HANDBOOK OF TABLEAU METHODS

Transcription

1 HANDBOOK OF TABLEAU METHODS

2 HANDBOOK OF TABLEAU METHODS Edited by MARCELLO D' AGOSTINO Universita di Ferrara, Ferrara, Italy DOV M. GABBAY King's College, London, United Kingdom REINER HAHNLE Universitiit Karlsruhe, Germany and JOACHIM POSEGGA Deutsche Telekom AG, Research Centre, Darmstadt, Germany SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

3 A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOI / Printed on acid-fru pa per AlI Rights Reserved I999 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1999 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, inc1uding photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

4 CONTENTS Preface Introduction Melvin Fitting Tableau Methods for Classical Propositional Logic Marcello D' Agostino First-order Tableau Methods Reinhold Letz Equality and other Theories Bernhard Beckert Tableaux for Intuitionistic Logics Arild Waaler and Lincoln Wallen Tableau Methods for Modal and Temporal Logics Rajeev Gore Tableau Methods for Substructural Logics Marcello D'Agostino, Dov Gabbay and Krysia Broda Tableaux for Nonmonotonic Logics Nicola Olivetti Tableaux for Many-valued Logics Reiner Hiihnle Implementing Semantic Tableaux Joachim Posegga and Peter Schmitt A Bibliography on Analytic Tableaux Theorem Proving Graham Wrightson Index vii

5 PREFACE Recent years have been blessed with an abundance of logical systems, arising from a multitude of applications. A logic can be characterised in many different ways. Traditionally, a logic is presented via the following three components: 1. an intuitive non-formal motivation, perhaps tie it in to some application area 2. a semantical interpretation 3. a proof theoretical formulation. There are several types of proof theoretical methodologies, Hilbert style, Gentzen style, goal directed style, labelled deductive system style, and so on. The tableau methodology, invented in the 1950s by Beth and Hintikka and later perfected by Smullyan and Fitting, is today one of the most popular, since it appears to bring together the proof-theoretical and the semantical approaches to the presentation of a logical system and is also very intuitive. In many universities it is the style first taught to students. Recently interest in tableaux has become more widespread and a community crystallised around the subject. An annual tableaux conference is being held and proceedings are published. The present volume is a Handbook a/tableaux presenting to the community a wide coverage of tableaux systems for a variety of logics. It is written by active members of the community and brings the reader up to frontline research. It will be of interest to any formal logician from any area. The first chapter by Melvin Fitting contains a general introduction to the subject which can help the reader in finding a route through the following chapters, but can also be read on its own as a 'crash course' on tableaux, concentrating on the key ideas and the historical background. In Chapter 2, focusing on classical propositional logic, Marcello D' Agostino explores and compares the main types of tableau methods which appear in the literature, paying special attention to variants and 'improvements' of the original method. Chapter 3, by Reinhold Letz, investigates the impact of tableaux on the challenging problems of classical quantification theory and in Chapter 4 Bernhard Beckert looks at various methodologies for equality reasoning. The treatment of non-classical tableaux is started in Chapter 5, by Lincoln Wallen and Arild Waaler, which deals with the fundamental topic of intuitionistic logic. An area in which the tableau methodology is proving particularly useful is that of modal logics. These, together with their temporal 'neighbours' are studied in Chapter 6 by Rajeev Gore. Chapter 7, by Marcello vii

6 viii D' Agostino, Dov Gabbay and Krysia Broda, discusses the new family of 'substructural logics' which include the well known relevance logic and linear logic. Another frontier of today's logical research where a tableau-style approach seems especially well-suited is that of non-monotonic logics which are treated in Chapter 8 by Nicola Olivetti. Chapter 9, by Reiner Halmle, offers a treatment of manyvalued logics, a traditional topic which has now re-gained popularity owing to its important computer science applications. Finally, in Chapter 10, Joachim Posegga and Peter Schmitt, present an intriguing 'minimalist' approach to the implementation of classical tableaux. The volume concludes with an extensive annotated bibliography compiled by Graham Wrightson. Some of the chapters overlap considerably in their contents. This was a deliberate choice, motivated primarily by the need for making each chapter selfcontained. However, we believe that different expositions of the same notions and ideas can also contribute to a deeper understanding of the subject. ACKNOWLEDGEMENTS We would like to thank all our friends and colleagues who helped with the Handbook project. Among them are the various chapter second readers: Roy Dyckhoff; Luis Farinas del Cerro; Jaakko Hintikka; Wilfrid Hodges; Donald Loveland; Daniele Mundici; and Paliath Narendran. We would also like to thank Mrs Jane Spurr for her usual efficient and dedicated production and administration work. Finally thanks are due to our Kluwer Editor, Mrs A. Kuipers for her patience and understanding while waiting for the completion of the book. The Editors M. D' Agostino, D. M. Gabbay, R. Hahnle and J. Posegga Ferrara, London, Karlsruhe and Darmstadt