Red Spider Meets a Rainworm: Conjunctive Query Finite Determinacy Is Undecidable

We solve a well known and long-standing open problem in database theory, proving that Conjunctive Query Finite Determinacy Problem is undecidable. The technique we use builds on the top of our Red Spider method which we developed in our paper [GM15] to show undecidability of the same problem in the "unrestricted case" -- when database instances are allowed to be infinite. We also show a specific instance $Q_0$, ${\cal Q}= \{Q_1, Q_2, \ldots Q_k\}$ such that the set $\cal Q$ of CQs does not determine CQ $Q_0$ but finitely determines it. Finally, we claim that while $Q_0$ is finitely determined by $\cal Q$, there is no FO-rewriting of $Q_0$, with respect to $\cal Q$, and we outline a proof of this claim