## Algebraic Computation

### InCLosure 3.0

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. "On Some Algebraic and Order-Theoretic Aspects of Machine Interval Arithmetic." Online Mathematics Journal 1, no. 2 (2019): 1-13. Abstractomj_01-02_p1-13_dawood.pdfWebsite

Interval arithmetic is a fundamental and reliable mathematical machinery for scientific computing and for addressing uncertainty in general. In order to apply interval mathematics to real life uncertainty problems, one needs a computerized (machine) version thereof, and so, this article is devoted to some mathematical notions concerning the algebraic system of machine interval arithmetic. After formalizing some purely mathematical ingredients of particular importance for the purpose at hand, we give formal characterizations of the algebras of real intervals and machine intervals along with describing the need for interval computations to cope with uncertainty problems. Thereupon, we prove some algebraic and order-theoretic results concerning the structure of machine intervals.

keywords: Interval mathematics, Machine interval arithmetic, Outward rounding, Floating-point arithmetic, Machine monotonicity, Dense orders, Orderability of intervals, Symmetricity, Singletonicity, Subdistributive semiring, S-semiring.

## InCLosure: A Language and Environment for Reliable Scientific Computing.InCLosure version 2.0

http://scholar.cu.edu.eg/henddawood/software/InCLosure
Copyright (c) 2018 by Hend Dawood.