Complexity lower bounds for succinct binary structures of bounded clique-width with restrictions

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 $\psi$ and an MSO restriction $\chi$, the problem is proven to be NP-hard or coNP-hard or P-hard, as long as $\psi$ is non-trivial on structures satisfying $\chi$ 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.