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Closures in formal languages and Kuratowski’s theorem
 Developments in Language Theory, 13th International Conference, DLT 2009
"... 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 reexamine this theorem in the setting of formal languages, where closure is either Kleene closure ..."
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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 reexamine 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. 1
A Term Rewriting System for Kuratowski’s ClosureComplement Problem ∗
"... We present a term rewriting system to solve a class of open problems that are generalisations of Kuratowski’s closurecomplement theorem. The problems are concerned with finding the number of distinct sets that can be obtained by applying combinations of axiomatically defined set operators. While th ..."
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We present a term rewriting system to solve a class of open problems that are generalisations of Kuratowski’s closurecomplement theorem. The problems are concerned with finding the number of distinct sets that can be obtained by applying combinations of axiomatically defined set operators. While the original problem considers only closure and complement of a topological space as operators, it can be generalised by adding operators and varying axiomatisation. We model these axioms as rewrite rules and construct a rewriting system that allows us to close some so far open variants of Kuratowski’s problem by analysing several million inference steps on a typical personal computer.
The Kuratowski ClosureComplement Theorem
"... topology, was first posed and proven by the Polish mathematician Kazimierz Kuratowski in 1922. Since then, Kuratowski’s Theorem and its related results, in particular, the structure of the Kuratowski monoid of a topological space, have been the subject of a plethora of papers. The formal statement o ..."
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topology, was first posed and proven by the Polish mathematician Kazimierz Kuratowski in 1922. Since then, Kuratowski’s Theorem and its related results, in particular, the structure of the Kuratowski monoid of a topological space, have been the subject of a plethora of papers. The formal statement of the theorem is as follows: Theorem 1: The Kuratowski ClosureComplement Theorem: Let (X, T) be a topological space and suppose A X. Then there are at most 14 distinct sets that can be formed by taking complements and closures of A. Moreover, this bound is attained for a subset of with the standard topology. As in the proof of Gardner and Jackson, we make use of operator notation. Given a topological space (X, T) define the complement operator a and the closure operator b on subsets A X by a(A) = X \ A and b(A) = closure(A). Notice that for any topological space (X, T) and subset A X, aa(A) = A. Then given any topological space (X, T), the set of all distinct operators on (X, T) produced by compositions of elements of the set {a, b} forms a monoid with identity element aa. This monoid is referred to as the Kuratowski monoid on (X, T). For any topological space (X, T), there is a natural partial order on the Kuratowski monoid on (X, T). If o1 and o2 are elements of the Kuratowski monoid on (X, T), we define the partial order ≤ as o1 ≤ o2 if for every A X, o1(A) o2(A). We are now ready to prove Theorem 1. Proof: Let (X, T) be a topological space. We have already seen that aa = id. Note also that bb = b. This immediately implies that any operator of the Kuratowski monoid on (X, The Kuratowski ClosureComplement Theorem