shown [6] that the simple similarity algorithms outperform structural similarity algorithms in biomedical ontologies. Therefore, we have based our mappings on the similarity distance [13] between labels and synonyms of classes in the modules. Finally, a new ontology is created where the mappings are represented by means of the OWL : EQUIVALENT C LASS axiom and small concise modules are imported.

1. Related Work The current mechanism of integration OWL ontologies is based on the OWL: IMPORTS axiom. This mechanism, however, contains several drawbacks. First, ontologies can be very large, and including a foreign ontology then leads to importing all ontologies in the transitive closure. The price for the import can be very high, because processing,

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querying and reasoning in the ontology tend to be time consuming operations. Second, developers are not necessarily experts in the domain of the ontology to be included, i.e. he or she can be only interested in a small subset of the domain. Furthermore, the import can damage the consistency of the including and included ontology, because the original concepts can be deﬁned with the foreign concepts, and the other way around, the concepts from foreign ontology can be redeﬁned in the including ontology [7]. Because of the disadvantages of the OWL: IMPORTS mechanism, there is a need for alternative strategies for ontology integration. To that end, there is ongoing research in the area of ontology modularity and integration. In [10], for example, the authors provide a mechanism and a tool support for reusing of ontology. They describe a method where a module of ontology can be extracted and reused in a safe and economic way. Based on the undecidability results for such problems as whether Q1 is an S module in Q [8], they have based a module extraction on the notion of locality [8]. An alternative direction for reuse of parts of ontology, however, without preserving the logical structure and inferences, was chosen by developers of OBI ontology [2] in their approach referred to as Minimum Information to Reference an External Ontology Term (MIREOT) [5]. MIREOT MIREOT is a set of guidelines which are used in order to create the Ontology of Biomedical Investigations (OBI [2]). These guidelines are based on importing of parts of foreign ontologies in the ontology developed. The imported part is simply the class of interest and its superclass with annotations (label, comment, deﬁnition). Although this method guarantees the minimal reuse, the logical inferences about the reused class, in the ﬁrst place, are not complete, because no axioms about this class are imported. In the second place, reuse of this class can lead to unintended inferences. We are, however, interested in a methodology that does not just reuse classes of interest, but also guarantees that logical inferences about these classes are conserved. Along the same lines of ontology reuse and integration the related theoretical formalisms, e.g. E-connections [12] and Distributed Description Logics [4], can be considered. We will, therefore, shortly discuss these formalisms. Distributed Description Logics In Distributed Description Logics (DDL) different knowledge systems are combined by means of a new set of axioms, so called bridge rules [9]. The DDL formalism with bridge rules is based on the idea of distributed and independent ontologies, which can be linked together. Next to the mechanism for ontology interconnection, there is also reasoning support available that provides the possibility to reason in distributed ontologies [18]. E−connections A formalism related to DDL is the E−connections. This technique can be used to combine different knowledge bases expressed in different languages. This technique provides a possibility to connect different ontologies by means of Link Properties, which are deﬁned as properties between classes of source ontology and classes of some foreign ontology. If, for example, the source ontology is from the domain of chemical substances, and a foreign ontology is from the domain of drugs with the classes chem:acetyl_salicilyc_acid and drug:aspirin respectively, then we can deﬁne a LinkProperty Is_U sed_In_Drug connecting two given classes. In our work we are not using DDL and E−connections. First, because these formalisms are currently in development and not standardized yet. Second, both these for-

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43

malisms require the detailed knowledge of the ontologies reused. The developer needs to be expert in all domains described in reused ontologies in order to interlink them whether with bridge rules or with Link Properties. Our methodology is, however, dedicated to the user which is not a specialist in biomedical domain. Finally, DDL and E−connections methods are developed for interlinking ontologies or for connecting autonomous ontological modules obtained from large ontologies. We are, however, interested in a method with which we can create a new ontology on the basis of terms of user interest and from ontologies which are available (on the I NTERNET).

2. Module Extraction There are two main directions in the area of module extraction. One of them is based on structural approaches [17,14]. A typical example of the structural approach is provided in [14] where the authors describe a method which can be used to extract self-contained parts (traversal views) from ontologies. In their research a traversal views is based on the set of concepts and relations of the user’s choice and the ontology graph is generated on basis of the hierarchy and chosen relations. As pointed out in [10] the structural approaches are not providing a guarantee that such module is complete and logically correct. Moreover, from studies [8] it follows that structural-based approaches generate signiﬁcantly larger modules than locality-based approaches. Therefore, we have based our module extraction on the well deﬁned notions of locality, conservative extensions and modularity as described in [8,10]. In [8] the concept of module is introduced where a module captures the meaning of a given set of terms; in addition the algorithm for module computing is presented. In this paper we use the deﬁnition of the signature and module as given in [8]. The signature is deﬁned as follows: Deﬁnition 1 (Signature). A signature Sig of a DL is the disjoint union of sets C of atomic concepts (A, B, . . .) representing sets of elements, R of atomic roles (r, s, . . .) representing binary relations, and I of individuals (a, b, c, . . .) representing constants. In other words, the signature of an ontology is a vocabulary or a set of symbols used in the knowledge base. The module is deﬁned as follows: Deﬁnition 2 (Module). Let L be a description logic, Q1 ⊆ Q be two ontologies expressed in L and S be a signature. Q1 is an S–module in Q w.r.t. L, if for every ontology P and every axiom α expressed in L with Sig(P ∪ α) ∩ Sig(Q) ⊆ S, we have P ∪ Q |= α iff P ∪ Q1 |= α. It means that all inferences that could be realized about symbols over signature S after whole ontology Q is imported in ontology P are the inferences that could be done after importing module Q1 in P. Hence, the import Q in P will not add extra information about the symbols from signature S compared to importing Q1 in P. In [10] the authors describe a system that, in a safe and economic way, can extract relevant parts from ontologies that can be further used in the development of a new ontology. We elaborate on these methods for our module extraction, because they seem to allow for generation of small and logically correct modules.

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2.1. Modules from Enriched Signature For our case study we are interested in the creation of an ontology concerning Toll-Like receptors in an pseudo-automated way. Therefore, we have used the following biomedical ontologies most of which are obtained from the OBO F OUNDRY repository/library: • • • • • • • • •

National Cancer Institute Ontology (NCI_T HESAURUS) GO Ontology (GO) Protein Ontology (PRO) Dendritic Cell ontology (DENDRITIC _ CELL) Pathway ontology (PATHWAY) Molecule Role Ontology (M OLECULE ROLE O NTOLOGY) Gene Regulation Ontology (GENE _ REGULATION) Medical Subject Heading ontology (M E SH) Chemical Entities of Biological Interest (C H EBI)

The Medical Subject Heading Ontology (M E S H), currently not a member of OBO F OUNDRY, was fetched from the resource [1]. A module comprises knowledge of a part of the domain that is dedicated to a set of terms of interest (seed terms). Let T1 be this set. In our case study we have used two terms Toll and TLR. Let S1 be a set of terms (signature) from the ontology O1 that represents the classes whose labels, descriptions, ID, or other annotation properties contain the symbols from T1 . The ﬁrst module that we have extracted is the module from NCI_T HESAURUS M1 . This is chosen because it is the largest ontology and it is expected to contain most matches with the seed terms. In order to generate a signature for the next ontology O2 , we are using not only the terms from T1 but we enrich this set with the terms from the module M1 . Consequently, the set of terms for the generation of the second module M2 will be the collection of two sets T2 = Sig(M1 ) ∪ T1 . The same procedure is applied to the rest of the ontologies, namely module Mi is extracted on the basis of the terms Ti = Sig(Mi−1 ) ∪ Ti−1 . Hence, during the module extraction more and more symbols are collected that can be matched and used for the signature extraction from a following ontology. This method, however, has two disadvantages. First, it depends on the order of ontologies. The symbols that are matched in ontology Oi are based on the terms from previously generated modules ∪i−1 k=1 Sig(Mk ) ∪ T1 . So, if for example, ontology Ok+1 contains symbols that have match only with the symbols from the module extracted from ontology Ok then these symbols will be never discovered if Ok+1 will be processed before Ok during the module extraction process. This because the module from Ok+1 will be empty. Second, with the generation of the new module Mi new symbols can be introduced that will match symbols from ontologies used in previous steps. In order to overcome these drawbacks we will introduce the generation of the ﬁxpoint. This will be discussed in section 2.2. The reason that we have used the symbols from the modules M1 , M2 , . . . , Mi−1 created in previous steps and not only seed terms T1 is that not each ontology contains entities that have matches with T1 , thus, only few ontologies can be considered for module extraction. All ontologies used in this experiment, however, contain overlapping symbols, thus the symbols from module extracted in previous step can match the symbols from ontology used in the next step.

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Figure 2. Schematic representation of ﬁxpoint ﬁnding algorithm. First, the set of terms T1 = T OLL ∪ T LR is used to ﬁnd matches in NCI_T HESAURUS ontology. Subsequently, for the extraction of the module Mi+1 the set of terms Ti is enriched with the symbols from the module Mi extracted in the previous step. The ﬁxpoint is reached when no new symbols are introduced.

generate the set of modules M1 , M2 , . . . , Mn from ontologies O1 , O2 , . . . , On T = {”Toll”, ”TLR”};//the set of seed terms T _is_changed = true; while(T_is_changed){ T _is_changed = false; for(each ontology Oi ){ Sigi = ﬁnd_matched_entities(Oi , T ); Mi = extract_module(Sigi ); for (each term t from Mi ){ if(t is not in T ){ add t to T ; T _is_changed = true;} } } } Figure 3. Algorithm that ﬁnds ﬁxpoint modules for different ontologies.

2.2. Fixpoint Modules We have investigated whether or not we will ﬁnd a ﬁxpoint1 with our method for module extraction. The schematic representation of ﬁxpoint ﬁnding procedure is depicted in Figure 2. The ﬁxpoint is reached at the moment the set of terms Ti which is used in order to generate modules during step ti does not change any more after another run with all ontologies. This can be written as ∪nk=1 Sig(Mk,i ) = ∪nk=1 Sig(Mk,i+1 ), where Mk,i is the module k created during the step ti . It can be formulated in a ”ﬁxpoint-like” way M od(T ) = T . The algorithm given in Figure 3 is used to ﬁnd a ﬁxpoint. In Table 1 the sizes of the modules with which the ﬁxpoint was reached are depicted. The content of these modules will not change any more after the next run of algorithm. 1 In mathematical terms, a ﬁxpoint (ﬁxed point) is a point that is mapped to itself by the function (f (x)

= x).

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Table 1. The size of the modules created on the basis of seed terms (Toll, TLR) after reaching ﬁxpoint. size in KB

original O size in KB

Module_from_gene_regulation Module_from_protein Module_from_chebi

88.7 23.4 218.6

418.8 9900 17800

Module_from_mesh Module_from_dendritic_cell

59.2 4.2

6200 57.5

module

Module_from_pathway

4.1

360.5

Module_from_cellular_component Module_from_molecular_function

35.4 11.4

18400 18400

Module_from_MoleculeRoleOntology Module_from_biological_process Module_from_Thesaurus

46.9 221.1 802.1

4100 18400 154700

2.3. Properties of Fixpoint In order to show that algorithm 3 ﬁnds a ﬁxpoint we have to assert: • the algorithm terminates after reaching of the ﬁxpoint, • the ﬁxpoint is unique and does not depend on the order of processing of the ontologies. Theorem 1 (Fixpoint Algorithm Terminates). Fixpoint algorithm 3 terminates when a ﬁxpoint is reached. Proof. Each module extraction procedure can be represented as a set of rules. Without loss of generality, we will consider the simplest case with only 2 ontologies O1 and O2 . Let Σ be a signature, and let the following rule set R1 correspond with the generation of new symbols by the module extraction procedure from the ontology O1 : Σ→A B→C D→E

(1)

G→H Then, the signature Σ will generate the set of symbols A. In the same manner, the set of symbols B will generate C and the set of symbols D will generate the set E. Regarding the modularity approach [8], the following property holds M odΣ∪M (O) = M, hence module extraction procedure over the symbols of module M will generate the same module. Thus, we can assert that there is no rule corresponding with the symbols generated on the right hand side (RHS) (A, C, E, H) for rule set R1 . In other words, insertion of a symbol that was previously generated does not introduce new symbols. Let the following set of rules R2 correspond with the generation of new symbols by the module extraction procedure from the ontology O2 :

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Σ→F A→B C→D

(2)

K→L The ﬁxpoint algorithm 3 iteratively generates new symbols, which are used as input for the module extraction in each iteration of the while loop. This procedure terminates because both sets of rules are ﬁnite. It follows from the fact that the module extraction procedure (M odS (O) = M) at each step generates a subset of an ontology M ⊆ O that itself contains a ﬁnite set of axioms/symbols. Hence, only a ﬁnite set of symbols can be used in the rule set. When all rules have ﬁred no new symbols will be introduced, stop condition will be satisﬁed, and the algorithm terminates. In the worst case, when each symbol on the right hand side (RHS) of one rule set will have corresponding symbol on the left hand side of another rule set, the algorithm requires n + m steps, where n and m are the sizes of rule sets. It leads to extraction of whole ontology as a module. This situation is, however, not realistic because it is not possible that all symbols on RHS of Oi will match symbols on LHS of Oj . It is important to prove that the ﬁxpoint found by the algorithm 3 is unique and independent of the order of processing of the ontologies in the module extraction process. Therefore, we introduce the following deﬁnition: Deﬁnition 3 (Chain of Fire). A chain of ﬁre CFΣ (O) (in short CF ) from an ontology O over a signature Σ is the sequence of rules triggered by signature Σ started from ontology O. For example, for the rules 1 and 2 introduced in Theorem 1 the chain of ﬁre started from ontology O1 is determined as follows: CFΣ (O1 ) = Σ → A → B → C → D → E,

(3)

and started from ontology O2 is determined as follows: CFΣ (O2 ) = Σ → F.

(4)

It is easy to see that each chain of ﬁre is uniquely determined by the given set of rules and Σ. When the number of ontologies is more than two (n > 2) we will have n rule sets. In this case Σ triggers n rules of the type Σ → S. In Figure 4 different chains of ﬁre (CF s) for ontologies O1 , O2 , O3 , O4 are represented. These CF s are constructed on the basis of the following rule sets: O1 Σ→A D→G C→F J →K

O2 Σ→B F →I D→H

O3 Σ→C A→E G→J

O4 Σ→D B→F

(5)

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Figure 4. Chains of Fire for ontologies O1 , O2 , O3 , O4 corresponding with the rule sets given in 5.

This example shows that there can be more than one CF generated from one ontology, because one symbol can match LHS of more than one rule. In this case CF will split in different subchains, for example in Figure 4, after generation of D, two rules D → G and D → H will be triggered and CF will split it two subchains. It is also possible that different CF s at some point generate the same symbol and then proceed with ﬁring along the same chain. This case is represented in Figure 4 for the symbols B and C that both generate F . Theorem 2 (Independence of Ontology Order). The ﬁxpoint generated by algorithm 3 is uniquely determined by the rule set {R1 , R2 , . . . , Rn } and independent of the order of ontologies {O1 , . . . , On }. Proof. During the ﬁrst iteration, the algorithm 3 triggers all rules of the type Σ → S, where S is a set generated after module extraction procedure from ontology O over signature Σ. Obviously, each CF will ﬁre because Σ triggers each chain from each ontology. It is obvious that every rule in each CF will ﬁre. Suppose that it is not true, thus the ﬁxpoint was reached and there is a rule Si → Sj that was not triggered. This is possible only when Si was not generated. It can happen only when all rules in the path Σ → S1 → . . . → Si were also not processed, thus the given chain was not triggered at all. It contradicts the fact that Σ triggers each chain. Because CF is uniquely determined by the given rule set R and each CF will ﬁre, ﬁxpoint will be uniquely determined by the rule set. Moreover, because each rule in each chain will ﬁre, the order of rules is not important, thus the ﬁxpoint (or modules generated at ﬁxpoint) is independent from the order of ontologies.

3. Ontology Mapping In our approach we are using a less stringent deﬁnition of the concept mapping compared with the deﬁnition given in [11] in which mapping is a morphism and determined in the following way:

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Deﬁnition 4 (Total Mapping). A total ontology mapping from O1 = (S1 , A1 ) to O2 = (S2 , A2 ) is a morphism f : S1 → S2 of ontological signatures, such that, A2 |= f (A1 ), where Ai is a set of axioms in ontology. In this paper mapping is not considered as a morphism, but as a partial function that maps from subset S1 ⊆ Sig(O1 ) to subset S2 ⊆ Sig(O2 ). We deliberatively reject the morphism requirement, thus, the structural dependencies will not be preserved after mapping. This is because we are interested in consequents of this mapping to the original ontologies, namely, whether and how the structural dependences will be broken. For our experimental prototype system we use our own mappings. Experiments with available alignment tools, such as Alignment API 2 do not give satisfactory results. Simply, we did not succeed to ﬁnd mappings in ontologies, although the similar concepts were present. The reason for this setback can be found in the fact that the Alignment API and corresponding alignment algorithms are more directed to ﬁnd the structural similarities, and not the syntactic similarities that are of interest to us. We are, however, aware that more thorough study is required in which different mapping algorithms can be compared with each other in order to ﬁnd the most suitable one. However, this study goes beyond the scope of this paper. Therefore, in order to ﬁnd similar concepts we apply the string similarity. It has been already shown [6] that in the case of biomedical ontologies simple mapping methods are sufﬁcient and outperform more complex methods. Our mapping algorithm is based on the Levenshtein Distance [13]. Here we will discuss how this algorithm works. Let us have a set of concepts from ontology O1 . For each concept we extract a number of characteristics. These characteristics will be further used in the calculation of the similarity, and these are: Label The label of the concept. This is a name of the concept. Synonyms Collection of names for the same term as present in different vocabularies. ID The concept identiﬁer. In some ontologies this is a unique string generated during serialization of the ontology with a dedicated tool. In other ontologies concept ID can be the same as the name of the concept. Because this property depends on how an ontology is serialized, it may get only a little weight during the calculation of the similarity. We compare these characteristics for all classes from ontology O1 with the same characteristics for all classes from ontology O2 . The comparison is based on the Levenshtein distance algorithm [13]. In order to have a metric that is independent of the length of the string and a metric which is normalized (in the range [0 . . . 1]), we have adapted the Levenshtein distance and introduce the metric Lev. For two strings A and B the metric Lev is calculated by the following equations: A&B = Lmax − Ld A \ B = LA − A&B B \ A = LB − A&B Lev = 2 Ontology

A&B , (A \ B + B \ A + A&B)

alignment API and implementation, http://alignapi.gforge.inria.fr

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where A&B is the common substring for A and B, Lmax is the length of the longest string, Ld is the Levenshtein distance, A \ B is the length of A without B, B \ A is the length of B without A, LA and LB are the lengths of A and B respectively, and Lev is a new metric satisfying our constrains. When two classes Ci and Cj from ontologies Oi and Oj are compared respectively, the metrics of label Levlabel , synonyms Levsyns and ID Levid can be combined to one metric Lev = w1 Levlabel + w2 Levsyns + w3 Levid , and used for the comparison.In this new metric a wi is a weight that determines the importance of the Levi and i wi = 1. Two classes Ci and Cj are considered to be similar if they have the maximum value for Lev metric and if this value is also higher than the threshold t. In our experiments we have used an empirically determined threshold value t = 0.95; the lower values of t generate less precise mappings.

4. Integration Information from Ontologies The ﬁnal step of the ontology creation is the integration of the modules into one ontology. This is done on the basis of mappings. If there is a mapping found between two classes Ci and Cj , from the modules Mi and Mj respectively, we add the equivalence relation OWL : EQUIVALENT C LASS between these classes in the new ontology. Besides the equivalence relationships the new ontology contains the OWL: IMPORTS axioms, by means of which all the created modules are imported. So far, this all seems rather straightforward. However, the problem with such integrated ontology O1...n is that it contains a lot of unsatisﬁable 3 classes. In order to understand the reason of this unsatisﬁability we have applied different experiments. First, we have merged all pairs of the modules, namely ∀i=j Oi,j ≡ Mi ∪ Mj . For each merged ontology Oi,j we have checked for unsatisﬁable classes. It was the case that already at this stage of integration different merged pairs contain unsatisﬁable classes. In order to reveal the reasons of unsatisﬁability we have used the explanation functionality of the Pellet [19] reasoner. 4.1. Solving Unsatisﬁable Classes in Merged Pairs Table 2 shows the number of unsatisﬁable classes appeared after integration of pairs of the modules. From this table follows that the module created from NCI_T HESAURUS and GENE _ REGULATION OT hes∪greg contains the largest amount of unsatisﬁable classes. For this reason we will use this integrated pair to explain how unsatisﬁabilities in merged ontologies are solved. In Figure 5 explanations for unsatisﬁability of the class Gene from the merged ontology OT hes∪greg are depicted. From this ﬁgure follows that the class Thes:Gene ≡ greg:Gene is subsumed via the chain of subclass relationships and via the equivalence Thes:Deoxyribonucleic_Acid ≡ greg:DNA by the class Thes:Drugs_and_Chemicals, but the classes Thes:Gene and Thes:Drugs_and_Chemicals are disjoint. This shows that the NCI_T HESAURUS ontology appears to be too restrictive. To repair this ﬂaw we therefore remove this restriction from the module OT hes . 3 From logical point of view a concept C is satisﬁable w.r.t. a knowledge base K iff there is an interpretation I with C I = ∅ that satisﬁes K and is unsatisﬁable otherwise.

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(1)

0

na

0 0 0

0 0 0

0 0 0

0 0 0

na 0 0

na 0

na

0 2 0 271

0 0 0 0

0 0 0 0

0 0 0 189

0 0 0 79

0 0 0 0

0 0 0 119

MT hes

0

MbioP r

na

54

MM olRole

na 0

Mmolf un

na 0 41

Mmesh

Mcellcom

MT hes

Mpw

Mcellcom Mmolf un MM olRole MbioP r

Mdenc

Mpw

Mchebi

Mchebi Mmesh Mdenc

Mprot

Mgreg Mprot

Mgreg

Table 2. Number of unsatisﬁable classes in the merged pairs of modules.

na 0 0 7

na 0 50

na 35

na

(2)

Figure 5. Explanation for unsatisﬁability of the class Gene for the module OT hes∪greg . The reason in (1) is that Thes:Gene and Thes:Drugs_and_Chemicals are disjoint. The reason in (2) is that greg:Chemical and greg:NucleicAcid are disjoint.

After removing this restriction from OT hes the class Gene was still unsatisﬁable. In this case the reason was the statement greg:Chemical disjointW ith greg:NucleicAcid, but via equivalences with classes in NCI_T HESAURUS, the greg:NucleicAcid was subsumed by greg:Chemical, see Figure 5 for an explanation. This points to a too restrictive modeling in the GENE _ REGULATION ontology or potential design errors. In Figure 6 the explanation for unsatisﬁability of the concept Chromatin from OT hes∪greg is presented. Here the concept Chromatin becomes subsumed under two concepts Thes:Anatomic_Structure_System_or_Substance and Thes:Drugs_ and_ Chemicals which are disjoint in NCI_T HESAURUS. The unsatisﬁabilities of the class Binding is also caused by disjointness in NCI_T HESAURUS, the explanation is depicted in Figure 6. After removing erroneous equivalences, such as Thes:Normal_Tissue equivalentT o greg:Tissue, and a number of disjointness axioms there are no unsatisﬁable classes left

52

(1)

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(2)

Figure 6. Explanation for unsatisﬁability of the classes Chromatin (1) and Binding (2) for the module OT hes∪greg .

in the integrated ontology OT hes∪greg . In other pairs of integrated modules the similar patterns of unsatisﬁability emerge, therefore we will not describe them here. 4.2. Solving Unsatisﬁable Classes in Integrated Ontology After we have repaired unsatisﬁable classes in the merged pairs of ontologies Oi,j we had to check satisﬁability of the integrated ontology. There were still 46 unsatisﬁable classes. First, we needed to remove the wrong assigned mapping Cell equivalentT o Cell_Space. After that, the class Chromatin was still unsatisﬁable. The explanation is given in Figure 7. The concept Chromatin here is not consistent as a result of the fact that the concept Nucleus is not consistent. The reason for unsatisﬁability of Nucleus is that after the integration it is subsumed by greg:CellComponent and greg:Cell at the same time, but these two classes are disjoint in GENE _ REGULATION ontology. This strange behavior is caused by modeling errors in M E SH ontology, where the concept organelle is a cellular_structure which is subsumed by the concept cell. This means that in M E SH the part_of relationship is mixed with the is_a relationship. At this point, we do not have the intention to repair the wrong subsumption in M E SH ontology, the easiest way in this case is just remove the disjointness between Cell and CellComponent in GENE _ REGULATION ontology. After removing other disjointness axioms of a similar kind, the ﬁnal integrated ontology contains only satisﬁable classes. In the current version of our method the unsatisﬁabilities are repaired manually. We are, however, aware that it makes this approach difﬁcult to use for biologists. Also, the strategy for repair is the most simple one, i.e. we remove disjointness axioms from modules. There exist, however, other strategies. For example, removing a modeling error from the original ontology, or rejecting the mapping causing the error. It is important to investigate how to generate and represent different repairing suggestions for the user in an automated way.

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Figure 7. Explanation for unsatisﬁability of the class Chromatin in the integrated ontology.

5. Conclusions We have described a method to generate a new ontology on the basis of the bio-ontologies available in OBO F OUNDRY. We have shown how to create modules on the basis of the terms of interest. The signature for the module extraction is enriched by the symbols from other modules with the ﬁxpoint as a stop criterion. We have integrated modules on the basis of mappings created using Levenshtein distance similarity. We have investigated how to solve unsatisﬁable classes that appear as a result of the integration of the modules. Although the number of unsatisﬁable classes was large, it was possible to solve unsatisﬁabilities with the help of explanations provided by the Pellet reasoner. We have investigated the unsatisﬁability patterns and concluded that the most frequent reason for the unsatisﬁability of the integrated ontologies is the disjointness of the concepts in NCI_T HESAURUS and GENE _ REGULATION ontologies. This may indicate that these ontologies are modeled too strictly. Moreover, we can conclude that in the M E SH ontology the part_of and is_a relationships are mixed. In this paper we have shown that the modularity and simple mappings provide a good foundation for the creation of a new ontology in an pseudo-automated way. This method can be used when an ontology engineer does not want to create a new ontology from scratch, but want to reuse knowledge already presented in other ontologies. Moreover, this is a strategy that should be preferred and has to be adapted more often as ontologies gain importance in life sciences.

