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 Pokrywczyński 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

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Query Inseparability and Module Extraction in OWL 2 QL Michael Zakharyaschev

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

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

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

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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

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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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S. Borgo / Goals of Modularity: A Voice from the Foundational Viewpoint

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.

[1]


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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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S. Schulz and P. López-García / Modularity Issues in Biomedical Ontologies

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

S. Schulz and P. López-García / Modularity Issues in Biomedical Ontologies

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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].

M. Zakharyaschev / Query Inseparability and Module Extraction in OWL 2 QL

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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,


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.


J(X, Y ) =

|X ∩ Y | |X ∪ Y |

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(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


23

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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[12] [13] [14]

[15] [16] [17]

Harith Alani and Christopher Brewster. Metrics for ranking ontologies. In Denny Vrandeˇcić, 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.


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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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C. del Vescovo et al. / The Modular Structure of an Ontology

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-


29

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

6

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1

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

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 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 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,

42


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-


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.


45

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 :


47

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)

48


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:


49

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

50


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.

51


(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)


(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.


53

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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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.

M. Grüninger et al. / Exploiting Modularity for Ontology Verification

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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,

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