Systèmes entiers d'équations sur un alphabet fini et conjecture d'ehrenfeucht

Abstract

The following conjecture has been posed by A. Ehrenfeucht. “For each language L over a finite alphabet Σ there exists a finite subset of F of L such that, for each pair (α, β) of morphisms of Σ ∗ into a free monoid, the equality α( χ) = β( χ) is true for all χ in L if and only if it is true for all χ in F”. Such a finite subset F of L is called a test set of L. A system S of equations over the alphabet Σ is called entire if there exists a morphism α of Σ ∗ into a free monoid such that S = α −1 ∘ α. We have introduced, for each morphism α of Σ ∗ in a free monoid such that Σ is finite and α −1(1) = {1} a generalized automaton, M (α)—called the shuffle automaton of α—which recognizes a rational language K( α) over a finite alphabet Δ and a pair (λ, μ) of morphisms of Δ ∗ into Σ ∗ such that α −1 ∘ α = {( λ ( χ), μ ( χ)); χ ϵ K ( α)}. Nivat's (1966) characterization of rational relations is a motivation for this construction and implies that all entire systems over a finite alphabet are rational. Culik, II and Salomaa (1978) have proved that every rational language over a finite alphabet admits a test set. Culik, II and Karhumäki (1983) have shown that if S is a system of equations over a finite alphabet Σ and if there exists a language L over a finite alphabet Δ, and morphisms λ and μ from Δ ∗ into Σ ∗ such that S = {( λ)( χ), μ( χ)); χ ϵ L}, then, if L admits a test set F, then, T = {( λ ( χ), μ ( χ)); χ ϵ F} is a finite equivalent subsystem of S. This gives an algorithm which determines, for each entire system of equations over a finite alphabet, a finite subsystem. This result can be improved by using the following theorem (Spehner, 1984): “Each finitely generated submonoid M of a free monoid has a finite m-presentation”, i.e., there exists a finite alphabet Σ and a finite part ϱ of Σ ∗ × Σ ∗ such that M is isomorphic to the quotient-monoid of Σ ∗ by the smallest Malcev-congruence containing ϱ. (π is a Malcev-congruence of Σ ∗ if the quotient-monoid Σ ∗/π is embeddable in a group.) The algorithm for determining the shuffle automaton M (α) and the finite equivalent subsystem of S = α −1 ∘ α works with a maximum efficiency because M (α) can be embedded in the minimal automaton of the language K ( α).