Fix a finite semigroup S and let a1​,…,ak​,b be tuples in a direct
power Sn. 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 letters or more 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