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Model Theoretic Characterisations of Description Logics

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Model Theoretic Characterisations of Description Logics Model Theoretic Characterisations of Description Logics esis submied in accordance with the requirements of the University of Liverpool for the degree of Doctor in Philosophy by Robert Edgar Felix Piro From Celle, Germany August 2012 Degree of Doctor in Philosophy P S Prof. Dr. Frank Wolter Department of Computer Science University of Liverpool S S Dr. Boris Konev Department of Computer Science University of Liverpool A Dr. Clare Dixon Department of Computer Science University of Liverpool I E Dr. Clare Dixon Department of Computer Science University of Liverpool E E Prof. Dr. Ulrike Saler School of Computer Science University of Manchester Abstract e growing need for computer aided processing of knowledge has led to an in- creasing interest in description logics (DLs), which are applied to encode know- ledge in order to make it explicit and accessible to logical reasoning. DLs and in particular the family around the DL have therefore been thor- oughly investigated w.r.t. their complexity theory and proof theory. e question arises which expressiveness these logics actually have. e expressiveness of a logic can be inferred by a model theoretic characterisa- tion. On concept level, these DLs are akin to modal logics whose model theoretic properties have been investigated. Yet the model theoretic investigation of the DLs with their TBoxes, which are an original part of DLs usually not considered in context of modal logics, have remained unstudied. is thesis studies the model theoretic properties of , , , as well as , , and . It presents model theoretic proper- ties, which characterise these logics as fragments of the first order logic (FO). e characterisations are not only carried out on concept level and on concept level extended by the universal role, but focus in particular on TBoxes. e prop- erties used to characterise the logics are ‘natural’ notions w.r.t. the logic under investigation: On the concept-level, each of the logics is characterised by an ad- apted form of bisimulation and simulation, respectively. TBoxes of , and are characterised as fragments of FO which are invariant under global bisimulation and disjoint unions. e logics , and , which incorporate individuals, are characterised w.r.t. to the class 핂 of all interpretations which interpret individuals as singleton sets. e characterisations for TBoxes of and both require, addi- tionally to being invariant under the appropriate notion of global bisimulation and an adapted version of disjoint unions, that an FO-sentence is, under certain circumstances, preserved under forward generated subinterpretations. FO-sentences equivalent to -TBoxes, are—due to ’s inverse roles—characterised similarly to and but have as third additional requirement that they are preserved under generated subinterpretations. as sub-boolean DL is characterised on concept level as the FO-fragment which is preserved under simulation and preserved under direct products. Equally valid is the characterisation by being preserved under simulation and having min- imal models. For -TBoxes, a global version of simulation was not sufficient but FO-sentences of -TBoxes are invariant under global equi-simulation, disjoint unions and direct products. For each of these description logics, the aracteristic concepts are explicated and the characterisation is accompanied by an investigation under which notion of saturation the logic in hand enjoys the Hennessy-and-Milner-Property. As application of the results we determine the minimal globally bisimilar com- panion w.r.t. -bisimulation and introduce the 1-to-2-rewritability prob- lem for TBoxes, where 1 and 2 are (description) logics. e laer is the problem to decide whether or not an 1-TBox can be equivalently expressed as 2-TBox. We give algorithms which decide -to--rewritability and -to-- rewritability. Acknowledgements At first, I would like express my gratitude to British society for its openness and fairness towards strangers which have made this thesis possible. In particular, I was warmly welcomed by the Renaissance Music Group and the Southport Bee- keepers which gave me close contact and insight into the British culture and ac- cepted me as one of their own. Just as important is the University of Liverpool and its Department of Com- puter Science, who created the opportunity for non-UK students to participate in a remunerated PhD-programme which allowed myself an adequate life in Liv- erpool for three years. I have greatly profited from the opportunities (ESSLLI09, in-house skills programmes, trips) and facilities (e.g. library and office) that the University and the Department provided. Special thanks go to my supervisors Boris Konev and in particular