Semisimple Synchronizing Automata and the Wedderburn-Artin Theory

We present a ring theoretic approach to Černý's conjecture via the Wedderburn-Artin theory. We first introduce the radical ideal of a synchronizing automaton, and then the natural notion of semisimple synchronizing automata. This is a rather broad class since it contains simple synchronizing automata like those in Černý's series. Semisimplicity gives also the advantage of “factorizing” the problem of finding a synchronizing word into the sub-problems of finding “short” words that are zeros into the projection of the simple components in the Wedderburn-Artin decomposition. In the general case this last problem is related to the search of radical words of length at most [Formula: see text] where n is the number of states of the automaton. We show that the solution of this “Radical Conjecture” would give an upper bound [Formula: see text] for the shortest reset word in a strongly connected synchronizing automaton. Finally, we use this approach to prove the Radical Conjecture in some particular cases and Černý's conjecture for the class of strongly semisimple synchronizing automata. These are automata whose sets of synchronizing words are cyclic ideals, or equivalently, ideal regular languages that are closed under taking roots.

The Cerný conjecture for automata respecting intervals of a directed graph

Automata, Logic and Semantics International audience The Cerný's conjecture states that for every synchronizing automaton with n states there exists a reset word of length not exceeding (n - 1)2. We prove this conjecture for a class of automata preserving certain properties of intervals of a directed graph. Our result unifies and generalizes some earlier results obtained by other authors.

A QUADRATIC UPPER BOUND ON THE SIZE OF A SYNCHRONIZING WORD IN ONE-CLUSTER AUTOMATA

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·ar = q·as 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.