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References [1] [2] [3] [4] [5]

[6] [7]

[8] [9] [10] [11] [12] [13] [14] [15]

[16] [17]

[18] [19]

The OBO Foundry Ontologies. http://berkeleybop.org/cgi-bin/obofoundry/ table.cgi. The Ontology for Biomedical Investigations. http://obi-ontology.org/page/Main_Page. Terminology Resources: NCI Enterprise Vocabulary Services (EVS), Dictionaries, FedMed, FDA, CDISC, and NCPDP terminology. Alex Borgida and Luciano Seraﬁni. Distributed Description Logics: Assimilating information from peer sources. Journal of Data Semantics, 1:2003, 2003. Melanie Courtot, Frank Gibson, Allyson L. Lister, James Malone, Daniel Schober, Ryan R. Brinkman, and Alan Ruttenberg. MIREOT: the Minimum Information to Reference an External Ontology Term. July 2009. Amir Ghazvinian, Natalya F. Noy, and Mark A. Musen. Creating mappings for ontologies in biomedicine: Simple methods work. In AMIA 2009 Symposium Proceedings, 2009. S. Ghilardi, C. Lutz, and F. Wolter. Did I damage my ontology? A case for conservative extensions in Description Logics. In Patrick Doherty, John Mylopoulos, and Christopher Welty, editors, Proceedings of the Tenth International Conference on Principles of Knowledge Representation and Reasoning (KR’06), pages 187–197. AAAI Press, 2006. Bernardo Cuenca Grau, Ian Horrocks, Yevgeny Kazakov, and Ulrike Sattler. Extracting modules from ontologies: A logic-based approach. In Modular Ontologies, pages 159–186. 2009. Bernardo Cuenca Grau, Bijan Parsia, and Evren Sirin. Working with multiple ontologies on the Semantic Web. Lecture Notes in Computer Science, 3298:620–634, 2004. Ernesto Jiménez-Ruiz, Bernardo Grau, Ulrike Sattler, Thomas Schneider, and Rafael Berlanga. Safe and economic re-use of ontologies: A logic-based methodology and tool support. pages 185–199. 2008. Yannis Kalfoglou and W. Marco Schorlemmer. Ontology mapping: The state of the art. In Semantic Interoperability and Integration, 2005. Oliver Kutz, Carsten Lutz, Frank Wolter, and Michael Zakharyaschev. E-Connections of abstract description systems. Artiﬁcial Intelligence, 156(1):1–73, 2004. Vladimir Levenshtein. Binary codes capable of correcting, deletions, insertions, and reversals. Soviet Physics-Doklady, 10(8):845–848, August 1965. Natalia F. Noy and Mark A. Musen. Specifying ontology views by traversal. LNCS, 3298:713–725, 2004. Natalya F. Noy, Nigam H. Shah, Patricia L. Whetzel, Benjamin Dai, Michael Dorf, Nicholas Grifﬁth, Clement Jonquet, Daniel L. Rubin, Margaret-Anne Storey, Christopher G. Chute, and Mark A. Musen. BioPortal: Ontologies and integrated data resources at the click of a mouse. Nucleic Acids Research, 37:W170–W173, 2009. OBO. The Open Biomedical Ontologies. http://www.obofoundry.org/. Julian Seidenberg and Alan Rector. Web ontology segmentation: Analysis, classiﬁcation and use. In WWW ’06: Proceedings of the 15th international conference on World Wide Web, pages 13–22, New York, NY, USA, 2006. ACM. L. Seraﬁni and A. Tamilin. DRAGO: Distributed reasoning architecture for the Semantic Web. Proc. of the Second European Semantic Web Conference (ESWC’05), pages 361–376, 2005. Evren Sirin, Bijan Parsia, Bernardo Cuenca Grau, Aditya Kalyanpur, and Yarden Katz. Pellet: A practical OWL-DL reasoner. J. Web Sem., 5(2):51–53, 2007.

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Exploiting Modularity for Ontology Veriﬁcation a ¨ Michael GRUNINGER , Torsten HAHMANN b , Megan KATSUMI a a Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada M5S 3G8 b Department of Computer Science, University of Toronto, Toronto, Ontario, Canada M5S 3G8

Abstract. Within knowledge representation, ontologies are logical theories that support software integration and decision support systems. Ontology veriﬁcation is concerned with the relationship between the intended structures for an ontology and the models of the axiomatization of the ontology. To verify a particular ontology, we ideally characterize all the models of the ontology up to elementary equivalence and prove that these models are equivalent to the intended structures for the ontology. In this paper, we investigate the use of automated theorem provers and model ﬁnders to assist in the interactive veriﬁcation of ﬁrst-order ontologies. We identify the reasoning tasks that are associated with different aspects of ontology veriﬁcation and discuss challenges for the application of automated reasoning systems to support these tasks. Keywords. ontology repository, ontology evaluation, ﬁrst-order logic, representation theorems

1. Introduction An ontology is a logical theory that axiomatizes the concepts in some domain, which can either be commonsense knowledge representation (such as time, process, and shape) or the representation of knowledge in more technical domains (such as biology and engineering). In current ontology research, the languages for formal ontologies (such as RDFS, OWL, and Common Logic) are fragments of ﬁrst-order logic, and many applications of ontologies, such as decision support and the semantic integration of software systems, rely on automated theorem proving or model generation. Within these applications, we need to make the claim that any inferences drawn by an automated reasoner using the ontology are actually entailed by the ontology’s intended structures. If an ontology’s axiomatization has unintended models, then it is possible to ﬁnd sentences that are entailed by the intended models, but which are not provable from the axioms of the ontology. Ontology veriﬁcation is concerned with proving that the intended structures for an ontology are equivalent to the models of the ontology’s axiomatization. In Section 3 we show how this is typically done in the metatheory, using a speciﬁcation of the models with respect to different classes of mathematical structures. As a result of this characterization, we can replace a proof about the relationships between two classes of models

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with automated reasoning tasks that prove theorems about the relationship between two ﬁrst-order theories. Many ontologies that require veriﬁcation are in an early development stage and thus lack a complete understanding of their models [6]. The relatively large size of ontologies (possibly containing many redundant axioms) when compared to traditional mathematical theories further complicates automated reasoning with ontologies. Traditional theorem provers are designed to reason with relatively small theories [7]. Proving particular properties in an ontology often requires only a small subset of its axioms, resulting in a great potential for inefﬁciencies. Since our theories are less structured than the knowledge bases examined in [1], we cannot easily use partition-based reasoning. However, ontologies have an advantage that many mathematical theories do not have: they can be modularized. Modularization is a well-known technique to deal with large artefacts such as software or theories. For mathematical theories modularization may be unnecessary or even impossible because of the strong interaction amongst all axioms of a particular theory. But ontologies can often be modularized into sets of axioms that are closely related (coherent), e.g. by restricting a certain predicate or function. At the same time, modularization tries to minimize dependencies between modules (coupling). In Section 4 we explore techniques for automated ontology veriﬁcation in which the relationships between the modules within an ontology repository play a key role. Although several approaches [4] to ontology modularity rely on conservative extensions, we can also use a repository that consists of sets of modules that are ordered by entailment, allowing for nonconservative extensions [5]. When verifying a particular ontology we consider the modules that axiomatize subtheories of the ontology; the reasoning problems for veriﬁcation of the ontology are solved by restricting them to the subtheories. In one approach, we search for the weakest modules in the repository that are required to ﬁnd a proof; the hypothesis is that such modules do not contain axioms that are intuitively unnecessary for the proof from the original ontology. In another approach, a set of modules can be used to develop potentially useful lemmas, which can then be reused by any other modules that are extensions. Conversely, a set of lemmas can be used to characterize new modules which contain the minimal set of axioms required to prove the lemmas. A key objective of this paper is to present the pragmatic challenges for automated ontology veriﬁcation and to show how modularity can, in principle, be leveraged to address those challenges. Despite their simplicity, the proposed veriﬁcation techniques that exploit modularity are often quite effective. We hope that our work here promotes the development of more sophisticated modularity-based techniques to support ontology veriﬁcation.

2. Modularity Modularization of ontologies is still a young area of research looking for best practices on how to modularize an ontology and there are many – often conﬂicting – criteria for good modularizations. Our focus here is not so much on how to modularize ontologies but how to use a given modularization to help with the task of ontology veriﬁcation. Thus, we assume that a modularization into fairly coherent modules with little coupling between modules is already given.

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A module consists of a set of axioms and a set of imports which in turn deﬁne a transitive import relation < – similar to modules in Common Logic [2]: Deﬁnition 1 A module M = (SM , IM ) is a set of axioms SM together with a set of imported modules IM . We say module M = (SM , IM ) imports module N = (SN , IN ) and write N < M if there is a chain of modules M0 , M1 , . . . , Mk with k ≥ 1, M = M0 , and N = Mk so that Mi ∈ IMi−1 for all 1 ≤ i ≤ k. If a module has an acyclic transitive import closure, it is a modular ontology: Deﬁnition 2 Let M be the set of all modules reachable from a module M , that is, N ∈ M iff N = M or N < M . We call the structure (M, <) a modular ontology iff M is a ﬁnite set and < is irreﬂexive, i.e., N ≮ N for all N ∈ M. We say the module M deﬁnes the ontology (M, <). The theory TM = N ∈M SN axiomatizes (M, <). Each module in a modular ontology deﬁnes itself a modular ontology. Theories of two modules in a modular ontology are related in the following way: Lemma 1 Let K, N ∈ M be modules of a modular ontology (M, <). If N < K then TN |= ϕ ⇒ TK |= ϕ for all sentences ϕ. Two axiomatizations with equivalent languages and models are not distinguished but considered alternative axiomatizations of one module or ontology. When designing ontologies, deﬁnitions are commonly used to capture intuitive concepts in the intended structures of the ontology. Theories of modules only containing deﬁnitions are known as deﬁnitional extensions; here, we do not treat them any different than theories deﬁned by other modules. Intuitively, modules closely resemble traditional mathematical theories; they are rather small, coherent axiomatizations of a single concept. While many ontology veriﬁcation tasks might work reasonably well for individual modules, they often fail for a complete ontology. However, the modular structure can be exploited in various ways to facilitate the automation of ontology veriﬁcation tasks. Though it is not guaranteed that the proposed procedures result in deﬁnite answers to speciﬁc veriﬁcation tasks, they are at least able to conﬁne a possible problem to a small set of modules requiring further manual inspection.

3. Reasoning Tasks for Ontology Veriﬁcation Our methodology for ontology veriﬁcation revolves around the application of modeltheoretic notions to the design and analysis of ontologies. The semantics of the ontology’s terminology can be characterized by a set of structures, which we refer to as the set of intended structures for the ontology. Intended structures are speciﬁed with respect to the models of well-understood mathematical theories (such as partial orderings, lattices, incidence structures, geometries, and algebra). The extensions of the relations in an intended structure are then speciﬁed with respect to properties of these models.

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Veriﬁcation is concerned with the relationship between the intended structures for an ontology and the models of the ontology’s axiomatization. In particular, we want to characterize the models of an ontology up to isomorphism and determine whether or not these models are elementarily equivalent to the intended structures of the ontology. From a mathematical perspective this is formalized by the notion of representation theorems. Representation theorems are proven in two parts – we ﬁrst prove every intended structure is a model of the ontology and then prove that every model of the ontology is elementary equivalent to some intended structure. Classes of structures for theories within an ontology are therefore axiomatized up to elementary equivalence – the theories are satisﬁed by any structure in the class, and any model of the theories is elementarily equivalent to a structure in the class. This approach ontology veriﬁcation rests upon the speciﬁcation of two classes of structures – the intended structures and the models of the ontology’s axiomatization. The primary challenge for proving representation theorems is that it can be quite difﬁcult to characterize the models of an ontology up to elementary equivalence. Ideally, since the classes of structures that are elementary equivalent to an ontology’s models often have their own axiomatizations, we should be able to reuse the characterizations of those other structures. Therefore, we replace a theorem about the relationships between two classes of models with automated reasoning tasks that focus on the relationship between two theories. 3.1. Representation Theorems We now show how a theorem about the relationship between the class of the ontology’s models and the class of intended structures can be replaced by a theorem about the relationship between the ontology (a theory) and the theory axiomatizing the intended structures (assuming that such axiomatization is known). We use automated reasoners to prove the latter relationship and thus verify an ontology in a (semi-)automated way. The mapping π is an interpretation of a theory TA with language LA into a theory TB with language LB if it preserves the theorems of TA . We say that the two theories TA and TB are deﬁnably equivalent iff they are mutually interpretable, i.e. TA is interpretable in TB and vice versa. Translation deﬁnitions for the interpretation are sentences in the language LA ∪ LB of the form (∀x) pi (x) ≡ ϕ(x) where pi (x) is a relation symbol in LA and ϕ(x) is a formula in LB with no variables occurring free except for those in x. The key to using theorem proving and model ﬁnding to support ontology veriﬁcation is the following theorem from [5]: Theorem 1 A theory T is deﬁnably equivalent with a set of theories T1 , ..., Tn iff the class of models M od(T ) can be represented by M od(T1 ) ∩ ... ∩ M od(Tn ). The necessary direction of a representation theorem (i.e. if a structure is intended, then it is a model of the ontology’s axiomatization) can be stated as M ∈ Mintended ⇒ M ∈ M od(Tonto )

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If we suppose that the theory that axiomatizes Mintended is the union of some previously known theories T1 , ..., Tn , then by Theorem 1 we need to show that Tonto interprets T1 ∪ ... ∪ Tn . If Δ is the set of translation deﬁnitions for this relative interpretation, then the necessary direction of the representation theorem is equivalent to the following reasoning task: Tonto ∪ Δ |= T1 ∪ ... ∪ Tn

(Rep-1)

The sufﬁcient direction of a representation theorem (any model of the ontology’s axiomatization is also an intended structure) can be stated as M ∈ M od(Tonto ) ⇒ M ∈ Mintended In this case, we need to show that T1 ∪...∪Tn interprets Tonto . If Π is the set of translation deﬁnitions for this relative interpretation, the sufﬁcient direction of the representation theorem is equivalent to the following reasoning task: T1 ∪ ... ∪ Tn ∪ Π |= Tonto

(Rep-2)

By Theorem 1, M od(Tonto ) is representable by Mintended iff T1 ∪ · · · ∪ Tn is deﬁnably equivalent to Tonto , which we can show by proving both of the above reasoning tasks. All subsequently discussed reasoning tasks for ontology veriﬁcation are related to the entailment problems Rep-1 and Rep-2.

4. Techniques for Automated Ontology Veriﬁcation Although we have speciﬁed a set of reasoning tasks that can be used to prove the representation theorems for an ontology, we still face challenges to automation arising from intractability and the semidecidability of ﬁrst-order logic in combination with little conﬁdence in the ontology’s axiomatization. If a theorem prover does not ﬁnd a proof, does a proof exist or is the theory in fact too weak to prove the goal? If a model ﬁnder does not construct a model, does a model exist or is the theory in fact inconsistent? In the context of ontology veriﬁcation, we are seeking techniques that can help us to solve the reasoning tasks Rep-1 and Rep-2 when the theorem prover fails to ﬁnd a proof. In this section, we discuss various techniques that can be used to address these issues based on the notion of ontology tuning, which alters the axiomatization of the theory so that it remains logically unchanged but is successful in proving a certain sentence. The modularity of an ontology is key to these techniques. With the technique of theory weakening, we are searching for the weakest subtheory that is required to prove the goal. Another technique provides guidance – in particular by exploiting relative interpretations – for the generation and reuse of lemmas that can assist a theorem prover. 4.1. Theory Weakening If the theorem prover fails to ﬁnd a proof, theory weakening can be used to remove axioms which are intuitively unnecessary for the proof and which might misguide the prover. Although this is a common approach in general, we can again use the modularity

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of the ontology to guide us in the identiﬁcation of the appropriate set of axioms. Suppose (M, <) is an ontology and that with the theory TN we are not able to prove a sentence ϕ. We can hypothesize that the theory TK deﬁned by a module K < N might still be able to prove ϕ, so that we have TN |= ϕ by the following relationship: Lemma 2 Let (M, <) be a modular ontology with modules K, N ∈ M that K < N . If TK ϕ, then TN ϕ. The challenge, of course, is to select good candidate modules, K, whose theory is likely strong enough to prove the property but also small enough (length of the axiomatization) to prove the property automatically. How to weaken the ontology’s theory Tonto depends on the speciﬁc reasoning task; it can be based on the relationship between Tonto and T1 ∪ ... ∪ Tn . We can approach it as follows. For a speciﬁc partial axiomatization Ti of the intended structures, ﬁnd the modules that axiomatize the subtheory Si ⊆ Tonto such that Ti ∪ Π |= Si Each such Si is equivalent to the axioms in Tonto that are interpretable by the theory Ti , so that we only need to use Ti rather than all of the theories T1 , ..., Tn when attempting to prove sentences of Tonto whose translations are provable in a particular subtheory Si . We may further conjecture that Ti and Si are deﬁnably equivalent, that is Si ∪ Δi |= Ti If for some i a counterexample to this can be generated, wecan ﬁnd the subtheory T ⊂ Tonto whose axioms are in the set difference of Tonto and i Si . We can then ﬁnish the reducibility proof by showing T1 ∪ ... ∪ Tn |= T The intuition here is that each of the theories Ti is deﬁnably equivalent to some subtheory of Tonto ; we only need to use all of the theories T1 , ..., Tn when attempting to prove sentences of Tonto that are not in any of the associated subtheories. If the subtheories Si cannot be extracted from the modular structure, we can still prove lemmas using successively stronger subtheories of the ontology by starting with the atomic (weakest) modules and including more and more modules until a proof is found or until no more counterexamples can be generated. Theory weakening equally applies to showing inconsistency of an ontology: it sufﬁces to show inconsistency of some weaker theory. Likewise, goals interrelating several deﬁned relations, such as a sentence stating that a set of relations is jointly exhaustive and pairwise disjoint (JEPD), can often be split into subgoals of which many may be provable in from a subset of the axioms and deﬁnitions. 4.2. Lemma Generation and Reuse The use of lemmas is a commonly accepted technique to improve theorem prover performance as it may reduce the number of steps required to obtain a proof. Previous work

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[6] has shown that lemmas can potentially assist with ontology veriﬁcation. As discussed earlier, ontology veriﬁcation requires ﬁnding proofs for both tasks Rep-1 and Rep-2. We will now explore how these tasks can be leveraged to develop potentially useful lemmas. Consider an ontology, Tonto that we are attempting to verify by proving that it is deﬁnably equivalent to a set of well-understood theories T1 , ..., Tn . Suppose that some Ti |= ϕ, and that ϕonto is the sentence expressed in the language of Tonto , using the translation deﬁnitions. If Tonto |= ϕonto we can now store and use ϕonto as a potentially useful lemma. With a similar intuition, we can consider weaker theories as sources for potential lemmas; any additional results of theories weaker than some Ti can be translated into lemmas expressed in the language of the ontology. More generally, if we can prove or already know that the ontology interprets some theory TA , we can assist in proving that the ontology also interprets a theory TB stronger than TA by translating all axioms of TA as well as any other results thereof into the language of the ontology to be used as lemmas. A similar approach of reusing lemmas has been implemented by the Interactive Mathematical Proof System (IMPS) [3]. For example, if we prove Tonto ∪ Δ |= TA then the translations of the axioms and lemmas in TA into the language Lonto of Tonto become lemmas to be used to show Tonto ∪ Δ |= TB This lends itself to an approach that begins with showing that Tonto interprets a very weak theory and then retaining the translated axioms as lemmas for proving interpretations of successively stronger theories. First, proofs of basic properties of relations should be attempted. These include, e.g., reﬂexivity, symmetry, anti-symmetry, or transitivity for binary relations or symmetry or anti-symmetry between certain places of a relation, acyclicity, transitivity, or orderability for ternary relations. Independent of the source of a lemma, for every lemma of an ontology there is some minimal module of the ontology whose theory proves the lemma: Lemma 3 Let (M, <) be a modular ontology. For any sentence Tonto |= ϕ, there is a weakest module W ∈ M so that TW |= ϕ and TN |= ϕ for all N < W . The lemma ϕ then characterizes the module W . Storing lemmas with their weakest module allows us to reuse the lemmas every time a module is used. Though different axiomatizations of such a weakest module exists, they have equivalent theories and thus we do not distinguish them.

5. Summary By specifying a set of theorem proving tasks that can be used to prove representation theorems for an ontology, we have shown how, in principle, to employ automated reasoners for rigorous ontology veriﬁcation. We identiﬁed the pragmatic issues associated with this method of veriﬁcation, and presented basic techniques geared towards making it work in

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practice. This leads to various open challenges for the further development of techniques to assist ontology veriﬁcation by exploiting the modular structure of ontologies: • How to design heuristics to be implemented in automated reasoners that maximally exploit the ontology tuning techniques and the modular structure of an ontology? How can ontology tuning be partly automated? • When the intended structures are not axiomatized, how to select good candidate modules whose theory is likely strong enough to prove a property of interest but small enough (length of the axiomatization) to prove the property automatically? • Can the set of lemmas entailed by a particular set of modules itself form a module? If so, could this be used as the basis for an approach to ontology modularization that is focused on entailed subtheories rather than subsets of axioms? At the moment, ontology veriﬁcation is primarily done manually – a user selects the ontology and tries to ﬁnd a model or an inconsistency proof using a reasoner. This is extremely tedious for large ontologies; veriﬁcation of ﬁrst-order ontologies often requires skillful use of theorem provers with extensive of manual tweaking of the input and options (parameters). The long-term goal is to support interactive ontology veriﬁcation with simple procedures that make extensive use of automated theorem provers and model ﬁnders. To this end, we conclude with a challenge to develop automated reasoning tools speciﬁcally for ontology veriﬁcation, i.e. that account for the special properties of modularized ontologies and their reasoning tasks and techniques, as presented here.

References [1] [2]

[3] [4] [5] [6] [7]

Amir, E., McIlraith, S.: Partition-based logical reasoning for ﬁrst-order and propositional theories. Artiﬁcial Intelligence 162(1–2): 49–88 (2005) Delugach, H. (ed.): Common Logic – a framework for a family of logic-based languages. ISO/IEC WD 24707 (Information Technology). standards.iso.org/ittf/ PubliclyAvailableStandards/c039175_ISO_IEC_24707_2007(E).zip. (2007) Farmer, W.M.: An infrastructure for intertheory reasoning. In: Proc. of Conf. of Automated Deduction (CADE-17). pp. 115–131. LNCS 1831, Springer (2000) B. Cuenca Grau, I. Horrocks, Y. Kazakov, U. Sattler: Modular reuse of ontologies: Theory and practice. Journal of Artiﬁcial Intelligence Research 31: 273–318 (2008) Gr¨uninger, M., Hahmann, T., Hashemi, A., Ong, D. Ontology veriﬁcation with repositories. In: Conf. on Formal Ontology in Inf. Systems (FOIS-10). pp. 317–33. IOS Press (2010) Katsumi, M., Gr¨uninger, M.: Theorem proving in the ontology lifecycle. In: Conf. on Knowledge Engineering and Ontology Design (KEOD). (2010) Pease, A., Sutcliffe, G.: First order reasoning on a large ontology. In: CADE-21 WS on Empirically Successful Automated Reasoning on Large Theories (ESARLT). (2007)

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Modularization Requirements in Bio-Ontologies: A Case Study of ChEBI Janna HASTINGS a,b,1 , Colin BATCHELOR c , Christoph STEINBECK a and Stefan SCHULZ d a Chemoinformatics and Metabolism, European Bioinformatics Institute, Hinxton, UK b Swiss Centre for Affective Sciences, University of Geneva, Switzerland c Informatics, Royal Society of Chemistry, Cambridge, UK d Institute for Medical Informatics, Statistics and Documentation, Medical University of Graz, Austria Abstract. Bio-ontologies such as the Gene Ontology and ChEBI are characterized by large sizes and relatively low expressivity. However, ongoing efforts aim to increase the formalisation of these ontologies by adding full deﬁnitions (equivalent classes). This increase in complexity results in a decrease of performance for standard reasoning tasks. In this paper, we explore the contribution which modularization can play in the evolution of bio-ontologies. In particular, we focus on ChEBI, the ontology of chemical entities of biological interest. ChEBI consists of around 25,000 classes, organised into a structure-based chemical classiﬁcation and enriched with a role-based classiﬁcation of their biologically properties. Ontology modularization – partitioning large ontologies into smaller, more manageable chunks – provides the only feasible mechanism for sustainably maintaining the large-scale and ever-growing ontologies in the biomedical domain. We evaluate available ontology partitioning tools. Keywords. modularity, modularization, bio-ontologies, performance, ChEBI

Introduction Biomedical ontologies such as the Gene Ontology (GO) [28], ChEBI [8], and other OBO Foundry ontologies [22] are characterised by large sizes and relative lack of complexity in their formal structure (measured in terms of both the complexity of the underlying Description Logic (DL) [3] used in the representation, and the number of fully deﬁned classes with necessary and sufﬁcient conditions). Historically, most bio-ontologies have been developed as directed acyclic graphs (DAGs) [2]. Efforts to increase their formalization are under way [15,23] with the goal to allow automated reasoning to support the curation effort, to automatically detect errors and to infer multiple parenthood so that only single inheritance hierarchies have to be explicitly maintained. However, with increasing size and complexity the performance of common ontology maintenance tools such as Protégé [1] and reasoners such as HermiT [21] suffers as a result. 1 Corresponding Author: European Bioinformatics Institute, Wellcome Trust Genome Campus, Hinxton, CB10 1SD, UK; E-mail: [email protected].

