Closures in Formal Languages and Kuratowski’s Theorem

Abstract

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.

The State Complexity of Star-Complement-Star

“…This proves that if each X i in S goes on a to a non-final state X ′ i in the NFA N 3 , then S goes on a in the DFA D min 3 to a set that is in the required form (1). Now consider the case that at least one X j in S goes to a final state X ′ j in the NFA N 3 .…”

“…Proof. As shown in [1], every such language can be expressed, up to inclusion of ε, as one of the following 5 languages and their complements:…”

Inverse star, borders, and palstars

“…Let L be a language such that L = L * . Then, following [3], we say that L is closed. Brzozowski [2] studied the the "smallest" language M such that L = M * .…”

Quotient Complexity of Closed Languages

“…Kuratowski's theorem was studied in the setting of formal languages in [5]. Positive closure and Kleene closure (star) are both closure operations.…”