We investigate the relationship between programming with constraints and database query languages. We show that efficient, declarative database programming can be combined with efficient constraint solving. The key intuition is that the generalization of a ground fact, or tuple, is a conjunction of constraints over a small number of variables. We describe the basic Constraint Query Language design principles and illustrate them with four classes of constraints: real polynomial inequalities, dense linear order inequalities, equalities over an infinite domain, and boolean equalities. For the analysis, we use quantifier elimination techniques from logic and the concept of data complexity from database theory. This framework is applicable to managing spatial data and can be combined with existing multidimensional searching algorithms and data structures.

Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton’s iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation

. The declarative programming paradigms used in constraint languages can lead to powerful extensions of Codd's relational data model. The development of constraint database query languages from logical database query languages has many similarities with the development of constraint logic programming from logic programming, but with the additional requirements of data efficient, set-at-a-time, and bottomup evaluation. In this overview of constraint query languages (CQLs) we first present the framework of [41]. The principal idea is that: "the k-tuple (or record) data type can be generalized by a conjunction of quantifier-free constraints over k variables". The generalization must preserve various language properties of the relational data model, e.g., the calculus/algebra equivalence, and have time complexity polynomial in the size of the data. We next present an algebra for dense order constraints that is simpler to evaluate than the calculus described in [41], and we sharpen some of...

: We study classes of infinite but finitely representable databases based on constraints, motivated by new database applications such as geographical databases. We formally define these notions and introduce the concept of query which generalizes queries over classical relational databases. We prove that in this context the basic properties of queries (satisfiability, containment, equivalence, etc.) are nonrecursive. We investigate the theory of finitely representable models and prove that it differs strongly from both classical model theory and finite model theory. In particular, we show that most of the well known theorems of either one fail (compactness, completeness, locality, 0/1 laws, etc.). An immediate consequence is the lack of tools to consider the definability of queries in the relational calculus over finitely representable databases. We illustrate this very challenging problem through some classical examples. We then mainly concentrate on dense order databases, and exhibit...

We define Recursive Markov Chains (RMCs), a class of finitely presented denumerable Markov chains, and we study algorithms for their analysis. Informally, an RMC consists of a collection of finite-state Markov chains with the ability to invoke each other in a potentially recursive manner. RMCs offer a natural abstract model for probabilistic programs with procedures. They generalize, in a precise sense, a number of well studied stochastic models, including Stochastic Context-Free Grammars (SCFG) and Multi-Type Branching Processes (MT-BP). We focus on algorithms for reachability and termination analysis for RMCs: what is the probability that an RMC started from a given state reaches another target state, or that it terminates? These probabilities are in general irrational, and they arise as (least) fixed point solutions to certain (monotone) systems of nonlinear equations associated with RMCs. We address both the qualitative problem of determining whether the probabilities are 0, 1 or in-between, and

We initiate the systematic study of algorithmic issues involved in finding equilibria (Nash and correlated) in games with a large number of players; such games, in order to be computationally meaningful, must be presented in some succinct, game-specific way. We develop a general framework for obtaining polynomial-time algorithms for optimizing over correlated equilibria in such settings, and show how it can be applied successfully to symmetric games (for which we actually find an exact polytopal characterization), graphical games, and congestion games, among others. We also present complexity results implying that such algorithms are not possible in certain other such games. Finally, we present a polynomial-time algorithm, based on quantifier elimination, for finding a Nash equilibrium in symmetric games when the number of strategies is relatively small.

We study topological queries over two-dimensional spatial databases. First, we show that the topological properties of semi-algebraic spatial regions can be completely specified using a classical finite structure, essentially the embedded planar graph of the region boundaries. This provides an invariant characterizing semi-algebraic regions up to homeomorphism. All topological queries on semi-algebraic regions can be answered by queries on the invariant whose complexity is polynomially related to the original. Also, we show that for the purpose of answering topological queries, semi-algebraic regions can always be represented simply as polygonal regions. We then study query languages for topological properties of two-dimensional spatial databases, starting from the topological relationships between pairs of planar regions introduced by Egenhofer. We show that the closure of these relationships under appropriate logical operators yields languages which are complete for topological prope...

This report introduces QEPCAD B, a program for computing with real algebraic sets using cylindrical algebraic decomposition (CAD). QEPCAD B both extends and improves upon the QEPCAD system for quantifier elimination by partial cylindrical algebraic decomposition written by Hoon Hong in the early 1990s. This paper briefly discusses some of the improvements in the implementation of CAD and quantifier elimination via CAD, and provides somewhat more detail on extensions to the system that go beyond quantifier elimination. The author is responsible for most of the extended features of QEPCAD B, but improvements to the basic CAD implementation and to the SACLIB library on which QEPCAD is based are the results of many people’s work, including: George E.

We investigate polynomial reductions and efficient branching rules for algorithms deciding propositional tautologies for DNF and coNP--complete subclasses. Upper bounds on the time complexity are given with exponential part 2 ff\Delta(F ) where (F ) is one of the measures n(F ) = #f variables g, `(F ) = #f literal occurrences g and k(F ) = #f clauses g. We start with a discussion of variants of the algorithms from [Monien/Speckenmeyer85] and [Luckhardt84] with the known upper bound 2 0:695\Deltan for 3-DNF and (roughly) (2 \Delta (1 \Gamma 2 \Gammap )) n for p-DNF, p 3, where p is the maximal clause length, giving now an uniform treatment for all p-DNF including the easy decidable case p 2. Recently for 3-DNF the bound has been lowered to 2 0:5892\Deltan ([K2]; see also [Sch2], [K3]). In this article further improvements are achieved by studying two additional characteristic groups of parameters. The first group differentiates according to the maximal numbers (a; b) of occ...

We consider s polynomials P 1 ; : : : ; P s in k ! s variables with coefficients in an ordered domain A contained in a real closed field R, each of degree at most d. We present a new algorithm which computes a point in each connected component of each non-empty sign condition over P 1 ; : : : ; P s . The output is the set of points together with the sign condition at each point. The algorithm uses s(s=k) k d O(k) arithmetic operations in A. The algorithm is nearly optimal in the sense that the size of the output can be as large as s(O(sd=k)) k . Previous algorithms of Canny and Renegar used (sd) O(k) operations [5, 7, 8, 15]. We use either these algorithms in the case s = 1 as a subroutine in our algorithm. As a bonus, our algorithm yields an independent proof of the bound on the number of connected components in all non-empty sign conditions ([14]) and also yields an independent proof of a theorem of Warren 1 Courant Institute of Mathematical Sciences, New York University, N...