Fixing monotone Boolean networks asynchronously

The asynchronous automaton associated with a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ is considered in many applications. It is the finite deterministic automaton with set of states $\{0,1\}^n$, alphabet $\{1,\dots,n\}$, where 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. This action is extended to words in the natural way. We then say that a word $w$ fixes $f$ if, for all states $x$, the result of the action of $w$ on $x$ is a fixed point of $f$. In this paper, we ask for the existence of fixing words, and their minimal length. Firstly, our main results concern the minimal length of words that fix monotone networks. We prove that, for $n$ sufficiently large, there exists a monotone network $f$ with $n$ components such that any word fixing $f$ has length $\Omega(n^2)$. For this first result we prove, using Baranyai's theorem, a property about shortest supersequences that could be of independent interest: there exists a set of permutations of $\{1,\dots,n\}$ of size $2^{o(n)}$, such that any sequence containing all these permutations as subsequences is of length $\Omega(n^2)$. Conversely, we construct a word of length $O(n^3)$ that fixes all monotone networks with $n$ components. Secondly, we refine and extend our results to different classes of fixable networks, including networks with an acyclic interaction graph, increasing networks, conjunctive networks, monotone networks whose interaction graphs are contained in a given graph, and balanced networks.