## Bulatov, Andrei and Mayr, Peter and Szendrei, Ágnes - The Subpower Membership Problem for Finite Algebras with Cube Terms

lmcs:4396 - Logical Methods in Computer Science, February 13, 2019, Volume 15, Issue 1
The Subpower Membership Problem for Finite Algebras with Cube Terms

Authors: Bulatov, Andrei and Mayr, Peter and Szendrei, Ágnes

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.

Source : oai:arXiv.org:1803.08019
Volume: Volume 15, Issue 1
Published on: February 13, 2019
Submitted on: March 22, 2018
Keywords: Computer Science - Logic in Computer Science,Primary: 68Q25, Secondary 08A30, 08A70