Solving Equation Systems in $\omega$-categorical Algebras

TL;DR

This paper investigates the computational complexity of solving equations in $oldsymbol{ extomega}$-categorical algebras, revealing complexity classifications for specific algebraic structures and introducing novel applications of the pseudo-Siggers polymorphism.

Contribution

It establishes complexity results for equation solving in $oldsymbol{ extomega}$-categorical algebras, notably applying the pseudo-Siggers identity to prove a new complexity dichotomy.

Findings

Equation solving is in P or NP-hard for certain algebra classes.

Undecidability occurs in some $oldsymbol{ extomega}$-categorical groups.

The pseudo-Siggers polymorphism is used to characterize complexity boundaries.

Abstract

We study the computational complexity of deciding whether a given set of term equalities and inequalities has a solution in an $\omega$-categorical algebra $\mathfrak{A}$. There are $\omega$-categorical groups where this problem is undecidable. We show that if $\mathfrak{A}$ is an $\omega$-categorical semilattice or an abelian group, then the problem is in P or NP-hard. The hard cases are precisely those where Pol$(\mathfrak{A},\neq)$ has a uniformly continuous minor-preserving map to the clone of projections on a two-element set. The results provide information about algebras $\mathfrak{A}$ such that Pol$(\mathfrak{A},\neq)$ does not satisfy this condition, and they are of independent interest in universal algebra. In our proofs we rely on the Barto-Pinsker theorem about the existence of pseudo-Siggers polymorphisms. To the best of our knowledge, this is the first time that the pseudo-Siggers identity has been used to prove a complexity dichotomy.