Theory resolution constitutes a set of complete procedures for incorporating theories into a resolution theorem-proving program, thereby making it unnecessary to resolve directly upon axioms of the theory. This can greatly reduce the length of proofs and the size of the search space. Theory resolution effects a beneficial division of labor, improving the performance of the theorem prover and increasing the applicability of the specialized reasoning procedures. Total theory resolution utilizes a decision procedure that is capable of determining unsatisfiability of any set of clauses using predicates in the theory. Partial theory resolution employs a weaker decision procedure that can determine potential unsatisfiability of sets of literals. Applications include the building in of both mathematical and special decision procedures, e.g., for the taxonomic information furnished by a knowledge representation system. Theory resolution is a generalization of numerous previously known resolution refinements. Its power is demonstrated by comparing solutions of "Schubert's Steamroller" challenge problem with and without building in axioms through theory resolution. 1 1

This article discusses the two incarnations of Otter entered in the CADE-13 Automated Theorem Proving Competition. Also presented are some historical background, a summary of applications that have led to new results in mathematics and logic, and a general discussion of Otter.

This paper shows how a question-answering system can use first-order logic as its language and an automatic theorem prover, based upon the resolution inference principle, as its deductive mechanism. The resolution proof procedure is extended to a constructive proof procedure. An answer construction algorithm is given whereby the system is able not only to produce yes or no answers but also to find or construct an object satisfying a specified condition. A working computer program, QA3, based on these ideas, is described. The performance of the program, illustrated by extended examples, compares favorably with several other question-answering programs. Methods are presented for solving state transformation problems. In addition to question-answering, the program can do automatic programming

Number 644

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ABSTRACT. Paramodulation can be used in conjunction with resolution for proving theoreIm in first-order logic with equality. Unit, input, and linear refutations using paramodulation and resolution are defined for theories with equality. It is proved that (a) if a set S of clauses has an input refutation, then S together with its unit factors and functionally reflexive axioms has a unit refutation; and (b) if C is a clause in an E-unsatisfiable set S of clauses including {x = x} and the functionally reflexive axioms and if (S- {C} ) is E-satisfiable, then S has a linear re-futation with top clause C. ("E-unsatisfiable " is called "R-unsatisfiable " by some authors.) The refutation completeness theorem proved by Wos and Robinson for paramodulation with set of support is a corollary of our result (b). Our result (b) provides a link between heuristk programs and theorem-proving programs for first-order theories with equality.

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