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Modularization – the decomposition of larger ontologies into smaller, more manageable subsets – appears to provide one sustainable approach to address this tools challenge [17]. Modular organisation of large ontologies allows better scalability through a ‘divide and conquer’ strategy, allowing reasoning tasks across different modules to be concurrently executed. And, if modules are well-designed, it is possible to edit them largely independently of one another, reducing the overhead on editing tools and human editors. As an important criterion for the quality of modularization it has been required that the modules extracted fulﬁl the safety requirements postulated by [6], i.e. whether they produce exactly the same entailments as the original ontology. This axiomatic approach has not seen much application in bio-ontologies, in contrast to computationally far less demanding graph-based approaches [16,30,20]. 1. Background 1.1. Ontology modularization Research surrounding ontology modules has progressed in several different directions. Automated approaches have been developed which are able to segment large ontologies into mutually non-overlapping parts (often called ‘segmentation’ or ‘partitioning’) [26], and to extract subsets of ontologies (called ‘module extraction’) [9,6]. Additional research involves the development of formalisms for representing modular ontologies [4,5] and for interrelating ontology modules which may be represented in different underlying languages [14]. We will use the term ‘ontology modularization’ to encompass all approaches which decompose an existing large ontology into smaller modules, which includes module extraction and segmentation. Module extraction generally refers to operations which retrieve modules with interesting properties where the complement of the module (i.e. the remainder of the ontology) is not relevant, while segmentation or partitioning approaches try to split up an ontology as a whole into modules which have the property that jointly they ‘cover’ the domain of the full original ontology. Our focus in this paper is on tools for which there are practical implementations available, rather than on theoretical approaches. For partitioning, we evaluate PATO2 [18] and SWOOP3 [12]. PATO partitions ontologies according to levels of dependencies between nodes in the source ontology using the algorithm described in [26]. SWOOP – the Semantic Web Ontology Editor – is an editing tool which partitions an ontology into a set of interconnected modules according to the algorithm described in [7]. 1.2. The ChEBI ontology ChEBI is an ontology for chemical entities of biological interest [8]. It represents molecules, ions and groups, organised into a structure-based classiﬁcation, where classes are deﬁned based on shared structural features. In addition, a role-based classiﬁcation deﬁnes classes based on possible activities of the associated chemicals in a biological or chemical context. An example of a structure class is carboxylic acid, which is deﬁned as all molecules containing a carboxy group. An example of a role-based class is cyclooxygenase inhibitor, deﬁned as inhibiting the action of a cyclooxygenase enzyme. 2 Downloadable from http://webrum.uni - mannheim.de/math/lski/Modularization/ pato.html. 3 Downloadable from http://code.google.com/p/swoop/.

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1.2.1. Ontology maintenance ChEBI is maintained through a custom editing tool which relies on an Oracle database and a web-based Java front end4 . Ontology curators are able to add ontology entries, edit the metadata such as names, synonyms and cross-references, and specify the relationships between entities. Validations are conducted automatically by the editing software in order to ensure the integrity of the overall ontology, including the following: • Checking that no deﬁnitional cycles have been created. • Checking for uniqueness (non-overlapping primary name or chemical structure). • Checking for prohibited combinations of relations, e.g. it is not allowed for A to be simultaneously subclassOf B and (has_part some B). • Checking that the sub-ontologies role and chemical entity are disjoint with respect to the is a relationship. 1.2.2. Ontology usage Applications provided by ChEBI which make use of the ontology include a powerful advanced search feature5 . In order to retrieve all entities which hold a speciﬁed relationship to some entity A the ID of A and the type of the relationship are entered in the ‘Ontology Filter’ box. This facility also retrieves all chemical entities which participate in a particular role, even though such entities may have vastly different structures. For example, searching for all entities linked by has_role to vasodilator agent retrieves 47 chemical entities (as of June 2011) including convallatoxin and amlodipine. 1.2.3. Increasing complexity to allow structural classiﬁcation The expressivity of the currently exported version of ChEBI in OWL is EL++ , a highly performant subset of OWL. However, efforts are underway to fully deﬁne structural chemical classes based on structural parts and properties to allow automated classiﬁcation. This automated classiﬁcation will reduce the maintenance overhead for curators by error checking and inferring multiple parenthood. Here are typical constructs referring to parts and properties of chemical entities • Value restriction: hydrocarbon equivalentTo molecule and has_atom only (hydrogen or carbon) • Existential quantiﬁcation: ‘carboxylic acid’ equivalentTo molecule and has_functional_group some ‘carboxy group’ • Cardinality restriction: ‘tricarboxylic acid’ equivalentTo molecule and has_functional_group exactly 3 ‘carboxy group’ • Data restriction: ‘peptide cation’ equivalentTo peptide and has_charge some double[>,0] Including these deﬁnitions results in an OWL expressivity of SIQ(D) for the ontology, and it is this version of the ontology, rather than the less expressive currently released version, that we will use in what follows. 4 The ontology is also provided in standard ontology ﬁle formats OBO [29] and OWL [25], which are exported from the underlying Oracle database as part of the monthly release procedure 5 http://www.ebi.ac.uk/chebi

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2. Performance of tools The ontology version used in this experiment contained 27,899 entities in total (a development version of the publicly available ontology). To estimate an upper bound for the number of deﬁned classes which would be included if all possible structure-based class deﬁnitions were incorporated, we use the number of classes that are not leaves, that is, they have asserted subclasses within the ontology, and they do not have a fully deﬁned chemical structure associated with them. This number is 9,218 in our development ontology version, i.e. approximately a third of the overall ontology size. To assess the time taken to reason over the ontology with different numbers of deﬁned classes, we automatically converted existing ChEBI relations to full class deﬁnitions (equivalent classes) in varying percentages of the overall number of relations. The results for reasoning time in seconds are illustrated in Figure 1. These classiﬁcation times are expected to have severe implications in the context of the maintenance and use of ChEBI with increasing complexity. Modularization provides a potential avenue to escape this dilemma.

Figure 1. Reasoner performance: results of running reasoning with HermiT using ChEBI with different numbers of deﬁned classes. On the left is the time taken to reason in seconds with different numbers of deﬁned classes included in the ontology. On the right is the number of deﬁned classes compared to the overall size of the ontology and to the estimated upper bound for the number of deﬁned classes which might be included.

2.1. Existing modularization techniques applied to ChEBI We attempted to evaluate the SWOOP and PATO partitioning implementations on the ChEBI ontology. Running SWOOP’s partitioning algorithm on ChEBI resulted in the construction of 465 partitions. However, the ﬁrst of these had a hugely disproportionate portion of the full ontology (ﬁle size 10Mb compared to 2Kb for the others). This large module contained 27,212 classes, and 2 object properties. This is almost entirely the full original ontology, which has 27,899 classes and 9 object properties. This modularization is therefore not likely to lead to any reduction in overall reasoning time for either editing or querying tasks. Besides which, there is a usability concern, in that the created modules are not organised semantically around subject modules, thus would not make any sense to the user or to ontology maintainers. It was unfortunately not possible to evaluate the PATO partitioning algorithm against ChEBI, since execution terminated in an OutOfMemory error, despite passing the maximum possible amount of memory to the Java virtual machine on a test machine with 4Gb of RAM.

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2.2. Requirements for bio-ontology modularization In this section we present an informal analysis of the requirements for ontology modularization in the context of ChEBI maintenance and usage as described in Section 1.2. Large bio-ontologies such as ChEBI and the Gene Ontology are actively maintained by teams of dedicated ontology curators. Although the OBO Foundry adopts a subject-delineated modular approach to ontology development beneath a shared, common, upper ontology, nevertheless individual ontologies have grown to sizes such that they have the potential to beneﬁt from further modularization. The modularization approach adopted should allow the following desiderata to be achieved for the modules which are created [16,23]. They should: • be comprehensible to ontology maintainers (e.g. by subject-speciﬁc delineation) • allow for automated classiﬁcation (inferred multiple parenthood) of entities based on logical deﬁnitions • allow modular reuse of parts of the ontology by neighbouring bio-ontologies • enable consistency checking and custom validations such as cycle detection Upper-level ontology distinctions are often used in modern ontology development efforts to create modular ontologies from the top down, for example in complex event processing [27]; and upper-level distinctions may be used to retroactively render ontology modules disjoint during ontology maintenance as was accomplished in the separation of ChEBI roles and chemical entities [8]. The disjointness of sub-ontologies beneath the upper levels suggests a type of modularization approach in which ontologies are decomposed into modules for each sub-ontology, and for purposes of certain reasoning tasks their inter-ontology links can be compressed into restrictions in terms of the upper levels themselves. This is because upper level distinctions usually form the domain and range of relations. For a simple example, We can collapse has_role some cyclooxygenase inhibitor into the underspeciﬁed has_role some role, and this underspeciﬁed formulation sufﬁces to perform validation that the overall ontology does not contain any cycles and that the relationships asserted are not incorrect with respect to their domains and ranges. Granularity is a dimension that is neither totally reducible to upper-level distinctions nor to domain distinctions (though it partially overlaps with both). For example, there are granularity distinctions within material objects within a single domain, e.g. in chemistry: chemical entities and chemical substances (portions of entities). It had been argued that, precisely because of the difﬁculty, granularity distinctions should not be the concern of upper level ontologies [19], such as between Object and ObjectAggregate [24], and between Object and FiatObjectPart in BFO. These distinctions become blurred when crossing granularity levels in domain ontologies. The relevance of granularity for modularization of ontologies has also been discussed in [13]. Granularity levels are often left implicit in bio-ontologies at present, hidden behind relationships such as has_part which can be interpreted both as granularity-crossing (when the whole consists of many parts of the given type) and not (when the whole consists of only a few parts of the given type). For example, the human body consists of many cells, thus a statement that human body has_part some cell can be considered a granularity-crossing statement. On the other hand, a human body has only two hands, therefore, a statement that human body has_part some hand is not granularity- crossing. For our purposes here, we will understand the level of granularity of an ontology in terms of the level of detail about the parts of an entity that are included in the ontology.

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For automatic classiﬁcation of chemical entities based on their structures, all the structural parts for each chemical entity have to be explicitly represented in the ontology in order to be available for reasoning for classiﬁcation. The relevant parts and properties can be computed from the chemical structure and automatically added to the ontology, thus do not involve a manual curation overhead, but they do lead to an explosion of the size of the ontology in terms of asserted parts, as illustrated in Figure 2. We have

Figure 2. Explosion of parts: illustrating the number of different parts in a single compound. The picture shows just a few of the parts of the cefpodoxime molecule that may be relevant for classiﬁcation. Parts of varying sizes from single atoms to nearly the whole of the molecule may be equally relevant.

previously evaluated description graphs as an alternative to explicitly including parts such as these in the ontology by including the full chemical structure instead [10], but as performance of reasoning was prohibitive with that approach we do not further consider it in this paper. These parts are often uninteresting to the casual browser of the ontology, thus may be mere ‘bloat’ once the classiﬁcation has been computed. They can be kept in a separate module which is only combined with the main (core) ontology when needed. For query answering, modules are needed which contain all the relevant information about a given entity in the ontology. Although different queries may be required in different applications, for most bio-ontologies, at least the immediate references to the entity in relationships in the ontology as well as the classiﬁcation hierarchy of the entity and of related entities is required. The queries which are most commonly performed on the ChEBI ontology via the web-based user interface are: • What are all the superclasses of a given entity (paths to the ontology root)? • What are all the subclasses of a given entity, and in particular, those with chemical structures associated? • What are all the roles for a given chemical entity (e.g. has it been used as a drug)? • What are all the chemical entities (in particular those with structures) associated with a given role? • What are the closely structurally related chemicals to a given chemical entity? • What are all the metadata, i.e. names, synonyms, database cross-references, and literature citations for a given chemical entity? Performance of question answering tasks such as these for widespread use of ontology applications is crucial. While the ChEBI approach thus far has made use of Lucene to precompute and index the relevant paths and properties needed for each entity to address

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these queries, future developments will enable the user to suggest more complex queries which are not pre-cached and therefore rapid ontology reasoning will be required. This type of question answering can be provided through the use of ontology simpliﬁcation strategies such as El Vira [11].

3. Conclusions In support of the development of modular bio-ontologies, tools are needed which are able to separate the ontologies into modules and recombine them into total ontologies, without subsequent loss of information or entailments and without adding redundancy. Furthermore, such modules need to be available for neighbouring ontologies to import and reason over [16]. With respect to ChEBI, one such neighbouring ontology is the Gene Ontology [28], within which many ChEBI cross-references have already been created [15]. What emerges from the above tentative discussion of requirements for modules in ChEBI, is that different and overlapping modules are required in support of different tasks in ontology maintenance and use. We therefore propose that ontology modularization be seen as a ﬂexible and task-oriented exercise, with different kinds of modules being delineated, at different times in the ontology maintenance and usage lifecycle, for different purposes. We understand this work as an informal contribution to bridging the divide between research in formal ontology modularization techniques within theoretical computer science and its application to bio-ontologies. Future work will focus on analysing the more formal properties of the module types that have been put forward here, and where feasible on developing implementations.

4. Acknowledgements This work was supported in part by the BBSRC, grant agreement number BB/G022747/1 within the “Bioinformatics and biological resources” fund.

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The Protégé ontology editing tool. http://protege.stanford.edu/. Gil Alterovitz, Michael Xiang, David P. Hill, Jane Lomax, Jonathan Liu, Michael Cherkassky, Jonathan Dreyfuss, Chris Mungall, Midori A. Harris, Mary E. Dolan, Judith A. Blake, and Marco F. Ramoni. Ontology engineering. Nature Biotechnology, 28(2):128–130, February 2010. Franz Baader, Diego Calvanese, Deborah L. McGuinness, Daniele Nardi, and Peter F. Patel-Schneider. The Description Logic Handbook: Theory, Implementation, and Applications, 2nd Edition. Cambridge University Press, 2 edition, September 2007. Jie Bao, George Voutsadakis, Giora Slutzki, and Vasant Honavar. Modular ontologies. chapter PackageBased Description Logics, pages 349–371. Springer-Verlag, Berlin, Heidelberg, 2009. A Borgida and L Seraﬁni. Distributed description logics: Assimilating information from peer sources. Journal of Data Semantics, 1:153–184, 2003. Bernardo Cuenca Grau, Ian Horrocks, Yevgeny Kazakov, and Ulrike Sattler. Modular reuse of ontologies: Theory and practice. J. Artif. Intell. Res. (JAIR), 31:273–318, 2008. Bernardo Cuenca Grau, Bijan Parsia, Evren Sirin, and Aditya Kalyanpur. Modularity and web ontologies. In Proceedings of KR-2006, Tenth International Conference on Principles of Knowledge Representation and Reasoning, June 2-5, 2006, pages 198–209. AAAI Press, 2006.

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[8] P de Matos, R Alcántara, A Dekker, M Ennis, J Hastings, K Haug, I Spiteri, S Turner, and C Steinbeck. Chemical Entities of Biological Interest: an update. Nucl. Acids Res., 38:D249–D254, 2010. [9] Paul Doran, Valentina Tamma, and Luigi Iannone. Ontology module extraction for ontology reuse: an ontology engineering perspective. In Proceedings of the sixteenth ACM conference on Conference on information and knowledge management, CIKM ’07, pages 61–70, New York, NY, USA, 2007. ACM. [10] Janna Hastings, Michel Dumontier, Duncan Hull, Matthew Horridge, Christoph Steinbeck, Ulrike Sattler, Robert Stevens, Tertia Hörne, and Katarina Britz. Representing chemicals using OWL, description graphs and rules. In Proc. of OWL: Experiences and Directions (OWLED 2010), 2010. [11] Robert Hoehndorf, Michel Dumontier, Anika Oellrich, Sarala Wimalaratne, Dietrich RebholzSchuhmann, Paul Schoﬁeld, and Georgios V. Gkoutos. A common layer of interoperability for biomedical ontologies based on OWL EL. Bioinformatics, 2011. [12] Aditya Kalyanpur, Bijan Parsia, Evren Sirin, Bernardo Cuenca Grau, and James Hendler. Swoop: A web ontology editing browser. Journal of Web Semantics, 4:2005, 2005. [13] C Maria Keet. Toward cross-granular querying over modularized ontologies. In Proceedings of the Workshop on Ontologies: Reasoning and Modularity (WORM-08), 2008. [14] Oliver Kutz, Dominik Lücke, Till Mossakowski, and Immanuel Normann. The OWL in the CASL designing ontologies across logics. In Workshop on OWL: Experiences and Directions, 2008. [15] Christopher J. Mungall, Michael Bada, Tanya Z. Berardini, Jennifer Deegan, Amelia Ireland, Midori A. Harris, David P. Hill, and Jane Lomax. Cross-Product Extensions of the Gene Ontology. Journal of biomedical informatics, February 2010. [16] Jyotishman Pathak, Thomas M. Johnson, and Christopher G. Chute. Survey of modular ontology techniques and their applications in the biomedical domain. Integr. Comput.-Aided Eng., 16:225–242, August 2009. [17] A L Rector, A Napoli, G Stamou, and et al. Report on Modularization of Ontologies. 2005. [18] Anne Schlicht and Heiner Stuckenschmidt. A ﬂexible partitioning tool for large ontologies. Web Intelligence and Intelligent Agent Technology, IEEE/WIC/ACM International Conference on, 1:482–488, 2008. [19] Stefan Schulz, Martin Boeker, Holger Stenzhorn, and JÃ˝urg Niggemann. Granularity issues in the alignment of upper ontologies. Methods of information in medicine, 48:184–189, 2009. [20] Stefan Schulz and Pablo Garcia-Lopez. Modularity issues in biomedical ontologies. In Oliver Kutz and Thomas Schneider, editors, Proceedings of the Workshop on Modular Ontologies 2011, Ljubljana, Slovenia, 2011. [21] R Shearer, B Motik, and I Horrocks. HermiT: A highly-efﬁcient OWL reasoner. In C. Dolbear, A. Ruttenberg, and U. Sattler, editors, Proceedings of the 5th Workshop on OWL: Experiences and Directions, Karlsruhe, Germany, 2008. [22] Barry Smith, Michael Ashburner, Cornelius Rosse, Jonathan Bard, William Bug, Werner Ceusters, Louis J Goldberg, Karen Eilbeck, Amelia Ireland, Christopher J Mungall, The OBI Consortium, Neocles Leontis, Philippe Rocca-Serra, Alan Ruttenberg, Susanna-Assunta Sansone, Richard H Scheuermann, Nigam Shah, Patricia L Whetzel, and Suzanna Lewis. The OBO Foundry: coordinated evolution of ontologies to support biomedical data integration. Nat Biotechnol, 25(11):1251–1255, Nov 2007. [23] Barry Smith and Mathias Brochhausen. Putting biomedical ontologies to work. Methods Inf Med, 49:135–140, 2010. [24] Barry Smith and Pierre Grenon. The cornucopia of formal ontological relations. Dialectica, 58:279–296, 2004. [25] Michael K. Smith, Chris Welty, and Deborah L. McGuinness. The Web Ontology Language, 2010. [26] J Stuckenschmidt and M Klein. Structure-based partitioning of large concept hierarchies. In Proc. of the International Semantic Web Conference (ISWC), 2004. [27] Kia Teymourian, Gökhan Coskun, and Adrian Paschke. Modular upper-level ontologies for semantic complex event processing. In O Kutz, editor, Modular Ontologies, 2010. [28] The Gene Ontology Consortium. Gene ontology: tool for the uniﬁcation of biology. Nat. Genet., 25:25– 9, 2000. [29] The Gene Ontology Consortium. The OBO language, version 1.2, 2010. [30] Pinar Wennerberg, Klaus Schulz, and Paul Buitelaar. Ontology modularization to improve semantic medical image annotation. Journal of Biomedical Informatics, 44:155–162, 2011.

Modular Ontologies O. Kutz and T. Schneider (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-799-4-71

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Modeling the Diversity of Spatial Information by Using Modular Ontologies and Their Combinations Joana HOIS a University of Bremen, Germany e-mail: [email protected] a

Abstract. Ontologies can specify spatial information based on different aspects, for different purposes, with different formalization details, with different granularity levels, and thus from different perspectives. Spatial systems that formally describe spatial information often have to take into account this diversity in spatial representations. We propose that such a system’s ontological representation should preserve the spatial diversity by making the different perspectives explicit. We use modularly designed spatial ontologies for reﬂecting different spatial perspectives and analyze which existing modularity techniques are relevant for combining the different modules. Keywords. Spatial Information, Modular Ontologies, Ontology Combinations

Introduction As space is a fundamental component (such as time) it is often abstractly formalized in upper-level or foundational ontologies. These ontologies formally describe entities and properties, such as physical objects, spatial relations and properties, movement, dimensionality, granularity, or spatial vagueness. Ontologies have also been developed specifically for the spatial domain and applied to a variety of spatial systems, such as geographical information systems [Kovacs et al., 2007], vision and image recognition systems [Schill et al., 2009], human-computer interaction and robotics [Kruijff et al., 2007]. General information, such as topology, orientation, distance, size, shape, morphology, spatial change and interaction, however, also describe relevant aspects of space, and thus have to be taken into account by ontological formalizations. As a consequence, an ontological speciﬁcation should aim at appropriately reﬂecting these spatial distinctions by complying with a speciﬁc spatial perspective that matches the spatial descriptions. To cope with these spatial distinctions and the different descriptions, modular ontologies can be used that reﬂect a selection of spatial information based on the module’s spatial perspective. For example, an ontology module that aims at specifying abstract spatial shapes of objects should avoid to specify movements or geographical entities and instead only represent a categorization of shapes and their characteristics from its ‘shapespeciﬁc’ perspective. Clearly, spatial systems may require more spatial information than provided by one perspective to achieve their application-speciﬁc tasks. For such systems

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and for a more extensive representation of spatial information, it is necessary to combine the different perspectives and their spatial descriptions. Hence, based on the perspectives of the modular spatial ontologies adequate techniques for their combinations have to be selected and applied. A distinction of different spatial perspectives for specifying modular spatial ontologies and their combinations is presented in this paper. In the next section, we introduce a classiﬁcation of spatial perspectives and characteristics that provides a basis for the spatial ontologies that have been modularly developed according to this classiﬁcation. Combinations of these spatial modules for application purposes is achieved by using different module combination techniques. In particular, these combinations are selected by means of the perspectives to be combined. Finally, an application example for spatial assistance that uses the spatial modular ontologies and their combinations is brieﬂy discussed.

1. Spatial Perspectives for Ontology Modules Although ontologies aim at specifying consensual knowledge of a certain domain, there is not automatically only one ontology that describes this domain in a correct way. In fact, there exist many ontologies of the same domain and they might even be incompatible with each other. This is caused either by different formalizations or thematically different perspectives [Gómez-Pérez et al., 2004,Euzenat and Shvaiko, 2007]. This situation, however, is natural in common-sense knowledge because terms can vary across societies or contexts or change over time and meanings are in many cases not permanently ﬁxed [Munn and Smith, 2008]. Also the deﬁnition of entities within one domain may vary according to different situations or contexts [Pike and Gahegan, 2007]. For spatial information, thematically distinct ontology modules can be used to describe particular spatial perspectives. These perspectives can be divided into the following four types, which is a revised version of the distinction presented in [Hois, 2010a], illustrated in Figure 1: Qualitative and Quantitative Space. Ontology modules that follow a quantitative or qualitative perspective are spatial speciﬁcations that are either closely related to spatial data formats or qualitative spatial models. Such information is relevant for a variety of spatial applications, in which spatial environments are represented that are closely connected to spatial data. Thus ontology modules can often be grounded in this spatial data. Ontology modules that comply with the qualitative and quantitative spatial perspective are speciﬁed as core or domain ontologies. Their categories and relations reﬂect basic spatial types of information, either qualitatively or geometrically. Examples are modules for orientations or regions. Abstract Space. Ontology modules that provide spatial speciﬁcations from an abstract space perspective address rather general axiomatizations of space, which are often already included in upper-level ontologies. Spatial applications, however, may need a reﬁned version of these representations that match their speciﬁc requirements, as the upperlevel speciﬁcations are too abstract. Nevertheless, these speciﬁcations provide a general categorization that can be reused for application-speciﬁc reﬁnements. Ontology modules with abstract spatial perspectives are speciﬁed as upper-level or foundational ontologies. Their categories describe general spatial aspects on a coarse granularity. Examples are 3D- or conceptual spaces modules.

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Figure 1. Spatial Perspectives. An ontological distinction of spatial perspectives showing the different spatial perspectives for qualitative and quantitative, abstract, domain-speciﬁc, and multimodal space.1

Domain-Speciﬁc Space. Ontology modules that specify spatial information from a domain-speciﬁc perspective are closely related to the requirements of spatial applications. They are designed to reﬂect spatial information in the application’s domain, mostly categorizations of environmental aspects. These modules may reuse ontology modules from the abstract space perspective and further reﬁne them. Ontology modules complying with the domain-speciﬁc spatial perspective are speciﬁed as task, domain, or application ontologies. They reﬂect context- or situation-speciﬁc types of spatial information on a ﬁne-grained level. Examples are modules for geography or actions. Multimodal Space. Ontology modules that describe spatial information from a multimodal perspective describe spatial information for one modality, e.g., spatial information available by gesture or language. These modules are primarily applicable for humancomputer interaction. Such speciﬁcations naturally depend on the cultural contexts for which they are designed or in which they are applied. Multimodal spatial ontology modules are speciﬁed as core ontologies that reﬂect modality-dependent spatial information. Examples are modules for language or vision.

2. Modular Ontologies and their Spatial Perspectives Based on the distinction of the different spatial perspectives an ontology module can comply with, spatial ontologies can be developed for speciﬁc use, purpose, domains, and tasks. In the following, some examples are presented that have been developed and applied in the context of spatial assistive systems. For application reasons, their speciﬁcations are formulated in OWL.2 Region-Based Ontology An example of a qualitative spatial perspective ontology module is the Region-Based Ontology (RBO), that speciﬁes ontological relations deﬁned by the RCC-8 qualitative calculus [Cohn et al., 1997]. The module’s aim is to provide a region-based set of RCC-8 relations, which can be reused by other modules to formulate relations among their domain-speciﬁc entities. As the RBO module describes a qual1 Cf.

