The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy

TL;DR

This paper proves that the word problem for omega-terms over each level of the Trotter-Weil Hierarchy is decidable, providing efficient algorithms and applications to separability problems in algebraic structures.

Contribution

It establishes decidability and efficient algorithms for the word problem over all levels of the Trotter-Weil Hierarchy, a significant advancement in algebraic language theory.

Findings

Decidability of the word problem for omega-terms at each hierarchy level.

Development of nondeterministic logarithmic space algorithms.

Existence of more efficient deterministic polynomial time algorithms.

Abstract

For two given $\omega$-terms $\alpha$ and $\beta$, the word problem for $\omega$-terms over a variety $\boldsymbol{\mathrm{V}}$ asks whether $\alpha=\beta$ in all monoids in $\boldsymbol{\mathrm{V}}$. We show that the word problem for $\omega$-terms over each level of the Trotter-Weil Hierarchy is decidable. More precisely, for every fixed variety in the Trotter-Weil Hierarchy, our approach yields an algorithm in nondeterministic logarithmic space (NL). In addition, we provide deterministic polynomial time algorithms which are more efficient than straightforward translations of the NL-algorithms. As an application of our results, we show that separability by the so-called corners of the Trotter-Weil Hierarchy is witnessed by $\omega$-terms (this property is also known as $\omega$-reducibility). In particular, the separation problem for the corners of the Trotter-Weil Hierarchy is decidable.