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. 2023 Mar 29;9:e1301. doi: 10.7717/peerj-cs.1301

Automatic differentiation of uncertainties: an interval computational differentiation for first and higher derivatives with implementation

Hend Dawood 1,, Nefertiti Megahed 1
Editor: Nicholas Higham
PMCID: PMC10280627  PMID: 37346667

Abstract

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.

Keywords: Interval analysis, Interval computations, Interval-valued functions, Interval automatic differentiation, Interval differentiability, Categorical differentiation arithmetic, Subdistributive semiring, Guaranteed interval enclosures, Quantifiable uncertainties, Verified computations


Dedication. In memory of Ramon Edgar Moore (1929–2015), the man who intervalized uncertainty.

Introduction

Uncertainty arises in all fields of modern science. It is a state of limited knowledge where “To know” means “To be uncertain of”. Acquiring reliable knowledge amidst uncertainty is the raison d’être of the present work. Motivated by an ever-increasing indeterminacy and complexity in physics and engineering and fueled by developments in computational and uncertainty mathematics, this work puts forward a categorical system of interval differentiation arithmetic that wholly addresses the computation of first and higher order automatic derivatives under uncertainty. Although scientists are fond of determinism, contemporary physical sciences have shown clearly that complete certainty is not reachable. The description of processes and states of physical systems discloses increasingly growing manipulations of uncertain quantifiable properties. Many features of the object world are rendered as numerical values that can either be measured or estimated by experts. Due to imperfection of our measuring methods, finiteness of our computations and lack of information, measured or estimated quantities can only be represented by finite approximations and thus are merely imprecise abstractions of reality (Dawood & Dawood, 2019a; Dawood & Dawood, 2020, and Dawood & Dawood, 2022).

In the effort to deal with the challenge of uncertainty, the subject of uncertainty mathematics has been developed in an extensive manner and many theoretical approaches have been introduced including fuzzy, probabilistic, and interval methods. A hot and fundamental topic of research that shades of into all approaches of uncertainty mathematics is interval analysis (see, e.g.Dawood, 2014; Dawood & Dawood, 2019a, and Dawood & Dawood, 2020). The key advantage of the interval methods is that they provide “guaranteed interval enclosures” of the exact values of quantifiable uncertainties. In practice, when modelling physical systems, we have two distinct approaches: getting guaranteed bounds of an uncertain quantity and computing a numerical approximation thereof. The two approaches are not equivalent: the former includes the latter, but the latter does not imply the former. For example, to guarantee stability under uncertainty in control systems and robotics, it is crucial to compute guaranteed enclosures of the quantifiable features of the system under consideration (Dawood, 2014 and Dawood & Dawood, 2020). Interval arithmetic brings forth a reliable way to cope with such problems. An interval number (a closed and bounded interval of real numbers) is a guaranteed enclosure of an imprecisely measured real-valued quantity, and an interval-valued function is consequently a guaranteed enclosure of a real-valued function under imprecision or uncertainty (or more generally, as we will see in this article, a reliable enclosure of the image of a family of real-valued functions). Historically speaking, the terms “interval arithmetic”, “interval analysis”, and “interval computations” are reasonably recent: they date from the fifties of the twentieth century. But the idea has been known since the third century BC, when Archimedes (287–212 BC) used lower and upper error bounds in the course of his computation of the constant π (Heath, 2009). In the dawning of the twentieth century, the first rigorous developments of the theory of intervals appeared in the works of Norbert Wiener, John Charles Burkill, Rosalind Cecily Young, and Mihailo Petrovic (see Wiener, 1921, Burkill, 1924; Young, 1931; Petrovic, 1932, and Petkovic, 2020). Later, several distinguished developments of interval arithmetic appeared in the works of Paul S. Dwyer, Mieczyslaw Jan Warmus, Teruo Sunaga, and others (see, e.g.Dwyer, 1951; Warmus, 1956, and Sunaga, 1958). However, it was not until 1959 that “interval analysis” in its modern sense was presented by the American mathematician and computer scientist, Ramon Edgar Moore (1929–2015), who was the first to recognize the power of interval arithmetic as a viable computational apparatus for coping with uncertainty and imprecision (Moore, 1959). Nowadays, interval mathematics is a bold enterprise that comprises many different kinds of problem and has many fruitful applications in diverse areas of science and engineering (see, e.g.Allahviranloo, Pedrycz & Esfandiari, 2022; Beutner, Ong & Zaiser, 2022, Dawood, 2019; Dawood & Dawood, 2020; Dawood & Dawood, 2022, IEEE 1788 Committee, 2018; Jiang, Han & Xie, 2021; Kearfott, 2021, Mahato, Rout & Chakraverty, 2020; Matanga, Sun & Wang, 2022, Shary & Moradi, 2021, and Zheng et al., 2020).

Two strands of research have led to the birth of the present work. The first strand starts from research in interval mathematics. The other strand stems from ordinary (real) automatic differentiation. Derivatives play an indispensable role in scientific computing. The expressions ‘automatic differentiation’, ‘auto-differentiation’, ‘computational differentiation’, ‘algorithmic differentiation’, and ‘differentiation arithmetic’ are in the just acceptation synonyms. They refer to a subtle and central tool to automatize the simultaneous computation of the numerical values of arbitrarily complex functions and their derivatives with no need for the symbolic representation of the derivative, only the function rule or an algorithm thereof is required (Dawood & Megahed, 2019). Auto-differentiation is thus neither numeric nor symbolic, nor is it a combination of both. It is also preferable to ordinary numerical methods: In contrast to the more traditional numerical methods based on finite differences, auto-differentiation is ‘in theory’ exact, and in comparison to symbolic algorithms, it is computationally inexpensive (Dawood & Megahed, 2019). The literature on algorithmic differentiation is immense and very diversified. For further reading, (see, e.g.Corliss & Rall, 1996; Dawood, 2014, Dawood & Megahed, 2019; Griewank & Walther, 2008; Moore, 1979, Neidinger, 2010, and Mitchell, 1991). Currently, for its efficiency and accuracy in computing first and higher order derivatives, auto-differentiation is a celebrated technique with diverse applications in scientific computing and mathematics. It should therefore come as no surprise that there are numerous computational implementations of auto-differentiation. Among these, we mention, without pretension to be complete, INTLAB, Sollya, and InCLosure (see, e.g.Rump, 1999, Chevillard, Joldes & Lauter, 2010, and Dawood, 2020). In practice, there are two types (modes) of algorithmic differentiation: a forward-type and a reversed-type (Dawood & Megahed, 2019). Presently, the two types are highly correlated and complementary and both have a wide variety of applications in, e.g., non-linear optimization, sensitivity analysis, robotics, machine learning, computer graphics, and computer vision (see, e.g.Abdelhafez, Schuster & Koch, 2019; Dawood, 2014, Dawood & Megahed, 2019; Fries, 2019, Sommer, Pradalier & Furgale, 2016, and Tingelstad & Egeland, 2017).

The use of ordinary auto-differentiation in the description and modeling of real world physical systems faces the problem of uncertainty. With the aid of interval mathematics, auto-differentiation can be intervalized to handle uncertainty in quantifiable properties of real world physical systems and accordingly provide the computational methods that suffice to deal with the important problem of “getting guaranteed bounds”. Interval differentiation arithmetic combines subtlety of ordinary algorithmic differentiation with power and reliability of interval mathematics. By integrating the complementary perspectives of both fields, interval differentiation arithmetic extends real differentiation arithmetic so markedly both in method and objective, and so far surpasses it in power and applicability. Real differentiation arithmetic, on the one hand, is concerned with the simultaneous calculation of the values of real functions and their derivatives with no requirement of the symbolic representation of the derivative. On the other hand, the subject matter of interval differentiation arithmetic is “interval functions” and its objective is the concurrent computation of guaranteed enclosures of images of real functions and their derivatives. This integration of interval and differentiation arithmetic is readily applicable to modelling and predicting the behaviour of real-world systems under uncertainty. Also, it has proved accuracy and efficiency in many scientific computations. As examples, we can mention enclosures of Taylor’s coefficients, gradients, integrals, bounding boxes in ray tracing, and solutions of ordinary differential equations.

Three main problems have motivated the research conducted in this article. In the first place, despite its major importance in both basic research and practical applications, to the best of our knowledge, the algebraic aspects of interval differentiation arithmetic are not in-depth investigated. In the second place, almost no attempt has been made so far to explicitly axiomatize the theory of interval differentiation arithmetic in terms of clear and distinct elementary logical notions. In the third place, although an interval function is naturally an extension of a family of real functions, to the best of our knowledge, in all interval literature, the notion of a family of functions is not considered, and an interval function is assumed to extend a single real function. This presumption introduces an unnecessary restriction to the semantic of an interval function in the general sense. Families of functions arise naturally in many real-life and physical applications. In economics, the Cobb–Douglas family of production functions is an example; in physics, electron models, dynamical systems, quantum models, Camassa–Holm and Novikov wave-breaking models, and many other physical phenomena are described by families of functions (see, e.g.Cobb & Douglas, 1928; Silberberg & Suen, 2001, Anco, da Silva & Freire, 2015, and Engesser, Gabbay & Lehmann, 2011). Providing the mathematical tools to get guaranteed enclosures of the images of families of real functions and their derivatives would provide an efficient way of predicting and controlling such physical systems and, thus, could have a substantial impact not only on theoretical research but also on many areas of applications. By the pursuit of this, formalizing the notion of a family of functions within the context of interval mathematics and interval differentiation arithmetic is one of the main motivations of this research.

Throughout the present text, we will understand by “interval differentiation arithmetic” (“interval differentiation algebra”, “ ΔJ-algebra”, or “ ΔJ-arithmetic”) the fundamental algebraic structure underlying interval auto-differentiation as it is currently practised and implemented. It is our object, in this article, to present a consistent and categorical formalization of a theory of dyadic interval differentiation numbers (ΔJ-numbers) that fully addresses first and higher order auto-derivatives of families of real functions. The fundamental significance of categoricity is that if an axiomatization of ΔJ-numbers is categorical, then it correctly accounts, up to isomorphism, for every structure of ΔJ-arithmetic. The notion of categoricity is a bedrock of contemporary mathematics. This is clearly described by John Corcoran in Corcoran (1980) and best reiterated in the words of Stewart Shapiro, “a categorical axiomatization is the best one can do” (Shapiro, 1985). In accordance with this categorical sense, the present article attempts to provide this “best” characterization. For this goal to be accomplished, we need to take a closer look at and formalize several fundamental analytic and algebraic concepts in the language of the theory to be axiomatized, so that one can establish the metatheoretic assertions of consistency and categoricity. This reformalization is mainly done in ‘On theories and structures: some metatheoretical fundamentals’ and ‘A differential interval algebra’. In ‘On theories and structures: some metatheoretical fundamentals’, we set the stage by establishing the mathematical terminology, notions, and definitions that will be used throughout the rest of this article. ‘Real differentiation arithmetic’ is devoted to describing briefly the basic elements of the theory TΔ of real differentiation arithmetic (Δ-arithmetic). In ‘A differential interval algebra’, we lay out an axiom system for the theory TδJ of 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. In ‘A categorical axiomatization of interval differentiation arithmetic’, we axiomatize a theory TΔJ of interval differentiation numbers (ΔJ-numbers) as a two-sorted extension of the theory TδJ of a differential interval algebra, and then we prove its consistency and categoricity. In order for the theory TΔJ to fully address and compute higher order and partial auto-derivatives using only dyadic ΔJ-numbers, in ‘Differentiation extension of interval functions and higher-order auto-differentiability’, we introduce the notion of a differentiation extension of interval functions, characterize differentiability for ΔJ-numbers, and establish their differentiability conditions. In ‘The algebraic structure of interval differentiation arithmetic’, we investigate the algebraic structure of ΔJ-arithmetic, establish its fundamental algebraic properties, and show that it forms a multiplicatively non-associative S-semiring in which multiplication is subalternative and flexible. Then, in ‘Monotonicity and isomorphism theorems for interval differentiation numbers’, we establish some monotonicity and isomorphism theorems for ΔJ-numbers and prove a result concerning the structure of Δ-numbers. Finally, in ‘Machine implementation of interval auto-differentiation’, we demonstrate the computational implementation of interval auto-differentiation and illustrate, by many numerical examples, how to concurrently compute guaranteed enclosures of images of both families of real functions and their first and higher order derivatives. The algorithms discussed in ‘Machine implementation of interval auto-differentiation’ are coded into reliable Common Lisp as a part of the software package, InCLosure1  (Dawood, 2020). The InCLosure commands to calculate the results of the numerical examples are described and InCLosure input and output files are accessible as a supplementary material to the article (see Dawood, 2020 and Dawood, 2023).

