The complexity of the equivalence problem for nonsolvable groups

Horväth, Gábor; Mérai, László; Szabó, Csaba; Lawrence, John W.

doi:10.1112/blms/bdm030

Cited by 26 publications

(20 citation statements)

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“…For matrix rings they proved a stronger theorem, that is the equivalence problem is coNPcomplete even if the input polynomials are restricted to only one monomial. To this problem they reduce the equivalence problem over the multiplicative subgroup of matrix rings, which is coNP-complete by [4]. For most matrix rings, arguments of Lawrence and Willard [6] establish coNP-completeness as well.…”

Section: Theorem 1 Let R Be a Finite Ring If R Is Nilpotent Then Tmentioning

confidence: 99%

The Complexity of the Equivalence Problem Over Finite Rings

Horväth¹

2011

Glasgow Math. J.

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Abstract. We investigate the complexity of the equivalence problem over a finite ring when the input polynomials are written as sum of monomials. We prove that for a finite ring if the factor by the Jacobson radical can be lifted in the centre, then this problem can be solved in polynomial time. This result provides a step in proving a dichotomy conjecture of Lawrence and Willard (J. Lawrence and R. Willard, The complexity of solving polynomial equations over finite rings (manuscript, 1997)).2010 Mathematics Subject Classification. 16Z05, 16P10. Introduction.Investigations into the algorithmic aspects of the equivalence problem for various finite algebraic structures were started in the early 1990s. The equivalence problem for a finite algebra A asks whether or not two expressions p and q are equivalent over A (denoted by A |= p ≈ q), i.e. whether p and q determine the same function over A. This question is decidable for a finite algebra A: checking all substitutions from A yields to an answer of this question. The equivalence problem is in coNP, since the 'no' answer can be verified by a substitution, where the two expressions differ. In this paper we investigate the computational complexity of the equivalence problem for finite rings. That is, for a given finite ring R what is the complexity of deciding whether or not two input polynomials determine the same function over R? First, Hunt and Stearnes [5] investigated the equivalence problem for finite rings. They proved that for finite nilpotent rings the polynomial equivalence problem could be solved in polynomial time in the length of the two input polynomials. Moreover, they proved that for commutative, non-nilpotent rings the equivalence problem is coNPcomplete. Later, Burris and Lawrence [2] generalised their result to non-commutative rings, and established a dichotomy theorem for rings.

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Section: Theorem 1 Let R Be a Finite Ring If R Is Nilpotent Then Tmentioning

confidence: 99%

The Complexity of the Equivalence Problem Over Finite Rings

Horväth¹

2011

Glasgow Math. J.

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“…As was mentioned in Section 1, Corollary 1 is an immediate consequence of Theorem 1 combined with the result of [8] that identity checking in each finite nonsolvable group is co-NP-complete.…”

Section: Proof Of Theoremmentioning

confidence: 78%

“…Namely, Burris and Lawrence [5] have proved that the problem Check-Id(G) is decidable in polynomial time whenever the group G is nilpotent or dihedral; the latter result has been obtained also by Horváth and Szabó [9] who have also established polynomial decidability of identity checking for some other types of metabelian groups. On the other hand, Horváth, Lawrence, Merai and Szabó [8] have discovered that for every nonsolvable finite group G the problem CheckId(G) is co-NP-complete. For finite semigroups beyond the class of groups, one has found so far only isolated examples in which identity checking is co-NP-complete, cf.…”

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confidence: 99%

Complexity of the identity checking problem for finite semigroups

Almeida

Volkov²,

Goldberg³

2009

J Math Sci

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We prove that the identity checking problem in a finite semigroup S is co-NP-complete whenever S has a nonsolvable subgroup or S is the semigroup of all transformations on a 3-element set. Motivation and Main ResultsMany basic algorithmic questions in algebra whose decidability is well known and/or obvious give rise to fascinating and sometimes very hard problems if one looks for the computational complexity of corresponding algorithms 1 . As an example, we mention the following question Var-Memb: given two finite algebras A and B of the same similarity type, does the algebra A satisfy all identities of the algebra B? (The notation Var-Memb comes from "variety membership" since in the language of variety theory the question is about the membership of the algebra A to the variety generated by the algebra B.) Clearly, the problem Var-Memb is of importance for universal algebra in which equational classification of algebras is known to play a central role. At the same time the problem is of interest for computer science: see, for instance, [3, Section 1] for a discussion of its relationships to formal specification theory. The fact that the problem Var-Memb is decidable easily follows from Tarski's HSP-theorem and has been already mentioned in Kalicki's paper [12] more than 50 years ago. However an investigation of the computational complexity of this problem has started only recently and has brought rather unexpected results. First, Bergman and Slutzki [3] gave an upper bound by showing that the problem Var-Memb belongs to the class 2-EXPTIME (the class of problems solvable in double exponential time). For some time it appeared that this bound was very rough but then Szekely [30] showed that the problem is NP-hard, and Kozik [17,18] proved that it is even EXPSPACE-hard. Finally, Kozik [19] has *

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“…They prove a stronger result for matrix rings: the equivalence problem is coNP-complete even if the input polynomials are restricted to only one monomial, that is the equivalence problem is coNPcomplete for the multiplicative semigroup of matrix rings. To this problem they reduce the equivalence problem over the multiplicative subgroup of matrix rings, which is coNP-complete by [15]. Almeida, Volkov and Goldberg proved an even more general result about semigroups (showing that the equivalence problem is coNP-complete for a semigroup if the equivalence problem is coNP-complete for the direct product of its maximal subgroups) yielding the same result for matrix rings [2].…”

Section: Introductionmentioning

confidence: 99%