Handbook of Logic and Proof Techniques for Computer Science

Transcription

1 Steven G. Krantz Handbook of Logic and Proof Techniques for Computer Science With 16 Figures BIRKHAUSER SPRINGER BOSTON * NEW YORK

2 Preface xvii 1 Notation and First-Order Logic The Use of Connectives Elementary Statements 1 1~1.2 Connectives Redundancy of the Connectives Additional Connectives Truth Values and Truth Tables Rules for Truth Values and Tables Multivalued Logics Modal Logic Compound Sentences and Truth Values Tautologies and Contradictions Contrapositives The Use of Quantifiers "For All" and "There Exists" Relations Between "For All" and "There Exists" The Propositional and the Predicate Calculus Derivability Semantics and Syntax A Consideration of First-Order Theories Herbrand's Theorem An Example from Group Theory Godel's Completeness Theorem Provable Statements and Tautologies Formulation of Godel's Completeness Theorem Additional Terminology Some More Formal Language Other Formulations of Godel Completeness... 15

3 viii Contents The Compactness Theorem Tautological Implication and Provability Second-Order Logic Semantics 17 2 Semantics and Syntax Elementary Symbols Formal Systems (Syntax) Well-Formed Formulas or wffs [Syntax] Free and Bound Variables (Syntax) The Semantics of First-Order Logic Interpretations Truth First-Order Theories A Proof System for First-Order Logic Two Fundamental Theorems 23 3 Axiomatics and Formalism in Mathematics Basic Elements Undefinable Terms Description of Sets Definitions Axioms Lemmas, Propositions, Theorems, and Corollaries Rules of Logic Proofs Models Definition of Model Examples of Models Finite Model Theory Minimality of Models Universal Algebra Consistency Definition of Consistency Godel's Incompleteness Theorem Introductory Remarks Godel's Theorem and Arithmetic Formal Enunciation of Arithmetic Some Standard Terminology Enunciation of the Incompleteness Theorem Church's Theorem Additional Formulations of Incompleteness Relative Consistency 36

4 ix 3.5 Decidability and Undecidability Introduction to Decidability Recursive Equivalence; Degrees of Recursive Unsolvability Independence Introduction to Independence Examples of Independence 38 4 The Axioms of Set Theory Introduction Axioms and Discussion Axiom of Extensionality Sum Axiom Power Set Axiom Axiom of Regularity Axiom for Cardinals Axiom of Infinity Axiom Schema of Replacement Axiom of Choice Concluding Remarks 42 5 Elementary Set Theory Set Notation Elements of a Set Set-Builder Notation The Empty Set Universal Sets and Complements Set-Theoretic Difference Ordered Pairs; the Product of Two Sets Sets, Subsets, and Elements The Elements of a Set Venn Diagrams Binary Operations on Sets Intersection and Union Properties of Intersection and Union Other Set-Theoretic Operations Relations and Equivalence Relations What Is a Relation? Partial and Full Orderings Equivalence Relations What Is an Equivalence Relation? Equivalence Classes 52

5 5.5.3 Examples of Equivalence Relations and Classes Construction of the Rational Numbers Number Systems Functions What Is a Function? Examples of Functions One-to-One or Univalent Onto or Surjective Set-Theoretic Isomorphisms Cardinal Numbers Comparison of the Sizes of Sets Cardinality and Cardinal Numbers An Uncountable Set Countable and Uncountable Comparison of Cardinalities The Power Set The Continuum Hypothesis Martin's Axiom Inaccessible Cardinals and Measurable Cardinals Ordinal Numbers Mathematical Induction Transfinite Induction A Word About Classes Russell's Paradox The Idea of a Class Fuzzy Set Theory Introductory Remarks About Fuzzy Sets Fuzzy Sets and Fuzzy Points An Axiomatic Theory of Operations on Fuzzy Sets Triangular Norms and Conorms The Lambda Calculus Free and Bound Variables in the A-Calculus Substitution Examples Sequences Bags 82

6 xi 6 Recursive Functions Introductory Remarks A System for Number Theory Primitive Recursive Functions Effective Computability Effectively Computable Functions and p.r. Functions General Recursive Functions Every Primitive Recursive Function Is General Recursive Turing Machines An Example of a Turing Machine Turing Machines and Recursive Functions Defining a Function with a Turing Machine Recursive Sets Recursively Enumerable Sets The Decision Problem Decision Problems with Negative Resolution The ^-Operator 93 7 The Number Systems The Natural Numbers Introductory Remarks Construction of the Natural Numbers Axiomatic Treatment of the Natural Numbers The Integers Lack of Closure in the Natural Numbers The Integers as a Set of Equivalence Classes Examples of Integer Arithmetic Arithmetic Properties of the Negative Numbers The Rational Numbers Lack of Closure in the Integers The Rational Numbers as a Set of Equivalence Classes Examples of Rational Arithmetic Subtraction and Division of Rational Numbers The Real Numbers Lack of Closure in the Rational Numbers Axiomatic Treatment of the Real Numbers The Complex Numbers Intuitive View of the Complex Numbers Definition of the Complex Numbers 102

