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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 \({{\mathrm{\mathsf {NECC}}}}\) graph built from the BAN and the update schedule. We show the relation between \(\kappa \) and the chromatic number of the \({{\mathrm{\mathsf {NECC}}}}\) graph. Thanks to this \({{\mathrm{\mathsf {NECC}}}}\) 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 \({{\mathrm{\mathsf {NECC}}}}\) 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 \(n/2+1\).

On the Cost of Simulating a Parallel Boolean Automata Network by a Block-Sequential One