Finitely Generated Ideal Languages and Synchronizing Automata

Abstract

We study representations of ideal languages by means of strongly connected synchronizing automata. For every finitely generated ideal language L we construct such an automaton with at most 2 n states, where n is the maximal length of words in L. Our constructions are based on the De Bruijn graph.

Ideal regular languages and strongly connected synchronizing automata

“…In case I is finitely generated is true that rdc(I) ≥ I + 1? The same problem in case I is a principal ideal language has been raised in [1]. This would give a better bound for the shortest synchronizing word for the class of finitely generated synchronizing automata with respect to the bound obtained in [9].…”

“…For instance in [1] it is shown an algorithm that given a principal ideal I = Σ * wΣ * with |w| = n in inputs, it returns a strongly connected synchronizing automaton with n + 1 states. Therefore in this case the bound is linear with respect to the state complexity of I R although it is not known whether or not it is tight.…”

“…Therefore in this case the bound is linear with respect to the state complexity of I R although it is not known whether or not it is tight. Even more recently in [2] the authors prove that in the case I is finitely generated, there is always a strongly connected synchronizing automaton with at most 2 I states and this bound is tight for ideals of the form Σ ≥n = {u ∈ Σ * : |u| ≥ n} for any n > 0. Similarly to [3], where the author has introduced the notion of reset complexity of an ideal I (rc(I)) as the number of states of the smallest synchronizing automaton A with Syn(A ) = I, we can also give a similar notion in the realm of strongly connected synchronizing automata/reset left regular decomposition.…”

“…In the same paper it is shown that the reset complexity can be exponentially smaller than the state complexity of the language. In the subsequent paper [1], the authors consider the special case of finitely generated synchronizing automata with the set of the reset words which is a principal ideal P = Σ * wΣ * generated by a word w ∈ Σ * , and it is presented an algorithm to generate a strongly connected synchronizing automaton B w with Syn(B w ) = P with the same number of states of A P . In the same paper the author address the question whether or not, for any ideal language I, there is always a strongly connected synchronizing automaton in SA(I).…”

Strongly connected synchronizing automata and the language of minimal reset words

“…For a general survey on synchronizing automata and Cerny's conjecture see [15,28]. In this paper we continue the language theoretic approach to synchronizing automata initiated in a series of recent papers [11,12,17,21,22,23,24]. The starting point of such an approach is a simple observation: the set of reset words is a two-sided ideal (ideal for short) of the free monoid Σ * that is also a regular language.…”