The attempted contribution of this article is therefore both a “logico-algebraic formalization” and an “extension” of interval differentiation arithmetic. The article gives an axiomatization of a comprehensive algebraic theory of interval differentiation arithmetic. Being based on clear and distinct elementary ideas of real and interval algebras, this formalized theory places the diverse approaches of interval auto-differentiation on a firm and unified mathematical basis. We extend this theory in two directions. On the one hand, to the best of our knowledge, in almost all computational differentiation literature, researchers tend to ‘borrow’ or ‘reinvent’ Clifford’s and Grassmann algebras2 as proposed algebraic characterizations respectively for first and higher-order algorithmic differentiation. Without resorting to defining any sort of Grassmann structures, our axiomatization of dyadic interval differentiation numbers extends to fully address interval auto-derivatives of first and higher order. On the other hand, from the very beginning, our axiomatic system includes the notion of an interval extension of a family of real functions and the differentiability criteria thereof. By virtue of introducing this notion, the theory is extended to provide the mathematical tools to get guaranteed enclosures of the images of families of real functions and their derivatives. Also noteworthy here is that with a few basic modifications, the categorical system TΔJ axiomatized in this text can be extended analogously to compute fuzzy auto-derivatives.

On theories and structures: some metatheoretical fundamentals

To achieve a rigorous formalization of the mathematical theory of this work, a specific formalized language and a particular logical apparatus are therefore required to attain all the results from obvious and distinct elementary mathematical concepts. So before we begin our axiomatization of interval auto-differentiation, we need to take a closer look at and formalize several preliminary mathematical concepts. To this aim, this section establishes the mathematical terminology, notions, and definitions that will be used throughout the rest of this article.

To make this article self-contained, we start by rehearsing some set-theoretical definitions. Let A be a set and let An be its n-th Cartesian power. A set ℜ is an n-ary (finitary) relation on A iff An and ℜ is a binary relation from An1 to A. Thus, for s=s1,,sn1An1 and tA, an n-ary relation ℜ is characterized to be An=s,t|sAn1tA. Accordingly, a finitary relation ℜ is a binary relation whose domain, range, field, and converse are characterized to be, respectively dom=sAn1|tAst, ran=tA|sAn1st, fld=domran, and ^=t,sAn|st. Obviously t ^sst and ^ ^= (Dawood & Dawood, 2019a and Dawood & Dawood, 2020).

Two indispensable definitions are those of images and preimages of finitary relations (see Dawood & Dawood, 2019a and Dawood & Dawood, 2020).

Definition 2.1 Images of Finitary Relations Dawood & Dawood, 2020

For 1 ≤ k ≤ n − 1, let be an n-ary relation on A, and for s,t, let s=s1,,sn1, with each sk is restricted to vary on SkA, that is, s is restricted to vary on SAn1. Then, the image of S (or the image of the sets Sk) with respect to, in symbols I, is characterized to be

T=IS=IS1,,Sn1=tA|sSst=tA|k=1n1skSks1,,sn1t.

The preimage S of T is characterized to be the image of T with respect to the converse ^ of. In other words

S=I ^T=sAn1|tTt ^s.

In consequence of the equivalence t ^sst, apparently T=ISS=I ^T.

In this sense, a general characterization of an n-ary (finitaryfunction can be introduced (Dawood & Dawood, 2019a). A set q is an n-ary function (a function of n variables) on a set A iff q is an n+1-ary relation on A, and sAnt,wAsqtsqwt=w. That is, an n-ary (finitary) function is an n+1-ary relation. Restricting ourselves to the particular case of functions, we can pass up the set-theoretical notation sqt in favor of the common notation t=qs. In accord with this formulation, the preceding definitions of domain, range, field, and converse also apply to finitary functions. We say that a function q is invertible or has an inverse q−1 iff the converse relation q ^ is also a function, in which case q1=q ^ (Dawood & Dawood, 2020). Hereon, functions will be denoted by the letters q, u, and v. With a few exceptions, from now onwards, we will usually consider only unary functions.

In order to achieve the overarching objective of this work, it is necessary first to take a closer look at several metamathematical3 concepts. A metalinguistic characterization of a formalized theory (an axiomatic theory) can be given. An axiomatic theory 𝔗 is characterized by an object formal language 𝔏 and a finite set of axioms Λ𝔗 (see Dawood & Dawood, 2020). Given an object formal language 𝔏 and a finite set Λ𝔗 of axioms (𝔏-sentences), let φ denote an 𝔏-sentence and let ⊨𝔏 denote the semantic consequence relation. The axiomatic 𝔏-theory 𝔗 of the set Λ𝔗 is the closure of Λ𝔗 under ⊨𝔏, that is 𝔗 = {φ ∈ 𝔏|Λ𝔗𝔏φ} (Dawood & Dawood, 2020). Next, the metatheoretical notions of a model, categoricity, and consistency of an axiomatic 𝔏-theory are characterized (see Dawood & Megahed, 2019).

Definition 2.2 Model of a Theory —

Let 𝔄 be a mathematical structure (interpretation). 𝔄 is said to be a model of an axiomatic 𝔏-theory 𝔗, in symbols 𝔄⊧𝔗, iff every formula φ of 𝔗 is satisfied by 𝔄. That is

ATφTAφ.

Definition 2.3 Categoricity of a Theory —

Let 𝔄 and 𝔅 be any two models of an axiomatic 𝔏-theory 𝔗.𝔗 is said to be categorical, in symbols CatT, iff 𝔄 and 𝔅 are isomorphic. That is

CatTABATBTAB.

Definition 2.4 Consistency of a Theory —

An axiomatic 𝔏-theory 𝔗 is said to be consistent, in symbols ConT, iff there is a model 𝔄 that satisfies the sentences of 𝔗. That is, ConTAAT.

A model of an axiomatic 𝔏-theory 𝔗 is a mathematical structure A=A,σA that makes the 𝔏-sentences of 𝔗 true. Particular mathematical structures are indispensable for the objective of this work. These are defined next (Dawood & Dawood, 2019a and Dawood & Dawood, 2020).

Definition 2.5 Ringoid Dawood & Dawood, 2019a

A ring-like structure (or a ringoid) is an algebraic structure A=A;+A,×A with +A and ×A are total binary operations on the universe set A. The operations +A and ×A are called respectively the addition and multiplication operations of the ringoid 𝔄.

Definition 2.6 S-Ringoid Dawood & Dawood, 2020

A subdistributive ringoid (or an S-ringoid) is a ringoid A=A;+A,×A that satisfies at least one of the following subdistributive properties.

  • (i)

    s,t,wAs×At+Aws×At+As×Aw,

  • (ii)

    s,t,wAt+Aw×Ast×As+Aw×As.

Properties (i) and (ii) in the previous definition are called respectively left and right S-distributivity (or subdistributivity) (Dawood & Dawood, 2020).

Definition 2.7 Semiring Dawood & Dawood, 2019a

A ringoid A=A;+A,×A is a semiring iff 𝔄 satisfies the following properties.

  • (i)

    A with +A forms a commutative monoid with 0A is an identity for +A,

  • (ii)

    A with ×A forms a monoid with 1A is an identity for ×A,

  • (iii)

    ×A is both left and right distributive over +A,

  • (iv)

    0A is an annihilating element for ×A.

If ×A is commutative, then 𝔄 is said to be a commutative semiring.

Definition 2.8 S-Semiring Dawood & Dawood, 2020

A subdistributive semiring (or an S-semiring) is an S-ringoid A=A;+A,×A that satisfies criteria (i), (ii), and (iv) in definition 2.7. A commutative S-semiring is one whose multiplication is commutative.

It is important here to point out that an S-semiring generalizes the notion of a near-semiring; a near-semiring is a structure satisfying the axioms of a semiring except that it is either left or right distributive (For further details on near-semirings and related concepts, the reader can refer to Clay, 1992; Pilz, 1983, and van Hoorn & van Rootselaar, 1967).

Lastly, we define two new algebraic structures.

Definition 2.9 NA Semiring —

A ringoid A=A;+A,×A is said to be an additively non-associative semiring (in short, +-NA semiring) iff 𝔄 satisfies (ii), (iii), and (iv) indefinition 2.7, and A;+A is a non-associative commutative monoid with identity element 0A. Similarly, 𝔄 is said to be a multiplicatively non-associative semiring (in short, ×-NA semiring) iff 𝔄 satisfies (i), (iii), and (iv) indefinition 2.7, and A;×A is a non-associative monoid with identity element 1A.

Definition 2.10 NA S-Semiring —

An S-ringoid A=A;+A,×A is said to be an additively non-associative S-semiring (in short, +-NA S-semiring) iff 𝔄 satisfies (ii) and (iv) indefinition 2.7, and A;+A is a non-associative commutative monoid with identity element 0A. Similarly, 𝔄 is said to be a multiplicatively non-associative S-semiring (in short, ×-NA S-semiring) iff 𝔄 satisfies (i) and (iv) indefinition 2.7, and A;×A is a non-associative monoid with identity element 1A.

It is clear that if multiplication is commutative in a NA S-semiring, then it is both left and right subdistributive.

Real Differentiation Arithmetic

Before setting forth the assertions of an axiomatic theory of interval differentiation arithmetic in the succeeding sections, we need to describe briefly the basic elements of the theory TΔ of real differentiation arithmetic (henceforth Δ-arithmetic). For further details and other constructions of Δ-arithmetic, the reader may consult, e.g.Dawood, 2014; Dawood & Megahed, 2019, Beda et al., 1959; Wengert, 1964; Moore, 1979, Rall, 1981, and Corliss & Rall, 1996.

We hereon use the letters q, u, and v as function symbols, and the letters s, t, and w as real variable symbols. Given a class σ = { + ,  × ;  − , −1; 0, 1,  ≤ } of descriptive (non-logical) signs, let R=R;σR be the field of real numbers, R1 be the set of unary real functions, and δ be the differential operator for elements of R1. For a q in R1, we use the predicate diffq,s0 to mean that q is differentiable at some s0 ∈ ℝ. We understand by a differential real field a structure Rδ=R;σR;δ constructed by equipping R with the operator δ and its basic axioms. It is natural to begin with the definition of a real differentiation number-number).

Definition 3.1 Real Differentiation Numbers —

The set of all real differentiation numbers-numbers, or Δ-pairs), with respect to a constant s0 ∈ ℝ, is defined to be

UR=qR2|qR1q =qs0,δ1qs0s0domqdiffq,s0.

That is, a Δ-number is an ordered pair of real numbers. Let the letters q, u, and v, or equivalently the pairs q,δqs0, u,δus0, and v,δvs0, be variable symbols ranging over the elements of U. Also, let a, b, and c, or equivalently a,0Rs0, b,0Rs0, and c,0Rs0, designate constants of U. In particular, we use 1U to denote the Δ-number 1R,0Rs0 and 0U to denote the Δ-number 0R,0Rs0.

The theory TΔ of a real differentiation algebra (or a Δ-algebra) can then be axiomatized as follows (Dawood & Megahed, 2019).

Definition 3.2 Theory of Real Differentiation Algebra —

Let q,δqs0 and u,δus0 be in U. A differentiation algebra over a differential real field Rδ=R;σR;δ, or a Δ-algebra, is a two-sorted structure UR=UR;R;σUR. The theory TΔ of 𝔘 is the deductive closure of the axioms of Rδ together with the following sentences.

  • (DA1)

    Δ-equality. q,δqs0=URu,δus0qs0=Rus0δqs0=Rδus0,

  • (DA2)

    Δ-addition. q,δqs0+URu,δus0=q+Ru,δq+Rδus0,

  • (DA3)

    Δ-multiplication. q,δqs0×URu,δus0=q×Ru,δq×Ru+Rq×Rδus0,

  • (DA4)

    Δ-negation. URq,δqs0=Rq,Rδqs0,

  • (DA5)

    Δ-reciprocal. q,δqs01UR=q1R,Rq2×Rδqs0.

Subtraction and division are defined as usual in terms of the four basic Δ-operations. For an economic exposition, we assert statements (DA2)–(DA5) as axioms but it should be mentioned that they are derivable from simpler statements. Hereafter, where no confusion is likely, the subscripts “ U”, “ ℝ”, and “ s0” will be omitted. Also, we will usually write the structure 𝔘 as UR;+UR,×UR;0UR,1UR, omitting the universe set ℝ.

Differentiable real functions can be extended to Δ-numbers via an extension principle (Dawood & Megahed, 2019). Let u be a real function differentiable at s0 ∈ ℝ, that is there is u =u,δus0UR, and let Q be a function rule. If QRu is differentiable at s0, then the differentiation extension of 𝒬 is defined to be QURu=QRu,δQRus0. For example, replacing Q by the “ sine” function, one obtains the trigonometric Δ-function sinu,δus0=sinu,δsinus0.

We will not discuss the algebraic properties of Δ-numbers further in the present section, for these will be considered later in ‘Monotonicity and isomorphism theorems for interval differentiation numbers’, in the general framework of the theory TΔJ of interval differentiation arithmetic.

