An algebraic representation of the fixed-point closure of *-continuous Kleene algebras – A categorical Chomsky–Schützenberger theorem

Abstract

Abstract The family ${\mathcal{R}} X^*$ of regular subsets of the free monoid $X^*$ generated by a finite set X is the standard example of a ${}^*$ -continuous Kleene algebra. Likewise, the family ${\mathcal{C}} X^*$ of context-free subsets of $X^*$ is the standard example of a $\mu$ -continuous Chomsky algebra, i.e. an idempotent semiring that is closed under a well-behaved least fixed-point operator $\mu$ . For arbitrary monoids M, ${\mathcal{C}} M$ is the closure of ${\mathcal{R}}M$ as a $\mu$ -continuous Chomsky algebra, more briefly, the fixed-point closure of ${\mathcal{R}} M$ . We provide an algebraic representation of ${\mathcal{C}} M$ in a suitable product of ${\mathcal{R}} M$ with $C_2'$ , a quotient of the regular sets over an alphabet $\Delta_2$ of two pairs of bracket symbols. Namely, ${\mathcal{C}}M$ is isomorphic to the centralizer of $C_2'$ in the product of ${\mathcal{R}} M$ with $C_2'$ , i.e. the set of those elements that commute with all elements of $C_2'$ . This generalizes a well-known result of Chomsky and Schützenberger (1963, Computer Programming and Formal Systems, 118–161) and admits us to denote all context-free languages over finite sets $X\subseteq M$ by regular expressions over $X\cup\Delta_2$ interpreted in the product of ${\mathcal{R}} M$ and $C_2'$ . More generally, for any ${}^*$ -continuous Kleene algebra K the fixed-point closure of K can be represented algebraically as the centralizer of $C_2'$ in the product of K with $C_2'$ .