一句话：证明在剩余小簇中，带立方项的有限代数集合的子幂成员问题是多项式时间可判定的。
子幂成员问题（SMP）是在固定有限代数集合 K 的代数直积中，判断给定元组是否属于由给定量生成的子代数；它是经典子代数成员问题的变体，并受约束满足问题（CSP）进展启发。方法上利用通用代数与 CSP 的代数理论，围绕 cube term 和 residually small variety 等结构条件进行分析。关键结果：若 K 为带 cube term 的有限代数集合且包含于 residually small variety，则 SMP(K)∈P；在带 cube term 的 variety 中，SMP(K)、SMP(HS K) 与寻找 K 中子幂的紧凑表示可相互多项式时间归约，且前两者属于 NP。
The subalgebra membership problem is the problem of deciding if a given element belongs to an algebra given by a set of generators. This is one of the best established computational problems in algebra. We consider a variant of this problem, which is motivated by recent progress in the Constraint Satisfaction Problem, and is often referred to as the Subpower Membership Problem (SMP). In the SMP we are given a set of tuples in a direct product of algebras from a fixed finite set $\mathcal{K}$ of finite algebras, and are asked whether or not a given tuple belongs to the subalgebra of the direct product generated by a given set. Our main result is that the subpower membership problem SMP($\mathcal{K}$) is in P if $\mathcal{K}$ is a finite set of finite algebras with a cube term, provided $\mathcal{K}$ is contained in a residually small variety. We also prove that for any finite set of finite algebras $\mathcal{K}$ in a variety with a cube term, each one of the problems SMP($\mathcal{K}$), SMP($\mathbb{HS} \mathcal{K}$), and finding compact representations for subpowers in $\mathcal{K}$, is polynomial time reducible to any of the others, and the first two lie in NP.
@article{paperbot827,
title = {The Subpower Membership Problem for Finite Algebras with Cube Terms},
author = {Andrei Bulatov and Peter Mayr and Ágnes Szendrei},
journal = {Logical Methods in Computer Science},
volume = {Volume 15, Issue 1},
year = {2019},
doi = {10.23638/lmcs-15(1:11)2019}
}