A Differential Interval Algebra

In order to axiomatize a categorical system of interval differentiation arithmetic (ΔJ-arithmetic) in the next sections, we need to lay out an axiom system for the theory TδJ of a differential interval algebra. The intended model of the axiomatic system TδJ is the differential S-semiring JR,R;+J,×J;δ, where 𝒥 is the set of real closed intervals (interval numbers, or J-numbers) and δ is the differential operator for unary interval functions (J-functions).

To be able to prove categoricity and consistency of ΔJ-arithmetic, the first step towards axiomatizing the theory TδJ necessitates dealing first with the notion of differentiability in a continuously ordered field in a purely syntactic way (leaving out any references to mathematical analysis or possible interpretations). For further details on the syntactic approaches to these notions, see, e.g.Dawood, 2012; Dawood & Dawood, 2020, Montague, Kalish & Mar, 1980; Robinson, 1951, and Tarski, 1983. The theory TF of continuously ordered fields (cofields) is characterized in the following definition Dawood & Megahed, 2019.

Definition 4.1 Theory of a Cofield —

Let F=F;+F,×F;0F,1F;F be a totally ordered field. The theory TF of a cofield (or a continuously ordered field) is the theory of 𝔉 together with the following axiom of continuity

  • (ACO)

    AFBFsAtBs<KtwFsAtBswtws<Kww<Kt.

We designate by F the converse of the non-strict total order F, and by “ F” and “ 1F” the unary F-operations of negation and reciprocal, respectively. Subtraction and division are defined as customary. From now onwards, when the context is clear, we may drop the subscript “ F”.

For n ≥ 1, let Kn designate the class of all n-ary F-functions. Hereon, the letters q, u, and v are used as variable symbols ranging over the elements of K1 (unary F-functions). The intended interpretation (model) of the theory TF corresponds the structure 𝔉 to the continuously4 (complete) ordered field R;+R,×R;0R,1R;R of real numbers and F1 is interpreted by the set R1 of unary ℝ-functions.

Toward formalizing a differential interval algebra, we first need to extend the theory TF of a cofield by axiomatizing some analytic concepts. Let qF1, and let s and l be, respectively, an F-variable symbol and an F-constant symbol. The ‘limit’ operator of the function qs with respect to m, denoted limsmqs, is defined thus (Dawood & Megahed, 2019):

limsmqs=Mϵ>0α>0sdomq0<sm<αqsM<ϵ,

where the one-place operation symbol , called an F-absolute value (or F-modulus), is defined by

tFwFt=w0tw=t¬0tw=t.

If there is no such MF, then the limit of q at m is said to be nonexistent in F. For an F-constant symbol s0domq, the ‘continuity’ predicate is a binary predicate, contq,s0, defined by

contq,s0qs0F limss0qs=qs0.

If contq,s0 is  true, then q is said to be continuous at s0. We also say that q is continuous on S0F iff it is continuous at all s0 ∈ S0, that is

contq,S0s0S0contq,s0.

Definition 4.2 n-DifferentialF -Operator Dawood & Megahed, 2019

Let n ≥ 0, and let s and β be F-variable symbols. The n-differential F-operator of a function qsF1, denoted δnqs, is characterized recursively by the following equations.

  • (i)

    δ0qs=qs,

  • (ii)

    δ1qs= limβ0Fqs+βqsβ=δ1δ0qs,

  • (iii)

    n1δnqs=δ1δn1qs.

Evidently if the limit in definition 4.2 exists, then the n-differential δnqs of q is consequently a unary F-function. Henceforth, we will usually write δnq and δq for δnqs and δ1qs, respectively.

Closely related to the differential operator is the n-differentiability predicate, which is characterized as follows.

Definition 4.3 n-DifferentiabilityF -Predicate Dawood & Megahed, 2019

Let n ≥ 0, let qsF1, and let s0domq be an F-constant symbol. The ternary n-differentiability F-predicate for the function q, denoted diffnq,s0, is defined by

diffnq,s0δnqs0F.

If diffnq,s0 is  true, then q is said to be n-differentiable at s0.

Since for s0domq, δ0qs0=qs0F, the predicate diff0q,s0 is always true and accordingly every qF1 is 0-differentiable at s0domq. Apparently, if diffnq,s0 is  true, then for 0 ≤ m < n, δmqs0F.

Definition 4.4 Continuous Differentiability F-Predicate —

Let n ≥ 0, let qsF1, and let s0domq be an F-constant symbol. The continuous n-differentiability F-predicate for the function q, denoted cdiffnq,s0, is characterized recursively by the following statements.

  • (i)

    cdiff0q,s0contq,s0,

  • (ii)

    cdiff1q,s0cdiff0q,s0contδ1q,s0,

  • (iii)

    n1cdiffnq,s0cdiffn1q,s0contδnq,s0.

If cdiffnq,s0 is  true, then q is said to be continuously n-differentiable at s0.

In a manner analogous to the differential operator, if cdiffnq,s0 is  true, then for 0 ≤ m < n, cdiffmq,s0 is true as well.

A theory TJ of an interval algebra or a classical5 interval algebra (henceforth a J-algebra) over a cofield can then be characterized as follows (Dawood & Dawood, 2020 and Dawood & Dawood, 2022).

Definition 4.5 Theory of Interval Algebra —

Let σ = { + ,  × ;  − , −1; 0, 1} be a class of descriptive (non-logical) signs, and let F=F;σF be a cofield. The theory TJ of an interval algebra (a J-algebra) over 𝔉 is the theory of a many-sorted algebraic structure JF=JF;F;σJF axiomatized by the following sentences.

  • (I1)

    SJFS=sF|s¯Fs¯Fs¯FsFs¯,

  • (I2)

    S,TJF+,×SJT=wF|sStTw=sFt,

  • (I3)

    SJF10JSJS=wF|sSw=Fs.

Axiom (I1) of the above definition characterizes what a J-number (an interval number, or an F-interval) is. Axioms (I2) and (I3) prescribe, respectively the binary operations of J-addition (“ +J”) and J-multiplication (“ ×J”), and the unary operations of J-negation (“ J”) and J-reciprocal (“ 1J”). The intended model of TJ corresponds the sets “F” and “ JF” to the sets “ ℝ” and “ 𝒥” (of real numbers and real closed intervals), respectively, and the symbols “ F”, and “ F” to the binary and unary ℝ-operations.

In the sequel, the upper-case letters S, T, and W, or equivalently s¯,s¯, t¯,t¯, and w¯,w¯, will be used as variable symbols ranging over the domain JF of J-numbers. A point (singleton, or degenerate) J-numbers {s} will be denoted by s. Also, the letters A, B, and C, or equivalently a¯,a¯, b¯,b¯, and c¯,c¯, will be used to designate constants of JF. In particular, we will use 1J and 0J to designate, respectively, the singleton J-numbers 1F and 0F. It is convenient here to single out the set Js of point J-numbers. This is defined thus:

Js=SJF|sFS=s,s.

Equality of J-numbers is an immediate consequence of the axiom of extensionality6 of set theory plus the fact that a J-number is a totally ≤-ordered subset of F. Precisely,

s¯,s¯=It¯,t¯s¯=Ft¯s¯=Ft¯.

The categoricity of the theory TJ of J-algebra is established by the following theorem.

Theorem 4.1 Categoricity of the Interval Theory —

The theory TJ of J-algebra is categorical. That is, CatTJ.

Proof

Let σ = { + ,  × ;  − , −1; 0, 1} be a class of descriptive (non-logical) signs of  L, and let J1=J1;F1;σJ1 and J2=J2;F2;σJ2 be two structures such that 𝔍1T𝒥∧𝔍2T𝒥. Accordingly, F1;σF1 and F2;σF2 are two cofields. A theory of cofields is categorical, that is, there is one and up to isomorphism only one cofield. The structure R;σR is characterized, up to isomorphism, as the only cofield.

Let i:F1F2 be the isomorphism from F1 onto F2. We can then define I:J1J2 by

IS=Is¯,s¯=is¯,is¯,

for all S=s¯,s¯ in J1 where s¯,s¯F1. By definition 4.5, It is straightforward to show that I is an isomorphism from J1 onto J2. This proves that TJ is categorical.

      □

That is, the theory TJ uniquely characterizes the algebra of J-numbers, and the structure 〈𝒥; ℝ; +𝒥, ×𝒥; 0𝒥, 1𝒥〉 is, up to isomorphism, the only possible model of TJ. Accordingly, in establishing our assertions about J-numbers, the properties of real numbers are assumed in advance.

By means of definition 4.5 and from the fact that J-numbers are ordered sets of ℝ, the following theorem is derivable (Dawood, 2012 and Dawood & Dawood, 2020).

Theorem 4.2 Interval Operations —

Let s¯,s¯ and t¯,t¯ be two J-numbers. The binary and unary J-operations (interval operations) are formulated thus:

  • (i)

    J-addition. s¯,s¯+Jt¯,t¯=s¯+Rt¯,s¯+Rt¯,

  • (ii)

    J-multiplication. s¯,s¯×Jt¯,t¯=minP,maxP,

  • (iii)

    J-negation. Js¯,s¯=Rs¯,Rs¯,

  • (iv)

    J-reciprocal. 0Js¯,s¯s¯,s¯1J=s¯1R,s¯1R,

where P=s¯×Rt¯,s¯×Rt¯,s¯×Rt¯,s¯×Rt¯, and min and max are respectively the-minimal and-maximal.

If no confusion is likely, we will often omit the subscripts J and ℝ. It is clear that interval addition, multiplication, and negation are total J-operations, while interval reciprocal is a partial J-operation. As customary, interval subtraction and division are defined respectively as ST=S+T and S÷T=S×T1.

The set-theoretic characterization of interval arithmetic brings to the fore a peculiar feature that seems strange at first. Definition 4.5 entails that a J-operation considers all occurrences of variables as independent (Dawood & Dawood, 2020). Let two J-variables S and T be assigned the same J-constant 1,0. Evidently, S×JS=S×JT=1,1 which is equal to the image, Iind, of the multivariate ℝ-function qinds,t=s×Rt, with s,t1,0. Now consider a unary ℝ-function qdeps=s×Rs, with s1,0. The image Idep of qdep is 0,1. Provided that images of ℝ-functions are inclusion monotonic (see, e.g., Dawood, 2012 and Dawood & Dawood, 2019b), we have the nice enclosure Idep⊆Iind, and therefore the result of a J-operation S×JT is a guaranteed interval enclosure of the image of the corresponding ℝ-function. Although this is typically appraised as one of the strengths of interval analysis, in many practical situations, interval enclosures might be too wide to be beneficial. The name of this crucial phenomenon is the interval dependency problem, a concept that we make precise in the next theorem (see Dawood & Dawood, 2019a and Dawood & Dawood, 2020).

Theorem 4.3 Dependency Problem —

Let Si be J-numbers, for 1 ≤ i ≤ n. Let qRs1,,sn be a continuous-function with si ∈ Si, and let qJS1,,Sn be a J-function defined by the same rule as q. The result of computing the image of the intervals Si under q, denoted IqS1,,Sn, using classical J-operations (definition 4.2), cannot be generally exact if some si are functionally dependent. That is,

  • (i)

    qIqS1,,SnqJS1,,Sn.

In general,

  • (ii)

    qIqS1,,SnqJS1,,Sn.

What this theorem shows is that the result obtained by the J-function qJ is usually overestimated due to the presence of functional dependence. Interval dependency is a ‘deep-rooted’ problem, dating back to the early works on interval arithmetic. A recent investigation of the logical underpinnings and some ways out of the problem can be found in Dawood & Dawood (2019a) and Dawood & Dawood (2020). A plausible definition and a graphical representation (dependency diagrams) of the dependence of interval variables were also proposed in Shary & Moradi (2021). Plenty of effort has been made to administer feasible remedies. With convenient refined techniques, the interval enclosure qJ can be made arbitrarily close to the image Iq. By noting regions of monotonicity, one technique is defining the elementary interval functions as the exact images of their corresponding real counterparts. Let n be a nonnegative integer and S=s¯,s¯ be a J-number. We can define as instances

eS=es¯,es¯,lnS=lns¯,lns¯ifs¯>0;
S=s¯,s¯ifs¯0,sinS=minsSsins,maxsSsins;
Sn=s¯n,s¯niffs¯>0ornis odd,s¯n,s¯niffs¯<0andnis even,0,Sniff0Sandnis even;

where S= maxs¯,s¯ is the J-absolute value(or J-modulus) of S.

Performing naive J-arithmetic (theorem 4.2) on these exact images we can get sharper enclosures of their algebraic combinations. Moreover, a diversity of interval methods has been devised to compute narrower interval enclosures. Without pretension to be complete, we can mention the subdivision method, centered forms, circular complex centered forms, generalized centered forms, Hansen’s method, remainder forms (see, e.g., Dawood & Dawood, 2019a; Moore, 1979, Rokne & Ratschek, 1984, Kulisch, 2013 and Alefeld & Mayer, 2000). For instance, the subdivision method presented by Moore in Moore, 1966 and Moore, 1979 is a celebrated method that can be described as follows. Let S=s¯,s¯ be a J-number. First, subdivision of S into n subintervals Si is applied such that

Si=s¯+i1S/n,s¯+iS/n,

where S=s¯s¯ and Si=S/n are respectively the widths (lengths) of S and Si. Consequently S=i=1nSi. Then, evaluating a J-function qJ for each subinterval Si yields the enclosure (Dawood, 2014)

IqSi=1nq JSiqJS.

