On fixable families of Boolean networks

The asynchronous dynamics associated with a Boolean network $f : \{0,1\}^n \to \{0,1\}^n$ is a finite deterministic automaton considered in many applications. 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. This action is extended to words in the natural way. We then say that a word $w$ {\em fixes} $f$ if, for all states $x$, the result of the action of $w$ on $x$ is a fixed point of $f$. A whole family of networks is fixable if its members are all fixed by the same word, and the fixing length of the family is the minimum length of such a word. In this paper, we are interested in families of Boolean networks with relatively small fixing lengths. Firstly, we prove that fixing length of the family of networks with acyclic asynchronous graphs is $\Theta(n 2^n)$. Secondly, it is known that the fixing length of the whole family of monotone networks is $O(n^3)$. We then exhibit two families of monotone networks with fixing length $\Theta(n)$ and $\Theta(n^2)$ respectively, namely monotone networks with tree interaction graphs and conjunctive networks with symmetric interaction graphs.