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

Abstract

Over finite words, there is a tight connection between the quantifier alternation hierarchy inside two-variable first-order logic $\mathsf {FO}^2$ and a hierarchy of finite monoids: the Trotter-Weil Hierarchy. The various ways of climbing up this hierarchy include Mal’cev products, deterministic and co-deterministic concatenation as well as identities of $\omega$-terms. We show that the word problem for $\omega$-terms over each level of the Trotter-Weil Hierarchy is decidable; this means, for every variety $\varvec{\mathrm {V}}$ of the hierarchy and every identity $u = v$ of $\omega$-terms, one can decide whether all monoids in $\varvec{\mathrm {V}}$ satisfy $u=v$. More precisely, for every fixed variety $\varvec{\mathrm {V}}$, our approach yields nondeterministic logarithmic space ($\text {NL}$) and deterministic polynomial time algorithms, which are more efficient than straightforward translations of the NL-algorithms. From a language perspective, the word problem for $\omega$-terms is the following: for every language variety $\mathcal {V}$ in the Trotter-Weil Hierarchy and every language variety $\mathcal {W}$ given by an identity of $\omega$-terms, one can decide whether $\mathcal {V} \subseteq \mathcal {W}$. This includes the case where $\mathcal {V}$ is some level of the $\mathsf {FO}^2$ quantifier alternation hierarchy. As an application of our results, we show that the separation problems for the so-called corners of the Trotter-Weil Hierarchy are decidable.

M. Kufleitner—The first author was supported by the German Research Foundation (DFG) under grants DI 435/5-2 and KU 2716/1-1.