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28
Efficient solving of quantified inequality constraints over the real numbers
 ACM Transactions on Computational Logic
"... Let a quantified inequality constraint over the reals be a formula in the firstorder predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In the ..."
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Cited by 25 (7 self)
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Let a quantified inequality constraint over the reals be a formula in the firstorder predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In the paper we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques. 1
Inner and outer approximations of existentially quantified equality constraints
 In Proceedings of the Twelfth International Conference on Principles and Practice of Constraint Programming, (CP 2006
, 2006
"... Abstract. We propose a branch and prune algorithm that is able to compute inner and outer approximations of the solution set of an existentially quantified constraint where existential parameters are shared between several equations. While other techniques that handle such constraints need some prel ..."
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Cited by 9 (5 self)
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Abstract. We propose a branch and prune algorithm that is able to compute inner and outer approximations of the solution set of an existentially quantified constraint where existential parameters are shared between several equations. While other techniques that handle such constraints need some preliminary formal simplification of the problem or only work on simpler special cases, our algorithm is the first pure numerical algorithm that can approximate the solution set of such constraints in the general case. Hence this new algorithm allows computing inner approximations that were out of reach until today. 1
Solving interval constraints by linearization in computeraided design. Reliable Computing
, 2006
"... Abstract. Current parametric CAD systems require geometric parameters to have fixed values. Specifying fixed parameter values implicitly adds rigid constraints on the geometry, which have the potential to introduce conflicts during the design process. This paper presents a soft constraint representa ..."
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Cited by 6 (5 self)
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Abstract. Current parametric CAD systems require geometric parameters to have fixed values. Specifying fixed parameter values implicitly adds rigid constraints on the geometry, which have the potential to introduce conflicts during the design process. This paper presents a soft constraint representation scheme based on nominal interval. Interval geometric parameters capture inexactness of conceptual and embodiment design, uncertainty in detail design, as well as boundary information for design optimization. To accommodate underconstrained and overconstrained design problems, a doubleloop GaussSeidel method is developed to solve linear constraints. A symbolic preconditioning procedure transforms nonlinear equations to separable form. Inequalities are also transformed and integrated with equalities. Nonlinear constraints can be bounded by piecewise linear enclosures and solved by linear methods iteratively. A sensitivity analysis method that differentiates active and inactive constraints is presented for design refinement. 1.
Imprecise probabilities with a generalized interval form
 Proc. 3rd Int. Workshop on Reliability Engineering Computing (REC'08
, 2008
"... Abstract. Di erent representations of imprecise probabilities have been proposed, such as behavioral theory, evidence theory, possibility theory, probability bound analysis, Fprobabilities, fuzzy probabilities, and clouds. These methods use intervalvalued parameters to discribe probability distrib ..."
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Cited by 4 (3 self)
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Abstract. Di erent representations of imprecise probabilities have been proposed, such as behavioral theory, evidence theory, possibility theory, probability bound analysis, Fprobabilities, fuzzy probabilities, and clouds. These methods use intervalvalued parameters to discribe probability distributions such that uncertainty is distinguished from variability. In this paper, we proposed a new form of imprecise probabilities based on generalized or modal intervals. Generalized intervals are algebraically closed under Kaucher arithmetic, which provides a concise representation and calculus structure as an extension of precise probabilities. With the separation between proper and improper interval probabilities, focal and nonfocal events are di erentiated based on the modalities and logical semantics of generalized interval probabilities. Focal events have the semantics of critical, uncontrollable, speci ed, etc. in probabilistic analysis, whereas the corresponding nonfocal events are complementary, controllable, and derived. A generalized imprecise conditional probability is de ned based on unconditional interval probabilities such that the algebraic relation between conditional and marginal interval probabilities is maintained. A Bayes ' rule with generalized intervals (GIBR) is also proposed. The GIBR allows us to interpret the logic relationship between interval prior and posterior probabilities.
