Automatic differentiation of uncertainties: an interval computational differentiation for first and higher derivatives with implementation
- Published
- Accepted
- Received
- Academic Editor
- Nicholas Higham
- Subject Areas
- Algorithms and Analysis of Algorithms, Optimization Theory and Computation, Scientific Computing and Simulation, Theory and Formal Methods
- 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
- Copyright
- © 2023 Dawood and Megahed
- Licence
- This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction and adaptation in any medium and for any purpose provided that it is properly attributed. For attribution, the original author(s), title, publication source (PeerJ Computer Science) and either DOI or URL of the article must be cited.
- Cite this article
- 2023. Automatic differentiation of uncertainties: an interval computational differentiation for first and higher derivatives with implementation. PeerJ Computer Science 9:e1301 https://doi.org/10.7717/peerj-cs.1301
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.
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”, “ -algebra”, or “ -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 (-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 -numbers is categorical, then it correctly accounts, up to isomorphism, for every structure of -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 TΔℝ of real differentiation arithmetic (Δℝ-arithmetic). In ‘A differential interval algebra’, we lay out an axiom system for the theory 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 of interval differentiation numbers (-numbers) as a two-sorted extension of the theory of a differential interval algebra, and then we prove its consistency and categoricity. In order for the theory to fully address and compute higher order and partial auto-derivatives using only dyadic -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 -numbers, and establish their differentiability conditions. In ‘The algebraic structure of interval differentiation arithmetic’, we investigate the algebraic structure of -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 -numbers and prove a result concerning the structure of Δℝ-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 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 be a set and let be its n-th Cartesian power. A set ℜ is an n-ary (finitary) relation on iff and ℜ is a binary relation from to . Thus, for and , an n-ary relation ℜ is characterized to be . Accordingly, a finitary relation ℜ is a binary relation whose domain, range, field, and converse are characterized to be, respectively , , , and . Obviously 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 ℜ be an n-ary relation on , and for , let , with each sk is restricted to vary on , that is, s is restricted to vary on . Then, the image of S (or the image of the sets Sk) with respect to ℜ, in symbols Iℜ, is characterized to be The preimage S of T is characterized to be the image of T with respect to the converse of ℜ. In other words
In consequence of the equivalence , apparently .
In this sense, a general characterization of an n-ary (finitary) function can be introduced (Dawood & Dawood, 2019a). A set q is an n-ary function (a function of n variables) on a set iff q is an -ary relation on , and . That is, an n-ary (finitary) function is an -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 . 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 is also a function, in which case (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|Λ𝔗⊨𝔏φ} (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
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 , iff 𝔄 and 𝔅 are isomorphic. That is
Definition 2.4: Consistency of a Theory
An axiomatic 𝔏-theory 𝔗 is said to be consistent, in symbols , iff there is a model 𝔄 that satisfies the sentences of 𝔗. That is, .
A model of an axiomatic 𝔏-theory 𝔗 is a mathematical structure 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 with and are total binary operations on the universe set . The operations and 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 that satisfies at least one of the following subdistributive properties.
-
,
-
.
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 is a semiring iff 𝔄 satisfies the following properties.
-
with forms a commutative monoid with is an identity for ,
-
with forms a monoid with is an identity for ,
-
is both left and right distributive over ,
-
is an annihilating element for .
If 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 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 is said to be an additively non-associative semiring (in short, +-NA semiring) iff 𝔄 satisfies (ii), (iii), and (iv) indefinition 2.7, and is a non-associative commutative monoid with identity element . 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 is a non-associative monoid with identity element .
Definition 2.10: NA S-Semiring
An S-ringoid 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 is a non-associative commutative monoid with identity element . 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 is a non-associative monoid with identity element .
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 TΔℝ of real differentiation arithmetic (henceforth Δℝ-arithmetic). For further details and other constructions of Δℝ-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 be the field of real numbers, be the set of unary real functions, and δ be the differential operator for elements of . For a q in , we use the predicate to mean that q is differentiable at some s0 ∈ ℝ. We understand by a differential real field a structure 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 (Δℝ-number).