[Gómez-Pérez et al., 2004] and [Palma et al., 2009] for a classiﬁcation of ontology types. OWL speciﬁcations are available at http://www.informatik.uni-bremen.de/~joana/ ontology/SpatialOntologies.html 2 The

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itative perspective on space it only deﬁnes a core set of entities and relations, namely the most-general type Region (speciﬁed as a subtype of Thing) and the RCC-8-related relations (ObjectProperties). Based upon the relation naming in RCC-8, the RBO module deﬁnes the relations: disconnectedFrom externallyConnectedTo, partiallyOverlaps, equalTo, tangentialProperPartOf, nonTangentialProperPartOf, inverseTangentialPartOf, and inverseNonTangentialProperPartOf. The RBO module extends the OWL speciﬁcation that has been introduced by [Grütter et al., 2008] as it provides more details and reﬁnements in the ontological speciﬁcation. Spatial Actions Ontology Complying with the abstract spatial perspective, the Spatial Action Ontology (SAO) is a module that speciﬁes spatial actions and entities to provide a high-level speciﬁcation for spatial actions. Aspects of actions contain the actor, actees, instruments, manners, and circumstantial information [Davidson, 1967]. An upper-level ontology that can be reused for this purpose, as it already speciﬁes the general information about actions and activities, is the Ontology of Descriptions and Situations (DnS) [Gangemi and Mika, 2003]. DnS provides a framework for representing contexts, methods, norms, theories, and situations. Among many other categories, it speciﬁes activities and planning. The SAO module introduces SpatialAction as a new subtype of DnS’s action category that requires the spatial action to provide information about the spatial change of objects. According to DnS, an activity is a set of actions that are part in this activity. Hence, the category SpatialActivity of the SAO module is introduced as a subclass of activity of the DnS ontology that can only consist of SpatialActions. It consequently has to deﬁne SpatialActions as parts of the activity. The actions of an activity can also be ordered by using the DnS relation sequenced-by. Home Automation Ontology An example of an ontology module with a domainspeciﬁc spatial perspective is the Home Automation Ontology (HAO), which formulates situations or states that are valid for certain smart home environments. The module’s aim is to support an automation system for assisted living environments and it is thus closely connected with the assisted living environment at hand. The module has to take into account objects and categories of the environment and their possible conditions together with information about the domain and its instances. It also relies on other spatial modules, such as the RBO and the SAO modules, by re-using their deﬁnitions for entities in indoor home environments and their spatial aspects. A speciﬁc example as part of the assisted living system is a monitoring system that supervises environmental conditions and user behaviors and adjusts or changes states of objects in the environment. If the system, for instance, needs to provide automatic lighting, it can use ontological deﬁnitions about lighting-related categories and constraints. Linguistic Spatial Ontology A module that complies with a multimodal spatial perspective is the Linguistic Spatial Ontology (GUM-Space) [Bateman et al., 2010], which provides spatial information as described by linguistic aspects. It speciﬁes the constructions that are used in natural language for describing spatial information. The module formulates semantic spatial categories and roles that can be found in language, and consequently this linguistic perspective is closely related to speciﬁc natural languages, e.g., English. For example, the ontology formalizes how paths in motion expressions can be used by deﬁning start, intermediate, and end points of the path. An example for an application that can apply this linguistic module is a spatial assistive system that communicates with its users by using natural language. In navigation-related tasks, for instance, a

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system has to understand the linguistically expressed destination [Ross, 2008]. Here, the natural language processing is based on several representational layers, one of which is the linguistic module.

3. Combining Spatial Ontology Modules Early work in modular ontologies has shown that modularity is seen as the crucial method for structuring large ontologies particularly to improve re-use, maintainability, and evolution [Rector, 2003]. Modularization also supports the use of ontologies in distributed systems, i.e., each system can use only those ontology modules necessary for its tasks and requirements [Stuckenschmidt and Klein, 2003]. A module can also deﬁne inter-module links to other sub-domain ontologies, which together yield the speciﬁcation of the whole domain [Parent and Spaccapietra, 2009]. In this section, we present and select existing module combinations techniques for combining spatial ontology modules. Extension and Reﬁnement Extensions and reﬁnements of ontologies are ways to use an existing ontology and to add new information to it. Either the contents of the ontology can be further extended by integrating the categories and relations of the existing ontology into new categories and relations, or the contents of the ontology can be further reﬁned by integrating new categories and relations into the categories and relations of the existing ontology. On a content-based comparison, extensions broaden or add scope and objectives of the extended ontology, whereas reﬁnements narrow or deﬁne more precisely the scope and objectives of the reﬁned ontology. On a technical-based comparison, the reﬁning ontology often has a stronger axiomatization than the extending ontology, which reﬂects a ﬁne-grained versus a coarse-grained category distinction [Kutz et al., 2010]. Extensions and reﬁnements are, for instance, used by the SAO module as it is a reﬁnement of the DnS ontology. Both methods can be used between any ontological modules, however, modules that comply with an abstract space perspective are primarily used as extensions whereas modules with qualitative, quantitative, or domain-speciﬁc spatial perspectives are used as reﬁnements. This is caused by their detail of speciﬁcation about the spatial entities they describe, as illustrated in Figure 1. Matching Matching provides a method to deﬁne so-called alignments between categories or relations from two different ontologies. An alignment deﬁnes a relation between two categories from different ontologies that are semantically equivalent. Several approaches can be used to automatically ﬁnd alignments between ontologies [Euzenat and Shvaiko, 2007]. These approaches use syntactic, structural, and extensional methods for the matching process. As alignments are used to deﬁne equivalence relations between ontology parts (categories or relations), a high number of alignments between ontologies in relation to their overall size (number of categories and relations) indicates a high similarity in terms of their contents. Hence, both ontologies are likely to share the same kind of information of the same domain, and alignments can be applied between spatial ontology modules that share the same spatial perspective. Furthermore, spatial ontology modules can also be aligned with other (non-ontological) representations of space, namely external sources for spatial information. For example, the RBO module is aligned with the hierarchically structured speciﬁcation of the RCC-8 relations.

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Figure 2. Combinations of Spatial Modules. Speciﬁcation of the spatial modules, their perspectives, and combinations. Arrows indicate reﬁnements, dashed lines indicate alignments, wide lines indicate link relations.

Connection A connection between two ontologies provides a method to combine ontology modules with each other, which are rather different and heterogeneous in their speciﬁcation and contents. The connections deﬁne link relations that reﬂect associations or similarities between categories and relations. The ontologies are kept separate and unaltered. A set of link relations can build an interface ontology that reconciles one ontology module with the other [Kutz et al., 2010]. The interface ontology can particularly be used to generate an overall ontology module entailing the two input ontologies and the link relations. In contrast to alignments, link relations do not deﬁne equivalence relations but counterpart relations between functionally related parts. Hence, connections identify link relations particularly between modules from different perspectives. The contents of the connected modules can highly differ. For using module connections not only the contents can be rather disjoint, but also the ontology languages in which the modules are deﬁned can be different. For spatial modules, link relations are relevant to connect, for instance, quantitative and domain-speciﬁc types of spatial information that reﬂect the same reality but from entirely different perspectives. An overall connection of the modules is achieved by E -connecting [Kutz et al., 2004] counterpart categories.

4. Application Example The following application example is an excerpt of an assisted living system that has been presented in [Hois, 2010b]. In summary, an assisted living environment uses spatial

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ontology modules for representing the environmental data. The assisted living system is supposed to provide building automation. We will here primarily focus on the modular representation of the different perspectives and their combinations. The application domain is characterized speciﬁcally for indoor environments in the context of assisted living, which is reﬂected by ontology modules with domain-speciﬁc spatial perspectives. Qualitative spatial information for topological relations is provided by ontology modules with qualitative spatial perspectives, namely the RBO module. Information about change and action in the environment is speciﬁed by using ontology modules with corresponding abstract and domain-speciﬁc spatial perspectives, namely a reﬁnement of the SAO module. Concrete control and monitoring information for the assisted living environment are speciﬁed by ontology modules complying with domainspeciﬁc spatial perspectives on the domain, namely the HAO module. Figure 2 illustrates the combinations of the different spatial ontology modules. An alignment is provided for the RBO module that anchors the region-based spatial relations in the external source RCC-8. The SAO module reﬁnes the DnS ontology by introducing speciﬁc spatial action and activity categories. Both modules are connected to be extended by the HAO module, which speciﬁes certain constraints that should be satisﬁed by the spatial assisted environment. For instance, constraints on the lightning can be formalized with regard to the topological position of users and their activities. The position-related information is deﬁned by speciﬁcations in the RBO module, the activity-related aspects are deﬁned in the SAO module, and the overall restriction on lightning is deﬁned in the HAO module by combining both representations.

5. Summary The four spatial perspectives provide a general distinction for spatial ontology modules. If an ontology is newly developed or reﬁned, it can be developed with regard to a certain perspective. The combination techniques used for individual modules depend on the spatial perspectives: extensions are particularly applicable for abstract perspectives whereas reﬁnements are applicable for qualitative, quantitative, or domain-speciﬁc spatial perspectives; matching ﬁts well the requirements for combining ontology modules with external spatial sources; and connections are applicable for combining ontology modules with incomparable spatial perspectives. In this paper, different ontology modules were presented that comply with the spatial perspective distinction. They have been combined on the basis of their perspectives for supporting an assisted living application. Although the ontological speciﬁcations are applicable for specifying requirements and control elements of the application [Hois, 2010b], an evaluation that analyzes the modular ontologies and their combinations is left for future work. In summary, a perspective determines how the domain is described and which aspects have to be taken into account. Applications can use those ontologies that satisfy their requirements. For example, an application providing HCI may need to apply an ontology complying with a multimodal perspective, or an assisted living application may deﬁne and re-use different ontologies for space from all perspectives.

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References [Bateman et al., 2010] Bateman, J. A., Hois, J., Ross, R., and Tenbrink, T. (2010). A linguistic ontology of space for natural language processing. Artiﬁcial Intelligence, 174(14):1027–1071. [Cohn et al., 1997] Cohn, A. G., Bennett, B., Gooday, J., and Gotts, N. M. (1997). Qualitative spatial representation and reasoning with the region connection calculus. GeoInformatica, 1:275–316. [Davidson, 1967] Davidson, D. H. (1967). The logical form of action sentences. In Rescher, N., editor, The Logic of Decision and Action, pages 81–95. [Euzenat and Shvaiko, 2007] Euzenat, J. and Shvaiko, P. (2007). Ontology Matching. [Gangemi and Mika, 2003] Gangemi, A. and Mika, P. (2003). Understanding the semantic web through descriptions and situations. In Int. Conference on Ontologies, Databases and Applications of Semantics. [Gómez-Pérez et al., 2004] Gómez-Pérez, A., Fernández-López, M., and Corcho, O., editors (2004). Ontological Engineering – with examples from the areas of Knowledge Management, e-Commerce and the Semantic Web. [Grütter et al., 2008] Grütter, R., Scharrenbach, T., and Bauer-Messmer, B. (2008). Improving an RCCderived geospatial approximation by OWL axioms. In Sheth, A. P., Staab, S., Dean, M., Paolucci, M., Maynard, D., Finin, T. W., and Thirunarayan, K., editors, 7th Int. Semantic Web Conference, pages 293– 306. [Hois, 2010a] Hois, J. (2010a). Formalizing diverse spatial information with modular ontologies. In Rapp, D. N., editor, Spatial Cognition 2010: Poster Presentations, pages 41–44. [Hois, 2010b] Hois, J. (2010b). Modularizing spatial ontologies for assisted living systems. In Bi, Y. and Williams, M.-A., editors, 4th Int. Conference on Knowledge Science, Engineering & Management, pages 424–435. [Kovacs et al., 2007] Kovacs, K., Dolbear, C., and Goodwin, J. (2007). Spatial concepts and OWL issues in a topographic ontology framework. In Geographical Information Systems Conference. [Kruijff et al., 2007] Kruijff, G.-J. M., Zender, H., Jensfelt, P., and Christensen, H. I. (2007). Situated dialogue and spatial organization: What, where. . .and why? Int. Journal of Advanced Robotic Systems, 4(1):125–138. [Kutz et al., 2004] Kutz, O., Lutz, C., Wolter, F., and Zakharyaschev, M. (2004). E-Connections of Abstract Description Systems. Artiﬁcial Intelligence, 156(1):1–73. [Kutz et al., 2010] Kutz, O., Mossakowski, T., and Lücke, D. (2010). Carnap, Goguen, and the Hyperontologies: Logical Pluralism and Heterogeneous Structuring in Ontology Design. Logica Universalis, 4:255– 333. [Munn and Smith, 2008] Munn, K. and Smith, B., editors (2008). Applied Ontology, volume 9 of Metaphysical Research. [Palma et al., 2009] Palma, R., Hartmann, J., and Haase, P. (2009). OMV - ontology metadata vocabulary for the semanticweb. Technical Report v2.4.1, OMV Consortium. [Parent and Spaccapietra, 2009] Parent, C. and Spaccapietra, S. (2009). An overview of modularity. In Stuckenschmidt, H., Parent, C., and Spaccapietra, S., editors, Modular Ontologies – Concepts, Theories and Techniques for Knowledge Modularization, pages 5–23. [Pike and Gahegan, 2007] Pike, W. and Gahegan, M. (2007). Beyond ontologies: Toward situated representations of scientiﬁc knowledge. Int. Journal of Man-Machine Studies, 65(7):674–688. [Rector, 2003] Rector, A. L. (2003). Modularisation of domain ontologies implemented in description logics and related formalisms including OWL. In 2nd Int. Conference on Knowledge Capture, pages 121–128. [Ross, 2008] Ross, R. (2008). Tiered models of spatial language interpretation. In Freksa, C., Newcombe, N., Gärdenfors, P., and Wölﬂ, S., editors, Spatial Cognition VI. Learning, Reasoning, and Talking about Space, pages 233–249. [Schill et al., 2009] Schill, K., Zetzsche, C., and Hois, J. (2009). A belief-based architecture for scene analysis: from sensorimotor features to knowledge and ontology. Fuzzy Sets and Systems, 160(10):1507–1516. [Stuckenschmidt and Klein, 2003] Stuckenschmidt, H. and Klein, M. C. A. (2003). Integrity and change in modular ontologies. In 18th Int. Joint Conference on Artiﬁcial intelligence, pages 900–908.

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Simple Reasoning for Contextualized RDF Knowledge1 a

Mathew Joseph a,b and Luciano Seraﬁni b FBK-IRST, Via Sommarive 18, 38050 Trento, Italy b DISI, University of Trento, Italy

Abstract. The standard semantic web languages and reasoning tools do not explicitly take into account the contextual dimension of knowledge, i.e., the fact that a certain statement (RDF triple) is not universally true, but true only in certain circumstances. Such contextual information, which includes for instance, the time interval, the spatial region, or the sub-domain in which a certain statement holds, are of foremost importance to determine correct answers for a user query. Rather than proposing a new standard, in this work, we introduce a framework for contextual knowledge representation and reasoning based on current RDF(S) standards, and we provide a sound and complete set of forward inference rules that support reasoning in and across contexts. The approach proposed in this paper has been implemented in a prototype, which will be brieﬂy described.

1. Introduction Although, RDF Knowledge that is available on the web from multiple distributed sources and servers, pertains to multiple heterogeneous domains, topics and time frames, the current search engines or RDF stores do not take into account this “contextual information” of the RDF data. Moreover, these systems answer the user queries forming a union/merge of all the knowledge they have. This approach ﬁrst of all leads to “inefﬁciency” because of having to search the whole knowledge store for every query. Another more fatal consequence that these systems ignore is the “inconsistencies” that result from the merge of knowledge that pertain to different/inconsistent domains. Where as, a Contextualized approach to knowledge representation brings modularity to the approach by segregating the relevant sources of knowledge for a query from irrelevant ones by the knowledge of contextual information of the query. Inconsistent sources of knowledge can be separated among different contexts and hence be reasoned separately. One of the signiﬁcant approaches that shed light in the direction of context based modeling is that of the one in [1]. The authors provide a context based semantics and a framework based on OWL2 and show by a sample use case, how a domain such as football world cup, can be modeled using some of the constructs like context classes, qualiﬁed predicates etc. Although OWL2 has become popular recently, some of the hin1 This work has been partially supported by the LiveMemories project (Active Digital Memories of Collective

Life). Many thanks to Andrei Tamilin for all the support recieved in implementation of the CKR system and Martin Homola, Francesco Corcoglionitti for all valuable suggestions.

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drances for a system based on OWL to be implementable are (i) decidability of conjunctive query answering is unknown [2] (ii) efﬁcient forward chained repositories cannot be built because of a proliferation of inferred statements. The proliferation effect in the latter point is caused due to its “if and only if” semantics, RDFS is the most popular semantic web language and it is based on the paradigm called “if” semantics [3]. Also, most of the knowledge currently available in the semantic web is in much weaker languages such as RDF, RDF(S) and OWL-Horst, which support efﬁcient forward chaining reasoning and query answering. The applicability of the approach proposed in [1] should therefore pass through its adaptation to weaker languages. In this paper, we following the basic principles in [1], adapt their framework based on OWL to RDFS, provide an RDFS based semantics suitable for contexts. Moreover, we provide sound/complete set of forward inference rules for computing logical consequence and provide a proof of completeness for the same. In developing our approach, we take into account its concrete implementability in state of the art RDF(S) knowledge stores, where contexts can be represented by named graphs, and contextual information as statements about graph names. Furthermore, contexts can be organized in broader/narrower hierarchical relation which supports the propagation of information across different contexts. The main impact of this work is that, using our framework one could create a semantic repository in which RDF statements can be contextualized with attributes such as time, topic, and location, and (s)he can query both on a single context or on a set of contexts. One could also state rules that propagate statements around different contexts. To check the feasibility of our theory we have implemented a prototype which will be shortly described in the ﬁnal section. With respect of previous works, the novel contribution of this paper is on the axiomatization of Contextualized knowledge repository on the bases of RDF(S) semantics and using the limited language of RDF, and the proof of soundness and completeness of such an axiomatization. 2. RDF(S) Background In this section we brieﬂy introduce RDF(S) constructs, but we assume familiarity with RDFS concepts and semantics as described in [4,5]. An RDF vocabulary V is composed of three mutually disjoint sets U, B, L representing respectively URI references, blank nodes, and literals. An RDF triple is any tuple (s, p, o) ∈ (U ∪ B) × (U) × (U ∪ B ∪ L). Any element of the set U ∪ L is called a name. Deﬁnition 1 (RDF graph) An RDF graph is a set of RDF triples. The universe, U (G), of a graph G is the set {s, p, o|(s, p, o) ∈ G}. The names, name(G), of a graph G is the set U (G)\ B. If S = {Gi }i∈I is a family of rdf graphs on the set of indices I, we deﬁne U (S) = i∈I U (Gi ) and name(S) = i∈I name(Gi ). A ground graph is an rdf graph without any blank node occurrences, i.e., a graph G such that U (G) = name(G). Deﬁnition 2 (Simple interpretation) A simple interpretation, of the vocabulary V = (U, B, L), is a tuple I = IR, IP, IC, IEXT, ICEXT, LV, IS where (1) IR is a nonempty set of objects, called the domain or universe of I; (2) IP ⊆ IR, is a set of objects denoting properties; (3) IC ⊆ IR, is a distinguished subset of IR

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denoting classes (4) IEXT : IP → 2IR×IR , a mapping that assigns an extension to each property name; (5) ICEXT : IC → 2IR , a mapping that assigns a set of objects to every object denoting a class; (6) LV ⊆ IR, the set of literal values, LV contains all plain literals; (7) IS : U ∪ L → IR ∪ IP, the interpretation mapping, a mapping that assigns an object in IR to each element in U ∪ L, such that IS is the identity function for plain literals and only assigns objects in LV to lexically valid literals in L. The class of RDF (resp. RDFS) interpretations is a subclass of the class of simple interpretations that satisfy additional constraints on the interpretation of the RDF (resp. RDFS) primitives. For instance an RDF interpretation should satisfy the condition: x ∈ IP

⇐⇒

x, IS(rdf:Property) ∈ IEXT(IP(rdf:type))

For lack of space, we don’t list all the conditions associated to RDF (resp. RDFS) interpretations and we refer the reader to [5]. Entailment between RDF(S) graphs can be deﬁned in the usual model theoretic way Deﬁnition 3 A simple (resp. RDF and RDFS) interpretation is a model of a graph G, in symbols I |=simple G (resp. I |=rdf G I |=rdfs G) if there is an assignment A to the blank nodes of G, such that for every triple (s, p, o) ∈ G, we have that IS+A(s), IS+A(o) ∈ IEXT(IS(p)), where IS + A(x) returns IS(x) if x ∈ U and A(x) if x ∈ B. Deﬁnition 4 A graph G simply (resp. RDF, RDFS) entails a graph H, in symbols G |=simple H (resp, G |=rdf H, G |=rdfs H) if for every simple (resp, RDF, RDFS) interpretation I that satisﬁes G, I also satisﬁes H. The provability relation is denoted by rdf s , rdf , simple . An RDFS class is denoted by symbols C, D etc; a property by R, S. X can be either a property or a class; individuals (denoted by symbols a, b etc.) are symbols that are instances of properties and classes. 3. Contextualized Knowledge Bases: A Formal Model In a contextualized knowledge base (CKB), each statement is qualiﬁed with a set of contextual parameters. Statements qualiﬁed with the same parameters are grouped as a unique graph which constitute a context. We call such contextual parameters as dimensions. Deﬁnition 5 (Contextual dimension) A contextual dimension Di , is a ground RDFS Graph deﬁned over a set of names V alSet(Di ) ⊆ U (Di ) and U (Di ) contains a property ≺i called cover, deﬁned over V alSet(Di ). Intuitively the relation cover provides a structure among the values of V alSet(Di ). An example of a contextual dimension is the Location dimension, whose Dlocation contains values of geographic locations, for instance milan, rome, italy etc. The relation ≺Location represent the coverage between geographical regions, for instance milan ≺Location italy, rome ≺Location italy. Other examples of dimensions are Time, Topic whose ≺time represents temporal interval containment (e.g., jan-2011 ≺Time 2011) and the relation ≺topic represents broader/narrower relation between domains

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(e.g., biology ≺Topic science). In the literature of the semantic web there are approaches that propose examples of dimension to qualify statements to encode for instance information about provenance, ownership and trust [6], or about time [7] and space [8]. In the literature of knowledge representation [9] proposes a set of 12 dimensions to model context dependent knowledge. The formal framework developed is general and admits an arbitrary, but ﬁxed number n ≥ 0 of contextual dimensions. A zero dimensional CKB can be thought of as a single RDFS knowledge repository without contextual information. Deﬁnition 6 (Context) A context, deﬁned over the n dimensions {Di }1≤i≤n , is a triple C, d(C), Graph(C), where 1. C is the context identiﬁer; 2. d(C) = d1 , . . . , dn where each di ∈ V alSet(Di ), i = 1, ..., n; 3. Graph(C) is an RDF graph Example 1 The following are two sample contexts in the 2-dimensional CKB in the dimension of Time and Topic. 2010, uefa_champions_legue Winner(inter) Defeats(inter, bayern_munich) Best-player(diego_milito) Happy(maurinho)

2010, french_open_tennis Winner(rafael_nadal) Defeats(rafael_nadal, robin_soderling) ... ...

It can be noted from the ﬁgure that Time and Topic are the two dimensions. For the ﬁrst context, the values taken by the context for these dimensions are respectively 2010 and uefa_champions_legue, where as in the second context values taken by these are respectively 2010 and french_open_tennis. Now consider the statement Winner(inter) which is true in the ﬁrst context. We can denote this fact by 2010, uefa_champions_legue : Winner(inter), the same statement is false in a context about 2010, french_open_tennis where as the statement Winner(nadal) is true. We occasionally use the symbol C to refer to a whole context whose context identiﬁer is C. An important component of a CKB is a knowledge base called meta knowledge where knowledge about dimensions and how their values are related to context identiﬁers are deﬁned. We deﬁne dimension space, D, as the set {V alSet(D1 ) × ... × V alSet(Dn )} Deﬁnition 7 (Meta Knowledge) A Meta Knowledge M, is an RDF graph composed of: 1. n rdf graphs {Di }1≤i≤n : the context dimensions; 2. a set C of URIs: the context identiﬁers; 3. for each C ∈ C, and each 1 ≤ i ≤ n, the assertion Di -Of(C, di ), for some di ∈ V alSet(Di ). Intuitively, Di -Of(.) is a function to associate a context C to a value in Di . For any two contexts C1 and C2 , we use the notation C1 ≺ C2 if Di -Of(C1 ) ≺i Di -Of(C2 ) for i = 1, ..., n. We also assume in the above deﬁnition that V alSet(Di )s are pairwise disjoint. One of the most important feature of contexts is that the same symbol may have different extensions in different contexts. For example, the class Winner in the context of 2010 UEFA champions legue denotes a different set w.r.t., the literally equal class in

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the context of 2010 French open tennis competition. This assumption however requires to have the possibility to distinguish the two classes when for instance one wants to integrate information contained in two or more contexts in a broader context. For this reason we introduce the mechanism of qualiﬁcation of a symbol w.r.t., a (set of) contextual dimensions. For instance, in a broader context about sport, the qualiﬁed symbols Winner2010,uefa_champions_legue and Winner2010,french_open_tennis are used to denote Winner in the two different contexts described above. Apart from resolving inconsistencies during data integration, qualiﬁcation can also be used in circumstances like: External Referencing To talk about classes and properties in a context from another context. For instance, one could use statements like 2010, sports : Winner2010,uefa_champions_legue (inter) to state that the fact Winner(inter) in the context of 2010, uefa_champions_legue from a context of 2010, sports. Lifting We can use this construct to deﬁne data transfer rules across contexts, for instance consider the two rules below stated in the context below: 2010, sports :

Winner2010,uefa_champions_legue rdfs:subClassOf Winner Winner2010,french_open_tennis rdfs:subClassOf Winner

Such rules can effectuate the movement of instances from the source class Winner, respectively from source contexts, 2010, uefa_champions_legue and 2010, french_open_tennis to destination class, Winner in destination context, 2010, sports A qualiﬁed class (property) in syntax is of the form Cd (Rd ), where d = d1 , . . . , dn is an n-dimensional vector representing a context. We call C (R), the base class (base property) of the qualiﬁed class Cd (qualiﬁed property Rd ). A description about how qualiﬁed class and properties are encoded using URIs is given in the section 6. We now formally deﬁne a system of multiple contexts which we call a contextualized knowledge repository. Deﬁnition 8 (Contextualized Knowledge Repository) A Contextualized Knowledge Repository (CKR) is a pair K = M, G where 1. M is a meta knowledge on the set of contexts C 2. G = {GC }C∈C is a family of rdfs graphs, one for each C ∈ C Before we deﬁne the model of a CKR, we state some of our assumptions and notations below. We put a restriction on our system that for any two distinct contexts C1 , C2 , d(C1 ) = d(C2 ). As a result of this contexts are uniquely determined by their dimension vectors and since each context has exactly one associated RDF graph (possibly empty). Hence from now on, for ease of notation and whenever there is no ambiguity, we use symbols d, f , g, h to also denote the corresponding graphs in G. Readers should note that whenever these symbols are used in association with ≺ operator (for instance d ≺ g) or when these are used as sufﬁxes of qualiﬁed symbols (for instance Cd ) they should be taken as dimension vectors. Deﬁnition 9 (Model of a CKR) model of a CKR K = M, G is a pair IM , IG where

