W. Older and A. Vellino. Constraint Arithmetic on Real Intervals. In Constraint Logic Programming: Selected Papers, F. Benhamou & A. Colmerauer eds., The MIT Press, Cambridge, MA, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Constraints are domain specific. They depend on the domain from which the values for the variables are taken. Popular domains for constraint programming are finite domains (the domain is a finite subset of the integers) 143] finite sets [40] trees [26] records [140] and real intervals [101]. Essential for constraints is constraint propagation. Constraint propagation excludes values for variables that are in conflict with a constraint. A constraint that connects several variables propagates information between its variables. Variables act as communication channels between several ....

William Older and Andre Vellino. Constraint arithmetic on real intervals. In Benhamou and Colmerauer [12], pages 175--196.

.... Theta : Theta Xn ) S : fC1 ; Cmg while S 6= and X 6= do C : choose one C i in S X : N i (X) if X 6= X then S : S [ fC j j 9vk 2 var(C j ) X k 6= Xkg X : X endif S : S n fCg endwhile end. Table 1. The Narrowing algorithm. We give without proof (see [3, 8]) the proposition below stating some of the properties of Nar to be used later: Proposition 1 (Narrowing algorithm) Nar is correct, confluent, contractant and monotone. 3 Structuring the store Preliminary Notations. We introduce two notations used in the following: Restriction. We define the ....

W. Older and A. Vellino. Constraint Arithmetic on Real Intervals. In F. Benhamou and A. Colmerauer, editors, Constraint Logic Programming: Selected Papers. The MIT Press, 1993.

.... projections, have been used extensively in the artificial intelligence community (under the name consistency techniques) to solve discrete combinatorial problems (e.g. 16, 15] They have been adapted to continuous problems (e.g. 5, 14] and used inside systems such as BNR Prolog and CLP(BNR) [21, 3] and many systems since then. The techniques used in systems like BNR Prolog and CLP(BNR) are weaker than box(1) consistency, since they decompose all constraints into ternary constraints on distinct variables before applying a form of box(1) consistency. They do not scale well on difficult ....

W. Older and A. Vellino. Constraint Arithmetic on Real Intervals. In Constraint Logic Programming: Selected Research. The MIT Press, Cambridge, Massachussetts, 1993.

....f1g 2 Delta (x = y) in f0g (x = y c) in f1g 2 Delta; c 6= 0 (x 6= y) in f1g (x = y c) in f1g 2 Delta; c 0 (x y) in f1g Table 2.9: Evaluating the axioms reasoning. The BNR Prolog language, for example, uses a narrowing algorithm to solve constraints over reals, integers and Booleans [OV93] Hard optimization problems [GJ79] are typically solved with techniques such as integer linear programming [Sch87] or local satisfaction, hill climbing techniques [SLM92] These techniques are tailored towards computing a particular solution rather than verifying all solutions. We do believe ....

....we also plan to extend the system with more rules. One set of rules will be for propagation based temporal reasoning, so that manyworlds models can be analyzed. Another set of rules will be added so that numerical methods for interval propagation and equation solving can be used as well [Cle87, OV93, BMvH94] This lets us model and analyze analog devices, as well as apply powerful methods for root finding from numerical analysis. 27 Chapter 3 The Evaluation of NP(FD) 3.1 Introduction This chapter summarizes results from running benchmark formulae under Logikkonsult s Prover version ....

W. Older and A. Vellino. Constraint arithmetic on real intervals. In Constraint Logic Programming: Selected Research (eds. Benhamou and Colmerauer). MIT Press, 1993.

....research was supported in part by the National Science Foundation grant No. CCR 0085949 and NASA Contract No. NAG2 1210. we also introduce extensions of the basic constraint language to allow more accurate analyses. For readers familiar with interval constraints for oating point computation [2, 13, 14, 23] based on interval arithmetic [20] integer range constraints are different. Such work deals primarily with rounding errors in real numbers, and the goal is to get an approximate real interval that includes all solutions to the original constraints. Range constraints deal with integer ranges, and ....

W.J. Older and A. Velino. Constraint arithmetic on real intervals. In Frederic Benhamou and Alain Colmerauer, editors, Constraint Logic Programming: Selected Research, pages 175-196. MIT Press, 1993.

....f1g 2 Delta (x = y) in f0g (x = y c) in f1g 2 Delta; c 6= 0 (x 6= y) in f1g (x = y c) in f1g 2 Delta; c 0 (x y) in f1g Table 2.9: Evaluating the axioms reasoning. The BNR Prolog language, for example, uses a narrowing algorithm to solve constraints over reals, integers and Booleans [OV93] Hard optimization problems [GJ79] are typically solved with techniques such as integer linear programming [Sch87] or local satisfaction, hill climbing techniques [SLM92] These techniques are tailored towards computing a particular solution rather than verifying all solutions. We do believe ....

