3234184
doi
10.5281/zenodo.3234184
oai:zenodo.org:3234184
user-mathscicu
user-omj
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Yasser Dawood
Department of Astronomy, Faculty of Science, Cairo University, Giza 12613, Egypt
A Logical Formalization of the Notion of Interval Dependency: Towards Reliable Intervalizations of Quantifiable Uncertainties
Hend Dawood
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
doi:10.5281/zenodo.2702404
doi:10.5281/zenodo.3466032
doi:10.5281/zenodo.2656089
info:eu-repo/semantics/openAccess
Creative Commons Attribution 4.0 International
https://creativecommons.org/licenses/by/4.0/legalcode
Interval Mathematics
Interval Dependency
Functional Dependence
Skolemization
Guaranteed Bounds
Interval Enclosures
Interval Functions
Quantifiable Uncertainty
Scientific Knowledge
Reliability
Fuzzy Mathematics
InCLosure
Subdistributive Semiring
S-Semiring
Interval Computations
Interval Analysis
Interval Arithmetic
Universal Intervals
Rounding Error
Real Functions
<p>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.</p>
Supplementary Material: http://doi.org/10.5281/zenodo.3466032
Download latest release of InCLosure via https://doi.org/10.5281/zenodo.2702404
Zenodo
2019-07-01
info:eu-repo/semantics/article
3234183
user-mathscicu
user-omj
user-inclosure
1579541035.166481
528768
md5:1fc878f2e9d4ae0d327a220c4f5325f5
https://zenodo.org/records/3234184/files/OMJ_01-03_p15-36_Dawood.pdf
public
10.5281/zenodo.2702404
Cites
doi
10.5281/zenodo.3466032
Cites
doi
10.5281/zenodo.2656089
Cites
doi
10.5281/zenodo.3234183
isVersionOf
doi
Online Mathematics Journal
01
03
15-36
2019-07-01