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Designing Programs That Check Their Work
, 1989
"... A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It d ..."
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Cited by 307 (17 self)
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A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It designs program checkers for a few specific and carefully chosen problems in the class FP of functions computable in polynomial time. Problems in FP for which checkers are presented in this paper include Sorting, Matrix Rank and GCD. It also applies methods of modern cryptography, especially the idea of a probabilistic interactive proof, to the design of program checkers for group theoretic computations. Two strucural theorems are proven here. One is a characterization of problems that can be checked. The other theorem establishes equivalence classes of problems such that whenever one problem in a class is checkable, all problems in the class are checkable.
Experiments in Coset Enumeration
 Groups and Computation III, Ohio State University Mathematical Research Institute Publications #8, Walter de Gruyter
, 2001
"... . Coset enumeration, based on the methods described by Todd and Coxeter, is one of the most important tools for investigating finitely presented groups. Such methods do not, in general, constitute an algorithm. Each problem has to be addressed individually, and determining whether or not an acce ..."
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Cited by 10 (9 self)
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. Coset enumeration, based on the methods described by Todd and Coxeter, is one of the most important tools for investigating finitely presented groups. Such methods do not, in general, constitute an algorithm. Each problem has to be addressed individually, and determining whether or not an acceptable solution for a difficult problem can be found using given resources requires an empirical approach (i.e., experimentation). We discuss some of the ideas involved, and illustrate with examples which emphasize some of the possibilities. 1991 Mathematics Subject Classification: primary 20F05; secondary 2004. 1. Introduction Coset enumeration is long established as a technique for the investigation of finitely presented groups. It was used well before the days of electronic computers, apparently first by Moore [14], and later popularised by Todd and Coxeter [17]. The first computer implementation was that of Haselgrove in 1953. This, along with other early implementations, is descr...
AndrewsCurtis And ToddCoxeter Proof Words
 in Oxford. Vol. I, London Math. Soc. Lecture Note Ser
, 2001
"... Andrews and Curtis have conjectured that every balanced presentation of the trivial group can be transformed into a standard presentation by a finite sequence of elementary transformations. It can be difficult to determine whether or not the conjecture holds for a particular presentation. We show th ..."
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Cited by 3 (2 self)
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Andrews and Curtis have conjectured that every balanced presentation of the trivial group can be transformed into a standard presentation by a finite sequence of elementary transformations. It can be difficult to determine whether or not the conjecture holds for a particular presentation. We show that the utility PEACE, which produces proofs based on ToddCoxeter coset enumeration, can produce AndrewsCurtis proofs.
PEACE 1.000: Proof Extraction after Coset Enumeration
, 2000
"... Contents Contents iii List of figures v List of tables vi 1 Introduction 1 1.1 Important notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Executive decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Notation & conventions . . . . . . . . . . . . . . . . . ..."
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Contents Contents iii List of figures v List of tables vi 1 Introduction 1 1.1 Important notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Executive decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Notation & conventions . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Background 3 2.1 Definition sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Coset enumeration 5 4 Definition sequences 6 5 Proof tables 7 5.1 Proof words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2 The proof table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.2.1 Coincidence processing . . . . . . . . . . . . . . . . . . . . . . 9 5.3 Command details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.4 Implementation details . . . . . . . . . . . . . . .
ADDENDUM TO AN ELEMENTARY INTRODUCTION TO COSET TABLE METHODS IN COMPUTATIONAL GROUP THEORY
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BEHIND AND BEYOND A THEOREM ON GROUPS RELATED TO TRIVALENT GRAPHS.
, 2007
"... In 2006 we completed the proof of a fivepart conjecture which was made in 1977 about a family of groups related to trivalent graphs. This family covers all 2generator, 2relator groups where one relator specifies that a generator is an involution and the other relator has three syllables. Our proo ..."
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In 2006 we completed the proof of a fivepart conjecture which was made in 1977 about a family of groups related to trivalent graphs. This family covers all 2generator, 2relator groups where one relator specifies that a generator is an involution and the other relator has three syllables. Our proof relies upon detailed but general computations in the groups under question. The proof is theoretical, but based upon explicit proofs produced by machine for individual cases. Here we explain how we derived the general proofs from specific cases. The conjecture essentially addressed only the finite groups in the family. Here we extend the results to infinite groups, effectively determining when members of this family of finitely presented groups are simply isomorphic to a specific quotient. 1.