Synchronizing Boolean networks asynchronously

Abstract

The asynchronous automaton of a Boolean network $f:{\{0,1\}}^n\to {\{0,1\}}^n$, considered in many applications, is the finite deterministic automaton where the set of states is ${\{0,1\}}^n$, the alphabet is $\left[n\right]$, and the action of letter i on a state x consists in either switching the ith component if $f_i\left(x\right)\ne x_i$ or doing nothing otherwise. In this paper, we ask for the existence of synchronizing words for this automaton, and their minimal length, when f is the and-net over an arc-signed digraph G on $\left[n\right]$: for every $i\in \left[n\right]$, $f_i\left(x\right)=1$ if and only if $x_j=1$ ($x_j\ne 0$) for every positive (negative) arc from j to i. Our main result is that if G is strongly connected and has no positive cycles, then either there exists a synchronizing word of length at most $10{(\sqrt{5}+1)}^n$ or G is a cycle and there are no synchronizing words. We also give complexity results showing that the situation is much more complex if one of the two hypothesis made on G is removed.