The Kuratowski Closure-Complement Problem

Fife, James

doi:10.2307/2691300

Cited by 7 publications

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“…Moreover, it is equivalent to define (S, ) via an interior operator satisfying (2). We now list some fundamental properties of closure systems; note the duality.…”

Section: Closure Operators and Closure Systemsmentioning

confidence: 99%

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Closures in Formal Languages and Kuratowski’s Theorem

Brzozowski

Grant

Shallit

2009

Developments in Language Theory

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Abstract. 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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“…Moreover, it is equivalent to define (S, ) via an interior operator satisfying (2). We now list some fundamental properties of closure systems; note the duality.…”

Section: Closure Operators and Closure Systemsmentioning

confidence: 99%

“…In 1922, Kuratowski proved that if S is any set in a topological space, then at most 14 distinct sets can be produced by repeatedly applying the operations of topological closure and complement to S [5,2]. Furthermore, there exist sets achieving this bound of 14 in many common topological spaces.…”

Section: Introductionmentioning

confidence: 99%

Closures in Formal Languages and Kuratowski’s Theorem

Brzozowski

Grant

Shallit

2009

Developments in Language Theory

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“…The famous Kuratowski 14-theorem states that, in a topological space, repeatedly applying the operations of closure and complement to any given set produces at most 14 distinct sets [7,18]. Kuratowski's theorem in the settings of formal languages has been studied by Brzozowski, Grant, and Shallit [2].…”

Section: Introductionmentioning

confidence: 99%

Kuratowski Algebras Generated by Prefix-, Suffix-, Factor-, and Subword-Free Languages Under Star and Complementation

Jirásek

Palmovský

Šebej

2019

Int. J. Found. Comput. Sci.

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We study Kuratowski algebras generated by prefix-, suffix-, factor-, and subword-free languages under the operations of star and complementation. We examine 12 possible algebras, and for each of them, we decide whether or not it can be generated by a prefix-, suffix-, factor-, or subword-free language. In each case when an algebra can be generated by such a language, we show that this language can be taken to be regular, and we compute upper bounds on the state complexities of all the generated languages. Finally, we find generators that maximize these complexities.

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“…The famous Kuratowski's 14-theorem states that, in a topological space, repeatedly applying the operations of closure and complement to any given set can produce at most 14 distinct sets [6,12]. Kuratowski's theorem in the settings of formal languages has been studied by Brzozowski, Grant, and Shallit [2].…”

Section: Introductionmentioning

confidence: 99%

Kuratowski Algebras Generated by Factor-, Subword-, and Suffix-Free Languages

Jirásek

Palmovský

Šebej

2017

Descriptional Complexity of Formal Systems

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We study Kuratowski algebras generated by suffix-, factor-, and subword-free languages under the operations of star and complementation. We examine 12 possible algebras, and for each of them, we provide an answer to the question whether or not it can be generated by a suffix-, factor-, or subword-free language. In each case when an algebra can be generated by such a language, we show that this language may be taken to be regular, and we compute upper bounds on the state complexities of all the generated languages. Finally, we find generators that maximize these complexities.

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The Kuratowski Closure-Complement Problem

Cited by 7 publications

References 0 publications

Closures in Formal Languages and Kuratowski’s Theorem

Closures in Formal Languages and Kuratowski’s Theorem

Kuratowski Algebras Generated by Prefix-, Suffix-, Factor-, and Subword-Free Languages Under Star and Complementation

Kuratowski Algebras Generated by Factor-, Subword-, and Suffix-Free Languages

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