Theory of Computing Systems

, Volume 62, Issue 3, pp 682–738 | Cite as

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



For two given ω-terms α and β, the word problem for ω-terms over a variety V asks whether α = β in all monoids in V. We show that the word problem for ω-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 ω-terms (this property is also known as ω-reducibility). In particular, the separation problem for the corners of the Trotter-Weil Hierarchy is decidable.


Regular language Finite monoid Pseudoidentity Omega-term Separation problem Trotter-Weil Hierarchy FO2 alternation hierarchy 


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institut für Formale Methoden der InformatikUniversity of StuttgartStuttgartGermany

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