  The {\em asynchronous automaton} associated with 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
$[n]$, and the action of letter $i$ on a state $x$ consists in either switching
the $i$th component if $f_i(x)\neq x_i$ or doing nothing otherwise. These
actions are extended to words in the natural way. A word is then {\em
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 {\em and-or-net} on $G$
if, for every $i\in [n]$, there is $a$ such that, for all state $x$, $f_i(x)=a$
if and only if $x_j=a$ ($x_j\neq a$) 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 $10(\sqrt{5}+1)^n$, much smaller than the
bound $(2^n-1)^2$ given by the well known \v{C}ern\'y'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.
