# Model Intervals

Model Intervals This paper summarizes the most important results and features of Modal Interval Analysis. The ground idea of Interval mathematics is that ordinary set-theoretical intervals are the consistent context for numerical computing. However, this interval context presents basic structural and semantic rigidity arising from its set-theoretical foundation. To correct this situation, Modal Interval Analysis defines intervals starting from the identification of real numbers with the sets of predicates they accept or reject. A modal interval is defined as a pair formed by a classical interval (i.e. a set of numbers) and a quantifier, following a similar method to that in which real numbers are associated in pairs having the same absolute value but opposite signs. Two different extensions for a continuous function (called semantic extensions, since both will have a meaning thanks to the important semantic theorems) are defined and their properties are established. The definition of the rational extension, and its relationships with the semantic extensions, it make possible to compute the semantic extensions and to give a logical meaning to the interval results of a rational computations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

# Model Intervals

36 pages

/lp/springer_journal/model-intervals-zy8Nm7AATd
Publisher
Springer Journals
Subject
Mathematics; Numeric Computing; Approximations and Expansions; Computational Mathematics and Numerical Analysis; Mathematical Modeling and Industrial Mathematics
ISSN
1385-3139
eISSN
1573-1340
D.O.I.
10.1023/A:1011465930178
Publisher site
See Article on Publisher Site

### Abstract

This paper summarizes the most important results and features of Modal Interval Analysis. The ground idea of Interval mathematics is that ordinary set-theoretical intervals are the consistent context for numerical computing. However, this interval context presents basic structural and semantic rigidity arising from its set-theoretical foundation. To correct this situation, Modal Interval Analysis defines intervals starting from the identification of real numbers with the sets of predicates they accept or reject. A modal interval is defined as a pair formed by a classical interval (i.e. a set of numbers) and a quantifier, following a similar method to that in which real numbers are associated in pairs having the same absolute value but opposite signs. Two different extensions for a continuous function (called semantic extensions, since both will have a meaning thanks to the important semantic theorems) are defined and their properties are established. The definition of the rational extension, and its relationships with the semantic extensions, it make possible to compute the semantic extensions and to give a logical meaning to the interval results of a rational computations.

### Journal

Reliable ComputingSpringer Journals

Published: Oct 3, 2004

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