## Interval Enclosures

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Dawood, Hend, and Yasser Dawood. "A Logical Formalization of the Notion of Interval Dependency: Towards Reliable Intervalizations of Quantifiable Uncertainties." Online Mathematics Journal 1, no. 3 (2019): 15-36. Abstractomj_01-03_p15-36_dawood.pdfWebsite

Progress in scientific knowledge discloses an increasingly paramount use of quantifiable properties in the description of states and processes of the real-world physical systems. Through our encounters with the physical world, it reveals itself to us as systems of uncertain quantifiable properties. One approach proved to be most fundamental and reliable in coping with quantifiable uncertainties is interval mathematics. A main drawback of interval mathematics, though, is the persisting problem known as the "interval dependency problem". This, naturally, confronts us with the question: Formally, what is interval dependency? Is it a meta-concept or an object-ingredient of interval and fuzzy computations? In other words, what is the fundamental defining properties that characterize the notion of interval dependency as a formal mathematical object? Since the early works on interval mathematics by John Charles Burkill and Rosalind Cecily Young in the dawning of the twentieth century, this question has never been touched upon and remained a question still today unanswered. Although the notion of interval dependency is widely used in the interval and fuzzy literature, it is only illustrated by example, without explicit formalization, and no attempt has been made to put on a systematic basis its meaning, that is, to indicate formally the criteria by which it is to be characterized. Here, we attempt to answer this long-standing question. This article, therefore, is devoted to presenting a complete systematic formalization of the notion of interval dependency, by means of the notions of Skolemization and quantification dependence. A novelty of this formalization is the expression of interval dependency as a logical predicate (or relation) and thereby gaining the advantage of deducing its fundamental properties in a merely logical manner. Moreover, on the strength of the generality of the logical apparatus we adopt, the results of this article are not only about classical intervals, but they are meant to apply also to any possible theory of interval arithmetic. That being so, our concern is to shed new light on some fundamental problems of interval mathematics and to take one small step towards paving the way for developing alternate dependency-aware interval theories and computational methods.

Keywords: Interval mathematics; Interval dependency; Functional dependence; Skolemization; Guaranteed bounds; Interval enclosures; Interval functions; Quantifiable uncertainty; Scientific knowledge; Reliability; Fuzzy mathematics; InCLosure.

Dawood, Hend, and Yasser Dawood. "Parametric Intervals: More Reliable or Foundationally Problematic?" Online Mathematics Journal 1, no. 3 (2019): 37-54. Abstractomj_01-03_p37-54_dawood.pdfWebsite

Interval arithmetic has been proved to be very subtle, reliable, and most fundamental in addressing uncertainty and imprecision. However, the theory of classical interval arithmetic and all its alternates suffer from algebraic anomalies, and all have difficulties with interval dependency. A theory of interval arithmetic that seems promising is the theory of parametric intervals. The theory of parametric intervals is presented in the literature with the zealous claim that it provides a radical solution to the long-standing dependency problem in the classical interval theory, along with the claim that parametric interval arithmetic, unlike Moore's classical interval arithmetic, has additive and multiplicative inverse elements, and satisfies the distributive law. So, does the theory of parametric intervals accomplish these very desirable objectives? Here it is argued that it does not.

Keywords: Interval mathematics, Classical interval arithmetic, Parametric interval arithmetic, Constrained interval arithmetic, Overestimation-free interval arithmetic, Interval dependency, Functional dependence, Dependency predicate, Interval enclosures, S-semiring, Uncertainty, Reliability.

Dawood, Hend. InCLosure (Interval enCLosure): A Language and Environment for Reliable Scientific Computing. 1.0 ed. Department of Mathematics, Faculty of Science, Cairo University, 2018. AbstractWebsite

InCLosure (Interval enCLosure) is a Language and Environment for Reliable Scientific Computing. InCLosure, provides rigorous and reliable results in arbitrary precision. From its name, InCLosure (abbreviated as "InCL") focuses on "enclosing the exact real result in an interval". The interval result is reliable and can be as narrow as possible.
InCLosure supports arbitrary precision in both real and interval computations. In real arithmetic, the precision is arbitrary in the sense that it is governed only by the computational power of the machine (default is 20 significant digits). The user can change the default precision according to the requirements of the application under consideration. Since interval arithmetic is defined in terms of real arithmetic, interval computations inherit the arbitrary precision of real arithmetic with an added property that the interval subdivision method is provided with an arbitrary number of subdivisions which is also governed only by the computational power of the machine. The user can get tighter and tighter guaranteed interval enclosures by setting the desired number of subdivisions to cope with the problem at hand.
All the computations defined in terms of real and interval arithmetic (e.g., real and interval automatic differentiation) inherit the same arbitrary precision.
InCLosure is written in Lisp, the most powerful and fast language in scientific computations. InCLosure provides easy user interface, detailed documentation, clear and fast results. Anyone can compute with InCLosure.

Dawood, Hend, and Yasser Dawood. Logical Aspects of Interval Dependency. Giza: Department of Mathematics, Faculty of Science, Cairo University, 2013. Abstract

By means of the most fundamental logical concepts of quantification theory, the notion of interval dependency is axiomatized and its fundamental properties are deduced.

Dawood, Hend, and Yasser Dawood. On the Metamathematics of the Theory of Interval Numbers. Giza: Department of Mathematics, Faculty of Science, Cairo University, 2010. Abstract

The aim of this paper is to provide a metamathematical investigation of the theory of intervals with the requisite predicate calculi and axiomatic set theory.

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