As the number n of subintervals gets larger, i=1nqJSi gets arbitrarily close to the exact image IqS. The subdivision method thus gives sharper enclosures than the naive evaluation qJS. In ‘Machine implementation of interval auto-differentiation’, we will deploy the subdivision method in order to compute reliable and realistic enclosures of families of real functions and their derivatives.

The characterization of the interval algebraic operations implies a number of familiar algebraic properties. However, being a particular kind of set arithmetic, interval arithmetic (J-arithmetic) has certain peculiar properties involving set inclusion. The singleton intervals 0J and 1J are identities for J-addition and J-multiplication, respectively; J-addition and J-multiplication are both associative and commutative; J-addition is cancellative; J-multiplication is cancellative only for all S0J; a J-number is invertible for J-addition (respectively, J-multiplication) if and only if it is a singleton J-number (respectively, a nonzero singleton J-number); and J-multiplication left and right S-distributes over J-addition (see definition 2.6 of ‘On theories and structures: some metatheoretical fundamentals’). In other words, in accordance with definition 2.8, the structure 〈𝒥; +𝒥, ×𝒥; 0𝒥, 1𝒥〉 of classical J-numbers can be shown to be a commutative S-semiring (Dawood & Dawood, 2019a and Dawood & Dawood, 2020).

Throughout this text we will make use of the following theorems (see Dawood, 2014 and Dawood & Dawood, 2020).

Theorem 4.4 Inclusion Monotonicity for J-Numbers —

Let S1, S2, T1, and T2 be J-numbers such that S1T1 and S2T2. Let J+,× be a binary J-operation and J,1 be a definable unary J-operation. Then

  • (i)

    S1JS2T1JT2,

  • (ii)

    JS1JT1.

From inclusion monotonicity, plus the fact that sSs,sS, if s ∈ S and t ∈ T, then for J+,× and J,1, we obviously have st ∈ S𝒥T and ♢s ∈ ♢𝒥S.

At this point, let us introduce an abbreviation that we will make use of. Let s=s1,,si,,sn be an ordered real n-tuple, and let S=S1,,Si,,Sn and T=T1,,Ti,,Tn be two ordered n-tuples of J-numbers, then

STiSiTi,
sSisiSi.

In the following theorem, let sJstsRt.

Theorem 4.5 Isomorphism Theorem for J-Numbers —

The structure Js;+J,×J;Is of point J-numbers is isomorphic to the ordered field 〈ℝ; +, ×; ≤ of real numbers.

Two further results we will need are stated below (Dawood, 2012).

Theorem 4.5 Algebraic Operations for Point J-Numbers —

Let S and T be two J-numbers. Then:

  • (i)
    The sum S + T is a point J-number iff each of S and T is a point J-number, that is
    S,TJS+TJsSJsTJs.
  • (ii)
    The product S × T is a point J-number iff each of S and T is a point J-number, or at least one of S and T is 0J, that is
    S,TJS×TJsSJsTJsS=0JT=0J.

Theorem 4.7 Zero Divisors in J-Numbers —

Nonzero zero divisors do not exist in J-arithmetic, that is

S,TJS×T=0JS=0JT=0J.

Before turning to the axioms of the theory TδJ of a differential J-algebra, it is necessary for our purpose to formalize some analytic concepts within the framework of the theory TJ of J-numbers.

Before proceeding any further, let us agree on some basic notation. By an n-ary real function (in short, ℝ-function) we will always mean a function q:𝒟⊆ℝn↦ℝ, and by an n-ary interval function (in short, J-function) we will always mean a function qJ:DJJnJ. The ℝ-subscripted symbols quv will designate ℝ-functions, while the J-subscripted symbols qJ,uJ,vJ will designate J-functions. For simplicity of notation, if the function type is apparent from the type of its variables(arguments), the subscripts “ ℝ” and “ J” will usually be dropped. For instance, whenever unambiguous, we use the notations qs1,,sn and qS1,,Sn for, respectively, an ℝ-function and a J-function, which are both defined by the same rule. For 1 ≤ i ≤ n and 1 ≤ j ≤ k , let Si and Aj be respectively J-variable symbols and J-constant symbols. We denote by qJSi:n;Aj:k an n-ary (or multivariate) J-function in the interval variables Si and the interval constants Aj. Similarly, we understand by qRsi:n;aj:k an n-ary ℝ-function in the real variables si and the real constants aj. For instance,

qJSi:2;Aj:2=qJS1,S2;A1,A2=A1S12+A2S2,

is a binary J-function whose variable arguments are S1 and S2, and whose constants are A1 and A2.

With a few exceptions, without loss of generality, the present discussion will be confined to unary functions only. For brevity, therefore, we will often adopt the standard notations qs and qS respectively for the unary functions qs;aj:k and qS;Aj:k. The sets of unary real and interval functions will be denoted by R1 and J1 respectively.

Next, we define the interval enclosure of a bounded set of real numbers.

Definition 4.6 Interval Enclosure of a Bounded Set —

Let A be a bounded subset of. The interval enclosure of A, denoted EJ, is defined to be

EJA=infA,supA.

Clearly, AEJA. For instance EJ3,4,2=2,4 and EJ1,3=1,3.

An important notion we will need is that of the image set of bounded subsets of ℝ, under an n-ary real-valued function. This notion is a special case of that of the corresponding n+1-ary relation on ℝ. More precisely, we have the following definition.

Definition 4.7 Image of Bounded Real Sets —

Let q be an n-ary function on, and for s,tq, let s=s1,,sn, with each si is restricted to vary on a bounded set 𝒮i ⊂ ℝ, that is, s is restricted to vary on a set S ⊂ ℝn. Then, the image of S (or the image of the sets Sk) with respect to q, in symbols Iq, is characterized to be

T=IqS=IqS1,,Sn=tR|sSsqt=tR|i=1nsiSit=qs1,,snR.

The preimage7 S of T is characterized to be the image of T with respect to the converse q ^ of q. In other words

S=Iq ^T=sRn|tTtq ^s.

Two notions essential for the investigation conducted in this article are those of a family of real functions and its image.

Definition 4.8 Real Family —

For 1 ≤ i ≤ n and 1 ≤ j ≤ k , an n-ary real family (a family of n-ary real functions, or in short, an-family), denoted QRsi:n;aj:k, is a set of real functions qRsi:n;aj:k subject to the following conditions

  • (i)

    q is a function rule,

  • (ii)

    si are variable symbols varying on bounded subsets Si of,

  • (iii)

    aj are constant symbols (coefficients) from bounded subsets Aj of, and

  • (iv)

    for each ajAj, qRsi:n;aj:k is continuous on the sets Si. We understand by the converse of Q, denoted Q ^R, the set of the converse relations q ^.

Note that a real family is generated by one function rule, that is, the functions qRsi:n;aj:k in Q all have the same rule q but different constant arguments. If the sets Aj are singletons, then the family Q reduces to exactly one n-ary real function. To clarify the matters, we give some examples.

Example 4.1 Real Families —

The following are instances of real families.

  • (i)
    Let Q be the family generated by the function rule qRsi:2;a=s12+as2, with the variables s1 and s2 vary respectively on the bounded sets 2,4 and 5,6 and the constant a is from the bounded set {3, 7}. The family Q has exactly the two binary functions
    qRsi:2;3=s12+3s2andqRsi:2;7=s12+7s2.
  • (ii)
    Let U be the family generated by the function rule uRs;a=as4, with the variable s varies on the bounded set 2,4 and the constant a is from the bounded set 1,2. The family U has an infinite number of unary functions. Among these are, for example
    uRs;1=s4,u Rs;32=3s42,...,etc.

We characterize the image of a real family as follows.

Definition 4.9 Image of a Real Family —

Let Q be a real family generated by a function rule t=qsi:n;aj:k, with siSi and ajAj. Then, the image of the family Q (or the image of the sets Si with respect to Q), denoted IQ, is the union of the images of S =S1,,SnRn with respect to each q in Q for all ajAj. That is

T=IQS=IQS1,,Sn=tR|i=1nsiSij=1kajAjt=qsi:n;aj:kR.

Obviously, for each q in Q, Iq⊆IQ. An immediate consequence of definition 4.9 and the well-known extreme value theorem (see Dawood, 2012) is the following important property.

Theorem 4.8 Main Theorem of Image Evaluation —

Let Q be a real family generated by a function rule qsi:n;aj:k, with siSi and ajAj. If Si and Aj are real closed intervals, then the image IQS1,,Sn of Si, with respect to the family Q, is in turn a real closed interval such that

IQS1,,Sn=minsiSiajAjqsi:n;aj:k,maxsiSiajAjqsi:n;aj:k.

If the sets Aj of coefficients are singletons, then the family is in turn a singleton and the image of IQ reduces to the usual image Iq of a real function q over real closed intervals

IqS1,,Sn=minsiSiqs1,,sn,maxsiSiqs1,,sn.

By referring to definition 4.6, we can characterize the important notion of the interval extension of a real family.

Definition 4.10 Interval Extension of a Real Family —

Let Q be an n-ary real family generated by a function rule qRsi:n;aj:k, with siSi and ajAj. We understand by an interval extension of Q an n-ary interval function qJSi:n;Aj:k of the same rule as q, and whose arguments are Si=EJSi and Aj=EJAj.

Clearly, if Si and Aj are real closed intervals, then Si=Si and Aj=Aj. We will henceforth deploy the predicate ExtqJ,QR to mean that an interval function qJ is the interval extension of the real family Q, or equivalently, the family Q is the real intension of the interval function qJ. If Aj are singletons, then the family Q is a singleton and we call qJ a simple extension of Q. If Si and Aj are singletons, then we call the point-valued interval function qJ a point extension of Q.

The following example will illustrate this point.

Example 4.2 Interval Extensions of Real Families —

Recall the real families Q and U of example 4.1. The interval extensions of Q and U are given respectively by

  • (i)

    qJSi:2;A=S12+AS2, with S1=EJ2,4=2,4, S2=EJ5,6=5,6, and A=EJ3,7=3,7.

  • (ii)

    uJS;A=AS4, with S=EJ2,4=2,4, and A=EJ1,2=1,2.

The previous discussion faces us with the reasonable question: does every interval function have a real intension? In order to answer this, we next define what a proper interval function is.

Definition 4.11 Proper Interval Function —

We say that an interval function qJ:DJJnJ is proper, in symbols PropqJ, iff it is set-theoretically definable in terms of a real function of the same rule. That is

PropqJqRqJS1,,Sn=wR|i=1nsiSiw=qRs1,,sn.

By definitions 4.11 and 4.5, the following result is derivable.

Theorem 4.9 Criteria for Proper Interval Functions —

Let ∘ ∈ { + ,  × } be a binary J-operation and ♢ ∈ { − , −1} be a definable unary J-operation. Then, the following statements are true.

  • (i)

    qJ,uJPropqJPropuJPropqJuJ,

  • (ii)

    qJPropqJPropqJ,

  • (iii)

    qJ,uJPropqJPropuJPropqJuJ.

In accordance with definition 4.11 and its previous consequence, we have then the following important result.

Theorem 4.10 Intensionality of an Interval Function —

An interval function is intensionable iff it is proper. In other words

PropqJQRExtqJ,QR.

For example all elementary interval functions are intensionable. On the contrary, degenerate functions such as the midpoint or radius of an interval are not proper and accordingly not intensionable. Definition 4.11 and the deductions from it can be easily generalized to proper Jm-valued functions, in which case their intensions will be families of ℝm-valued functions.

Toward axiomatizing a theory of a differential interval algebra, it remains to formalize the notions of differentiability of a real family and of an interval function. Henceforth, we will consider only families of unary real functions and their interval extensions. Accordingly, when there is no potential for ambiguity, we will write Qs, or simply Q, for the unary real family Qs;aj:k.

Next, we extend the differential operator to families of unary real functions.

Definition 4.12 Differential Operator for a Real Family —

Let QRs;aj:k be a unary real family in the real variable s and the real constants aj. For a nonnegative integer n, the n-differential operator of QRs;aj:k, denoted δnQRs;aj:k, is defined to be the set of all real functions δnqs;aj:k for every q ∈ Q and every constant aj.

We have yet nothing to tell us if a real family is differentiable. The following two definitions introduce, respectively, the notions of differentiability and continuous differentiability of a unary real family Qs;aj:k.

Definition 4.13 Differentiability of a Real Family —

A unary real family QRs;aj:k is n-differentiable at a real constant s0, in symbols diffnQ,s0, iff for every q in Q, s0domq, and q is n-differentiable at s0. That is

diffnQ,s0qQs0domqdiffnq,s0.

