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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 138 (20 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.
Solving Polynomial Systems Using a Branch and Prune Approach
 SIAM Journal on Numerical Analysis
, 1997
"... This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists ..."
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Cited by 114 (8 self)
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This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists in enforcing at each node of the search tree a unique local consistency condition, called boxconsistency, which approximates the notion of arcconsistency wellknown in artificial intelligence. Boxconsistency is parametrized by an interval extension of the constraint and can be instantiated to produce the HansenSegupta's narrowing operator (used in interval methods) as well as new operators which are more effective when the computation is far from a solution. Newton has been evaluated on a variety of benchmarks from kinematics, chemistry, combustion, economics, and mechanics. On these benchmarks, it outperforms the interval methods we are aware of and compares well with stateoftheart continuation methods. Limitations of Newton (e.g., a sensitivity to the size of the initial intervals on some problems) are also discussed. Of particular interest is the mathematical and programming simplicity of the method.
Revising Hull and Box Consistency
 INT. CONF. ON LOGIC PROGRAMMING
, 1999
"... Most intervalbased solvers in the constraint logic programming framework are based on either hull consistency or box consistency (or a variation of these ones) to narrow domains of variables involved in continuous constraint systems. This paper rst presents HC4, an algorithm to enforce hull consist ..."
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Cited by 107 (14 self)
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Most intervalbased solvers in the constraint logic programming framework are based on either hull consistency or box consistency (or a variation of these ones) to narrow domains of variables involved in continuous constraint systems. This paper rst presents HC4, an algorithm to enforce hull consistency without decomposing complex constraints into primitives. Next, an extended denition for box consistency is given and the resulting consistency is shown to subsume hull consistency. Finally, BC4, a new algorithm to eciently enforce box consistency is described, that replaces BC3the original solely Newtonbased algorithm to achieve box consistencyby an algorithm based on HC4 and BC3 taking care of the number of occurrences of each variable in a constraint. BC4 is then shown to signicantly outperform both HC3 (the original algorithm enforcing hull consistency by decomposing constraints) and BC3. 1 Introduction Finite representation of numbers precludes computers from exactly solv...
The Essence of Constraint Propagation
 CWI QUARTERLY VOLUME 11 (2&3) 1998, PP. 215 { 248
, 1998
"... We show that several constraint propagation algorithms (also called (local) consistency, consistency enforcing, Waltz, ltering or narrowing algorithms) are instances of algorithms that deal with chaotic iteration. To this end we propose a simple abstract framework that allows us to classify and comp ..."
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Cited by 104 (6 self)
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We show that several constraint propagation algorithms (also called (local) consistency, consistency enforcing, Waltz, ltering or narrowing algorithms) are instances of algorithms that deal with chaotic iteration. To this end we propose a simple abstract framework that allows us to classify and compare these algorithms and to establish in a uniform way their basic properties.
Complete search in continuous global optimization and constraint satisfaction
 ACTA NUMERICA 13
, 2004
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Interval arithmetic: From principles to implementation
 J. ACM
"... We start with a mathematical definition of a real interval as a closed, connected set of reals. Interval arithmetic operations (addition, subtraction, multiplication and division) are likewise defined mathematically and we provide algorithms for computing these operations assuming exact real arithme ..."
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Cited by 89 (12 self)
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We start with a mathematical definition of a real interval as a closed, connected set of reals. Interval arithmetic operations (addition, subtraction, multiplication and division) are likewise defined mathematically and we provide algorithms for computing these operations assuming exact real arithmetic. Next, we define interval arithmetic operations on intervals with IEEE 754 floating point endpoints to be sound and optimal approximations of the real interval operations and we show that the IEEE standard’s specification of operations involving the signed infinities, signed zeros, and the exact/inexact flag are such as to make a correct and optimal implementation more efficient. From the resulting theorems we derive data that are sufficiently detailed to convert directly to a program for efficiently implementing the interval operations. Finally we extend these results to the case of general intervals, which are defined as connected sets of reals that are not necessarily closed. 1
Constraint propagation
 Handbook of Constraint Programming
, 2006
"... Constraint propagation is a form of inference, not search, and as such is more ”satisfying”, both technically and aesthetically. —E.C. Freuder, 2005. Constraint reasoning involves various types of techniques to tackle the inherent ..."
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Cited by 76 (5 self)
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Constraint propagation is a form of inference, not search, and as such is more ”satisfying”, both technically and aesthetically. —E.C. Freuder, 2005. Constraint reasoning involves various types of techniques to tackle the inherent
Safety verification of hybrid systems by constraint propagation based abstraction refinement
, 2005
"... This paper deals with the problem of safety verification of nonlinear hybrid systems. We start from a classical method that uses interval arithmetic to check whether trajectories can move over the boundaries in a rectangular grid. We put this method into an abstraction refinement framework and impr ..."
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Cited by 75 (11 self)
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This paper deals with the problem of safety verification of nonlinear hybrid systems. We start from a classical method that uses interval arithmetic to check whether trajectories can move over the boundaries in a rectangular grid. We put this method into an abstraction refinement framework and improve it by developing an additional refinement step that employs interval constraint propagation to add information to the abstraction without introducing new grid elements. Moreover, the resulting method allows switching conditions, initial states and unsafe states to be described by complex constraints instead of sets that correspond to grid elements. Nevertheless, the method can be easily implemented since it is based on a welldefined set of constraints, on which one can run any constraint propagation based solver. Tests of such an implementation are promising.
Consistency Techniques for Continuous Constraints
 Constraints
, 1996
"... We consider constraint satisfaction problemswith variables in continuous,numerical domains. Contrary to most existing techniques, which focus on computing one single optimal solution, we address the problem of computing a compact representation of the space of all solutions admitted by the constrai ..."
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Cited by 59 (7 self)
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We consider constraint satisfaction problemswith variables in continuous,numerical domains. Contrary to most existing techniques, which focus on computing one single optimal solution, we address the problem of computing a compact representation of the space of all solutions admitted by the constraints. In particular, we show how globally consistent (also called decomposable) labelings of a constraint satisfaction problem can be computed.
Universally Quantified Interval Constraints
 PROCEEDINGS OF THE 6TH INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING
, 2000
"... Nonlinear real constraint systems with universally and/or existentially quantified variables often need be solved in such contexts as control design or sensor planning. To date, these systems are mostly handled by computing a quantifierfree equivalent form by means of Cylindrical Algebraic Decompo ..."
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Cited by 54 (0 self)
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Nonlinear real constraint systems with universally and/or existentially quantified variables often need be solved in such contexts as control design or sensor planning. To date, these systems are mostly handled by computing a quantifierfree equivalent form by means of Cylindrical Algebraic Decomposition (CAD). However, CAD restricts its input to be conjunctions and disjunctions of polynomial constraints with rational coefficients, while some applications such as camera control involve systems with arbitrary forms where time is the only universally quantified variable. In this paper, the handling of universally quantified variables is first related to the computation of innerapproximation of real relations.