Our main result is a succinct counterpoint to Courcelle's meta-theorem as follows: every cw-nontrivial 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. Cw-nontrivial properties are those which have infinitely many models and infinitely many countermodels with bounded cliquewidth. Moreover, we explore what happens when the cw-nontriviality 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：我们的主要结果是对Courcelle元定理的一个简洁对应点，表述如下：对于由简洁表示给出的图，每个cw-非平凡的一元二阶（MSO）性质要么是NP-难的，要么是coNP-难的。简洁表示是计算邻接关系的布尔电路。Cw-非平凡性质是指那些具有无限多个模型和无限多个反模型，且这些模型和反模型的团宽度均有界的性质。此外，我们探讨了当放弃cw-非平凡性条件时会发生什么，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，前述二分法也不再成立。