Fr'ed'eric Benhamou and William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming, 32:1--24, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[9] the BNR Prolog team [5, 20, 21] showed that the consistency method also has important advantages in solving numerical problems. That is, in solving constraint problems where the unknowns are reals. A theoretical foundation was established in a report dated 1993, which was recently published [4]. BNR Prolog is now available as ALS Prolog. Other implementations are Prolog IV [3] and Numerica [13] Independently of this, and preceding it, there is a long tradition in Russia of subdefinite calculation of which Unicalc [25] is a result. 2 Redundant constraints Redundant clauses may not ....

....method only removes inconsistent values and may not remove all such values, it may be that the resulting intervals contain no solution. Thus results in the consistency method have the meaning: if a solution exists, then it is in the intervals found. Some references to interval constraints are [4, 5, 28, 13]. 4 Unstable recurrences In one of the first quantum mechanical investigations of the water molecule, Coolidge [10] needed to evaluate c(n) R 1 0 x n e x dx for n = 1; 2; Integration by parts yields the recurrence relation c(n 1) e Gamma n c(n) Applying this to the known ....

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Fr'ed'eric Benhamou and William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming, 32:1--24, 1997.

....of misfortune. The advent of interval constraints is yet another one. Interval constraints Independently, Davis [9] and Cleary [7] arrived at a relational generalization of interval arithmetic, which is now known as interval constraints. Shortly after, Cleary s work was used in BNR Prolog [6, 5] to obtain software that can be viewed in two different ways. 1. BNR Prolog as a version of Prolog where soundness is preserved for queries involving real numbers. 2. BNR Prolog as interval constraint system that happens to have Prolog as programming language front end. Neither of these ....

Fr'ed'eric Benhamou and William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming, 32:1--24, 1997.

....taking precedence over security. Perhaps Java is not just an anomaly, but a harbinger of change. In this changed climate, interval arithmetic may be more readily accepted than before. 4. The rise of interval constraints. A recent application of interval arithmetic is interval constraints [2, 9]. The Numerica package [9] shows that in applications such as nonlinear systems of algebraic equations and non convex global optimization, interval constraints is easier to use and gives higher performance than algorithms that are programmed directly in interval arithmetic. Many benchmark results ....

....Like Pascal XSC, BNR Prolog optimally determines [ Gamma0:5; 0:5] 1; 1] Gamma1; 1] to be empty. Unlike Pascal XSC code in [5] BNR Prolog optimally determines [ Gamma1; 0] 1; 1] 1; 1] to be empty because the quotient does not contain its greatest lower bound. Unfortunately, apart from [15, 2] nothing seems to have been published about the arithmetic of BNR Prolog. 8 Conclusions We should emphasize that we do not attempt to decree that the full detail of interval division, as revealed in this paper, be implemented. In any implementation, efficiency and simplicity of code have to be ....

Fr'ed'eric Benhamou and William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming, 32:1--24, 1997.

....Finally, the source code for the system is publicly available, which is a clear prerequisite for veri ability. 1 Introduction Historically, there have been two approaches to implementing Interval Arithmetic constraint solvers, represented by the two systems: CLP(BNR) and Newton. In CLP(BNR) [6, 23, 22, 5, 2], each constraint is decomposed into primitive constraints (similar to compiling to 3 address code) and then a general constraint solving engine is invoked to repeatedly contract each primitive constraint until some termination condition is satis ed. In Newton, Numerica, and similar systems [3, ....

....an interesting and important one but takes us too far a eld from the main concerns of this paper. Nevertheless, by examining the table in Figure 4 one sees that most of the operators are for relatively simple operations (max(a; b) or boolean operators where false=true are represented by 0=1 as in [5]) The only complex constraints are the arithmetic and special functions and hence the job of verifying the correctness of the SILK system is again of manageable size. 2.4.4 Performance As we will see, later in this paper, the CLIP system is able to handle constraint systems with hundreds of ....

Frederic Benhamou and William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming, 32:1-24, 1997.

....numbers. The first problem was addressed by CHIP [7] which incorporated a consistency algorithm by modifying unification and by adding inference rules. The success of CHIP has been the main cause of the current interest in constraint logic programming. To address the second problem, BNR Prolog [3, 2] also combined a consistency algorithm with logic programming. The method of combination with the consistency algorithm is different from the one used in CHIP: unification was not modified and no inference rules were added. An obvious approach to the incorporation of consistency methods in logic ....