Definition 4.14 Continuous Differentiability of a Real Family —

A unary real family QRs;aj:k is continuously n-differentiable at a real constant s0, in symbols cdiffnQ,s0, iff for every q in Q, s0domq, and q is continuously n-differentiable at s0. That is

cdiffnQ,s0qQs0domqcdiffnq,s0.

In accordance with the above concepts, the differential operator for interval functions is then definable.

Definition 4.15 Interval Differential Operator —

Let n ≥ 0, and let qS be a unary interval function that has a real intension the family Qs. The n-differential J-operator of qS, denoted δnqS, is characterized to be the interval extension of δnQs. In other words, let δnQs=Us, then δnqS=uS.

In a manner analogous to differentiability in ℝ, the interval differentiability predicate is definable as follows.

Definition 4.16 Interval Differentiability Predicate —

Let n ≥ 0, let qJ1, and let S0domq be a J-constant symbol. The ternary n-differentiability J-predicate, denoted diffnq,S0, is defined by

diffnq,S0δnqS0J.

If diffnq,S0 is  true, then the interval function q is said to be n-differentiable at the closed interval S0.

Throughout this article, we will employ the following abbreviation.

diffnq1,q2,,qkS0diffnq1,S0diffnq2,S0...diffnqk,S0.

By means of definitions 4.15 and 4.16 plus a simple continuity argument, we have the following theorem that establishes the criteria for interval differentiability.

Theorem 4.11 Interval Differentiability Criteria —

An interval function qJJ1 is n-differentiable at a J-number S0 if and only if

  • (i)

    qJS is proper with a real intension Qs, and

  • (ii)

    Qs is continuously n-differentiable at every s0 ∈ S0.

From the fact that images of ℝ-functions are inclusion isotonic (Dawood, 2012), we have the next key result concerning interval enclosures of ℝ-families.

Theorem 4.12 Image Enclosure of a Real Family —

Let qs be a real function in a family Qs, with s is restricted to vary on a real closed interval S0, and let qS0 be the interval extension of Qs at S0. The following two sentences are true.

  • (i)

    qQIqS0IQS0qS0,

  • (ii)

    δqδQIδqS0IδQS0δqS0.

Moreover, finer enclosures of real families can be obtained via the subdivision method. The following corollary is implied by theorem 4.12.

Corollary 4.1 Subdivision Enclosure of a Real Family —

Recall the notation used intheorem 4.12, and let S0 be subdivided into n ≥ 1 subintervals. Then

IQS0i=1nqSiqS0.

Obviously, IQS0=limni=1nqSi.

To the best of our knowledge, in all interval literature, an interval-valued function is assumed to have singleton (real) constants and accordingly an interval function might be only an extension of a single real function. An interesting and important observation from the above discussion is that this presumption introduces an unnecessary restriction to the semantic of an interval function in the general sense. As above characterized, a proper interval function qJSi:n;Aj:k is an extension of a whole family of real functions and this family is a singleton if, and only if, the interval constants Aj are singletons.

With the aid of the notions now at hand, we can then axiomatize the theory TδJ of a differential interval algebra (henceforth a differential J-algebra).

Definition 4.17 Theory of a Differential Interval Algebra —

Let σ = { + ,  × ;  − , −1; 0, 1} be a class of non-logical signs, and let TJ be the theory of an interval S-semiring J=J;σJ. The theory TδJ of a differential J-algebra Jδ=J;σJ;δ is the deductive closure of TJ together with the following two axioms.

  • (i)

    q,uJ1δq+u=δq+δu,

  • (ii)

    q,uJ1δq×u=q×δu+u×δq.

Consider the constant interval functions qS=0J and uS=1J. With the aid of definition 4.15, obviously δ0J=δ1J=0J. More generally, for any interval constant symbol A, δA=0J and δAS=A. Accordingly, the set J can be defined thus: J=qJ1|δq=0J. On grounds of definition 4.15 and axioms (i) and (ii) of the preceding definition, further properties of interval differentiation can be derived analogously.

A Categorical Axiomatization of Interval Differentiation Arithmetic

Building on the system TδJ of a differential J-algebra axiomatized in the previous section, the present section provides a rigorous mathematical foundation for interval differentiation arithmetic (henceforth ΔJ-arithmetic). We are almost ready to lay out an axiom system for the theory TΔJ of interval differentiation numbers (henceforth ΔJ-numbers) as a two-sorted extension of TδJ. By virtue of the mathematical underpinnings presented in ‘A differential interval algebra’, we axiomatize, in the present section, the basic operations of TΔJ and prove some of their fundamental properties. Moreover, we prove categoricity and consistency of ΔJ-arithmetic.

An obvious starting point is to define interval differentiation n-tuples.

Definition 5.1 Interval Differentiation n-Tuples —

Let Jδ=J;σJ;δ be a differential J-algebra, let q be a unary J-function, and for an integer n ≥ 0, let Jn be the n-th Cartesian power of J. The set of all interval differentiation n-tuples over J, with respect to an individual J-constant S0J, is characterized to be

nU J=QJn+1|qJ1Q =δ0qS0,δ1qS0,,δnqS0S0domqdiffnq,S0.

An interval differentiation n-tuple is thus an ordered n-tuple of J-constants. Hereafter, we will usually write q, δq,..., δnq for δ0qS0, δ1qS0, …, δnqS0, respectively. The present article is concerned with dyadic interval differentiation tuples, that is n-tuples with n = 1; and we will hereon adopt the name “interval differentiation numbers” (“ ΔJ-numbers”, or “ ΔJ-pairs ”) for dyadic interval differentiation tuples. Let UJ designate the set of ΔJ-numbers at some J-constant S0, and let the letters Q, U, and V, or equivalently the pairs q,δqS0, u,δuS0, and v,δvS0, be variable symbols varying on the set UJ of ΔJ-pairs. Also, let the letters A, B, and C, or equivalently a,0JS0, b,0JS0, and c,0JS0, designate constants of UJ. In particular, we use 1UJ to designate the ΔJ-number 1J,0JS0 and 0UJ to designate the ΔJ-number 0J,0JS0. Moreover, it is convenient for our purpose to define a proper subset of UJ as

UJ,0=QUJ|Q =q,0JS0.

We are now ready to axiomatize the theory TΔJ of an interval differentiation algebra (or a ΔJ-algebra) over an interval S-semiring.

Definition 5.2 Theory of Interval Differentiation Algebra —

Let σ = { + ,  × ;  − , −1; 0, 1} be a class of non-logical signs, and let q,δqS0, u,δuS0, and v,δvS0 be in UJ. An interval differentiation algebra (or, in short, a ΔJ-algebra) over a differential J-algebra Jd=J;σJ;δ is a two-sorted structure UJ=UJ;J;σUJ. The theory TΔJ of 𝔘𝒥 is the deductive closure of the system TδJ of 𝔍d and the following set of axioms.

  • (IDA1)

    ΔJ-equality. q,δqS0=UJu,δuS0qS0=JuS0δqS0=JδuS0,

  • (IDA2)

    Binary ΔJ-operations. +,×q,δqS0UJu,δuS0=qJu,δqJuS0,

  • (IDA3)

    Unary ΔJ-operations. 10JqS0UJq,δqS0=Jq,δJqS0.

The intended model of the theory TΔJ corresponds the sets “ J” and “ UJ” to the sets of J-numbers and ΔJ-numbers, respectively, and the symbols “ J”, and “ J” to the binary and unary J-operations. When the context is clear, for simplicity henceforth, we will drop the subscripts “ UJ”, “ J”, and “ S0”. Also, we will usually write the algebraic structure 𝔘𝒥 as UJ;+UJ,×UJ;0UJ,1UJ, omitting the set J.

The inclusion and membership relations for ΔJ-numbers can be defined as follows.

Definition 5.3 Inclusion Relation on ΔJ-Numbers —

The inclusion relation on ΔJ-numbers, denoted UJ, is defined as follows.

Q,UUJQUJUqJS0uJS0δqJS0δuJS0.

Definition 5.4 Membership Relation in ΔJ-Numbers —

The membership relation in ΔJ-numbers, denoted UJ, is defined as follows.

qURQUJqUJQqRs0qJS0δqRs0δqJS0.

An important notion for our purposes is that of a point ΔJ-number.

Definition 5.5 Point ΔJ-Number —

By a point (or singleton) ΔJ-number, denoted q=q,δqS0, we understand a ΔJ-number whose all components are point intervals, that is qS0 and δqS0 are in Js.

The set of all point ΔJ-numbers will be denoted by Uq. In the sequel, we will make use of the following theorem that establishes the criteria when a ΔJ-number is a singleton.

Theorem 5.1 Criteria for Point ΔJ-Numbers —

A ΔJ-number q,δqS0 is point iff

  • (i)

    q is a constant point-valued function, that is q=cJs, or

  • (ii)

    S0=s0Js and each constant in the rule of q is a point interval.

Proof

The proof is immediate from theorem 4.6.

      □

By means of definitions 4.15 plus the rules of differential sum and product, axiomatized in definitions 4.17, the following theorem is easily derivable from the theory TΔJ.

Theorem 5.2 Algebraic Operations of ΔJ-Numbers —

Let q,δqS0 and u,δuS0 be two ΔJ-numbers. Then, the binary and unary ΔJ-operations are formulated as follows.

  • (i)

    ΔJ-addition. q,δq+UJu,δu=q+Ju,δq+Jδu,

  • (ii)

    ΔJ-multiplication. q,δq×UJu,δu=q×Ju,δq×Ju+Jq×Jδu,

  • (iii)

    ΔJ-negation. UJq,δq=Jq,Jδq.,

  • (iv)

    ΔJ-reciprocal. 0JqS0q,δq1UJ=q1J,Jq2×Jδq.

To complete our characterization of ΔJ-arithmetic, we define as customary ΔJ-subtraction and ΔJ-division respectively as QU=Q+U and Q÷U=Q×U1.

With the aid of the meta-theoretic notions characterized in definitions 2.2–2.4, we are able to proceed towards proving three important meta-theorems about the theory TΔJ of ΔJ-numbers, concerning respectively existence, categoricity and consistency of a ΔJ-algebra.

Theorem 5.3 Existence of aΔJ-Algebra

There exists at least one ΔJ-algebra.

Proof

Since the theory TJ of a J-algebra has the model JR;σJ of J-numbers, it follows that the theory TΔJ has a model UJ;J;σUJ, and thus existence of a ΔJ-algebra is proved.

      □

Theorem 5.4 Categoricity of ΔJ-Arithmetic —

The theory TΔJ of ΔJ-numbers is categorical.

Proof

The theorem follows from the categoricity of the theory TJ of interval algebra by an argument analogous to the one used in theorem 4.1.

      □

That is, the theory TΔJ uniquely characterizes the algebra of ΔJ-numbers, and the structure UJ;+UJ,×UJ;0UJ,1UJ is, up to isomorphism, the only possible model of TΔJ. To reiterate, in accord to theorem 5.4, the system TΔJ, axiomatized in definition 5.2, is the “best” axiomatization of ΔJ-numbers, in the sense that it rightly accounts, up to isomorphism, for every structure of ΔJ-arithmetic.8

The next theorem establishes the consistency of the theory TΔJ of ΔJ-numbers.

Theorem 5.5 Consistency of ΔJ-Arithmetic —

The theory TΔJ of ΔJ-numbers is consistent.

Proof

In accord to definition 2.4, the proof is immediate from theorem 5.3. The theory TΔJ is satisfiable by the model UJ;+UJ,×UJ;0UJ,1UJ and thus is consistent.

    □

Owing to the categoricity theorem for TΔJ, the algebraic properties of J-numbers are naturally assumed priori. Therefore, whenever unambiguous, hereon we will use these properties without further mention.

Noteworthy, by virtue of the theory developed so far, we have the profound results that each ΔJ-number represents a guaranteed interval enclosure of the image of a whole family of ℝ-functions and their derivatives and accordingly that a ΔJ-number is an interval extension of every Δ-number that corresponds to each function in the real family (See ‘Machine implementation of interval auto-differentiation’ for clarifying numerical examples). In consequence of theorem 4.12, these important results are made precise in the following immediate theorem and its corollary.

Theorem 5.6 Differential Enclosure of a Real Family —

Let Q be a unary real family continuously differentiable on a real closed interval S0 and let qJ be its interval extension. Then, for every q in Q

IqRS0,IδqRS0UJIQS0,IδQS0UJqJS0,δqJS0.

Corollary 5.1 Interval Extension of a Δ-Number —

Let q be a real function continuously differentiable on a real closed interval S0. Then, for every s0 ∈ S0,

qR,δqRs0UJqJ,δqJS0.

Finally, let us note that we can get sharper enclosures of the pair IQS0,IδQS0 with the aid of the subdivision method. In consequence of theorems 5.2 and 5.6 we are led to the following theorem.

Theorem 5.7 Subdivision Theorem for ΔJ-Numbers —

Recall the notation used intheorem 5.6, and let S0 be subdivided into n ≥ 1 subintervals. Then

IQS0,IδQS0UJi=1nq JSi,δqJSiUJqJS0,δqJS0.

