Two hard meta-problems at the core of the infinite-domain CSP dichotomy conjecture

The infinite-domain CSP dichotomy conjecture extends the finite-domain CSP dichotomy theorem to reducts of finitely bounded homogeneous structures. Every such structure is uniquely described by a particular sentence of the logic SNP. We show that the question whether a given SNP sentence describes a structure within the scope of the conjecture is undecidable even if the sentence comes from the connected Datalog fragment, and that the closely related problem of testing the amalgamation property for universal sentences is EXPSPACE-hard. We also discuss some philosophical implications of these results for the infinite-domain CSP dichotomy conjecture.