Sequentialization and procedural complexity in automata networks

Abstract

In this article we consider finite automata networks and make some connections between the two most classical deterministic update schedules: the parallel one (all automata are updated all together) and the sequential ones (the automata are updated periodically one at a time according to a total order). Given a network h with n automata, where each automaton has q possible states, the cost of sequentialization of h is the minimum number of additional automata required for another automata network to simulate h with the same q states thanks to a sequential update schedule. We prove that this cost is at most ⌈n/2+logq⁡(n/2+1)⌉ and, conversely, we construct a network h with cost at least n/2−logq⁡(n)/2−2 for q≥4 and ⌊n/3⌋ for q≥2. Furthermore, we prove that, except in some marginal well identified cases, n plus the cost of sequentialization of h is exactly the procedural complexity of h with unlimited memory. Finally, we prove that the cost of sequentialization of h is at most the directed pathwidth of its interaction graph.