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.

翻译：有限代数结构A的子幂成员问题SMP(A)询问：从A^k到A的给定部分函数是否可由A的项运算插值得到。虽然该问题在一般情况下可能是EXPTIME-完全的，但Willard提出疑问：若A为Mal'tsev代数，该问题是否总能多项式时间内求解。这类代数包含抽象代数中研究的许多重要结构，如群、拟群、环、布尔代数。本文对一大类2-幂零Mal'tsev代数给出了Willard问题的肯定答案，并进一步开发了未来可能对一般幂零Mal'tsev代数问题求解至关重要的工具。