CLEARY J.G., Logical Arithmetic. In Future computing Systems 2(2):125149, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....other constraints to propagate and infinite domains could be refined indefinitely. Note that this is not a problem in practice with well known domains such as the real domain since the precision of any real number is bound so that the domain is effectively finite. This is the argument given in [CLE87, LEE93] and [BEN97] Our system allows (possibly infinite) user defined domains. For this reason, we incorporate a generic predicate P recision : Bool which limits domain refinement. Predicate P recision defines the condition that the glb and the lub of a range must hold to get the ....

....used in Table 6. The process terminates when the solution fd S1 g is found. User definitions for maximum and minimum are used in the narrowing of the interval constraints (that is, when is applied) 6 Related work Accounts of conventional interval arithmetic can be found in [MOO66] CLE87] and [SID92] describe several proposals to incorporate arithmetic into logic programming. Benhamou and Older [BEN97] and Lee and Van Emden [LEE93] apply interval reasoning to numerical computations on the real domain. MAJ97] presents a survey of the existing work in the area of interval based ....

CLEARY J.G., Logical Arithmetic. In Future computing Systems 2(2):125149, 1987.

....such that # V k (r) verifies c k for every k # 1, m . A constraint constraint system is said to be inconsistent if it has no solution (otherwise, it is consistent) Constraint satisfaction over R is generally intractable due to the limitations of machine arithmetic. Hence, J. G. Cleary [7] has proposed to implement constraint satisfaction over I. The idea is to consider an interval extension of each term composing a constraint (typically, the natural interval extension) and the possibly interval interpretation for each relation symbol ## from R: let I, J # I, we have I ## J ## ....

....operators that locally perform domain pruning (w.r.t. a constraint) before global splitting (w.r.t. the constraint system) The next section reviews the notions of box consistency and hull consistency. GranvilliersRC2000.tex; 5 05 2000; 9:05; p.4 5 2.3. Local consistency techniques Hull [7] and box [4] consistency define a consistency property of the bounds of some variables domains w.r.t. a constraint (see [8] for a discussion of partial consistencies over continuous domains) A narrowing operator enforcing such a property discards from the bounds of the underlying domain some of ....

J. G. Cleary. Logical Arithmetic. Future Comp. Sys., 2(2):125--149, 1987.

....Nedialkov s VNODE package [8] To overcome the above difficulties, research has been carried out to take advantage of the efficiency and soundness of constraint techniques. 3.3. Validated Interval Constraint Approaches The application of the interval constraints framework (introduced by Cleary [9]) for validated ODE solving was suggested by Older [10] and Hickey [11] The idea is to represent a problem by a constraint network whose nodes are the trajectory values on the discrete points of t, and imposing constraints on them according to the approximation step (as in the interval arithmetic ....

Cleary, J.G.: Logical Arithmetic. In Future Generation Computing Systems, 2(2) (1987) 125-149.

....complex systems, namely that described in the previous section, requires the use of more complex constraints with variables over the reals and nonlinear relations between them, and to represent the uncertainty of biophysical phenomena. The Interval Constraints framework (introduced by Cleary [13]) is adequate for handling non linear constraints over continuous variables. Its propagation mechanism starts from a set of constraints over real variables, using Interval Arithmetic [14] to reduce the domains of some variables. This is propagated to other constraints that further reduce the ....

Cleary J.G.: Logical Arithmetic. In Future Generation Computing Systems, 2(2) (1987) 125-149.

....Nedialkov s VNODE package [8] To overcome the above difficulties, research has been carried out to take advantage of the efficiency and soundness of constraint techniques. 3.3. Validated Interval Constraint Approaches The application of the interval constraints framework (introduced by Cleary [9]) for validated ODE solving was suggested by Older [10] and Hickey [11] The idea is to represent a problem by a constraint network whose nodes are the trajectory values on the discrete points of t, and imposing constraints on them according to the approximation step (as in the interval arithmetic ....

Cleary, J.G.: Logical Arithmetic. In Future Generation Computing Systems, 2(2) (1987) 125-149.

....(c =CTInfeasible) I.DisplayGlobalSolutions( return 0; Figure 5. An example of programming with ICE InC . All solutions to a set of tree equations in three variables are solved and displayed. The key technology underlying Interval Solver is Interval Constraint Satisfaction (Davis, 1987; Cleary, 1987, Hyvnen, 1989) developed in the fields of artificial intelligence, constraint) logic programming, and interval analysis (Interval, 1998) The interval constraint satisfaction problem (ICSP) corresponding to a sheet is represented as a C object of class Ice included in the ICE InC library. ....

Cleary, J. (1987) Logical Arithmetic. Future Computing Systems 2 (2), 1987.

