Solving promise equations over monoids and groups

TL;DR

This work delivers a complete complexity dichotomy for promise systems of equations over finite monoids and, as a special case, finite groups, using an algebraic PCSP framework based on polymorphisms and minions. The central vehicle is the monoidal minion $\mathscr{M}_{M,a}$, which yields tractability via $\mathrm{BLP}+\mathrm{AIP}$ precisely when $a$ is regular and the image is a union of subgroups, and hardness otherwise. For groups, $\mathrm{AIP}$ suffices whenever an Abelian endomorphism extending the given map exists. The paper also shows a general reduction from PCSPs to promise equations over semigroups and constructs explicit templates linking polymorphisms to monoidal minions, highlighting both the reach and limitations of the algebraic approach in PCSPs.

Abstract

We give a complete complexity classification for the problem of finding a solution to a given system of equations over a fixed finite monoid, given that a solution over a more restricted monoid exists. As a corollary, we obtain a complexity classification for the same problem over groups.

Figures (1)

• Figure 1: The semilattice $\widehat{S_D}$, where lines indicate the order.

Theorems & Definitions (45)

• Lemma 1
• proof
• Theorem 1: BBKO21
• Theorem 2: BGWZ20
• Definition 1
• Definition 2
• Theorem 3: Main
• Corollary 1
• Corollary 2
• Corollary 3