## Revising Hull and Box Consistency (1999)

Venue: | INT. CONF. ON LOGIC PROGRAMMING |

Citations: | 76 - 13 self |

### BibTeX

@INPROCEEDINGS{Benhamou99revisinghull,

author = {Frédéric Benhamou and Frédéric Goualard and Laurent Granvilliers and Jean-François Puget},

title = {Revising Hull and Box Consistency},

booktitle = {INT. CONF. ON LOGIC PROGRAMMING},

year = {1999},

pages = {230--244},

publisher = {MIT press}

}

### Years of Citing Articles

### OpenURL

### Abstract

Most interval-based 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 Newton-based 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...

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Citation Context ...nd solving a particular constraint then lies in eliminating some of the values of these domains for which the constraint does not hold (inconsistency), using local consistency techniques and ltering [=-=9-=-]. In practice, enforced consistencies only approximate perfect local consistency since some solutions may be unrepresentable with oating-point numbers. Two worth mentioning approximate consistencies ... |

878 |
Interval analysis
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Citation Context ...precludes computers from exactly solving continuous problems. Interval constraint solvers such as Prolog IV, Numericas[14], and ILOG Solver [12], tackle this problem by relying on interval arithmetics=-=[10-=-] to compute veried approximations of the solutions to constraint systems. Domains are associated to every variable occurring in the problem, and solving a particular constraint then lies in eliminati... |

170 | Numerica : A Modeling Language for Global Optimization
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Citation Context ...decomposing constraints) and BC3. 1 Introduction Finite representation of numbers precludes computers from exactly solving continuous problems. Interval constraint solvers such as Prolog IV, Numericas=-=[14-=-], and ILOG Solver [12], tackle this problem by relying on interval arithmetics[10] to compute veried approximations of the solutions to constraint systems. Domains are associated to every variable oc... |

169 | Older: Applying Interval Arithmetic to Real, Integer and Bolean Constraints - Benhamou, W - 1995 |

123 | A C++ Implementation of CLP
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(Show Context)
Citation Context ...s) and BC3. 1 Introduction Finite representation of numbers precludes computers from exactly solving continuous problems. Interval constraint solvers such as Prolog IV, Numericas[14], and ILOG Solver =-=[12-=-], tackle this problem by relying on interval arithmetics[10] to compute veried approximations of the solutions to constraint systems. Domains are associated to every variable occurring in the problem... |

121 | Hentenryck. CLP(intervals) revisited
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(Show Context)
Citation Context ...proximate perfect local consistency since some solutions may be unrepresentable with oating-point numbers. Two worth mentioning approximate consistencies are hull consistencys[1] and box consistency [=-=2]-=-. Most interval constraint solvers are based on either one of them. Enforcing hull consistency usually requires decomposing the user's constraints into so-called primitive constraints [4]. A well know... |

89 |
Logical arithmetic
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(Show Context)
Citation Context ... consistency [2]. Most interval constraint solvers are based on either one of them. Enforcing hull consistency usually requires decomposing the user's constraints into so-called primitive constraints =-=[4-=-]. A well known drawback of this method is that the introduction of new variables induced by the decomposition hinders ecient domain tightening. On the other hand, the original algorithm enforcing box... |

67 | Constraint arithmetic on real intervals
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(Show Context)
Citation Context ...constraint system included in the input box; the algorithm is conuent: the output is independent of the reinvocation order of constraints. These properties are proved in the same way as for HC3 (see [=-=11]-=-) once HC4revise has been proved to be a constraint narrowing operator (see Prop. 1). Algorithm 1: HC4 algorithm HC4(in {c 1 , . . . , c m } ; inout B = I 1 I n ) begin S # {c 1 , . . . , c m } while ... |

47 | Interval constraint logic programming
- Benhamou
- 1994
(Show Context)
Citation Context ...ed consistencies only approximate perfect local consistency since some solutions may be unrepresentable with oating-point numbers. Two worth mentioning approximate consistencies are hull consistencys[=-=1]-=- and box consistency [2]. Most interval constraint solvers are based on either one of them. Enforcing hull consistency usually requires decomposing the user's constraints into so-called primitive cons... |

22 | Comparing partial consistencies
- Collavizza, Delobel, et al.
- 1999
(Show Context)
Citation Context ...ox consistency is given, that no longer solely relies on the natural interval extension of constraints and captures both the original denition of box consistency [2] and the one by Collavizza et al. [=-=5-=-]. A new algorithms(BC4) permitting to eciently enforce box consistency is then given. BC4 adapts the computation method to the number of occurrences of each variable in a constraint: domain narrowing... |

15 |
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Citation Context ...ccessful, the domain of each variable x i in the constraint is intersected with the one in x i .bwd . Note: Backward propagation in a term is quite similar to automatic dierentiation in reverse mode [=-=7]-=- for computing the partial derivatives of a real function: after the forward evaluation, the aim is either to evaluate the projection narrowing operators, or the partial derivatives for each node cont... |

10 |
An extension of the WAM for hybrid interval solvers
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Citation Context ...n the other hand, the original algorithm enforcing box consistency processes constraints without decomposing them but is not at best with constraints involving many variables with few occurrences. In =-=-=-[6], some of the authors have presented DecLIC, a CLP language allowing the programmer to choose the best tted consistency to use for each constraint of a system; however, deferring the choice of the ... |

9 |
Standard for binary #oating-point arithmetic
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Citation Context ...g are presented. Let R be the set of reals compactied with the innities {-#,+#} in the obvious way, and F # R a nite subset of reals corresponding to binary oating-point numbers in a given format [8]. Let F # be F#{-#,+#}. For every 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 conventions: (+#) + = +#, (-#) - = -#, (... |

5 |
Canonical extensions as common basis for interval constraints and interval arithmetic
- Emden
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(Show Context)
Citation Context ...y # z = #, x + # = t} with the addition of the new variable #. Formally, given a real n-ary constraint c and a box B, let # (k) c (B) be the k-th canonical extension of # c w.r.t. B dened as follows [=-=13]-=-: # (k) c (B) = {r k # R | #r 1 # I 1 , . . . , #r k-1 # I k-1 , #r k+1 # I k+1 , . . . , #r n # I n s.t. (r 1 , . . . , r n ) # # c } 2 HC3 is our own denomination for Algorithm Nar given in [2] and ... |