Cycle Uniication

Two-literal clauses of the form L R occur quite frequently in logic programs, deductive databases, and { disguised as an equation { in term rewriting systems. These clauses deene a cycle if the atoms L and R are weakly uniiable, ie. if L uniies with a new variant of R. The obvious problem with cycles is to control the number of iterations through the cycle. In this paper we consider the cycle uniication problem of unifying two literals G and F modulo a cycle. We review the state of the art of cycle uniication and give some new results for a special type of cycles called matching cycles, ie. cycles LR for which there exists a substitution such that L = R or L = R. Altogether, these results show how the deductive process can be eeciently controlled for special classes of cycles without losing completeness.

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

[2]  Jack Minker,et al.  On recursive axioms in deductive databases , 1983, Inf. Syst..

[3]  Hans Jürgen Ohlbach.  Abstraction Tree Indexing for Terms , 1990, ECAI.

[4]  Laurent Vieille,et al.  Recursive Query Processing: The Power of Logic , 1989, Theor. Comput. Sci..

[5]  Jörg Würtz,et al.  Unifying Cycles , 1992, ECAI.

[6]  Wolfgang Bibel.  Perspectives on Automated Deduction , 1991, Automated Reasoning: Essays in Honor of Woody Bledsoe.

[7]  J. Lloyd.  Foundations of Logic Programming , 1984, Symbolic Computation.

[8]  Manfred Schmidt-Schauß.  Implication of Clauses is Undecidable , 1988, Theor. Comput. Sci..

[9]  Frank Pfenning.  Single Axioms in the Implicational Propositional Calculus , 1988, CADE.

[10]  Hans Jürgen Ohlbach.  Compilation of Recursive Two-Literal Clauses into Unification Algorithms , 1990, AIMSA.

[11]  Maurice Bruynooghe,et al.  A Practical Technique for Detecting Non-terminating Queries for a Restricted Class of Horn Clauses, Using Directed, Weighted Graphs , 1990, ICLP.

[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]  Graham Wrightson,et al.  Solving a Problem in Relevance Logic with an Automated Theorem Prover , 1984, CADE.

[14]  Elmar Eder.  Properties of Substitutions and Unifications , 1983, GWAI.

[15]  Lawrence J. Henschen,et al.  What Is Automated Theorem Proving? , 1985, J. Autom. Reason..

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