Logics of time and computation

LOGICS OF TIME AND COMPUTATION

CSLI Lecture Notes No. 7

LOGICS OF TIME AND COMPUTATION Second Edition Revised and Expanded

Robert Goldblatt

CSLI

CENTER FOR THE STUDY OF LANGUAGE AND INFORMATION

Copyright © 1992 Center for the Study of Language and Information Leland Stanford Junior University Printed in the United States CIP data and other information appear at the end of the book

To my daughter Hannah

Preface to the First Edition These notes are based on lectures, given at Stanford in the Spring Quarter of 1986, on modal logic, emphasising temporal and dynamic logics. The main aim of the course was to study some systems that have been found relevant recently to theoretical computer science. Part One sets out the basic theory of normal modal and temporal prepositional logics, covering the canonical model construction used for completeness proofs, and the filtration method of constructing finite models and proving decidability results and completeness theorems. Part Two applies this theory to logics of discrete (integer), dense (rational), and continuous (real) time; to the temporal logic of henceforth, next, and until, as used in the study of concurrent programs; and to the prepositional dynamic logic of regular programs. Part Three is devoted to first-order dynamic logic, and focuses on the relationship between the computational process of assignment to a variable, and the syntactic process of substitution for a variable. A completeness theorem is obtained for a proof theory with an infinitary inference rule. There is more material here than was covered in the course, partly because I have taken the opportunity to gather together a number of observations, new proofs of old theorems etc., that have occurred to me from time to time. Those familiar with the subject will observe, for instance, that in Part Two proofs of completeness for various logics of discrete and continuous time, and for the temporal logic of concurrency, as well as the discussion of Bull's theorem on normal extensions of S4.3, all differ from those that appear in the literature. In order to make the notes effective for classroom use, I have deliberately presented much of the material in the form of exercises (especially in Part One). These exercises should therefore be treated as an integral part of the text. Acknowledgements. My visit to Stanford took place during a period of sabbatical leave from the Victoria University of Wellington which was supported by both universities, and the Fulbright programme. I would like to thank Solomon Feferman and Jon Barwise for the facilities that were made available to me at that time. The CSLI provided generous access to its excellent computer-typesetting system, and the Center's Editor, Dikran Karagueuzian, was particularly helpful with technical advice and assistance in the preparation of the manuscript.

vu

Preface to the Second Edition The text for this edition has been increased by more than a third. Major additions are as follows. • §7, originally concerned with incompleteness, now discusses a number of other metatheoretic topics, including first-order definability, (in)validity in canonical frames, failure of the finite model property, and the existence of undecidable logics with decidable axiomatisation. • §9 now includes a study of the " branching time" system of Computational Tree Logic, due to Clarke and Emerson, which introduces connectives that formalise reasoning about behaviour along different branches of the tree of possible future states. Completeness and decidability are shown by the method of filtration in an adaptation of ideas due to Emerson and Halpern. • In §10 dynamic logic is extended by the concurrency command a fl/3, interpreted as "a and (3 executed in parallel". This is modelled by the use of "reachability relations", in which the outcome of a single execution is a set of terminal states, rather than a single state. This leads to a semantics for [ a ] and < a > which makes them independent (i.e. not interdefinable via negation). The resulting logic is shown to be finitely axiomatisable and decidable, by a new theory of canonical models and filtrations for reachability relations. A significant conceptual change involves the definition of a "logic" (p. 16), which no longer includes the rule of Uniform Substitution. Logics satisfying this rule are called Uniform, and are discussed in detail on page 23. The change causes a number of minor adaptations throughout the text. A notable technical improvement concerns the completeness proof for S4.3Dum in §8 (pp. 73-75). The original Dwm-Lemma has been replaced by a direct proof that non-last clusters in the filtration are simple. This has resulted in some re-arrangement of the material concerning Bull's Theorem, and a simplification of the completeness theorem for the temporal logic of concurrency in §9 (pp. 95-96). Other small changes include additional material about the Diodorean modality of spacetime (p. 45), and a rewriting of the basic filtration construction for dynamic logic (p. 114) using a uniform method of proving the first filtration condition that obviates the need to establish any standardmodel conditions for the canonical model. Reformatting the text has provided the opportunity to make numerous changes in style and expression, as well as te, correct typos. I will be thankful for, if not pleased by, information about any further such errors. rob @math. vuw. ac.nz Vlll

Contents

Preface to the First Edition Preface to the Second Edition

vii viii

Part One: Prepositional Modal Logic 1. Syntax and Semantics 3 2. Proof Theory 16 3. Canonical Models and Completeness 24 4. Filtrations and Decidability 31 5. Multimodal Languages 37 6. Temporal Logic 40 7. Some Topics in Metatheory 48

1

Part Two: Some Temporal and Computational Logics 8. Logics with Linear Frames 65 9. Temporal Logic of Concurrency 84 10. Prepositional Dynamic Logic 109 Part Three: First-Order Dynamic Logic 11. Assignments, Substitutions, and Quantifiers 12. Syntax and Semantics 146 13. Proof Theory 154 14. Canonical Model and Completeness 162 Bibliography Index

175

169

141 143

63

Part One

Prepositional Modal Logic

1

Syntax and Semantics

BNF The notation of Backus-Naur form (BNF) will be used to define the syntax of the languages we will study. This involves specifying certain syntactic categories, and then giving recursive equations to show how the members of those categories are generated. The method can be illustrated by the syntax of standard propositional logic, which has one main category, that of the formulae. These are generated from some set of atomic formulae (or propositional variables), together with a constant _L (the falsum), by the connective —> (implication). In BNF, this is expressed in one line as < formula > ::= < atomic formula > | ± | < formula >->< formula > The symbol ::= can be read "comprises", or "consists of", or simply "is". The vertical bar | is read "or". Thus the equation says that a formula is either an atomic formula, the falsum, or an implication between two formulae. The definition becomes even more concise when we use individual letters for members of syntactic categories, in the usual way. Let $ be a denumerable set of atomic formulae, with typical member denoted p. The set of all formulae generated from $ will be denoted Fma(