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27
CLP(Intervals) Revisited
, 1994
"... The design and implementation of constraint logic programming (CLP) languages over intervals is revisited. Instead of decomposing complex constraints in terms of simple primitive constraints as in CLP(BNR), complex constraints are manipulated as a whole, enabling more sophisticated narrowing procedu ..."
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Cited by 121 (18 self)
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The design and implementation of constraint logic programming (CLP) languages over intervals is revisited. Instead of decomposing complex constraints in terms of simple primitive constraints as in CLP(BNR), complex constraints are manipulated as a whole, enabling more sophisticated narrowing procedures to be applied in the solver. This idea is embodied in a new CLP language Newton whose operational semantics is based on the notion of boxconsistency, an approximation of arcconsistency, and whose implementation uses Newton interval method. Experimental results indicate that Newton outperforms existing languages by an order of magnitude and is competitive with some stateoftheart tools on some standard benchmarks. Limitations of our current implementation and directions for further work are also identified.
Interval constraint logic programming
 CONSTRAINT PROGRAMMING: BASICS AND TRENDS, VOLUME 910 OF LNCS
, 1995
"... Abstract. In this paper, we present anoverview on the use of interval arithmetic to process numerical constraints in Constraint Logic Programming. The main principle is to approximate nary relations over IR with Cartesian products of intervals whose bounds are taken in a nite subset of I R.Variabl ..."
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Cited by 47 (5 self)
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Abstract. In this paper, we present anoverview on the use of interval arithmetic to process numerical constraints in Constraint Logic Programming. The main principle is to approximate nary relations over IR with Cartesian products of intervals whose bounds are taken in a nite subset of I R.Variables represent real values whose domains are intervals de ned in the same manner. Narrowing operators are de ned from approximations. These operators compute, from an interval and a relation, aset included in the initial interval. Sets of constraints are then processed thanks to a local consistency algorithm pruning at each stepvalues from initial intervals. This algorithm is shown to be correct and to terminate, on the basis of a certain number of properties of narrowing operators. We focus here on the description of the general framework based on approximations, on its application to interval constraint solving over continuous and discrete quantities, we establish a strong link between approximations and local consistency notions and show that arcconsistency is an instance of the approximation framework. We nally describe recentwork on di erent variants of the initial algorithm proposed by John Cleary and developed by W. Older and A. Vellino which havebeen proposed in this context. These variants address four particular points: generalization of the constraint language, improvement of domain reductions, e ciency of the computation and nally, cooperation with other solvers. Some open questions are also identi ed. 1
A New Representation for Exact Real Numbers
, 1997
"... We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infiite product of matrices, with nonnegative coefficients. Any rational interval in the one point compactification of the rea ..."
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Cited by 42 (8 self)
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We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infiite product of matrices, with nonnegative coefficients. Any rational interval in the one point compactification of the real line, represented by the unit circle S¹, is expressed as the image of the base interval [0�1] under an lft. A sequence of shrinking nested intervals is then represented by an infinite product of matrices with integer coefficients such that the first socalled sign matrix determines an interval on which the real number lies. The subsequent socalled digit matrices have nonnegative integer coe cients and successively re ne that interval. Based on the classi cation of lft's according to their conjugacy classes and their geometric dynamics, we show that there is a canonical choice of four sign matrices which are generated by rotation of S¹ by =4. Furthermore, the ordinary signed digit representation of real numbers in a given base induces a canonical choice of digit matrices.
Enhanced Interval Analysis for Phase Stability: Cubic Equation of State Models
 IND. ENG. CHEM. RES
, 1998
"... The reliable prediction of phase stability is a challenging computational problem in chemical process simulation, optimization and design. The phase stability problem can be formulated either as a minimization problem or as an equivalent nonlinear equation solving problem. Conventional solution meth ..."
