## Universally Quantified Interval Constraints (2000)

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Venue: | PROCEEDINGS OF THE 6TH INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING |

Citations: | 52 - 0 self |

### BibTeX

@INPROCEEDINGS{Benhamou00universallyquantified,

author = {Frédéric Benhamou and Frédéric Goualard},

title = {Universally Quantified Interval Constraints},

booktitle = {PROCEEDINGS OF THE 6TH INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING},

year = {2000},

pages = {67--82},

publisher = {Springer Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

Non-linear 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 quantifier-free 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 inner-approximation of real relations.

### Citations

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Citation Context ... shown to be e#cient tools for solving some challenging non-linear constraint systems in various application areas (e.g. robotics, chemistry [16], or electronics [19]). Relying on interval arithmetic =-=[17], these to-=-ols guarantee completeness (all solutions in the input are retained), and permit isolating punctual solutions with an "arbitrary" accuracy. Taking as input a constraint system and a Cartesia... |

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Citation Context ... 4. INTERVAL CONSTRAINT SOLVING Finite representation of numbers by computers prevents them solving accurately real constraints. Interval constraint solving relies on interval arithmetic [Moore 1966; =-=Alefeld and Herzberger 1983-=-] to compute verified approximate solutions of real constraint systems. Underlying real relations may be approximated by considering one of their computer-representable superset or subset. This sectio... |

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Citation Context ...m out. A variation of the camera’s focal length. To our knowledge, existing declarative camera movement generators compute camera animation frame by frame [Drucker 1994] or use calculated key frames [=-=Shoemake 1985-=-] (fixed camera location and orientation) and interpolation of the camera parameters in-between.Universally Quantified Interval Constraint Solving · 7 Horizontal panoramic Travelling (tracking shot o... |

397 |
Quantifier elimination for real closed fields by cylindrical algebraic decomposition
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Citation Context ... constraint systems with universally/existentially quantified variables have mainly be handled by symbolic methods, among which one may single out Cylindrical Algebraic Decomposition (CAD) by Collins =-=[11]-=-. CAD is quite a powerful method since it permits handling more than one quantified variable for disjunctions/conjunctions of constraints. However, it has strong requirements on the form of the constr... |

181 | Numerica: A Modeling Language for Global Optimization
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Citation Context ...camera motion using a cinematic language is sketched, and results from a prototype are presented. 1 Introduction Interval constraint-based solvers such as clp(BNR) [8], ILOG Solver [18], and Numericas=-=[24]-=- have been shown to be e#cient tools for solving some challenging non-linear constraint systems in various application areas (e.g. robotics, chemistry [16], or electronics [19]). Relying on interval a... |

177 | Applying interval arithmetic to real, integer and Boolean constraints
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Citation Context ...declarative modeller for expressing camera motion using a cinematic language is sketched, and results from a prototype are presented. 1 Introduction Interval constraint-based solvers such as clp(BNR) =-=[8]-=-, ILOG Solver [18], and Numericas[24] have been shown to be e#cient tools for solving some challenging non-linear constraint systems in various application areas (e.g. robotics, chemistry [16], or ele... |

139 | A C++ Implementation of CLP
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Citation Context ...ler for expressing camera motion using a cinematic language is sketched, and results from a prototype are presented. 1 Introduction Interval constraint-based solvers such as clp(BNR) [8], ILOG Solver =-=[18]-=-, and Numericas[24] have been shown to be e#cient tools for solving some challenging non-linear constraint systems in various application areas (e.g. robotics, chemistry [16], or electronics [19]). Re... |

139 | Interval analysis for computer graphics
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Citation Context ...er-defined constraints while manipulating camera parameters. Camera motion in an animated scene is not treated. Yet another approach [Jardillier and Languénou 1998]—inspired by Snyder’s seminal work [=-=Snyder 1992-=-]—to the camera motion computation problem relies on interval arithmetic [Moore 1966] to take into account multiple constraints and screen-space properties. Satisfying camera movement parameters—with ... |

134 | Through-the-lens camera control
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Citation Context ...the camera parameters through the use of sliders), thereby impeding an out-of-the-field user from predicting the exact behaviour of the controlled device. On the other hand, some authors [Blinn 1988; =-=Gleicher and Witkin 1992-=-; Drucker et al. 1992] from computer graphics and cinematography fields worked on determining camera parameters from given properties of a desired scene. Once more, most of these works are concerned w... |

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Citation Context ...hm whose output is a set of sound boxes of variable domains for some constraint system. Soundness is achieved by computingsinner approximations of the underlying real relations, using box consistencys=-=[7]-=----a well-known, e#cient, local consistency---on the negation of the involved constraints. An algorithm is then applied to the solving of constraint systems where one variable is universally quantifie... |