7 xii Contents The Distinguished Complex Numbers 1 and i Examples of Complex Arithmetic Algebraic Closure of the Complex Numbers The Quaternions Algebraic Definition of the Quaternions A Basis for the Quaternions The Cayley Numbers Algebraic Definition of the Cayley Numbers Nonstandard Analysis The Need for Nonstandard Numbers Filters and Ultrafilters A Useful Measure An Equivalence Relation An Extension of the Real Number System Methods of Mathematical Proof Axiomatics Undefinables Definitions Axioms Theorems, ModusPonendoPonens, and ModusTollens Proof by Induction Mathematical Induction Examples of Inductive Proof Complete or Strong Mathematical Induction Proof by Contradiction Examples of Proof by Contradiction Direct Proof Examples of Direct Proof Other Methods of Proof Examples of Counting Arguments The Axiom of Choice Enunciation of the Axiom Examples of the Use of the Axiom of Choice Zorn's Lemma The Hausdorff Maximality Principle The Tukey-Tychanoff Lemma A Maximum Principle for Classes Consequences of the Axiom of Choice Paradoxes The Countable Axiom of Choice Consistency of the Axiom of Choice 126

8 xiii 9.7 Independence of the Axiom of Choice Proof Theory General Remarks Cut Elimination Propositional Resolution Interpolation Finite Type Universes Conservative Systems Beth's Definability Theorem Introductory Remarks The Theorem of Beth Category Theory Introductory Remarks Metacategories and Categories Metacategories Operations in a Category Commutative Diagrams Arrows Instead of Objects Metacategories and Morphisms Categories Categories and Graphs Elementary Examples of Categories Discrete Examples of Categories Functors UNatural Transformations Algebraic Theories Complexity Theory Preliminary Remarks Polynomial Complexity Exponential Complexity Two Tables for Complexity Theory Table Illustrating the Difference Between Polynomial and Exponential Complexity Problems That Can Be Solved in One Hour Comparing Polynomial and Exponential Complexity Problems of Class P Polynomial Complexity Tractable Problems 149

9 xiv Contents Problems That Can Be Verified in Polynomial Time Problems of Class NP Nondeterministic Turing Machines NP Contains P The Difference Between NP and P Foundations of NP-Completeness Limits of the Intractability of NP Problems NP-Completeness Polynomial Equivalence Definition of NP-Completeness Intractable Problems and NP-Complete Problems Structure of the Class NP The Classes Pspace and Log-Space Cook's Theorem The Satisfiability Problem Enunciation of Cook's Theorem Examples of NP-Complete Problems Problems from Graph Theory Problems from Network Design Problems from the Theory of Sets and Partitions.' Storage and Retrieval Problems Sequencing and Scheduling Problems Problems from Mathematical Programming Problems from Algebra and Number Theory Game and Puzzle Problems Problems of Logic Miscellaneous Problems MoreonP/NP NPC and NPI Problems in NPI NP-Hard Problems Descriptive Complexity Theory Boolean Algebra Description of Boolean Algebra A System of Encoding Information Axioms of Boolean Algebra Boolean Algebra Primitives Axiomatic Theory of Boolean Algebra Boolean Algebra Interpretations Theorems in Boolean Algebra 170

10 xv Properties of Boolean Algebra A Sample Proof Illustration of the Use of Boolean Logic Boolean Algebra Analysis The Word Problem Introductory Remarks What Is a Group? First Consequences Subgroups and Generators Homomorphisms What Is a Free Group? The Definition Words The Word Problem Extensions of Homomorphisms An Illustrative Example Relations and Generators Consequences Generators and Relations Amalgams Free Product with Amalgamation The Free Product Finitely Presented Groups Description of the Word Problem The Word Problem and Recursion Theory Recursively Presented Groups Solvability of the Word Problem Novikov's Theorem 182 List of Notation from Logic 183 Glossary of Terms from Mathematical and Sentential Logic 189 A Guide to the Literature 219 Bibliography 231 Index 237