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• IM is an rdfs-interpretation for a meta knowledge space M, with the additional semantic conditions IM |=rdfs M. • IG = {Ig }g∈D is a family of RDFS Interpretations, Ig = IRg , IPg , ICg , IEXTg , ICEXTg , LVg , ISg when the following conditions are satisﬁed: 1. IRg ⊆ IRh if g ≺ h; 2. ISg (a) = ISh (a) if ISh (a) ∈ IRg , g ≺ h, for every a ∈ ΣU ; 3. ISg (a) = ISh (a) if ISg (a) ∈ IRh , g ≺ h, for every a ∈ ΣU ; 4. ICEXTg (ISg (C)) = ICEXTg (ISg (Cg )); 5. ICEXTg (ISg (R)) = ICEXTg (ISg (Rg )); 6. ICEXTh (ISh (Cg )) = ICEXTg (ISg (Cg )) if g ≺ h; 7. IEXTh (ISh (Rg )) = IEXTg (ISg (Rg )) if g ≺ h; 8. ICEXTg (ISg (Cd )) = ICEXTh (ISh (Cd )) ∩ IRg , if g ≺ h; 9. IEXTg (ISg (Rd )) = IEXTh (ISh (Rd )) ∩ IR2g , if g ≺ h; 10. For every g ∈ D, Ig |=rdfs Gg Let’s analyze IG of the deﬁnition 9. Condition 1 imposes that domains increase when the contexts get broader. Condition 2 and 3 imposes that URIs and literals referred in different contexts denotes the same resource, this assumption has been made mainly in motivation with the initiatives regarding standardization of use of URIs. Also in classical rdfs entailment between any two graphs, URIs and Literals in these graphs are treated as denoting the same resources. Conditions 4,5 constraints the denotation of a class (property) in a context to be equivalent to the qualiﬁed class (property) obtained by making explicit the dimensional values of the context in which it appears. Conditions 6,7 imposes that the extension of a qualiﬁed class (property) qualiﬁed with the parameters of the context in which it appears (say d) remain the same in all the broader contexts (i.e., with dimensional values equal to e ! d). Conditions 8,9 impose that the interpretation of a qualiﬁed predicate in a narrower context (i.e., with dimensional values d ≺ e) is obtained by restricting the interpretation of the same predicate in the broader context e. Condition 10 enforces that every local model of a CKR interpretation satisﬁes that relative context. 4. Reasoning in CKR The objective of this section is to deﬁne a set of inference rules (IR), along with RDF, RDFS inference rules that are sound and complete w.r.t. the semantics described above. The initial objective of this axiomatization is to provide a naive forward decision procedure for for query answering for the CKR system. Reasoning in CKR is performed at two distinct layers, at the meta level and at the object level. Meta level reasoning is performed in the metaknowledge and has the main objective of inferring coverage relation between contexts. Object level reasoning is performed within and across the set of contexts and has the main objective of inferring knowledge within a context and propagating knowledge across contexts connected via coverage relation. In a CKR model IM , IG , the interpretation of the meta-knowledge IM is a standard RDF(S) interpretation with the additional semantic conditions imposing functionality on the property Di -Of. An RDFS semantic extension and sound/complete axiomatization with such a construct is already available thanks to OWL-Horst [3]. For lack of

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space we don’t report these set of rules. For object knowledge, there is principally two kinds of reasoning local entailment and cross-context entailment. The former happens by virtue of the fact that Ig , g ∈ D in deﬁnition 9 is also an RDFS interpretation and hence, we suppose to have the following “macro rule” for every rdf graph G. g : (s1 , p1 , o1 ), . . . , g : (sn , pn , on ) and (s1 , p1 , o1 ), . . . , (sn , pn , on ) g : (s, p, o)

rdfs

(s, p, o)

LR

As an implication of local reasoning, our axiomatization must be an extension of the set of RDFS inference rules. The inference rules for Blank nodes, RDF and RDFS, given in [4], are hence assumed to be included in the axiomatization in addition to the inference rules which capture the additional conditions on RDFS interpretation given in the deﬁnition of CKR model. Cross-context entailment allows to deduce conclusions in one of the contexts based on the evidence from other contexts. This set of inference rules for this type of entailment has different context symbols in premises and conclusion (for example, see table 2). In the following, p, q denotes URIs for properties; u, v denotes a URIs or a blank node, i.e any subject of an rdf triple; a, b denote URIs or Literals, and x, y denote URIs, blank nodes or Literals, i.e they can any possible objects. l stands for literals. _ : k, _ : l, _ : m, _ : n are blank node identiﬁers. alloc(x,_:b) , as in standard RDFS inference rules, is used with inference rules that introduce new blank nodes for existing names or blank nodes. Intuitively it means that the blank node _ : b must have been created by an earlier rule application on the blank node for symbol x, or if there is no such blank node, _:b is a “fresh” node which does not occur in the graph. Also the relation allocated_to between blank nodes follow transitivity across contexts i.e, if a blank node _:l is allocated in a context d for another blank node _:m in context e, which it self was allocated for another blank node _:n in context f , then _:l is allocated for _:n. Table 1 reports the set of rules to infer that a qualiﬁed class (property) have the same extension as a base class (property) in their home context. These set of inference rules make sure that condition 4 and 5 of deﬁnition 9 is satisﬁed. The next sets of rule Table 1. Local completion rules 2a: 2b: 3a: 3b:

g g g g

: C rdf:type rdfs:Class : Cg rdf:type rdfs:Class : R rdf:type rdf:Property : Rg rdf:type rdf:Property

⇒ ⇒ ⇒ ⇒

g g g g

: Cg rdfs:subClassOf C : C rdfs:subClassOf Cg : Rg rdfs:subPropertyOf R : R rdfs:subPropertyOf Rg

formalize how instances of various classes and properties and their qualiﬁed counterparts move around between various contexts of the system. For this we introduce the notation presentIn(g, a) that holds when the name a, occurs in a triple (s, p, o) that can be derived in context g, i.e g : (s, p, o) and s = a or p = a or o = a. Conventionally for literals we assume that presentIn(g, l) is always true. Domain expansion The rules reported in Table 2 formalize the fact that if a qualiﬁed triple, i.e triple of the form (a rdf:type Cd ) or (a Rd b) holds in a narrower context it should hold also in

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the broader context. This is by virtue of the fact that domain of contexts expand as one moves to broader contexts. A special remark is necessary for blank nodes. Since in RDF Table 2. Domain expansion rules 4a: 4b: 5: 6a: 6b: 7:

g : (a rdf:type Cd ) & M : g ≺ h g : (a Rd b) & M : g ≺ h

⇒

h : (a rdf:type Cd )

⇒

g : (_ : m rdf:type Cd ) & M : g ≺ h

⇒

h : (a Rd b) h : (_ : n rdf:type Cd )

g : (_ : m Rd b) & M : g ≺ h g : (a Rd _ : m) & M : g ≺ h

⇒ ⇒

g : (_ : m Rd _ : n) & M : g ≺ h

⇒

alloc(_ : m, _ : n) h : (_ : n Rd b) (alloc(_ : m, _ : n)) h : (a Rd _ : n) (alloc(_ : m, _ : n))

h : (_ : k Rd _ : l) alloc(_ : m, _ : k), alloc(_ : n, _ : l)

blank nodes are treated like existential variables, and therefore, they cannot be shared across contexts. This implies that when we shift up a triple with a blank node, we have to allocate a new fresh node. Example 2 An illustration of a derivation using the above mentioned inference rules: (i) g : (a rdf : type C), M : g ≺ h h : (a rdf : type Cg ) g : (a rdf : type C) IR 2b, rdfs 9 g : (a rdf : type Cg ) h : (a rdf : type Cg ) (ii) g : (a rdf : type Ch ), M : g ≺ h

M:g≺h

IR 4a

h : (a rdf : type C)

M:g≺h g : (a rdf : type Ch ) IR 4a h : (a rdf : type Ch ) IR 2a, rdfs 9 h : (a rdf : type C) Domain contraction Table 3 reports the set of rules that allow to propagate triple from broader contexts into narrower context whenever the property of the triple is qualiﬁed with dimensions smaller than or equal to the dimensions of the narrower context, i.e to derive d : Cd (a) from e : Cd (a) given d ≺ e. This kind of reasoning that happens as a consequence of semantic condition 6 and 7 of the deﬁnition 9. Also in this case a remark on blank nodes is necessary. The presence of a blank node in a broader context might not correspond to a resource in the narrower context, as there could be resources in a larger domain that are not present in smaller domains. Hence the conditional execution of 10, 13a, 13b and 14. They only operate on blank nodes that are allocated in the larger domain corresponding to blank nodes in smaller domain and hence sure of their presence in smaller domain.

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g : (a rdf:type Cd )

⇒

g : (a Rd b)

h : (a rdf:type Cg ) & M : g ≺ h h : (a Rg b) & M : g ≺ h

⇒ ⇒

g : (a rdf:type C) g : (a R b)

h : (_ : m rdf:type Cg ) & M : g ≺ h h : (_ : m rdf:type Cd ) & M : g ≺ h

⇒

9: 10: 11a: 11b: 12:

h : (a Rd b) & M : g ≺ h presentIn(g, a), presentIn(g, b)

alloc(_ : n, _ : m) h : (_ : m Rg b) & M : g ≺ h h : (a Rg _ : m) & M : g ≺ h h : (_ : m Rg _ : n) & M : g ≺ h

13a: 13b: 14:

h : (a rdf : type Cd ) & M : g ≺ h presentIn(g, a)

8b: 9a: 9b:

Table 3. Domain contraction rules ⇒

8a:

87

⇒

g : (_ : n rdf:type C) alloc(_ : m, _ : n)

g : (_ : n rdf:type Cd )

⇒

g : (_ : n R b) (alloc(_ : m, _ : n))

⇒

g : (a R _ : n) (alloc(_ : m, _ : n)) ⎧ ⎪ g : (_ : k R _ : l) ⎨ alloc(_ : m, _ : k) ⎪ ⎩ alloc(_ : n, _ : l)

⇒

h : (_ : m Rd b) & M : g ≺ h alloc(_ : n, _ : m), presentIn(g, b)

⇒

g : (_ : n Rd b)

h : (a Rd _ : m) & M : g ≺ h alloc(_ : n, _ : m), presentIn(g, a)

⇒

g : (a Rd _ : n)

⇒

g : (_ : k Rd _ : l)

h : (_ : m Rd _ : n) & M : g ≺ h alloc(_ : k, _ : m), alloc(_ : l, _ : n)

5. Completeness of CKR entailment In this section, we will prove soundness and completeness of the inference rules w.r.t., consequence relation in CKR. We say that a triple (s, p, o) logically follows from K in the context g, in symbols K |= g : (s, p, o) if, for all models (IM , IG ) of K, Ig |=rdfs (s, p, o). The completeness proof, relies on the construction of an interpretation called herbrand interpretation as is done in the RDFS completeness proof in [4]. It is an interpretation constructed from the lexical items in G. The strategy, we will adopt for the completeness proof, is similar to that of [4]. In brief the steps are as follows • form the closure(described further) of the CKR K w.r.t to the inference rules. • construct a herbrand interpretation from this closure in which there is a reversible mapping from it’s vocabulary to the elements in it’s domain. • prove that the herbrand interpretation is a model for the CKR K Before proving completeness, we need to introduce some technicalities. In RDF inferencing, the following two inference rules are used for graph entailement. We also need them for completeness proof. lg : s p o ⇒ s p _:n(alloc(o,_:n)) gl : s p _:n(alloc(o,_:n)) ⇒ s p o

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which allocate a blank node for every object element of a triple. We also suppose, w.l.o.g that, the set of blank nodes of the graphs of a CKR are pairwise disjoint. On the basis of the above rule, we deﬁne the mapping sur, which maps every URI and blank node into itself, and every literal l of U (G) into the blank node allocated via the rule “lg”. Finally we deﬁne the operator, closure of a graph. Given a CKR K, the closure operator in symbols, D(K), is a set of labeled triples g : (s, p, o) deﬁned as follows: 1. if (s, p, o) ∈ Gg then g : (s, p, o) ∈ D(K); 2. If (s, p, o) is an RDF(S) axiomatic triple, then g : (s, p, o) ∈ (K) for any g; 3. D(K) is closed under LR, the rules of table 1, 2 and 3 and the “lg” and “gl” rules. The set cl(K, g) is deﬁned as {(s, p.o) | g : (s, p.o) ∈ D(K)}. The herbrand interpretation IGH , for the object knowledge base, G of a CKR, is given by IGH = {IgH }g∈G where H H H H H H each IgH = IRH g , IPg , ICg , IEXTg , ICEXTg , LVg , ISg for the graph g is given by the following construction. For the construction of herbrand interpretation and for the completeness proof, the approach is an adaptation of the one in [4]. LVH g = {l|cl(K, g) contains sur(l) rdf:type rdfs:Literal} H IRg = U (cl(K, g)) IPH g = {p | (sur(p), rdf:type, rdf:Property) ∈ cl(K, g)} ICgH = {s | (sur(s), rdf:type, rdfs:Class) ∈ cl(K, g)} H for every p ∈ IPH g , IEXTg (p) = {(s, o) | (sur(s), p, sur(o)) ∈ cl(K, g)} H for every o ∈ ICg , ICEXTH g (o) = {s|(sur(s), rdf:type, sur(o)) ∈ cl(K, g)} xml(x), if x is a well typed XML literal in cl(K, g), • ISH g (x) = x otherwise where xml(.) is the xml datatype function, that maps valid xml literals to xml value space.

• • • • • •

Let B be a function blank node mapping such that if x is blank node allocated to an xml literal l then B(x) = xml(l); if it is allocated to a plain literal l, then B(x) = l otherwise B(x) = x. Lemma 1 If CKR does not have an XML clash2 , then IGH |= G. Proof By hypothesis, for every g ∈ G does not have an XML Clash. Since each IgH is closed with respect to RDF, RDFS inference rules, IgH |=rdf s g [4]. Now in order to prove that IGH |= G, we need to prove that IGH = {IgH }g∈G satisﬁes each of the semantic conditions of IG in deﬁnition 9. Below we enumerate each condition and prove that they are actually satisﬁed 1. Suppose if g ≺ h and if IRg ⊆ IRh , this means that there exists a element v ∈ IRg and v ∈ IRh . This means that there exists a triple (s, p, o) ∈ cl(g) where s = v or p = v or o = v and (s, p, o) ∈ cl(h). But this cannot be the case by the virtue of execution of inference rules (4,5,6,7) to completion as these rules make sure that there is a triple (s, p, o) in cl(h) for every (s, p, o) in cl(g). 2 a graph has an XML clash, when it contains the statement, l rdf:type rdf:XMLLiteral, and l is a lexically invalid XML literal

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2. Since every v ∈ U (cl(g)) is interpreted to it self by the ISH g except the valid XML literals that are interpreted to the XML value space, this condition is trivially satisﬁed. 3. Similar to the above 4. Taken care by inference rules 2a,2b 5. Taken care by inference rules 3a,3b 6. In order to prove the condition ICEXTg (ISg (Cg )) = ICEXTh (ISh (Cg )) if g ≺ h. We could prove this by showing that (i) ICEXTg (ISg (Cg )) ⊆ ICEXTh (ISh (Cg )) and (ii) ICEXTh (ISh (Cg )) ⊆ ICEXTg (ISg (Cg )) holds case (i) Suppose an element v ∈ ICEXTg (ISg (Cg )) then there exists a triple (sur(v) rdf:type sur(Cg )) in cl(g). Suppose sur(v) and sur(Cg ) are URIs then by virtue of inference rule 4a, we have a triple (sur(v) rdf:type sur(Cg )) in cl(h). Suppose if they are blank nodes (or one of them) then by rules 5,6a,6b,7 then there exists a triple (sur(v) rdf:type sur(Cg )) in cl(h) case (ii) Suppose an element v ∈ ICEXTh (ISh (Cg )) then there exists a triple (sur(v) rdf:type sur(Cg )) in cl(h). Suppose sur(v) and sur(Cg ) are URIs then by virtue of inference rule 8a, we have a triple (sur(v) rdf:type sur(Cg )) in cl(g). Suppose if they are blank nodes (or one of them) then by rules 9,11a,11b then there exists a triple (sur(v) rdf:type sur(Cg )) in cl(g) 7. Similar to the above 8. In order to prove the condition ICEXTg (ISg (Cd )) = ICEXTh (ISh (Cd )) ∩ IRg if g ≺ h. We prove this by showing that both (i) ICEXTg (ISg (Cd )) ⊆ ICEXTh (ISh (Cd ))∩IRg and (ii)ICEXTh (ISh (Cd ))∩IRg ⊆ ICEXTg (ISg (Cd )) holds case(i) Suppose an element v ∈ ICEXTg (ISg (Cd )) then there exists a triple (sur(v) rdf:type sur(Cd )) in cl(g). Suppose sur(v) and sur(Cg ) are URIS then by virtue of inference rule 4a, we have a triple (sur(v) rdf:type sur(Cg )) in cl(h). Suppose if they are blank nodes (or one of them) then by rules 5,6a,6b,7 then there exists a triple (sur(v) rdf:type sur(Cg )) in cl(h) case(ii) We prove this by contradiction Suppose if v ∈ ICEXTg (ISg (Cd )) and if there exists a element v ∈ IRg and v ∈ IEXTh (ISh (Cd )) then there exists a triple (sur(v) rdf:type sur(Cd )) in cl(h). suppose sur(v) and sur(Cd )) are uris then there exists a triple (sur(v) rdf:type sur(Cd )) by inference rules 8a. Suppose if they are blank nodes (or one of them) then by rule 10,13a,13b,14 there exists a triple (sur(v) rdf:type sur(Cg )) in cl(g) since v ∈ IRg (i.e sur(v) is allocated in cl(h) for sur(v) in cl(g). Hence this implies that v ∈ IEXTg (ISg (Cd )) (contradiction) 9. Similar to the above 10. By hypothesis we have that g does not have an XML clash for every g ∈ G, each IgH is closed with respect to RDF, RDFS inference rules and is an rdfs model for g[4]. Theorem 1 (completeness) Suppose if CKR does not have an XML clash and CKR |= g : (s, p, o) then CKR g : (σ(s), p, σ(o)) where σ is a renaming function for blank nodes, for every g ∈ G in CKR.

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Figure 1. Implementation architecture.

Proof suppose CKR |= g : (s, p, o) then this means that IgH |= (s, p, o). This means H that there exists a mapping A from blank nodes to IRH g such that Ig + A |= (s, p, o). H H This implies that Ig + A |=rdf s (s, p, o). This implies that Ig + A |=rdf (s, p, o). This implies that IgH + A |=simple (s, p, o) (Since semantic conditions of simple interpretations are contained in RDF interpretations which themselves are contained in RDFS interpretations). Let us now extend A such that A(l) = xml(l) for any valid xml literal l else A(x) = x for any other x that is not a blank node or xml literal. now IgH + B |= (A(s), p, A(o)). where B is the identity mapping for blank nodes. This implies that there exists a triple (sur(A(s)) p sur(A(o)) in cl(K, g) which is obtained only by the set of inference rules for rdf, rdfs, lg, gl and ckr inference rules.

6. Implementation The formal framework described previously is implemented as an extension of Sesame RDF store with OWLIM plugin3 In the system, contexts are represented as named graph (NG). The NG identiﬁer is used for attaching dimension values to the context that corresponds to dimension-value pairs outside the box. The actual data in the graph pertains to the knowledge about the domain corresponding to what is inside the box. The standard approach of taking the union of all the graphs in the repository and interpreting as a single RDF interpretation is not applicable since data in different contexts can have inconsistencies mentioned before. Hence, we store and reason data in each context separately from the data in other contexts. The overall architecture of the contextualized knowledge repository is graphically depicted in Fig. 1. The repository is principally divided into two components: Meta Knowledge Base Meta-knowledge base, as is a unique graph that contains all the information about dimensions, their values, cover relations between these values and deﬁnition of qualiﬁed classes/properties, it is implemented as a single Sesame/OWLLim Repository. In the current prototype, we limit to the following three main contextual di3 See

http://openrdf.org and http://www.ontotext.com/owlim

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mensions: Time Dimensions values are time intervals of the form (start-time, end-time). Cover relation is the standard containment relation between intervals. There is presently no serialization for this structure, because, the relation between two values can be determined by the start time and end time values of the interval. Location dimension determines the geographical region in which the set of statements in a context is true. This structure is represented using OWL-Horst serialization. Topic dimension determines the subject topic that a context pertains to. Similarly to locations, the current structure of this dimension is also is represented as an OWL-Horst ontology. Qualiﬁed Classes and Properties are speciﬁed to the system by means of the following sample syntax and is part of the meta knowledge space.

Qualiﬁed Classes and Properties are instances of two designated meta classes called qualified:Class and qualified:Property as shown in the above rdf snippet. In this snippet, http://ns#Winner_In_FrenchOpen_2010 is a qualiﬁed class whose base class is http://ns#Winner and intuitively represents an abstract concept of Winner in context of 2010, French_Open_Tennis. Object Knowledge Base Actual RDFS content of each context deﬁned in meta knowledge base is materialized as an individual Sesame/OWLim repository. The repositories are logically organized as hierarchies based on cover relation of their corresponding dimension values. We currently have implemented conjunctive query answering on the whole CKR that is sound and complete with respect to it’s semantics. Moreover, individual contexts can by queried with SPARQL. 7. Related Work One of the approaches that tried to add a single temporal dimension to standard RDFS was the work in [7]. The authors deﬁne the notion of a ‘temporal rdf graph’ in which is triple is augmented of the form (s, p, o) : t, where t is a time point. A temporal graph g1 temporally entails another temporal graph g2 if at every instant t, g1(t) entails g2(t), where g(t) = {(s, p, o)|(s, p, o) : t ∈ g}. The authors provide a deductive system and query language that motivates the usefulness of temporal RDF graphs. although the semantics and inference rules are suitable to time dimension, it cannot be applied to any of the other dimension in the literature of contexts like locations, topics, provenance etc. Annotated RDF [10] is a theoretical framework and a concrete proposal to extend RDF triples with a set of annotations taken from an n-tuple of partially ordered set. Considering partially ordered meta-attributes is what makes the two approaches quite similar. The main difference however is that here we do the meta-tagging at the level of an RDF graph, rather for each single triple. For efﬁciency reasons, graph level annotation is

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preferable w.r.t., triple level annotation in formalizing context, since usually the context is ﬁxed for a group of statements. In [11], the authors, also give a semantics for graphs annotated with values from a lattice structure. The authors give a semantics based on interpretation structure which is popularly used in many-valued logics. The main difference from our work is that the semantics in [11] impose a restriction on dimensions used, so that it is a complete lattice with join(∨), meet(∧) operators with also and ⊥ values, where as, we do not have such restrictions as, we assume the use of existing ontologies of time, topic, locations etc. for dimensions which necessarily are not in lattice structure. Another difference is that, the extension of the top class (rdfs:Resource) is the same across all contexts, where as for us, it’s extension is context-dependent and moreover, for any contexts d and e with d ≺ e, then extension of rdfs:Resource in context d is a subset of rdfs:Resource in e . Also any symbol a is interpreted as the same resource across different contexts in [11], where as for us, it is context dependent. We also have the mechanism of qualiﬁed predicates, which mainly used to refer to other classes and properties in other contexts and can be used to state lifting rules as described in section 3, such constructs are absent in [11]. Jie Bao et.al. [12] provide a more concrete formalization of theory of context by McCarthy. They extend/modify the theory described in [13] with a predicate isin(c, φ) representing the fact that the triple φ is in the context c. This approach is particularly suited for manipulating contexts as ﬁrst class objects. A considerable number of constructs were introduced for combining contexts (like c1 ∧ c1 , c1 ∨ c2 and ¬c) and for relating contexts (like c ⇒ c2 , and c → c2 ). The work, however, does not give an axiomatization of all these operators. the same family of the one proposed here. The main difference between the two are that we provide a set of propagation patterns, a sound and complete axiomatization and an implementation based on Sesame 2 named graphs. The authors in [14] describe a detailed architecture of Yars, a semantic repository and with a search/query engine which stores rdf data in the form of ((s, p, o), c) where (s, p, o) is an RDF triple in context c. The architecture contains modules that includes the crawler, index manager and indexer, query processing and a query distribution module. The indexing module contains an keyword based indexer and a quad index that is distributed over multiple servers that also includes a context identiﬁer as an index key. Although the work motivates the use of context in semantic repository, it does not provide a semantics for various contexts or do not show how contexts have be related to one another or how queries can be forwarded to the right context or how to ﬁnd answers that are distributed across multiple contexts. CKR is a special case of quantiﬁed Multi Context Systems (a.k.a distributed ﬁrst order logic) [15] in which contexts interfere only via coverage relation. In CKR, bridge rules are not listed explicitly as in MCS, but they are "automatically" determined by the coverage relation formalized in the metaknowledege. The bridge rules "generated" by the coverage statements are described in the section about inference rules (section 4). 8. Conclusion We have presented a framework for context based knowledge representation that is based on RDFS. We introduce a mechanism called qualiﬁcation that is useful in a data integration scenario. We provide a semantics that we found intuitive for contexts. We pro-

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vide a set of inference rules that can be used to construct a forward chained semantic repository and provide completeness proof for the inference rules. We further describe the architecture of our implemented system that is built by extending the Sesame RDF store.

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M.Homola, L.Seraﬁni, and A.Tamilin, “Modeling contextualized knowledge,” in Procs. of the 2nd Workshop on Context, Information and Ontologies (V. Ermolayev, J. M. Gomez-Perez, P. Haase, and P. Warren, eds.), vol. 626 of CEUR-WS, 2010. P. Hitzler, M. Krötzsch, B. Parsia, P. F. Patel-Schneider, and S. Rudolph, “OWL 2 Web Ontology Language Primer,” W3C Recommendation, World Wide Web Consortium, October 2009. H. J. ter Horst, “Completeness, decidability and complexity of entailment for rdf schema and a semantic extension involving the owl vocabulary,” Web Semant., vol. 3, no. 2-3, pp. 79–115, 2005. P. Hayes, “Rdf semantics,” Feb. 2004. P. Hitzler, R. Sebastian, and M. Krötzsch, Foundations of Semantic Web Technologies. London: Chapman & Hall/CRC, 2009. R. Q. Dividino, S. Sizov, S. Staab, and B. Schueler, “Querying for provenance, trust, uncertainty and other meta knowledge in rdf,” J. Web Sem., vol. 7, no. 3, pp. 204–219, 2009. C. Gutierrez, C. A. Hurtado, and A. A. Vaisman, “Temporal rdf,” in ESWC, pp. 93–107, 2005. M. Stocker and E. Sirin, “Pelletspatial: A hybrid rcc-8 and rdf/owl reasoning and query engine,” in Proceedings of the 5th International Workshop on OWL: Experiences and Directions (OWLED 2009), Chantilly, VA, United States, October 23-24, 2009, vol. 529 of CEUR Workshop Proceedings, 2009. D.Lenat, “The Dimensions of Context Space,” tech. rep., CYCorp, 1998. O. Udrea, D. R. Recupero, and V. S. Subrahmanian, “Annotated rdf,” ACM Trans. Comput. Logic, vol. 11, no. 2, pp. 1–41, 2010. U. Straccia, N. Lopes, G. Lukacsy, and A. Polleres, “A general framework for representing and reasoning with annotated semantic web data,” in Proceedings of the 24th AAAI Conference on Artiﬁcial Intelligence (AAAI 2010), Special Track on Artiﬁcial Intelligence and the Web, July 2010. J. Bao, J. Tao, and D. McGuinness, “Context representation for the semantic web.” In Web Science Conference. Online at http://www.websci10.org/, 2010. J. McCarthy, “Notes on formalizing context,” in IJCAI’93, 1993. A. Harth, J. Umbrich, A. Hogan, and S. Decker, “Yars2: a federated repository for querying graph structured data from the web,” in Proceedings of the ISWC/ASWC-2007, 2007. C. Ghidini and L. Seraﬁni, “Distributed ﬁrst order logics,” in Frontiers of Combining Systems, 1998.

