Rice-like complexity lower bounds for Boolean and uniform automata networks

TL;DR

This paper establishes that many questions about automata networks with limited states are computationally hard, especially for non-trivial cases, highlighting a sharp complexity gap in their analysis.

Contribution

It proves Rice-like complexity lower bounds for automata networks with bounded alphabets, including Boolean networks, and explores the complexity gap between trivial and non-trivial problems.

Findings

Non-trivial monadic second order logic questions are NP-hard or coNP-hard.

Trivial questions in deterministic automata networks are solvable in constant time.

Non-triviality in non-deterministic networks relates to bounded cliquewidth.

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.