On Non-Determinism in Machines and Languages

this paper, we compare various means to define non-deterministic constructs, both in computing devices and in logical languages. Our goal is to establish correspondences between different formalisms, and to better understand the expressive power of the constructs. A deterministic query is a mapping from databases to relations. A non-deterministic query is a relation which associates possibly several distinct relations to each database. We define the class NQ of all computable non-deterministic queries. Our purpose is to characterize subclasses of NQ with query languages on one side and classes of queries computable on machines of various complexity classes on the other side. Non-deterministic queries correspond to multivalued functions in complexity theory [Sel94]. Classes of deterministic queries computable on Turing machines with bounded resources are defined with respect to the recognition problem as usual [Kan90]. In the case of non-deterministic queries, we followed the definition of [ASV90], and considered the computation problem, which is more appropriate in the later case. We consider classical complexity classes in the polynomial range, PTIME, NP, and their complement and intersections, NP"co-NP, etc. We also consider less usual classes, such as in particular the class UP of unambiguous computations [Val76], the class DP, of problems whose computation requires in parallel a branch in NP and a branch in co-NP [PY82], and the class