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 *

We exhibit a 6-element semigroup that has no finite identity basis but nevertheless generates a variety whose finite membership problem admits a polynomial algorithm.