Sequentialization and procedural complexity in automata networks

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