Definition 3.1: Real Differentiation Numbers
The set of all real differentiation numbers (Δℝ-numbers, or Δℝ-pairs), with respect to a constant s0 ∈ ℝ, is defined to be
That is, a Δℝ-number is an ordered pair of real numbers. Let the letters q, u, and v, or equivalently the pairs , , and , be variable symbols ranging over the elements of Uℝ. Also, let a, b, and c, or equivalently , , and , designate constants of Uℝ. In particular, we use 1Uℝ to denote the Δℝ-number and 0Uℝ to denote the Δℝ-number .
The theory TΔℝ of a real differentiation algebra (or a Δℝ-algebra) can then be axiomatized as follows (Dawood & Megahed, 2019).
Definition 3.2: Theory of Real Differentiation Algebra
Let and be in Uℝ. A differentiation algebra over a differential real field , or a Δℝ-algebra, is a two-sorted structure . The theory TΔℝ of 𝔘ℝ is the deductive closure of the axioms of Rδ together with the following sentences.
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Δℝ-equality. ,
-
Δℝ-addition. ,
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Δℝ-multiplication. ,
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Δℝ-negation. ,
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Δℝ-reciprocal. .
Subtraction and division are defined as usual in terms of the four basic Δℝ-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 𝔘ℝ as , omitting the universe set ℝ.
Differentiable real functions can be extended to Δℝ-numbers via an extension principle (Dawood & Megahed, 2019). Let u be a real function differentiable at s0 ∈ ℝ, that is there is , and let be a function rule. If is differentiable at s0, then the differentiation extension of 𝒬ℝ is defined to be . For example, replacing by the “ sine” function, one obtains the trigonometric Δℝ-function .
We will not discuss the algebraic properties of Δℝ-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 of interval differentiation arithmetic.
A Differential Interval Algebra
In order to axiomatize a categorical system of interval differentiation arithmetic (-arithmetic) in the next sections, we need to lay out an axiom system for the theory of a differential interval algebra. The intended model of the axiomatic system is the differential S-semiring , where 𝒥ℝ is the set of real closed intervals (interval numbers, or -numbers) and δ is the differential operator for unary interval functions (-functions).
To be able to prove categoricity and consistency of -arithmetic, the first step towards axiomatizing the theory 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 of continuously ordered fields (cofields) is characterized in the following definition Dawood & Megahed, 2019.
Definition 4.1: Theory of a Cofield
Let be a totally ordered field. The theory of a cofield (or a continuously ordered field) is the theory of 𝔉 together with the following axiom of continuity
-
.
We designate by the converse of the non-strict total order , and by “ ” and “ ” the unary -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 “ ”.
For n ≥ 1, let designate the class of all n-ary -functions. Hereon, the letters q, u, and v are used as variable symbols ranging over the elements of (unary -functions). The intended interpretation (model) of the theory corresponds the structure 𝔉 to the continuously4 (complete) ordered field of real numbers and is interpreted by the set of unary ℝ-functions.
Toward formalizing a differential interval algebra, we first need to extend the theory of a cofield by axiomatizing some analytic concepts. Let , and let s and l be, respectively, an -variable symbol and an -constant symbol. The ‘limit’ operator of the function with respect to m, denoted , is defined thus (Dawood & Megahed, 2019): where the one-place operation symbol , called an -absolute value (or -modulus), is defined by If there is no such , then the limit of q at m is said to be nonexistent in . For an -constant symbol , the ‘continuity’ predicate is a binary predicate, , defined by If is true, then q is said to be continuous at s0. We also say that q is continuous on iff it is continuous at all s0 ∈ S0, that is
Definition 4.2: n-Differential -Operator Dawood & Megahed, 2019
Let n ≥ 0, and let s and β be -variable symbols. The n-differential -operator of a function , denoted , is characterized recursively by the following equations.
-
,
-
,
-
.
Evidently if the limit in definition 4.2 exists, then the n-differential of q is consequently a unary -function. Henceforth, we will usually write δnq and δq for and , respectively.
Closely related to the differential operator is the n-differentiability predicate, which is characterized as follows.
