Synchronizing automata and the language of minimal reset words

TL;DR

This paper explores the relationship between synchronizing automata and minimal reset words, providing bounds on reset word lengths, and constructing automata with prescribed reset word sets using novel structural tools.

Contribution

It introduces the tail structure of ideals and constructs automata with specific reset word sets, establishing existence and complexity bounds.

Findings

Automata with certain minimal reset word properties have bounded reset lengths.

Existence of a universal automaton for a given ideal as reset words set.

An algorithm to compute automata with prescribed reset words within exponential time.

Abstract

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 $\frac{1}{4}\min\{|u|: u\in I\}+\frac{1}{16}$ has a reset word of length at most $(n-\frac{1}{2})^{2}$ 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 $I=\Sigma^{*}M\Sigma^{*}$. With this tool, we first show the existence of an infinite strongly connected synchronizing automaton $\mathcal{A}$ 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 $\mathcal{A}$. 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 $(km^{k})2^{km^{k}n}$ states, where $k=|\Sigma|$, $m$ 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 $\mathcal{O}((k^{2}m^{k})2^{km^{k}n})$.