Asynchronous simulation of Boolean networks by monotone Boolean networks

TL;DR

This paper demonstrates how the asynchronous dynamics of certain Boolean networks can be simulated by monotone Boolean networks with twice as many components, revealing complex update strategies and contrasting with known properties of monotone networks.

Contribution

It proves that asynchronous dynamics of Boolean networks without negative loops can be simulated by monotone networks with 2n components, and shows existence of networks with complex update strategies.

Findings

Asynchronous dynamics can be simulated by monotone networks with 2n components.

Existence of monotone networks requiring exponentially many updates to reach fixed points.

Contrasts with known properties where fewer updates suffice for monotone networks.

Abstract

We prove that the fully asynchronous dynamics of a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ 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 $f:\{0,1\}^n\to\{0,1\}^n$, 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 $f:\{0,1\}^n\to\{0,1\}^n$ 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.