Equality control methods in machine theorem proving

ConclusionIn this survey we have not considered any of the issues related to the design of problemoriented machine proving systems and the use of heuristic methods. This does not mean that these issues are ignored by the scientific community. On the contrary, combination of mathematical-logic rules with heuristics in machine proving procedures is a necessary condition for achieving useful results. An important example of such combination is the mixed unification considered in subsec. 2.4. A logical conclusion of this approach should involve replacing the general unification algorithm with special-purpose equation solvers, assuming that this is allowed by the relevant theory. (Such an approach has already been sketched in logic programming [56].) There are examples of such heuristics, but their development is not always accompanied by sufficiently complete mathematical argument, which unfortunately is characteristic of many propositions in the heuristic framework.

[1]  Jieh Hsiang,et al.  Refutational Theorem Proving Using Term-Rewriting Systems , 1985, Artif. Intell..

[2]  Philip T. Cox,et al.  Deduction Plans: A Basis for Intelligent Backtracking , 1981, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[3]  Hisao Tamaki,et al.  Semantics of a Logic Programming Language with a Reducibility Predicate , 1984, SLP.

[4]  Larry Wos,et al.  The Concept of Demodulation in Theorem Proving , 1967, JACM.

[5]  James B. Morris,et al.  E-Resolution: Extension of Resolution to Include the Equality Relation , 1969, IJCAI.

[6]  Vincent J. Digricoli.  The Management of Heuristic Search in Boolean Experiments with Rue Resolution , 1985, IJCAI.

[7]  Mark E. Stickel,et al.  Complete Sets of Reductions for Some Equational Theories , 1981, JACM.

[8]  Alberto Martelli,et al.  An Efficient Unification Algorithm , 1982, TOPL.

[9]  J. S. Moore,et al.  Proof-Checking, Theorem Proving, and Program Verification. , 1983 .

[10]  Jean-Pierre Jouannaud,et al.  Termination of a Set of Rules Modulo a Set of Equations , 1984, CADE.

[11]  Vincent J. Digricoli.  Automatic deduction and equality , 1979, ACM '79.

[12]  Jia-Huai You,et al.  Conceptual Basis and Evaluation Strategies for Integrating Functional and Logic Programming , 1984, SLP.

[13]  M. Stone.  The theory of representations for Boolean algebras , 1936 .

[14]  J. Robinson.  An Overview of Mechanical Theorem Proving , 1970 .

[15]  Etienne Paul,et al.  A New Interpretation of the Resolution Principle , 1984, CADE.

[16]  G. Huet,et al.  Equations and rewrite rules: a survey , 1980 .

[17]  Michael Gelfond,et al.  Theory of deductive systems and its applications , 1987 .

[18]  Richard C. T. Lee,et al.  Symbolic logic and mechanical theorem proving , 1973, Computer science classics.

[19]  James R. Slagle,et al.  Automated Theorem-Proving for Theories with Simplifiers Commutativity, and Associativity , 1974, JACM.

[20]  Gerald E. Peterson,et al.  A Technique for Establishing Completeness Results in Theorem Proving with Equality , 1980, SIAM J. Comput..

[21]  Malcolm C. Harrison,et al.  Another Generalization of Resolution , 1978, JACM.

[22]  Robert Anderson.  Completeness results for e-resolution , 1970, AFIPS '70 (Spring).

[23]  Jean-Marie Hullot,et al.  Canonical Forms and Unification , 1980, CADE.

[24]  Claude Kirchner,et al.  Incremental Construction of Unification Algorithms in Equational Theories , 1983, ICALP.

[25]  Viktor Mikhaĭlovich Glushkov,et al.  An Introduction to Cybernetics , 1957, The Mathematical Gazette.

[26]  Lawrence J. Henschen,et al.  Semantic Paramodulation for Horn Sets , 1983, IJCAI.

[27]  Lawrence J. Henschen,et al.  A New Hyperparamodulation Strategy for the Equality Relation , 1985, IJCAI.

[28]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[29]  Laurent Fribourg.  Oriented Equational Clauses as a Programming Language , 1984, J. Log. Program..

[30]  Jörg H. Siekmann,et al.  Universal Unification , 1982, GWAI.

[31]  Lawrence J. Henschen,et al.  Hyperparamodulation: A Refinement of Paramodulation , 1980, CADE.

[32]  Nachum Dershowitz,et al.  Orderings for term-rewriting systems , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[33]  Werner Dilger,et al.  ASSIP-T: A Theorem-Proving Machine , 1985, IJCAI.

[34]  Gérard P. Huet,et al.  A Complete Proof of Correctness of the Knuth-Bendix Completion Algorithm , 1981, J. Comput. Syst. Sci..

[35]  Nachum Dershowitz,et al.  Rewrite Methods for Clausal and Non-Clausal Theorem Proving , 1983, ICALP.

[36]  José Meseguer,et al.  Equality, Types, Modules, and (Why not ?) Generics for Logic Programming , 1984, J. Log. Program..

[37]  Daniel Brand,et al.  Proving Theorems with the Modification Method , 1975, SIAM J. Comput..