Abstract
This paper presents a new result in the equational theory of regular languages, which emerged from lively discussions between the authors about Stone and Priestley duality. Let us call lattice of languages a class of regular languages closed under finite intersection and finite union. The main results of this paper (Theorems 5.2 and 6.1) can be summarized in a nutshell as follows:
A set of regular languages is a lattice of languages if and only if it can be defined by a set of profinite equations.
The product on profinite words is the dual of the residuation operations on regular languages.
In their more general form, our equations are of the form u →v, where u and v are profinite words. The first result not only subsumes Eilenberg-Reiterman’s theory of varieties and their subsequent extensions, but it shows for instance that any class of regular languages defined by a fragment of logic closed under conjunctions and disjunctions (first order, monadic second order, temporal, etc.) admits an equational description. In particular, the celebrated McNaughton-Schützenberger characterisation of first order definable languages by the aperiodicity condition x ω = x ω + 1, far from being an isolated statement, now appears as an elegant instance of a very general result.
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Gehrke, M., Grigorieff, S., Pin, JÉ. (2008). Duality and Equational Theory of Regular Languages. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5126. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70583-3_21
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DOI: https://doi.org/10.1007/978-3-540-70583-3_21
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