  The subpower membership problem SMP(A) of a finite algebraic structure A asks
whether a given partial function from A^k to A can be interpolated by a term
operation of A, or not. While this problem can be EXPTIME-complete in general,
Willard asked whether it is always solvable in polynomial time if A is a
Mal'tsev algebras. In particular, this includes many important structures
studied in abstract algebra, such as groups, quasigroups, rings, Boolean
algebras. In this paper we give an affirmative answer to Willard's question for
a big class of 2-nilpotent Mal'tsev algebras. We furthermore develop tools that
might be essential in answering the question for general nilpotent Mal'tsev
algebras in the future.
