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FAQs: Why Generalized Interval?

This page answers some questions on our generalized interval.

What is generalized interval

A traditional interval [X]=[a,b] is defined as a set of real numbers that are between the lower bound a and upper bound b, i.e. [X]:= {x in R | a≤x≤b}. In constrast, a generalized interval is defined by a pair of real numbers as [X]:=[a,b] where (a,b in R, real numbers).

Generalized interval has been given different names, including modal interval, Kaucher interval, directed interval, and generalized interval.

What is the difference between the generalized interval and the traditional interval in interval analysis?

There are two major differences between generalized interval and classical interval, which are its algebraic and semantic extensions respectively.

(1) Algebraic extension:

The basic calculus (+,-,*,/) of generalized interval is based on Kaucher arithmetic instead of the traditional one. In classical interval analysis, interval does not shrink. [1,2]-[1,2]=[-1,1]≠0. In generalized interval, a dual operator is introduced, which 'flips' interval bounds, e.g. dual[1,2]=[2,1]. If a dual is associated with "-", [1,2]-dual[1,2]=0.

In the traditional interval analysis, intervals are sets of real numbers. Therefore, during calculation based on the interval arithmetic, the widths of intervals always increase. This leads to the problem of range over-estimation. That is, in [A]+[B]=[C], the width of [C] cannot be smaller than that of [A] or [B].

More importantly, when solving the equation [A]+[X]=[C] to find [X], [X] is not the algebraic solution, meaning that when plugging the solution [X] back in the equation, we will find the equation does not hold any more. For example, [2,3]+[X]=[4,6], the solution is [X]=[4,6]-[2,3]=[1,4]. However, when plugged in, [2,3]+[1,4]=[3,7] which is not the original right-hand-side [4,6]. In addition, there is lack of invertibility to form a group structure. [X]-[X] ≠ 0, unless [X] is degerated and becomes a real number. For example, [1,2]-[1,2]=[-1,1]≠0. Therefore, intervals form a semi-group, instead of group.

In contrast, a concept of improper is introduced in generalized interval, as opposed to the traditional interval, which now is called proper. For example, [1,2] is a proper interva, whereas [2,1] is an improper interval. The proper and improper intervals have a dual relationship. [1,2]=dual[2,1]. Simlarly, [2,1]=dual[1,2]. The introduction of improper interval is in the same sense as the introduction of negative numbers in real analalysis. The topological group structure is better and simpler when we have negative numbers, even though we do not observe negative numbers naturally. A "zero" exists in real analysis, e.g. 2-2=0. Similarly, a "zero" now exists in the arithmetic of generalized interval, also called Kaucher arithmetic. [X]-dual[X]=0. Generalized interval form a group.

The widths of generalized intervals now could reduce during calculation. For example, [1,3]+[2,1]=[3,4]. Furthermore, [1,3]+dual[1,3]=[1,3]+[3,1]=[4,4] has a zero width. That is, [X]+dual[X] is always a real number!

(2) Semantic extension:

In classical interval analysis, [1,2]+[2,4]=[3,6] is the worst-case enclosure, which only has the meaning of "∀x∈[1,2], ∀y∈[2,4], ∃z∈[3,6] such that x+y=z". In generalized interval, more meanings can be expressed out of the syntax of algebra. For instance, [1,2]+[4,2]=[5,4] can be interpreted as "∀x∈[1,2], ∀ z∈[4,5], ∃y∈[2,4] such that x+y=z". [2,1]+[2,4]=[4,5] can be interpreted as "∀y∈[2,4], ∃x∈[1,2], ∃z∈[4,5] such that x+y=z". etc.

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