....concerning open bounds is lost and results are always closed. A more proper approach is to generalize IA function evaluation rules for dealing with interval open ends (and with infinities ) by considering different interval limit combinations separately for each arithmetic operation (see e.g. Cleary (1987)) For example, two half open intervals (a,b] can be added by rule: x,y] u,v] x u,y v] However, in this approach rounding errors may cause problems. For example, when computing (0.1,1] 0.2,2] 0.3 ,3] the open minimum, denoted by 0.3 = 0.1 0.2 cannot be represented precisely by a machine ....

....arithmetic, too large and too small values can be represented by special infinity values and . Interval functions can then be modified by generalizing arithmetical operators for . For example, interval addition (x,y) u,v) x u,y v) can be generalized by the following rules for adding (Cleary, 1987): x=x ( x=x = Since discontinuous IA can be reduced to IA, this technique can be used for generalizing discontinuous interval arithmetic rules (1.7) as well. Cleary argues that in his arithmetic system, cases and ( and some other corresponding ill defined cases with ....

Cleary, J., (1987) Logical arithmetic, Future Computing Systems 2 (2) 125-149.

....consistency of an interval situation (solution) means that all variables are locally consistent. Locally consistent solutions can be determined by various local filtering or propagation algorithms (Davis, 1987) for global solutions additional numerical or algebraic techniques are usually needed (Cleary, 1987; Hyv nen, 1992) 1.3 EXAMPLES For example, 1.2) represents the problem of evaluating the actual range of values of a 5th order polynomial. By using initial situation y=0, x ( 1e100,1e100) 1.5) the global solution would be the set of the zeros of the polynomial, in this case x 1,2,3,4,5 . ICSP ....

....the actual solutions, not only in bounding them, an additional solution generation algorithm should be programmed on top of the library. For example, solution generation by iteratively splitting selected intervals can used like in various interval extensions of logic programming (see e.g. Cleary, 1987; LHomme, 1993; Older, Vellino, 1993) 2.2 ALGEBRAIC AND NUMERICAL TECHNIQUES The main technical novelty of ICE InC library is to apply a combination of interval analysis, computer algebra and tolerance propagation techniques (Hyv nen, 1992) for determining local and, especially, global interval ....

Cleary, J.C. (1987) Logical arithmetic, Future Computing Systems 2 (2) 125149.

....in CLP. Functional interval arithmetic has been introduced by R. Moore [17] to deal with the incorect behaviours of finite precision arithmetic. To provide a relational model for numeric processing on intervals in Prolog, relational arithmetic on real intervals has been proposed by John Cleary in [5]. The two major drawbacks of Cleary s model are the constraint solving restriction to interval convex relations (relations built from continuous, monotonic functions) and the use of nonlogical variables which tends to separate constraint solving on intervals from the CLP scheme. W. Older and A. ....

.... On one hand, if F intervals are defined as being floating point intervals, this definition is closely akin to the ideas at the basis of relational interval arithmetic and, in the case where ae is reduced to a singleton, introduces the approximation of real numbers by floating point intervals (see [5]) On the other hand, the generalization which allows ae to be any relation on (ae is not restricted to be an interval) will be used to deal with non interval convex relations 2 . The approx function is then naturally extended to any n ary relation on in the following way: Definition 1 For ....

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J. G. Cleary, "Logical Arithmetic", Future Computing Systems, Vol 2,No 2, p 125--149, 1987.

....have been used. These reasons have motivated the development of numerous CLP systems based on interval arithmetic (e.g. BNR Prolog [20] Newton [1] CLP(BNR) 3] Interlog [12, 5, 14] Prolog IV [4] All these systems use an arc consistency like algorithm [17] adapted for numeric constraints [8, 7]. The standard interval narrowing algorithm has two main limitations : ffl the so called problem of early quiescence [8] i.e. the algorithm stops before reaching a good approximation of the set of possible values. This problem is due to the fact that interval narrowing algorithm guarantees ....

....do not occur very often. It is true that early quiescence of interval narrowing algorithm is far more frequent than slow convergence. However, when the interval narrowing algorithm ends prematurely, a kind of enumeration interleaved with this algorithm is generally performed (e.g. domain splitting [7] or stronger consistencies [14] During this interleaved process, slow convergence phenomena have a great chance to occur and to increase the required computing time considerably. Slow convergence phenomena move very often into cyclic phenomena after a transient period (a kind of stabilization ....

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J. Cleary, `Logical arithmetic,' Future Computing Systems, vol. 2, no. 2, pp. 125--149, 1987.

