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Automated Deduction by Theory Resolution
 Journal of Automated Reasoning
, 1985
"... Theory resolution constitutes a set of complete procedures for incorporating theories into a resolution theoremproving 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 resoluti ..."
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Cited by 121 (1 self)
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Theory resolution constitutes a set of complete procedures for incorporating theories into a resolution theoremproving 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
Otter: The CADE13 Competition Incarnations
 JOURNAL OF AUTOMATED REASONING
, 1997
"... This article discusses the two incarnations of Otter entered in the CADE13 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. ..."
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Cited by 44 (3 self)
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This article discusses the two incarnations of Otter entered in the CADE13 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.
The Applications of Theorem Proving to QuestionAnswering Systems
, 1969
"... This paper shows how a questionanswering system can use firstorder 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 ..."
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Cited by 27 (0 self)
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This paper shows how a questionanswering system can use firstorder 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 questionanswering programs. Methods are presented for solving state transformation problems. In addition to questionanswering, the program can do automatic programming
Completeness of linear refutation for theories with equality
 J. ACM
, 1971
"... ABSTRACT. Paramodulation can be used in conjunction with resolution for proving theoreIm in firstorder 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 refuta ..."
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ABSTRACT. Paramodulation can be used in conjunction with resolution for proving theoreIm in firstorder 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 Eunsatisfiable set S of clauses including {x = x} and the functionally reflexive axioms and if (S {C} ) is Esatisfiable, then S has a linear refutation with top clause C. ("Eunsatisfiable " is called "Runsatisfiable " 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 theoremproving programs for firstorder theories with equality.
The Problem of Hyperparamodulation
"... Abstract. This article is the thirtysecond of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks one to find a variant of the inference rule hyperparamodulation that avoids generating many of the binary paramodulants ordina ..."
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Abstract. This article is the thirtysecond of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks one to find a variant of the inference rule hyperparamodulation that avoids generating many of the binary paramodulants ordinarily deduced. Key words. Automated reasoning, hyperparamodulation, inference rule, unsolved research problem. Question: What is the appropriate definition of hyperparamodulation that avoids generating all paramodulants? (This question is the ninth of 33 problems proposed for research in [6] and will be referred to as Research Problem 9 throughout this article. All references to sections, chapters, test problems, and such also refer to [6].) An automated reasoning program employing paramodulation [4, 11] treats equality as ‘‘understood’’ or ‘‘built in’’; this inference rule generalizes the usual notion of equality substitution. A reasoning program using hyperresolution [5] combines a number of small (binary resolution) deduction steps into one. Observing the successes obtained with these inference rules naturally suggested research whose goal was to find an inference rule that would possess the good features of both. The result was
Problem Corner." Reasoning about Equality*
, 1985
"... Abstract. This note contains a set of six theorems that can be used to assess the ability of a theoremproving system to reason about equality. The six theorems are graduated in terms of difficulty: they range from fairly trivial to quite difficult. They do not cover all aspects of equality reasoning ..."
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Abstract. This note contains a set of six theorems that can be used to assess the ability of a theoremproving system to reason about equality. The six theorems are graduated in terms of difficulty: they range from fairly trivial to quite difficult. They do not cover all aspects of equality reasoning, but they have proved useful to us in developing our system.