Metamathematics

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Dawood, Hend, and Yasser Dawood. "Universal Intervals: Towards a Dependency-Aware Interval Algebra." In Mathematical Methods in Interdisciplinary Sciences. Hoboken, New Jersey: John Wiley and Sons, 2020. Abstractuniversal_intervals_abstract_dawood_wiley_2020.pdf

Interval computations are most fundamental in addressing uncertainty and imprecision. The intended status of this chapter is to be both an introduction and a treatise on some theoretical and practical aspects of interval mathematics. In the body of the work, there is room for novelties which may not be devoid of interest to researchers and specialists. The theories of classical intervals and parametric intervals are formally constructed and their mathematical structures are uncovered. By means of the logical concepts of Skolemization and quantification dependence, the notion of interval dependency is formalized by putting on a systematic basis its meaning, and thus gaining the advantage of indicating formally the criteria by which it is to be characterized and, accordingly, deducing its fundamental properties in a merely logical manner. Moreover, with a view to treating some problems of the present interval theories, a new alternate theory of intervals, namely the "theory of universal intervals", is presented and proved to have a nice S-field algebra, which extends the ordinary field of the reals. Our approach is formal by the pursuit of formulating the mathematical concepts in a strictly accurate manner, our perspective is systematic by taking the passage from the informal treatments to the formal technicalities of mathematical logic, and our concern is to take one small step towards paving the way for developing dependency-aware interval methods.

Keywords: Interval mathematics, Classical interval arithmetic, Parametric interval arithmetic, Universal interval arithmetic, Interval dependency, Functional dependence, Guaranteed enclosures, S-Semiring, S-Field, Skolemization.

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, and Nefertiti Megahed. "A Consistent and Categorical Axiomatization of Differentiation Arithmetic Applicable to First and Higher Order Derivatives." Punjab University Journal of Mathematics 51, no. 11 (2019): 77-100. Abstractpujm_51-11_p77-100_dawood.pdfWebsite

Differentiation arithmetic is a principal and accurate technique for the computational evaluation of derivatives of first and higher order. This article aims at recasting real differentiation arithmetic in a formalized theory of dyadic real differentiation numbers that provides a foundation for first and higher order automatic derivatives. After we set the stage by putting on a systematic basis certain fundamental notions of the algebra of differentiation numbers, we begin by setting up an axiomatic theory of real differentiation arithmetic, as a many-sorted extension of the theory of a continuously ordered field, and then establish the proofs for its consistency and categoricity. Next, we carefully construct the algebraic system of real differentiation arithmetic, deduce its fundamental properties, and prove that it constitutes a commutative unital ring. Furthermore, we describe briefly the extensionality of the system to an interval differentiation arithmetic and to an algebraically closed commutative ring of complex differentiation arithmetic. Finally, a word is said on machine realization of real differentiation arithmetic and its correctness, with an addendum on how to compute automatic derivatives of first and higher order.

Keywords: Automatic differentiation; Categorical differentiation arithmetic; Consistent differentiation arithmetic; Commutative unital ring; Interval differentiation arithmetic; Algebraically closed commutative rings.

Dawood, Hend, and Yasser Dawood. On the Mathematical Foundations of Algorithmic Differentiation. Giza: Department of Mathematics, Faculty of Science, Cairo University, 2017. Abstract

In this report, we set up an axiomatic system of algorithmic differentiation and deduce its fundamental algebraic properties.

Dawood, Hend, and Yasser Dawood. Interval Algerbras: A Formalized Treatment. Giza: Department of Mathematics, Faculty of Science, Cairo University, 2016. Abstract

The theories of classical intervals, modal intervals, and constraint intervals are formally constructed and their mathematical structures are in-depth investigated.

Dawood, Hend, and Yasser Dawood. The Form of the Uncertain: On the Mathematical Structures of Uncertainty. Giza: Department of Mathematics, Faculty of Science, Cairo University, 2016. Abstract

In this report, we investigate into the algebraic structures underlying the mathematics of uncertainty.

Dawood, Hend. "Interval Mathematics as a Potential Weapon against Uncertainty." In Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems. Hershey, PA: IGI Global, 2014. Abstractinterval_mathematics_as_a_potential_weapon_against_uncertainty.pdf

This chapter is devoted to introducing the theories of interval algebra to people who are interested in applying the interval methods to uncertainty analysis in science and engineering. In view of this purpose, we shall introduce the key concepts of the algebraic theories of intervals that form the foundations of the interval techniques as they are now practised, provide a historical and epistemological background of interval mathematics and uncertainty in science and technology, and finally describe some typical applications that clarify the need for interval computations to cope with uncertainty in a wide variety of scientific disciplines.

Keywords: Interval mathematics, Uncertainty, Quantitative Knowledge, Reliability, Complex interval arithmetic, Machine interval arithmetic, Interval automatic differentiation, Computer graphics, Ray tracing, Interval root isolation.

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. Interval Mathematics: Foundations, Algebraic Structures, and Applications. Cairo: Cairo University, 2012. Abstractinterval_mathematics_msc_thesis_by_hend_dawood.pdf

We begin by constructing the algebra of classical intervals and prove that it is a nondistributive abelian semiring. Next, we formalize the notion of interval dependency, along with discussing the algebras of two alternate theories of intervals: modal intervals, and constraint intervals. With a view to treating some problems of the present interval theories, we present an alternate theory of intervals, namely the "theory of optimizational intervals", and prove that it constitutes a rich S-field algebra, which extends the ordinary field of the reals, then we construct an optimizational complex interval algebra. Furthermore, we define an order on the set of interval numbers, then we present the proofs that it is a total order, compatible with the interval operations, dense, and weakly Archimedean. Finally, we prove that this order extends the usual order on the reals, Moore's partial order, and Kulisch's partial order on interval numbers.

Keywords:
Classical interval arithmetic, Machine interval arithmetic, Interval dependency, Constraint intervals, Modal intervals, Classical complex intervals, Optimizational intervals, Optimizational complex intervals, S-field algebra, Ordering subsets of the reals, Interval arithmetic, Ordering interval numbers, Ordinal power, Total order, Well order, Order compatibilty, Weak Archimedeanity, Dedekind completeness, Interval lattice, Interval order topology, Moore's partial order, Kulisch's partial order.

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