Definition 4.3: n-Differentiability -Predicate Dawood & Megahed, 2019
Let n ≥ 0, let , and let be an -constant symbol. The ternary n-differentiability -predicate for the function q, denoted , is defined by If is true, then q is said to be n-differentiable at s0.
Since for , , the predicate is always true and accordingly every is 0-differentiable at . Apparently, if is true, then for 0 ≤ m < n, .
Definition 4.4: Continuous Differentiability -Predicate
Let n ≥ 0, let , and let be an -constant symbol. The continuous n-differentiability -predicate for the function q, denoted , is characterized recursively by the following statements.
-
,
-
,
-
.
If is true, then q is said to be continuously n-differentiable at s0.
In a manner analogous to the differential operator, if is true, then for 0 ≤ m < n, is true as well.
A theory of an interval algebra or a classical5 interval algebra (henceforth a -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 be a cofield. The theory of an interval algebra (a -algebra) over 𝔉 is the theory of a many-sorted algebraic structure axiomatized by the following sentences.
-
,
-
,
-
.
Axiom (I1) of the above definition characterizes what a -number (an interval number, or an -interval) is. Axioms (I2) and (I3) prescribe, respectively the binary operations of -addition (“ ”) and -multiplication (“ ”), and the unary operations of -negation (“ ”) and -reciprocal (“ ”). The intended model of corresponds the sets “” and “ ” to the sets “ ℝ” and “ 𝒥ℝ” (of real numbers and real closed intervals), respectively, and the symbols “ ”, and “ ” to the binary and unary ℝ-operations.
In the sequel, the upper-case letters S, T, and W, or equivalently , , and , will be used as variable symbols ranging over the domain of -numbers. A point (singleton, or degenerate) -numbers {s} will be denoted by . Also, the letters A, B, and C, or equivalently , , and , will be used to designate constants of . In particular, we will use and to designate, respectively, the singleton -numbers and . It is convenient here to single out the set of point -numbers. This is defined thus:
Equality of -numbers is an immediate consequence of the axiom of extensionality6 of set theory plus the fact that a -number is a totally ≤-ordered subset of . Precisely,
The categoricity of the theory of -algebra is established by the following theorem.
Theorem 4.1: Categoricity of the Interval Theory
The theory of -algebra is categorical. That is, .
Let σ = { + , × ; − , −1; 0, 1} be a class of descriptive (non-logical) signs of , and let and be two structures such that 𝔍1⊧T𝒥∧𝔍2⊧T𝒥. Accordingly, and are two cofields. A theory of cofields is categorical, that is, there is one and up to isomorphism only one cofield. The structure is characterized, up to isomorphism, as the only cofield.
Let be the isomorphism from onto . We can then define by for all in where . By definition 4.5, It is straightforward to show that I is an isomorphism from onto . This proves that is categorical.
That is, the theory uniquely characterizes the algebra of -numbers, and the structure 〈𝒥ℝ; ℝ; +𝒥, ×𝒥; 0𝒥, 1𝒥〉 is, up to isomorphism, the only possible model of . Accordingly, in establishing our assertions about -numbers, the properties of real numbers are assumed in advance.
By means of definition 4.5 and from the fact that -numbers are ordered sets of ℝ, the following theorem is derivable (Dawood, 2012 and Dawood & Dawood, 2020).
Theorem 4.2: Interval Operations
Let and be two -numbers. The binary and unary -operations (interval operations) are formulated thus:
-
-addition. ,
-
-multiplication. ,
-
-negation. ,
-
-reciprocal. ,
where , and min and max are respectively the ≤ℝ-minimal and ≤ℝ-maximal.
If no confusion is likely, we will often omit the subscripts and ℝ. It is clear that interval addition, multiplication, and negation are total -operations, while interval reciprocal is a partial -operation. As customary, interval subtraction and division are defined respectively as and .
