1 research outputs found
Synchronizing automata and the language of minimal reset words
We study a connection between synchronizing automata and its set 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
has a reset word of length at
most In the last part of the paper we focus on the
existence of synchronizing automata with a given ideal 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 . With this
tool, we first show the existence of an infinite strongly connected
synchronizing automaton having as the set of reset words and
such that every other strongly connected synchronizing automaton having as
the set of reset words is an homomorphic image of . Finally, we
show that for any non-unary regular ideal there is a strongly connected
synchronizing automaton having as the set of reset words with at most
states, where , is the length of a
shortest word in , and is the dimension of the smallest automaton
recognizing (state complexity of ). This automaton is computable and we
show an algorithm to compute it in time .Comment: 17 page