....of conventional floating point arithmetic. These problems are, of course, not new ones, but have plagued numerical computation since the earliest days of digital computers. One mechanism for overcoming these problems by applying a Prolog like mechanism to intervals was first suggested by Cleary [4]; these ideas were first fully implemented at Bell Northern Research (BNR) in BNR Prolog in 1987 and have been successfully applied to problems far more complex than those described in [4] Although similar in intent, Cleary s mechanism differs substantially from other constraint logic programming ....

.... for overcoming these problems by applying a Prolog like mechanism to intervals was first suggested by Cleary [4] these ideas were first fully implemented at Bell Northern Research (BNR) in BNR Prolog in 1987 and have been successfully applied to problems far more complex than those described in [4]. Although similar in intent, Cleary s mechanism differs substantially from other constraint logic programming languages such as CLP( 9] Prolog III [5] and CHIP [7] in that it does not use term rewriting or symbolic equation solving techniques. In many repects the interval constraint system of ....

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Cleary, J. C. "Logical Arithmetic", Future Computing Systems,2 (2), pp.125--149, 1987.

....finite manipulatable structure. CHIP s numeric domains are always finite integer sets 2 . BNR Prolog s domains are real intervals and it efficiently implements many constraints on real numbers by using consistency algorithms to tighten those intervals closer to actual solutions to the constraints (Cleary, 1987). To compare numeric CLP systems which use consistency case analysis algorithms, it is useful to distinguish between solutions to a query and answers to that query given by a CLP language. A solution for a query Q with respect to a program P is a substitution s of terms for Q s variables such that ....

....analysis provides a divide and conquer method for finding solutions to the CSP. Arc consistency is interleaved with case analysis algorithms to further reduce the search space. Case analysis is implemented by the built in predicate, split(Vars) which is similar to predicates described elsewhere (Cleary, 1987; Older and Vellino, 1990; Van Hentenryck, 1989) Split(Vars) repeatedly cycles through the list Vars of variables in a roundrobin fashion removing approximately half the values in each variable s domain. Upon backtracking, split restores half of a domain and removes the other half. Echidna s real ....

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Cleary, J. G. 1987. Logical Arithmetic. Future Computing Systems. 2 (2). 125-149.

....integer domains (CHIP) 5,11] rationals (Prolog III) 3] or floating point numbers (CLP( 7] In these systems the set of constraints is effectively restricted to linear equations or inequalities, for which there are well known solution algorithms. Another approach suggested by John Cleary in [2] and also proposed independently by Hyvonen in [6] considers relational arithmetic on continuous domains by adapting techniques from interval arithmetic. These ideas for constraint interval Y g f X (a) Y g f X (b) Figure 1: Conventional Fixed Point Iteration arithmetic were first fully ....

Cleary, J. C. "Logical Arithmetic", Future Computing Systems, 2 (2), pages 125--149, 1987.

....a logical form of arithmetic for use in Prolog are well known. The difficulties of doing correct computations with floating point are also well known. One mechanism for overcoming both of these problems applying a Prolog like narrowing mechanism to intervals was first suggested by Cleary [Cleary 1987]. These ideas were first fully implemented at Bell Northern Research (BNR) in BNR Prolog in 1987 [BNR Prolog 1988] and have been successfully applied to problems far more complex than those described in [Cleary 1987] Although similar in intent, Cleary s mechanism differs substantially from other ....

.... a Prolog like narrowing mechanism to intervals was first suggested by Cleary [Cleary 1987] These ideas were first fully implemented at Bell Northern Research (BNR) in BNR Prolog in 1987 [BNR Prolog 1988] and have been successfully applied to problems far more complex than those described in [Cleary 1987]. Although similar in intent, Cleary s mechanism differs substantially from other constraint logic programming languages such as CLP( Jaffar and Michaylov 1987] and Prolog III [Colmerauer 1990] in that it is not based on term rewriting or symbolic equation solving techniques. In many respects ....

Cleary, J. C. "Logical Arithmetic", Future Computing Systems, 2 (2), pp.125--149, 1987.

.... Conventional programming systems for computing on intervals, such as enhanced Pascal and Fortran compilers have existed for many years [6] More recently, Cleary suggested that a relational form of interval arithmetic could be seamlessly integrated into a logic programming language (Prolog)[3]. This idea is qualitatively different from other forms of automated interval arithmetic because the combined power of a symbolic, logic programming language on the one hand and a mathematically correct, logically sound and computationally efficient programming system (constraint arithmetic on ....

....language and the presence of conventional, procedural arithmetic in Prolog spoils the logical character of the language. The relational model for computing on real intervals centers around the notion of narrowing. The first proposal for a relational arithmetic using intervals by John Cleary in [3] was designed to address the problem that arithmetic in Prolog is evaluated functionally (whereas Prolog is otherwise a relational language) Just as the Prolog unification mechanism narrows the space of possible instantiations of a variable as computations proceed so, in this model, the ....

