Give a digital computer a problem in arithmetic, and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer. The cause of this sorry situation is not that software is full of bugs—although that is very likely true as well— nor is it that hardware is unreliable. The problem is simply that computers are discrete and finite machines, and they cannot cope with some of the continuous and infinite aspects of mathematics. Even an innocent-looking number like 1 ⁄10 can cause no end of trouble: In most cases, the computer cannot even read it in or print it out exactly, much less perform exact calculations with it. Errors caused by these limitations of digital machines

We define value constraints, a method for incorporating constraint propagation into logic programming. It is a subscheme of the CLP scheme and is applicable wherever one has an efficient method for representing sets of possible values. As examples we present: small finite sets, sets of ground instances of a term, and intervals of reals with floating-point numbers as bounds. Value constraints are defined by distinguishing two storage management strategies in the CLP scheme. In value constraints the infer step of the CLP scheme is implemented by Waltz filtering. We give a semantics for value constraints in terms of set algebra that gives algebraic characterizations of local and global consistency. the existing extremal fixpoint characterization of chaotic iteration is shown to be applicable to prove convergence of Waltz filtering. Postscript version

Interval constraints can be used to solve problems in numerical analysis. In this paper we show that one can improve the performance of such an interval constraint program by the declarative use of constraints that are redundant in the sense of not needed to define the problem. The first example shows that computation of an unstable recurrence relation can be improved. The second example concerns a solver of nonlinear equations. It shows that, by adding as redundant constraints instances of Taylor's theorem, one can obtain convergence that appears to be quadratic.

The work advances a new class of global optimization methods, called graph subdivision methods, that are based on the simultaneous adaptive subdivision of both the function's domain of definition and the range of values.

Although CLP(R) is a promising application of the logic programming paradigm to numerical computation, it has not addressed what has long been known as ``the pitfalls of [numerical] computation''. These show that rounding errors induce a severe correctness problem wherever floating-point computation is used. Independently of logic programming, constraint processing has been applied to problems in terms of real-valued variables. By using the techniques of interval arithmetic, constraint processing can be regarded as a computer-generated proof that a certain real-valued solution lies in a narrow interval. In this paper we propose a method for interfacing this technique with CLP(R). This is done via a real-valued analogy of Apt's proof-theoretic framework for constraint processing.

this paper to present a unified framework in which the graph subdivision method can be appreciated. Foremost among these is the venerable principle of considering not only the primal formulation of the optimization problem, but also its dual. Hence the title

The main goal of this introduction is to make the book more accessible to readers who are not familiar with interval computations: to beginning graduate students, to researchers from related fields, etc. With this goal in mind, this introduction describes the basic ideas behind interval computations and behind the applications of interval computations that are surveyed in the book.

Give a digital computer a problem in arithmetic, and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer. The cause of this sorry situation is not that software is full of bugs—although that is very likely true as well— nor is it that hardware is unreliable. The problem is simply that computers are discrete and finite machines, and they cannot cope with some of the continuous and infinite aspects of mathematics. Even an innocent-looking number like 1 ⁄10 can cause no end of trouble: In most cases, the computer cannot even read it in or print it out exactly, much less perform exact calculations with it. Errors caused by these limitations of digital machines are usually small and inconsequential, but sometimes every bit counts. On February 25, 1991, a Patriot missile battery assigned to protect a military installation at Dahrahn, Saudi Arabia, failed to intercept a Scud missile, and the malfunction was blamed on an error in computer arithmetic. The Patriot’s control system kept track of time by counting tenths of a second; to convert the count into full seconds, the computer multiplied by 1 ⁄10. Mathematically, the procedure is unassailable, but computationally it was disastrous. Because the decimal fraction 1 ⁄10 has no exact finite representation in binary notation, the computer had to approximate. Apparently, the conversion constant stored in the program was the 24-bit binary fraction 0.00011001100110011001100, which is too small by a factor of about one ten-millionth. The discrepancy sounds tiny, but over four days it built up to about a third of a second. In combination with other peculiarities of the control software, the inaccuracy caused a miscalculation of almost 700 meters in the predicted position of the incoming missile. Twenty-eight soldiers died. Of course it is not to be taken for granted that better arithmetic would have saved those 28 lives. (Many other Patriots failed for unrelated reasons; some analysts doubt whether any Scuds were stopped by Patriots.) And surely the underlying problem was not the slight drift in the clock but a design vulnerable to such minor timing