A Quadratic Upper Bound on the Size of a Synchronizing Word in One-Cluster Automata

Abstract

Abstract.Černý's conjecture asserts the existence of a synchronizing word of length at most (n − 1) 2 for any synchronized n-state deterministic automaton. We prove a quadratic upper bound on the length of a synchronizing word for any synchronized n-state deterministic automaton satisfying the following additional property: there is a letter a such that for any pair of states p, q, one has p · a r = q · a s for some integers r, s (for a state p and a word w, we denote by p · w the state reached from p by the path labeled w). As a consequence, we show that for any finite synchronized prefix code with an n-state decoder, there is a synchronizing word of length O(n 2 ). This applies in particular to Huffman codes.

Synchronizing Automata of Bounded Rank

“…Though theČerný conjecture is open in general, it has been confirmed for some restricted classes of synchronizing automata, see [1,6,8,13,16]. For some classes quadratic upper bound is established, see [4,11].…”

On incomplete and synchronizing finite sets

“…In particular, in [3] (see also [4]), Béal and Perrin have proved that a complete synchronizing prefix code X on an alphabet of d letters with n code-words has a synchronizing word of length O(n 2 ). In this paper we are interested in finding upper bounds to the minimal lengths of incompletable and synchronizing words of a finite set X in terms of the size of X.…”

On the Synchronizing Probability Function and the Triple Rendezvous Time for Synchronizing Automata

“…It has been proven to hold for several families of automata (see [2,4,9,10,13,14,20,27]), including cyclic and Eulerian. However, the best general upper bound on the reset threshold of an automaton with n states is equal to (n 3 − n)/6, obtained by Pin and Frankl [15,23], and rediscovered independently in [21].…”