  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.
