Query Languages for Semi - Algebraic Databasesbased on Descriptive Complexity over Rklaus

We propose the study of query languages for databases involving real numbers as data (called semi-algebraic databases in the sequel). As main new aspect our approach is based on real number complexity theory as introduced in 8] and descriptive complexity for the latter developed in 17]. Using this formal framework a uniform treatment of semi-algebraic query languages is obtained. Precise results about both the data-and the expression-complexity of several such query languages are proved. More explicitly, relying on descriptive complexity theory over R gives the possibility to derive a hierarchy of complete languages for most of the important real number complexity classes. A clear correspondence between diierent logics and such complexity classes is established. In particular, it is possible to formalize queries involving in a uniform manner real spaces of diierent dimensions. This can be done in such a way that the logical description exactly reeects the computational complexity of a query. The latter might circumvent a problem appearing in some of the former approaches dealing with semi-algebraic databases (see 20] , 18]), where the use of rst-order logic over real-closed elds can imply inef-ciency as soon as the dimension of the underlying real space is not xed-no matter whether the query under consideration is easy to compute or not. 1. Introduction Semi-algebraic databases have raised increasing interest in recent years. Inspired by problems of computational and semi-algebraic geometry in the meanwhile there can be found many approaches dealing with real number data and real polynomial inequality constraints. To give a (by no means complete) list of references consider for example 20], 24], 19], 4], 5], 6], the survey paper 18] as well as the literature cited in there. An important task in database theory is to study the complexity of queries asked to databases. In relation with nite model theory and descriptive complexity theory this led to numerous results on the complexity of several query languages described by logical means (see 1] for a survey). For the present work the papers 9] and 25] are of extreme interest. Here the notions of data-and expression-complexity were introduced and studied; important complexity classes in the Turing model were captured by the data-resp. expression-complexity of several such query languages. Our goal is to set up a similar theory for semi-algebraic databases. However, in order to deal in a comparable manner with