Synchronizing Boolean networks asynchronously

Abstract

The asynchronous automaton of a Boolean network f:{0,1}n→{0,1}n, considered in many applications, is the finite deterministic automaton where the set of states is {0,1}n, the alphabet is [n], and the action of letter i on a state x consists in either switching the ith component if fi(x)≠xi 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 [n]: for every i∈[n], fi(x)=1 if and only if xj=1 (xj≠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(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.