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Modular Ontologies O. Kutz and T. Schneider (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-799-4-94

The Onto-Logical Translation Graph Till MOSSAKOWSKI a,1 and Oliver KUTZ b DFKI GmbH Bremen and University of Bremen b Research Center on Spatial Cognition, University of Bremen a

Abstract. We present an overview of the landscape of ontology languages, mostly pertaining to the ﬁrst-order paradigm. In particular, we present a uniform formalisation of these languages based on the institution theoretical framework, allowing a systematic treatment and analysis of the translational relationships between the various languages and a general analysis of properties of such translations. We also discuss the importance of language translation from the point of view of ontological modularity and logical pluralism, and for the borrowing of tools and reasoners between languages. Keywords. Ontology languages, logic translations, institution theory

Introduction and Motivation Ontologies are nowadays being employed in a large number of diverse information-rich domains. While the OWL standard has lead to an important uniﬁcation of notation and semantics, still many distinct formalisms are used for writing ontologies. Some of these, as RDF, OBO and UML, can be seen more or less as fragments and notational variants of OWL, while others, like F-logic and Common Logic (CL), clearly go beyond the expressiveness of OWL.2 Moreover, not only the underlying logics are different, but also the modularity constructs. In this paper, we face this diversity not by proposing yet another ontology language that would subsume all the others, but by accepting the diverse reality and formulating means (on a sound and formal semantic basis) to compare and integrate ontologies that are written in different formalisms. This view is a bit different from that of unifying languages such as OWL and CL, which are meant to be “universal” formalisms (for a certain domain/application ﬁeld), into which everything else can be mapped and represented. While such “universal” formalisms are clearly important and helpful for reducing the diversity of formalisms, it is still a matter of fact that no single formalism will be the Esperanto that is used by everybody. It is therefore important to both accept the existing diversity of formalisms and to provide means of organising their coexistence in a way that enables formal interoperability among ontologies. In this work, we lay the foundation for a distributed ontology language (DOL), which will allow users to use their own preferred ontology formalism while becoming 1 Corresponding

Author: Till Mossakowski, DFKI GmbH Bremen, Enrique-Schmidt Strasse 8, 28359 Bremen, Germany; E-mail: [email protected]. 2 For uniformity, we here typeset all logics in the same font, slightly deviating from common usage.

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interoperable with other formalisms (see [23] for further details). The DOL language is in particular intended to be at the core of a new ISO standardisation effort on ontology interoperability (proposed in ISO/TC 37/SC 3). At the heart of our approach is a graph of ontology languages and translations. This graph will enable users to • relate ontologies that are written in different formalisms (e.g. prove that the OWL version of the foundational ontology D OLCE is logically entailed by the ﬁrst-order version); • re-use ontology modules even if they have been formulated in a different formalism; • re-use ontology tools like theorem provers and module extractors along translations between formalisms. The paper is organised as follows. In Section 1, we recall institutions, which formalise the notion of logical system, and in Section 2, we will cast many known ontology languages as institutions (largely following [24], but casting e.g. F-logic as an institution for the ﬁrst time). This is then repeated for institution comorphisms (formalising the notion of logic translation) in Sections 3 and 4. The latter section contains the key contribution of this work: a graph of comorphisms among logics used for ontologies, together with their main properties.

1. Institutions: Formalising the Notion of Logical System When relating different ontology formalisms, it is helpful to use a meta-notion that formalises the intuitive notion of logical system. Goguen and Burstall have introduced institutions [15] exactly for this purpose. We assume some acquaintance with the basic notions of category theory and refer to [1] or [26] for an introduction. Deﬁnition 1. An institution is a quadruple I = (Sign, Sen, Mod, |=) consisting of the following: • a category Sign of signatures and signature morphisms, • a functor Sen : Sign −→ Set3 giving, for each signature Σ, the set of sentences Sen(Σ), and for each signature morphism σ : Σ −→ Σ , the sentence translation map Sen(σ) : Sen(Σ) −→ Sen(Σ ), where often Sen(σ)(ϕ) is written as σ(ϕ), • a functor Mod : Signop −→ CAT 4 giving, for each signature Σ, the category of models Mod(Σ), and for each signature morphism σ : Σ −→ Σ , the reduct functor Mod(σ) : Mod(Σ ) −→ Mod(Σ), where often Mod(σ)(M ) is written as M σ , and M σ is called the σ-reduct of M , while M is called a σ-expansion of M σ , • a satisfaction relation |=Σ ⊆ |Mod(Σ)| × Sen(Σ) for each Σ ∈ |Sign|, such that for each σ : Σ −→ Σ in Sign the following satisfaction condition holds: () 3 Set

M |=Σ σ(ϕ) iff M σ |=Σ ϕ

is the category having all small sets as objects and functions as arrows. is the category of categories and functors. Strictly speaking, CAT is not a category but only a socalled quasicategory, which is a category that lives in a higher set-theoretic universe. 4 CAT

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for each M ∈ |Mod(Σ )| and ϕ ∈ Sen(Σ), expressing that truth is invariant under change of notation and context.5 $ A theory in an institution is a pair Th = Σ, Γ consisting of a signature Σ and a set Γ of sentences over Σ. The models of Th are those Σ-models that satisfy Γ. Satisﬁability and logical consequence are deﬁned in the standard way. Moreover, the following kernel language of modular speciﬁcations [32] can be interpreted in any institution: SP ::= Σ, Γ | SP1 ∪ SP2 | σ(SP ) | σ −1 (SP ) with the following semantics: Mod(Σ, Γ) = {M ∈ Mod(Σ) | M |= Γ} Mod(SP1 ∪ SP2 ) = Mod(SP1 ) ∩ Mod(SP2 ) Mod(σ(SP )) = {M | M |σ ∈ Mod(SP )} Mod(σ −1 (SP )) = {M |σ | M ∈ Mod(SP )} Most modularity concepts used for ontologies can be mapped into this kernel language.

2. Ontology Languages as Institutions We now cast a rather comprehensive list of well-known ontology languages as institutions, largely following [24], but also extending the list of formalisms given there by including F-logic, OBO, RDFS, and some modular ontology languages, but leaving out some of the less often used formalisms, such as fuzzy and paraconsistent DL. Deﬁnition 2 (Propositional Logic). The institution PL of propositional logic has sets Σ (propositional symbols) as signatures, and functions σ : Σ1 → Σ2 between such sets as signature morphisms. A Σ-model M is a mapping from Σ to {true, false}. The reduct of a Σ2 -model M2 along σ : Σ1 → Σ2 is the Σ1 -model given by the composition M2 ◦ σ. Σ-sentences are built from Σ with the usual propositional connectives, and sentence translation along a signature morphism just replaces the propositional symbols along the morphism. Finally, satisfaction of a sentence in a model is deﬁned by the standard truth-table semantics. It is straightforward to see that the satisfaction condition holds. $ Propositional reasoning is at the core of ontology design. Boolean expressivity is sufﬁcient to axiomatise the taxonomic structure of an ontology by imposing disjointness and sub- or super-concept relationships via implication and negation, as well as e.g. nonempty overlap of concepts. Deﬁnition 3 (Untyped First-order Logic). In the institution FOL= of untyped ﬁrst-order logic with equality, signatures are ﬁrst-order signatures, consisting of a set of function symbols with arities, and a set of predicate symbols with arities. Signature morphisms map symbols such that arities are preserved. Models are ﬁrst-order structures, and sentences are ﬁrst-order formulas. Sentence translation means replacement of the translated 5 Note, however, that non-monotonic formalisms can only indirectly be covered this way, but compare, e.g., [18].

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symbols. Model reduct means reassembling the model’s components according to the signature morphism. Satisfaction is the usual satisfaction of a ﬁrst-order sentence in a ﬁrst-order structure. $ Deﬁnition 4 (Many-sorted First-order Logic). The institution FOLms= of many-sorted ﬁrst-order logic with equality is similar to FOL= , the main difference being that signatures are many-sorted ﬁrst-order signatures, consisting of sorts and typed function and predicate symbols, and that formulas need to be well-typed. For details, see [15]. $ Although not strictly more expressive than untyped FOL= , introducing a sort structure allows a cleaner and more principled design of ﬁrst-order ontologies. Moreover, axioms involving different sorts can be stated more succinctly, and static type checking gives more control over correct modelling. Deﬁnition 5 (Common Logic - CL). Common Logic (CL) has ﬁrst been formalised as an institution in [24]. A CL signature Σ (called vocabulary in CL terminology) consists of a set of names, with a subset called the set of discourse names, and a set of sequence markers. A signature morphism consists of two maps between these sets, such that the property of being a discourse name is preserved and reﬂected.6 A Σ-model consists of a set UR, the universe of reference, with a non-empty subset UD ⊆ UR, the universe of discourse, and four mappings: • rel from UR to subsets of UD ∗ = {< x1 , . . . , xn > |x1 , . . . , xn ∈ UD} (i.e., the set of ﬁnite sequences of elements of UD); • fun from UR to total functions from UD ∗ into UD; • int from names in Σ to UR, such that int(v) is in UD if and only if v is a discourse name; • seq from sequence markers in Σ to UD ∗ . Model reducts leave UR, UD, rel and fun untouched, while int and seq are composed with the appropriate signature morphism component. A Σ-sentence is a ﬁrst-order sentence, where predications and function applications are written in a higher-order like syntax as t(s). Here, t is an arbitrary term, and s is a sequence term, which can be a sequence of terms t1 . . . tn , or a sequence marker. However, a predication t(s) is interpreted like the ﬁrst-order formula holds(t, s), and a function application t(s) like the ﬁrst-order term app(t, s), where holds and app are ﬁctitious symbols denoting the semantic objects rel and fun. In this way, CL provides a ﬁrst-order simulation of a higherorder language. Quantiﬁcation variables are partitioned into those for individuals and those for sequences. Sentence translation along signature morphisms is done by simple replacement of names and sequence markers. Interpretation of terms and formulae is as in ﬁrst-order logic, with the difference that the terms at predicate resp. function symbol positions are interpreted with rel resp. fun in order to obtain the predicate resp. function, as discussed above. A further difference is the presence of sequence terms (namely sequence markers and juxtapositions of terms), which denote sequences in UD ∗ , with term juxtaposition interpreted by sequence concatenation. Note that sequences are essentially a second-order feature. For details, see [11]. As an example, consider the D OLCE formula ∀φ(φ(x)), corresponding to ψ∈Π (ψ(x)), where predicate variables φ, ψ range 6 That

is, a name is a discourse name if and only if its image under the signature morphism is.

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over a ﬁnite set Π of explicitly introduced universals. In CL, this is written, using standard logical syntax (note that CL is agnostic about concrete syntax) ∀φ.Π(φ) −→ φ(x) or in the often used Lisp-like syntax of the CL Interchange Format CLIF: (forall (?phi) (if (pi ?phi) (?phi ?x))) Sequence markers add even more ﬂexibility. For example, it is possible to express that a list of predicates is mutually disjoint as follows (using the sequence marker “. . .”): mutually-disjoint(P ) mutually-disjoint(P Q . . .) ←→ (∀x.¬(P (x) ∧ Q(x))) ∧ mutually-disjoint(P . . .) ∧ mutually-disjoint(Q . . .) $ For the rationale and methodology of CL and the possibility to deﬁne dialects covering different ﬁrst-order languages, see [11]. Deﬁnition 6 (CASL). CASL (the Common Algebraic Speciﬁcation Language, [4, 30]) provides an extension of many-sorted ﬁrst-order logic with partial functions, subsorting and so-called sort-generation constraints. While partial functions and subsorting do not essentially add expressivity (they can be coded out), sort-generation constraints do: they are many-sorted induction axioms (of a second-order nature) that can be used for the deﬁnition of datatypes like natural numbers, lists, trees etc. Deﬁnition 7 (Relational Schemes - Rel-S). This logic, ﬁrst introduced in [21], is about schemes for relational databases and their integrity constraints. A signature in this institution consists of a set of sorts and a set of relation symbols, where each relation symbol is indexed with a string of sorted ﬁeld names as in: paper(key id : integer, title : string, published in : integer) journal(key id : integer, name : string, impact factor : ﬂoat)

Some sorts for the relational schema as integer, ﬂoat and string are predeﬁned and equipped with default interpretations. The identiﬁer key can be used as a preﬁx to sorted ﬁeld names to specify the primary (compound) key of the schema. Signature morphisms map sorts, relation symbols and ﬁeld names in a compatible way, such that primary keys are preserved. A model consists of a carrier set for each sort, where some sorts have predeﬁned carrier sets, and an n-ary relation for each relation symbol with n ﬁelds. Model reduction is like that of many-sorted ﬁrst-order logic. A sentence is a link (integrity constraint) between two ﬁeld names of two relation symbols. For example, the link paper[published in] → journal[id] one to many

requires that the ﬁeld published in of any paper coincides with the id of at least one journal (the many-one character of this relationship is expressed by the keyword

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one to many). Other possible relationships are one to one and many to many. Sentence translation is just renaming of relation symbols and of sorts. A link r[f ] → s[g] t is satisﬁed in case of t = one to many if for each element in r[f ] there are zero or more occurrences of this element in s[g], but for each element in s[g] there is at most one occurrence of an element in r[f ]. For t = one to one in both cases only one occurrence is allowed, and for many to many there is no restriction on the number of occurrences. $ Deﬁnition 8 (Description Logics: OWL and its proﬁles EL, QL, RL). Signatures of the description logic ALC consist of a set A of atomic concepts, a set R of roles and a set I of individual constants, while signature morphisms provide respective mappings. Models are single-sorted ﬁrst-order structures that interpret concepts as unary and roles as binary predicates. Sentences are subsumption relations C1 C2 between concepts, where concepts follow the grammar C ::= A | | ⊥ | C1 C2 | C1 C2 | ¬C | ∀R.C | ∃R.C These kind of sentences are also called TBox sentences. Sentences can also be ABox sentences, which are membership assertions of individuals in concepts (written a : C for a ∈ I) or pairs of individuals in roles (written R(a, b) for a, b ∈ I, R ∈ R). Sentence translation and reduct is deﬁned similarly as in FOL= . Satisfaction is the standard satisfaction of description logics. The logic SROIQ [19], which is the logical core of the Web Ontology Language OWL 2 DL7 extends ALC with the following constructs: (i) complex role boxes (denoted by SR): these can contain: complex role inclusions such as R ◦ S S as well as simple role hierarchies such as R S, assertions for symmetric, transitive, reﬂexive, asymmetric and disjoint roles (called RBox sentences), as well as the construct ∃R.Self (collecting the set of ‘R-reﬂexive points’); (ii) nominals (denoted by O); (iii) inverse roles (denoted by I); qualiﬁed and unqualiﬁed number restrictions (Q). For details on the rather complex grammatical restrictions for SROIQ (e.g. regular role inclusions, simple roles) compare [19], and see the example given below. SROIQ can be straightforwardly rendered as an institutions following the previous examples, but compare also [25]. The OWL 2 speciﬁcation contains three further DL fragments of SROIQ, called proﬁles, namely EL, QL, and RL.8 These are obtained by imposing syntactic restrictions on the language constructs and their usage, with the motivation that these fragments are of lower expressivity and support speciﬁc computational tasks. For instance, RL is designed to make it possible to implement reasoning systems using rule-based reasoning engines, QL to support query answering over large amounts of data, and EL is a subBoolean fragment sufﬁciently expressive e.g. for dealing with very large biomedical ontologies such as the NCI thesaurus. To sketch one of these proﬁles in some more detail, the (sub-Boolean) description logic EL underlying EL has the same sentences as ALC but restricts the concept language of ALC as follows: C ::= B | | C1 C2 | ∃R.C Given that EL, QL, and RL are obtained via syntactic restrictions but leaving the overall SROIQ semantics intact, it is obvious that they are subinstitutions of SROIQ. $ 7 See 8 See

also http://www.w3.org/TR/owl2-overview/ http://www.w3.org/TR/owl2-profiles/ for details of the speciﬁcations.

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Apart from some exceptions9 , description logics can be seen as fragments of ﬁrstorder logic via the standard translation [2] that translates both the syntax and semantics of various DLs into untyped ﬁrst-order logic. A similar situation obtains in the case of the OBO language designed for biomedical ontologies: Deﬁnition 9 (OBO). OBO is a very popular ontology language in the area of biomedical ontology engineering. On the syntactic side, it is straightforward to describe the language’s signatures and sentences in an institution theoretic style, but we here have to leave out the details of such a description. On the semantic side, OBO is a curious case of a language that has been used extensively and for which editors and even reasoners have been successfully implemented, relying initially on only informally speciﬁed semantics. Beginning with OBO version 1.2, and building on an agreement with the OBO community concerning the informal semantics, it was realised that formal semantics could be (mostly10 ) borrowed from OWL, see [17]. In the most recent version of OBO, version 1.4, the translation to OWL 2 as providing the formal semantics is now an ofﬁcial part of the draft speciﬁcation.11 This is an instance of borrowing model theory in the sense of [9], by which an institution OBOOWL is obtained. Since the translation is still partial, OBOOWL is only a subset of the full OBO 1.4. For instance, in order to preserve decidability, SROIQ prohibits cardinality constraints on transitive object properties, whilst the full OBO 1.4 allows this. To render the full OBO 1.4 as an institution, the translation that deﬁnes the satisfaction relation between OBO sentences and the derived semantics has to be extended beyond SROIQ. This is straightforward: the added constructs in OBO such as Boolean constructors on roles have a clear correspondent in DL semantics, which makes it straightforward to complete the mapping of the semantics. $ Deﬁnition 10 (RDF and RDFS). Following [25], we deﬁne the institutions for the Resource Description Framework (RDF) and RDF-Schema (RDFS), respectively. These are based on a logic called bare RDF (bRDF), which consists of triples only (without any predeﬁned resources). A signature Rs in bRDF is a set of resource references. For sub, pred, obj ∈ Rs , a triple of the form (sub, pred, obj) is a sentence in bRDF, where sub, pred, obj represent subject name, predicate name, object name, respectively. An Rs -model M = Rm , Pm , Sm , EXTm consists of a set Rm of resources, a set Pm ⊆ Rm of predicates, a mapping function Sm : Rs → Rm , and an extension function EXTm : Pm → P(Rm × Rm ) mapping every predicate to a set of pairs of resources. Satisfaction is deﬁned as follows: M |=Rs (sub, pred, obj) ⇔ (Sm (sub), (Sm (obj)) ∈ EXTm (Sm (pred)). Both RDF and RDFS are built on top of bRDF by ﬁxing a certain standard vocabulary both as part of each signature and in the models. Actually, the standard vocabulary is given by a certain theory. In case of RDF, it contains e.g. resources rdf:type 9 For instance, adding transitive closure of roles or ﬁxpoints to DLs makes them decidable fragments of second-order logic [5]. 10 Some language constructs, such as ‘being necessarily false’ were seen to not have sufﬁciently clear semantics, and were subsequently dropped from the OBO language. 11 http://www.geneontology.org/GO.format.obo-1_4.shtml#OWL

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and rdf:Property and rdf:subject, and sentences (rdf:type, rdf:type, rdf:Property), and (rdf:subject, rdf:type, rdf:Property). In the models, the standard vocabulary is interpreted with a ﬁxed model. Moreover, for each RDF-model M = Rm , Pm , Sm , EXTm , if p ∈ Pm , then it must hold (p, Sm (rdf:Property)) ∈ EXTm (rdf:type). For RDFS, similar conditions are formulated (here, for example also the subclass relation is ﬁxed). In the case of RDFS, the standard vocabulary contains more elements, like rdf:domain, rdf:range, rdf:Resource, rdf:Literal, rdf:Datatype, rdf:Class, rdf:subClassOf, rdf:subPropertyOf, rdf:member, rdf:Container, rdf:ContainerMembershipProperty. There is also RDFSOWL , an extension of RDFS with resources like owl:Thing and owl:oneOf, tailored towards the representation of OWL. Deﬁnition 11 (Modular Ontology Languages: E-connections and DDL). E-connections can be considered as many-sorted heterogeneous theories: component ontologies can be formulated in different logics, but have to be built from many-sorted vocabulary, and link relations are interpreted as relations connecting the sorts of the component logics. The main difference between distributed description logics (DDLs) [6] and various E-connections now lies in the expressivity of the ‘link language’ L connecting the different ontologies. While the basic link language of DDL is a certain sub-Boolean fragment of many sorted ALC, the basic link language of E-connections is ALCI ms .12 The idea to ‘connect’ logics can be elegantly generalised to the institutional level (compare [3] who note that their ‘connections’ are an instance of a more general cocomma construction). Without giving the full details of such a generalisation, it should be clear that, intuitively, we need to formalise the idea that an abstract connection of two logics S1 and S2 is obtained by deﬁning a bridge language L(E), where the elements of E go across the sort-structure of the respective logics, and where theory extensions (containing the bridge axioms) are deﬁned over a new language deﬁned from the disjoint union of the original languages together with L(E), containing certain expressive means applied (inductively) to the vocabulary of E. Note that this generalises the E-connections of [22], the DDLs of [6], as well as the connections of Baader and Ghilardi [3] in two important respects: ﬁrst, the institutional level generalises the term-based abstract description languages (ADS) that are an abstraction of modal and description logics, and second, the rather general deﬁnition of bridge theory similarly abstracts from the languages previously employed for linking that were similarly inspired by modal logic operators. Given this, the phrasing of DDL and E-connections as institutions is easily obtained from the component institutions: the institution of DDLOWL is the institution whose component logics are OWL based and whose bridge rules follow DDL restrictions, the institution of ECoOWL allows only OWL-based components, but allows the more general = bridge expressivity of E-connections,13 and ECoFOL is the institution whose components can be build from full ﬁrst-order logic, and whose bridge rules allow full ﬁrst-order logic over link relations. 12 More precisely, ALCI ms here comprises existential and universal restrictions on link relations (and their inverses) with Booleans belonging to the components. This can be weakened to e.g. sub-Boolean DL, or strengthened to more expressive many-sorted DLs involving e.g. number restrictions or Boolean operators on links, see [22] for details. 13 Note that allowing full OWL expressivity on the link language leads to undecidability also for OWL-based components.

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It should then be rather clear that e.g. E-connections of OWL ontologies can be encoded within ‘many-sorted’ SROIQ, with additional syntactic restrictions capturing the allowed bridge axioms, see also Section 4. $ Deﬁnition 12 (F-Logic). F-logic [20] is an object-oriented extension of ﬁrst-order logic. For simplicity, we here treat the monotonic part of F-logic only, since this is a logic in the classical sense and can hence be formalised as an institution. The non-monotonic part should be formalised with methods like logic programming over an institution, see [13]. F-logic inherits signatures from FOL= . Sentences are ﬁrst-order sentences with equality, with the following additional formulas: • is-a assertions O : C expressing membership of an object in a class, • subclass assertions C :: D, • object atoms of the form O[me], where the me is a method expression14 . Method expression have the following forms: • • • • • •

non-inheritable scalar expressions ScalarMethod @ t1 , . . . tn → t, non-inheritable set-valued expressions SetMethod @ t1 , . . . tn {u1 , . . . , um }, inheritable scalar expressions ScalarMethod @ t1 , . . . tn •→ t, inheritable set-valued expressions SetMethod @ t1 , . . . tn • {u1 , . . . , um }, scalar signature expressions ScalarMethod @ t1 , . . . tn ⇒ (u1 , . . . , um ), set-valued signature expressions SetMethod @ t1 , . . . tn ⇒ ⇒ (u1 , . . . , um ).

Here, ScalarMethod, SetMethod and the ti and ui are terms. Models are ﬁrst-order structures (unsorted, that is, over a universe U ) equipped with additional components serving for the interpretation of the additional syntax: • : is interpreted with a binary relation ε, and :: with an irreﬂexive partial order ≺, such that aεB and b c imply aεc, • for u ∈ U and each n ≥ 0, there are n n ∗ partial functions I→ (u), I•→ (u) : U n+1 −→ ◦ U, n n ∗ partial functions I (u), I• (u) : U n+1 −→ ◦ P(U ), n n n+1 ∗ partial anti-monotonic functions I⇒ (u), I⇒ −→ ◦ P↑ (U ), where ⇒ (u) : U P↑ (U ) is the set of upward-closed (w.r.t. ≺) subsets of U .

Satisfaction is deﬁned like for ﬁrst-order logic, where : and :: are interpreted with ε and ≺, respectively. An object atom O[ScalarMethod @ t1 , . . . tn → t] holds in a model M n under a variable valuation ν, if I→ (ν(ScalarMethod))(ν(O), ν(t1 ), . . . , ν(tn )) is deﬁned and equal to ν(t); similarly for •→. O[SetMethod @ t1 , . . . tn {u1 , . . . , um }] holds in n M w.r.t. ν, if I (ν(SetMethod))(ν(O), ν(t1 ), . . . , ν(tn )) is deﬁned and contains the set {ν(u1 ), . . . , ν(um )}; similarly for •, ⇒ and ⇒ ⇒. Having so many different arrows with the same semantics seems superﬂuous at ﬁrst sight; their use will become clear when looking at type-checking and non-monotonic inference, which are deﬁned on top of the logic given here. For this, as well as the rationale and the methodology for the use of F-logic in the ﬁeld of object-oriented modelling, see [20]. $ 14 object

molecules O[me1 ; . . . ; men ] abbreviate conjunctions of object atoms.