Moreover, IQS0,IδQS0=limni=1nqJSi,δqJSi.

Differentiation Extension of Interval Functions and Higher-Order Auto-Differentiability

We aim to fully address and compute higher order and partial auto-derivatives using only dyadic ΔJ-numbers (ΔJ-pairs), and without resorting to defining any sort of n-dimensional Grassmann algebras. Towards this end, we are to extend the theory TΔJ, by introducing the notion of a differentiation extension of J-functions, characterizing differentiability for ΔJ-numbers, and establishing their differentiability conditions.

In view of our definition of ΔJ-numbers, the following alternate characterization of interval differentiability is at our disposal.

diff1q,S0δqS0JRq,δqS0UJ.

In order to have ΔJ-functions beyond the rational functions defined in ‘A categorical axiomatization of interval differentiation arithmetic’, an extension principle should be introduced. Thus we require to extend J-functions to ΔJ-functions. In accord to the above characterization, we have the next definition.

Definition 6.1 Differentiation Extensions of J-Functions —

For k ∈ {1, …, n}, let ukJ1 be differentiable at S0domuk, that is for each uk there is Uk=ukS0,δukS0UJ. Let QJu1,,un be an n-place J-function of u1, …, un which is differentiable at S0. A differentiation extension of QJ is an n-place ΔJ-function QUJU1,,Un defined to be

QUJU1,,Un=QJu1,,un,δQJu1,,un,

and obtained from QJ by replacing, in QJ, each occurrence of a J-function symbol uk by the corresponding ΔJ-variable symbol Uk.

The definition is so framed that since diff1QJ,S0 is true, the differentiation extension QUJ of QJ is in UJ. Thus, QJ and QUJ are both defined by the same symbolic rule but with different types of arguments (variables); QJ is a J-function whereas QUJ is a ΔJ-function. By analogy with rational J-functions, a rational ΔJ-function is a (multivariate) ΔJ-function obtained by the application of a finite number of the binary and unary algebraic ΔJ-operations UJ+UJ,×UJ and UJUJ,1UJ. Hereon, if the function type is apparent from the context, the subscripts J and UJ will be omitted. For instance, whenever unambiguous, we use the notations Qu1,,un and QU1,,Un for, respectively, a J-function and its differentiation extension.

Here it will suffice to give an example. Let the J-functions u1S= cosS and u2S=S3 be both differentiable at some S0, and let QJu1,u2 be differentiable at S0 such that

QJu1,u2=u1S+u2S= cosS+S3.

The differentiation extension of QJ is then

QUJU1,U2=u1,δu1S0+u2,δu2S0=cosS,δcosSS0+S3,δS3S0=cosS+S3,δcosS+S3S0=QJu1,u2,δQJu1,u2S0

By virtue of our definition of the extension principle for J-functions (definition 6.1), we are able to define fundamental ΔJ-functions. For example, replacing Q by the “cos” function, one obtains the trigonometric ΔJ-function cosu,δuS0=cosu,δcosuS0. In ‘Machine implementation of interval auto-differentiation’, we will give further discussion on differentiation extensions of J-functions as well as more illustrative numerical examples.

Here, let us stress that restricting our discussion to single-variable J-functions is not a loss of generality, since an n-variable J-function can be viewed as a class of n single-variable J-functions. What is noteworthy in addition is that higher-order interval auto-derivatives can be computed in the framework of our system TΔJ of dyadic ΔJ-numbers (ΔJ-pairs). With the aid of definition 6.1, we next characterize the n-differential operator and the n-differentiability predicate for ΔJ-pairs.

Definition 6.2 n-Differential Operator of a ΔJ-Number —

For an integer n ≥ 0, the n-differential operator of a ΔJ-pair U =u,δuUJ, in symbols δnU, can be characterized recursively by

  • (i)

    δ0U = U,

  • (ii)

    δ1U=δu,δδu=δu,δ2u=δ1δ0U,

  • (iii)

    n1δnU=δnu,δn+1u=δ1δn1U.

Definition 6.3 n-Differentiability Predicate for a ΔJ-Number —

Let n ≥ 0. The n-differentiability predicate for a ΔJ-pair U =u,δuUJ, in symbols diffnU, is characterized by diffnUδnUUJ.

Consequently, the next theorem, concerning higher-order auto-differentiability of J-functions, is provable.

Theorem 6.1 n-Differentiability Condition for a ΔJ-Number —

Let n ≥ 0. Then for a ΔJ-pair U =u,δuUJ, we have diffnUdiffn+1u,S0.

Proof

It is clear that if the J-function u is n+1-differentiable at S0, then, for n ≥ 0, δnu,δn+1uS0UJ, and the proof follows by definition 6.3.

      □

Our main objective in this section is to show that higher-order interval auto-derivatives are computable using only dyadic ΔJ-numbers (ΔJ-pairs). Towards this end, we need readily available Leibniz’s rules for ΔJ-numbers. By definitions 6.2 and 4.17, plus theorem 5.2, the following theorem is derivable.

Theorem 23 Leibniz’s Rules for ΔJ-Numbers —

Let Q and U be ΔJ-numbers. Then

  • (i)

    δQ+U=δQ+δU,

  • (ii)

    δQ×U=Q×δU+U×δQ.

By virtue of this theorem, and applying induction, the general Leibniz rule for ΔJ-numbers can be easily established. Let Q and U be n-times differentiable ΔJ-numbers. Then

δnQU=k=0nn!k!nk!δnkQδkU.

A nice consequence that we wish to point out is that with the general Leibniz rule for ΔJ-numbers at our disposal, and once we have in our machine implementation differentiation kernels (seeds) for the higher order dyads u,δu, δu,δ2u, …, δnu,δn+1u, it is readily possible to compute higher order auto-derivatives by doing only dyadic ΔJ-arithmetic. In other words, within the framework of the theory TΔJ of dyadic ΔJ-numbers, higher order auto-differentiation is directly realizable without resorting to defining a Grassmann algebra for n-ary vectors of the form u,δu,,δnu. This, along with some illustrative examples, will be discussed further in ‘Machine implementation of interval auto-differentiation’.

Note also that considering only single-variable J-functions is not a loss of generality, since an n-variable J-function can be viewed as a class of n single-variable J-functions. Accordingly, partial auto-derivatives, gradients and Hessians are readily computable.

Finally, let us conclude this section with a few additional comments. In accord to definition 5.5, in the theory TΔJ of ΔJ-numbers, a singleton ΔJ-number defines a Δ- number. That is, all the results of this section apply to Δ- arithmetic as well. Moreover, computing the ΔJ- number q,δqS0 is very useful in practice. In engineering and physical sciences, a recurring problem is to compute the derivatives under parametric uncertainty (For further details, the reader may consult, e.g., Dawood, 2014, Dawood & Dawood, 2022; Kulisch, 2013; Moore, 1966, Neidinger, 2010; Sommer, Pradalier & Furgale, 2016, and Tingelstad & Egeland, 2017). Also noteworthy here is that with a few basic modifications, the categorical system TΔJ axiomatized in this text can be extended analogously to compute fuzzy auto-derivatives (For further details on fuzzy analysis, see, e.g., Goetschel & Voxman, 1986 and Puri & Ralescu, 1983).

The Algebraic Structure of Interval Differentiation Arithmetic

Building on the parts of the theory established in ‘A differential interval algebra’ and ‘A categorical axiomatization of interval differentiation arithmetic’, this section provides a detailed investigation of the algebraic structure of ΔJ-numbers. By virtue of the categoricity of the theory TΔJ (theorem 5.4), the properties of J-numbers are assumed priori.

We commence this section by establishing the algebraic properties of ΔJ-addition and ΔJ-multiplication.

Theorem 7.1 Algebraic Properties of ΔJ-Addition —

The following algebraic properties hold for ΔJ-addition.

  • (i)

    Identity element for +.QUJ0UJ+Q=Q+0UJ=Q,

  • (ii)

    Inverses for +.Q,UUJQ+U=0UJQUqU=Q,

  • (iii)

    Cancellativity for +.Q,U,VUJQ+V=U+VQ=U,

  • (iv)

    Commutativity for +.Q,UUJQ+U=U+Q,

  • (v)

    Associativity for +.Q,U,VUJQ+U+V=Q+U+V.

Proof

The proof for (i) follows from theorem 5.2. (ii) follows from theorems 4.6 and 5.2. By cancellativity, commutativity and associativity of J-addition, (iii), (iv) and (v) are easily provable by theorem 5.2.

      □

Theorem 7.2 Algebraic Properties of ΔJ-Multiplication —

The following algebraic properties hold for ΔJ-multiplication.

  • (i)

    Annihilating element for ×.QUJ0UJ×Q=Q×0UJ=0UJ,

  • (ii)

    Identity element for ×.QUJ1UJ×Q=Q×1UJ=Q,

  • (iii)

    Inverses for ×.Q,UUJQ×U=1UJQUqU=Q10qS0,

  • (iv)

    Cancellativity for ×.Q,U,VUJQ×V=U×VQ=U0vS0,

  • (v)

    Commutativity for ×.Q,UUJQ×U=U×Q.

Proof

The proof for (i) and (ii) follows immediately from theorem 5.2. For (iii), assume that Q×U=1UJ=1,0, which yields, by theorems 5.2, 4.6, and the invertibility properties of J-arithmetic, that QUqU=Q10qS0. The converse direction is easily derivable by assuming the right hand side. By the cancellative properties of J-arithmetic, (iv) follows from theorems 5.2 and 4.2. By commutativity of J-multiplication, (v) is easily derivable from theorem 5.2.

      □

Thus, not all elements of UJ are invertible for addition or multiplication. A ΔJ-number Q is invertible for addition if, and only if, it is a point ΔJ-number and is invertible for multiplication if, and only if, it is a point ΔJ-number with 0qS0. Also, unlike interval arithmetic, ΔJ-arithmetic has nonzero zero divisors, since 0,α¯,α¯×0,β¯,β¯=0UJ. Moreover, ΔJ-multiplication is not associative, which is figured in the following theorem.

Theorem 7.3 Associativity of ΔJ-Multiplication —

In general, ΔJ-multiplication is not associative. That is

Q,U,VUJQ×U×VQ×U×V.

Proof

We prove the statement by a counter example. Let q, u, and v be J-functions defined respectively as qS=S2+S, uS=S, and vS=3×S2. Then, for S0=1,1, we have the corresponding ΔJ-numbers Q=q,δq=1,2,1,3, U=u,δu=1,1,1, and V=v,δv=0,3,6,6. Now, by theorem 5.2, we have Q×U×V=6,6,21,27 and Q×U×V=6,6,24,27. Hence Q×U×VQ×U×V.

      □

However not associative, ΔJ-multiplication satisfies two weak associativity properties namely subalternativity and flexibility. These are established in the next two theorems.

Theorem 7.4 Subalternativity of ΔJ-Multiplication —

ΔJ-multiplication is subalternative, that is

Q,UUJQ × UQ × Q×U.

Proof

Let Q and U be in UJ. Then, from the associative and subdistributive properties of J-arithmetic, we have

Q × U=q×q×u,δq×q×u+q×δq×u+q×δu=q×q×u,δq×q×u+q×δq×u+q×δuq×q×u,δq×q×u+q×δq×u+q×q×δu=q×q×u,2×q×δq×u+q×q×δu=q×q,2×q×δq×u,δu=q,δq×q,δq×u,δu=Q × Q×U.

Therefore, multiplication is subalternative in UJ.

      □

Theorem 7.5 Flexibility of ΔJ-Multiplication —

ΔJ-multiplication is flexible, that is

Q,UUJQ×U×Q = Q×U×Q.

Proof

The theorem follows by the fact that ΔJ-multiplication is commutative (theorem 7.2).

      □

Now, we turn to the algebraic property of distributivity. Like the case with J-arithmetic, ΔJ-multiplication is not distributive over ΔJ-addition. For example, consider the three ΔJ-numbers Q, U, and V given in the proof of theorem 7.3. Then we have Q×U+V=4,8,14,26, while Q×U+Q×V =5,8,19,26, and hence Q×U+VQ×U+Q×V. In contrast, ΔJ-arithmetic is subdistributive (S-distributive). This is established in the next theorem.

Theorem 7.6 Subdistributivity in ΔJ-Numbers —

ΔJ-arithmetic is subdistributive, that is

Q,U,VUJV×Q+UV×Q+V×U.

Proof

Let Q, U, and V be any three ΔJ-numbers. According to theorem 5.2, and assuming the properties of interval operations, we have

V×Q+U=v×q+u,δv×q+u+v×δq+δuv×q+v×u,δv×q+δv×u+v×δq+v×δu=v×q+v×u,δv×q+v×δq+δv×u+v×δu=v×q,δv×q+v×δq+v×u,δv×u+v×δu=V×Q+V×U,

and therefore multiplication subdistributes over addition in UJ.

