# Counting symbol switches in synchronizing automata

## TL;DR

This paper explores the switch count in synchronizing automata, showing it has similar complexity to synchronizing word length, and constructs automata with quadratic switch counts, revealing new insights into automata synchronization properties.

## Contribution

It introduces the concept of switch count in automata, proves its complexity matches that of synchronizing word length, and provides automata constructions with quadratic switch counts.

## Key findings

- Switch count can be quadratic in automata with many symbols.
- Binary automata on up to 9 states have maximal switch counts determined.
- More symbols can increase switch count beyond what is possible with fewer symbols.

## Abstract

Instead of looking at the lengths of synchronizing words as in \v{C}ern\'y's conjecture, we look at the switch count of such words, that is, we only count the switches from one letter to another. Where the synchronizing words of the \v{C}ern\'y automata $\mathcal{C}_n$ have switch count linear in $n$, we wonder whether synchronizing automata exist for which every synchronizing word has quadratic switch count. The answer is positive: we prove that switch count has the same complexity as synchronizing word length. We give some series of synchronizing automata yielding quadratic switch count, the best one reaching $\frac{2}{3} n^2 + O(n)$ as switch count. We investigate all binary automata on at most 9 states and determine the maximal possible switch count. For all $3\leq n\leq 9$, a strictly higher switch count can be reached by allowing more symbols. This behaviour differs from length, where for every $n$, no automata are known with higher synchronization length than $\mathcal{C}_n$, which has only two symbols. It is not clear if this pattern extends to larger $n$. For $n\geq 12$, our best construction only has two symbols.