....we also plan to extend the system with more rules. One set of rules will be for propagation based temporal reasoning, so that manyworlds models can be analyzed. Another set of rules will be added so that numerical methods for interval propagation and equation solving can be used as well [Cle87, OV93, BMvH94] This lets us model and analyze analog devices, as well as apply powerful methods for root finding from numerical analysis. 27 Chapter 3 The Evaluation of NP(FD) 3.1 Introduction This chapter summarizes results from running benchmark formulae under Logikkonsult s Prover version ....

W. Older and A. Vellino. Constraint arithmetic on real intervals. In Constraint Logic Programming: Selected Research (eds. Benhamou and Colmerauer). MIT Press, 1993.

....1 x 2 2x 2 3 5x 6 11x 7 0: Fig. 7. The Constrained Optimization Problem for h100. 5 Related Work The introduction of a relational form of interval arithmetic in logic programming has been proposed by Cleary in [3] These ideas have been developed and made popular by the CLP system BNR Prolog [29] and generalized to constraint solving over discrete quantities in its successor CLP(BNR) 27,2] Many other systems (e.g [15,35] have been developed on similar principles. The key idea behind CLP(Intervals) languages is to let users state arbitrary constraints over reals and to narrow down the ....

W. Older and A. Vellino. Constraint Arithmetic on Real Intervals. In Constraint Logic Programming: Selected Research. The MIT Press, Cambridge, Massachussetts, 1993.

....analysis problems [13, 49] that need lower and upper bounds on the values of integer variables. To reduce this imprecision, we also introduce extensions of the basic constraint language to allow more accurate analyses. For readers familiar with interval constraints for oatingpoint computation [6, 25, 26, 37] based on interval arithmetic [34] integer range constraints are di erent. Such work deals primarily with rounding errors in real numbers, and the goal is to get an approximate real interval that includes all solutions to the original constraints. Range constraints deal with integer ranges, and ....

W.J. Older and A. Velino. Constraint arithmetic on real intervals. In Frederic Benhamou and Alain Colmerauer, editors, Constraint Logic Programming: Selected Research, pages 175-196. MIT Press, 1993.

....executes the following query and gets an interval solution guaranteed to contain the answer. cos(X) X . X = 0.73908513321515, 0. 73908513321517] Historically, there have been two approaches to implementing the constraint solvers inside CLP(Intervals) languages: the RISC approach of CLP(BNR) [7, 33, 32, 6, 2], and the CISC approach of Newton and similar systems [3, 39, 38, 4, 15, 27] In the RISC approach 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 select a primitive constraint ....

.... An early version [19] of this paper appeared in the Second International Workshop on Practical Aspects of Declarative Languages (PADL 00) 2 Overview of the CLP(Intervals) language: CLIP Although there are several RISC style CLP(Intervals) systems that are currently available (e.g. BNRPROLOG [33], clp(BNR) 32] Prolog IV [35] ECLiPSe RIA[11] DECLIC [13] our interest was in a system whose underlying constraint solver was very fast and publicly accessible (so we could analyze it) and which used algorithms that had been proved correct. None of the systems we had seen at the time had ....

W. Older and A. Vellino, Constraint Arithmetic on Real Intervals, in Constraint Logic Programming: Selected Research. Colmerauer, A. and Benhamou, F. (eds), MIT Press 1993.

....Clearly, fairness is the weakest possible condition on the sequence f . If for any i = 1; m, p i would have a last occurrence in f , then examples can be found where the sequence converges to a fixpoint common to the other operators that is not a fixpoint of p i . Older and Vellino [OV93] derive the result independently of [LM84] for a more restricted class of fair sequences f , which consists of repetitions of a single permutation of f1; mg. 7 The Waltz algorithm as chaotic iteration of consistency operators To summarize section 6, if the number of representable sets ....

....application of dormant operators. In the typical situation, where each constraint involves only a small subset of all variables, the fair sequence obtained from ordering the constraints according to a permutation of their indexes, and then repeating this same permutation forever, as considered in [OV93] has the disadvantage that unnecessarily many dormant operators are applied. For, whenever a live operator is applied, it always becomes dormant, by the idempotency of the consistency operator. It will surely remain dormant as long as no other operator has contracted the set associated with at ....

William Older and Andr'e Vellino. Constraint arithmetic on real intervals. In Fr'ed'eric Benhamou and Alain Colmerauer, editors, Constraint Logic Programming, pages 175--195. MIT Press, 1993.

