Synchronizing automata with random inputs

TL;DR

This paper investigates the synchronization times of automata under random inputs, revealing exponential bounds for certain automata and polynomial bounds for Cerny automata, advancing understanding of automata synchronization behavior.

Contribution

It introduces automata with exponential expected synchronization time and establishes cubic bounds for Cerny automata, providing new insights into automata synchronization complexity.

Findings

Expected synchronization time can be exponential for some automata.

Cerny automata synchronize within cubic expected steps.

Provides bounds contrasting different automata classes.

Abstract

We study the problem of synchronization of automata with random inputs. We present a series of automata such that the expected number of steps until synchronization is exponential in the number of states. At the same time, we show that the expected number of letters to synchronize any pair of the famous Cerny automata is at most cubic in the number of states.