🤖 AI Summary
This paper investigates the computational complexity of model checking monadic second-order (MSO) properties on Boolean and finite-alphabet automata networks. Determining the hardness of nontrivial MSO queries over dynamical graphs—specifically, whether such properties are decidable or tractable under structural constraints. Method: The analysis integrates MSO logic, computational complexity theory, treewidth-based decomposition, and finite dynamical systems modeling, applied to bounded-alphabet (including Boolean) and nondeterministic networks of bounded treewidth. Contribution/Results: First Rice-type complexity lower bounds for nontrivial MSO problems over bounded-alphabet and bounded-treewidth networks are established; it is proven that every nontrivial MSO property is either NP-hard or coNP-hard on Boolean automata networks—revealing a sharp complexity dichotomy between trivial and nontrivial properties; and classical Rice’s undecidability theorem is tightened to the finite-domain setting. These results establish fundamental intractability barriers for algorithmic analysis of constrained-state systems, thereby extending theoretical boundaries of formal verification and decidability in complex dynamical systems.
📝 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 non-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 treewidth, which offers a structure to establish metatheorems of complexity lower bounds.