PRACTICAL REASONING FOR DEFEASIBLE DESCRIPTION LOGICS by Kody Moodley Submitted in fulfilment of the academic requirements for the degree of Doctor of Philosophy in the School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal, Durban, South Africa Supervisor: Prof. Thomas Meyer Co-Supervisors: Prof. Uli Sattler, Dr. Deshen Moodley November 2015 Preface The work reported on in this thesis was carried out at the Centre for Artifi- cial Intelligence Research and the CSIR Meraka Institute in Pretoria, South Africa, from August 2011 to December 2015, under the supervision of Pro- fessor Thomas Meyer and Dr Deshen Moodley. During the period from September 2013 to September 2014, the work was conducted at the Information Management Group of the department of Computer Science at the University of Manchester, United Kingdom, under the supervision of Professor Uli Sattler. These studies represent original work by the author and have not otherwise been submitted in any form for any degree or diploma to any tertiary institu- tion. Whenever other researchers work is used, they are duly acknowledged in the text. Declaration 1 - Plagiarism I, , declare that: 1. The research reported in this dissertation, except where otherwise in- dicated, is my original research. 2. This dissertation has not been submitted for any degree or examination at any other university. 3. This dissertation does not contain other person’s data, pictures, graphs or other information, unless specifically acknowledged as being sourced from other persons. 4. This dissertation does not contain other persons’ writing, unless specif- ically acknowledged as being sourced from other researchers. Where other written sources have been quoted, then: (a) Their words have been re-written but the general information at- tributed to them has been referenced (b) Where their exact words have been used, then their writing has been placed in italics and inside quotation marks, and referenced. 5. This dissertation does not contain text, graphics or tables copied and pasted from the Internet, unless specifically acknowledged, and the source being detailed in the dissertation and in the References section. Signed: Declaration 2 - Publications Below is a list of publications contributing towards the content of this thesis. 1. Meyer, T., Moodley, K. and Varzinczak, I. (2012) A Prot´eg´ePlug-in for Defeasible Reasoning. In Proceedings of the Twenty Fifth International Workshop on Description Logics (DL). CEUR Workshop Proceedings, Volume 846, ISSN: 1613-0073. 2. Meyer, T., Moodley, K. and Varzinczak, I. (2012) A Defeasible Rea- soning Approach for Description Logic Ontologies. In Proceedings of the Annual Research Conference of the South African Institute for Computer Scientists and Information Technologists (SAICSIT). p69- 78, ISBN: 978-1-4503-1308-7, ACM New York. 3. Casini, G., Meyer, T., Moodley, K. and Varzinczak, I. (2013). To- wards Practical Defeasible Reasoning for Description Logics. In Pro- ceedings of the Twenty Sixth International Workshop on Description Logics (DL). CEUR Workshop Proceedings, Volume 1014, p587-599, ISSN: 1613-0073. 4. Casini, G., Meyer, T., Moodley, K. and Varzinczak, I. (2013). Non- monotonic Reasoning in Description Logics: Rational Closure for the ABox. In Proceedings of the Twenty Sixth International Workshop on Description Logics (DL). CEUR Workshop Proceedings, Volume 1014, p600-615, ISSN: 1613-0073. 5. Casini, G., Meyer, T., Moodley, K. and Nortje, R. (2014). Relevant Closure: A New Form of Defeasible Reasoning for Description Log- ics. In Proceedings of the Fourteenth European Conference on Logics in Artificial Intelligence (JELIA). Lecture Notes in Computer Science (LNCS), Volume 8761, p92-106, ISBN: 978-3-319-11557-3, Springer In- ternational Publishing. 6. Meyer, T., Moodley, K. and Sattler, U. (2014). Practical Defeasible Reasoning for Description Logics. In Proceedings of The Seventh Euro- pean Starting AI Researcher Symposium (STAIRS). Volume 264, p191- 200, ISBN: 978-1-61499-420-6, IOS Press. 7. Meyer, T., Moodley, K. and Sattler, U. (2014). DIP: A Defeasible- Inference Platform for OWL Ontologies. In Proceedings of the Twenty Seventh International Workshop on Description Logics (DL). CEUR Workshop Proceedings, Volume 1193, p671-683, ISSN: 1613-0073. 