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. 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.

翻译：我们的主要结果是针对库尔切勒元定理的一个简洁反论：对于由简洁表示给出的图，每个树状单子二阶性质要么是NP难解的，要么是coNP难解的。简洁表示是计算邻接关系的布尔电路。树状性质是指具有无限多个树宽有界的模型与反模型的性质。此外，我们探讨了当树状条件被移除时的情况，并证明在合理的复杂性假设下，即使对于一阶逻辑可表达的问题，前述二分性也不再成立。