Journal of Computer and System Sciences
Finite automata, formal logic, and circuit complexity
Finite automata, formal logic, and circuit complexity
Uniform and nonuniform recognizability
Theoretical Computer Science
Duality and Equational Theory of Regular Languages
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part II
A Mezei-Wright theorem for categorical algebras
Theoretical Computer Science
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We propose a new approach to the notion of recognition, which departs from the classical definitions by three specific features. First, it does not rely on automata. Secondly, it applies to any Boolean algebra (BA) of subsets rather than to individual subsets. Thirdly, topology is the key ingredient.We prove the existence of a minimum recognizer in a very general setting which applies in particular to any BA of subsets of a discrete space. Our main results show that this minimum recognizer is a uniform space whose completion is the dual of the original BA in Stone-Priestley duality; in the case of a BA of languages closed under quotients, this completion, called the syntactic space of the BA, is a compact monoid if and only if all the languages of the BA are regular. For regular languages, one recovers the notions of a syntactic monoid and of a free profinite monoid. For nonregular languages, the syntactic space is no longer a monoid but is still a compact space. Further, we give an equational characterization of BA of languages closed under quotients, which extends the known results on regular languages to nonregular languages. Finally, we generalize all these results from BAs to lattices, in which case the appropriate structures are partially ordered.