Frank Wolter with his keen and unmistakable sense for scientifically interesting problems. He was a very knowledgeable and patient contact. I would like to thank Clare Dixon, who as my adviser gave me valuable feedback and cared for my progress towards the goal. I am also very grateful for all the opportunities to present many of the results of my thesis at international workshops and conferences. e results of Chapter 5 were published in [91] and presented at the European Conference on Artificial Intelligence 2010 (ECAI2010). e results of Chapter 3, Section 4.4 and Chapter 6 were published in [92] and presented at the International Joint Conferences on Artificial Intelligence (IJCAI2011). Results of this thesis were also presented at the DL-Workshop 2010 in Water- loo, Canada and at the British Colloquium for eoretical Computer Science in Manchester in 2012. anks go also to my fellow PhD-colleagues and friends at the Department of Computer Science who created an enjoyable atmosphere and who entrusted me with the appointment as representative for the PhD-students in the Department. I would like thank Michel Ludwig, my flatmate and dear colleague for the good start he gave me when I arrived in Liverpool, his patience and support over the two years of flatmateship. I also would like to thank Helen Parkinson who refined my qualities as flatmate during the 8 month we shared the house together. Last but not least, I would like to give big thanks to my partner James Furlong for all his love and support especially during the last gruelling months of my doctorate. To my mother Gisela Elisabeth Johanna Piro Contents 1 Introduction 12 1.1 Description Logics in Context .................... 12 1.1.1 Description Logics: A First Glimpse ............. 12 1.1.2 Applications of Description Logic .............. 14 1.1.3 The Trade-Off between Expressivity and Complexity ... 16 1.1.4 The Language Family around ............. 17 1.1.5 Contribution ......................... 18 1.1.6 Structure of the Thesis .................... 19 1.2 Preliminaries .............................. 23 1.2.1 Signature and Interpretation ................ 23 1.2.2 Syntax and Semantics of -concepts .......... 24 1.2.3 Translation into Standard Modal Logic ........... 25 1.2.4 Translation into First Order Logic .............. 27 1.2.5 Abstract Logic ......................... 28 2 The characterisation of 31 2.1 The Characterisation of -Concepts ............... 31 2.1.1 Bisimulation and Tree-Unravelling ............. 31 2.1.2 Finite Bisimulation and Locality of -Concepts .... 34 2.1.3 Characteristic Concepts ................... 38 2.1.4 Saturation and the Theorem of Hennessy and Milner ... 41 2.1.5 -Concepts as Bisimulation Invariant Fragment of FO 46 2.2 The Characterisation of u-Concepts .............. 48 2.2.1 Syntax, Semantics and Normal Form of u-Concepts . 49 2.2.2 Global Bisimulation, Finite Global Bisimulation and Characteristic u-Concepts ............... 51 xi 2.2.3 u-Concepts as Global Bisimulation Invariant Fragment of FO ........................ 54 2.3 The Characterisation of -TBoxes ................ 56 2.3.1 Invariance under Disjoint Unions .............. 58 2.3.2 -TBoxes as FO-Fragment ................ 60 3 The Characterisation of and 63 3.1 ................................. 63 3.1.1 The Characterisation of -Concepts ......... 63 3.1.2 The Characterisation of u-Concepts ......... 69 3.1.3 The Characterisation of -TBoxes ........... 73 3.2 ................................. 75 3.2.1 The Characterisation of -Concepts ......... 76 3.2.2 The Characterisation of u-Concepts ........ 88 3.2.3 The Characterisation of -TBoxes .......... 99 4 The Characterisation of , and 102 4.1 The Model Class 핂 .......................... 103 4.2 ................................. 105 4.2.1 The Characterisation of -Concepts ......... 106 4.2.2 The Characterisation of u-Concepts ........ 109 4.2.3 The Characterisation of -TBoxes .......... 112 4.3 ................................ 124 4.3.1 The Characterisation of -Concepts ........ 124 4.3.2 The Characterisation of u-Concepts ....... 126 4.3.3 The Characterisation of -TBoxes ......... 128 4.3.4 Minimal u1- and u-Bisimilar Companions . 145 4.4 ............................... 153 4.4.1 The Characterisation of -Concepts ....... 153 4.4.2 The Characterisation of u-Concepts ....... 158 4.4.3 The Characterisation of -TBoxes ......... 165 5 The characterisation of 172 5.1 ................................... 172 5.1.1 Syntax and Semantics .................... 172 5.1.2 -Simulation ........................ 173 5.1.3 The ⊔ Extension of .................. 176 5.1.4 Characterisation of as FO-Fragment .......... 177 5.1.5 Characterisation of ⊔-TBoxes and -TBoxes ..... 182 6 Application 196 6.1 The -to- Rewritability Problem ............. 196 6.2 The -to- Rewritability Problem ............... 209 6.2.1 Algorithm to
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