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 -operation considers all occurrences of variables as independent (Dawood & Dawood, 2020). Let two -variables S and T be assigned the same -constant . Evidently, which is equal to the image, Iind, of the multivariate ℝ-function , with . Now consider a unary ℝ-function , with . The image Idep of qdep is . 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 -operation 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 -numbers, for 1 ≤ i ≤ n. Let be a continuous ℝ-function with si ∈ Si, and let be a -function defined by the same rule as qℝ. The result of computing the image of the intervals Si under qℝ, denoted , using classical -operations (definition 4.2), cannot be generally exact if some si are functionally dependent. That is,
-
.
In general,
-
.
What this theorem shows is that the result obtained by the -function 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 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 be a -number. We can define as instances where is the -absolute value(or -modulus) of S.
Performing naive -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 be a -number. First, subdivision of S into n subintervals Si is applied such that where and are respectively the widths (lengths) of S and Si. Consequently . Then, evaluating a -function for each subinterval Si yields the enclosure (Dawood, 2014) As the number n of subintervals gets larger, gets arbitrarily close to the exact image . The subdivision method thus gives sharper enclosures than the naive evaluation . 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 (-arithmetic) has certain peculiar properties involving set inclusion. The singleton intervals and are identities for -addition and -multiplication, respectively; -addition and -multiplication are both associative and commutative; -addition is cancellative; -multiplication is cancellative only for all ; a -number is invertible for -addition (respectively, -multiplication) if and only if it is a singleton -number (respectively, a nonzero singleton -number); and -multiplication left and right S-distributes over -addition (see definition 2.6 of ‘On theories and structures: some metatheoretical fundamentals’). In other words, in accordance with definition 2.8, the structure 〈𝒥ℝ; +𝒥, ×𝒥; 0𝒥, 1𝒥〉 of classical -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 -Numbers
Let S1, S2, T1, and T2 be -numbers such that S1⊆T1 and S2⊆T2. Let be a binary -operation and be a definable unary -operation. Then
-
,
-
.
From inclusion monotonicity, plus the fact that , if s ∈ S and t ∈ T, then for and , we obviously have s∘ℝt ∈ S∘𝒥T and ♢ℝs ∈ ♢𝒥S.
At this point, let us introduce an abbreviation that we will make use of. Let be an ordered real n-tuple, and let and be two ordered n-tuples of -numbers, then
In the following theorem, let .
Theorem 4.5: Isomorphism Theorem for -Numbers
The structure of point -numbers is isomorphic to the ordered field 〈ℝ; +ℝ, ×ℝ; ≤ℝ〉 of real numbers.
Two further results we will need are stated below (Dawood, 2012).
Theorem 4.5: Algebraic Operations for Point -Numbers
Let S and T be two -numbers. Then:
-
The sum S + T is a point -number iff each of S and T is a point -number, that is
-
The product S × T is a point -number iff each of S and T is a point -number, or at least one of S and T is , that is
Theorem 4.7: Zero Divisors in -Numbers
Nonzero zero divisors do not exist in -arithmetic, that is
Before turning to the axioms of the theory of a differential -algebra, it is necessary for our purpose to formalize some analytic concepts within the framework of the theory of -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 qℝ:𝒟ℝ⊆ℝn↦ℝ, and by an n-ary interval function (in short, -function) we will always mean a function . The ℝ-subscripted symbols qℝ, uℝ, vℝ will designate ℝ-functions, while the -subscripted symbols will designate -functions. For simplicity of notation, if the function type is apparent from the type of its variables(arguments), the subscripts “ ℝ” and “ ” will usually be dropped. For instance, whenever unambiguous, we use the notations and for, respectively, an ℝ-function and a -function, which are both defined by the same rule. For 1 ≤ i ≤ n and 1 ≤ j ≤ k , let Si and Aj be respectively -variable symbols and -constant symbols. We denote by an n-ary (or multivariate) -function in the interval variables Si and the interval constants Aj. Similarly, we understand by an n-ary ℝ-function in the real variables si and the real constants aj. For instance, is a binary -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 and respectively for the unary functions and . The sets of unary real and interval functions will be denoted by and respectively.
Next, we define the interval enclosure of a bounded set of real numbers.