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Cited by 30 (20 self)
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The reliable prediction of phase stability is a challenging computational problem in chemical process simulation, optimization and design. The phase stability problem can be formulated either as a minimization problem or as an equivalent nonlinear equation solving problem. Conventional solution methods are initialization dependent, and may fail by converging to trivial or nonphysical solutions or to a point that is a local but not global minimum. Thus there has been considerable recent interest in developing more reliable techniques for stability analysis. Recently we have demonstrated, using cubic equation of state models, a technique that can solve the phase stability problem with complete reliability. The technique, which is based on interval analysis, is initialization independent, and if properly implemented provides a mathematical guarantee that the correct solution to the phase stability problem has been found. However, there is much room for improvement in the computational efficiency of the technique. In this paper we consider two means of enhancing the efficiency of the method, both based on sharpening the range of interval function evaluations. Results indicate that by using the enhanced method, computation times can be reduced by nearly an order of magnitude in some cases.
Consistency Techniques in Ordinary Differential Equations
, 2000
"... This paper takes a fresh look at the application of interval analysis to ordinary differential equations and studies how consistency techniques can help address the accuracy problems typically exhibited by these methods, while trying to preserve their efficiency. It proposes to generalize interval t ..."
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Cited by 16 (1 self)
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This paper takes a fresh look at the application of interval analysis to ordinary differential equations and studies how consistency techniques can help address the accuracy problems typically exhibited by these methods, while trying to preserve their efficiency. It proposes to generalize interval techniques intoatwostep process: a forward process that computes an enclosure and a backward process that reduces this enclosure. Consistency techniques apply naturally to the backward (pruning) step but can also be applied to the forward phase. The paper describes the framework, studies the various steps in detail, proposes a number of novel techniques, and gives some preliminary experimental results to indicate the potential of this new research avenue.
Combining local consistency, symbolic rewriting and interval methods
 In Procs. of AISMC3, volume 1138 of LNCS
, 1996
"... Abstract. This paper is an attempt to address the processing of nonlinear numerical constraints over the Reals by combining three di erent methods: local consistency techniques, symbolic rewriting and interval methods. To formalize this combination, we de ne a generic twostep constraint processing ..."
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Cited by 14 (4 self)
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Abstract. This paper is an attempt to address the processing of nonlinear numerical constraints over the Reals by combining three di erent methods: local consistency techniques, symbolic rewriting and interval methods. To formalize this combination, we de ne a generic twostep constraint processing technique based on an extension of the Constraint Satisfaction Problem, called Extended Constraint Satisfaction Problem (ECSP). The rst step is a rewriting step, in which the initial ECSP is symbolically transformed. The second step, called approximation step, is based on a local consistency notion, called weak arcconsistency, de ned over ECSPs in terms of xed point ofcontractant monotone operators. This notion is shown to generalize previous local consistency concepts de ned over nite domains (arcconsistency) or in nite subsets of the Reals (arc Bconsistency and interval, hull and boxconsistency). A ltering algorithm, derived from AC3, is given and is shown to be correct, con uent and to terminate. This framework is illustrated by the combination of Grobner Bases computations and Interval Newton methods. The computation of Grobner Bases for subsets of the initial set of constraints is used as a rewriting step and operators based on Interval Newton methods are used together with enumeration techniques to achieve weak arcconsistency on the modi ed ECSP. Experimental results from a prototype are presented, as well as comparisons with other systems.
A SymbolicNumerical Branch and Prune Algorithm for Solving Nonlinear Polynomial Systems
 Journal of Universal Computer Science
, 1998
"... : This paper discusses the processing of nonlinear polynomial systems using a branch and prune algorithm within the framework of constraint programming. We propose a formalism for a kind of branch and prune algorithm implementing symbolic and numerical methods to reduce the systems with respect to ..."
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Cited by 12 (0 self)
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: This paper discusses the processing of nonlinear polynomial systems using a branch and prune algorithm within the framework of constraint programming. We propose a formalism for a kind of branch and prune algorithm implementing symbolic and numerical methods to reduce the systems with respect to a relation defined from both inclusion of variable domains and inclusion of sets of constraints. The second part of the paper presents an instantiation of this general scheme. The pruning step is implemented as a cooperation of factorizations, substitutions and partial computations of Grobner bases to simplify the systems, and interval Newton methods address the numerical, approximate solving. The branching step creates a partition of domains or generates disjunctive constraints from equations in factorized form. Experimental results from a prototype show that interval methods generally benefit from the symbolic processing of the initial constraints. Key Words: Branch and prune algorithm, n...
