Abstract-We solve a well known, long-standing open problem in relational databases theory, showing that the conjunctive query determinacy problem (in its "unrestricted" version) is undecidable.

We show that all-instances termination of chase is undecidable. More precisely, there is no algorithm deciding, for a given set T consisting of Tuple Generating Dependencies (a.k.a. Datalog ∃ program), whether the 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. 2 Φ and Ψ are positive, without equality. Our negative results hold for single head TGDs, which means that Ψ is a single atom.

We study the description logic SQ with number restrictions applicable to transitive roles, extended with either nominals or inverse roles. We show tight 2EXPTIME upper bounds for unrestricted entailment of regular path queries for both extensions and finite entailment of positive existential queries for nominals. For inverses, we establish 2EXPTIME-completeness for unrestricted and finite entailment of instance queries (the latter under restriction to a single, transitive role).

The chase procedure is a fundamental algorithmic tool in database theory with a variety of applications. A key problem concerning the chase procedure is all-instances termination: for a given set of tuple-generating dependencies (TGDs), is it the case that the chase terminates for every input database? In view of the fact that this problem is undecidable, it is natural to ask whether known well-behaved classes of TGDs ensure decidability. We consider here the main paradigms that led to robust TGD-based formalisms, that is, guardedness and stickiness. Although all-instances termination is well-understood for the oblivious version of the chase, the more subtle case of the restricted (a.k.a. the standard) chase is rather unexplored. We show that all-instances restricted chase termination for guarded and sticky single-head TGDs is decidable.

We solve a problem, stated in [CGP10a], showing that Sticky Datalog ∃ , defined in the cited paper as an element of the Datalog ± project, has the Finite Controllability property. In order to do that, we develop a technique, which we believe can have further applications, of approximating Chase(T , D), for a database instance D and a set of tuple generating dependencies and Datalog rules T , by an infinite sequence of finite structures, all of them being models of T and D.