A quadratic upper bound on the size of a synchronizing word in one-cluster automata

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(n2). This applies in particular to Huffman codes. © 2009 Springer Berlin Heidelberg.