Deductive Generalization and Meta-Reasoning or How to Formalize Genesis

When using resolution for showing the unsatisfiability of logic formulas one often encounters infinite sequences of structurally similar clauses. We suggest to use deductive generalization to derive a small number of meta-clauses subsuming these infinite sequences. Resolution is extended by meta-unification in order to resolve meta-clauses instead of ordinary ones. As a step towards a unification procedure for general meta-terms we investigate the unification of monadic ones. We describe an algorithm yielding a finite, com­ plete and orthonormal set of unifiers. First experiments show the usefulness of our approach: theorem provers save space as well as time, proofs become considerably shorter.

[1]  J. A. Robinson,et al.  A Machine-Oriented Logic Based on the Resolution Principle , 1965, JACM.

[2]  Wolfgang Bibel.  Advanced Topics in Automated Deduction , 1987, Advanced Topics in Artificial Intelligence.

[3]  Michael Clausen,et al.  Efficient Solution of Linear Diophantine Equations , 1989, J. Symb. Comput..

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

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

[6]  Maurice Bruynooghe,et al.  On the Existence of Nonterminating Queries for a Restricted Class of PROLOG-Clauses , 1989, Artif. Intell..

[7]  Donald W. Loveland,et al.  Automated theorem proving: a logical basis , 1978, Fundamental studies in computer science.

[8]  J. A. Robinson,et al.  Automatic Deduction with Hyper-Resolution , 1983 .

[9]  Wolfgang Bibel,et al.  Automated Theorem Proving , 1987, Artificial Intelligence / Künstliche Intelligenz.

[10]  G. Makanin.  The Problem of Solvability of Equations in a Free Semigroup , 1977 .

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

[12]  Philippe Devienne.  Weighted Graphs: A Tool for Studying the Halting Problem and Time Complexity in Term Rewriting Systems and Logic Programming , 1990, Theor. Comput. Sci..

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

[14]  H. Brown,et al.  Computational Problems in Abstract Algebra , 1971 .

[15]  Paul Walton Purdom,et al.  Detecting Looping Simplifications , 1987, RTA.