We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixed-precision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next we survey some recent applications of this paradigm. Finally, we outline some basic theory and techniques in this paradigm. 1 This paper will appear as a chapter in the 2nd edition of Computing in Euclidean Geometry, edited by D.-Z. Du and F.K. Hwang, published by World Scientific Press, 1994. 1 1 Two Numerical Computing Paradigms Computation has always been intimately associated with numbers: computability theory was early on formulated as a theory of computable numbers, the first computers have been number crunchers and the original mass-produced computers were pocket calculators. Although one's first exposure to computers today is likely to be some non-numerical application, numeri...

Traditionally, dependency theory has been developed for uninterpreted data. Specifically, the only assumption that is made about the data domains is that data values can be compared for equality. However, data is often interpreted and there can be advantages in considering it as such, for instance obtaining more compact representations as done in constraint databases. This paper considers dependency theory in the context of interpreted data. Specifically, it studies constraint-generating dependencies. These are a generalization of equality-generating dependencies where equality requirements are replaced by constraints on an interpreted domain. The main technical results in the paper are a general decision procedure for the implication and consistency problems for constraint-generating dependencies, and complexity results for specific classes of such dependencies over given domains. The decision procedure proceeds by reducing the dependency problem to a decision problem for the constraint theory of interest, and is applicable as soon as the underlying constraint theory is decidable. The complexity results are, in some cases, directly lifted from the constraint theory; in other cases, optimal complexity bounds are obtained by taking into account the specific form of the constraint decision problem obtained by reducing the dependency implication problem.

We study the complexity of quantifier elimination and decision in first-order theories of temporal constraints. With the exception of Ladkin, AI researchers have largely ignored this problem. We consider the first-order theories of point and interval constraints over two time structures: the integers and the rationals. We show that in all cases quantifierelimination can be done in PSPACE. We also show that the decision problem for arbitrarily quantified sentences is PSPACE-complete while for 9 k sentences it is \Sigma p k -complete. Our results must be of interest to researchers working on temporal constraints, computational complexity of logical theories, constraint databases and constraint logic programming. 1 INTRODUCTION The study of temporal constraints has recently received much attention from the AI community [All83, LM88, Lad88, VKvB89, vBC90, DMP91, KL91, Mei91, vB92, Kou92, GS93, SD93]. Much of this work draws upon concepts and techniques from the literature of general co...

. Constraint databases generalize relational databases by finitely representable infinite relations. This paper surveys the state of the art in constraint databases: known results, remaining open problems and current research directions. The paper also describes a new algebra for databases with integer order constraints and a complexity analysis of evaluating queries in this algebra. In memory of Paris C. Kanellakis 1 Introduction There is a growing interest in recent years among database researchers in constraint databases, which are a generalization of relational databases by finitely representable infinite relations. Constraint databases are parametrized by the type of constraint domains and constraint used. The good news is that for many parameters constraint databases leave intact most of the fundamental assumptions of the relational database framework proposed by Codd. In particular, 1. Constraint databases can be queried by constraint query languages that (a) have a semantics ba...

In previous work we have developed the scheme of indefinite L-constraint databases where L, the parameter, is a first-order constraint language. This scheme extends the constraint database proposal of Kanellakis, Kuper and Revesz to include indefinite (or uncertain) information in the style of Imielinski and Lipski. In this paper we study the complexity of query evaluation in an important instance of this abstract scheme: indefinite temporal constraint databases. Our results indicate that the data/combined complexity of query evaluation does not change when we move from queries in relational calculus over relational databases, to queries in relational calculus with temporal constraints over temporal constraint databases. This fact remains true even when we consider query evaluation in relational databases with indefinite information vs. query evaluation in indefinite temporal constraint databases. In the course of our work, we provide precise bounds on the complexity of decision/quanti...

Abstract. Extension of the relational database model to represent complex data has been a focus of much research in recent years. At the same time, an alternative extension of the relational database model has proposed using constraint databases that finitely describe infinite relations. This paper attempts to combine these two divergent approaches. In particular a query language called Datalog with set order constraints, or Datalog ⊂ P(Z) , is proposed. This language can express many natural problems with sets, including reasoning about inheritance hierarchies. Datalog ⊂ P(Z) queries over set constraint databases are shown to be evaluable bottom-up in closed form and to have DEXPTIME-complete data complexity. 1

. Guaranteeing termination of programs on all valid inputs is important for database applications. Termination cannot be guaranteed in Stratified Datalog with integer (gap)-order, or Datalog :;! Z , programs on generalized databases because they can express any Turingcomputable function [23]. This paper introduces a restriction of Datalog :;! Z that can express only computable queries. The restricted language has a high expressive power and a non-elementary data complexity. 1 Introduction Constraint logic programming [14, 15, 27, 12, 10, 9] has a great potential for being adapted for database use. A successful adaptation of constraint logic programming has to meet usual database requirements. In the constraint query languages framework [19] two requirements are identified as especially important: (a) closed-form evaluation and (b) bottom-up processing. Closed-form evaluation means that all possible tuple answers to a query are represented finitely by an output constraint database ...

An important requirement of advanced temporal applications is the ability to deal with definite, indefinite, finite and infinite temporal information. There is currently no database model which offers this functionality in a single unified framework. We argue that the combination of relational databases and temporal constraints offers a powerful framework which addresses these needs. In this thesis we develop the foundations of a theory of temporal constraint databases and indefinite temporal constraint databases. At first, we study a hierarchy of parameterized database models: ML -relational databases, L-constraint databases and indefinite L-constraint databases. The language L, the parameter, defines the constraint vocabulary and ML is the structure over which L-constraints will be interpreted. The models of temporal constraint databases and indefinite temporal constraint databases can then be studied as instances of the last two of the above parameterized models. In the course of...

. This paper describes the implementation of a constraint database system with integer and set of integers data types. The system called DISCO allows Datalog queries and input databases with both integer gap-order [30] and set order constraints [31]. The DISCO query language can easily express many complex problems involving sets. The paper also presents efficient running times for several sample queries. 1 Introduction Recently there has been much interest in constraint databases that generalize relational databases by allowing infinite relations that are finitely represented using constraint tuples (ex., [23, 3, 4, 8, 17, 21, 25, 28]). DISCO (short for Datalog with Integer and Set order COnstraints) is a constraint database system being developed at the University of Nebraska. DISCO implements a particular case of constraint query languages for which a general framework was proposed in [23] analogously to the constraint logic programming framework of Jaffar and Lassez [18]. The part...

. We lay the foundations of a theory of constraint databases with indefinite information based on the relational model. We develop the scheme of indefinite L-constraint databases where L, the parameter, is a first-order constraint language. This scheme extends the proposal of Kanellakis, Kuper and Revesz to include indefinite information in the style of Imielinski and Lipski. We propose declarative and procedural query languages for the new scheme and study the semantics of query evaluation. 1 Introduction In this paper we lay the foundations of a theory of indefinite constraint databases based on the relational model [Mai83]. As a starting point of our investigation, we take the model of constraint databases proposed in [KKR90]. This model is useful for the representation of unrestricted (i.e., finite or infinite) definite information. However, indefinite information is also important in many applications e.g., planning and scheduling, medical expert systems, geographical informatio...