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2009

Closures in Formal Languages and Kuratowski's Theorem

10 years 11 months ago
Closures in Formal Languages and Kuratowski's Theorem
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. There are precisely 9 such algebras in the case of positive closure, and 12 in the case of Kleene closure. We study how the properties of being open and closed are preserved under concatenation. We investigate analogues, in formal languages, of the separation axioms in topological spaces; one of our main results is that there is a clopen partition separating two words if and only if the words do not commute. We can decide in quadratic time if the language specified by a DFA is closed, but if the language is specified by an NFA, the problem is PSPACE-complete.
Janusz A. Brzozowski, Elyot Grant, Jeffrey Shallit
Added 17 Feb 2011
Updated 17 Feb 2011
Type Journal
Year 2009
Where DLT
Authors Janusz A. Brzozowski, Elyot Grant, Jeffrey Shallit
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