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**Closures in formal languages and Kuratowski’s theorem.**
*(English)*
Zbl 1246.68139

Summary: 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 by “closure” we mean either Kleene closure or positive closure. We classify languages according to the structure of the algebras 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.

### MSC:

68Q45 | Formal languages and automata |

54A05 | Topological spaces and generalizations (closure spaces, etc.) |

68Q42 | Grammars and rewriting systems |

### Keywords:

closure system; complement; complexity; concatenation; decision problem; formal language; Kleene closure; Kuratowski’s theorem; positive closure; separating two words
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\textit{J. Brzozowski} et al., Int. J. Found. Comput. Sci. 22, No. 2, 301--321 (2011; Zbl 1246.68139)

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