Synchronizing automata and the language of minimal reset words

We study a connection between synchronizing automata and its set M of minimal reset words, i.e., such that no proper factor is a reset word. We first show that any synchronizing automaton having the set of minimal reset words whose set of factors does not contain a word of length at most 1 4 min{|u| : u ∈ I} + 1 16 has a reset word of length at most (n − 1 2 ) 2. In the last part of the paper we focus on the existence of synchronizing automata with a given ideal I that serves as the set of reset words. To this end, we introduce the notion of the tail structure of the (not necessarily regular) ideal I = Σ∗MΣ∗. With this tool, we first show the existence of an infinite strongly connected synchronizing automaton A having I as the set of reset words and such that every other strongly connected synchronizing automaton having I as the set of reset words is an homomorphic image of A . Finally, we show that for any non-unary regular ideal I there is a strongly connected synchronizing automaton having I as the set of reset words with at most (km)2 n states, where k = |Σ|, m is the length of a shortest word in M , and n is the dimension of the smallest automaton recognizing M (state complexity of M). This automaton is computable and we show an algorithm to compute it in time O((k2mk)2km kn).