.... by local consistency methods have been proposed, the most famous being arc consistency [19] and path consistency [23] Since the introduction of local consistency in Constraint Logic Programming [29] various extensions have been proposed, among which methods to solve socalled interval constraints [10, 8, 26, 15, 14, 5, 18, 3, 4, 28, 27]. More recently several authors have studied various combinations of solvers in the case of continuous real constraints [20, 22, 13] and, in particular, combinations of techniques from computer algebra (Grobner bases) Interval Constraint methods and techniques from numerical analysis. Similar ....

....by set inclusion, constitute complete lattices in which the meet operation is defined by set intersection. The join of sets is defined as the smallest set larger than all of them, i.e. the approximation of their union. For a discussion on a lattice theoretic view of interval constraints, see [26]) Given an approximate domain A, and an n ary relation # over IR, a narrowing operator for the relation # is a correct, contractant, monotone, idempotent function N : A n # A n . More formally, Definition 2 Let A be an approximate domain. Let # be an n ary relation over IR. The function N ....

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W. Older and A. Vellino. Constraint Arithmetic on Real Intervals. In F. Benhamou and A. Colmerauer, editors, Constraint Logic Programming: Selected Research. MIT Press, 1993.

....propagation [10,7,5] CP) is a cornerstone algorithm of constraint programming, mainly devoted to the computation of local consistency properties of constraint satisfaction problems. The abstract formulation of CP is the combination of a set of reduction functions (black box solvers) on a domain [8,9,2,1]. Intuitively, there is a dependence relation between functions and domains, such that a function must be applied if a domain it depends on is reduced. The essential property of CP is confluence or strategy independence. In other words, the order solvers are applied does not influence the output ....

William Older and Andre Vellino. Constraint Arithmetic on Real Intervals. In Frederic Benhamou and Alain Colmerauer, editors, Constraint Logic Programming: Selected Research. MIT Press, 1993.

....correct: the output Cartesian product of domains is a superset of the declarative semantics of the constraint system included in the input box; the algorithm is con uent: the output is independent of the reinvocation order of constraints. These properties are proved in the same way as for HC3 (see [11]) once HC4revise has been proved to be a constraint narrowing operator (see Prop. 1) Algorithm 1: HC4 algorithm HC4(in c 1 , c m ; inout B = I 1 I n ) begin S # c 1 , c m while (S #= # and B #= #) do c # choose one c i in S B # # HC4Revise(c,B,I # ....

....HC4 and HC3 use the same narrowing operators, and that the node attributes in HC4revise mimic the new variables in HC3. The proof then follows since the application order of the same constraint narrowing operators does not in uence the xed point (con uency property of propagation algorithms [11]) 4 Box consistency and related algorithms Box consistency [2] has been introduced to avoid decomposing constraints, thus tackling the dependency problem for variables with many occurrences. Section 4.1 generalizes its de nition and surveys the original method used to enforce it. Some ....

W. J. Older and A. Vellino. Constraint arithmetic on real intervals. In Frdric Benhamou and Alain Colmerauer, editors, Constraint Logic Programming: Selected Papers. The MIT Press, 1993.

....1. Introduction and related work Interval methods for solving nonlinear systems were adapted from numerical analysis techniques in the last thirty years. One may cite Taylor expansions, Newton methods and Gauss Seidel iterations. In the last decade, some authors (Cleary [4] Hyvonen [9] Older [12], Benhamou [2] Lhomme [10] Faltings [6] Van Hentenryck [15] have shown that constraint satisfaction techniques are a powerful alternative to traditional interval methods. One of the most e#cient solver, Numerica [16] is based on a combination of local consistency (domain pruning) and ....

W. Older and A. Vellino. Constraint Arithmetic on Real Intervals. In Frederic Benhamou and Alain Colmerauer, editors, Constraint Logic Programming: Selected Research. MIT Press, 1993.

....constraint C i and of a constraint narrowing operator N i for C i , and X is an n ary Cartesian product of elements of A. The algorithm for processing such systems is essentially an adaptation of AC 3 [14] and of the filtering algorithms used in interval constraint based systems like BNR Prolog [19], CLP(BNR) 5] or Newton [4, 20] Given an initial ECS (S, X) it works by iterating the application of constraint narrowing operators until reaching a stable state as shown below. Propagation(in (C 1 , N 1 ) Cm , Nm ) inout X = n i=1 X i ) begin BenGran97rc final.tex; ....

W. Older and A. Vellino. Constraint Arithmetic on Real Intervals. In F. Benhamou and A. Colmerauer, editors, Constraint Logic Programming: Selected Research. MIT Press, 1993.