Quantified Set Inversion Algorithm with Applications to Control
"... Abstract. In this paper, a new algorithm based on Set Inversion techniques and Modal Interval Analysis is presented. This algorithm allows solving problems involving quantified constraints over the reals through the characterization of their solution sets. The presented methodology can be applied ov ..."
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Cited by 4 (4 self)
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Abstract. In this paper, a new algorithm based on Set Inversion techniques and Modal Interval Analysis is presented. This algorithm allows solving problems involving quantified constraints over the reals through the characterization of their solution sets. The presented methodology can be applied over a wide range of problems involving uncertain (non)linear systems. Finally, an advanced application is solved.
A Generalized Interval LU Decomposition for the Solution of Interval Linear Systems
, 2007
"... Generalized intervals (intervals whose bounds are not constrained to be increasingly ordered) extend classical intervals providing better algebraic properties. In particular, the generalized interval arithmetic is a group for addition and for multiplication of zero free intervals. These properties ..."
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Cited by 3 (0 self)
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Generalized intervals (intervals whose bounds are not constrained to be increasingly ordered) extend classical intervals providing better algebraic properties. In particular, the generalized interval arithmetic is a group for addition and for multiplication of zero free intervals. These properties allow one constructing a LU decomposition of a generalized interval matrix A: the two computed generalized interval matrices L and U satisfy A = LU with equality instead of the weaker inclusion obtained in the context of classical intervals. Some potential applications of this generalized interval LU decomposition are investigated.
Solving Existentially Quantified Constraints with One Equality and Arbitrarily Many Inequalities
 In Proc. of CP’03, LNCS 2833
, 2003
"... This paper contains the first algorithm that can solve disjunctions of constraints of the form g1 . . . ..."
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Cited by 3 (0 self)
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This paper contains the first algorithm that can solve disjunctions of constraints of the form g1 . . .
INNER APPROXIMATION OF THE RANGE OF VECTORVALUED FUNCTIONS
"... Abstract. No method for the computation of a reliable subset of the range of vectorvalued functions is available today. A method for computing such inner approximations is proposed in the specific case where both domain and codomain have the same dimension. A general sufficient condition for the i ..."
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Cited by 3 (1 self)
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Abstract. No method for the computation of a reliable subset of the range of vectorvalued functions is available today. A method for computing such inner approximations is proposed in the specific case where both domain and codomain have the same dimension. A general sufficient condition for the inclusion of a box inside the image of a box by a continuously differentiable vectorvalued is first provided. This sufficient condition is specialized to a more efficient one, which is used in a specific bisection algorithm that computes reliable inner and outer approximations of the image of a domain defined by constraints. Some experimentations are presented.
Semantic Tolerancing with Generalized Intervals
"... A new tolerance modeling scheme, semantic tolerance modeling, was recently developed to enable interpretable tolerance analysis. In this paper, a new dimensioning and tolerancing practice, semantic tolerancing, is proposed with the theoretical support of semantic tolerance models. Following principl ..."
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Cited by 2 (1 self)
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A new tolerance modeling scheme, semantic tolerance modeling, was recently developed to enable interpretable tolerance analysis. In this paper, a new dimensioning and tolerancing practice, semantic tolerancing, is proposed with the theoretical support of semantic tolerance models. Following principles of interpretability, this new tolerancing approach captures more design intent, including flexible material selection, component sorting in selective assembly, rigidity of constraints, and assembly sequence.
Generating and Applying Rules for Interval Valued Fuzzy Observations
 Lecture Notes in Computer Science
"... Abstract. One of the objectives of intelligent data engineering and automated learning is to develop algorithms that learn the environment, generate rules, and take possible courses of actions. In this paper, we report our work on how to generate and apply such rules with a rule matrix model. Since ..."
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Cited by 2 (1 self)
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Abstract. One of the objectives of intelligent data engineering and automated learning is to develop algorithms that learn the environment, generate rules, and take possible courses of actions. In this paper, we report our work on how to generate and apply such rules with a rule matrix model. Since the environments can be interval valued and rules often fuzzy, we further study how to obtain and apply rules for interval valued fuzzy observations. 1