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33 Basic Test Problems: A Practical Evaluation of Some Paramodulation Strategies
, 1996
"... Introduction Many researchers who study the theoretical aspects of inference systems believe that if inference rule A is complete and more restrictive than inference rule B, then the use of A will lead more quickly to proofs than will the use of B. The literature contains statements of the sort "ou ..."
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Cited by 24 (5 self)
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Introduction Many researchers who study the theoretical aspects of inference systems believe that if inference rule A is complete and more restrictive than inference rule B, then the use of A will lead more quickly to proofs than will the use of B. The literature contains statements of the sort "our rule is complete and it heavily prunes the search space; therefore it is efficient". 2 These positions are highly questionable and indicate that the authors have little or no experience with the practical use of automated inference systems. Restrictive rules (1) can block short, easytofind proofs, (2) can block proofs involving simple clauses, the type of clause on which many practical searches focus, (3) can require weakening of redundancy control such as subsumption and demodulation, and (4) can require the use of complex checks in deciding whether such rules should be applied. The only way to determ
The Hot List Strategy
, 1997
"... Experimentation strongly suggests that, for attacking deep questions and hard problems with the assistance of an automated reasoning program, the more effective paradigms rely on the retention of deduced information. A significant obstacle ordinarily presented by such a paradigm is the deduction and ..."
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Cited by 9 (4 self)
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Experimentation strongly suggests that, for attacking deep questions and hard problems with the assistance of an automated reasoning program, the more effective paradigms rely on the retention of deduced information. A significant obstacle ordinarily presented by such a paradigm is the deduction and retention of one or more needed conclusions whose complexity sharply delays their consideration. To mitigate the severity of the cited obstacle, I formulated and feature in this article the hot list strategy. The hot list strategy asks the researcher to choose, usually from among the input statements characterizing the problem under study, one or more statements that are conjectured to play a key role for assignment completion. The chosen statementsconjectured to merit revisiting, again and againare placed in an input list of statements, called the hot list. When an automated reasoning program has decided to retain a new conclusion Cbefore any other statement is chosen to initiat...
Hilbert’s twentyfourth problem
 American Mathematical Monthly
, 2001
"... 1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Cong ..."
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Cited by 9 (4 self)
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1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of
Conquering the Meredith Single Axiom
 J. Automated Reasoning
, 2000
"... For more than three and onehalf decades beginning in the early 1960s, a heavy emphasis on proof finding has been a key component of the Argonne paradigm, whose use has directly led to significant advances in automated reasoning and important contributions to mathematics and logic. The theorems t ..."
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Cited by 8 (5 self)
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For more than three and onehalf decades beginning in the early 1960s, a heavy emphasis on proof finding has been a key component of the Argonne paradigm, whose use has directly led to significant advances in automated reasoning and important contributions to mathematics and logic. The theorems that have served well range from the trivial to the deep, even including some that corresponded to open questions. Often the paradigm asks for a theorem whose proof is in hand but that cannot be obtained in a fully automated manner by the program in use. The theorem whose hypothesis consists solely of the Meredith single axiom for twovalued sentential (or propositional) calculus and whose conclusion is the Lukasiewicz threeaxiom system for that area of formal logic was just such a theorem. Featured in this article is the methodology that enabled the program OTTER to find the first fully automated proof of the cited theorem, a proof with the intriguing property that none of its steps...
Experiments concerning the Automated Search for Elegant Proofs
 Technical Memorandum ANL/MCSTM221, Mathematics and Computer Science Division, Argonne National Laboratory
, 1997
"... ..."
Universitaet Leipzig
"... Throughout the twentieth century, the worlds of logic and mathematics were well aware of Hilbert’s twentythree problems and the challenge they offered. Although not known until very recently, there existed yet one more challenge offered by Hilbert, his twentyfourth problem. This problem focuses on ..."
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Throughout the twentieth century, the worlds of logic and mathematics were well aware of Hilbert’s twentythree problems and the challenge they offered. Although not known until very recently, there existed yet one more challenge offered by Hilbert, his twentyfourth problem. This problem focuses on finding simpler proofs, on the criteria for measuring simplicity, and on the ‘‘development of a theory of the method of proof in mathematics in general’’. Of the three themes of Hilbert’s twentyfourth problem, the first two are central to this article. We visit various areas of logic, showing that some of the studies of the masters are indeed strongly connected to this newly discovered problem. We also demonstrate that the use of an automated reasoning program (specifically, W. McCune’s OTTER) enables one to address this challenging problem. We offer questions that remain unanswered.
If
"... Gibbard [2] presents an argument to the effect that any conditional satisfying certain principles must be equivalent to the material (viz., classical) conditional. Here is one rendition of Gibbard’s (informal) argument. Let be the classical material conditional, and let be the indicative conditional ..."
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Gibbard [2] presents an argument to the effect that any conditional satisfying certain principles must be equivalent to the material (viz., classical) conditional. Here is one rendition of Gibbard’s (informal) argument. Let be the classical material conditional, and let be the indicative conditional. Suppose that the indicative satisfies the importexport law. Thatis, suppose (IE) A B C is logically equivalent to A & B C.