A conjecture on the concatenation product

Jean-Eric Pin (1) , Pascal Weil (2)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

In a previous paper, the authors studied the polynomial closure of a variety of languages and gave an algebraic counterpart, in terms of Mal'cev products, of this operation. They also formulated a conjecture about the algebraic counterpart of the boolean closure of the polynomial closure --- this operation corresponds to passing to the upper level in any concatenation hierarchy ---. Although this conjecture is probably true in some particular cases, we give a counterexample in the general case. Another counterexample, of a different nature, was independently given recently by Steinberg. Taking these two counterexamples into account, we propose a modified version of our conjecture. We show in particular that a solution to our new conjecture would give a solution of the decidability of the levels 2 of the Straubing-Thérien hierarchy and of the dot-depth hierarchy. Consequences for the other levels are also discussed.

Mots clés: regular language; finite automaton; concatenation product

Domaines: Autre [cs.OH]; Mathématique discrète [cs.DM]

Jean-Eric Pin, Pascal Weil. A conjecture on the concatenation product. RAIRO - Theoretical Informatics and Applications (RAIRO: ITA), 2001, 35, pp.597-618. ⟨hal-00112623⟩