Dynamic Logic David Harel, The Weizmann Institute Dexter Kozen, Cornell University Jerzy Tiuryn, University of Warsaw The MIT Press, Cambridge, Massac

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1 Dynamic Logic David Harel, The Weizmann Institute Dexter Kozen, Cornell University Jerzy Tiuryn, University of Warsaw The MIT Press, Cambridge, Massachusetts, 2000 Among the many approaches to formal reasoning about programs, Dynamic Logic enjoys the singular advantage of being strongly related to classical logic. Its variants constitute natural generalizations and extensions of classical formalisms. For example, Propositional Dynamic Logic (PDL) can be described as a blend of three complementary classical ingredients: propositional calculus, modal logic, and the algebra of regular events. In First-Order Dynamic Logic (DL), the propositional calculus is replaced by classical rst-order predicate calculus. Dynamic Logic is a system of remarkable unity that is theoretically rich as well as of practical value. It can be used for formalizing correctness specications and proving rigorously that those specications are met by a particular program. Other uses include determining the equivalence of programs, comparing the expressive power of various programming constructs, and synthesizing programs from specications. This book provides the rst comprehensive introduction to Dynamic Logic. It is divided into three parts. The rst part reviews the appropriate fundamental concepts of logic and computability theory and can stand alone as an introduction to these topics. The second part discusses PDL and its variants, and the third part discusses DL and its variants. Examples are provided throughout, and exercises and a short historical section are included at the end of each chapter. Contents I Fundamental Concepts Mathematical Preliminaries Notational Conventions Sets...3 A Note on Foundations...4 Sets and Classes Relations Binary Relations...6 Equivalence Relations...8 Functions....9 Partial Orders Well-Foundedness and Induction Graphs and Dags Lattices Transnite Ordinals...13 Set-Theoretic Denition of Ordinals...14 Transnite Induction Zorn's Lemma and the Axiom of Choice Set Operators...16 Monotone, Continuous, and Finitary Operators Prexpoints and Fixpoints...18 Closure Operators...19

2 The Knaster{Tarski Theorem Bibliographical Notes Exercises Computability and Complexity Machine Models Deterministic Turing Machines Nondeterministic Turing Machines...33 Alternating Turing Machines...34 Universal Turing Machines and Undecidability Complexity Classes Time and Space Complexity...38 Oracle Machines and Relative Computability...40 Recursive and R.E. Sets...42 The Arithmetic Hierarchy...42 The Analytic Hierarchy Reducibility and Completeness Reducibility Relations Completeness Tiling Problems Bibliographical Notes Exercises Logic What is Logic? Languages...68 Models, Satisfaction, and Validity...68 Deduction Propositional Logic...71 Syntax...71 Semantics...73 Set-Theoretic Representation...76 A Deductive System The Deduction Theorem...79 Completeness Compactness...81 An Equational System Equational Logic...86 Syntax...86 Semantics...88 The Quotient Construction A Deductive System The HSP Theorem Predicate Logic Syntax Scope, Bound and Free Variables Semantics A Deductive System...111

3 Completeness with Equality Compactness The Lowenheim{Skolem Theorem Undecidability Ehrenfeucht{Frasse Games Innitary Logic Syntax An Innitary Deductive System The Downward Lowenheim{Skolem Theorem Complexity Modal Logic Propositional Modal Logic Multimodal Logic Unwinding Modal Logic and Programs Bibliographical Notes Exercises Reasoning About Programs What are Programs? States and Executions Programming Constructs While Programs Regular Programs Recursion R.E. Programs Nondeterminism Program Verication Partial and Total Correctness Hoare Logic Exogenous and Endogenous Logics Bibliographical Notes Exercises II Propositional Dynamic Logic Propositional Dynamic Logic Syntax Semantics Computation Sequences Satisability and Validity A Deductive System Basic Properties Properties Inherited from Modal Logic Properties of [, ;, and? The Converse Operator, The Iteration Operator Reexive Transitive Closure and Induction

4 5.7 Encoding Hoare Logic Bibliographical Notes Exercises Filtration and Decidability The Fischer{Ladner Closure Filtration and the Small Model Theorem Filtration over Nonstandard Models Bibliographical Notes Exercises Deductive Completeness Deductive Completeness Logical Consequences Bibliographical Notes Exercises Complexity ofpdl A Deterministic Exponential-Time Algorithm ALower Bound Compactness and Logical Consequences Bibliographical Notes Exercises Nonregular PDL Context-Free Programs Basic Results Undecidable Extensions Two-Letter Programs One-Letter Programs Decidable Extensions Tree Models Pushdown Automata on Innite Trees Decidability for Simple-Minded Languages Other Decidable Classes More on One-Letter Programs A Decidable Case Cases with no Finite Model Property Bibliographical Notes Exercises Other Variants of PDL Deterministic PDL and While Programs Restricted Tests Representation by Automata Complementation and Intersection Converse Well-Foundedness and Total Correctness...271

5 10.7 Concurrency and Communication Bibliographical Notes III First-Order Dynamic Logic First-Order Dynamic Logic Basic Syntax Richer Programs Seqs and R.E. Programs Arrays and Stacks Wildcard Assignment Semantics States as Valuations Assignment Statements Programs and Formulas Satisability and Validity Bibliographical Notes Exercises Relationships with Static Logics The Uninterpreted Level Uninterpreted Reasoning: Schematology Failure of Classical Theorems Expressive Power The Interpreted Level Interpreted Reasoning: Arithmetical Structures Expressive Power over N Bibliographical Notes Exercises Complexity The Validity Problem The Uninterpreted Level: Validity The Interpreted Level: Validity over N Spectral Complexity Coding Finite Structures Spectra Bibliographical Notes Exercises Axiomatization The Uninterpreted Level Completeness for Termination Assertions Innitary Completeness for the General Case The Interpreted Level Relative Completeness for Correctness Assertions Arithmetical Completeness for the General Case Bibliographical Notes

6 Exercises Expressive Power The Unwind Property Spectra and Expressive Power Bounded Nondeterminism Regular Programs Boolean Stacks Algebraic Stacks and Beyond Unbounded Memory Polyadic Vocabulary Monadic Vocabulary The Power of a Boolean Stack Unbounded Nondeterminism Bibliographical Notes Exercises Variants of DL Algorithmic Logic Nonstandard Dynamic Logic Well-Foundedness Dynamic Algebra Probabilistic Programs Concurrency and Communication Bibliographical Notes Other Approaches Logic of Eective Denitions Temporal Logic The Inductive Assertions Method The Temporal Approach Expressiveness The Until Operator Concurrency and Nondeterminism Complexity and Deductive Completeness Embedding TL in DL Process Logic Axiomatization The -Calculus Kleene Algebra Kleene Algebra with Tests References Notation and Abbreviations Index

EXTENSIONS OF FIRST ORDER LOGIC

EXTENSIONS OF FIRST ORDER LOGIC Maria Manzano University of Barcelona CAMBRIDGE UNIVERSITY PRESS Table of contents PREFACE xv CHAPTER I: STANDARD SECOND ORDER LOGIC. 1 1.- Introduction. 1 1.1. General