Modal Logics of Ordered Trees Ulrich Endriss Modal Logics of Ordered Trees Ulrich Endriss 2003 A thesis submitted to the University of London for the degree of Doctor of Philosophy King’s College London Department of Computer Science Revised Edition (March 2003) Abstract This thesis makes a contribution to the area of modal and temporal logics, an impor- tant family of non-classical logics that have found numerous applications in computer science and several other disciplines. Temporal logics, in particular, have been applied very successfully in the area of systems specification and verification. However, these logics often do not support the notion of refinement in a natural manner. There is no simple way to extend a given system specification, expressed in a standard temporal logic such as, for instance, propositional linear temporal logic, by the specification of a new subsystem. It is therefore important to extend available formalisms in ways that allow for the representation of complex systems evolving over time in a modular fashion. To this end, we propose to equip propositional linear temporal logics with a ‘zoom feature’ in the following manner. If we extend the class of modal frames of this logic by adding a vertical accessibility relation to connect every time point with a new time line, we obtain a tree-like structure where the children of each node form a linear order. More precisely, we obtain a modal logic with frames that are ordered trees. The horizontal accessibility relation is still given a temporal interpretation, while the vertical allows us to move between different levels of abstraction, that is, between a system and its subsystems. Our logic provides modal operators working both along the branches of a tree and along the order declared over the children of a node. The body of this work is devoted to the theoretical study of this modal logic of ordered trees. In particular, we are concerned with an axiomatic characterisation of our logic complementing its semantics-driven definition and with proving its decidability. Acknowledgements Many people have, in one way or another, played their part in the genesis of this thesis. Individual contributions range from very concrete support with my scientific endeavours to just being nice and making my years as a PhD student a very enjoyable time. In what follows, I shall mostly concentrate on the former type of contribution, but hope everyone else will get the message as well. First and foremost, I should like to thank Dov Gabbay, my thesis supervisor, for his encouragement and support throughout my studies, for being a great inspiration from Day One, for letting me do things my way, and for always sharing his insights on logic and people. The Group of Logic and Computation at King’s College (now the Group of Logic, Language and Computation) has provided the perfect environment for my studies. Everyone in the group, our numerous visitors, and many others in the Department of Computer Science have contributed to a very pleasant and stimulating working atmosphere. This includes, in particular, my first generation of office mates in 542: Stefan Schlobach, Odinaldo Rodrigues, and Wilfried Meyer-Viol. I’m also indebted to Hans J¨urgen Ohlbach for his help and support whilst being my second thesis supervisor at the beginning of my PhD studies, before he moved on to Munich. Stefan Schlobach, Wilfried Meyer-Viol, and Raquel Fern´andezhave read large parts of various earlier incarnations of this thesis and their feedback has substantially im- proved the final product. Thanks are also due to Misha Zakharyaschev, Agi Kurucz, and Valentin Shehtman for answering a plethora of questions on modal logic; and to the regulars at our logic reading group: David Gabelaia, Roman Kontchakov, Corinna Elsen- broich, and George Metcalfe. I have also greatly benefited from discussing my work with the many people I met along the way, in particular Martina Kullmann, Carsten Lutz, Enrico Franconi, Alessandro Artale, Anuj Dawar, and Ian Hodkinson. Naza Aguirre deserves a special mention for showing me that the software engineering people still need a lot more temporal logic. Finally, I would like to thank Howard Barringer and Robin Hirsch for having agreed to examine this work. On a more personal note, this seems like a good place to acknowledge the part my parents, Micha and Moni Endriß, have had in all of this. London, January 2003 U.E. The research reported here has been generously funded by the UK Engineering and Phys- ical Sciences Research Council (EPSRC) through a number of projects: GR/L91818/01 (grant holders: D. Gabbay and H. J. Ohlbach), GR/R45369/01 (M. Zakharyaschev, D. Gabbay, and I. Hodkinson), and GR/N23028/01 (D. Gabbay), all at King’s College London. The writing-up of the thesis was completed while the author was employed at Imperial College London on a research project funded by the European Union under grant reference number IST-2001-32530 (grant holders at Imperial: F. Toni and F. Sadri). Contents 1 Zooming in 1 1.1 Adding a Zoom to Linear Temporal Logic . 