Synchronizing Boolean networks asynchronously

The asynchronous automatonof 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 ion a state xconsists in either switching the ith component if f(i)(x) not equal 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 [n]: for every i is an element of[n], f(i)(x) = 1 if and only if x(j)= 1( xj not equal 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(root 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 complexif one of the two hypothesis made on G is removed. (c) 2023 Elsevier Inc. All rights reserved.