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The TPTP Problem Library
, 1999
"... This report provides a detailed description of the TPTP Problem Library for automated theorem proving systems. The library is available via Internet, and forms a common basis for development of and experimentation with automated theorem provers. This report provides: ffl the motivations for buildin ..."
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Cited by 100 (6 self)
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This report provides a detailed description of the TPTP Problem Library for automated theorem proving systems. The library is available via Internet, and forms a common basis for development of and experimentation with automated theorem provers. This report provides: ffl the motivations for building the library; ffl a discussion of the inadequacies of previous problem collections, and how these have been resolved in the TPTP; ffl a description of the library structure, including overview information; ffl descriptions of supplementary utility programs; ffl guidelines for obtaining and using the library; Contents 1 Introduction 2 1.1 Previous Problem Collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 What is Required? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Inside the TPTP 6 2.1 The TPTP Domain Structure . . . . . . . . . . . . . . . . . . . . . ...
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.
Automatic generation of classification theorems for finite algebras
 In Proc. of IJCAR 2004, volume 3097 of LNAI
, 2004
"... Classifying finite algebraic structures has been a major motivation behind much research in pure mathematics. Automated techniques have aided this process, but this has largely been at a quantitative level, e.g., to prove that there are no quasigroups of a given type for a given size, or to count th ..."
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Cited by 20 (15 self)
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Classifying finite algebraic structures has been a major motivation behind much research in pure mathematics. Automated techniques have aided this process, but this has largely been at a quantitative level, e.g., to prove that there are no quasigroups of a given type for a given size, or to count the number of groups of a particular order. Classification theorems of a more qualitative nature are often more interesting. For example, Kronecker's classification of finite Abelian groups [1] states that every Abelian group, G, of size n can be expressed as a direct product of cyclic groups, G = C s1 \Theta \Delta \Delta \Delta \Theta C sm, where n = s
WALDMEISTER: Development of a High Performance CompletionBased Theorem Prover
, 1996
"... : In this report we give an overview of the development of our new Waldmeister prover for equational theories. We elaborate a systematic stepwise design process, starting with the inference system for unfailing KnuthBendix completion and ending up with an implementation which avoids the main dise ..."
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Cited by 14 (0 self)
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: In this report we give an overview of the development of our new Waldmeister prover for equational theories. We elaborate a systematic stepwise design process, starting with the inference system for unfailing KnuthBendix completion and ending up with an implementation which avoids the main diseases today's provers suffer from: overindulgence in time and space. Our design process is based on a logical threelevel system model consisting of basic operations for inference step execution, aggregated inference machine, and overall control strategy. Careful analysis of the inference system for unfailing completion has revealed the crucial points responsible for time and space consumption. For the low level of our model, we introduce specialized data structures and algorithms speeding up the running system and cutting it down in size  both by one order of magnitude compared with standard techniques. Flexible control of the midlevel aggregation inside the resulting prover is made po...
Single Axioms for Odd Exponent Groups
 J. Automated Reasoning
, 1995
"... With the aid of automated reasoning techniques, we show that all previously known short single axioms for odd exponent groups are special cases of one general schema. We also demonstrate how to convert the proofs generated by an automated reasoning system into proofs understandable by a human. x0. I ..."
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Cited by 14 (6 self)
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With the aid of automated reasoning techniques, we show that all previously known short single axioms for odd exponent groups are special cases of one general schema. We also demonstrate how to convert the proofs generated by an automated reasoning system into proofs understandable by a human. x0. Introduction. There are two eras in the history of single axioms for groups and varieties of groups. The early results, by Neumann and others [7], often produced single axioms which were larger than the minimal possible size, but which were constructed via some scheme which made them easy to verify by hand. Although not optimal, these results had the virtue that a person could conceptually grasp their proofs. The second era began with the advent of McCune's automated reasoning system OTTER [4]; now one could produce shorter and simpler single axioms, which often required much more complex verifications. Short single axioms for groups and some varieties of groups were found by McCune and Wos [...
Automated Reasoning about Cubic Curves
 Computers and Mathematics with Applications
, 1993
"... It is well known that the nary morphisms defined on projective algebraic curves satisfy some strong localtoglobal equational rules of derivation not satisfied in general by universal algebras. For example, every rationally defined group law on a cubic curve must be commutative. Here we extract fr ..."
