[Paper Review] Closures in Formal Languages and Kuratowski's Theorem

Janusz Brzozowski, Elyot Grant+1 more|ArXiv.org|Jan 23, 2009

Advanced Algebra and Logic3 references4 citations

TL;DR

This paper investigates the closure properties of formal languages under Kleene and positive closure operations, extending Kuratowski's classical theorem on topological closure and complement to formal language theory. It classifies languages based on the algebras generated by repeated application of complement and closure, proving there are exactly 12 distinct algebras under Kleene closure and 9 under positive closure, with tight bounds on the number of generated languages (up to 14).

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.

Motivation & Objective

• To extend Kuratowski’s classical 14-set theorem from topological spaces to formal languages by replacing topological closure with Kleene or positive closure.
• To classify all possible algebras generated by a language under repeated application of complement and closure operations in formal language settings.
• To determine the exact number of distinct algebras and the maximum number of languages generated under each closure type.
• To provide explicit language examples for each algebraic type, demonstrating the tightness of the bounds.

Proposed method

• Define Kleene closure (L∗= ⋃i≥0 Li) and positive closure (L+= ⋃i≥1 Li) as closure operators on the set of all finite words Σ∗.
• Use duality between closure and interior operators: L⊛ = L−∗− (Kleene interior), L⊕ = L−+− (positive interior).
• Classify languages based on whether they are open, closed, clopen, or neither, using the closure and interior operators.
• Analyze the structure of the algebra E(L) generated by a language L under closure, complement, and their iterations.
• Use a homomorphism φ to map languages in E(L) to their base sets in B(L), reducing complexity and enabling classification.
• Apply case analysis based on the status of the empty word ǫ and the openness/closedness of L and its derived sets to enumerate all possible algebras.

Experimental results

Research questions

• RQ1 How many distinct languages can be generated by repeatedly applying complement and Kleene closure to a single formal language?
• RQ2 What structural properties of a language determine the size and composition of the algebra generated under closure and complement operations?
• RQ3 How does the presence or absence of the empty word ǫ affect the algebraic structure of the generated language set?
• RQ4 Are there distinct algebraic types under positive closure versus Kleene closure, and if so, how many?
• RQ5 Can tight upper bounds on the number of generated languages be established, and are these bounds achievable?

Key findings

• There are exactly 12 distinct algebras generated by any language under repeated application of Kleene closure and complement, with a maximum of 14 distinct languages.
• For positive closure, there are exactly 9 distinct algebras, with a maximum of 14 generated languages, though the structure differs from the Kleene case.
• The bound of 14 languages under Kleene closure is tight, as demonstrated by the example language a ∪ ab ∪ bb in case (9), which generates exactly 14 distinct languages.
• The structure of the algebra depends critically on whether the language is open, closed, clopen, or neither, and on the presence or absence of the empty word ǫ.
• When a language L is clopen and contains ǫ, the generated algebra has size 2; when L is clopen but does not contain ǫ, the size is also 2, but the algebra is dual to the first case.
• In cases where L is neither open nor closed, the algebra size ranges from 4 to 14, depending on the closure and openness properties of L+ and L⊕, with the maximum size 14 achieved when both L+ and L⊕ are neither closed nor open and their closures differ.