Complexity of the identity checking problem for finite semigroups

Abstract

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 *

The complexity of the equivalence and equation solvability problems over nilpotent rings and groups

“…For recent results and detailed references see e.g. [1], [12], [13], [14], [17], [18], [21]. Although the literature is fairly extensive for monoids, the equivalence and equation solvability problems even for the simplest case, the case of nite groups, proved to be a far more challenging topic than for nite rings.…”

Identity checking problem for transformation monoids

“…To give a full classification of the monoids of partial transformations with respect to the complexity of identity checking, we can apply the result from [1] mentioned in the introduction to state that the problem CHECK-ID(PT n ) is coNP-complete for every n ≥ 5. It is possible to prove the coNP-completeness of CHECK-ID(PT n ) also in the cases n = 2 or 3 (see [11]), while the problem CHECK-ID(PT 1 ) is decidable in polynomial time (the monoid PT 1 is 2-element semilattice).…”

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

“…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]. For most matrix rings, arguments of [25] establish coNPcompleteness, as well.…”

“…[8,9,10]) or for nite semigroups and monoids (e.g. [2,6,20,21,22,23,24,29,31,33]). Just to mention some of the most recent results: following up on [2], Klíma nished the characterization for the transformation monoids [24], or Gorazd and Krzaczkowski characterized the complexity of these problems for all two-element general algebras [9,10].…”

“…[2,6,20,21,22,23,24,29,31,33]). Just to mention some of the most recent results: following up on [2], Klíma nished the characterization for the transformation monoids [24], or Gorazd and Krzaczkowski characterized the complexity of these problems for all two-element general algebras [9,10]. Although the literature is fairly extensive for monoids, the equivalence and equation solvability problems even for the simplest case, the case of nite groups, proved to be a far more challenging topic than for nite rings.…”