Termination of oblivious chase is undecidable

We show that all--instances termination of chase is undecidable. More precisely, there is no algorithm deciding, for a given set $\cal T$ consisting of Tuple Generating Dependencies (a.k.a. Datalog$^\exists$ program), whether the $\cal T$-chase on $D$ will terminate for every finite database instance $D$. Our method applies to Oblivious Chase, Semi-Oblivious Chase and -- after a slight modification -- also for Standard Chase. This means that we give a (negative) solution to the all--instances termination problem for all version of chase that are usually considered. The arity we need for our undecidability proof is three. We also show that the problem is EXPSPACE-hard for binary signatures, but decidability for this case is left open. Both the proofs -- for ternary and binary signatures -- are easy. Once you know them.