127 |
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Citation Context ...ed by our tool though). Due to lack of space, the following description of camera movements (see Figure 2) is restricted to primitive ones, and the reader is referred to the excellent book by Arijon [=-=Arijon 1976-=-] for a thorough presentation of the others. Panoramic shot. A panoramic shot may be horizontal or vertical (i.e. around vertical or horizontal axis). The camera location is usually constant; Travelli... |

106 | D.: Solving polynomial systems using a branch and prune approach - Hentenryck, McAllister, et al. - 1997 |

95 |
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Citation Context ... (brackets) are introduced: let L = {(, [} and U = {), ]} be respectively the set of left and right brackets. Let B = L # U be the set of brackets totally ordered by the ordering # defined as follows =-=[9]-=-: ) # [#] # (. The set of floating-point bounds F # is defined from B and F as follows: F # = F # # F # where # F # = (FsL # {# -#, ( #, # +#, ( #}) F # = (FsU # {# -#, ) #, # +#, ) #}) Real bounds se... |

86 |
The complexity of linear problems in fields
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Citation Context ...mposition (CAD) by Collins [Collins 1975], Cylindrical Trigonometric Decomposition (CTD) by Pau and Schicho [Pau and Schicho 1999], and virtual substitution of parametric test points by Weispfenning [=-=Weispfenning 1988-=-; Dolzmann et al. 1997]. Some numerical methods based on a bisection scheme also exist that solve constraint systems with universally quantified variables by computing an inner approximation of the un... |

85 |
Cinematic Primitives for Multimedia
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(Show Context)
Citation Context ...omputer graphics (visualization in virtual environments [Blinn 1988]) to Robotics (sensor planning [Abrams andUniversally Quantified Interval Constraint Solving · 5 Allen 1997]), and cinematography [=-=Davenport et al. 1991-=-]. Whatever the activity may be, the objective is always to provide the user with an adequate view of some point(s) of interest in a scene for a predefined duration. In sensor planning [Tarabanis 1990... |

82 | Revising hull and box consistency
- Benhamou, Goualard, et al.
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(Show Context)
Citation Context ..., box consistency is first formally presented. Therefrom, an operator based on it is given. The definition for box consistency given below is an instance of the extended one stated by Benhamou et al. =-=[5]-=- that slightly di#ers from the original definition [7] in that it is parameterized by approximation operators. Definition 1 (Box consistency [5]) Let c be a real constraint, C an interval extension fo... |

73 |
Where am i? what am i looking at
- Blinn
- 1988
(Show Context)
Citation Context ...orithm will be presented in Section 5. 3. CAMERA CONTROL AND THE VIRTUAL CAMERAMAN PROBLEM Camera control is of interest to many fields, from computer graphics (visualization in virtual environments [=-=Blinn 1988-=-]) to Robotics (sensor planning [Abrams andUniversally Quantified Interval Constraint Solving · 5 Allen 1997]), and cinematography [Davenport et al. 1991]. Whatever the activity may be, the objective... |

60 |
Interval analysis in the extended interval space
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(Show Context)
Citation Context ...enceforth reducing the number of variables to consider. Following the work of Markov [Markov 1995] and Shary [Shary 1995], a direction for future research is to compare the use of Kaucher arithmetic [=-=Kaucher 1980-=-] to compute inner approximations of relations with the use of outer-contracting operators. Yet, this approach requires algebraizing trigonometric constraints, an operation known to slow down computat... |

58 | Consistency Techniques for Continuous Constraints
- Sam-Haroud, Faltings
- 1996
(Show Context)
Citation Context ...rifying: #j # {1, . . . , p} : eval {c1 ##cm} (I 1 I n I j v ) = true (2) In this paper, we will call this algorithm EIA4. This process is, in some way, related to the work of Sam-Haroud and Faltings =-=[21]-=- where true, false, or unknown boxes are organized into 2 k -trees to ease the computation of global consistencies. Equation (2) implies that each retained box is included in the inner-approximation o... |

50 |
Output feedback stabilization and related problems-solutions via decision methods
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(Show Context)
Citation Context ...at least one solution in each box. The foregoing solvers ensure only, at best, the second property. Furthermore, problems originating from camera control [12], sensor plannings[1], and control design =-=[2]-=-, not only require the output boxes to contain only solution points, but also that some input variables be universally quantified. To date, constraint systems with universally/existentially quantified... |

