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EATCS Monographs on Theoretical Computer Science Volume 24 Editors: W. Brauer G.Rozenberg A.Salomaa Advisory Board: G.Ausiello M.Broy S.Even J.Hartmanis N.Jones T.Leighton M.Nivat C.Papadimitriou D.Scott Tamäs Gergely LäszlöÜry First-Order Programming Theories With 10 Figures Springer-Verlag Berlin Heidelberg GmbH Authors Prof. Dr. Tamäs Gergely Applied Logic Laboratory P. O. Box 675,1539 Budapest ,Hungary Dr.Läszlö Üry Applied Logic Laboratory P. O. Box 675,1539 Budapest, Hungary Editors Prof. Dr. Wilfried Brauer Institut für Informatik, Technische Universität München Arcisstrasse 21, D-8000 München 2, FRG Prof. Dr. Grzegorz Rozenberg Institute of Applied Mathematics and Computer Science University of Leiden, Niels-Bohr-Weg 1, P. 0. Box 9512 NL-2300 RA Leiden, The Netherlands Prof. Dr. Arto Salomaa Department of Mathematics, University of Turku SF-20500 Turku 50, Finland ISBN 978-3-642-63503-8 ISBN 978-3-642-58205-9 (eBook) DOI 10.1007/978-3-642-58205-9 Library of Congress Cataloging-in-Publication Data Gergely, T. (Tamäs) First-order programming theories / by Tamäs Gergely and Laszlö Üry. p. cm. - (EATS mono graphs on theoretical computer science: v. 24) Includes bibliographical references and index. 1. Electronic digital computers - Programming. I. Üry, Läszlö, 1953- . II. Title III. Series. QA76.6.G4744 1991 005.1-dc20 91-27370 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and a permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Softcover reprint of the hardcover 1st edition 1991 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Data coversion by authors 45/3140 - 543210 - Printed on acid-free paper Preface This work presents a purely classical first-order logical approach to the field of study in theoretical computer science sometimes referred to as the theory of programs, or programming theory. This field essentially attempts to provide a precise mathematical basis for the common activities involved in reasoning about computer programs and programming languages, and it also attempts to find practical applications in the areas of program specification, verification and programming language design. Many different approaches with different mathematical frameworks have been proposed as a basis for programming theory. They differ in the mathe matical machinery they use to define and investigate programs and program properties and they also differ in the concepts they deal with to understand the programming paradigm. Different approaches use different tools and viewpoints to characterize the data environment of programs. Most of the approaches are related to mathe matical logic and they provide their own logic. These logics, however, are very eclectic since they use special entities to reflect a special world of programs, and also, they are usually incomparable with each other. This Babel's mess irritatedus and we decided to peel offthe eclectic com ponents and try to answer all the questions by using classical first-order logic. In our opinion, this is the logic even programmers and pragmatically oriented peoplearefamiliar with, and ifthis logic becomes appropriate to serve as a ba sis for programming theory, then this theory will satisfy the requirement that it speak about the world under consideration as simply as Possi9le. This work reflects our journey from the eclectic programming logics to the simple classical first-order logic, which furthermore turned out to be more powerful than the non-classical ones. Putting this long journey into a book we had B. Mahr's full support and encouragement, for which we would like to thank him. We also acknowledge Springer-Verlag for their kind, willing and supportive cooperation and the unknown referees for their invaluable comments on our book. We are espe cially grateful to J. A. Ross, Copy Editor, for his scrupulous inspection ofthe manuscript and whose work hopefully led to a smooth reading ofthis book. VI Preface We are indebted to H. Andreka, I. Nemeti and I. Sain for our fruitful discussions of the subject in the past sixteen years. Special thanks go to our colleagues M. Erdelyi-Szab6, A. Juhos, A. Kurucz and M. Szots at the Ap plied Logic Laboratory, Budapest for their helpful comments and criticisms on reading thepenultimateversionofourmanuscript. Finally, we thank ourwives Lydia and Agnes for their support and passionate involvement in bringing our book to this final shape. Budapest, August 1991 Tamas Gergely, Laszlo Dry Contents Introduction 1 Mathematical Background 1. Logic and Model Theory 15 1.1 Basic Mathematical Notions 15 1.2 Derivation Systems 18 1.3 Basic Notions ofMany-Sorted First-Order Language 21 1.4 Relations in Many-Sorted Structures 28 1.5 Ultraproducts 29 1.6 Arithmetic 30 2. Inductive Deflnability 37 Part I Computability 3. Introduction to Part I 49 4. Main Properties ofProgram Schemas 53 4.1 Basic Definitions ofProgram Schemas 54 4.2 Flowchart Schemas 58 4.3 Main Properties ofO"-type Programs 63 4.4 Canonical Form ofPrograms 70 5. Extension ofProgram Schemas 77 5.1 Main Properties ofExtensions 78 5.2 Set Extension 84 5.3 Halting Problem for Extensions 88 6. Program Schemas with Stacks 91 6.1 On the Number ofStacks 91 6.2 Deterministic Programming Languages 99 7. Computability 103 7.1 Computability on First-Order Structures 103 7.2 Church's Thesis 112 8. On Inductive Deflnability of1- and 2-Computable Relations 119 8.1 Logical Characterization of I-Computable Relations 120 8.2 On the Role of Natural Numbers 127 VIn Contents Part II Extended Dynamic Logics 9. Introduction to Part II 135 10. Description ofProgram Properties 139 11. Den-based Descriptive Languages 143 12. The ProblemofCompleteness 155 12.1 Incompleteness Results 156 12.2 On Restricted Partial Correctness Theories 161 12.3 Den-Completeness 165 13. Dynamic Logic Generated by Extension 171 14. Continuous Denotational Semantics 179 14.1 Transitive Extension 180 14.2 Non-standard Dynamic Logic 193 14.3 Axiomatic Calculus for the Extended Dynamic Logic 202 14.4 On Program Equivalence 205 14.5 Constructing Transitive Extensions 206 15. Definable Denotational Semantics 211 Part III Temporal Characterization of Programs 16. Introduction to Part III 229 17. Temporal Logic 233 18. Temporal Logical Description ofProgram Properties 239 19. Is Temporal Logic Expressible in Dynamic Logic? 