....and passive constraints in the CLP scheme has the advantage that one rediscovers the consistency method in a very natural way. I show that this not only holds for the general idea, but also for the consistency algorithm itself and several of the algebraic properties found by Benhamou and Older [2]. 2. Review of the CLP scheme The CLP scheme is based on the observation that in logic programming the Herbrand base can be replaced by any of many other semantic domains. Hence the scheme has as parameter a tuple h Sigma; D; L; T i, where Sigma is a signature, D is a Sigma structure, L is a ....

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Fr'ed'eric Benhamou and William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming. To appear.

....directly from mathematical formulae without having to rst prove results about interval arithmetic operators. 1 Introduction Historically, there have been two approaches to implementing Interval Arithmetic constraint solvers, represented by the two systems: CLP(BNR) and Newton. In CLP(BNR) [5, 21, 20, 4, 2], each constraint is decomposed into primitive constraints (similar to compiling to 3 address code) and then a general constraint solving engine is invoked to repeatedly contract each primitive constraint until some termination condition is satis ed. In Newton, Numerica, and similar systems [3, ....

Frederic Benhamou and William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming, 32:1-24, 1997.

....and does not provide any guarantee that the stated precision bounds will be achieved. Either an interval arithmetic approach or a detailed line by line numerical analysis will be required if one wants a provably correct estimate of the error. The key idea behind interval arithmetic constraints [BO97] is to view numeric computing problems as constraint systems that relate a set of real (or complex) variables or functions. The variables whose values are to be computed are initially unbounded in this model (i.e. they have the value [ Gamma1; 1] The goal of the computation is to shrink the ....

....applying various contraction operators which are automatically generated from the constraint set. In the full version of this paper [HW99] we present the The Constraint Contractor Method. The basic idea is to define a general family of validated contractors, based on arithmetic constraints[BO97] and to experiment with various combinations of these contractors. When a reasonably good sequence of contractions is found, one can then generate a procedure to implement these contractors and to signal an exception if the width of the computed intervals exceeds the required error bounds. This ....

Fr'ed'eric Benhamou and William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming, 32, 1997.

....(and validated) bounds on the solutions of a wide class of ODEs [16] The best system currently available is Lohner s AWA system [13] Interval Arithmetic Constraints. A more recent approach to numerical computation is to build a general intervalbased constraint solver into the language itself [2, 17]. In these languages, the mathematical model appears exactly as a set of constraints interspersed throughout the program. Moreover, since the built in solver will likely be used to solve many problems, considerable effort can be put into verifying its correctness and optimizing it on various ....

Fr'ed'eric Benhamou and William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming, 32:1--24, 1997.

....taking precedence over security. Perhaps Java is not just an anomaly, but a harbinger of change. In this changed climate, interval arithmetic may be more readily accepted than before. 4. The rise of interval constraints. A recent application of interval arithmetic is interval constraints [2, 9]. The Numerica package [9] shows that in applications such as nonlinear systems of algebraic equations and nonconvex global optimization, interval constraints is easier to use and gives higher performance than algorithms that are programmed directly in interval arithmetic. Many benchmark results ....

....can have. Like Pascal XSC, BNR Prolog optimally determines [ 0:5; 0:5] 1; 1] 1; 1] to be empty. Unlike Pascal XSC code in [5] BNR Prolog optimally determines [ 1; 0] 1; 1] 1; 1] to be empty because the quotient does not contain its greatest lower bound. Unfortunately, apart from [15, 2] nothing seems to have been published about the arithmetic of BNR Prolog. 8 Conclusions We should emphasize that we do not attempt to decree that the full detail of interval division, as revealed in this paper, be implemented. In any implementation, e ciency and simplicity of code have to be ....

Frederic Benhamou and William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming, 32:1-24, 1997.

....directly from mathematical formulae without having to rst prove results about interval arithmetic operators. 1 Introduction Historically, there have been two approaches to implementing Interval Arithmetic constraint solvers, represented by the two systems: CLP(BNR) and Newton. In CLP(BNR) [4, 19, 3, 2], each constraint is decomposed into primitive constraints (similar to compiling to 3 address code) and then a general constraint solving engine is invoked to repeatedly contract each primitive constraint until some termination condition is satis ed. In Newton, Numerica, and similar systems [21, ....