50 | Interval Constraint Logic Programming
- Benhamou
- 1995
(Show Context)
Citation Context ...nt systems solved. The reader is referred to the above-mentionned references for a thorough presentation of interval arithmetic. A great part of what is exposed in the following is drawn from [8] and =-=[4]-=-. Proofs not given here may be found in these papers. 2.1 Approximation of a relation Solving real constraints requires the ability to represent the underlying relations. The approximate representatio... |

43 | Cinema: a system for procedural camera movements
- Drucker, Galyean, et al.
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(Show Context)
Citation Context ...ugh the use of sliders), thereby impeding an out-of-the-field user from predicting the exact behaviour of the controlled device. On the other hand, some authors [Blinn 1988; Gleicher and Witkin 1992; =-=Drucker et al. 1992-=-] from computer graphics and cinematography fields worked on determining camera parameters from given properties of a desired scene. Once more, most of these works are concerned with the computation o... |

41 |
An improvement of the projection operator in cylindrical algebraic decomposition
- Hong
- 1998
(Show Context)
Citation Context .... Christie As far as the authors know, one may distinguish two methods for solving Ex. 1: —quantifier elimination by Cylindrical Algebraic Decomposition (CAD), due to Collins [Collins 1975] and Hong [=-=Hong 1990-=-], or Cylindrical Trigonometric Decomposition [Pau and Schicho 1999]; —evaluation and bisection, for which we will only consider the works by Garloff and Graf [Garloff and Graf 1999] based on Bernstei... |

38 |
On the numerical condition of polynomials in Bernstein form. Computer Aided Geometric Design
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(Show Context)
Citation Context ...one can express p in terms of Bernstein coefficients: p(x) = ∑ bI(D)BN,I(x) I∈S where BN,I(x) is the Ith Bernstein polynomial of degree N. It is then well known that the following property does hold [=-=Farouki and Rajan 1987-=-]: ∀a ∈ D: min I∈S bI(D) � p(a) � max I∈S bI(D) (convex hull property) As a consequence, it is possible to accurately bound the range of p(x) on every box, and then to know whether an inequality such ... |

34 | Real quantifier elimination in practice
- Dolzmann, Sturm, et al.
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(Show Context)
Citation Context ...Collins [Collins 1975], Cylindrical Trigonometric Decomposition (CTD) by Pau and Schicho [Pau and Schicho 1999], and virtual substitution of parametric test points by Weispfenning [Weispfenning 1988; =-=Dolzmann et al. 1997-=-]. Some numerical methods based on a bisection scheme also exist that solve constraint systems with universally quantified variables by computing an inner approximation of the underlying relation [Gar... |

31 |
IEEE standard for binary floating-point arithmetic
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(Show Context)
Citation Context ...rounding: ⌈↾↿⌉ : R ⊲ −→ F ⊲ β ↦−→ min{γ ∈ F ⊲ | γ � β} For each g ∈ F ∞ , let g + be the smallest element in F ∞ greater than g, and g − the greatest element in F ∞ smaller than g (with the IEEE 754 [=-=IEEE 1985-=-] conventions: (+∞) + = +∞, (−∞) − = −∞, (+∞) − = max(F), (−∞) + = min(F). Bounds are used to construct intervals as follows: let I◦ = F ⊳ × F ⊲ be the set of closed/open floating-point intervals (hen... |

30 | Intelligent Camera Control in Graphical Environments
- Drucker
- 1994
(Show Context)
Citation Context ...ent in any box is a solution, or there exists at least one solution in each box. The foregoing solvers ensure only, at best, the second property. Furthermore, problems originating from camera control =-=[12]-=-, sensor plannings[1], and control design [2], not only require the output boxes to contain only solution points, but also that some input variables be universally quantified. To date, constraint syst... |

26 |
and Eric Languènou. Screen-space constraints for camera movements: the virtual cameraman
- Jardillier
- 1998
(Show Context)
Citation Context ... then applied to the solving of constraint systems where one variable is universally quantified. An application to temporal constraints arising from camera motion modelling (virtual cameraman problems=-=[14]-=-) is sketched: following the work of Jardillier and Languenou, a prototype for a declarative modeller has been devised, which should eventually allow a non-technician user to control the positioning o... |

24 | Application of quantifier elimination theory to control system design
- Abdallah, Dorato, et al.
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(Show Context)
Citation Context ...the sound output domain to have some precise bounds. Such a problem is presented in the following, and some of the methods used to solve it are briefly surveyed. Example 1 (A feedback control system [=-=Abdallah et al. 1995-=-]). Consider a temperature regulating system in a room modelled as an unstable first order system whose transfer function is G(s, p) = p1/(1 −s/p2), with 0.8 � p1, p2 � 1.25, and simple output feedbac... |