243 20. Is Dynamic Logic Expressible in Temporal Logic? 247 21. The Case ofEnumerable Models 263 22. Temporal Axiomatization ofProgramVerification Methods 267 Part IV Programming Logic with Explicit Time 23. Introduction to Part IV 285 24. Time Logic 289 24.1 Time Extension 289 24.2 On Regular Time Theories 292 25. Definability in Regular Time Theories 297 25.1 Finitary Fixed-Point Theory 298 25.2 Inductive Extensions Generated by Regular Theories 300 25.3 A Non-Finitary Fixed-Point Theory 305 26. Expressive Power ofTime 309 26.1 Embedding Temporal Logic 309 26.2 Embedding Transitive and Inductive Extensions 313 26.3 Description ofProgram Extensions 320 Contents IX Epilogue 325 References 335 Notations , 341 Subject Index 347 Introduction Progranuning Theory. The field ofstudy dealing with programs and program ming languages in theoretical computer science is usually referred to as pro gramming theory, or the theory of programs. This field essentially deals with the formal investigation of the world of programs, including characterization of programs, description of program properties and checking whether a given program satisfies certainproperties. Inorder tosupport this investigation, pro gramming theory attempts essentially to provide an appropriate mathematical foundation. This theory alsoattempts toprovideappropriate, theoretically well based methods and tools to support practical applications, e.g. in the field of program specification and verification and also in program design and in pro gramming language development. Conditionally we can divide programming theory into two main subfields. Programming languages and programs play the main role in the first subfield, while in the second one the languages which describe program properties have the leading role. We assign to the first area the research that aims to develop formal tools for the characterization ofprogramming languages and programs from different aspects. First of all we should provide appropriate formal tools to define programs as formal objects. These tools consider programs from the syntacticaspect. Thisaspectdealswiththe "how" specificationofprograms,i.e. it analyses how a program is to be given, how it is constructed, howoptimalor complex it is, etc. The programs, however, represent algorithms and areformal objects of a special kind. Namely, beyond the syntactic aspect they have a meaningconnectedwiththeirexecution. Executionofrealprogramstakesplace in real computers and results in changes in the contents of the storage units (registers) ofthe computers, which, in turn, form the data environment ofthe program under execution. Programming theory, however, deals with abstract programminglanguagesand, accordingly, theexecutionofprogramstakesplace inanappropriate "abstract" dataenvironment whichrepresents thecomputers. Therefore, the semantics of a programming language is connected with the characterization ofchanges caused by the program's execution in the data environment. Thesemanticsofprogramminglanguagesdeals withquestionswe may call "what"-questions, regarding what a program does or what properties it has. Having defined the program semantics we can investigate the computa tional power ofa programming language. Moreover, it is an important task to compare different programming languages with respect to their computational 2 Introduction power. To carryout this task, theformal characterizationofprogramminglan guages must be continued to the point where purely formal objects can be introduced to replace programming languages. The main problems the second subfield deals with are connected with the selection ofprogram properties that are interesting from theoretical and prag matical points of view. Having selected the interesting properties, a formal language must be developed which facilitates the description of these proper ties and theirinvestigation. Wealsorequireaformal calculusfor thedescriptive language which allows us to prove whether the program to be considered pos sesses any interesting properties. MathematicalFramework.Programmingtheoryshouldpossessappropriatefor mal methods and tools to deal successfully with the problems arising in both subfields. Moreover, it would be desirable to have a mathematical foundation which could provide a unique framework and view for both subfields. Differ ent approaches with different mathematical frameworks have been proposed as a basis for programming theory. They differ in the mathematical machinery they use to define and investigate programs and program properties and in the concepts they use to understand the programming paradigm. What mathematical framework will be adequate to characterize programs and programming languages and to reason about them from both syntactic and semantic perspectives? Mathematical logic iJ the only diJcipline which haJ a well developed culture to deal with the Jyntax and JemanticJ ofJome formally defined languageJ. However,inmathematicallogicthe meaningofa formal sen tence is characterized by its interpretation in the appropriate models and by analysing its validity with respect to these models. Logic systems by nature are Jtatic, while the meaningofalgorithms and so ofprograms connected with execution require dynamic consideration. Therefore, if we intend to use logic for developing a programming theory we have to find a solution for this con tradictory situation which provides appropriate logical tools to represent the dynamic aspect ofprograms, namely their execution. While we are looking for the required solution the following questions about the handling of program meaning should be answered: • What does a logic formulation characterize if it speaks about program execution? • What computationprocessesappearintheselectedlogicduringtheformal characterization? • Does program execution coincide with these computational processes? Thesequestionsshould beansweredwhile dealingwith thefirst subfieldof programming theory when we intend to represent and characterize programs in the scope of mathematical logic and to develop a corresponding theory of computation. Ofcourse,consideringthesecondsubfieldofprogrammingtheory, above all program properties should be selected which are interesting from a theoretical or pragmatic point ofview, and an appropriate language should be developed to describe and, possibly, to prove them. Ifwe wish to develop this