Frederic Benhamou and William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming, 32:1-24, 1997.

....constraints only remove inconsistent values and may not remove all such values, it may be that the resulting intervals contain no solution. Thus results in interval constraints have the meaning: if a solution exists, then it is in the intervals found. Some references to interval constraints are [4, 5, 18, 10]. 3 Machine Numbers and Intervals In numerical analysis, algorithms are studied in an idealized setting, where the operations are applied to real numbers and yield the correct real valued results. The theorems that are proved in numerical analysis concern properties of the idealized algorithms ....

....interval, and then the spuriously non empty right hand side contains at most one machine number, which is either r or s. Similarly for [t; u] Lemma 12. 8r; s ae R n ; we have r ae s ) bx(r) ae bx(s) monotonicity of bx) 8r ae R n ; we have bx(r) bx(bx(r) idempotence of bx) See [4]. We have now the tools to describe sets of reals in a way that is practical on a computer: as a pair of machine numbers. A great advantage of the IEEE floating point numbers is that they are machine numbers in our sense because they include the infinities. 4 Interval Constraint Systems It is ....

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Fr'ed'eric Benhamou and William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming, 32:1--24, 1997.

....not address most of the common blunders in scientific computing and still requires the working scientist to use either tools for solving very specific types of numerical problems or to use validated libraries. 1. 4 Interval Arithmetic Constraints The key idea behind interval arithmetic constraints [7, 10, 23, 31, 36, 40, 44, 45] is to view scientific computing problems as constraint systems that relate a set of real (or complex) variables or functions. The variables whose values are to be computed are initially unbounded in this model (i.e. they have the value [ Gamma1; 1] The goal of the computation is to shrink the ....

.... In the previous grant we built prototype libraries of validated contraction operators for the primitive arithmetic constraints (x y = z; x y = z; x 2 = z) and the the primitive elementary constraints (y = exp(x) y = sin(x) y = cos(x) y = tan(x) y = x n , y = x z ) as well as the BNR [7] boolean and integer constraints (x = y z, y = y z, integer(x) etc. of Benhamou and Older. The elementary functions were computed by combining interval arithmetic and classical techniques[13, 14, 15, 49] in novel ways. These libraries were written in C, ported to SGI and HP workstations, ....

Fr'ed'eric Benhamou and William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming, 32:1--24, 1997.

....of interval propagation to division propagation. A division is a union of ordered, non overlapping intervals. Another active area of research is the incorporation of interval constraints into constraint logic programming. Examples of such systems include CLP(BNR) Older Benhamou 93, Benhamou Older 96] and Newton [Benhamou et al. 94] see [Benhamou 95] for a recent survey. Most of these systems use local propagation to narrow the intervals that describe the permitted values for a variable (see for example the local tolerance propagation procedure in [Hyvonen 92] These techniques are similar ....

Fr'ed'eric Benhamou and William Older. Applying interval arithmetic to real, integer and boolean constraints. Journal of Logic Programming, 1996. Forthcoming.

....opportunities for blunders, and in some sense may aggravate this situation since it requires scientists to be aware of the difference between interval variables and numerical variables and when to use which. 2. 1 Interval Arithmetic Constraints The key idea behind interval arithmetic constraints [BO97, Cle87, Hic94, HvEW98, Hyv89, Ju98, OV93, vE97a, vE97b] is to view numeric computing problems as constraint systems that relate a set of real (or complex) variables or functions. The variables whose values are to be computed are initially unbounded in this model (i.e. they have the value [ Gamma1; 1] The goal of the computation is to shrink the ....

Fr'ed'eric Benhamou and William J. Older. Applying interval arithmetic to real, integer, and Boolean constraints. Journal of Logic Programming, 32, 1997.

....point representation of numbers) containing the intersection of the initial box and of the relation add = fz 2 R I j x y = zg. Then, the overall computation over the store is expressed in terms of fixed points of the constraint narrowing operators. For example, in most interval based system [BO97, OV93, OB93, Gou95, Col96, BT95] a floating point interval (that is a real valued interval whose bounds are floating point numbers) is associated to every variable and constraints are decomposed in primitive constraints (fresh variables are introduced when necessary) These constraints are ....

Fr'ed'eric Benhamou and William J. Older. Applying Interval Arithmetic to Real, Integer and Boolean Constraints. Journal of Logic Programming, 1997. (Forthcoming).

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