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Deﬁnition 13 (HOL). [7] presents an institution for a higher-order logic extending Church’s type theory [10] with polymorphism; this is basically the higher-order logic used in modern interactive theorem provers like Isabelle/HOL [31] (one additional feature of Isabelle are type classes). 3. Institution Comorphisms: Formalising Logic Translations We will formalise ontology languages (logics) as institutions and ontology language translations as so-called institution comorphisms, see [16, 29]: Deﬁnition 14 (Institution

Comorphism). Given two institutions I and J with I = I I I I Sign , Mod , Sen , |= and J = SignJ , ModJ , SenJ , |=J , an institution comorphism from I to J consists of a functor Φ : SignI −→ SignJ , and natural transformations β : ModJ ◦ Φ =⇒ ModI and α : SenI =⇒ SenJ ◦ Φ, such that M |=JΦ(Σ) αΣ (ϕ) ⇔ βΣ (M ) |=IΣ ϕ. holds, called the satisfaction condition. Here, Φ(Σ) is the translation of signature Σ from institution I to institution J, αΣ (ϕ) is the translation of the Σ-sentence ϕ to a Φ(Σ)-sentence, and βΣ (M ) is the translation (or perhaps better: reduction) of the Φ(Σ)-model M to a Σ-model. A simple theoroidal comorphism is like a comorphism, except that the signature translation functor Φ maps to the category of theories over the target institution. A simple example is given by considering the well-known translation of OWL into untyped ﬁrst-order logic, mapping concepts to unary and roles to binary predicates. We will give the details of this paradigmatic case in Section 4. The practical usefulness of institution comorphisms grows with their properties. An important property is model expansion, which is similar to conservative extension and ensures that logical consequence is represented faithfully. The amalgamation property ensures good interaction with modular speciﬁcations. Finally, subinstitution comorphisms capture the notion of sublogic. Deﬁnition 15. An institution comorphism is model-expansive, if each model translation βΣ is surjective on objects. Let ρ = (Φ, α, β) : I −→ J be an institution comorphism and let D be a class of signature morphisms in I. Then ρ is said to have the (weak) D-amalgamation property, if for each signature morphism σ : Σ1 −→ Σ2 ∈ D, the diagram ModI (Σ2 ) o

βΣ2

ModJ (Φ(Σ2 ))

ModI (σ)

ModI (Σ1 ) o

ModJ (Φ(σ))

βΣ1

ModJ (Φ(Σ1 ))

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admits (weak) amalgamation, i.e. any for any two models M2 ∈ ModI (Σ2 ) and M1 ∈ ModJ (Φ(Σ1 )) with M2 |σ = βΣ1 (M1 ), there is a unique (not necessarily unique) M2 ∈ ModJ (Φ(Σ2 )) with βΣ2 (M2 ) = M2 and M2 |Φ(σ) = M1 . In case that D consists of all signature morphisms, the (weak) D-amalgamation property is also called (weak) D-exactness. If we omit D, we understand it to consist of all monomorphisms (typically, these are the injective morphisms). An institution comorphism ρ = (Φ, α, β) : I −→ J is said to be model-isomorphic if for each Σ ∈ SignI , βΣ is an isomorphism. It is a subinstitution comorphism [27], if moreover Φ is an embedding and each αΣ is injective. The intuition is that theories should be embedded, while models should be represented exactly (such that modeltheoretic results carry over). It is easy to see that a model-isomorphic comorphism also is model-expansive and exact. Examples will be given in Section 4. The crucial results for comorphisms with good properties are “borrowing” results, that is, a proof calculus or theorem prover capturing logical consequence in the target logic of the comorphism can be borrowed for deciding logical consequence also in the source logic. Proposition 16 (Borrowing [9]). Let ρ = (Φ, α, β) : I −→ J be an institution comorphism, Σ a signature in I and Γ ∪ {ϕ} a set of Σ-sentences. Then Γ |=IΣ ϕ =⇒ αΣ (Γ) |=JΦ(Σ) αΣ (ϕ), Γ satisﬁable ⇐= αΣ (Γ) satisﬁable, and if ρ is model-expansive, also the converse directions hold. Moreover, if SP is a modular speciﬁcation and ρ is exact, then [8] SP |=IΣ ϕ ⇐⇒ ρ(SP ) |=JΦ(Σ) αΣ (ϕ), SP satisﬁable ⇐⇒ ρ(SP ) satisﬁable, where ρ(SP ) is the translation of SP using Φ and α.

4. The Onto-Logical Translation Graph Little work has been devoted to the general problem of translation between ontologies formulated in different logical languages and/or vocabularies. One such approach is given in [14], who discuss translations between OWL ontologies. They use so-called bridging axioms (formulated in ﬁrst-order) to relate the meaning of terms in different ontologies,15 and present an algorithm to ﬁnd such translations. More prominent in the ontology engineering world are of course the standard translation into ﬁrst-order logic, 15 Not

to be confused with the ‘bridge axioms’ in DDL [6].

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Figure 1. Logic translations between ontology languages

which essentially ‘coincides’ with the direct semantics of OWL, and more interestingly the case of OBO discussed above, where the logic translation delivers a deﬁnition of formal semantics for the OBO language (which it itself does not have). We here present an overview of logic-translations between the common ontology languages as introduced above. Note that the resulting graph of logics and translations can be used in several ways: one way is to take some logic high up in the graph, like HOL, and map every ontology into it. While this universal, all-purpose approach may make sense from a semantic point of view, it makes little sense from a practical point of view: a more distributed, multilateral and pluralist approach has the advantage that specialised tools can be used, whilst still interfacing ontologies written in different languages. Subinstitutions: EL → OWL, QL → OWL and RL → OWL and FOL= → FOLms= are obvious subinstitutions. OWL → FOL= is a straight-forward extension of the standard translation [5] mapping individuals to constants, classes to unary predicates and roles to binary predicates. = OWL → ECoFOL uses OWL → FOL= twice, at the level of the base logic ECo and at the level of the bridge rules. PL → FOLms= is a subinstitution by mapping propositional variables to nullary predicates.

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OBOOWL → OWL: signatures and sentences are translated according to the OBO standard, whereas the model translation is the identity (due to borrowing of model theory). OBO1.4 → FOL= extends the composition OBOOWL → OWL → FOL= by an explicit straight-forward coding of the additional features not present in OBOOWL . bRDF → FOL= The subinstitution comorphism from bRDF to FOL= maps a a bRDF signature Rs to the FOL= signature Φ(Rs ) which has Rs as set of constants, and moreover is equipped with a unary predicate P and a ternary predicate EXT . A bRDF-sentence (sub, pred, obj) is translated to EXT (sub, pred, obj). Finally, a FOL= -model of Φ(Rs ) is translated to the bRDF which has the model’s universe as set of resources Rm , while Pm is given by the interpretation of P and Sm by the interpretation of the constants. EXTm can be easily constructed from the interpretation of EXT . FOL= → F-logic is an obvious subinstitution. bRDF → RDF: this is an obvious inclusion, except that bRDF resources need to be renamed if they happen to have a predeﬁned meaning in RDF. The model translation needs to forget the ﬁxed parts of RDF models, since this part can always reconstructed in a unique way, we get an isomorphic model translation. RDF → RDFS and RDFS → RDFSOWL are similar. DDLOWL → ECoOWL is a subinstitution, because all DLL bridge rules are ECoOWL bridge rules. = OWL → DDLOWL and FOL= → ECoFOL are obvious subinstitutions: everything is mapped into one component. = ECoFOL → FOLms= maps each component to a sort, and function and predicates symbols are typed with the sort of their respective component. Simple theoroidal subinstitutions: RDF → FOL= : this is a straightforward extension of bRDF → RDF, axiomatising explicitly the extra conditions imposed on models. RDFS → FOL= and RDFSOWL → FOL= are similar. The theory of the ﬁxed part is (after translation to FOL= ) added to the translations of signatures. FOLms= → FOL= is a theoroidal subinstitution comorphism: a many-sorted signature is translated to an unsorted one by turning each sort into a unary predicate (these are called sort predicates), and each function and predicate symbol is translated by erasing its typing information in the signature, while turning it into a sentence, using the sort predicates. A sentence is translated by erasing the type information and relativising quantiﬁers to the sort predicates. A model is translated by turning the interpretations of sort predicates into carrier sets, and keeping functions and predicates. ECoOWL → OWL uses a similar technique: the different components are mapped into classes, which are then used to relativise (using intersection with these classes) sentences. F-logic → FOL= : the additional ingredients of F-logic are two binary relations and a bunch of partial functions; all these can be coded as (suitably axiomatised) predicates in a straightforward way. Note that the translated signatures become inﬁnite due to the parameterisation of I→ etc. over the natural numbers.

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CL → CASL: speciﬁes the theory of lists and the implicit components of CL models explicitly in CASL. CASL → HOL codes out partiality and subsorting using standard methods, while induction axioms are translated to their explicit second-order Peano-style formulation, see [28] for details. Rel-S → FOLms= : database tables are mapped to predicates, and the involved datatypes are speciﬁed in FOL= 16 . Integrity constraints are expressible as ﬁrst-order sentences, and given a ﬁrst-order model, its predicates are construed as database tables. Simultaneously exact and model-expansive comorphisms: PL → FOL= translates propositional variables to nullary predicates. The model translation forgets the universe (and is hence not an isomorphism). A theoroidal variant adds (to the signature translation) the axiom ∀x, y . x = y enforcing a singleton universe (then, the model translation is at least an equivalence of categories). The translation PL → CL is similar. PL → OWL is a each propositional variable in a signature is mapped to an atomic OWL class. Additionally, the signature translation globally adds one individual a and the axiom {a} expressing that the domain consists of a single point. A propositional sentence (i.e. a Boolean combination of propositional variables) is mapped to membership of a in the corresponding OWL class term (i.e. a Boolean combination of atomic classes) — note that this can be expressed either as ABox statement a : C or as TBox statement {a} ⊆ C. In order to translate an OWL model, for each atomic class A (resulting from a propositional variable A), a : A is evaluated, and the result is assigned to the propositional variable A. The satisfaction condition is straightforward. FOL= → CL: the signature translation maps constants, function symbols and predicates to names. Sentences are left untouched. From a CL-model, it is possible to extract a FOL= -model by restricting functions and predicates to those sequences that have the length of the arity of the symbol (note that this restriction is the reason for not getting an isomorphism). bRDF → OWL: a bRDF signature is translated to OWL by providing a class P and three roles sub, pred and obj (these reify the extension relation), and one individual per bRDF resource. A bRDF triple (s, p, o) is translated to the OWL sentence

∃U.(∃sub.{s} ∃pred.{p} ∃obj.{o}). From an OWL model I, obtain a bRDF model by inheriting the universe and the interpretation of individuals (then turned into resources). The interpretation P I of P gives Pm , and EXTm is obtained by de-reifying, i.e. EXTm (x) := {(y, z)|∃u.(u, x) ∈ predI , (u, y) ∈ subI , (u, z, ) ∈ obj I }. RDF → OWL is deﬁned similarly. The theory of RDF built-ins is (after translation to OWL) added to any signature translation. This ensures that the model translation can add the built-ins. 16 Strictly speaking, for a complete speciﬁcation of inductive datatypes, second-order logic is needed; in this case, the translation ends in HOL.

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OWL → RDFSOWL : this is the RDF serialisation of OWL, formalised as a comorphism in [25]. Model-expansive comorphisms: OWL → F-logic: translations from OWL to F-logic are discussed in [12].

5. Conclusion We argued that there is a multitude of logics and languages in practical use for the speciﬁcation of ontologies that calls for logical pluralism, understood pragmatically. In order to achieve ontology interoperability despite of this pluralism, it is crucial to establish and formalise translations among these logics. We have done this, using so-called institution comorphisms. As Proposition 16 shows, problems of logical consequence and satisﬁability can be translated along such translations in a sound and complete way, opening the door for re-use of tools like theorem provers and model ﬁnders. It turns out that this is the case even if logical consequence and satisﬁability of modular ontologies is concerned: by Proposition 16, nearly all of our translations (with the exception of some translations of OWL to F-logic) interact well with modularity. While we have clariﬁed and summarised the relations among different ontology languages at the semantic level, we have not touched the methodological level. Methodology concerns the way certain features are formalised using logic, as well as the pragmatic level of logic. In order to make ontologies interoperable across different logics, their methodologies (which also may vary within one logic) have to be considered as well. Moreover, some methodologies may also lead to further logic translations that need to be considered. This is left for future work, together with the study of more languages, such as UML as well as some non-classical formalisms that are being used for ontologies. Also, different modularity concepts should be studied and compared.

Acknowledgement Work on this paper has been supported by the DFG-funded Research Center on Spatial Cognition (SFB/TR 8).

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[7] B ORZYSZKOWSKI , T. Higher-order logic and theorem proving for structured speciﬁcations. In WADT (1999), D. Bert, C. Choppy, and P. D. Mosses, Eds., vol. 1827 of Lecture Notes in Computer Science, Springer, pp. 401–418. [8] B ORZYSZKOWSKI , T. Logical systems for structured speciﬁcations. Theoretical Computer Science 286 (2002), 197–245. [9] C ERIOLI , M., AND M ESEGUER , J. May I borrow your logic? (transporting logical structures along maps). Theoretical Computer Science 173 (1997), 311–347. [10] C HURCH , A. A Formulation of the Simple Theory of Types. Journal of Symbolic Logic 5, 1 (1940), 56–69. [11] C OMMON L OGIC W ORKING G ROUP. Common Logic: Abstract syntax and semantics. Tech. rep., 2003. [12] DE B RUIJN , J., AND H EYMANS , S. On the Relationship between Description Logic-based and F-Logicbased Ontologies. Fundam. Inf. 82 (August 2008), 213–236. [13] D IACONESCU , R. Institution-independent Model Theory. Birkhäuser, 2008. [14] D OU , D., AND M C D ERMOT, D. Towards theory translation. In Declarative Agent Languages and Technologies IV (2007), Springer. [15] G OGUEN , J. A., AND B URSTALL , R. M. Institutions: Abstract Model Theory for Speciﬁcation and Programming. Journal of the ACM 39 (1992), 95–146. ¸ , G. Institution morphisms. Formal aspects of computing 13 (2002), 274– [16] G OGUEN , J. A., AND RO SU 307. [17] G OLBREICH , C., H ORRIDGE , M., H ORROCKS , I., M OTIK , B., AND S HEARER , R. OBO and OWL: Leveraging Semantic Web Technologies for the Life Sciences. In Proc. of ISWC-07 (Busan, Korea, November 11-15 2007), K. A. et al., Ed., vol. 4825 of LNCS, Springer, pp. 169–182. [18] G UERRA , S. Composition of Default Speciﬁcations. J. Log. Comput. 11, 4 (2001), 559–578. [19] H ORROCKS , I., K UTZ , O., AND S ATTLER , U. The Even More Irresistible SROIQ. In Proc. of the 10th Int. Conf. on Principles of Knowledge Representation and Reasoning (KR2006) (June 2006), AAAI Press, pp. 57–67. [20] K IFER , M., L AUSEN , G., AND W U , J. Logical Foundations of Object-Oriented and Frame-Based Languages. Journal of the ACM 42 (July 1995), 741–843. [21] K UTZ , O., L ÜCKE , D., AND M OSSAKOWSKI , T. Heterogeneously Structured Ontologies—Integration, Connection, and Reﬁnement. In Advances in Ontologies. Proc. of the KR-08 Ontology Workshop (KROW 2008) (Sydney, Australia, 2008), T. Meyer and M. A. Orgun, Eds., vol. 90 of CRPIT, ACS, pp. 41–50. [22] K UTZ , O., L UTZ , C., W OLTER , F., AND Z AKHARYASCHEV, M. E-Connections of Abstract Description Systems. Artiﬁcial Intelligence 156, 1 (2004), 1–73. [23] K UTZ , O., M OSSAKOWSKI , T., G ALINSKI , C., AND L ANGE , C. Towards a Standard for Heterogeneous Ontology Integration and Interoperability. In Proc. of the First International Conference on Terminology, Languages and Content Resources (LaRC-11) (Seoul, South Korea, June 2011). [24] K UTZ , O., M OSSAKOWSKI , T., AND L ÜCKE , D. Carnap, Goguen, and the Hyperontologies: Logical Pluralism and Heterogeneous Structuring in Ontology Design. Logica Universalis 4, 2 (2010), 255–333. Special Issue on ‘Is Logic Universal?’. [25] L UCANU , D., L I , Y.-F., AND D ONG , J. S. Semantic Web Languages—Towards an Institutional Perspective. In Algebra, Meaning, and Computation, Essays Dedicated to Joseph A. Goguen on the Occasion of His 65th Birthday (2006), K. Futatsugi, J.-P. Jouannaud, and J. Meseguer, Eds., vol. 4060 of Lecture Notes in Computer Science, Springer, pp. 99–123. [26] M AC L ANE , S. Categories for the Working Mathematician, 2nd ed. Springer, Berlin, 1998. [27] M ESEGUER , J. General logics. In Logic Colloquium 87. North Holland, 1989, pp. 275–329. [28] M OSSAKOWSKI , T. Relating CASL with other speciﬁcation languages: the institution level. Theoretical Computer Science 286 (2002), 367–475. [29] M OSSAKOWSKI , T., TARLECKI , A., AND D IACONESCU , R. What is a logic translation? Logica Universalis 3, 1 (2009), 95–124. Winner of the Universal Logic 2007 Contest. [30] M OSSES , P. D., Ed. CASL Reference Manual, vol. 2960 of Lecture Notes in Computer Science. Springer, 2004. Freely available at http://www.cofi.info. [31] N IPKOW, T., PAULSON , L. C., AND W ENZEL , M. Isabelle/HOL—A Proof Assistant for Higher-Order Logic, vol. 2283 of LNCS. Springer, 2002. [32] S ANNELLA , D., AND TARLECKI , A. Speciﬁcations in an arbitrary institution. Information and Computation 76 (1988), 165–210.

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Constructing an Ontology Repository: A Case Study with Theories of Time Intervals a

Darren ONG a , Michael GRÜNINGER a Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada M5S 3G8 Abstract. Ontology repositories stand to beneﬁt through the connecting of stored ontologies via the meta-theoretic relationships they share. Creating this repository framework facilitates ontology reuse and design by allowing users to integrate different ontologies related in this manner. In this paper we construct such a repository by utilizing an automated theorem prover to identify and verify the relationships between three different ontologies of time intervals (two introduced by Hayes in his Catalog of Temporal Theories and one by van Benthem introduced in A Logic of Time). We identify the translation axioms and provide an account of the relationships found between ontologies. Keywords. ontology repository, relationships between theories, ﬁrst-order logic, theories of time intervals, automated theorem proving

1. Introduction The relationships identiﬁed between modules in an ontology repository integrate stored ontologies in a manner that allows for functionality that would otherwise be lost if the repository were just loose collections of axioms. In such a repository, users gain the ability to traverse stored ontologies in a more efﬁcient and directed manner by having those sharing a similar domain explicitly linked. Users would be able to explore a single hierarchy composed of all related ontology modules, rather than the separate hierarchies within each ontology. Having the theories ordered by their relative strength within the hierarchy presents to users the theories that exist in each direction (stronger or weaker) so they can recognize which theory should be explored next in order to ﬁnd one that more precisely captures their intended models. Since each module of an ontology represents a different set of ontological commitments, having the repository connect all ontologies that share logical similarities would also increase the number of choices presented to the user for ontology design and reuse. Each ontology would now have available to it a set of new extensions through the translation of modules belonging to other ontologies connected in the repository. For example, when two ontologies are connected through the repository, they are able to use translation deﬁnitions within the repository to effectually share their modules.

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An ontology repository of this nature also serves to facilitate the semantic integration of systems using different ontologies. If the systems are using ontologies that are related in the repository, it would be possible to use that relationship together with the stored translation deﬁnitions to understand what information can be accurately shared between them. Therefore, as the repository grows so do the semantic integration possibilities between stored ontologies. After reviewing the framework proposed in [6] and [7] to construct such a ﬁrstorder ontology repository, we consider the different ontologies for time intervals (Periods, Approximate-Point, and Interval-Meeting) and describe each of their modules 1 . All ontologies are speciﬁed using Common Logic (ISO 24707), which is a standardized logical language for the speciﬁcation of ﬁrst-order ontologies and knowledge bases. In the ﬁnal section, we generate translation deﬁnitions between the three ontologies and use them to determine the relationships between their modules. In building the repository, the logical relationships between each ontology are established, resulting in a hierarchy that encompasses all modules stored in the repository. We also address the secondary objective of this paper, namely, to investigate the viability of using automated reasoning (theorem provers and model builders) to assist in the construction of an ontology repository by identifying the metatheoretical relationships between the theories that are modules within the repository.

2. Building the Repository’s Logical Framework 2.1. Hierarchies and Repositories All theories within the repository are organized into hierarchies (as proposed in [7]): Deﬁnition 1 A hierarchy H = H, < is a partially ordered, ﬁnite set of theories H = T1 , ..., Tn such that 1. L(Ti ) = L(Tj ), for all i, j 2 ; 2. Ti ≤ Tj iff for any σ ∈ L(Ti ), Ti |= σ ⇒ Tj |= σ If Ti and Tj are theories in the same hierarchy such that Ti < Tj , then Tj is a non-conservative extension of Ti . All theories in a particular hierarchy share the same set of non-logical lexicon, and are ordered by nonconservative extension such that the extensions restrict the set of models of the theory it is extending. With respect to this ordering relation, we say that a theory Ti is stronger than a theory Tj if it is a nonconservative extension of Tj [6]. 2.2. Deﬁnable Interpretations In order to build the logical framework of the repository, we need to identify and verify the relationships between stored ontologies. In order to integrate modules of different 1 The terms modules and theories are used interchangeably and with an ontology being the set of one or more modules. 2 For any theory T , the language of T is denoted by L(T ).

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ontologies into the same repository framework the set of translation deﬁnitions needs to be speciﬁed. Deﬁnition 2 Translation deﬁnitions for an interpretation of T0 into T1 are sentences in the language L0 ∪ L1 of the form (∀x) pi (x) ≡ Φ(x) where pi (x) is a relation symbol in L0 and Φ(x) is a formula in L1 whose only free variables are x. Translation deﬁnitions can be considered to be an axiomatization of the interpretation of T0 into T1 . Theorem 1 T1 is interpretable in T2 iff there exist a set of translation deﬁnitions Δ for T1 into T2 such that T2 ∪ Δ |= T1 Proof: If T1 is interpretable in T2 , then there exists an interpretation π such that π assigns to each n-place relation symbol P a formula πP of L1 in which at most the variables v1 , ..., vn occur free [4]. It is easy to see that πP is the consequent of a translation deﬁnition and that the literal P (v1 , ..., vn ) is the consequent of a translation deﬁnition. Conversely, suppose we are given a set of translation deﬁnitions Δ such that T2 ∪ Δ |= T1 , that is, T1 |= σ ⇒ T2 ∪ Δ |= σ For each translation deﬁnition, (∀x) pi (x) ≡ Φi (x) specify the mapping π(pi ) = Φi (x) For any sentence σ ∈ T1 , we know that π(σ) is logically equivalent to σ by substitution of the consequents of the translation deﬁnitions in Δ, so that T2 ∪ Δ |= π(σ) Since the translation deﬁnitions are conservative, this is equivalent to T2 |= π(σ) and hence π is an interpretation. In this section we outline the procedure for our use of automated theorem provers in the process of verifying the meta-level relationships between modules. We will use this procedure in the next section to establish the relationships between modules across the different ontologies of time intervals added to the repository. By adding new ontologies to the repository, we end up constructing a unifying underlying hierarchy that relates the relative strength of stored ontology modules.

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2.3. Determining Deﬁnable Equivalence Between Theories The methodology for proving the relationship between two theories of different hierarchies follows from the steps used to determine deﬁnable equivalence of the theories. Suppose that Σ12 are the translation deﬁnitions for T1 into T2 , and that Σ21 are the translation deﬁnitions for T2 into T1 . The process of verifying that two theories T1 , T2 are deﬁnably equivalent can be broken down into three reasoning problems: 1. 2. 3. 4.

T1 ∪ T2 ∪ Σ12 is consistent; T1 ∪ Σ12 |= T2 ; T1 ∪ T2 ∪ Σ21 is consistent; T2 ∪ Σ21 |= T1 .

Success of all four reasoning problems proves that the theories T1 and T2 are are deﬁnably equivalent, so that they can be considered to be alternative axiomatizations of the same set of models. If the ﬁrst or third step fails, so that the axioms of the two theories together with the translation deﬁnitions are inconsistent, then the two theories have disjoint sets of models and they are not translatable into one another. If only the second step fails, then T1 might be weaker than T2 . If only the fourth step fails, then T2 might be weaker than T1 . In the last two scenarios, we can only say that one theory is strictly weaker than the other if we are able to ﬁnd a deﬁnably equivalent theory of the stronger theory, in the core hierarchy of the weaker, and show that it non-conservatively extends the weaker theory. The ultimate goal of this work is to automate the procedure of adding new theories into a hierarchy and deriving the relationships among theories in the same hierarchy as well as among theories in different hierarchies. Automated theorem provers are used for steps 2 and 4, while model builders are used for steps 1 and 3. This approach is not restricted to a particular theorem prover, but for the present paper, Prover9 and Mace4 [13] were chosen. Prover9 is a ﬁrst-order logic automated theorem prover that uses resolution to prove that goal sentences are entailed by the background theory (assumptions). Mace4 is a ﬁnite-model generator used to ﬁnd counter-examples of the goal. 2.4. Related Work The use of relative interpretations between ﬁrst-order axiomatized theories as a means to combine smaller theories has been implemented by the Interactive Mathematical Proof System. IMPS is a repository of mathematical theories that utilizes relative interpretations for the transportation of theorems between theories of different lexicon for the purposes of theorem proving [5]. However, our repository focuses on relating stored theories to facilitate ontology design and understanding with the scope of stored ontologies being broader than just mathematical theories. In [12], HETS (Heterogeneous Tool Set) is used to manage libraries of ontologies speciﬁed in CASL. These CASL speciﬁcations allow users to group parts of an ontology into smaller sub-theories while deﬁning translations, unions,reductions and extensions between them. HETS visualizes these relationships via development graphs by denoting the dependencies between the theories. While the focus of HETS is developing ontologies (libraries) through the combination of smaller theories, we also want to examine the relationships between similar libraries (sets) of theories. In [10], proving consistency of a large ontology is done by breaking

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it down into smaller, easier consistency proofs and proofs of conservativity of theory extensions. Relative consistency proofs are used by providing theory interpretations into another theory that is known or assumed to be consistent. Kutz and Mossakowski design the architechural speciﬁcation that modularizes the DOLCE ontology and the theory interpretations into conservative extensions of those theories to establish the consistency of DOLCE via relative consistency proofs. However, more than consistency, we are interested in determining if sets of models of two differently axiomatized ontologies are equivalent. ε-connections is used to combine OWL ontology modules with independent domains where the relations between modules are strictly those of conservative extensions [3]. This is insufﬁcient as ontology hierarchies in our repository are built around nonconservative extensions of theories. This allows our repository to store extensions of the same theory with slightly different sets of ontological commitments. In [6], Gruninger et al. use the notion of deﬁnable equivalence to construct representation theorems as a means to characterize the models of one theory using another. The relationships between theories proven in this paper is the ﬁrst step towards such an application. 3. Theories of Time Intervals A key objective of this paper is to explore a set of time ontologies in different hierarchies as a source of case studies for constructing and updating an ontology repository. In particular, we we want to show automated reasoning tools can be used to determine relationships among theories in the same hierarchy as well as to determine relationships among theories in different hierarchies. In this section, we introduce the three hierarchies of time interval ontologies that are the focus of this papers – • the hierarchy HP eriods , whose theories were introduced in [2], • the hierarchy HApproximate−P oint presented in [9], • the hierarchy HInterval−M eeting , which has been explored in [1], [9], and [11]. With respect to relationships between these time ontologies, Ladkin [11] fully characterizes the models of another of Hayes’ time interval theories, Tim , that uses the meets relation. In doing so he is able to extend the theory to one that is equivalent to TIN T (Q) . However, the process is manual and only proof sketches are provided in comparison to the semi-automated procedure described in this paper that utilizes an automated theorem prover to provide detailed machine proofs. There are other ﬁrst-order time ontologies which are not considered in this paper [14]. Ontologies of timepoints (also discussed in [2] and [9]) are all in the same hierarchy (that is, they all have the same language); as a result, they would not be useful to illustrate the methodology for determining relationships between hierarchies. Ontologies that axiomatize relations between both timepoints and time interval are introduced in [9]; the relationships between theories in these hierarchies is characterized in [8] using the same methodology as the current paper. 3.1. Theories of the Hierarchy HP eriods Here we introduce the four ﬁrst-order theories in the hierarchy HP eriods (see Figure 1) whose axioms were provided by van Benthem in [2]. We begin by describing the

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weakest theory of Periods and then work our way up the hierarchy looking at each of its extensions.

