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.

翻译：与布尔网络 $f:\{0,1\}^n\to\{0,1\}^n$ 相关的{\em 异步自动机}（广泛应用于各类场景）是一类有限确定性自动机：其状态集为 $\{0,1\}^n$，字母表为 $[n]$，字母 $i$ 作用于状态 $x$ 的结果为：若 $f_i(x)\neq x_i$ 则切换第 $i$ 个分量，否则不进行任何操作。这些作用以自然方式扩展至单词。若某单词对所有状态的执行结果均相同，则称该单词为{\em 同步化}的。本文针对一类称为“与或网络”的布尔网络基础类别，研究同步化单词的存在性及其最小长度问题：给定 $[n]$ 上的带符号有向图 $G$，若对于每个 $i\in [n]$，存在 $a$ 使得对所有状态 $x$，$f_i(x)=a$ 当且仅当对所有从 $j$ 到 $i$ 的正（负）弧有 $x_j=a$（$x_j\neq a$），则称 $f$ 为 $G$ 上的{\em 与或网络}；因此若 $a=1$（$a=0$），则 $f_i$ 是正负文字的合取（析取）。我们的主要结果是：若 $G$ 强连通且无正环，则要么 $G$ 上所有与或网络均存在长度不超过 $10(\sqrt{5}+1)^n$ 的同步化单词——该值远小于经典Černý猜想给出的界限 $(2^n-1)^2$——要么 $G$ 是环且其上不存在任何与或网络具有同步化单词。这一结论与以下复杂性结果形成鲜明对比：即使 $G$ 强连通或无正环，判断 $G$ 上所有与或网络是否均存在同步化单词的问题仍是 coNP-难的。