Slowly Synchronizing Automata with Idempotent Letters of Low Rank

TL;DR

This paper constructs specific synchronizing automata with idempotent letters of low rank, demonstrating that their reset thresholds grow quadratically with the number of states, using a semigroup-theoretic approach.

Contribution

It introduces a novel construction of automata with idempotent, low-rank input letters that achieve near-quadratic reset thresholds, expanding understanding of automata synchronization.

Findings

Automata with n states and 3 letters have reset thresholds ~ n^2/2.

All input letters act as idempotent selfmaps of rank n/2.

Construction applies for each even n, demonstrating quadratic growth.

Abstract

We use a semigroup-theoretic construction by Peter Higgins in order to produce, for each even $n$, an $n$-state and 3-letter synchronizing automaton with the following two features: 1) all its input letters act as idempotent selfmaps of rank $\dfrac{n}2$; 2) its reset threshold is asymptotically equal to $\dfrac{n^2}2$. In the revised version a few inaccuracies (spotted by the anonymous referees of the previous version) have been removed and several relevant references have been added.