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Cleary, J. C. "Logical Arithmetic", Future Computing Systems,2 (2), pp.125-- 149, 1987.

....uncertain data. Applications of interval computations include management of rounding errors (Moore, 1966, 1979) sensitivity and worst case analysis (Skelboe, 1979) global optimization (Ratschek, Rokne, 1988; Hansen, 1992) interval constraint reasoning (Davis, 1987; Hyv nen 1992) and programming (Cleary, 1987; Older, Vellino, 1993) spreadsheets (Hyv nen 1991, 1994) graphical interfaces (Borning, 1979; 1986) scheduling and planning systems (Stefik, 1981; Fox, 1983) qualitative reasoning (Kuipers, Berleant, 1988) design systems (Steinberg, 1987; Murtagh, Shimura, 1990) A fundamental task to be ....

Cleary, J.C. (1987) Logical arithmetic, Future Computing Systems 2 (2) 125149.

....could be refined indefinitely. Thus, in this case, the procedure may not terminate. Note that this is not a problem in practice with well known domains such as the real domain since the precision of any real number is bound so that the domain is effectively finite. This is the argument given in [CLE87,LEE93] and [BEN97] We therefore incorporate in our system a generic predicate P recision defined as P recision : a a Bool. which limits domain refinement and is applied on simple interval constraints (i.e.without using indexicals) 10 Let X 2 fc; dg be a simple interval constraint ....

CLEARY J.G., Logical Arithmetic. In Future computing Systems 2(2):125149, 1987.

....in such applications typically results in tedious trial and error sessions where the user tries to find a feasible solution (output values) by iteratively guessing argument values (input values) Problems such as (1. 4) can be approached by using interval constraint satisfaction techniques [8, 10, 15, 17] in which function sets (or more generally equation sets) are treated as symmetric relational constraints without committing to any input output distinction. In addition to using the trial and error scheme, interval constraint techniques can also support top down problem solving in which a ....

....open bounds is lost and results are always closed. A more proper approach is to generalize IA function evaluation rules for dealing with interval open ends (and with infinities Sigma1) by considering different interval limit combinations separately for each arithmetic operation (see e.g. [8]) For example, two half open intervals (a; b] can be added by rule: x; y ] u; v ] x u; y v ] However, in this approach rounding errors may cause problems. For example, when computing (0:1; 1 ] 0:2; 2 ] 0:3 Sigma; 3 ] the open minimum, denoted by 0:3 Sigma = 0:1 0:2 cannot be ....

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Cleary, J. C., "Logical Arithmetic", Future Computing Systems, 2(2), 1987, pp. 125--149.

....step. The solution to this is to allow facts to contain constraints such as 2T. This approach ensures that the cost of assignments is proportional only to the number of assignments not to the time gap between them. Current Starlog implementations allow general and powerful arithmetic constraints [Cle87]. While these fix some large scale problems they can be expensive to maintain and to propagate through the bottom up deduction process. This is particularly true when it is necessary to perform constructive negation on such constraints. The results in [Cle97] provide an effective but relatively ....

Cleary, J.G.(1987) "Logical Arithmetic," Future Computing Systems, 2(2), pp. 125-149.

....which I will not attempt to summarise here. They key idea of using explicit time stamps is presaged by work in (Kowalski and Sergot, 1986) and more recently in (Kesim and Sergot, 1993) The use of constraints for arithmetic (which is the basis for the handling of time stamps)was first described in (Cleary, 1987). The use of a close variant of Starlog in an object oriented deductive database context can be found in a series of papers by Liu (Liu, 1992; Liu and Cleary, 1992, 1994, 1995) More detailed descriptions of Starlog and its application can be found in (Cleary, 1990, 1993) and (Kaushik, 1991) ....

....approach is closely related to the classical approach to interval arithmetic (Alefield and Herzberger, 1983) where new operations are defined on intervals. Thus it is possible to add, subtract and (with some difficulty) divide two intervals. The approach to relational arithmetic advocated in (Cleary,1987) is somewhat different. All statements there are about individual numbers and their relationships to each other. No operations on intervals per se are defined and the logic is a logic of pure arithmetic. The underlying implementation, however, does use intervals to represent the deduced results ....

Cleary, J.G.(1987) "Logical Arithmetic," Future Computing Systems, 2(2), pp. 125149. Galton, A. Ed. (1987) Temporal Logics and their Applications, Academic Press, London.

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CLEARY J.G., Logical Arithmetic. In Future computing Systems 2(2):125149, 1987.

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