Extending Modal Logic

Extending Modal Logic Maarten de Rijke Extending Modal Logic ILLC Dissertation Series 1993-4 institute [brlogic, language andcomputation For further information about ILLC-publications, please contact: Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam phone: +31-20-5256090 fax: +31—20-5255101 e-mail: [email protected] Extending Modal Logic Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam, op gezag van de Rector Magnificus Prof. Dr. P.W.M. de Meijer in het openbaar te verdedigen in de Aula der Universiteit (Oude Lutherse Kerk, ingang Singel 411, hoek Spui) op vrijdag 3 december 1993 te 15.00 uur door Maarten de Rijke geboren te Vlissingen. Promotor: Prof.dr. J.F.A.K. van Benthem Faculteit Wiskunde en Informatica Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam The investigations were supported by the Philosophy Research Foundation (SWON), which is subsidized by the Netherlands Organization for Scientific Research (NWO). Copyright © 1993 by ‘.\/Iaarten de Rijke Cover design by Han Koster. Typeset in LATEXbythe author. Camera-ready copy produced at the Faculteit der Wiskunde en Informatica, Vrije Universiteit Amsterdam. Printed and bound by Haveka B.V., Alblasserdam. ISBN: 90-800 769—9—6 Contents Acknowledgments vii I Introduction 1 1 What this dissertation is about 3 1.1 Modal logic . 3 1.2 Extending modal logic . 3 1.3 A look ahead . 4 2 What is Modal Logic? 6 2.1 Introduction . 6 2.2 A framework for modal logic 7 2.3 Examples . 10 2.4 Questions and comments . 12 2.5 Concluding remarks 15 II Three Case Studies 17 3 The Modal Logic of Inequality 19 3.1 Introduction . 19 3.2 Some comparisons 23 3.3 Axiomatics 29 3.4 Definability . 42 3.5 Concluding remarks . 46 4 A System of Dynamic Modal Logic 48 4.1 Introduction . 48 4.2 Preliminaries . 50 4.3 Using D./VIE . 52 4.4 The expressive power of D./VIE . 54 4.5 Decidability . 60 4.6 Completeness . 64 4.7 Concluding remarks . 68 vi Contents 5 The Logic of Peirce Algebras 71 5.1 Introduction . 71 5.2 Preliminaries . 71 5.3 Modal preliminaries . 76 5.4 Axiomatizing the set equations . 82 5.5 Set equations and relation equations . 82 5.6 Expressive power . 101 5.7 Concluding remarks . 104 III Two General Themes 105 6 Modal Logic and Bisimulations 107 6.1 Introduction . 107 6.2 Preliminaries . 108 6.3 Basic bisimulations . 109 6.4 Modal equivalence and bisimulations . 113 6.5 Definability and characterization . 118 6.6 Preservation . 126 6.7 Beyond the basic pattern 132 6.8 Concluding remarks . 135 7 Correspondence Theory for Extended Modal Logic 137 7.1 Introduction . 137 7.2 Preliminaries . 138 7.3 Reducibility . 139 7.4 Finding the right instances . 141 7.5 Reduction algorithms . 152 7.6 Applying the algorithms . 159 7.7 Another perspective: global restrictions 165 7.8 Concluding remarks . 167 Appendix. Background material 169 Bibliography 171 Index 176 List of symbols 179 Samenvatting 181 Acknowledgments It takes four ingredients to cook an agreeable PhD project: the right supervi sion, (ii) the right co-authors and colleagues, (iii) sufficiently many projects and side activities running in parallel, and (iv) plenty of trips abroad. I would like to thank several people and organizations for providing these ingredients and making my four years as a PhD student a valuable experience. When I arrived as a PhD student it was agreed that Dick de Jongh would be my supervisor for the first six months. With his questions and insights Dick helped me deliver my first reports (which, by the way, are still among my favorite ‘children’). While Dick was, so to say, the person who taught me how to fly, Johan van Benthem, who took over from Dick, showed me exotic destinations I could fly to. His special style of ‘long-distance supervision’ was the right thing for me at the right time, leaving me enough elbow-room to follow side routes, but also making sure that I didn’t forget about the main route: writing this dissertation. Thank you Dick and Johan! One of the most rewarding academic things must be co-working on one project or another. I had the pleasure of co-writing a paper and/or book with a number of people. Thank you Andrei Arsov, Patrick Blackburn, Claire Gardent, Edith Hemaspaandra, Wiebe van der Hoek, Dick de Jongh, Yde Venema, and Roel Wieringal Further people I would like to thank for contributing to my scientific well-being during the past four years include Valentin Goranko, David Israel, Marcus Kracht, Valentin Shehtman, Dimiter Vakarelov, Frank Veltman, Heinrich Wansing, and all the colleagues from the Department of Mathematics and Computer Science of the University of Amsterdam. An important maxim for PhD students that is often overlooked is that the best way to