An Order Theory Resolution Calculus

In this paper we present an ordered theory resolution calculus and prove its completeness. Theory reasoning means to relieve a calculus from explicitly drawing inferences in a given theory by special purpose inference rules (e.g. E-resolution for equality reasoning). We take advantage of orderings (e.g. simplification orderings) by disallowing to resolve upon clauses which violate certain maximality constraints; stated positively, a resolvent may only be built if all the selected literals are maximal in their clauses. By this technique the search space is drastically pruned. As an instantiation for theory reasoning we show that equality can be built in by rigid E-unification.

[1]  Paliath Narendran,et al.  Rigid E-Unification: NP-Completeness and Applications to Equational Matings , 1990, Inf. Comput..

[2]  Neil V. Murray,et al.  Theory Links: Applications to Automated Theorem Proving , 1987, J. Symb. Comput..

[3]  Mark E. Stickel.  Theory Resolution: Building in Nonequational Theories , 1983, AAAI.

[4]  Harald Ganzinger,et al.  Completion of First-Order Clauses with Equality by Strict Superposition (Extended Abstract) , 1990, CTRS.

[5]  Nachum Dershowitz,et al.  Termination of Rewriting , 1987, J. Symb. Comput..

[6]  Uwe Petermann.  Towards a Connection Procedure with Built in Theories , 1990, JELIA.

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

[8]  Hector J. Levesque,et al.  Krypton: A Functional Approach to Knowledge Representation , 1983, Computer.

[9]  Peter Baumgartner.  A Model Elimination Calculus with Built-in Theories , 1992, GWAI.

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

[11]  Jörg H. Siekmann,et al.  The Markgraf Karl Refutation Procedure , 1980, IJCAI.

[12]  Deepak Kapur,et al.  First-Order Theorem Proving Using Conditional Rewrite Rules , 1988, CADE.

[13]  Peter B. Andrews.  Theorem Proving via General Matings , 1981, JACM.

[14]  W. W. Bledsoe,et al.  A Linear Format for Resolution With Merging and a New Technique for Establishing Completeness , 1970, JACM.

[15]  Michaël Rusinowitch,et al.  A New Method for Establishing Refutational Completeness in Theorem Proving , 1986, CADE.