23 | P.: A constraint satisfaction approach to a circuit design problem
- Puget, Hentenryck
- 1998
(Show Context)
Citation Context ...lver [18], and Numericas[24] have been shown to be e#cient tools for solving some challenging non-linear constraint systems in various application areas (e.g. robotics, chemistry [16], or electronics =-=[19]). Relying-=- on interval arithmetic [17], these tools guarantee completeness (all solutions in the input are retained), and permit isolating punctual solutions with an "arbitrary" accuracy. Taking as in... |

21 |
Chemical Equilibrium Systems as Numerical Test Problems
- Meintjes, Morgan
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(Show Context)
Citation Context ...clp(BNR) [8], ILOG Solver [18], and Numericas[24] have been shown to be e#cient tools for solving some challenging non-linear constraint systems in various application areas (e.g. robotics, chemistry =-=[16], or elect-=-ronics [19]). Relying on interval arithmetic [17], these tools guarantee completeness (all solutions in the input are retained), and permit isolating punctual solutions with an "arbitrary" a... |

20 | Extending consistent domains of numeric CSP
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(Show Context)
Citation Context ...Nevertheless, a comforting idea is that most of the traditional camera movements involve but few of the degrees of freedom, thenceforth reducing the number of variables to consider. Collavizza et al. =-=[10] devised a-=- scheme for computing inner-approximations of the relation underlying a real constraint system: starting from a "seed" known to be included in the relation, they expand the domain of the var... |

20 | Extending the Constraint Propagation of Intervals
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- 1990
(Show Context)
Citation Context ...irement. Consider, for instance, a civil engineering problem [20] such as floor design where retaining non-solution points may lead to a physically unfeasible structure. As pointed out by Ward et al. =-=[25]-=- and Shary [23], one may expect di#erent properties from the boxes composing S o depending on the problem at hand, namely: every element in any box is a solution, or there exists at least one solution... |

20 | Solving strict polynomial inequalities by Bernstein expansion’. The Use
- Garloff, Graf
- 1999
(Show Context)
Citation Context ...997]. Some numerical methods based on a bisection scheme also exist that solve constraint systems with universally quantified variables by computing an inner approximation of the underlying relation [=-=Garloff and Graf 1999-=-; Kutsia and Schicho 1999; Sam-Haroud and Faltings 1996]. However, all these methods have strong requirements on the form of the constraints they process: for the most part, they are limited to polyno... |

18 | ªComputing Camera Viewpoints in an Active Robot Workcell,º Int'l
- Abrams, Allen, et al.
- 1999
(Show Context)
Citation Context ...lution, or there exists at least one solution in each box. The foregoing solvers ensure only, at best, the second property. Furthermore, problems originating from camera control [12], sensor plannings=-=[1]-=-, and control design [2], not only require the output boxes to contain only solution points, but also that some input variables be universally quantified. To date, constraint systems with universally/... |

16 | On Directed Interval Arithmetic and its Applications
- Markov
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(Show Context)
Citation Context ...t least the two following definitions for an inner approximation Inner(ρ) of ρ: (1) Def. A. Inner(ρ) = B1, where B1 ∈ {B ∈ I n ◦ | B ⊆ ρ} (an inner-approximation is any box included in the relation) [=-=Markov 1995-=-; Armengol et al. 1998]; (2) Def. B. Inner(ρ) = B1, where B1 ∈ {B = I1 × · · · × In | B ⊆ ρ ∧ ∀j ∈ {1, . . ., n}, ∀I ′ j ⊇ Ij : B|Ij,I ′ j ⊆ ρ ⇒ I′ j = Ij} (an inner-approximation is a box included in... |

8 |
Interval Gauss-Seidel method for generalized solution sets to interval linear systems. Reliable computing
- Shary
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(Show Context)
Citation Context ...er, for instance, a civil engineering problem [20] such as floor design where retaining non-solution points may lead to a physically unfeasible structure. As pointed out by Ward et al. [25] and Shary =-=[23]-=-, one may expect di#erent properties from the boxes composing S o depending on the problem at hand, namely: every element in any box is a solution, or there exists at least one solution in each box. T... |

7 |
Constraint Consistency Techniques for Continuous Domains", thèse de l’Ecole Polytechnique Fédérale de Lausanne
- Sam
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(Show Context)
Citation Context ...o of boxes approximating each solution contained in the input box. However, soundness is not guaranteed while it is sometimes a strong requirement. Consider, for instance, a civil engineering problem =-=[20]-=- such as floor design where retaining non-solution points may lead to a physically unfeasible structure. As pointed out by Ward et al. [25] and Shary [23], one may expect di#erent properties from the ... |