      □

The preceding theorem establishes left S-distributivity. Right S-distributivity follows from commutativity of ΔJ-multiplication.

We will now make use of the previous results to prove the next theorem about the algebraic structure of ΔJ-numbers.

Theorem 7.7 Commutative NA S-Semiring of ΔJ-Numbers —

The structure 𝔘𝒥 = 〈U𝒥; +U𝒥, ×U𝒥0U𝒥1U𝒥 is a commutative ×-NA S-semiring in which ×UJ is subalternative and flexible.

Proof

By theorem 7.1, the additive structure UJ;+UJ;0UJ is a cancellative commutative monoid. By theorems 7.2 and 7.3, the multiplicative structure UJ;×UJ;1UJ is a noncancellative commutative NA-monoid. In consequence of theorem 7.6, ×UJ subdistributes over +UJ. By theorem 7.2, 0UJ is an absorbing element for ×UJ. According to definition 2.10, the structure 𝔘𝒥, of ΔJ-arithmetic, is therefore a commutative ×-NA S-semiring. Lastly, by theorems 7.4 and 7.5, ×UJ is subalternative and flexible.

    □

Lastly, we prove two special results on the structure Uq of point ΔJ-numbers.

Theorem 7.8 Sub-Algebraicity of Point ΔJ-numbers —

The structure Uq of point ΔJ-numbers is a subalgebra of the structure 𝔘𝒥 of ΔJ-numbers. In symbols UqUJ.

Proof

By definition of ΔJ-numbers, 0UJ and 1UJ are both elements of the set UqUJ. By theorems 4.6 and 5.1, for +UJ,×UJ and for any q and u in Uq, qu is in turn in Uq. Then, the criteria for the subalgebraicity of Uq is established and therefore UqUJ.

      □

Theorem 7.9 Commutative Ring of Point ΔJ-Numbers —

The structure Uq=Uq;+UJ,×UJ;0UJ,1UJ is a commutative unital ring in which every element whose first component is nonzero has an inverse for ×UJ.

Proof

Restricting the operations +UJ and ×UJ to the set Uq in theorems 7.3, 7.6, and 7.2, it follows respectively that ×UJ is associative, ×UJ distributes over +UJ, and every element whose first component is nonzero has a ×UJ-inverse in Uq. The proof is thus established in consequence of theorem 7.7.

      □

That is, the multiplicatively non-associative S-semiring 𝔘𝒥 of ΔJ-numbers has as a subalgebra a commutative unital ring Uq.

Monotonicity and Isomorphism Theorems for Interval Differentiation Numbers

In this section, some monotonicity and isomorphism theorems for ΔJ-numbers are established, and finally a corollary concerning the structure of Δ-numbers is entailed. A first key result we will next prove is the inclusion monotonicity theorem for ΔJ-arithmetic, which establishes that the inclusion relation is compatible with the algebraic ΔJ-operations.

Theorem 8.1 Inclusion Monotonicity for ΔJ-Numbers —

Let Q1, Q2, U1, and U2 be ΔJ-numbers such that Q1U1 and Q2U2. Let +,× be a binary ΔJ-operation and ,1 be a definable unary ΔJ-operation. Then

  • (i)

    Q1Q2U1U2,

  • (ii)

    Q1⊆♢U1.

Proof

By hypothesis, we have Q1U1 and Q2U2. Then, according to definition 5.3 and theorem 4.4, we have

Q1+Q2=q1,δq1+q2,δq2=q1+q2,δq1+δq2u1+u2,δu1+δu2=U1+U2.

Analogously, Q1 × Q2U1 × U2 and ♢Q1⊆♢U1. This completes the proof.

      □

In consequence of this theorem, from the fact that qs0,δqs0QqQ, we have the following important special case.

Corollary 8.1 Membership Monotonicity for ΔJ-Numbers —

Let Q and U be ΔJ-numbers with q ∈ Q and u ∈ U. Let +,× be a binary ΔJ-operation and ,1 be a definable unary ΔJ-operation. Then

  • (i)

    qu ∈ QU,

  • (ii)

    q ∈ ♢Q.

Two important results, concerning isomorphism theorems for ΔJ-arithmetic, are figured in the following theorems.

Theorem 8.2 Isomorphicity to Δ-Numbers —

The structure Uq=Uq;+UJ,×UJ is isomorphic to the structure 𝔘 = 〈U; +U, ×U of Δ-numbers. In symbols UqUR.

Proof

Let ι:URUq be the mapping from U to Uq given by

ιq=q=q,δq.

By means of the fact that point intervals are isomorphic to real numbers (theorem 4.5), it is straightforward to show that ι is an isomorphism from U onto Uq.

      □

Theorem 8.3 Isomorphicity to J-Numbers —

The structure UJ,0=UJ,0;+UJ,×UJ is isomorphic to the commutative S-semiring 𝔍 = 〈𝒥; +𝒥, ×𝒥 of interval numbers.

Proof

Let ι:JRUJ,0 be the mapping from 𝒥 to UJ,0 defined by ια¯,α¯=α¯,α¯,0. Obviously, ι is an isomorphism from 𝒥 onto UJ,0 and therefore UJ,0JR.

      □

Accordingly, up to isomorphism, the sets 𝒥 and UJ,0 are equivalent, and therefore the structure UJ,0 is a commutative S-semiring.

In consequence of theorems 7.9 and 8.2, we have the following corollary concerning the structure of Δ-numbers.

Corollary 8.2 Commutative Ring of Δ-Numbers —

The algebra 𝔘 = 〈U; +U, ×U0U1U of Δ-numbers is a commutative unital ring in which every element whose first component is nonzero has an inverse for ×U.

Machine Implementation of Interval Auto-Differentiation

In this last section, we consider some aspects of the computational implementation of interval auto-differentiation in the framework of our theory TΔJ of ΔJ-numbers (ΔJ-pairs). The algorithm of the theory is coded in Common Lisp as a part of the software package InCLosure (InCL) (Dawood, 2020 and Dawood, 2023). After providing a mathematical flavor of the algorithm,9 we offer insights of the theory by giving some simple examples that illustrate how to concurrently compute guaranteed enclosures of images of families of real functions and their derivatives, then we deal with a more sophisticated problem whose result values will be calculated to an arbitrary precision using InCL commands, and finally, we give a brief account of how to calculate higher order interval auto-derivatives using the theory TΔJ of dyadic ΔJ-arithmetic.

In a way analogous to that of Δ-arithmetic (see, e.g., Dawood, 2014 and Dawood & Megahed, 2019), ΔJ-arithmetic can be machine realized. Toward calculating the ΔJ-pair of a differentiable J-function at S0J, we begin with a minimal class of symbolic rules of differentiable J-functions and their derivatives which acts as ΔJ-seeds (ΔJ-kernels) for carrying out the computation. As examples of ΔJ-kernels, one can start with the following elementary ΔJ-pairs.

ASb,AbSb1,lnS,1/S,eS,eS,sinS,cosS,cosS,sinS, and so forth.

The class of unary ΔJ-kernels will be denoted by P1, and we will understand by J-kernels a class K1=qJ1|q,δqP1. Accordingly, the first-order interval auto-derivative of a J-function  qS, at S0J, can be viewed as

qS,S0InputΔJ-KernelsChain RuleΔJ-AlgebraOutputq,δqS0.

To further illustrate, we next give some examples that can be worked by hand.

Example 9.1 ΔJ-Number for the Cosine Function —

Consider the J-function

qS= cosSwiths¯0.

Computing the ΔJ-pair q,δq1,4 yields

q,δq1,4=cos1,4,sin1,4/21,41,4=cos1,4,sin1,4/21,41,4=cos1,2,sin1,2/21,21,4=cos2,cos1,sin1,1/2,41,4=cos2,cos1,1/2,1/4sin1,11,4=cos2,cos1,1/2,1/4sin11,4.

The first component of the resulting ΔJ-pair, cos2,cos1, is a guaranteed enclosure of the image of the real function qs= coss over the interval 1,4, while the second component, 1/2,1/4sin1, is a guaranteed enclosure of the image of the real function δqs over the same interval. For example,

q4= cos2cos2,cos1=q1,4,δq4=sin2/41/2,1/4sin1=δq1,4.

Example 9.2 ΔJ-Number for a Family of Real Functions —

Let Q be the family of real functions

qRs,a,b,c=as3+bs2+c,

where the variable sS=1,2 and the constants a, b, and c are respectively in 1,1, 0,1, and 0,2. We want to compute enclosures of the images of the family Q and its derivative δQ. The interval extension of Q is the interval function

qS,1,1,0,1,0,2=1,1S3+0,1S2+0,2.

The ΔJ-pair q,δq1,2 at the interval S0=1,2 is computed as follows.

q,δq1,2=1,11,23+0,11,22+0,2,31,11,22+20,11,21,2=1,11,8+0,10,4+0,2,3,30,4+0,21,21,2=8,8+0,4+0,2,12,12+2,41,2=8,14,14,161,2.

The intervals 8,14 and 14,16 are guaranteed enclosures of the images of the family Q and its derivative δQ over the interval 1,2 respectively. In fact, the interval 8,14 is the exact image, IQ, of Q over the interval 1,2.

The previous example clearly embodies that the theory TΔJ presented in this text is powerful and reliable for simultaneously providing guaranteed enclosures of families of real functions and their derivatives. For instance, the following real functions and their derivatives are members of the families Q and δQ of example 9.2 respectively.

q1s=s3+s2+2,q2s=s3δq1s=3s2+2s,δq2s=3s2.

The exact images of these functions are included in the result of example 9.2 as follows.

Iq11,2=2,148,14=q1,2,Iq21,2=8,18,14=q1,2,Iδq11,2=13,1614,16=δq1,2,Iδq21,2=12,014,16=δq1,2.

The overestimation of the exact image of δQ in example 9.2 naturally arises from the interval dependency problem. One noteworthy virtue of the subdivision theorem for ΔJ-numbers (theorem 5.7) is that one can deploy the subdivision method to decrease the overestimation and hence obtain arbitrarily sharper intervals that get closer to the exact image. The next example clarifies the matters.

Example 9.3 ΔJ-Number with Subdivision —

Consider again the J-function of example 9.2 given by

qS,1,1,0,1,0,2=1,1S3+0,1S2+0,2.

We desire to compute the ΔJ-pair q,δq1,2 at the interval S0=1,2 by subdividing the interval 1,2 into the three subintervals 1,0, 0,1, and 1,2 of width 1. Then, we compute the ΔJ-numbers q,δq1,0, q,δq0,1, and q,δq1,2 as follows.

q,δq1,0=1,11,03+0,11,02+0,2,31,11,02+20,11,01,0=1,11,0+0,10,1+0,2,3,30,1+0,21,01,0=1,1+0,1+0,2,3,3+2,01,0=1,4,5,31,0,

Similarly, q,δq0,1=1,4,3,50,1 and q,δq1,2=8,14,12,161,2. Hence the resulting ΔJ-number by the subdivision technique is given by

1,41,48,14,5,33,512,16=8,14,12,16.

Compared to the naive result in example 9.2, the resulting ΔJ-pair with the subdivision method gives a better enclosure, fortunately it is the exact result, for the image of the first derivative. That is

8,14,12,168,14,14,16.

In the previous example, the subdivision technique with only three subintervals yields the exact images. In some problems, when the result is far away from the exact images, increasing the number of subintervals could give arbitrarily better enclosures of the images but with the disadvantage of increased computational time. Now, we move on to compute, to an arbitrarily sharper intervals, the result of a more sophisticated example using the software package InCLosure.

Example 9.4 Interval Auto-Differentiation using InCLosure —

Let qS be a J-function defined by

qS=0.5,0.4S103,2S6 sine5/8sincose3S+4eS5/30,2.

The package InCLosure guarantees arbitrarily sharper intervals which are restricted only by the machine’s computational capabilities. The ΔJ-pair q,δq1,2 for the J-function q at 1,2 can be computed using the following InCL command.

 
IAD "[-0.5,-0.4]*X^10-[-3,-2]*X^6-sin(e^(5/8*(sin(cos(e^(3^X+e^-X/5/3*4))))))-[0,2]" 
 1 "X=[-1,2]" 

This will result in

515.0,191.4427985505345723919,147294.44809204868892190371,145297.44809204868892190371.

The second parameter, “1”, in the preceding InCL command, is the number of subdivisions (which means no subdivisions). To get sharper results, one can increase the number of subdivisions arbitrarily.Table 1 shows InCLosure values for the ΔJ-pair q,δq1,2 computed by subdividing 1,2 into 1, 5, 10, 20, 50, and 100 subintervals.

Table 1. InCLosure values for interval automatic differentiation.