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Cited by 12 (10 self)
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It is well known that the nary morphisms defined on projective algebraic curves satisfy some strong localtoglobal equational rules of derivation not satisfied in general by universal algebras. For example, every rationally defined group law on a cubic curve must be commutative. Here we extract from the geometry of curves a firstorder property (gL) satisfied by all morphisms defined on these curves such that the equational consequences known for projective curves can be derived automatically from a set of six rules (stated within the firstorder logic with equality). First, the rule (gL) is implemented in the theoremproving program Otter. Then we use Otter to automatically prove some incidence theorems on projective curves without any further reference to the underlying geometry or topology of the curves. AMS Subject Classification (1991). Primary: 68T15, 08B05. Secondary:14H52, 20N05. 1 Introduction The term "equational logic" refers to the study of various metalogical notions rel...
Single Identities for Ternary Boolean Algebras
 Computers and Mathematics with Applications
, 1993
"... this paper, we show that the equational theory of TBAs is onebased. Our methods for finding a single identity for the theory of TBAs are interesting from two distinct points of view. First, from the algebraic, since TBAs enjoy both permutable and distributive congruences, they admit a single ternar ..."
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Cited by 9 (6 self)
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this paper, we show that the equational theory of TBAs is onebased. Our methods for finding a single identity for the theory of TBAs are interesting from two distinct points of view. First, from the algebraic, since TBAs enjoy both permutable and distributive congruences, they admit a single ternary polynomial p(x; y; z), the socalled Pixley polynomial [1, p. 405]. We first find such a polynomial p(x; y; z) and use a technique of R. Padmanabhan and R. W. Quackenbush [7] to construct a single identity for the equational theory in question. This is done in Section 2. Second, from the viewpoint of automated reasoning, we use the program Otter to discover new single identities based upon the results of the algebraic view. Actually we obtain here three new identities shorter in length than those obtained by the formal algebraic process of Section 2each characterizing the equational theory of TBAs. The relevant Otter proofs are also included. 2 The Algebraic View
Application of Automated Deduction to the Search for Single Axioms for Exponent Groups
 in Logic Programming and Automated Reasoning
, 1995
"... We present new results in axiomatic group theory obtained by using automated deduction programs. The results include single axioms, some with the identity and others without, for groups of exponents 3, 4, 5, and 7, and a general form for single axioms for groups of odd exponent. The results were obt ..."
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Cited by 8 (5 self)
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We present new results in axiomatic group theory obtained by using automated deduction programs. The results include single axioms, some with the identity and others without, for groups of exponents 3, 4, 5, and 7, and a general form for single axioms for groups of odd exponent. The results were obtained by using the programs in three separate ways: as a symbolic calculator, to search for proofs, and to search for counterexamples. We also touch on relations between logic programmingand automated reasoning. 1 Introduction A group of exponent n is a group in which for all elements x, x n is the identity e. Groups of exponent 2, xx = e, are also called Boolean groups. A single axiom for an equational theory is an equality from which the entire theory can be derived by equational reasoning. We are concerned with single axioms for groups of exponent n, n 2. B. H. Neumann [6, p.83] gives a general form for single axioms for certain subvarieties of groups, including exponent groups. The a...
The Shortest Single Axioms for Groups of Exponent 4
 Computers and Mathematics with Applications
, 1993
"... We study equations of the form (ff = x) which are single axioms for groups of exponent 4, where ff is a term in product only. Every such ff must have at least 9 variable occurrences, and there are exactly three such ff of this size, up to variable renaming and mirroring. These terms were found by an ..."
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Cited by 7 (2 self)
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We study equations of the form (ff = x) which are single axioms for groups of exponent 4, where ff is a term in product only. Every such ff must have at least 9 variable occurrences, and there are exactly three such ff of this size, up to variable renaming and mirroring. These terms were found by an exhaustive search through all terms of this form. Automated techniques were used in two ways: to eliminate many ff by verifying that (ff = x) true in some nongroup, and to verify that the group axioms do indeed follow from the successful (ff = x). We also present an improvement on Neumann's scheme for single axioms for varieties of groups. x0. Introduction. If n 1 is an integer, a group of exponent n is a group in which x n is the identity for all elements x. We study equations of the form (ff = x) which are single axioms for groups of exponent n, where ff is a term in product only. Note that in our definition of "exponent n", we do not require that n is the smallest exponent, so, for ...