Automata for the commutative closure of regular sets

TL;DR

This work addresses computing the commutative closure ${\\frak C}(L)$ of a regular language $L$ when the closure is regular. It develops an algebraic pipeline that maps words to their Parikh images in $\\mathbb{N}^k$, represents commutative images via semi-simple and resimple expressions, and then constructs a deterministic automaton for ${\\frak C}(L)$ using shuffle automata. A key contribution is a simpler, fully constructive algorithm that yields a complete automaton for the commutative closure and a rigorous complexity analysis, including state- and time-bounds, grounded in the theory of recognizable sets and formal power series. The approach provides a practical path from RegLangs to their commutative closures when regular, and clarifies the role of Gohon's decision procedure within this automata-theoretic construction.

Abstract

Consider $ A^* $, the free monoid generated by the finite alphabet $A$ with the concatenation operation. Two words have the same commutative image when one is a permutation of the symbols of the other. The commutative closure of a set $ L \subseteq A^* $ is the set $ {C}(L) \subseteq A^* $ of words whose commutative image coincides with that of some word in $ L $. We provide an algorithm that, given a regular set $ L $, produces a finite state automaton that accepts the commutative closure $ {C}(L) $, provided that this closure set is regular. The problem of deciding whether $ {C}(L) $ is regular was solved by Ginsburg and Spanier in 1966 using the decidability of Presburger sentences, and by Gohon in 1985 via formal power series. The problem of constructing an automaton that accepts $ {C}(L) $ has already been studied in the literature. We give a simpler algorithm using an algebraic approach.

Figures (12)

• Figure 1: Finite state automaton accepting ${\EuScript C}({b(a^2\cup b^2)}^*)$.
• Figure 2: Semi-simple and semi-linear expressions denoting the same set
• Figure 3: On the left, an infinite set of ${\mathbb N}^2$ that is not recognizable, because it is impossible to characterize it as a finite union of products of rational sets in $\mathbb{N}$. On the right, an infinite set of ${\mathbb N}^2$ that is recognizable: it can be described as the product between the even and the odd numbers, both rational sets in $\mathbb{N}$.
• Figure 4: Finite state automaton for $\varphi^{-1}(R)$ when $R$ is an atomic resimple expression of $\mathbb{N}$ ($k=1$ and $A=\{a\}$).
• Figure 5: Original set and the automata construction of \ref{['prop:resimple automaton']}.

Theorems & Definitions (71)

• Definition 1: Commutative Closure
• Proposition 1: GINSBURGGOHON
• Theorem 1
• Example 1: Regular commutative closure
• Definition 2: Rational expression
• Definition 3: Rational set
• Definition 4: Star height of a rational expression
• Example 2
• Definition 5: Star height of a rational set
• Proposition 2