By Gabriel Kuper, Leonid Libkin, Jan Paredaens

This e-book is the 1st finished survey of the sphere of constraint databases. Constraint databases are a reasonably new and energetic region of database learn. the most important proposal is that constraints, akin to linear or polynomial equations, are used to symbolize huge, or perhaps endless, units in a compact manner. the power to house countless units makes constraint databases quite promising as a expertise for integrating spatial and temporal information with common re lational databases. Constraint databases convey suggestions from quite a few fields, corresponding to common sense and version idea, algebraic and computational geometry, in addition to symbolic computation, to the layout and research of knowledge versions and question languages. The e-book is a collaborative attempt related to many authors who've con tributed chapters on their fields of craftsmanship. regardless of this, the publication is designed to be learn as a complete, rather than a set of person surveys. In par ticular, the terminology and the fashion of presentation were standardized, and there are a number of cross-references among the chapters. the belief of constraint databases is going again to the overdue Paris Kanellakis.

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This e-book is the 1st finished survey of the sphere of constraint databases. Constraint databases are a reasonably new and energetic quarter of database learn. the main suggestion is that constraints, equivalent to linear or polynomial equations, are used to symbolize huge, or perhaps endless, units in a compact method.

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Its arity is given by SC. - If e 1 and e2 are RAEs of arities k1 and k 2 respectively, then the cartesian product (e1 x e2) is a RAE of arity k1 + k2, and, provided that k1 = k 2, the union (e 1 U e2) and the difference (e 1 - e 2 ) are RAEs of arity k1. - If e is a RAE of arity k, and i 1, .. , ip E {1 , . , k }, then the projection 1t"i 1 , . . ,ip(e) is a RAE of arity p. - Finally, if e is a RAE of arity k, and () is a quantifier-free formula over [} on the variables x 1, ... , Xk, then the selection ae(e) is a RAE of arity k .

Algebra/calculus Fig. 1. The constraint database framework The goal is to define query languages for constraint databases that have these properties. We focus here on first-order languages, that generalize t he relational algebra and calculus. The basic idea is that a query over a constraint database is a first-order formula over the constraint domain. What is the answer of such a query? On one level, the answer is clear: As a constraint relation represents a (possibly infinite) set of points, the semantics of the query can be defined as the mapping that applies the given formula to this set of points, with the result as the answer to the query.

What about the constraint setting? Let us begin with a precise definition. Let M be a fixed structure. 1. Two relational calculus formulae over M are called equivalent over M if they express the same unrestricted query over M. As far as effective decidability is concerned, the equivalence problem is equivalent (sic) to the satisfiability problem, defined as follows. 2. A relational calculus sentence over M is called constraintsatisfiable if it is satisfied on at least one definable database over M.