Complexity of the Identity Checking Problem for Finite Semigroups

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. 1 Motivation and Main Results Many 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 algorithms1. 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 first author acknowledges the support of the Centro de Matemática da Universidade do Porto, financed by FCT through the programmes POCTI and POSI, with Portuguese and European Community structural funds, as well as the support of the FCT project PTDC/MAT/65481/2006. The second and the third authors have been supported by the Russian Foundation for Basic Research, grant 05-01-00540. In this paper complexity is understood in the sense of the monographs [7, 21]; the reader can find there the definitions of the complexity classes NP, co-NP, EXPSPACE, etc that are