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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 57 (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.
Single axioms for groups and Abelian groups with various operations
 Preprint MCSP2701091, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL
, 1991
"... This paper summarizes the results of an investigation into single axioms for groups, both ordinary and Abelian, with each of following six sets of operations: fproduct, inverseg, fdivisiong, fdouble division, identityg, fdouble division, inverseg, fdivision, identityg, and fdivision, inverseg. In al ..."
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Cited by 34 (11 self)
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This paper summarizes the results of an investigation into single axioms for groups, both ordinary and Abelian, with each of following six sets of operations: fproduct, inverseg, fdivisiong, fdouble division, identityg, fdouble division, inverseg, fdivision, identityg, and fdivision, inverseg. In all but two of the twelve corresponding theories, we present either the rst single axioms known to us or single axioms shorter than those previously known to us. The automated theoremproving program Otter was used extensively to construct sets of candidate axioms and to search for and nd proofs that given candidate axioms are in fact single axioms. 1
Short Single Axioms for Boolean Algebra
 J. Automated Reasoning
, 2002
"... We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke tha ..."
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Cited by 23 (10 self)
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We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke than the ones we present. Automated deduction techniques were used for several different aspects of the work. Keywords: Boolean algebra, Sheffer stroke, single axiom 1. Background and
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 22 (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
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 13 (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 [...
OTTER experiments in a system of combinatory logic
 Journal of Automated Reasoning
, 1995
"... Abstract. This paper describes some experiments involving the automated theoremproving program OTTER in the system TRC of illative combinatory logic. We show how OTTER can be steered to find a contradiction in an inconsistent variant of TRC, and present some experimentally discovered identities in T ..."
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Abstract. This paper describes some experiments involving the automated theoremproving program OTTER in the system TRC of illative combinatory logic. We show how OTTER can be steered to find a contradiction in an inconsistent variant of TRC, and present some experimentally discovered identities in TRC. 1. Introduction. OTTER [5] is a resolution/paramodulation theoremproving program for firstorder logic with equality. It has been used successfully in several areas of logic and algebra [8], [9], [6], [7], [4]. In this paper we describe our experiments with OTTER in the system TRC
Computers, Reasoning and Mathematical Practice
"... ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every e ..."
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Cited by 7 (3 self)
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ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of r of R then R is commutative. Special cases of this, for example f(x) is x 2 \Gamma x or x 3 \Gamma x, can be given a first order proof in a few lines of symbol manipulation. The usual proof of the general result [20] (which takes a semester's postgraduate course to develop from scratch) is a corollary of other results: we prove that rings satisfying the condition are semisimple artinian, apply a theorem which shows that all such rings are matrix rings over division rings, and eventually obtain the result by showing that all finite division rings are fields, and hence commutative. This displays von Neumann's architectural qualities: it is "deep" in a way in which the symbol manipulati...
Automatic construction and verification of isotopy invariants
 IN PROC. OF IJCAR 2006, LNAI
, 2006
"... We extend our previous study of the automatic construction of isomorphic classification theorems for algebraic domains by considering the isotopy equivalence relation, which is of more importance than isomorphism in certain domains. This extension was not straightforward, and we had to solve two ma ..."
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We extend our previous study of the automatic construction of isomorphic classification theorems for algebraic domains by considering the isotopy equivalence relation, which is of more importance than isomorphism in certain domains. This extension was not straightforward, and we had to solve two major technical problems, namely generating and verifying isotopy invariants. Concentrating on the domain of loop theory, we have developed three novel techniques for generating isotopic invariants, by using the notion of universal identities and by using constructions based on substructures. In addition, given the complexity of the theorems which verify that a conjunction of the invariants form an isotopy class, we have developed ways of simplifying the problem of proving these theorems. Our techniques employ an intricate interplay of computer algebra, model generation, theorem proving and satisfiability solving methods. To demonstrate the power of the approach, we generate an isotopic classification theorem for loops of size 6, which extends the previously known result that there are 22. This result was previously beyond the capabilities of automated reasoning techniques.
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 6 (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 ...