Preferential Description Logics: Reasoning in the ... I, Graham Deane, declare that this thesis is my

  • View
    0

  • Download
    0

Embed Size (px)

Text of Preferential Description Logics: Reasoning in the ... I, Graham Deane, declare that this thesis is...

  • Imperial College of Science, Technology and Medicine

    Department of Computing

    Preferential Description Logics: Reasoning in the presence of inconsistencies

    Graham Deane

    Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in Computing of Imperial College London and

    the Diploma of Imperial College, June 2016

  • Abstract

    The Web Ontology Languages define a rich machine readable language for knowledge repre-

    sentation on the world wide web. The current generation are based on Description Logics, a

    family of decidable logics that offer low complexity of reasoning. Due to the principle of explo-

    sion, from a contradiction anything follows, inconsistencies prevent meaningful inference from

    classical reasoners. However, locating and repairing inconsistencies can be resource intensive.

    This thesis presents an inconsistency tolerant semantics for the Description Logic ALC called

    Preferential ALC (p-ALC). A p-ALC knowledge base is comprised of defeasible and non-

    defeasible axioms. The defeasible ABox and TBox are labelled with weights that reflect a

    relative measure of confidence in each axiom and provide a mechanism to “arbitrate” between

    inconsistencies. Entailment is defined through the notion of preferred interpretations which

    minimise the total weight of the unsatisfied axioms. We introduce a tableau algorithm for

    p-ALC in which the open branches correspond to preferred interpretations. We prove that the

    algorithm is sound, complete and terminates for any input knowledge base and show how it

    can be used to compute p-ALC entailment by refutation. The proposed p-ALC differs from

    existing semantics that obtain inferences from inconsistent knowledge bases designed for clas-

    sical reasoners. For instance: the para-consistent and the repair semantics, lack a mechanism

    for “arbitration” of inconsistency; and the mechanism included in possibilistic logic results in

    a logic with a weak consequence relation.

    We present an implementation of the algorithm using answer set programming (ASP) that is

    solved incrementally and exploits the optimisation of answer sets to identify preferred interpre-

    tations of p-ALC. The well defined semantics of ASP enabled us to prove that the implementa-

    tion is correct. The implementation is also verified empirically and the performance is evaluated

    against a set of synthetically generated inconsistent knowledge bases in which inconsistency is

    introduced.

    i

  • ii

  • Statement of Originality

    I, Graham Deane, declare that this thesis is my own work and has not been submitted in any

    form for another degree or diploma at any university or other institute of tertiary education.

    Information derived from the published and unpublished work of others has been acknowledged

    in the text and a list of references is given in the bibliography.

    The copyright of this thesis rests with the author and is made available under a Creative

    Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to

    copy, distribute or transmit the thesis on the condition that they attribute it, that they

    do not use it for commercial purposes and that they do not alter, transform or build upon

    it. For any reuse or redistribution, researchers must make clear to others the licence

    terms of this work.

    iii

  • iv

  • Acknowledgements

    I would like to express my deepest gratitude to my co-supervisors Dr Krysia Broda and Professor

    Alessandra Russo who have provided their continuous expert guidance, encouragement and

    support during the work presented in, and the writing of, this thesis.

    I would also like to thank my PhD examiners Professor Marek Sergot (internal) and Dr Antonis

    Bikakis (external) from University of London for their very constructive feedback and detailed

    comments during the finalisation of this thesis.

    I am most grateful to all the staff and the researchers in the Department of Computing at

    Imperial College for their advice and support during my time in the department.

    I would like to thank Nicolas Matentzoglu for his assistance in searching for knowledge bases

    within the corpora at Manchester and Kody Moodley for sharing his views on the generation

    of synthetic knowledge bases.

    The work presented in this thesis was funded by the Engineering and Physical Sciences Research

    Council.

    v

  • vi

  • Contents

    Acknowledgements v

    1 Introduction 1

    1.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    1.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    2 An introduction to Description Logics 9

    2.1 The Description Logic ALC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    2.1.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    2.1.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    2.1.3 Reasoning tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    2.2 The extended family of Description Logics . . . . . . . . . . . . . . . . . . . . . 17

    2.2.1 The EL family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

    2.2.2 The DL-Lite family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    2.3 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    2.3.1 Complexity of Description Logic reasoning tasks . . . . . . . . . . . . . . 27

    2.4 Inconsistency in Description Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    vii

  • viii CONTENTS

    2.4.1 A survey of existing approaches for handling inconsistencies . . . . . . . 30

    2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

    3 Preferential ALC 39

    3.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

    3.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

    3.3 Properties of p-ALC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

    3.3.1 Existence of preferred interpretations . . . . . . . . . . . . . . . . . . . . 48

    3.3.2 Refutation-style proofs can be used to show p-ALC entailment . . . . . . 53

    3.4 Reasoning in the presence of inconsistencies . . . . . . . . . . . . . . . . . . . . 55

    3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

    4 A Tableau Algorithm for p-ALC 61

    4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

    4.2 The tableau algorithm for p-ALC . . . . . . . . . . . . . . . . . . . . . . . . . . 72

    4.3 General properties of the p-ALC tableau algorithm . . . . . . . . . . . . . . . . 82

    4.3.1 Static blocking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

    4.3.2 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

    4.3.3 All branches close . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

    4.3.4 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

    4.4 Soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    4.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

  • CONTENTS ix

    4.6 Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

    4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

    5 Implementation 109

    5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

    5.1.1 Syntax and Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

    5.1.2 Reasoning tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

    5.1.3 Features specific to clingo . . . . . . . . . . . . . . . . . . . . . . . . . . 118

    5.1.4 Properties of Answer Set Programs . . . . . . . . . . . . . . . . . . . . . 121

    5.2 Encoding the p-ALC tableau algorithm in ASP . . . . . . . . . . . . . . . . . . 127

    5.2.1 The representation of a p-ALC knowledge base . . . . . . . . . . . . . . . 128

    5.2.2 Encoding the expansion rules . . . . . . . . . . . . . . . . . . . . . . . . 131

    5.3 Properties of the translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

    5.4 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

    5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

    6 Evaluation 166

    6.1 Introduction and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

    6.2 Verification of correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

    6.3 Knowledge bases for performance evaluation . . . . . . . . . . . . . . . . . . . . 171

    6.3.1 Inconsistent knowledge bases . . . . . . . . . . . . . . . . . . . . . . . . . 172

    6.3.2 Corpora of knowledge bases . . . . . . . . . . . . . . . . . . . . . . . . . 17