On two combinatorial problems arising from automata theory

Jean-Eric Pin (1)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

We present some partial results on the following conjectures arising from automata theory. The first conjecture is the triangle conjecture due to Perrin and Schützenberger. Let A = {a, b} be a two-letter alphabet, d a positive integer and let B_d = {a^iba^j | 0 <= i+j <= d}. If X \subset B_d is a code, then |X| <= d+1. The second conjecture is due to Cerný and the author. Let A be an automaton with n states. If there exists a word of rank <= k in A, there exists such a word with length <= (n-k)^2.

Mots clés: finite automata; synchronizing word; Cerny's conjecture

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

Jean-Eric Pin. On two combinatorial problems arising from automata theory. Combinatorial mathematics (Marseille-Luminy, 1981), 1983, Marseille-Luminy, pp.535-548. ⟨hal-00143937⟩