## Quantifier Elimination versus Generalized Interval Evaluation: A comparison on a Special Class of Quantified Constraints

Venue: | in "Proc. of the 11th Information Processing and Management of Uncertainty International Conference, IPMU 2006 |

Citations: | 2 - 0 self |

### BibTeX

@INPROCEEDINGS{Grandón_quantifierelimination,

author = {Carlos Grandón and Projet Coprin},

title = {Quantifier Elimination versus Generalized Interval Evaluation: A comparison on a Special Class of Quantified Constraints},

booktitle = {in "Proc. of the 11th Information Processing and Management of Uncertainty International Conference, IPMU 2006},

year = {},

pages = {786--793}

}

### OpenURL

### Abstract

This paper presents and compares two methods for checking if a box is included inside the solution set of an equality constraint with existential quantification of its parameters. We focus on distance constraints, where each existentially quantified parameter has only one occurrence, because of their usefulness and their simplicity. The first method relies on a specific quantifier elimination based on geometric considerations whereas the second method relies on computations with generalized intervals— interval whose bounds are not constrained to be ordered. We show that on two dimension problems, the two methods yield equivalent results. However, when dealing with higher dimensions, generalized intervals are more efficient.

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Citation Context ..., generalized intervals, continuous domains. 1 Introduction The interval theory ([13, 6]) was born in the 60’s aiming rigorous computations with uncertain quantities. Interval constraint propagation (=-=[1, 2]-=-) is a widely used technique that allows one to reduce the domains of variables involved in a numerical constraint without losing any solution. When this technique Alexandre Goldsztejn University of N... |

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Citation Context ..., the distance constraint fixes the distance between a and x to be equal to r. The approximation of such constraints can be useful in many contexts, e.g. GPS localization or parallel robots modeling (=-=[15, 12]-=-). We propose and compare two different methods for checking if a box is included inside the solution set of a distance equation with existentially quantified parameters. On one hand, the quantified d... |

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Citation Context ... is changed to an equivalent non quantified disjunction/conjunction of constraints which can be checked using interval arithmetic. On the other hand, the Kaucher arithmetic of generalized intervals (=-=[9, 4]-=-), which represents a new formulation of the modal intervals theory ([14, 7]), allows one to verify the inclusion through a generalized interval evaluation of the constraint. These two tests for inner... |

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Citation Context ...tervals are more efficient. Keywords: Inner approximation, distance constraint, AE-solution set, quantifier elimination, generalized intervals, continuous domains. 1 Introduction The interval theory (=-=[13, 6]-=-) was born in the 60’s aiming rigorous computations with uncertain quantities. Interval constraint propagation ([1, 2]) is a widely used technique that allows one to reduce the domains of variables in... |

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Citation Context ...he constraint. These two tests for inner boxes are implemented in a branch and prune algorithm and experiments have been carried out on academic examples in order to compare them. Notations Following =-=[10]-=-, intervals are denoted by boldface letters. Integral intervals are denoted by [m..n]. Let E = {e1, ..., en} be an ordered set of indices, the vector (xe1 , ..., xen) is denoted by xE, so that (x1, . ... |

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Citation Context ...he speedup of computations, such inner boxes often have interesting interpretations. There are different situations where the solution set of a CSP has a non-null volume, e.g. inequality constraints (=-=[11]-=-) or constraints with existentially quantified parameters, e.g. a constraint on variable x ∈ � like (∃a ∈ a) (c(a, x)) where a is an interval ([7]). In this paper, we focus on quantified distance cons... |

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Citation Context ... is changed to an equivalent non quantified disjunction/conjunction of constraints which can be checked using interval arithmetic. On the other hand, the Kaucher arithmetic of generalized intervals (=-=[9, 4]-=-), which represents a new formulation of the modal intervals theory ([14, 7]), allows one to verify the inclusion through a generalized interval evaluation of the constraint. These two tests for inner... |

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Citation Context ..., the distance constraint fixes the distance between a and x to be equal to r. The approximation of such constraints can be useful in many contexts, e.g. GPS localization or parallel robots modeling (=-=[15, 12]-=-). We propose and compare two different methods for checking if a box is included inside the solution set of a distance equation with existentially quantified parameters. On one hand, the quantified d... |

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Citation Context ...[3, 3.5], 0) r (2) = [2.95, 3.05] a (3) = ([−2.5, −2.25], 2) r (3) = [3.25, 3.5] P3 (3D, three equations) x = ([0, 100], [−100, 100], [0, 100]) a (1) = ([−0.1, 0.1], [−0.1, 0.1], [−0.1, 0.1]) r (1) = =-=[4, 5]-=- a (2) = ([4.9, 5.1], [−0.1, 0.1], [−0.1, 0.1]) r (2) = [3, 4] a (3) = ([1.8, 2.2], [3.95, 4.05], [0.8, 1.2]) r (3) = [4, 5] A branch and prune algorithm combining 2Bconsistency and bisection techniqu... |

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Citation Context ... has a non-null volume, e.g. inequality constraints ([11]) or constraints with existentially quantified parameters, e.g. a constraint on variable x ∈ � like (∃a ∈ a) (c(a, x)) where a is an interval (=-=[7]-=-). In this paper, we focus on quantified distance constraints where the variables are the coordinates of a point x ∈ � n . As existentially quantified parameters, we have the coordinates of another po... |

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Citation Context ...straints which can be checked using interval arithmetic. On the other hand, the Kaucher arithmetic of generalized intervals ([9, 4]), which represents a new formulation of the modal intervals theory (=-=[14, 7]-=-), allows one to verify the inclusion through a generalized interval evaluation of the constraint. These two tests for inner boxes are implemented in a branch and prune algorithm and experiments have ... |