Rice-like complexity lower bounds for Boolean and uniform automata networks
Abstract
Automata networks are a versatile model of finite discrete dynamical systems composed of interacting entities (the automata), able to embed any directed graph as a dynamics on its space of configurations (the set of vertices, representing all the assignments of a state to each entity). In this world, virtually any question is decidable by a simple exhaustive search. We lever the Rice-like complexity lower bound, stating that any non-trivial monadic second order logic question on the graph of its dynamics is NP-hard or coNP-hard (given the automata network description), to bounded alphabets (including the Boolean case). This restriction is particularly meaningful for applications to "complex systems", where each entity has a restricted set of possible states (its alphabet). For the deterministic case, trivial questions are solvable in constant time, hence there is a sharp gap in complexity for the algorithmic solving of concrete problems on them. For the non-deterministic case, non-triviality is defined at bounded cliquewidth, which offers a structure to establish metatheorems of complexity lower bounds.
Cite
@article{arxiv.2409.08762,
title = {Rice-like complexity lower bounds for Boolean and uniform automata networks},
author = {Aliénor Goubault-Larrecq and Kévin Perrot},
journal= {arXiv preprint arXiv:2409.08762},
year = {2025}
}