Definition 4.6: Interval Enclosure of a Bounded Set
Let be a bounded subset of ℝ. The interval enclosure of , denoted , is defined to be
Clearly, . For instance and .
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 -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 ℝ, and for , let , 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 ) with respect to q, in symbols Iq, is characterized to be The preimage7 S of is characterized to be the image of with respect to the converse of q. In other words
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 ℝ-family), denoted , is a set of real functions subject to the following conditions
-
q is a function rule,
-
si are variable symbols varying on bounded subsets of ℝ,
-
aj are constant symbols (coefficients) from bounded subsets of ℝ, and
-
for each , is continuous on the sets . We understand by the converse of Qℝ, denoted , the set of the converse relations .
Note that a real family is generated by one function rule, that is, the functions in Qℝ all have the same rule q but different constant arguments. If the sets are singletons, then the family Qℝ 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.
-
Let Qℝ be the family generated by the function rule , with the variables s1 and s2 vary respectively on the bounded sets and and the constant a is from the bounded set {3, 7}. The family Qℝ has exactly the two binary functions
-
Let Uℝ be the family generated by the function rule , with the variable s varies on the bounded set and the constant a is from the bounded set . The family Uℝ has an infinite number of unary functions. Among these are, for example
We characterize the image of a real family as follows.
Definition 4.9: Image of a Real Family
Let Qℝ be a real family generated by a function rule , with and . Then, the image of the family Qℝ (or the image of the sets with respect to Qℝ), denoted IQ, is the union of the images of with respect to each q in Qℝ for all . That is
Obviously, for each q in Qℝ, 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 Qℝ be a real family generated by a function rule , with and . If and are real closed intervals, then the image of , with respect to the family Qℝ, is in turn a real closed interval such that
If the sets 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
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 Qℝ be an n-ary real family generated by a function rule , with and . We understand by an interval extension of Qℝ an n-ary interval function of the same rule as qℝ, and whose arguments are and .
Clearly, if and are real closed intervals, then and . We will henceforth deploy the predicate to mean that an interval function is the interval extension of the real family Qℝ, or equivalently, the family Qℝ is the real intension of the interval function . If are singletons, then the family Qℝ is a singleton and we call a simple extension of Qℝ. If and are singletons, then we call the point-valued interval function a point extension of Qℝ.
The following example will illustrate this point.
Example 4.2: Interval Extensions of Real Families
Recall the real families Qℝ and Uℝ of example 4.1. The interval extensions of Qℝ and Uℝ are given respectively by
-
, with , , and .
-
, with , and .
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 is proper, in symbols , iff it is set-theoretically definable in terms of a real function of the same rule. That is
By definitions 4.11 and 4.5, the following result is derivable.
Theorem 4.9: Criteria for Proper Interval Functions
Let ∘ ∈ { + , × } be a binary -operation and ♢ ∈ { − , −1} be a definable unary -operation. Then, the following statements are true.
-
,
-
,
-
.
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
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 -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 , or simply Q, for the unary real family .
Next, we extend the differential operator to families of unary real functions.
Definition 4.12: Differential Operator for a Real Family
Let 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 , denoted , is defined to be the set of all real functions for every q ∈ Qℝ 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 .
Definition 4.13: Differentiability of a Real Family
A unary real family is n-differentiable at a real constant s0, in symbols , iff for every q in Qℝ, , and q is n-differentiable at s0. That is
Definition 4.14: Continuous Differentiability of a Real Family
A unary real family is continuously n-differentiable at a real constant s0, in symbols , iff for every q in Qℝ, , and q is continuously n-differentiable at s0. That is
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 be a unary interval function that has a real intension the family . The n-differential -operator of , denoted , is characterized to be the interval extension of . In other words, let , then .
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 , and let be a -constant symbol. The ternary n-differentiability -predicate, denoted , is defined by If 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.
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 is n-differentiable at a -number S0 if and only if
-
is proper with a real intension , and
-
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 be a real function in a family , with s is restricted to vary on a real closed interval S0, and let be the interval extension of at S0. The following two sentences are true.
-
,
-
.
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
Obviously, .
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 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 of a differential interval algebra (henceforth a differential -algebra).
