On the cost of simulating a parallel Boolean automata network by a block-sequential one

TL;DR

This paper investigates the minimal additional automata needed to simulate a parallel Boolean automata network with a block-sequential schedule, introducing a graph-based approach to bound this cost.

Contribution

It introduces the NECC graph and relates its chromatic number to the simulation cost, providing bounds and conjectures for the minimal automata needed.

Findings

Bounds the simulation cost between n/2 and 2n/3+2

Shows the clique number of NECC graphs is at most n/2

For bijective BANs, the cost is at most n/2+1

Abstract

In this article we study the minimum number $\kappa$ of additional automata that a Boolean automata network (BAN) associated with a given block-sequential update schedule needs in order to simulate a given BAN with a parallel update schedule. We introduce a graph that we call $\mathsf{NECC}$ graph built from the BAN and the update schedule. We show the relation between $\kappa$ and the chromatic number of the $\mathsf{NECC}$ graph. Thanks to this $\mathsf{NECC}$ graph, we bound $\kappa$ in the worst case between $n/2$ and $2n/3+2$ ($n$ being the size of the BAN simulated) and we conjecture that this number equals $n/2$. We support this conjecture with two results: the clique number of a $\mathsf{NECC}$ graph is always less than or equal to $n/2$ and, for the subclass of bijective BANs, $\kappa$ is always less than or equal to $n/2+1$.