Symbolic Logic and Mechanical Theorem Proving

This book contains an introduction to symbolic logic and a thorough discussion of mechanical theorem proving and its applications. The book consists of three major parts. Chapters 2 and 3 constitute an introduction to symbolic logic. Chapters 4-9 introduce several techniques in mechanical theorem proving, and Chapters 10 an 11 show how theorem proving can be applied to various areas such as question answering, problem solving, program analysis, and program synthesis.

Inhalt
Preface

Acknowledgments


1. Introduction


1.1 Artificial Intelligence, Symbolic Logic, and Theorem Proving


1.2 Mathematical Background


References


2. The Propositional Logic


2.1 Introduction


2.2 Interpretations of Formulas in the Propositional Logic


2.3 Validity and Inconsistency in the Propositional Logic


2.4 Normal Forms in the Propositional Logic


2.5 Logical Consequences


2.6 Applications of the Propositional Logic


References


Exercises


3. The First-Order Logic


3.1 Introduction


3.2 Interpretations of Formulas in the First-Order Logic


3.3 Prenex Normal Forms in the First-Order Logic


3.4 Applications of the First-Order Logic


References


Exercises


4. Herbrand's Theorem


4.1 Introduction


4.2 Skolem Standard Forms


4.3 The Herbrand Universe of a Set of Clauses


4.4 Semantic Trees


4.5 Herbrand's Theorem


4.6 Implementation of Herbrand's Theorem


References


Exercises


5. The Resolution Principle


5.1 Introduction


5.2 The Resolution Principle for the Propositional Logic


5.3 Substitution and Unification


5.4 Unification Algorithm


5.5 The Resolution Principle for the First-Order Logic


5.6 Completeness of the Resolution Principle


5.7 Examples Using the Resolution Principle


5.8 Deletion Strategy


References


Exercises


6. Semantic Resolution and Lock Resolution


6.1 Introduction


6.2 An Informal Introduction to Semantic Resolution


6.3 Formal Definitions and Examples of Semantic Resolution


6.4 Completeness of Semantic Resolution


6.5 Hyperresolution and the Set-of-Support Strategy: Special Cases of Semantic Resolution


6.6 Semantic Resolution Using Ordered Clauses


6.7 Implementation of Semantic Resolution


6.8 Lock Resolution


6.9 Completeness of Lock Resolution


References


Exercises


7. Linear Resolution


7.1 Introduction


7.2 Linear Resolution


7.3 Input Resolution and Unit Resolution


7.4 Linear Resolution Using Ordered Clauses and the Information of Resolved Literals


7.5 Completeness of Linear Resolution


7.6 Linear Deduction and Tree Searching


7.7 Heuristics in Tree Searching


7.8 Estimations of Evaluation Functions


References


Exercises


8. The Equality Relation


8.1 Introduction


8.2 Unsatisfiability under Special Classes of Models


8.3 Paramodulation-An Inference Rule for Equality


8.4 Hyperparamodulation


8.5 Input and Unit Paramodulations


8.6 Linear Paramodulation


References


Exercises


9. Some Proof Procedures Based on Herbrand's Theorem


9.1 Introduction


9.2 The Prawitz Procedure


9.3 The V-Resolution Procedure


9.4 Pseudosemantic Trees


9.5 A Procedure for Generating Closed Pseudosemantic Trees


9.6 A Generalization of the Splitting Rule of Davis and Putnam


References


Exercises


10. Program Analysis


10.1 Introduction


10.2 An Informal Discussion


10.3 Formal Definitions of Programs


10.4 Logical Formulas Describing the Execution of a Program


10.5 Program Analysis by Resolution


10.6 The Termination and Response of Programs


10.7 The Set-of-Support Strategy and the Deduction of the Halting Clause


10.8 The Correctness and Equivalence of Programs


10.9 The Specialization of Programs


References


Exercises


11. Deductive Question Answering, Problem Solving, and Program Synthesis


11.1 Introduction


11.2 Class A Questions


11.3 Class B Questions


11.4 Class C Questions


11.5 Class D Questions


11.6 Completeness of Resolution for Deriving Answers


11.7 The Principles of Program Synthesis


11.8 Primitive Resolution and Algorithm A (A Program-Synthesizing Algorithm)


11.9 The Correctness of Algorithm A


11.10 The Application of Induction Axioms to Program Synthesis


11.11 Algorithm A (An Improved Program-Synthesizing Algorithm)


References


Exercises


12. Concluding Remarks


References


Appendix A


A.1 A Computer Program Using Unit Binary Resolution


A.2 Brief Comments on the Program


A.3 A Listing of the Program


A.4 Illustrations


References


Appendix B


Bibliography


Index

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