....way to combine deduction and constraint solving, for the goal of reasoning with or solving non trivial problems in a declarative way. During last years, several important CLP systems have been developed such as (only to mention a few) Prolog III [6, 7] CLP(R) 13, 9] CHIP [8] CAL [1] CLP(BNR) [16, 2], RISC CLP [10, 11] See also [4] for an extension) The computation domain provided by these systems are trees, finite sets (domains) booleans, real numbers, and various combinations. In this paper, we describe a new CLP system for the domain of complex functions. A complex function is a ....

W. Older and A. Vellino. Constraint arithmetic on real intervals. In F. Benhamou and A. Colmerauer, editors, Constraint Logic Programming: Selected Research. MIT Press, 1993.

....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 related one ....

William Older and Andr'e Vellino. Constraint Arithmetic on Real Intervals. In Fr'ed'eric Benhamou and Alain Colmerauer, editors, Constraint Logic Programming: Selected Research. MIT Press, 1993. Reactive LTR N o 22532 DiSCiPl D.WP1.1.M1.1-Part 2 35

....algorithm depend on the choices we make In this paper, we assume that somehow the algorithm has been already shown to be terminating. Thus, we address the remaining issue: confluency. A simpler case when each sub solver produces only one constraint has been studied by Older and several others [4] in the context of the interval based constraint solving algorithm, which is used for the constraint logic programming system CLP(BNR) They showed that the confluency is ensured if 2 each sub solvers (narrowing operators in their context) are contracting and monotone with respect to certain ....

W. Older and A. Vellino. Constraint arithmetic on real intervals. In F. Benhamou and A. Colmerauer, editors, Constraint Logic Programming: Selected Research. MIT Press, 1993. 6

....Monfroy in [20] also describes a CLP language that deals with nonlinearity over the complex numbers using Grobner bases method. Examples of other CLP languages are CHIP over finite domains [1] CLP( Sigma ) over regular sets [24] Echidna over discrete sets [8] BNR Prolog over real intervals [21]. Whereas most of the well known implementation of the CLP scheme are concerned with finite domains or only with linear constraints, RISC CLP(Real) is able to deal with nonlinearity over the real numbers by employing partial cylindrical algebraic decomposition as quantifier elimination procedure ....

William Older and Andr`e Vellino. Constraint arithmetic on real intervals. In Fr`ed`eric Benhamou and Alain Colmerauer, editors, Constraint Logic Programming: Selected Research. The MIT Press, Cambridge, 1992.

....which can be numerically solved. However it does not take conditions into account. The second approach is based on interval reasoning. It stems from the observation that target value computation problems form a subset of interval constraint satisfaction problems. Interval constraint reasoning (Older and Vellino 1993), a set of powerful techniques for solving constraint satisfaction problems is hence a good candidate for solving target value computation problems. These techniques cope with under constrained problems. However, up until now, they lack efficiency for tackling large models. In this paper, we ....

....set of rules of consistency which is fired whenever a change occurs in the arguments of the constraint. The range of constraint arguments is computed using interval arithmetic rules given in (Moore 1966) These principles are common to most existing interval constraint solvers, including CLP(BNR) (Older and Vellino 1993) and Inc (Hyvnen, De Pasquale and Lehtola 1993) Interval computations are performed with outward ranging (rounding the left endpoint down and the right endpoint up) so that bounds are always correct. Whenever it is possible, the evaluated range of a constraint expression is further narrowed ....

Older W. and Vellino A. Constraint Arithmetic on Real Intervals. In: Constraint Logic Programming: Selected Research, F. Benhamou and A. Colmerauer eds., pp. 175-196, MIT Press, 1993.

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W. Older and A. Vellino. Constraint Arithmetic on Real Intervals. In Constraint Logic Programming: Selected Papers, F. Benhamou & A. Colmerauer eds., The MIT Press, Cambridge, MA, 1993.

No context found.

W. Older, A. Vellino, Constraint arithmetic on real intervals, in \Constraint logic programming : selected research", Ed. F. Benhamou and A. Colmerauer ed., MIT Press, 1993.

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W.J. Older and A. Velino. Constraint arithmetic on real intervals. In Frederic Benhamou and Alain Colmerauer, editors, Constraint Logic Programming: Selected Research, pages 175--196. MIT Press, 1993.

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W. Older and A. Vellino. Constraint Arithmetic on Real Intervals. In Frdric Benhamou and Alain Colmerauer, editors, Constraint Logic Programming: Selected Research. MIT Press, 1993.

No context found.

W. J. Older and A. Velino. Constraint arithmetic on real intervals. In Frederic Benhamou and Alain Colmerauer, editors, Constraint Logic Programming: Selected Research, pages 175-196. MIT Press, 1993.

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