# Symbolic Logic and Mechanical Theorem Proving

**By**

- Chin-Liang Chang, Lockheed Missiles & Space Company, Inc., Menlo Park, CA
- Richard Lee, National Tsing Hua University, Hsinchu, Taiwan

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.

View full description### Audience

Senior college students, and first-year graduate students studying mathematics.

### Book information

- Published: May 1973
- Imprint: ACADEMIC PRESS
- ISBN: 978-0-12-170350-9

### Table of Contents

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