A Topological Approach to Recognition

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A Topological Approach to Recognition
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 a...
Mai Gehrke, Serge Grigorieff, Jean-Eric Pin
Added 19 Jul 2010
Updated 19 Jul 2010
Type Conference
Year 2010
Authors Mai Gehrke, Serge Grigorieff, Jean-Eric Pin
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