Definition 4.17: Theory of a Differential Interval Algebra
Let σ = { + , × ; − , −1; 0, 1} be a class of non-logical signs, and let be the theory of an interval S-semiring . The theory of a differential -algebra is the deductive closure of together with the following two axioms.
-
,
-
.
Consider the constant interval functions and . With the aid of definition 4.15, obviously . More generally, for any interval constant symbol A, and . Accordingly, the set can be defined thus: . 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 of a differential -algebra axiomatized in the previous section, the present section provides a rigorous mathematical foundation for interval differentiation arithmetic (henceforth -arithmetic). We are almost ready to lay out an axiom system for the theory of interval differentiation numbers (henceforth -numbers) as a two-sorted extension of . By virtue of the mathematical underpinnings presented in ‘A differential interval algebra’, we axiomatize, in the present section, the basic operations of and prove some of their fundamental properties. Moreover, we prove categoricity and consistency of -arithmetic.
An obvious starting point is to define interval differentiation n-tuples.
Definition 5.1: Interval Differentiation n-Tuples
Let be a differential -algebra, let q be a unary -function, and for an integer n ≥ 0, let be the n-th Cartesian power of . The set of all interval differentiation n-tuples over , with respect to an individual -constant , is characterized to be
An interval differentiation n-tuple is thus an ordered n-tuple of -constants. Hereafter, we will usually write q, δq,..., δnq for , , …, , 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” (“ -numbers”, or “ -pairs ”) for dyadic interval differentiation tuples. Let designate the set of -numbers at some -constant S0, and let the letters Q, U, and V, or equivalently the pairs , , and , be variable symbols varying on the set of -pairs. Also, let the letters A, B, and C, or equivalently , , and , designate constants of . In particular, we use to designate the -number and to designate the -number . Moreover, it is convenient for our purpose to define a proper subset of as
We are now ready to axiomatize the theory of an interval differentiation algebra (or a -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 , , and be in . An interval differentiation algebra (or, in short, a -algebra) over a differential -algebra is a two-sorted structure . The theory of 𝔘𝒥 is the deductive closure of the system of 𝔍d and the following set of axioms.
-
-equality. ,
-
Binary -operations. ,
-
Unary -operations. .
The intended model of the theory corresponds the sets “ ” and “ ” to the sets of -numbers and -numbers, respectively, and the symbols “ ”, and “ ” to the binary and unary -operations. When the context is clear, for simplicity henceforth, we will drop the subscripts “ ”, “ ”, and “ S0”. Also, we will usually write the algebraic structure 𝔘𝒥 as , omitting the set .
The inclusion and membership relations for -numbers can be defined as follows.
Definition 5.3: Inclusion Relation on -Numbers
The inclusion relation on -numbers, denoted , is defined as follows.
Definition 5.4: Membership Relation in -Numbers
The membership relation in -numbers, denoted , is defined as follows.
An important notion for our purposes is that of a point -number.
Definition 5.5: Point -Number
By a point (or singleton) -number, denoted , we understand a -number whose all components are point intervals, that is and are in .
The set of all point -numbers will be denoted by . In the sequel, we will make use of the following theorem that establishes the criteria when a -number is a singleton.
Theorem 5.1: Criteria for Point -Numbers
A -number is point iff
-
q is a constant point-valued function, that is , or
-
and each constant in the rule of q is a point interval.
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 .
Theorem 5.2: Algebraic Operations of -Numbers
Let and be two -numbers. Then, the binary and unary -operations are formulated as follows.
-
-addition. ,
-
-multiplication. ,
-
-negation. .,
-
-reciprocal. .
To complete our characterization of -arithmetic, we define as customary -subtraction and -division respectively as and .
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 of -numbers, concerning respectively existence, categoricity and consistency of a -algebra.
Theorem 5.3: Existence of a-Algebra
There exists at least one -algebra.
Since the theory of a -algebra has the model of -numbers, it follows that the theory has a model , and thus existence of a -algebra is proved.
Theorem 5.4: Categoricity of -Arithmetic
The theory of -numbers is categorical.
