Synchronizing Times for $k$-sets in Automata

TL;DR

This paper improves bounds on the shortest words that synchronize subsets of states in automata, providing new upper bounds for triples, quadruples, and quintuples, and demonstrating large bounds in non-synchronizing cases.

Contribution

It introduces tighter upper bounds for the minimal length of words mapping k states to one in automata, advancing understanding of synchronization complexity.

Findings

Improved upper bound for triples to approximately 0.19n^2

Extended bounds for 4 and 5 states

Example showing large minimal lengths in non-synchronizing automata

Abstract

An automaton is synchronizing if there is a word that maps all states onto the same state. \v{C}ern\'{y}'s conjecture on the length of the shortest such word is probably the most famous open problem in automata theory. We consider the closely related question of determining the minimum length of a word that maps $k$ states onto a single state. For synchronizing automata, we improve the upper bound on the minimum length of a word that sends some triple to a a single state from $0.5n^2$ to $\approx 0.19n^2$. We further extend this to an improved bound on the length of such a word for 4 states and 5 states. In the case of non-synchronizing automata, we give an example to show that the minimum length of a word that sends $k$ states to a single state can be as large as $\Theta\left(n^{k-1}\right)$.