Figure 1. Hierarchy of Period theories from [2]. Dashed lines denote nonconservative extensions.

3.1.1. Minimal Theory of Periods (Tperiods 3 ) This theory constitutes the minimal set of conditions that must be met by any period structure [2] and has two relations (precedence and inclusion) and two conservative definitions (glb and overlaps) as its non-logical lexicon. A total of eight axioms are contained within this theory. Transitivity (TRANS) and irreﬂexivity (IRREF) axioms for the precedence relation make it a strict partial order, and TRANS, reﬂexivity (REF), and anti-symmetry (ANTIS) axioms for the inclusion relation make it a partial order. While the axioms of monotonicity (MON) enforce correct interplay between the precedence and inclusion relations. Van Benthem further includes in this minimal theory the axiom CONJ, that guarantees the existence of greatest lower bounds (glb) between overlapping intervals. No sorts are speciﬁed; hence, every element in the domain is a “time interval”. Also, the axioms of this theory are satisﬁed by both inﬁnite and ﬁnite models. 3.1.2. Theory of Mixed Periods (Tmixed_periods 4 ) The theory of mixed periods extends Tperiods with eleven more axioms that force an inﬁnite linear ordering of intervals at both ends of the line(SUCC, LIN*), the joining and interaction of neighbouring intervals (NEIGH, MOND), intervals are uninterrupted stretches (CONV), the relationship between overlap and inclusion of intervals (FREE), and the coverage of any pair of intervals by a larger interval (DIR, DISJ). The lexicon is expanded with two new conservative deﬁnitions (lub and underlaps), but no new relations. This theory reﬁnes the models of time intervals from Tperiods to capture some common intuitions about time, but enforces neither density nor discreteness of intervals. 3.1.3. Theory of Intervals Over Rational Numbers (TIN T (Q) 5 ) The theory TIN T (Q) is the ﬁrst-order axiomatization of rational intervals where the only countable model (up to isomorphism) is characterized by the set of intervals that exist between the rational numbers [2]. This theory extends the Tmixed_periods by adding two 3 http://stl.mie.utoronto.ca/colore/time/periods.clif 4 http://stl.mie.utoronto.ca/colore/time/mixed_periods.clif 5 http://stl.mie.utoronto.ca/colore/time/periods_over_rationals.clif

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axioms – one for density, that states that any interval can be divided into two smaller intervals, and a second axiom enforcing Allen’s temporal relations. 3.1.4. Theory of Intervals Over Integers (TIN T (Z) 6 ) The theory of TIN T (Z) , on the other hand, is the ﬁrst-order axiomatization of discrete intervals whose models are the sets of intervals over the integers. The axiomatization provided by van Benthem includes the second-order principle of well-foundedness. However, he conjectures that this theory is elementary equivalent to one in which the principle of well-foundedness is replaced by an axiom for atomicity of intervals (ATOM). This modiﬁed theory constitutes the theory of discrete periods TIN T (Z) . The theories TIN T (Q) and TIN T (Z) both extend Tmixed_periods , but are inconsistent with one another as one takes on the ontological commitment of density for its models of intervals and the other the commitment of discreteness. 3.2. Theories of the Approximate-Point Hierarchy The ontologies in the hierarchy HApproximate−P oint (see Figure 2) represent intervals as approximations of points. As presented in [9], the hierarchy consists of three theories; once again, we begin by examining the axioms of the weakest theory and then move on to its extensions.

Figure 2. Hierarchy of Approximate-Point theories from [9]. Dashed lines denote nonconservative extensions.

3.2.1. Theory of Approximate-Point (Tap 7 ) Hayes’ ﬁrst-order theory Tap consists of two relations, where precedes is the relation between sufﬁciently distinct intervals, and where ﬁner is the relation of sub-intervals. The precedes relation is a strict partial ordering, while the ﬁner relation is a partial ordering. Also, there is the ncdf (not clearly distinguishable from) relation, a conservative deﬁnition that relates two intervals that share a common interval. Transitivity and irreﬂexivity axioms exist for the precedes relation, while transitivity, reﬂexivity, and anti-symmetry axioms exist for the ﬁner relation. 3.2.2. Theory of Dense Approximate-Point (Tdense_ap 8 ) Since the theory Tap enforces neither density or discreteness for elements in its domain (time intervals), this theory Tdense_ap exists as an extension of Tap that adds an axiom to enforce density of time intervals. This axiom states that ﬁner intervals always exist. 6 http://stl.mie.utoronto.ca/colore/time/periods_over_integers.clif 7 http://stl.mie.utoronto.ca/colore/time/approximate_point.clif 8 http://stl.mie.utoronto.ca/colore/time/approximate_dense_point.clif

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3.2.3. Theory of Discrete Approximate-Point (Tdiscrete_ap 9 ) In this extension of Tap , two conservative deﬁnitions are added – one for the meets relation, that states that an interval meets another if there does not exist an interval between them, and the other for the moment class of elements that deﬁnes them as the smallest interval (an interval is a moment iff it contains no smaller intervals). The axiom for discreteness then stipulates that every moment meets another moment, both in its past and in its future. 3.3. Theory of Interval-Meeting (Tim 10 ) The theory of Interval-Meeting axiomatizes models of time intervals meeting at points. This theory’s lexicon consists of only one relation (meets) and three conservative deﬁnitions (starts, during, and ﬁnishes). Five axioms enforce inﬁnite linear ordering between intervals at both ends of the line, the uniqueness of meeting points between intervals, and the existence of a union of two intervals. 4. Relationship between the Hierarchies HApproximate−P oint and HP eriods The ﬁrst step in constructing the repository is the determine the relationships between the theories in the hierarchies HApproximate−P oint and HP eriods . We identify the translation deﬁnitions between their root theories and proceed to determine which theories in each hierarchy are deﬁnably equivalent to each other. 4.1. Translation Deﬁnitions Σp_ap The translation deﬁnitions, Σp_ap , between these two hierarchies are straight-forward as they use the same set of relations to axiomatize time intervals. In more complex cases, there would exist a different set of deﬁnitions for each direction of translation between core hierarchies, but the one-to-one mapping of relations here makes Σp_ap ≡ Σap_p . These translation deﬁnitions are required in order to relate the two ontologies in the repository and compare the similarities, differences, and relative strengths of each of their modules. Relations that have conservative deﬁnitions in ontology do not require explicit translation deﬁnitions of their own since their deﬁnitional axioms can be reused in conjunction with the translation deﬁnitions when translating between ontologies. However, the simple translation deﬁnition between the conservative deﬁnitions of overlaps and ncdf are added to increase the efﬁciency of the automated theorem prover. Deﬁnition 3 The translation deﬁnitions Σp_ap for the interpretation of theories in HApproximate−P oint to theories in HP eriods is the set of sentences (forall (x y) (iff (forall (x y) (iff (forall (x y) (iff

(precedence x y) (precedes x y))) (inclusion x y) (finer x y))) (overlaps x y) (ncdf x y)))

The translation deﬁnitions Σap_p for the interpretation of theories in HP eriods to theories in HApproximate−P oint is equivalent to Σp_ap . 9 http://stl.mie.utoronto.ca/colore/time/approximate_discrete_point.clif 10 http://stl.mie.utoronto.ca/colore/time/interval_meeting.clif

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4.2. Relationship Between Tperiods and Tap The ﬁrst relationship to be determined between these two hierarchies will be between their weakest theories, Tperiods and Tap . Lemma 1

Tperiods ∪ Σp_ap |= Tap

Proof: Using Mace, one can construct models of Tperiods ∪ Σp_ap that falsify each of the following sentences: (forall (forall (forall (forall

(x y) (or (ncdf x y) (precedes x y) (precedes y x))) (x) (exists (y) (precedes y x))) (x) (exists (y) (precedes x y))) (x y) (exists (z) (and (finer x z) (finer y z))))

Lemma 2

Tap ∪ Σap_p |= Tperiods

f inite Proof: Let Tap be the subtheory of Tap without the axioms that force the existence of inﬁnite sets of intervals. f inite ∪ Σap_p that falsiﬁes the Using Mace, one can construct a model M of Tap axiom CONJ of Tperiods . This model can be extended to construct a model of Tap ∪ Σap_p that falsiﬁes the axiom CONJ.

We can use these results to identify the subtheory Tap_root of Tap that is deﬁnably equivalent to the subtheory Tperiods_root of Tperiods 11 . Theorem 2 Tap_root is deﬁnably equivalent to Tperiods_root . 4.3. Relationship Between Tmixed_periods and Tap Intuitively, the above results show that Tperiods is not strong enough to deﬁnably interpret Tap , so we move to Tmixed_periods , which is the next theory within the hierarchy HP eriods . Prover9 was used to show the following in a straightforward manner: Lemma 3

Tmixed_periods ∪ Σp_ap |= Tap

Since Tmixed_periods is an extension of Tperiods , we have Lemma 4

Tap ∪ Σap_p |= Tmixed_periods

Nevertheless, we can specify an extension Tap_interval 12 of Tap such that Tap_interval ∪ Σap_p |= Tmixed_periods Tmixed_periods ∪ Σp_ap |= Tap_interval so that we have Theorem 3 Tap_interval is deﬁnably equivalent to Tmixed_periods . 11 http://stl.mie.utoronto.ca/colore/time/ap_root.clif

http://stl.mie.utoronto.ca/colore/time/periods_root.clif 12 http://stl.mie.utoronto.ca/colore/time/ap_interval.clif

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4.4. Relationship Between TIN T (Q) and Tdense_ap It is at this point that the proofs become less trivial as the density axioms differ between the two ontologies. Lemma 5

TIN T (Q) ∪ Σp_ap |= Tdense_ap

Proof: The density axiom of Tdense_ap : (forall (x) (exists (y) (and

(finer y x) (not (finer

x y)))))

is proven using the following subset of axioms of TIN T (Q) : (forall (x) (not (precedence x x))) (forall (x y z) (if (and (precedence x y) (inclusion z y)) (precedence x z))) (forall (x y z) (iff (lub x y z) (and (inclusion x z) (inclusion y z) (forall (u) (if (and (inclusion x u) (inclusion y u)) (inclusion z u)))))) (cl-comment "DENS* Axiom") (forall (x) (exists (y z) (and

(precedence y z) (lub y z x))))

Lemma 6

Tdense_ap ∪ Σap_p |= TIN T (Q)

f inite Proof: Using Mace, one can construct a model M of Tap ∪ Σap_p that falsiﬁes the axiom

all x all y all z ((precedes(x,y) & precedes(y,z)) -> (all u ((finer(x,u) & finer(z,u)) -> finer(y,u)))).

of TIN T (Q) . This model can be extended to construct a model of Tap_dense ∪Σap_p that falsiﬁes the axiom. As before, we can extend Tdense_ap to specify a new theory in the hierarchy which is deﬁnably equivalent to TIN T (Q) : Theorem 4 Let Tap_rational = Tdense_ap ∪ Tap_interval is deﬁnably equivalent to TIN T (Q)

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Proof: Prover9 can be used to show that Tdense_ap ∪ Σap_p |= DEN S∗ so that we have Tap_rational ∪ Σap_p |= TIN T (Q) 4.5. Relationship Between TIN T (Z) and Tdiscrete_ap Theorem 5

TIN T (Z) ∪ Σp_ap |= Tdiscrete_ap

f inite Proof: Let TIN T (Z) be the subtheory of TIN T (Z) without the axioms that force the existence of inﬁnite sets of intervals. f inite Using Mace, one can construct a model M of TIN T (Z) ∪ Σap_p that falsiﬁes the axiom

(forall (x) (exists (y) (and

(meets x y) (not (exists (y) (and (finer y x) (not (= x y)))))))

of Tdiscrete_ap . This model can be extended to construct a model of TIN T (Z) ∪ Σap_p that falsiﬁes the axiom. Lemma 7

Tdiscrete_ap ∪ Σap_p |= TIN T (Z)

f inite Proof: Using Mace, one can construct a model M of Tap ∪ Σap_p that falsiﬁes the axiom used in the proof of Lemma 6, which is also entailed by TIN T (Q) . This model can be extended to construct a model of Tap_discrete ∪ Σap_p that falsiﬁes the axiom.

Theorem 6 Tap_integer = Tdiscrete_ap ∪ Tap_interval is deﬁnably equivalent to TIN T (Z) . 5. Integrating the Hierarchy HInterval_M eeting with the Repository Next, we consider the case of integrating the ontology Tim into the repository. To do so we identify the translation deﬁnitions between Tim and the theories in the hierarchy HApproximate_P oint as we attempt to ﬁnd deﬁnably equivalent theories. Since the relationships between theories in the hierarchies HApproximate_P oint and HP eriods have already been established, we can determine the relationship between theories in HInterval_M eeting and HApproximate_P oint by composition of the translation deﬁnitions.

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5.1. Adding New Theories to the Hierarchy In addition to the axioms presented in [9], there are two other time interval ontologies that have been discussed in the literature and which are in the same hierarchy as Tim . The ﬁrst is the set of axioms introduced in [1], which we refer to as Tallen−hayes 13 The second is an ontology14 characterized by Ladkin in [11], which includes the axioms of Tallen−hayes and extends it so that it axiomatizes T h(IN T (Q)). Lemma 8 Tallen−hayes is a nonconservative extension of Tim . f inite be the subtheory of Tim without the axioms that force the existence Proof: Let Tim f inite of inﬁnite sets of intervals. Using Mace, one can construct a model M of Tim ∪ Σim_ap that falsiﬁes the additional axiom. This model can be extended to construct a model of Tim that falsiﬁes this axiom.

5.2. Translation Deﬁnitions Σap_im Translation of the lexicon between these two hierarchies is not one-to-one like those between theories in the hierarchies HApproximate_P oint and HP eriods , so more complex translation deﬁnitions are required. Deﬁnition 4 The translation deﬁnitions Σap_im for the interpretation of theories in HInterval−M eeting by theories in HApproximate−P oint is the set of sentences (forall (x y) (iff

(meets x y) (and (precedes x y) (not (exists (z) (and (precedes x z) (precedes z y)))))))

Deﬁnition 5 The translation deﬁnitions Σim_ap for the interpretation of theories in HApproximate−P oint by theories in HInterval−M eeting is the set of sentences (forall (x y) (iff

(precedes x y) (or (meets x y) (exists (z) (and (meets x z) (meets z y))))))

(forall (x y) (iff

(finer x y) (or (starts x y)(during x y)(finishes x y)(= x y))))

Note that starts, ﬁnishes, and during are conservative deﬁnitions of theories in HInterval−M eeting and, therefore, do not require translation deﬁnitions to theories in HApproximate−P oint . 13 http://stl.mie.utoronto.ca/colore/time/allen_hayes.clif 14 http://stl.mie.utoronto.ca/colore/time/ladkin_intq.clif

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5.3. Relationship Between Tim and Tap An interesting discovery was that these two theories are in fact not deﬁnably equivalent: Lemma 9

Tim ∪ Σim_ap |= Tap

f inite Proof: Mace constructed models of Tim ∪Σim_ap that falsiﬁed each of the following two axioms of Tap :

(forall (x y) (or

(ncdf i j) (precedes i j) (precedes j i)))

(forall (x y) (exists (z) (and (finer x z) (finer y z))))

The next lemma illustrates the challenges related to the secondary objective of the paper, namely, the investigation of the feasibility of using an automated theorem prover in the discovery of the relationships between modules within the repository. Lemma 10

Tim ∪ Σim_ap |= Tap_root

The theorem prover easily found proofs for the Tap axioms for inﬁnite intervals in both directions, reﬂexivity of the ﬁner relation, and irreﬂexivity of the precedes relation; however, getting the theorem prover to provide proofs for the seven remaining axioms of Tap required user assistance in the form of lemmas. For example, in proving the entailment of the transitivity axiom of precedes, the user had to manually translate the axiom into the lexicon of the background theory (in this case Tim ) using the translation deﬁnitions for the theorem prover to ﬁnd the proof. (cl-comment "original axiom") (forall (x y z) (if (and (precedes x y) (precedes y z)) (precedes x z))) (cl-comment "translated axiom") (forall (x y z) (if (and (or (meets x y) (exists (u) (and (meets x u) (meets u y)))) (or (meets y z) (exists (w) (and (meets y w) (meets w z))))) (or (meets x z) (exists (p) (and (meets x p) (meets p z))))

The next set of axioms that required the use of lemmas were axioms that contained the relation ﬁner. For the theorem prover to ﬁnd proof for the entailment of the antisymmetry axiom of ﬁner, both the use of lemmas and the need to select only a subset of axioms for use in the background theory were required. Since the translation deﬁnition of ﬁner is a disjunction of the relations starts, during, ﬁnishes and equivalence, the lemmas required were axioms that represented each of the possible cases of the translation of ﬁner to the lexicon of Tim . The remaining sentences are as follows:

D. Ong and M. Grüninger / Constructing an Ontology Repository

(forall (i j) (if (forall (i j) (if (forall (i j) (if

123

(during i j) (not (during j i))))) (finishes i j) (not (finishes j i))))) (starts i j) (not (starts j i)))))

To obtain entailment proofs for the transitivity axiom of ﬁner and the axioms for separation and orthogonality between the relations ﬁner and precedes similar lemmas that considered each case for the translation of ﬁner individually (nine lemmas for transitivity, four lemmas for separation, and four lemmas for orthogonality), and the careful selection of axioms used for the background theory were once again required. Selecting the appropriate subset of axioms of the background theory for the theorem prover to use was signiﬁcant as this was often the difference between the theorem prover timing out, and a proof being found. In all of the cases we found that the deﬁnitional axioms and the axiom for inﬁnite intervals: (cl-comment "infinite intervals") (forall (i) (exists (j k) (and (meets j i) (meets i k))))

were prime candidates for removal from the background theory when they were not required for the proof (for example, removing the deﬁnitional axiom of starts when starts was not needed to prove the lemma). Theorem 7

Tap ∪ Σap_im |= Tim

Proof: Mace constructs a model that falsiﬁes the following axiom of Tim : (forall (i j k m) (if (and (meets i j) (meets k l)) (or (meets i l) (exists (n) (or

(and (and

(meets (meets (meets (meets

i n k n

n) l)) n) j)))))))

Theorem 8

Tap ∪ Σap_im |= Tmeets_root

Proof: In order to prove that the axiom of “unique meeting-places”: (forall (i j k m) (if

(and (meets i j) (meets i k) (meets i m)) (meets j m)))

is entailed, the following lemma was needed (forall (i j k m) (if

(and (meets i j) (meets i k) (meets l j)) (precedes l k)))

By examining the relationship between Tap and Tmeets_root , we identiﬁed a new ontology Tm_exist 15 in the hierarchy HApproximate−P oint such that Tm_exist is a nonconservative extension of Tap and Tm_exists deﬁnably interprets Tim . 15 http://stl.mie.utoronto.ca/colore/time/m_exist.clif

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Tladkin intq

?

TINT (Q)

Tap rational Tap integer

Tap dense TINT (Z)

Tap discrete

Tallen hayes

Tap interval

Tmixed

periods

Tm exist

Tim

Tap

Tperiods

Tmeets root

Tap root

Tperiods root

Figure 3. Hierarchy of interval theories in the repository. Dashed lines denote nonconservative extensions, dotted lines denote deﬁnable interpretation, and solid lines denote deﬁnable equivalence. The relationship between Tladkin_intq and Tap_rational is conjectured by Ladkin in [11], but this has not be veriﬁed.

6. Summary We have been able to construct an ontology repository containing three different time interval ontologies by identifying the meta-level relationships that exist between them (see Figure 3). In doing so, we have effectively created a common hierarchy linking all ontology modules stored in the repository through the translation deﬁnitions presented. Building this network of relationships for the repository beneﬁts ontology design by making explicit the relationships shared by different ontologies in the same domain. The user is able to compare modules across ontologies to better judge the ﬁt of an ontology to their needs. For example, a user deciding between the two ontologies of dense time intervals, TIN T (Q) and Tdense_ap , would be made aware that TIN T (Q) is the stronger theory with the set of additional axioms displayed for analysis. The repository framework also allows user navigation within the repository to be more directed as users are now able to traverse a single hierarchy of all related theories rather than smaller, isolated hierarchies within separate ontologies. For instance, users browsing the current repository would tra-

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125

verse a single hierarchy composed of all time interval theories rather than three isolated hierarchies of the individual ontologies. Translation deﬁnitions between ontologies can also be used to convert modules of one language into modules of another giving insight into the level of semantic integration between them. For example, a system using Tap as its ontology and another system using Tmixed_periods is guaranteed to interpret only sentences entailed by Tap the same since Tap < Tmixed_periods , whereas systems using Tap and Tim are fully integrated. Furthermore, we have shown that it is feasible to use an automated theorem prover and a generic process to uncover and verify the meta-level relationships between modules across hierarchies. In each of the module comparisons, an automated theorem prover was able to help in determining whether one module was deﬁnably equivalent, deﬁnably interpretable by (weaker than), deﬁnably interpretable in (stronger than), or inconsistent with another and in the process identifying the axioms responsible for those relationships. However, more work can be done in making the theorem proving process more streamlined; in particular, the process of determining which lemmas or subsets of axioms in the background theory are needed to obtain a proof is still very much ad-hoc. In the future, we look to extend the repository to include more ontologies of varying domains and generality such as mereotopology and manufacturing. We also want to explore the application of the repository framework to the design of new ontologies by integrating and extending existing ontologies.

References [1] [2] [3] [4] [5] [6] [7]

[8] [9] [10] [11] [12] [13] [14]

Allen, J. and Hayes, P. (1989) Moments and points in an interval-based temporal logic, Computational Intelligence 5:225-238. van Benthem, Johan F. A. K. (1991) The Logic of Time: A Model-Theoretic Investigation into the Varieties of Temporal Ontology and Temporal Discourse, Springer; 2nd edition. Cuenca Grau, B., Parsia, B., Sirin, E. (2009) Ontology Integration Using E-connections, pp.293-320, Modular Ontologies, Stuckenschmidt et al. (eds.). Springer-Verlag, Berlin. Enderton, H. (1972) Mathematical Introduction to Logic,Academic Press. Farmer, W. M. (2000) An Infrastructure for Intertheory Reasoning. In: CADE-17, Lecture Notes in Computer Science (LNCS), D. McAllester, ed., 1831:115-131. Gruninger, M., Hashemi, A., and Ong, D. (2010) Ontology Veriﬁcation with Repositories, Formal Ontologies and Information Systems 2010, Toronto, Canada. Gruninger, M., Hahmann, T., Hashemi, A., Ong, D., and Ozgovde, A. (2011) Modular First-Order Ontologies via Repositories, submitted to International Journal Applied Ontology: Modularity in Ontologies. Gruninger, M. and Ong, D. (2011) Veriﬁcation of Time Ontologies with Points and Intervals, 18th International Symposium on Temporal Representation and Reasoning. Hayes, P. (1996) A Catalog of Temporal Theories, Tech Report UIUC-BI-AI-96-01, University of Illinois. Kutz, O., Mossakowski, T. (2011) A Modular Consistency Proof for Dolce, Association for the Advancement of Artiﬁcial Intelligence 2011, San Francisco, USA. Ladkin, P. B. (1987) Models for Axioms of Time Intervals, pp. 234-239, Proceedings AAAI-87, Seattle, WA. Lüttich, K., Mossakowski, T. (2004) Speciﬁcation of Ontologies in CASL, Formal Ontologies and Information Systems 2004, Torino, Italy. McCune, W. (2005-2010) Prover9 and Mace4, http://www.cs.unm.edu/∼mccune/Prover9. Vila, L. (2005) Formal Theories of Time and Temporal Incidence, in Handbook of Temporal Reasoning in Artiﬁcial Intelligence, Fisher, Gabbay, Vila: eds. Elsevier, 2005.

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127

Subject Index automated theorem proving biomedical ontologies bio-ontologies ChEBI concept grouping decomposition first-order logic institution theory locality-based modules logic translations modular ontologies modularity modularization module extraction ontology combinations ontology comprehension

110 7 63 63 12 25 55, 110 94 25 94 71 63 7, 63 10, 40 71 25

ontology evaluation 55 ontology integration 40 ontology languages 94 ontology mapping 40 ontology partitioning 12 ontology repository 55, 110 ontology reuse 40 ontology summarization 12 OWL 2 QL 10 performance 63 relationships between theories 110 representation theorems 55 SNOMED CT 7 spatial information 71 theories of time intervals 110 Σ-query inseparability 10

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129

Author Index Batchelor, C. Borgo, S. Coskun, G. del Vescovo, C. Dmitrieva, J. Grüninger, M. Hahmann, T. Hastings, J. Hois, J. Joseph, M. Katsumi, M. Kutz, O. López-García, P.

63 1 12 25 40 55, 110 55 63 71 79 55 v, 94 7

Mossakowski, T. Ong, D. Parsia, B. Paschke, A. Rothe, M. Sattler, U. Schneider, T. Schulz, S. Serafini, L. Steinbeck, C. Teymourian, K. Verbeek, F.J. Zakharyaschev, M.

94 110 25 12 12 25 v, 25 7, 63 79 63 12 40 10

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