On incomplete and synchronizing finite sets

TL;DR

This paper explores bounds on the minimal lengths of synchronizing and incompletable words in finite automata and codes, relating to longstanding conjectures, and provides a quadratic bound under certain conditions.

Contribution

It establishes a quadratic upper bound for the minimal length of synchronizing pairs in finite complete codes assuming Restivo's conjecture.

Findings

Provides a quadratic bound related to the length of words in finite codes

Connects the problem to Cerny's and Restivo's conjectures

Advances understanding of synchronization in automata and codes

Abstract

This paper situates itself in the theory of variable length codes and of finite automata where the concepts of completeness and synchronization play a central role. In this theoretical setting, we investigate the problem of finding upper bounds to the minimal length of synchronizing words and incompletable words of a finite language X in terms of the length of the words of X. This problem is related to two well-known conjectures formulated by Cerny and Restivo, respectively. In particular, if Restivo's conjecture is true, our main result provides a quadratic bound for the minimal length of a synchronizing pair of any finite synchronizing complete code with respect to the maximal length of its words.