We present a Rice-like complexity lower bound for any MSO-definable problem on binary structures succinctly encoded by circuits. This work extends the framework recently developed as a counterpoint to Courcelle's theorem for graphs encoded by circuits, in two interplaying directions: (1) by allowing multiple binary relations, and (2) by restricting the interpretation of new symbols. Depending on the pair of an MSO problem $ψ$ and an MSO restriction $χ$, the problem is proven to be NP-hard or coNP-hard or P-hard, as long as $ψ$ is non-trivial on structures satisfying $χ$ with bounded clique-width. Indeed, there are P-complete problems (for logspace reductions) in our extended context. Finally, we strengthen a previous result on the necessity to parameterize the notion of non-triviality, hence supporting the choice of clique-width.

翻译：本文针对通过电路简洁编码的二元结构上任何MSO可定义问题，提出了一个类似Rice定理的复杂性下界。这项工作扩展了最近发展起来的、作为电路编码图的Courcelle定理之对立面的理论框架，并沿两个相互交织的方向推进：(1) 允许多元二元关系；(2) 限制新符号的解释。根据MSO问题$ψ$与MSO限制$χ$的组合，只要$ψ$在满足$χ$且具有有界团宽的结构上是非平凡的，该问题即被证明是NP难的或coNP难的或P难的。事实上，在我们扩展的语境中，存在对对数空间归约而言的P完全问题。最后，我们强化了先前关于非平凡性概念必须参数化的结果，从而支持了选择团宽作为参数的合理性。