Re: Motion 52: final "Expressions" text for vote

["G. William (Bill) Walster" <bill@xxxxxxxxxxx>]

Vladik,

The difficulty is that a given interval can contain values that are 
outside the natural domain of definition of a given function or 
operation.  For example what is the smallest set of intervals that must 
be included in the result of the following operations on the interval X 
= [-1, +1]:

1/X
1/(1/(x))
ln(X)
1/ln(X)
X/X
X/X - X/X

The answers have nothing whatever to do with implementation or a 
standard for it.  The answers do depend on the definition of the 
universal set of intervals and the real numbers contained therein and 
the natural domain of any given expression and/or function. For example, 
in addition to real numbers in the open interval (-oo, +oo), shall 
intervals include {-oo, +oo} and/or oo = 1/0 and their reciprocals?  If 
so, how?

Shall there be an empty interval that is not the same as the empty set 
because the empty interval is not a subset of every interval? If so, 
when must the empty interval be included in the result of an interval 
expression evaluation to prevent a containment failure?

Are there any conditions under which X - X = 0 for given non-degenerate 
interval variables X?

Cheers,

Bill


On 11/22/13 8:04 PM, Kreinovich, Vladik wrote:
> For intervals proper, the defiition is the same always, f([x1],...,[xn]) is the smallest interval that contains the range of the given function f(x1,...,xn) on given intervals [x1], ..., [xn], i.e., the set of all the values
> {f(x1,...,xn):x1 is in [x1], ..., xn is in [xn]}.
>
> The questions arise when we want to extend this to situations when [xi] are not intervals
>
> (there are also definitions for decorations)
>
> ________________________________________
> From: Bill Walster [billwalster@xxxxxxxxx] on behalf of G. William (Bill) Walster [bill@xxxxxxxxxxx]
> Sent: Friday, November 22, 2013 2:31 PM
> To: Kreinovich, Vladik; Michel Hack; stds-1788
> Subject: Re: Motion 52: final "Expressions" text for vote
>
> Vladik,
>
> Then please point me to the purely mathematical foundation for computing
> with "well defined" intervals.
>
> Cheers,
>
> Bill