Ideal regular languages and strongly connected synchronizing automata

Abstract

Abstract. We introduce the notion of reset left regular decomposition of an ideal regular language and we prove that the category formed by these decompositions with certain morphisms is equivalent to the category of strongly connected synchronizing automata. We show that each ideal regular language has at least a reset left regular decomposition. As a consequence, each ideal regular language is the set of synchronizing words of some strongly connected synchronizing automaton. Furthermore, this one-to-one correspondence allows us to introduce the notion of reset decomposition complexity of an ideal. This notion allows the reformulation ofČerný's conjecture in pure language theoretic terms.

Semisimple Synchronizing Automata and the Wedderburn-Artin Theory

“…On the other hand, it is well known that ifČerný's conjecture is solved for the class of strongly connected synchronizing automaton, then this conjecture holds in general. Thus, the approach of studying synchronizing automata via their languages of synchronizing words is supported by the following result presented in a first version in [14] and then improved in [15] which we partially report here in the following theorem.…”

Synchronization of finite automata

“…For |Σ| ⩾ 2 any regular ideal in Σ * can be interpreted as a language Sync A for an appropriate strongly connected synchronizing automaton A -this (highly non-trivial) refinement was obtained by Reis and Rodaro [144].…”