Closures in Formal Languages and Kuratowski's Theorem

Details Closures in Formal Languages and Kuratowski's Theorem Closures in Formal Languages and Kuratowski's Theorem

2009-01-23

arxiv.org/abs/0901.3761v1

Computer Science

Computational Complexity

submitted to DLT 2009

Scientific paper

A famous theorem of Kuratowski states that in a topological space, at most 14 distinct sets can be produced by repeatedly applying the operations of closure and complement to a given set. We re-examine this theorem in the setting of formal languages, where closure is either Kleene closure or positive closure. We classify languages according to the structure of the algebra they generate under iterations of complement and closure. We show that there are precisely 9 such algebras in the case of positive closure, and 12 in the case of Kleene closure.

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