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Re: "natural interval extension"

Dear Frédéric,

The clearest definition I am aware of is given in the book by
Ratschek and Rokne, Computer Methods for the Range of Functions
(1984):

If f(x) is an expression and X an interval of the same dimension as
x, then the (interval-)expression which is obtained by replacing
each occurrence of x in f(x) by X is denoted by f(X). The
expression f(X) is then called the natural interval extension of
f(X) on X.

Three expressions, no functions involved.

Au contraire, this is the typical definition found in most books, and
it is severely lacking in several aspects as soon as you need
computing with intervals. Nothing is said about the replacement of the
operators, included the 0-ary ones---aka constants. Consider the
expression:

f(x) = x + 0.1

For all *practical* purposes, you cannot merely replace the
real-valued variable by an interval-valued one to get the "natural
interval extension" of f(x). You also would have to replace the
addition by its interval extension (definition?), and the constant
"0.1" by the smallest interval---in the sense of the inclusion---that
contains "0.1". Ratschek and Rokne's definition is not good enough to
specify all this. As are many (most) other definitions in the literature
In "real" interval arithmetic (intervals with real bounds, no roundoff), 0.1 and + are well defined. The same holds for rational expressions. For roots or transcendental functions, f(X) = range(f,X) if X \subset domain(f) can be used (I'm aware that there are alternatives to this).
For "machine" interval arithmetic, you have to specify the basic arithmetic operations, transfer functions such as x \in R to X \in IR, and the elementary functions *first*. You also have to impose conventions like x+y*z meaning x+(y*z) for interval operations, that is X+Y*Z meaning X+(Y*Z). Nevertheless, once these atoms are defined, the definition of Ratschek and Rokne applies to expressions. Am I missing something?
BTW: Ratschek and Rokne credit Moore (1979) for the above definition. However, the wording in Moore's 1979 book is different. 

Regards,

Markus

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