Hend Dawood

Acquiring reliable knowledge amidst uncertainty is a topical issue of modern science. Interval mathematics has proved to be of central importance in coping with uncertainty and imprecision. Algorithmic differentiation, being superior to both numeric and symbolic differentiation, is nowadays one of the most celebrated techniques in the field of computational mathematics. In this connexion, laying out a concrete theory of interval differentiation arithmetic, combining subtlety of ordinary algorithmic differentiation with power and reliability of interval mathematics, can extend real differentiation arithmetic so markedly both in method and objective, and can so far surpass it in power as well as applicability. This article is intended to lay out a systematic theory of dyadic interval differentiation numbers that wholly addresses first and higher order automatic derivatives under uncertainty. We begin by axiomatizing a differential interval algebra and then we present the notion of an interval extension of a family of real functions, together with some analytic notions of interval functions. Next, we put forward an axiomatic theory of interval differentiation arithmetic, as a two-sorted extension of the theory of a differential interval algebra, and provide the proofs for its categoricity and consistency. Thereupon, we investigate the ensuing structure and show that it constitutes a multiplicatively non-associative S-semiring in which multiplication is subalternative and flexible. Finally, we show how to computationally realize interval automatic differentiation. Many examples are given, illustrating automatic differentiation of interval functions and families of real functions.

InCLosure (Interval enCLosure) is a Language and Environment for Reliable Scientific Computing. Interval computations are radically different from traditional numerical approximation methods and reliable computing under uncertainty is a key focus for modern research in mathematics, computer science, physics, and engineering. InCLosure is a system for carrying out reliable and self-validated computations in arbitrary precision. From its name, InCLosure (abbreviated as "InCL") focuses on "computing guaranteed interval enclosures", that is, "enclosing the exact real result in an interval".
InCLosure is powerful enough to carry out computations ranging from simple real and interval arithmetic, through symbolic and numeric differentiation to an arbitrary order, real automatic differentiation, and interval automatic differentiation, up to and including interval enclosures of integrals and Taylor model computations. No matter how complicated the problem under consideration is, InCLosure provides arbitrary precisions and reliable interval results that can be as narrow as possible by the computational power of the hosting machine.
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, Taylor model computations, and so forth) inherit the same arbitrary precision.
InCLosure is designed to support both interactive and batch modes. In the InCLosure interactive interface, the user can input an InCL command and see its result before moving on to the next command. InCLosure can also be used in batch mode in which case sequences of InCL commands can be given to InCLosure via InCL input files with the results saved in simple and intuitively formatted output files.
InCLosure is coded entirely in Lisp, arguably the fastest and most powerful language for scientific computations. InCLosure provides a friendly and easy-to-use user interface, a simple and intuitive language, a detailed documentation, and clear and fast results. Anyone can compute with InCLosure.

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.

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.

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.

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.

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.

This standard is a simplified version and a subset of the IEEE Std 1788TM-2015 for Interval Arithmetic and includes those operations and features of the latter that in the the editors view are most commonly used in practice. IEEE Std 1788.1-2017 specifies interval arithmetic operations based on intervals whose endpoints are IEEE Std 754TM-2008 binary64 floating-point numbers and a decoration system for exception-free computations and propagation of properties of the computed results.A program built on top of an implementation of IEEE Std 1788.1-2017 should compile and run, and give identical output within round off, using an implementation of IEEE Std 1788-2015, or any superset of the former.Compared to IEEE Std 1788-2015, this standard aims to be minimalistic, yet to cover much of the functionality needed for interval computations. As such, it is more accessible and will be much easier to implement, and thus will speed up production of implementations.

The dynamically combined deductive and hypothetico-deductive method has been available to objective investigators since ancient times. Only in the last half-century has it been taught in courses on scientific method and critical thinking. The below chart for teaching and applying it is only about thirty years old.

Readers of this chart will note its similarity to another which charts a very different method with very different goals.This method aims at knowledge of validity and invalidity; the other aims at knowledge of truth and falsity.

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.

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.

In this report, we define an algebra of dyadic intervals. Thereupon, we present a formalization of a generalized theory of interval differentiation and investigate into some of the analytic and algebraic properties thereof.

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

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

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

This standard specifies basic interval arithmetic (IA) operations selecting and following one of the commonly used mathematical interval models. This standard supports the IEEE 754 floating point formats of practical use in interval computations. Exception conditions are defined, and standard handling of these conditions is specified. Consistency with the interval model is tempered with practical considerations based on input from representatives of vendors, developers and maintainers of existing systems. The standard provides a layer between the hardware and the programming language levels. It does not mandate that any operations be implemented in hardware. It does not define any realization of the basic operations as functions in a programming language.

In this report, we study some of the order-theoretic and algebraic aspects of interval mathematics.

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.

A new dependency-aware interval arithmetic is presented and its algebraic system is investigated.

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

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.

Reviews

"This new book by Hend Dawood is a fresh introduction to some of the basics of interval computation. It stops short of discussing the more complicated subdivision methods for converging to ranges of values, however it provides a bit of perspective about complex interval arithmetic, constraint intervals, and modal intervals, and it does go into the design of hardware operations for interval arithmetic, which is something still to be done by computer manufacturers."
   - Ramon E. Moore, (The Founder of Interval Computations)
   Professor Emeritus of Computer and Information Science, Department of Mathematics, The Ohio State University, Columbus, U.S.A.

"A popular math-oriented introduction to interval computations and its applications. This short book contains an explanation of the need for interval computations, a brief history of interval computations, and main interval computation techniques. It also provides an impressive list of main practical applications of interval techniques."
   - Vladik Kreinovich, (International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems)
   Professor of Computer Science, University of Texas at El Paso, El Paso, Texas, U.S.A.

"I am delighted to see one more Egyptian citizen re-entering the field of interval mathematics invented in this very country thousands years ago."
   - Marek W. Gutowski,
   Institute of Physics, Polish Academy of Sciences, Warszawa, Poland

Book Description

Scientists are, all the time, in a struggle with uncertainty which is always a threat to a trustworthy scientific knowledge. A very simple and natural idea, to defeat uncertainty, is that of enclosing uncertain measured values in real closed intervals. On the basis of this idea, interval arithmetic is constructed. The idea of calculating with intervals is not completely new in mathematics: the concept has been known since Archimedes, who used guaranteed lower and upper bounds to compute his constant Pi. Interval arithmetic is now a broad field in which rigorous mathematics is associated with scientific computing. This connection makes it possible to solve uncertainty problems that cannot be efficiently solved by floating-point arithmetic. Today, application areas of interval methods include electrical engineering, control theory, remote sensing, experimental and computational physics, chaotic systems, celestial mechanics, signal processing, computer graphics, robotics, and computer-assisted proofs. The purpose of this book is to be a concise but informative introduction to the theories of interval arithmetic as well as to some of their computational and scientific applications.

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.