Testing for Synchronization

TL;DR

This paper investigates the probability and algorithms for automaton synchronization, showing that for strongly connected automata the probability of being synchronizing approaches one with a linear-time algorithm, while for reachable partial automata the problem is NP-complete.

Contribution

It generalizes synchronization probability results to strongly connected partial automata and introduces a linear-time algorithm, also proving NP-completeness for reachable partial automata.

Findings

Probability of synchronization approaches 1 for strongly connected automata as n grows.

A linear expected time algorithm for checking synchronization in strongly connected automata.

Synchronization testing for reachable partial automata is NP-complete.

Abstract

We consider the first problem that appears in any application of synchronizing automata, namely, the problem of deciding whether or not a given $n$-state $k$-letter automaton is synchronizing. First we generalize results from \cite{RandSynch},\cite{On2Problems} for the case of strongly connected partial automata. Specifically, for $k>1$ we show that an automaton is synchronizing with probability $1-O(\frac{1}{n^{0.5k}})$ and present an algorithm with linear in $n$ expected time, while the best known algorithm is quadratic on each instance. This results are interesting due to their applications in synchronization of finite state information sources. After that we consider the synchronization of reachable partial automata that has application for splicing systems in computational biology. For this case we prove that the problem of testing a given automaton for synchronization is NP-complete.