focus on a large project is to have plenty of side activities for distrac tion. One of my main and most rewarding side routes has been the organization of the Seminar on Intensional Logic. I would like to thank the many people that attended the Seminar, the people that presented a paper before the Seminar, and the people that were involved with one of the “spin-offs” of the Seminar. In addition I would like to thank the Dutch Network for Logic, Language and Information (TLI), the European Community, and the Department of Mathe matics and Computer Science of the University of Amsterdam for making it all financially possible. As for the fourth essential ingredient for enjoyable PhD projects, I am very grateful to the Department of Mathematics and Computer Science of the Uni versity of Amsterdam, the ERASMUS project, NWO, SWON, the TEMPUS vii viii Acknowledgments project, and TLI for financing my trips to Belgium, France, Sweden, England, the USA, Hungary, Bulgaria, Spain, Portugal, and Germany. This dissertation has benefitted from valuable input by Johan van Benthem, Patrick Blackburn, Peter van Emde Boas, Wiebe van der Hoek (who generously read and re-read the entire manuscript), and Yde Venema. A special thanks to Vincent van Oostrom for his help in printing the camera-ready copy. At a more personal level I owe a lot to a small number of people. It’s best that I try and settle my large debt to them in a private setting. Amstelveen Maarten de Rijke October, 1993. Part I Introduction 1 What this dissertation is about This dissertation is about extending modal logic. It tells you what a system of extended modal logic is, it gives you three case studies of systems of modal logic, and it gives you very general approaches to two important themes in modal logic. This Chapter offers a brief overview of the dissertation. 1.1 MODAL LOGIC According to the general framework presented in Chapter 2 below, a modal language is first and foremost a description language for relational structures. Its distinguishing features are 1. the use of simple restricted relational patterns to describe the truth con ditions of its operators, 2. the use of multiple sorts, 3. a concern for the fine structure of model theory. To illustrate these points, recall that the standard modal language .M£(<>) has its formulas built up from proposition letters p, the usual Boolean connec tives n and A, and a unary modal operator 0. Models for the language .M£(<>) assign subsets P of the universe to proposition letters, and use a binary relation R to interpret the modal operator: :1:l: Op iff for some y both Ray and Py hold. Observe that this truth definition for the diamond 0 involves two sorts of objects (a relation and a set), and that it involves only two individual vari ables. Indeed, as was first observed by Dov Gabbay, when interpreted on models the standard modal language is essentially a restricted fragment of a first-order language employing only two variables. Moreover, due to the restricted quantification in the truth definition of O, the satisfiability problem for .M£(<>) is decidable, and the model theory of the language can be analyzed quite elegantly with bisimulations. This restricted quantification also makes sure that we have a decent proof theory for /\/l£(<>). 1.2 EXTENDING MODAL LOGIC Quite often the need arises to go beyond a given description language. In the case of modal logic viewed as a tool for talking about relational structures this is usually because of one of the following two reasons: 4 What this dissertation is about [l.3 —it may be necessary to capture more aspects of the structures we work with than we are able to with our current modal language: we need to increase the expressive power of our description language, —as interests and research directions change new kinds of structures come into focus, calling for novel languages that employ appropriate vocabular ies. An extensive list of examples illustrating both of the above phenomena is given in Chapter 2. Chapters 3, 4 and 5 below contain case studies of formalisms whose raison d’étre may be traced back to one (or both) of the above points. The standard modal language cannot express several natural properties of its underlying semantic structures. Chapter 3 analyzes a simple proposal to overcome some of the most striking of these defects: the addition of a difierence operator D whose semantic definition reads: Dd) is true at a state iff (bis true at a dzfierent state. In recent years modal logic has found novel applications in artificial intel ligence, natural language analysis, knowledge representation, and theoretical computer science at large.

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