Subdivisions InCLosure values for the pair q,δq1,2
1 (no subdivision) 515.0,191.4427985505345723919,147294.44809204868892190371,145297.44809204868892190371
5 499.940928,179.8726123514945723919,77358.50282798795561838428,75356.39752085195561838428
10 466.724862,110.8030425325745723919,75987.53606834857190112981,73699.56740236057190112981
20 434.82104971875,39.8387755903470333294,75400.15661633499832211557,72660.87036293595925961557
50 408.379583369088,19.78286094370217251581,75067.10983026439761984353,71855.91048800864805542753
100 398.096543381742,14.8675821888729383479,74958.36726177567900543373,71542.87733701227696496573

Example 9.4 shows that computing auto-derivatives under interval uncertainty, with the arbitrarily sharper and guaranteed results of InCLosure, is competitive and obviously preferable to the ordinary numerical approximation methods.

In closing, let us get a grip on how to compute higher-order interval auto-derivatives in the framework of our theory TΔJ of dyadic ΔJ-numbers (ΔJ-pairs). By virtue of Leibniz’s rules for ΔJ-numbers (theorem 6.2), and once we have extended the class P1 of ΔJ-kernels by including the higher order dyads u,δu, δu,δ2u, …, δnu,δn+1u, for an arbitrary n, we can compute higher order interval auto-derivatives by doing only dyadic ΔJ-arithmetic. Consequently, within our own development, one can implement higher order interval auto-differentiation without resorting to defining any sort of n-dimensional Grassmann algebra for n-ary vectors of the form u,δu,,δnu. With this in mind, let us consider the J-function

qS= cosS2+ lnS.

We desire to compute the ΔJ-pairs δ1q,δ2q and δ2q,δ3q at some S0J. Then, the class K1 of unary J-kernels should include ΔJ-pairs up to the third order. That is, for q1S= cosS, q2S=S2, and q3S= lnS, we should have respectively the following ΔJ-kernels

cosS,sinS,sinS,cosS,cosS,sinS;S2,2S,2S,2,2,0;lnS,1S,1S,1S2,1S2,2S3.

Now to compute δq,δq=δq,δ2q at S0, we have the following dyadic ΔJ-pairs for respectively cosS2 and lnS

δq1q2,δ2q1q2S0=δq1q2S0×δq2S0,δ2q1q2S0×δq2S02+δq1q2S0×δ2q2S0=2S0 sinS02,4S02 cosS022sinS02,δq3,δ2q3S0=1S0,1S02,

where all the values in the above ΔJ-pairs are computed by direct evaluation of the ΔJ-kernels. Having now the required ΔJ-numbers for cosS2 and lnS, one can simply add the resultant ΔJ-pairs by ΔJ-addition to get δq,δ2q at S0.

Likewise, to compute the dyad δ2q,δ3q, we have the following dyadic ΔJ-numbers for respectively cosS2 and lnS

δ2q1q2,δ3q1q2S0=δ2q1q2S0×δq2S02+δq1q2S0×δ2q2S0,δ3q1q2S0×δq2S03+2δq2S0×δ2q2S0×δ2q1q2S0+δ2q1q2S0×δq2S0×δ2q2S0+δq1q2S0×δ3q2S0=4S02 cosS022sinS02,8S03 sinS0212S0 cosS02,δ2q3,δ3q3S0=1S02,2S03,

and by ΔJ-addition of the resultant ΔJ-numbers we get δ2q,δ3q at S0. This illustrates that the theory TΔJ of dyadic ΔJ-numbers (ΔJ-pairs) is completely sufficient for computing interval auto-derivatives of first and higher orders.

Conclusion

As we detailed in the introduction and elsewhere, combining subtlety of ordinary automatic differentiation with reliability of interval mathematics results in an intervalized version of algorithmic differentiation, namely “interval differentiation arithmetic”, which so markedly surpasses its ordinary counterpart in power and reliability. With the aid of interval mathematics, automatic differentiation can be intervalized to handle uncertainty in quantifiable properties of real world physical systems and accordingly provide the computational methods that suffice to deal with the important problem of “getting guaranteed bounds” of images of real functions and their derivatives; and so, this article has been devoted to recasting interval differentiation arithmetic in a formalized theory, by putting into a systematic form its fundamental notions, and thus attaining the advantage of a concrete algebraic foundation that has then enabled us to extend the theory in such a manner that adds to its power, reliability, and applicability.

In the first place, after formalizing some set-theoretical and logical notions of particular importance for our purpose, we gave an axiomatization of a theory of a differential interval algebra and then we presented the notion of an interval extension of a family of real functions, together with some analytic notions of interval functions. Secondly, we set up an axiomatic theory of interval differentiation arithmetic, as a two-sorted extension of the theory of a differential interval algebra, and then we gave the proofs for its categoricity and consistency. We consequently constructed the algebraic system of interval differentiation arithmetic, deduced its fundamental properties, and showed that it constitutes a multiplicatively non-associative S-semiring in which multiplication is subalternative and flexible. Then, we established some monotonicity and isomorphism theorems for interval differentiation numbers and proved a result concerning the structure of real differentiation numbers. And, lastly, we gave a brief account of the computational implementation of interval differentiation arithmetic and showed how to concurrently compute guaranteed enclosures of images of both families of real functions and their first and higher order derivatives.

From the very beginning, our axiomatic system included the notion of an interval extension of a family of real functions and the differentiability criteria thereof. Also, our construction differs in that we did not make use of Clifford’s dual numbers or Grassmann numbers that are repeatedly ‘borrowed’ or ‘reinvented’ in the literature as proposed algebraic characterizations respectively for first and higher-order algorithmic differentiation. Moreover, a well-known fact of logic is that a “categorical” formalization of a theory is the “best” characterization thereof. By dint of being categorical, the axiomatic theory presented in this work lays serious claim to being “best” in the sense that it rightly accounts, up to isomorphism, for all structures of interval differentiation arithmetic. Furthermore, along the course to the main business of this study, a number of useful notions have been introduced and formalized within the context of the proposed theory. Among these, we can mention interval enclosure of a bounded set, interval extension of a real family, proper interval functions and the criteria thereof, differentiability and continuous differentiability of a real family, interval differentiability criteria, differential enclosure of a real family, differentiation extension of an interval function, and differentiability criterion for interval differentiation numbers.

We would also remark that, on the strength of our axiomatization, many nice consequences come for free: categoricity and consistency of both the theory of interval algebra and the theory of interval differentiation algebra follow immediately, criteria for differentiability of families of real functions and their interval extensions are easily established, and the algebras of intervals and real differentiation numbers are both isomorphically embedded in the algebra of interval differentiation numbers.

The main contribution is therefore both a “logico-algebraic formalization” and an “extension” of interval differentiation arithmetic. The article provides an axiomatization of a comprehensive algebraic theory of interval differentiation arithmetic based on clear and distinct elementary ideas of real and interval algebras. We extend this formalized theory in two directions. On the one hand, although we made use of neither Clifford’s dual numbers nor Grassmann hyper-dual numbers, our new formalization of dyadic interval differentiation numbers fully addresses interval auto-derivatives of first and higher order. On the other hand, by virtue of introducing the notion of an interval extension of a family of real functions, the theory is extended to provide the mathematical tools to get guaranteed enclosures of the images of families of real functions and their derivatives. Noteworthy also is that with a few basic modifications, the categorical system axiomatized in this text can be extended analogously to compute fuzzy auto-derivatives. Nevertheless, despite all the aforementioned advantages, guaranteed interval enclosures come at a price: the interval subdivision method could be computationally inefficient when manipulating problems involving thousands of uncertain quantities. Fortunately, there are many ways out of this problems. Among these, we mention, without pretension to be complete, Hansen’s centered forms, remainder forms, Kulisch’s complete intervals, Kaucher intervals, and Dawood’s universal intervals (For further details, see, e.g., Dawood & Dawood, 2019a; Dawood & Dawood, 2020; Dawood & Dawood, 2022, and Shary & Moradi, 2021).

In conclusion, the “self-validating” feature of interval automatic differentiation makes it useful and applicable in a wide range of scientific fields. In engineering and physical sciences, a recurring problem is to compute the derivatives under parametric uncertainty. In this regard, an intervalized theory of algorithmic differentiation is believed to be very useful for manipulating problems involving quantifiable uncertainties. The system proposed in this article provides a rigorous and extended mathematical foundation for both real and interval automatic differentiation. Being both a formalization and an extension of differentiation arithmetic as it is currently practised, the authors believe that such a formalization, hopefully, might have a worthwhile impact on both theoretical research and real world applications, with computational advantages for the solutions of new types of practical problems which can be expressed in terms of the mathematical machinery presented in the body of this article.

Acknowledgments

We would like to express our sincere thanks to the academic editor and the reviewers for their constructive and insightful comments that greatly contributed to improving this work.

Funding Statement

The authors received no funding for this work.

Footnotes

1

InCLosure(Interval enCLosure) is an environment and a language for reliable scientific computing. Latest version of the software package InCLosure is freely accessible from CERN’s archive via https://doi.org/10.5281/zenodo.2702404.

2

Grassmann algebras (or exterior algebras) were introduced by Hermann Grassmann in Grassmann (1844). Clifford’s algebras, introduced in Clifford (1873) by William Clifford, are Grassmann algebras in one dimension. Exterior algebras have nowadays many applications (see, e.g.Colombaro, Font-Segura & Martinez, 2020; Dawood & Megahed, 2019, and Trindade, Pinto & Floquet, 2019).

3

Metamathematics (“epitheory”, or “metatheory”) is the study of formalized mathematical theories and languages, and the interpretations thereof. A metatheoretical investigation produces metatheorems about the object theory under consideration (see, e.g.Curry, 1977; Hunter, 1971, and Kleene, 1952).

4

Many mathematicians use the term “complete ordered field” as a synonymous substitute for “continuously ordered field”. Following Alfred Tarski (see, e.g.Tarski, 1994), we pass up the adjective ‘complete’ in favor of the word ‘continuously’. We reserve the word ‘complete’ for different logical uses.

5

There are so many systems of interval algebras (see, e.g., Hansen, 1975, Kulisch, 2013; Gardenyes, Mielgo & Trepat, 1985; Markov, 1995, Kaucher, 1980; Shary, 2002; Piegat & Dobryakova, 2020, Dawood, 2011; Dawood, 2012; Dawood & Dawood, 2019b, and Dawood & Dawood, 2020). Here we axiomatize classical (naive) interval algebra as introduced in, e.g., Alefeld & Mayer, 2000; Moore, 1979, Dawood, 2014, and Dawood, 2019. An axiom system for an interval differentiation algebra over a different theory of intervals will be fundamentally the same as the axiom system presented in this text, but it might differ in the resulting algebraic structure.

6

For two sets A and B, A=BwwAwB.

7

From the fact that the converse relation q ^ is always definable, the preimage of a function q is always definable, regardless of the definability of the inverse function q−1.

8

Categoricity is a bedrock of mathematics. For further details on the key role of categoricity in logic and mathematics, see, e.g., Corcoran, 1980, Dawood & Megahed, 2019, and Shapiro, 1985.

9

The two types (modes) of algorithmic differentiation are both realizable in the theory TΔJ of ΔJ-algebra. Here, we consider only the forward-type. With a few basic modifications, the reversed-type is implementable as well.

Additional Information and Declarations

Competing Interests

The authors declare there are no competing interests.

Author Contributions

Hend Dawood conceived and designed the experiments, performed the experiments, analyzed the data, performed the computation work, prepared figures and/or tables, authored or reviewed drafts of the article, conceptualization, formal analysis, and methodology, and approved the final draft.

Nefertiti Megahed conceived and designed the experiments, performed the experiments, analyzed the data, authored or reviewed drafts of the article, formal analysis and methodology, and approved the final draft.

Data Availability

The following information was supplied regarding data availability:

To reproduce the results of the numerical examples of the ‘Machine implementation of interval auto-differentiation’ section, the latest version of InCLosure is available for download from CERN’s archive at Zenodo: Hend Dawood. (2020). InCLosure (Interval enCLosure): A Language and Environment for Reliable Scientific Computing (3.0). Zenodo. https://doi.org/10.5281/zenodo.3903993.

An InCLosure input file and its corresponding output containing, respectively, the code and results of the examples are also available at Zenodo: Hend Dawood. (2023). InCLosure Code for Interval Automatic Differentiation [Data set]. Zenodo. https://doi.org/10.5281/zenodo.7595449.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Data Citations

  1. Dawood H. 2023. InCLosure Code for Interval Automatic Differentiation [Data set] Zenodo. [DOI]

Data Availability Statement

The following information was supplied regarding data availability:

To reproduce the results of the numerical examples of the ‘Machine implementation of interval auto-differentiation’ section, the latest version of InCLosure is available for download from CERN’s archive at Zenodo: Hend Dawood. (2020). InCLosure (Interval enCLosure): A Language and Environment for Reliable Scientific Computing (3.0). Zenodo. https://doi.org/10.5281/zenodo.3903993.

An InCLosure input file and its corresponding output containing, respectively, the code and results of the examples are also available at Zenodo: Hend Dawood. (2023). InCLosure Code for Interval Automatic Differentiation [Data set]. Zenodo. https://doi.org/10.5281/zenodo.7595449.


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