Asynchronous Simulation of Boolean Networks by Monotone Boolean Networks

Abstract

We prove that the fully asynchronous dynamics of a Boolean network $f:\{0,1\}^n\rightarrow \{0,1\}^n$ without negative loop can be simulated, in a very specific way, by a monotone Boolean network with 2n components. We then use this result to prove that, for every even n, there exists a monotone Boolean network $f:\{0,1\}^n\rightarrow \{0,1\}^n$, an initial configuration x and a fixed point y of f such that: (i) y can be reached from x with a fully asynchronous updating strategy, and (ii) all such strategies contains at least $2^{\frac{n}{2}}$ updates. This contrasts with the following known property: if $f:\{0,1\}^n\rightarrow \{0,1\}^n$ is monotone, then, for every initial configuration x, there exists a fixed point y such that y can be reached from x with a fully asynchronous strategy that contains at most n updates.