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50
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.
Efficient solving of large nonlinear arithmetic constraint systems with complex boolean structure,”
 Journal on Satisfiability, Boolean Modeling, and Computation,
, 2007
"... Abstract In order to facilitate automated reasoning about large Boolean combinations of nonlinear arithmetic constraints involving transcendental functions, we provide a tight integration of recent SAT solving techniques with intervalbased arithmetic constraint solving. Our approach deviates subst ..."
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Cited by 90 (12 self)
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Abstract In order to facilitate automated reasoning about large Boolean combinations of nonlinear arithmetic constraints involving transcendental functions, we provide a tight integration of recent SAT solving techniques with intervalbased arithmetic constraint solving. Our approach deviates substantially from lazy theorem proving approaches in that it directly controls arithmetic constraint propagation from the SAT solver rather than delegating arithmetic decisions to a subordinate solver. Through this tight integration, all the algorithmic enhancements that were instrumental to the enormous performance gains recently achieved in propositional SAT solving carry over smoothly to the rich domain of nonlinear arithmetic constraints. As a consequence, our approach is able to handle large constraint systems with extremely complex Boolean structure, involving Boolean combinations of multiple thousand arithmetic constraints over some thousands of variables.
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.
Efficient constraint propagation engines
 Transactions on Programming Languages and Systems
"... This paper presents a model and implementation techniques for speeding up constraint propagation. Three fundamental approaches to improving constraint propagation based on propagators as implementations of constraints are explored: keeping track of which propagators are at fixpoint, choosing which p ..."
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Cited by 62 (9 self)
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This paper presents a model and implementation techniques for speeding up constraint propagation. Three fundamental approaches to improving constraint propagation based on propagators as implementations of constraints are explored: keeping track of which propagators are at fixpoint, choosing which propagator to apply next, and how to combine several propagators for the same constraint. We show how idempotence reasoning and events help track fixpoints more accurately. We improve these methods by using them dynamically (taking into account current domains to improve accuracy). We define prioritybased approaches to choosing a next propagator and show that dynamic priorities can improve propagation. We illustrate that the use of multiple propagators for the same constraint can be advantageous with priorities, and introduce staged propagators that combine the effects of multiple propagators with priorities for greater efficiency. 1
Solving Mixed and Conditional Constraint Satisfaction Problems
 Constraints
, 2003
"... Constraints are a powerful general paradigm for representing knowledge in intelligent systems. The standard constraint satisfaction paradigm involves variables over a discrete value domain and constraints which restrict the solutions to allowed value combinations. This standard paradigm is inapplica ..."
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Cited by 22 (2 self)
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Constraints are a powerful general paradigm for representing knowledge in intelligent systems. The standard constraint satisfaction paradigm involves variables over a discrete value domain and constraints which restrict the solutions to allowed value combinations. This standard paradigm is inapplicable to problems which are either: (a) mixed, involving both numeric and discrete variables, or (b) conditional? containing variables whose existence depends on the values chosen for other variables, or (c) both, conditional and mixed.
The rough guide to constraint propagation
 In 5th International Conference on Principles and Practice of Constraint Programming (CP’99
, 1999
"... Abstract. We provide here a simple, yet very general framework that allows us to explain several constraint propagation algorithms in a systematic way. In particular, using the notions commutativity and semicommutativity, we show how the wellknown AC3, PC2, DAC and DPC algorithms are instances o ..."
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Cited by 19 (2 self)
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Abstract. We provide here a simple, yet very general framework that allows us to explain several constraint propagation algorithms in a systematic way. In particular, using the notions commutativity and semicommutativity, we show how the wellknown AC3, PC2, DAC and DPC algorithms are instances of a single generic algorithm. The work reported here extends and simplifies that of Apt [1]. 1
Automatic Generation of Numerical Redundancies for Nonlinear Constraint Solving
 RELIABLE COMPUTING
, 1997
"... In this paper we present a framework for the cooperation of symbolic and propagationbased numerical solvers over the real numbers. This cooperation is expressed in terms of fixed points of closure operators over a complete lattice of constraint systems. In a second part we instantiate this framewor ..."
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Cited by 16 (4 self)
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In this paper we present a framework for the cooperation of symbolic and propagationbased numerical solvers over the real numbers. This cooperation is expressed in terms of fixed points of closure operators over a complete lattice of constraint systems. In a second part we instantiate this framework to a particular cooperation scheme, where propagation is associated to pruning operators implementing interval algorithms enclosing the possible solutions of constraint systems, whereas symbolic methods are mainly devoted to generate redundant constraints. When carefully chosen, it is well known that the addition of redundant constraint drastically improve the performances of systems based on local consistency (e.g. Prolog IV or Newton). We propose here a method which computes sets of redundant polynomials called partial Grobner bases and show on some benchmarks the advantages of such computations. Keywords: Numerical constraints, interval constraints, approximate solving, local consist...
Interval Constraint Solving for Camera Control and Motion Planning
 Bell Northern Research
, 2004
"... Many problems in robust control and motion planning can be reduced to either find a sound approximation of the solution space determined by a set of nonlinear inequalities, or to the “guaranteed tuning problem ” as defined by Jaulin and Walter, which amounts to finding a value for some tuning parame ..."
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Cited by 14 (2 self)
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Many problems in robust control and motion planning can be reduced to either find a sound approximation of the solution space determined by a set of nonlinear inequalities, or to the “guaranteed tuning problem ” as defined by Jaulin and Walter, which amounts to finding a value for some tuning parameter such that a set of inequalities be verified for all the possible values of some perturbation vector. A classical approach to solve these problems, which satisfies the strong soundness requirement, involves some quantifier elimination procedure such as Collins ’ Cylindrical Algebraic Decomposition symbolic method. Sound numerical methods using interval arithmetic and local consistency enforcement to prune the search space are presented in this paper as much faster alternatives for both soundly solving systems of nonlinear inequalities, and addressing the guaranteed tuning problem whenever the perturbation vector has dimension one. The use of these methods in camera control is investigated, and experiments with the prototype of a declarative modeller to express camera motion using a cinematic language are reported and commented.
Advisors for Incremental Propagation
 THIRTEENTH INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING
, 2007
"... While incremental propagation for global constraints is recognized to be important, little research has been devoted to how propagatorcentered constraint programming systems should support incremental propagation. This paper introduces advisors as a simple and efficient, yet widely applicable metho ..."
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Cited by 14 (2 self)
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While incremental propagation for global constraints is recognized to be important, little research has been devoted to how propagatorcentered constraint programming systems should support incremental propagation. This paper introduces advisors as a simple and efficient, yet widely applicable method for supporting incremental propagation in a propagatorcentered setting. The paper presents how advisors can be used for achieving different forms of incrementality and evaluates cost and benefit for several global constraints.
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 13 (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...