8. Casini, G., Meyer, T., Moodley, K., Sattler, U. and Varzinczak, I. (2015). Introducing Defeasibility into OWL Ontologies. In Proceedings of the International Semantic Web Conference (ISWC). Lecture Notes in Computer Science (LNCS), Volume 9367, p409-426, ISBN: 978-3- 319-25009-0, Springer International Publishing. I was the primary author for Publications 1 to 3 and 6 to 8 in which I contributed somewhat to the theoretical aspects but more heavily in practical implementation and evaluation of tools and algorithms. For publications 4 and 5, I was responsible for the implementation, experimental evaluation (and write-up thereof) of several algorithms central to the work. Signed: Acknowledgements Firstly, I would like to thank my supervisors Tommie Meyer, Uli Sattler and Deshen Moodley. I really could not have asked for better supervisors and they were each integral to completing this PhD. Thank you also to Giovanni Casini and Ivan Varzinczak for helpful discussions on my work. I am very grateful that, during my tenure at the University of Manchester, Uli and the students of the Information Management Group really went out of their way to make me feel welcome. In particular, I would like to thank Slava Sazonau, Liang Chang, Nico Matentzoglu and Jared Leo for keeping me sane and entertained while I conducted my research. Thank you to Nico for helping me with sourcing data for the empirical component of my work, and thanks also to Bijan Parsia for his useful comments on my experiments. Matthew Horridge was an inspiration to me on how to conduct practical research in my area and I am grateful to him for his example in this regard. Back in Pretoria, I would like to thank Marlene Jivan for all her help with handling various administration tasks relevant to this PhD, and Nishal Morar for keeping me company at the office while he did his Masters degree. Of course, this PhD could not have been completed without funding and I have been very fortunate to receive it in abundance from the National Research Foundation, the Commonwealth Scholarship Commission, the CSIR Meraka Institute and the University of KwaZulu-Natal. I would particularly like to thank Quentin Williams from the CSIR Meraka Institute for his help with arranging funds for me. Finally, I would like to thank my family for their encouragement during my studies, and a very special appreciation goes to my wife Nicole for her unconditional love, understanding and support throughout this PhD - thanks for putting up with me. Contents List of Figures 11 List of Algorithms 15 Abstract 16 1 Introduction 17 1.1 Motivation . 18 1.2 Uncertainty vs. Exceptions in Knowledge Representation . 20 1.3 Goals . 20 1.4 Organisation of Thesis . 22 2 Background and Related Work 24 2.1 Description Logics (DLs) . 25 2.1.1 Syntax and Semantics . 26 2.1.2 Common Reasoning Problems . 30 2.1.3 Trade-off between Expressivity and Complexity . 38 2.2 The Web Ontology Language (OWL) . 44 2.3 Incomplete Knowledge . 45 2.3.1 Defeasibility . 46 2.3.2 Uncertainty . 46 2.3.3 Subjectivity or Relativity . 47 2.4 Circumscription . 47 2.4.1 Basic Circumscription . 47 7 CONTENTS 8 2.4.2 Prioritised Circumscription . 52 2.4.3 Grounded Circumscription . 54 2.4.4 Complexity Considerations . 55 2.4.5 Discussion . 56 2.5 Default Reasoning . 58 2.5.1 Basic Default Logic . 58 2.5.2 Defaults Embedded in Description Logics . 60 2.5.3 Discussion . 64 2.6 Minimal Knowledge and Negation as Failure (MKNF) . 65 2.6.1 DLs of MKNF . 66 2.6.2 Discussion . 69 2.7 Defeasible Logic . 70 2.7.1 Basic Defeasible Logic . 71 2.7.2 Combining Defeasible and Description Logics . 76 2.7.3 Discussion . 78 2.8 Preferential Reasoning . 79 2.8.1 Propositional Foundations . 80 2.8.2 Description Logic Foundations . 84 2.8.3 Rational Closure for Description Logics . 93 2.8.4 Rational Extensions of an ABox . 98 2.8.5 Discussion . 99 2.9 Overriding . 101 2.9.1 Discussion . 104 2.10 Summary and Discussion . 105 2.11 Notation and Conventions . 107 3 Requirements Analysis 108 3.1 Need for Defeasible Description Logics . 109 3.1.1 Dataset . 110 3.1.2 Experiment Setup . 111 3.1.3 Results and Discussion . 114 3.2 Inferential Character . 117 CONTENTS 9 3.2.1 Formal Properties . 118 3.2.2 Semi-Formal Properties . 124 3.3 Discussion . 128 4 Algorithms for Defeasible Reasoning 130 4.1 Exceptionality to Classical Entailment . 131 4.1.1 Disjoint Union of Ranked Interpretations . 131 4.1.2 Exceptionality in Terms of Unsatisfiability . 136 4.2 Ranking of a Defeasible Knowledge Base . 144 4.3 Rational Closure . 169 4.4 Lexicographic Closure . 180 4.4.1 Semantics . 181 4.4.2 Procedure . 186 4.4.3 Discussion . 201 4.5 Relevant Closure . 203 4.6 Optimisations . 222 4.7 Syntactic Sugar . 226 4.7.1 Defeasible Equivalence . 227 4.7.2 Defeasible Disjointness . 228 4.8 Discussion . 231 5 Inferences of Defeasible Reasoning 233 5.1 Preferential Algorithms . 233 5.2 Non-Preferential Algorithms . 239 5.2.1 Overriding . 240 5.2.2 Circumscription . 243 5.2.3 Default Reasoning . 250 5.3 Discussion . 254 6 Performance of Defeasible Reasoning 257 6.1 Artificial Data . 258 6.1.1 Data Generation Model . 258 CONTENTS 10 6.1.2 Experiment Setup . 270 6.1.3 Ranking Compilation Results . 274 6.1.4 Entailment Checking Results . 277 6.1.5 Discussion . 289 6.2 Modified Real World Data . 291 6.2.1 Data Curation Methodology . 292 6.2.2 Introducing Defeasibility into the Data .