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A Prolog Technology Theorem Prover: Implementation by an Extended Prolog Compiler
 Journal of Automated Reasoning
, 1987
"... A Prolog technology theorem prover (PTTP) is an extension of Prolog that is complete for the full firstorder predicate calculus. It differs from Prolog in its use of unification with the occurs check for soundness, the modelelimination reduction rule that is added to Prolog inferences to make the ..."
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A Prolog technology theorem prover (PTTP) is an extension of Prolog that is complete for the full firstorder predicate calculus. It differs from Prolog in its use of unification with the occurs check for soundness, the modelelimination reduction rule that is added to Prolog inferences to make the inference system complete, and depthfirst iterativedeepening search instead of unbounded depthfirst search to make the search strategy complete. A Prolog technology theorem prover has been implemented by an extended PrologtoLISP compiler that supports these additional features. It is capable of proving theorems in the full firstorder predicate calculus at a rate of thousands of inferences per second. 1 This is a revised and expanded version of a paper presented at the 8th International Conference on Automated Deduction, Oxford, England, July 1986, and is to appear in Journal of Automated Reasoning. This research was supported by the Defense Advanced Research Projects Agency under Co...
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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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.
Experiments with DiscriminationTree Indexing and Path Indexing for Term Retrieval
 JOURNAL OF AUTOMATED REASONING
, 1990
"... This article addresses the problem of indexing and retrieving firstorder predicate calculus terms in the context of automated deduction programs. The four retrieval operations of concern are to find variants, generalizations, instances, and terms that unify with a given term. Discriminationtree ..."
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Cited by 47 (0 self)
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This article addresses the problem of indexing and retrieving firstorder predicate calculus terms in the context of automated deduction programs. The four retrieval operations of concern are to find variants, generalizations, instances, and terms that unify with a given term. Discriminationtree indexing is reviewed, and several variations are presented. The pathindexing method is also reviewed. Experiments were conducted on large sets of terms to determine how the properties of the terms affect the performance of the two indexing methods. Results of the experiments are presented.
Automated Reasoning and Bledsoe's Dream for the Field
"... In one sense, this article is a personal tribute to Woody Bledsoe. As such, the style will in general be that of private correspondence. However, since this article is also a compendium of experiments with an automated reasoning program, researchers interested in automated reasoning, mathematics, an ..."
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In one sense, this article is a personal tribute to Woody Bledsoe. As such, the style will in general be that of private correspondence. However, since this article is also a compendium of experiments with an automated reasoning program, researchers interested in automated reasoning, mathematics, and logic will find pertinent material here. The results of those experiments strongly suggest that research frequently benefits greatly from the use of an automated reasoning program. As evidence, I select from those results some proofs that are better than one can find in the literature, and focus on some theorems that, until now, had never been proved with an automated reasoning program, theorems that Hilbert, Church, and various logicians thought significant. To add spice to the article, I present challenges for reasoning programs, including questions that are still open. 1 This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Depa...
Automated theorem proving: mapping logic into AI
 Proceedings of the International Symposium on Methodologies for Intelligent Systems
, 1986
"... ABSTRACT. Logic can be defined as the formal study of reasoning; if we replace "formal " by "mechanical " we can place almost the entire set of methodologies used in the field of automated theorem proving (ATP) within the scope of logic. Because of the goals of A ..."
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ABSTRACT. Logic can be defined as the formal study of reasoning; if we replace &quot;formal &quot; by &quot;mechanical &quot; we can place almost the entire set of methodologies used in the field of automated theorem proving (ATP) within the scope of logic. Because of the goals of ATP, if not always the methodologies, ATP has been considered to be within the domain of AI. We explore the methodologies of ATP, including the logics that underlie the theorem provers, and discuss some of the mechanisms that utilize these logics. These include term rewriting systems, mathematical induction, inductionless induction and even mixed integer programming. The ATP field, via resolution, has even provided the foundation for an exciting AI and database programming language, PROLOG. We conclude with a new method for extending the PROLOG system to work with nonHorn clause sets within a positive logic format, particularly simple for &quot;slightly nonHorn &quot; clause sets.
Automated reasoning: Real uses and . . .
"... An automated reasoning program has provided invaluable assistance in answering certain previously open questions in mathematics and in formal logic. These questions would not have been answered, at least by those who obtained the results, were it not for the program's contribution. Others have ..."