4 |
Quantifier elimination for trigonometric polynomials by cylindrical trigonometric decomposition
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- 1999
(Show Context)
Citation Context ...been handled by symbolic methods, among which one may single out Cylindrical Algebraic Decomposition (CAD) by Collins [Collins 1975], Cylindrical Trigonometric Decomposition (CTD) by Pau and Schicho [=-=Pau and Schicho 1999-=-], and virtual substitution of parametric test points by Weispfenning [Weispfenning 1988; Dolzmann et al. 1997]. Some numerical methods based on a bisection scheme also exist that solve constraint sys... |

4 |
On maximal inner estimation of the solution sets of linear systems with interval parameters
- Sharaya
(Show Context)
Citation Context ...relation, they expand the domain of the variables as much as possible until obtaining a “maximal” subset of the innerapproximation (with maximality to be understood in the sense of Shary [Shary 1995; =-=Sharaya 1998-=-]). A drawback of their method lies in that they do not provide any means to compute the seed. An interesting direction for research would be to try using our algorithm to quickly isolate such a seed ... |

3 |
Modal interval analysis for error-bounded semiqualitative simulation
- Armangol, Travé-Massuyès, et al.
- 1998
(Show Context)
Citation Context ...t least the two following definitions for an inner approximation Inner(#) of #: 1. Def. A. Inner(#) = B 1 , where B 1 # {B # I n # | B # #} (an innerapproximation is any box included in the relation) =-=[15, 3]-=-; 2. Def. B. Inner(#) = B 1 , where B 1 # {B = I 1 I n | B # # # #j # {1, . . . , n}, #I # j # I j : B| I j ,I # j # # # I # j = I j } (an inner-approximation is a box included in the relation that ca... |

3 |
Algebraic solutions to interval linear equations and their applications
- Shary
- 1995
(Show Context)
Citation Context ...uded in the relation, they expand the domain of the variables as much as possible until obtaining a "maximal" subset of the inner-approximation (with maximality to be understood in the sense=-= of Shary [22]-=-). A drawback of their method lies in that they do not provide any means to compute the seed. An interesting direction for research would be to try using our algorithm to quickly isolate such a seed f... |

3 |
Automated sensor planning and modeling for robotic vision tasks
- Tarabanis
- 1990
(Show Context)
Citation Context ...rt et al. 1991]. Whatever the activity may be, the objective is always to provide the user with an adequate view of some point(s) of interest in a scene for a predefined duration. In sensor planning [=-=Tarabanis 1990-=-] for instance, one is interested in positioning a camera over a robot in order to be able to watch its work whatever the position of its manipulating “hand” may be. The targeted applications are most... |

2 | Universally quantified constraint solving: an application to camera control
- Benhamou, Goualard, et al.
- 2000
(Show Context)
Citation Context ...2.2 for the negation of the involved constraints is then described, and compared to the one of Jardillier and Languenou. Due to lack of space, the reader is referred to the associated research report =-=[6]-=- for the proofs of the propositions to be stated below. 3.1 Computing inner sets From Def. C above, an inner-approximation operator may be defined as follows: Definition 3 (Inner approximation operato... |

1 | Collision problems by an improved CAD-based quantifier elimination algorithm
- Hong
- 1991
(Show Context)
Citation Context ...ng # and # to the same value leads to nearly the same solution sets for both algorithms EIA4 and ICAb5. 5 Conclusion Unlike the methods used to deal with universally quantified variables described in =-=[13], the algo-=-rithms presented in this paper are purely numerical ones (except for the negation of constraints). Since they rely on "traditional" techniques used by most of the interval constraint-based s... |

1 |
Interval constraint logic programming. In A. Podelski Ed., Constraint programming: basics and trends
- Benhamou
- 1995
(Show Context)
Citation Context ...l relations are approximated and interval constraint systems solved. The reader is referred to the above-mentionned references for a thorough presentation of interval arithmetic, and to [Cleary 1987; =-=Benhamou 1995-=-] for descriptions of interval constraints and interval-based solving methods. A great part of what � 010 · F. Benhamou, F. Goualard, É. Languénou, and M. Christie is exposed in the following is draw... |

1 |
Numerical solving of constraints of multivariate polynomial strict inequalities
- Kutsia, Schicho
- 1999
(Show Context)
Citation Context ...thods based on a bisection scheme also exist that solve constraint systems with universally quantified variables by computing an inner approximation of the underlying relation [Garloff and Graf 1999; =-=Kutsia and Schicho 1999-=-; Sam-Haroud and Faltings 1996]. However, all these methods have strong requirements on the form of the constraints they process: for the most part, they are limited to polynomial constraints (except ... |