Hardness Results for the Subpower Membership Problem

TL;DR

This paper investigates the computational complexity of the subpower membership problem in finite algebras, establishing hardness results linked to certain algebraic conditions and providing complexity classifications.

Contribution

It characterizes Maltsev conditions that do not imply cube terms and constructs algebras with EXPTIME-complete subpower membership problems.

Findings

Subpower membership problem is at least as hard as for any algebra satisfying certain Maltsev conditions.

Finite algebras in congruence distributive and k-permutable varieties can have EXPTIME-complete subpower membership problems.

Characterization of Maltsev conditions not implying cube terms.

Abstract

The main result of this paper shows that if $\mathcal{M}$ is a consistent strong linear Maltsev condition which does not imply the existence of a cube term, then for any finite algebra $\mathbb{A}$ there exists a new finite algebra $\mathbb{A}_\mathcal{M}$ which satisfies the Maltsev condition $\mathcal{M}$, and whose subpower membership problem is at least as hard as the subpower membership problem for $\mathbb{A}$. We characterize consistent strong linear Maltsev conditions which do not imply the existence of a cube term, and show that there are finite algebras in varieties that are congruence distributive and congruence $k$-permutable ($k \geq 3$) whose subpower membership problem is EXPTIME-complete.