The subpower membership problem for semigroups

Fix a finite semigroup S and let a1, . . . , ak , b be tuples in a direct power S. The subpower membership problem (SMP) asks whether b can be generated by a1, . . . , ak. If S is a finite group, then there is a folklore algorithm that decides this problem in time polynomial in nk. For semigroups this problem always lies in PSPACE. We show that the SMP for a full transformation semigroup on 3 or more letters is actually PSPACE-complete, while on 2 letters it is in P. For commutative semigroups, we provide a dichotomy result: if a commutative semigroup S embeds into a direct product of a Clifford semigroup and a nilpotent semigroup, then SMP(S) is in P; otherwise it is NP-complete.