Synchronizing Boolean networks asynchronously
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The asynchronous automaton associated with 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 f_i(x)≠ x_i or doing nothing otherwise. These actions are extended to words in the natural way. A word is then synchronizing if the result of its action is the same for every state. In this paper, we ask for the existence of synchronizing words, and their minimal length, for a basic class of Boolean networks called and-or-nets: given an arc-signed digraph G on [n], we say that f is an and-or-net on G if, for every i∈ [n], there is a such that, for all state x, f_i(x)=a if and only if x_j=a (x_j≠ a) for every positive (negative) arc from j to i; so if a=1 (a=0) then f_i is a conjunction (disjunction) of positive or negative literals. Our main result is that if G is strongly connected and has no positive cycles, then either every and-or-net on G has a synchronizing word of length at most 10(√(5)+1)^n, much smaller than the bound (2^n-1)^2 given by the well known Černý's conjecture, or G is a cycle and no and-or-net on G has a synchronizing word. This contrasts with the following complexity result: it is coNP-hard to decide if every and-or-net on G has a synchronizing word, even if G is strongly connected or has no positive cycles.
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