1 1.2 Thesis Overview . 10 2 Modal and Temporal Logics 13 2.1 Introducing Modalities . 13 2.2 Possible World Semantics . 18 2.3 Axioms and Correspondences . 24 2.4 Temporal Logic . 31 2.5 Bibliographic Notes . 37 3 Semantic Characterisation 39 3.1 Syntax and Semantics . 39 3.2 Some Defined Operators . 48 3.3 Correspondences . 50 3.4 Ontological Considerations . 55 3.5 General Models and P-morphisms . 60 3.6 Loosely Ordered Trees . 67 3.7 Summary . 94 4 Axiomatic Characterisation 97 4.1 Axioms and Rules . 97 4.2 Some Derived Theorems . 108 4.3 Completeness without Descendant Modalities . 113 4.4 Alternatives . 141 4.5 Summary . 145 5 Decidability 147 5.1 Monadic Second-order Logic . 147 5.2 Finite Binary Trees . 159 5.3 Discretely Ordered Trees . 163 5.4 General Ordered Trees . 169 5.5 Summary . 182 vii viii Contents 6 Zooming out 185 6.1 Extensions . 185 6.2 Summary of Main Results . 192 6.3 Open Problems . 193 Appendices 197 A Relations and Orders 199 A.1 Relations . 199 A.2 Linear Orders . 200 B Derivations 203 B.1 Proof of Lemma 4.8 .............................. 203 Bibliography 205 List of Figures 211 List of Tables 213 Index 215 Chapter 1 Zooming in In this thesis we develop a new modal logic to represent complex systems evolving over time in a modular fashion. As will be explained in this chapter, this new logic may be regarded as the result of extending well-known linear temporal logic by a second dimension that allows us to zoom into states and thereby model events associated with them in more detail. In this sense, our logic may be described as an extended temporal logic. From a more abstract point of view, our logic is best characterised as a modal logic of ordered trees. In this first chapter we motivate the enterprise of developing such a logic in the first place and give an informal outline of some of the main results reported in the sequel. The final section also provides a compact overview of the thesis as a whole, organised by chapters. 1.1 Adding a Zoom to Linear Temporal Logic Why? What do we mean by adding a zoom to temporal logic and why would anyone want to do something like this? We begin our explanation with a story, the story of Mary. Mary was an aspiring young software engineer who was working for an up-and-coming information technology company offering state-of-the-art software solutions to its clients and profitable revenue to its shareholders. One day, not that long ago, the company’s young and dynamic marketing department managed to secure a major contract with a new heavy-weight client. It was their biggest software development contract yet and a lot (if not everything) depended on its successful completion. Mary was appointed Chief Engineer on the project. Most people would have been intimidated by that much responsibility. Not Mary though. She was not only a true professional, but had also benefited from an excellent educational system which had provided her with a very respectable academic background. Mary had completed a degree in Computer Science at one of the country’s finest univer- sities. She was fluent in all the major programming languages, knew her computer inside out, and commanded an intimate knowledge of all phases of the software development 1 2 Chapter 1. Zooming in process, be it design, implementation, or evaluation of the final product. She was even good at maths. But was all of this going to be enough for this new and highly critical project? The central question seemed to be: How do you make sure the system will really do what it is supposed to? Mary was lucky. In her final year at university, besides all the programming lan- guages, software engineering, databases, and computer architecture, she had also at- tended a class on non-classical logics, which was at the time taught by a very famous logic professor. In fact, she probably did not even realise just how lucky she had been, because most of the time the famous professor was tied up in research, which regularly involved writing books, proving ground-breaking new results, and even developing whole new schools of thought. But this particular year he did teach and Mary enjoyed the course very much. One of the subjects that made a lasting impression on her was the field of temporal logics. She could remember it very well. Temporal logics were aston- ishingly simple, yet extremely powerful; full of mathematical depth and beauty, but also — as she had realised by now — of immediate industrial relevance.