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An automated reasoning program has provided invaluable assistance in answering certain previously open questions in mathematics and in formal logic. These questions would not have been answered, at least by those who obtained the results, were it not for the program's contribution. Others have used such a program to design logic circuits, many of which proved superior (with respect to transistor count) to the existing designs, and to validate the design of other circuits. These successes establish the value of an automated reasoning program for research and suggest the value for practical applications. We thus conclude that the field of automated reasoning is on the verge of becoming one of the more significant branches of computer science. Further, we conclude that the field has already advanced from stage 1, that of potential usefulness, to stage 2, that of actual usefulness. To pass to stage 3, that of wide acceptance and use, requires, among other things, easy access to an automated reasoning program and an understanding of the various aspects of automated reasoning. In fact, an automated reasoning program is available that is portable and can be run on relatively inexpensive machines. Moreover, a system exists for producing a reasoning program tailored to given specifications. As for the requirement of understanding the aspects of automated reasoning, much research remains—research aided by access to a reasoning program. A large obstacle has thus been removed, permitting many to experiment with and find uses for a computer program that can be relied upon as a most valuable automated reasoning assistant.
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"... We present a new strategy for semantic paramodulation for Horn sets and prove its completeness The strategy requires for each paramoduiation that either both parents be false positive units or that one parent and the paramodulant both be false relative to an interpretation We also discuss some of th ..."
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We present a new strategy for semantic paramodulation for Horn sets and prove its completeness The strategy requires for each paramoduiation that either both parents be false positive units or that one parent and the paramodulant both be false relative to an interpretation We also discuss some of the issues involved in choosing an interpretation that has a chance of giving better performance that simple setofsupport paramoduiation. 1.
A New Hyperparamodulatlon Strategy for the Equality Relation*
"... Equality is an Important relation and many theorems can be easily symbolized through it's use. A proposed Inference rule called HLresolution Is intended to have the benefits of hyper steps while controlling the application of paramodulation. It generates a resolvent by building a paramodulatio ..."
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Equality is an Important relation and many theorems can be easily symbolized through it's use. A proposed Inference rule called HLresolution Is intended to have the benefits of hyper steps while controlling the application of paramodulation. It generates a resolvent by building a paramodulation and demodulation link between two terms using a pre pro ceased plan as a guide. The rule Is complete for Eunsatisfiable Horn sets. The linking process makes use of an equality graph which is constructed once at the beginning of the run. Once a pair of candidate terms for HLresolutlon is chosen in the search, potential linkages can be found and tested for compatibility efficiently by looking at the paths in the graph. The method was implemented on an existing theoremproving system. A number of algebra and a comparison with setofsupport paramodulation was made. 1.
A Prolog Technology Term Rewriter
, 1994
"... Term rewriting is basic to a number of programming methods, including functional programming, symbolic mathematics, theorem proving, and constraint normalization. In this paper we approach term rewriting in the spirit of Stickel's Prolog Technology Theorem Prover (PTTP): to exploit the combin ..."
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Term rewriting is basic to a number of programming methods, including functional programming, symbolic mathematics, theorem proving, and constraint normalization. In this paper we approach term rewriting in the spirit of Stickel's Prolog Technology Theorem Prover (PTTP): to exploit the combination of high level programming and efficient implementation provided by Prolog. We show that there is an elegant and simple way to exploit Prolog technology in implementing term rewriting systems. The method is very flexible, permitting us to vary the rewriting strategy (from outermost to innermost to ad hoc), kinds of unification (reduction, narrowing, or higherorder rewriting), and the compilation techniques used (from partial evaluation to surgery on the underlying rewrite engine, which is specified in terms of axioms of equality). In this approach, innermost and outermost reduction, as well as many combinations of these, are obtained by merely rearranging the clauses that express eq...
Otterlambda, A Theoremprover WITH UNTYPED LAMBDAUNIFICATION
, 2004
"... Support for lambda calculus and an algorithm for untyped lambdaunification has been implemented, starting from the source code for Otter. The result is a new theorem prover called Otterλ. This is the first time that a resolutionbased, clauselanguage prover (that accumulates deduced clauses and u ..."
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Support for lambda calculus and an algorithm for untyped lambdaunification has been implemented, starting from the source code for Otter. The result is a new theorem prover called Otterλ. This is the first time that a resolutionbased, clauselanguage prover (that accumulates deduced clauses and uses strategies to control the deduction and retention of clauses) has been combined with a lambdaunification algorithm to assist in the deductions. The resulting prover combines the advantages of the proofsearch algorithm of Otter and the power of higherorder unification. We describe the untyped lambda unification algorithm used by Otterλ and give several example theorems.