Reliable Computation of Phase Stability and Equilibrium from the SAFT Equation of State
 Industrial & Engineering Chemistry Research
, 2001
"... In recent years, molecularlybased equations of state, as typified by the SAFT (statistical associating fluid theory) approach, have become increasingly popular tools for the modeling of phase behavior. However, whether using this, or even much simpler models, the reliable calculation of phase behav ..."
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Cited by 9 (6 self)
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In recent years, molecularlybased equations of state, as typified by the SAFT (statistical associating fluid theory) approach, have become increasingly popular tools for the modeling of phase behavior. However, whether using this, or even much simpler models, the reliable calculation of phase behavior from a given model can be a very challenging computational problem. A new methodology is described that is the first completely reliable technique for computing phase stability and equilibrium from the SAFT model. The method is based on interval analysis, in particular an interval Newton/generalized bisection algorithm, which provides a mathematical and computational guarantee of reliability, and is demonstrated using nonassociating, selfassociating, and crossassociating systems. New techniques are presented that can also be exploited when conventional pointvalued solution methods are used. These include the use of a volumebased problem formulation, in which the core thermodynamic function for phase equilibrium at constant temperature and pressure is the Helmholtz energy, and an approach for dealing with the internal iteration needed when there are association effects. This provides for direct, as opposed to iterative, determination of the derivatives of the internal variables. 1
A parametric representation of fuzzy numbers and their arithmetic operators. Fuzzy sets and systems
 vol
, 1997
"... Abstract: Direct implementation of extended arithmetic operators on fuzzy numbers is computationally complex. Implementation of the extension principle is equivalent to solving a nonlinear programming problem. To overcome this difficulty many applications limit the membership functions to certain sh ..."
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Cited by 9 (2 self)
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Abstract: Direct implementation of extended arithmetic operators on fuzzy numbers is computationally complex. Implementation of the extension principle is equivalent to solving a nonlinear programming problem. To overcome this difficulty many applications limit the membership functions to certain shapes, usually either triangular fuzzy numbers (TFN) or trapezoidal fuzzy numbers (TrFN). Then calculation of the extended operators can be performed on the parameters defining the fuzzy numbers, thus making the calculations trivial. Unfortunately the TFN shape is not closed under multiplication and division. The result of these operators is a polynomial membership function and the triangular shape only approximates the actual result. The linear approximation can be quite poor and may lead to incorrect results when used in engineering applications. We analyze this problem and propose six parameters which define parameterized fuzzy numbers (PFN), of which TFNs are a special case. We provide the methods for performing fuzzy arithmetic and show that the PFN representation is closed under the arithmetic operations. The new representation in conjunction with the arithmetic operators obeys many of the same arithmetic properties as TFNs. The new method has better accuracy and similar accepted by Fuzzy Sets and Systems: Special Issue on Fuzzy Arithmeticcomputational speed to using TFNs and appears to have benefits when used in engineering applications.
Algebraic Solutions to a Class of Interval Equations
 J. UNIVERSAL COMPUTER SCIENCE
, 1998
"... The arithmetic on the extended set of proper and improper intervals is an algebraic completion of the conventional interval arithmetic and thus facilitates the explicit solution of certain interval algebraic problems. Due to the existence of inverse elements with respect to addition and multiplicati ..."
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Cited by 4 (4 self)
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The arithmetic on the extended set of proper and improper intervals is an algebraic completion of the conventional interval arithmetic and thus facilitates the explicit solution of certain interval algebraic problems. Due to the existence of inverse elements with respect to addition and multiplication operations certain interval algebraic equations can be solved by elementary algebraic transformations. The conditionally distributive relations between extended intervals allow that complicated interval algebraic equations, multiincident on the unknown variable, be reduced to simpler ones. In this paper we give the general type of "pseudolinear" interval equations in the extended interval arithmetic. The algebraic solutions to a pseudolinear interval equation in one variable are studied. All numeric and parametric algebraic solutions, as well as the conditions for nonexistence of the algebraic solution to some basic types pseudolinear interval equations in one variable are found. Some examples leading to algebraic solution of the equations under consideration and the extra functionalities for performing true symbolicalgebraic manipulations on interval formulae in a Mathematica package are discussed.