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

(Extended Abstract)
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)

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.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institut für Formale Methoden der InformatikUniversität StuttgartStuttgartGermany

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