Handbook of Automated Reasoning

Part I. HistoryChapter 1. The Early History of Automated Deduction (Martin Davis).1. Presburger's Procedure.2. Newell, Shaw & Simon, and H. Gelernter.3. First-Order Logic.Bibliography. Index.Part II. Classical Logic.Chapter 2. Resolution Theorem Proving (Leo Bachmair, Harald Ganzinger).1. Introduction.2. Preliminaries.3. Standard Resolution.4. A Framework for Saturation-Based Theorem Proving.5. General Resolution.6. Basic Resolution Strategies.7. Refined Techniques for Defining Orderings and Selection Functions.8. Global Theorem Proving Methods.9. First-Order Resolution Methods.10. Effective Saturation of First-Order Theories.11. Concluding Remarks.Bibliography. Index.Chapter 3. Tableaux and Related Methods (Reiner Hähnle).1. Introduction.2. Preliminaries.3. The Tableau Method.4. Clause Tableaux.5. Tableaux as a Framework.6. Comparing Calculi.7. Historical Remarks & ResourcesBibliography. Notation. Index.Chapter 4. The Inverse Method (Anatoli Degtyarev, Andrei Voronkov).1. Introduction.2. Preliminaries.3. Cooking classical logic.4. Applying the recipe to nonclassical logics.5. Naming and connections with resolution.6. Season your meal: strategies and redundancies7. Path calculi8. Logics without the contraction rules9. ConclusionBibliography. IndexChapter 5. Normal Form Transformations (Matthias Baaz, Uwe Egly, Alexander Leitsch).1. Introduction.2. Notation and Definitions.3. On the Concept of Normal Form.4. Equivalence-Preserving Normal Forms.5. Skolem Normal Form.6. Conjunctive Normal Form.7. Normal Forms in Nonclassical Logics.8. Conclusion.Bibliography. Index.Chapter 6. Computing Small Clause Normal Forms (Andreas Nonnengart, Christoph Weidenbach).1. Introduction.2. Preliminaries.3. Standard CNF-Translation.4. Formula Renaming.5. Skolemization.6. Simplification.7. Bibliographic Notes.8. Implementation Notes.Bibliography. Index.Part III. Equality and other theories.Chapter 7. Paramodulation-Based Theorem Proving (Robert Nieuwenhuis, Albert Rubio).1. About this chapter.2. Preliminaries.3. Paramodulation calculi.4. Saturation procedures.5. Paramodulation with constrained clauses.6. Paramodulation with built-in equational theories.7. Symbolic constraint solving.8. Extensions.9. Perspectives.Bibliography. Index.Chapter 8. Unification Theory (Franz Baader, Wayne Snyder).1. Introduction.2. Syntactic unification.3. Equational unification.4. Syntactic methods for E-unification.5. Semantic approaches to E-unification.6. Combination of unification algorithms.7. Further topics.Bibliography. Index.Chapter 9. Rewriting (Nachum Dershowitz, David A. Plaisted).1. Introduction.2. Terminology.3. Normal Forms and Validity.4. Termination Properties.5. Church-Rosser Properties.6. Completion.7. Relativized Rewriting.8. Equational Theorem Proving.9. Conditional Rewriting.10. Programming.Bibliography. Index.Chapter 10. Equality Reasoning in Sequent-Based Calculi (Anatoli Degtyarev, Andrei Voronkov).1. Introduction.2. Translation of logic with equality into logic without equality.3. Free variable systems.4. Early history.5. Simultaneous rigid E-unification.6. Incomplete procedures for rigid E-unification.7. Sequent-based calculi and paramodulation.8. Equality elimination.9. Equality reasoning in nonclassical logics.10. Conclusion and open problems.Bibliography. Calculi and inference rules. Index.Chapter 11. Automated Reasoning in Geometry (Shang-Ching Chou, Xiao-Shan Gao).1. A history review of automated reasoning in geometry.2. Algebraic approaches to automated reasoning in geometry.3. Coordinate-free approaches to automated reasoning in geometry.4. AI approaches to automated reasoning in geometry.5. Final remarks. Bibliography. Index.Chapter 12. Solving Numerical Constraints (Alexander Bockmayr, Volker Weispfenning).1. Introduction.2. Linear constraints over fields.3. Linear diophantine constraints.4. Non-linear constraints over continuous domains.5. Non-linear diophantine constraints.Bibliography. Index.Part IV. Induction.Chapter 13. The ARobinson, Alan is the author of 'Handbook of Automated Reasoning' with ISBN 9780444829498 and ISBN 0444829490.