Imperial College of Science, Technology and Medicine (University of London) Department of Computing A Generic Logic Environment by William Mark Grant Dawson A thesis presented to the University of London in partial fulfilment of the requirements for the degree of Doctor of Philosophy (Ph.D.) and the Diploma of Imperial College (D.I.C.) in Computing Science 3 A bstract The thesis presents a framework for the development and manipulation of a wide variety of logical systems. The approach is original in its aim of providing tools to assist the construction of logics, and the framework has been realised on a computer workstation as a logic development environment. Both the framework and the design and implementation of the prototype environment are described. The definition of a logical system is taken to have two components: the language from which the syntactic elements—including judgements—of the system are constructed; and a deductive part, in which the judgements can be interpreted as relations, expressed using inference rules. A method is given for transforming different syntactic presentations of logical systems into the all-introduction style required. This method is applied to Axiomatic and Natural Deduction presentations of several systems, including intuitionistic, relevance, linear, classical, modal and many-valued logics. It is shown that the framework is sufficiently expressive to capture presentations of these systems. A number of tools are provided for the analysis of systems presented within the environment. One of these checks whether a judgement holds within a given system, by constructing a derivation using inference rules. As it is intended that the environment be used during the development of logical systems, a simple strategy language is provided to select between inference rules. A second tool allows a user of the environment to conduct case studies which reveal relationships between connectives. In particular, it is shown how the distinct modalities of some classical and intuitionistic modal systems can be found, and their inter-relationships discovered. A third, and more fundamental, type of analysis is also discussed. When a new inference rule is added to a system, it may not allow anything new to be proved. When this is the case, the new rule is said to be derived with respect to the other rules. If a property of a judgement is expressed as an inference rule, and this rule is found to be derived with respect to the other—more computationally useful—rules, then the property was previously implicitly present. Such knowledge can be used to give results about the presentation of the system as a whole, for example whether the system is consistent. To my parents Tom and Jane 5 Contribution of the Thesis The thesis makes several contributions to the study of proof theoretic presentations of logical systems made in the spirit of Gentzen’s sequent calculi. A new, simple, but expressive framework is presented that supports the presentations of such systems in a natural way. This is done in Chapter 1, with aspects concerned with quantification treated in Chapter 4. The framework can also be readily implemented on a computer as is shown in Chapter 7. The next aspect of our contribution is a systematic study which shows how a variety of other, proof theoretic, presentations of logical systems can be transformed into a form suitable for the framework. This forms Chapter 2 and the first part of Chapter 3. Of particular interest is the way in which our approach reveals the nature and manipulation of assumptions in a system, and thereby which structural properties are required in the framework. The methodology is applied Classical and Intuitionistic logic in Chapter 2, where Linear and Relevant systems are also briefly discussed. The analysis is extended in Chapter 3 to a number of modal systems. A way of Gentzenising many-valued systems is also shown, as well as examples that illustrate how the framework treats hypersequents and defeasible systems. The final aspect of our contribution is an environment implemented on a computer workstation which supports the framework. The environment provides its users with access to a number of activities concerned with the manipulation and analysis of systems. The environment’s interface has novel aspects that are described in Chapter 6. The environment supports a user’s preferred syntactic presentation directly, and derivations made in a system are shown graphically as proof trees. One of the activities that is supported by the environment is the construction of a derivation in a system. The environment translates the user’s representation of a system into an efficient internal form that is used to construct such derivations. Chapter 7 gives details of this translation. The activity of making many such derivations in a structured way is illustrated in the second part of Chapter 3 for the case of Modal systems, where the relationship between strings of modalities is often of interest. We illustrate this analysis by showing how the environment was used to discover modalities in Modal systems based on Classical and Intuitionistic logics. Finally, Chapter 5 presents a novel discussion on the role of derived rules in a system presented using the framework. The argument made there yields a systematic search for schematic proofs that can replace derived rules. Examples are given of the application of this approach. 6 Acknowledgements First and foremost I would like to extend my heartfelt thanks to my supervisor Krysia Broda. Krysia has shown great patience, and given me much advice and assistance throughout my studies. I am greatly in her debt. It is a pleasure to have worked with Martin Sadler when he was at Imperial College. Martin’s influence was pervasive here, and he was always a source of inspiration and encouragement throughout the GENESIS project. I hope his influence can be seen in this thesis as well. I would like to thank Tom Maibaum and Samson Abramsky for their continuing support and encouragement. Friends, colleagues and students of the Department of Computing have been a constant source of stimulation. Paul Taylor deserves particular thanks for writing his excellent T^X macros that have greatly improved the presentation of the Thesis, as has his dedication to the task of providing a modern TgX environment at Imperial College. Special thanks must go to Jan-Simon Pendry for his work converting oj-prolog to C++, writing its module language and introducing many enhancements. David James’ TEXpertise fathomed major difficulties with the technology used to print the figures without scissors and glue. Mark Ryan, Krysia, Tom, Martin, Paul and David have kindly read drafts of this thesis and made many thoughtful comments and corrections; needless to say, any errors that remain are mine. My thanks go to them all, and to the many other people at the College who have helped me in one way or another. C ontents 1 Introduction 13 1.1 Logics in Computer Science........................................................................... 15 1.2 Thesis .................................................................................................................. 18 1.3 The Framework.................................................................................................. 19 1.4 A ctivities .................................................................................... 29 1.5 The environm ent...................................................................................................33 1.6 Structure of the th e sis......................................................................................... 33 2 Methodology 34 2.1 Introduction............................................................................................................ 34 2.2 Axiomatic Presentations......................................................................................35 2.3 Natural Deduction Presentations......................................................................38 2.4 From Axioms to Natural Deduction Rules ..................................................40 2.5 Introduction and Elimination Presentations..................................................42 2.6 ; From Natural Deduction to Sequents ........................................................43 2.7 All-introduction Sequent R u le s .........................................................................48 2.8 From Sequents to All-introduction R u le s ..................................................... 49 2.9 Equivalence of Axiomatic and Sequent P resentations .............................. 57 2.10 Conclusion................................................................................................................62 7 8 CONTENTS 3 Non-standard logics 64 3.1 Introduction............................................................................................................. 64 3.2 Modal L o g ic s ..........................................................................................................65 3.3 Determining Modalities..........................................................................................80 3.4 Logics of Knowledge andBelief............................................................................91 3.5 Three and more valued logics .......................................................... 94 3.6 Hypersequents........................................................................................................104 3.7 Defeasible R easoning...........................................................................................109