The Complexity of Checking Identities Over Finite Groups

Horväth, Gábor; Szabó, Csaba

doi:10.1142/s0218196706003256

Cited by 25 publications

(22 citation statements)

References 5 publications

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“…Namely, that the equivalence problem for G is solvable in polynomial time if G is solvable, and coNP-complete otherwise. This conjecture has been veried for G A B, where A and B are Abelian groups such that the exponent of A is squarefree and (|A| , |B|) = 1 in [17], and for nonsolvable groups in [15].…”

Section: Introductionmentioning

confidence: 92%

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The complexity of the equivalence and equation solvability problems over meta-Abelian groups

Horväth

2015

Journal of Algebra

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Section: Introductionmentioning

confidence: 92%

“…In [7] Goldmann and Russel explicitly ask for the complexity of the equation solvability problem for S 3 . In [17] it is proved that this problem is in P for groups of order pq for primes p and q. Furthermore, the equation solvability problem is in P for the group A 4 , as well [18].…”

Section: Introductionmentioning

confidence: 98%

The complexity of the equivalence and equation solvability problems over meta-Abelian groups

Horväth

2015

Journal of Algebra

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“…Groups with feasible identity checking still are not completely described but recently one has obtained considerable advances towards such a description. 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.…”

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confidence: 87%

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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“…We show the tractability of the POL-EQ(A 1 2 ) problem in Lemma 8. Recently in [3,6], it was proved that the POL-EQ(S 3 ) problem is decidable in polynomial time. Hence B 1 2 is the only monoid with at most six elements and hard CHECK-ID and POL-EQ problems.…”

Section: Proposition 5 ([15]mentioning

confidence: 99%

“…For that reason some papers (e.g. [6]) refer to the preprint which had contained a few additional results, namely examples of monoids with different complexity of the studied problems.…”

Section: Historical Remarkmentioning

confidence: 99%