The theorem follows from the categoricity of the theory of interval algebra by an argument analogous to the one used in theorem 4.1.
That is, the theory uniquely characterizes the algebra of -numbers, and the structure is, up to isomorphism, the only possible model of . To reiterate, in accord to theorem 5.4, the system , axiomatized in definition 5.2, is the “best” axiomatization of -numbers, in the sense that it rightly accounts, up to isomorphism, for every structure of -arithmetic.8
The next theorem establishes the consistency of the theory of -numbers.
Theorem 5.5: Consistency of -Arithmetic
The theory of -numbers is consistent.
In accord to definition 2.4, the proof is immediate from theorem 5.3. The theory is satisfiable by the model and thus is consistent.
Owing to the categoricity theorem for , the algebraic properties of -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 -number represents a guaranteed interval enclosure of the image of a whole family of ℝ-functions and their derivatives and accordingly that a -number is an interval extension of every Δℝ-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 be its interval extension. Then, for every qℝ in Q
Corollary 5.1: Interval Extension of a Δℝ-Number
Let q be a real function continuously differentiable on a real closed interval S0. Then, for every s0 ∈ S0,
Finally, let us note that we can get sharper enclosures of the pair 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 -Numbers
Recall the notation used intheorem 5.6, and let S0 be subdivided into n ≥ 1 subintervals. Then Moreover, .
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 -numbers (-pairs), and without resorting to defining any sort of n-dimensional Grassmann algebras. Towards this end, we are to extend the theory , by introducing the notion of a differentiation extension of -functions, characterizing differentiability for -numbers, and establishing their differentiability conditions.
In view of our definition of -numbers, the following alternate characterization of interval differentiability is at our disposal.
In order to have -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 -functions to -functions. In accord to the above characterization, we have the next definition.
Definition 6.1: Differentiation Extensions of -Functions
For k ∈ {1, …, n}, let be differentiable at , that is for each uk there is . Let be an n-place -function of u1, …, un which is differentiable at S0. A differentiation extension of is an n-place -function defined to be and obtained from by replacing, in , each occurrence of a -function symbol uk by the corresponding -variable symbol Uk.
The definition is so framed that since is true, the differentiation extension of is in . Thus, and are both defined by the same symbolic rule but with different types of arguments (variables); is a -function whereas is a -function. By analogy with rational -functions, a rational -function is a (multivariate) -function obtained by the application of a finite number of the binary and unary algebraic -operations and . Hereon, if the function type is apparent from the context, the subscripts and will be omitted. For instance, whenever unambiguous, we use the notations and for, respectively, a -function and its differentiation extension.
Here it will suffice to give an example. Let the -functions and be both differentiable at some S0, and let be differentiable at S0 such that The differentiation extension of is then
By virtue of our definition of the extension principle for -functions (definition 6.1), we are able to define fundamental -functions. For example, replacing by the “cos” function, one obtains the trigonometric -function . In ‘Machine implementation of interval auto-differentiation’, we will give further discussion on differentiation extensions of -functions as well as more illustrative numerical examples.
Here, let us stress that restricting our discussion to single-variable -functions is not a loss of generality, since an n-variable -function can be viewed as a class of n single-variable -functions. What is noteworthy in addition is that higher-order interval auto-derivatives can be computed in the framework of our system of dyadic -numbers (-pairs). With the aid of definition 6.1, we next characterize the n-differential operator and the n-differentiability predicate for -pairs.
Definition 6.2: n-Differential Operator of a -Number
For an integer n ≥ 0, the n-differential operator of a -pair , in symbols δnU, can be characterized recursively by
-
δ0U = U,
-
,
-
.
Definition 6.3: n-Differentiability Predicate for a -Number
Let n ≥ 0. The n-differentiability predicate for a -pair , in symbols , is characterized by .
Consequently, the next theorem, concerning higher-order auto-differentiability of -functions, is provable.
Theorem 6.1: n-Differentiability Condition for a -Number
Let n ≥ 0. Then for a -pair , we have .
It is clear that if the -function u is -differentiable at S0, then, for n ≥ 0,