A bound for the length of the shortest reset words for semisimple synchronizing automata via the packing number

Abstract

We show that if a semisimple synchronizing automaton with n states has a minimal reachable non-unary subset of cardinality r ≥ 2, then there is a reset word of length at most (n − 1)D(2, r, n), where D(2, r, n) is the 2-packing number for families of r-subsets of [1, n].

Synchronization of finite automata

“…Investigations of synchronizing automata using tools of representation theory have produced a number of informative results. The pioneering studies in this direction were apparently carried out by Rystsov [152]; among other publications in which synchronizing automata were investigated using methods of representation theory we should mention the papers [3]- [5], [16], [147], and [172] by Almeida, Rodaro, Steinberg, and their coauthors.…”