## Preprint 2018-04

## Julio Aracena, Maximilien Gadouleau, Adrien Richard, Lilian Salinas:

### Fixing monotone Boolean networks asynchronously

### Abstract:

The asynchronous automaton associated with a Boolean network f:B^n-->B^n is considered in many applications. It is the finite deterministic automaton with set of states B^n, alphabet {1,...,n}, where the action of letter i on a state x consists in either switching the ith component if f_i(x) is not equal to 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 Theorem, a property about shortest supersequences that could be of independent interest: there exists a set of permutations of {1,...,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.