Careful synchronisation and the diameter of transformation semigroups with few generators

TL;DR

This paper improves lower bounds on the length of carefully synchronising words in partial automata and explores the diameter of transformation semigroups, extending previous results with simpler constructions and broader implications.

Contribution

It provides stronger lower bounds for carefully synchronising words and analyzes the diameter of partial automata, extending existing bounds to larger alphabets with simpler methods.

Findings

Lower bounds improved to $2^{(n - 4)/3}$ and $4^{(n - 4)/5}$ for small alphabets.

Large alphabet size matches upper bounds on the diameter asymptotically.

Application to the diameter of finite semigroups of nonnegative matrices.

Abstract

A word is called carefully synchronising for a partial deterministic finite semi-automaton if it maps all states to the same state. Equivalently, it is a composition of partial transformations equal to a constant total transformation. There is a sequence of several papers providing stronger and stronger lower bounds on the length of shortest carefully synchronising words for $n$-state partial DFAs over small alphabets. It resulted in the lower bounds of $\Omega(\frac{2^{n/3}}{n\sqrt{n}})$ and $\Omega(\frac{4^{n/5}}{n})$ for alphabets of two and three letters respectively, obtained by de Bondt, Don, and Zantema. Using a significantly simpler construction, we improve these lower bounds to $2^{(n - 4)/3}$ and $4^{(n - 4)/5}$ respectively. We then consider a tightly related question of the diameter of a partial DFA, which is the smallest $\ell \ge 0$ such that words of length at most $\ell$ express all the transformations induced by words in this DFA. We show that an alphabet of large enough constant size already asymptotically matches the upper bound on the diameter for arbitrary alphabet size, extending the construction of Panteleev that requires an alphabet of size exponential in the number of states. We then discuss an application to the diameter of finite semigroups of nonnegative matrices, and some open problems.