Logic and its Applications

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1 Logic and its Applications Edmund Burke and Eric Foxley PRENTICE HALL London New York Toronto Sydney Tokyo Singapore Madrid Mexico City Munich

Contents Preface xiii Propositional logic 1 1.1 Informal introduction 1 1.2 Logical connectives 2 1.2.1 Negation (not) 2 1.2.2 Conjunction (and) 2 1.2.3 Disjunction (or) 4 1.2.4 Implication 5 1.2.5 Equivalence 1 1.

2 Contents Preface xiii Propositional logic Informal introduction Logical connectives Negation (not) Conjunction (and) Disjunction (or) Implication Equivalence Sum and product notations Priorities of Operators Truth-tables of formulae How to construct the truth-table of a formula Identical truth-tables Interpretations and modeis Tautologies, absurdities and mixed formulae Other logical connectives Truth functions Monodie Operators Dyadic Operators Triadic Operators Representing truth functions in terms ofdyadic and monadic Operators Manipulating propositional formulae Standard identities Complete sets of connectives Other complete sets of connectives Sheffer functions 23 V

Contents 1.5.5 Normalforms 1.6 The negation of propositional formulae 1.6.1 Definition 1.6.2 Generalized De Morgan's law 1.6.3 Extended disjunction and cönjunction 1.6.4 Duality 1.

3 Contents Normalforms 1.6 The negation of propositional formulae Definition Generalized De Morgan's law Extended disjunction and cönjunction Duality 1.7 Arguments and argument forms Some definitions associated with formulae Some rules for propositional formulae The validity of an argument Mathematical if-and-only-if proofs A theorem Another theorem 1.8 Summary 1.9 Worked examples 1.10 Exercises Formal approach to propositional logic 2.1 Introduction Formal Systems of propositional logic Proofs and deductions Constructing formal Systems ie relationship between formal Systems and interpretations 2.2 The formal propositional logic system L The construction of system L Proofs in system L Deductions in system L Derived rules of inference in system L Examples Notation for rules 2.3 The soundness and completeness theorems for system Introduction The soundness theorem for system L The completeness theorem for system L 2.4 Independence of axioms and rules 2.5 Lemmon's system of propositional logic An introduction to the system Proofs and deductions in Lemmon's system Examples of deductions in Lemmon's system 2.6 Summary 2.7 Worked examples 2.8 Exercises

Contents vii 3 Applications to logic design 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Introduction Simplification techniques 3.2.1 A simple example 3.2.2 Karnaugh maps 3.2.3 Quine-McClusky minimization Universal decision elements (UDEs) 3.

4 Contents vii 3 Applications to logic design Introduction Simplification techniques A simple example Karnaugh maps Quine-McClusky minimization Universal decision elements (UDEs) Definition A few four-variable universal decision elements Logic design Binary arithmetic adders Sequential logic Summary Worked examples Exercises 4 Predicate logic 4.1 Informal introduction Background Universal and existential quantifiers Translating between first-order languages and the English language Hints for translating from English to logic Examples Summary Exercises 4.2 The semantics of predicate logic First-order languages Interpretations Satisfaction Truth-tables of interpretations Herbrand interpretations Summary Worked examples Exercises 4.3 Syntactical Systems of predicate logic The system K of predicate logic Discussion of the system K First-order theories Summary Worked example Exercises 4.4 Soundness and completeness

viii Contents 4.4.1 Introduction 175 4.4.2 The soundness of System K 176 4.4.3 Consistency 180 4.4.4 The completeness of System K 180 4.4.5 Summary 185 4.4.6 Worked examples 185 4.4.7 Exercises 188 5 Logic programming 190 5.

5 viii Contents Introduction The soundness of System K Consistency The completeness of System K Summary Worked examples Exercises Logic programming Introduction Programming with propositional logic Definitions for propositional logic Propositional resolution Refutation and deductions Negation in logic programming SLD-resolution Clausal form for predicate logic Prenexform Clausal form Hörn clauses The semantics of logic programming Hörn clauses and their Herbrand modeis Logic programs and their Herbrand modeis Least Herbrand modeis Construction of least Herbrand modeis Unification and answer substitutions Substitutions Unification Practicalities Programming with predicate logic The resolution rule The proof strategy of Prolog: SLD-resolution Negation in logic programming: the closed-world assumption Concluding remarks Worked examples Exercises Formal System specification Introduction A simple example A State Schema Operations or events and their Schema 235

Contents ix 6.1.4 Pre-and post-conditions 241 6.2 Notational differences 241 6.3 The Z specification language 243 6.3.1 Basic type definitions 244 6.3.2 Free type definitions 244 6.3.3 Schema inclusion 245 6.

6 Contents ix Pre-and post-conditions Notational differences The Z specification language Basic type definitions Free type definitions Schema inclusion Schema types Example: a Computer file system Axiom schema Schema algebra Linearnotation Schema extension Some other types of definition Schema inclusion The tuple and pred Operators Ornamentation of schema names Logical Operations on schema Schema quantification Identifier renaming Identifier hiding Schema pre-condition Schema composition Schema piping Axiomatic descriptions Summary Worked examples Some simple examples Case study: a video-rental shop Case study: a car-ferry terminal Exercises 282 Appendix A: Mathematical background 284 AI Induction proofs 284 A2 Set theory 286 A2.1 Comprehensive specification of a set 286 A2.2 Operations involving sets 287 A3 Bags 288 A4 Relations 289 A4.1 Domain and ränge 289 A4.2 Composition 290 A4.3 Domain and ränge Operations 290 A4.4 Override Operation 292 A4.5 Set Image 292

7 x Contents A5 A6 A4.6 Equivalence relations Functions Sequences Appendix B: Other notations Bl Alternative notations B2 Polish notation B3 Worked examples B4 Exercises Appendix C: Symbols used in the book 302 Index 305

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