Hardness of monadic second-order formulae over succinct graphs
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Our main result is a succinct counterpoint to Courcelle's meta-theorem as follows: every arborescent monadic second-order (MSO) property is either NP-hard or coNP-hard over graphs given by succinct representations. Succint representations are Boolean circuits computing the adjacency relation. Arborescent properties are those which have infinitely many models and countermodels with bounded treewidth. We actually prove this result in the terminology of automata network, which is a generalization of finite cellular automata over arbitrary graphs. This model arose from the biological modelization of neural networks and gene regulation networks. Our result states that every arborescent MSO property on the transition graph of automata networks is either NP-hard or coNP-hard. Moreover, we explore what happens when the arborescence condition is dropped and show that, under a reasonable complexity assumption, the previous dichotomy fails, even for questions expressible in first-order logic.
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