Equations over free inverse monoids with idempotent variables

TL;DR

This paper studies equations over free inverse monoids with idempotent variables, proving decidability and complexity bounds, and extending results to broader classes of equations.

Contribution

It introduces idempotent variables for inverse monoid equations and establishes decidability and complexity results, improving previous bounds and covering new equation families.

Findings

Decidable in DEXPTIME whether equations have solutions

Problem is DEXPTIME hard when the quotient group rank is at least two

Decidability extends to systems with one irreducible variable and unbalanced equations

Abstract

We introduce the notion of idempotent variables for studying equations in inverse monoids. It is proved that it is decidable in singly exponential time (DEXPTIME) whether a system of equations in idempotent variables over a free inverse monoid has a solution. The result is proved by a direct reduction to solve language equations with one-sided concatenation and a known complexity result by Baader and Narendran: Unification of concept terms in description logics, 2001. We also show that the problem becomes DEXPTIME hard , as soon as the quotient group of the free inverse monoid has rank at least two. Decidability for systems of typed equations over a free inverse monoid with one irreducible variable and at least one unbalanced equation is proved with the same complexity for the upper bound. Our results improve known complexity bounds by Deis, Meakin, and Senizergues: Equations in free inverse monoids, 2007. Our results also apply to larger families of equations where no decidability has been previously known.