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Granular Computing: An Emerging Paradigm
, 2001
"... We provide an overview of Granular Computing a rapidly growing area of information processing aimed at the construction of intelligent systems. We highlight the main features of Granular Computing, elaborate on the underlying formalisms of information granulation and discuss ways of their developme ..."
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We provide an overview of Granular Computing a rapidly growing area of information processing aimed at the construction of intelligent systems. We highlight the main features of Granular Computing, elaborate on the underlying formalisms of information granulation and discuss ways of their development. We also discuss the concept of granular modeling and present the issues of communication between formal frameworks of Granular Computing. © 2007 World Academic Press, UK. All rights reserved.
Using Directed Acyclic Graphs to Coordinate Propagation and Search for Numerical Constraint Satisfaction Problems
 In Proceedings of the 16th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2004
, 2004
"... A. NEUMAIER [1] has given the fundamentals of interval analysis on directed acyclic graphs (DAGs) for global optimization and constraint propagation. We show in this paper how constraint propagation on DAGs can be made efficient and practical by: (i) working on partial DAG representations; and (ii) ..."
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Cited by 14 (5 self)
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A. NEUMAIER [1] has given the fundamentals of interval analysis on directed acyclic graphs (DAGs) for global optimization and constraint propagation. We show in this paper how constraint propagation on DAGs can be made efficient and practical by: (i) working on partial DAG representations; and (ii) enabling the flexible choice of the interval inclusion functions during propagation. We then propose a new simple algorithm which coordinates constraint propagation and exhaustive search for solving numerical constraint satisfaction problems. The experiments carried out on different problems show that the new approach outperforms previously available propagation techniques by an order of magnitude or more in speed, while being roughly the same quality w.r.t. enclosure properties. I.
Interval Computations and IntervalRelated Statistical Techniques: Tools for Estimating Uncertainty of the Results of Data Processing and Indirect Measurements
"... In many practical situations, we only know the upper bound ∆ on the (absolute value of the) measurement error ∆x, i.e., we only know that the measurement error is located on the interval [−∆, ∆]. The traditional engineering approach to such situations is to assume that ∆x is uniformly distributed on ..."
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Cited by 4 (1 self)
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In many practical situations, we only know the upper bound ∆ on the (absolute value of the) measurement error ∆x, i.e., we only know that the measurement error is located on the interval [−∆, ∆]. The traditional engineering approach to such situations is to assume that ∆x is uniformly distributed on [−∆, ∆], and to use the corresponding statistical techniques. In some situations, however, this approach underestimates the error of indirect measurements. It is therefore desirable to directly process this interval uncertainty. Such “interval computations” methods have been developed since the 1950s. In this chapter, we provide a brief overview of related algorithms, results, and remaining open problems.
Combining Multiple Inclusion Representations in Numerical Constraint Propagation
 Publications Three Representative Papers
"... Abstract — This paper proposes a novel generic scheme enabling the combination of multiple inclusion representations to propagate numerical constraints. The scheme allows bringing into the constraint propagation framework the strength of inclusion techniques coming from different areas such as inter ..."
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Cited by 4 (4 self)
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Abstract — This paper proposes a novel generic scheme enabling the combination of multiple inclusion representations to propagate numerical constraints. The scheme allows bringing into the constraint propagation framework the strength of inclusion techniques coming from different areas such as interval arithmetic, affine arithmetic and mathematical programming. The scheme is based on the DAG representation of the constraint system. This enables devising finegrained combination strategies involving any factorable constraint system. The paper presents several possible combination strategies for creating practical instances of the generic scheme. The experiments reported on a particular instance using interval constraint propagation, interval arithmetic, affine arithmetic and linear programming illustrate the flexibility and efficiency of the approach. I.
Modal Intervals Revisited Part 1: A Generalized Interval Natural Extension
 Reliable Computing
"... Modal interval theory is an extension of classical interval theory which provides richer interpretations (including in particular inner and outer approximations of the ranges of real functions). In spite of its promising potential, modal interval theory is not widely used today because of its origin ..."
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Modal interval theory is an extension of classical interval theory which provides richer interpretations (including in particular inner and outer approximations of the ranges of real functions). In spite of its promising potential, modal interval theory is not widely used today because of its original and complicated construction. The present paper proposes a new formulation of modal interval theory. New extensions of continuous real functions to generalized intervals (intervals whose bounds are not constrained to be ordered) are defined. They are called AEextensions. These AEextensions provide the same interpretations as the ones provided by modal interval theory, thus enhancing the interpretation of the classical interval extensions. The construction of AEextensions strictly follows the model of classical interval theory: starting from a generalization of the definition of the extensions to classical intervals, the minimal AEextensions of the elementary operations
Interval Computations as an Important Part of Granular Computing: An Introduction
 in Handbook of Granular Computing, Chapter 1
, 2008
"... This chapter provides a general introduction to interval computations, especially to interval computations as an important part of granular computing. ..."
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This chapter provides a general introduction to interval computations, especially to interval computations as an important part of granular computing.
and the art of dividing by zero
"... • intervals replace real numbers. Why use validated numerics? • provides rigorous error bounds; • models uncertainty; • may produce faster numerical methods. ..."
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• intervals replace real numbers. Why use validated numerics? • provides rigorous error bounds; • models uncertainty; • may produce faster numerical methods.
COMPUTING SCIENCE A LUCID INTERVAL
"... Give a digital computer a problem in arithmetic, and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer. The cause of this sorry situation is not that software is full of bugs—although that is very likely true as well— nor is it that hardwa ..."
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Give a digital computer a problem in arithmetic, and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer. The cause of this sorry situation is not that software is full of bugs—although that is very likely true as well— nor is it that hardware is unreliable. The problem is simply that computers are discrete and finite machines, and they cannot cope with some of the continuous and infinite aspects of mathematics. Even an innocentlooking number like 1 ⁄10 can cause no end of trouble: In most cases, the computer cannot even read it in or print it out exactly, much less perform exact calculations with it. Errors caused by these limitations of digital machines are usually small and inconsequential, but sometimes every bit counts. On February 25, 1991, a Patriot missile battery assigned to protect a military installation at Dahrahn, Saudi Arabia, failed to intercept a Scud missile, and the malfunction was blamed on an error in computer arithmetic. The Patriot’s control system kept track of time by counting tenths of a second; to convert the count into full seconds, the computer multiplied by 1 ⁄10. Mathematically, the procedure is unassailable, but computationally it was disastrous. Because the decimal fraction 1 ⁄10 has no exact finite representation in binary notation, the computer had to approximate. Apparently, the conversion constant stored in the program was the 24bit binary fraction 0.00011001100110011001100, which is too small by a factor of about one tenmillionth. The discrepancy sounds tiny, but over four days it built up to about a third of a second. In combination with other peculiarities of the control software, the inaccuracy caused a miscalculation of almost 700 meters in the predicted position of the incoming missile. Twentyeight soldiers died. Of course it is not to be taken for granted that better arithmetic would have saved those 28 lives. (Many other Patriots failed for unrelated reasons; some analysts doubt whether any Scuds were stopped by Patriots.) And surely the underlying problem was not the slight drift in the clock but a design vulnerable to such minor timing