Rump intervals ARE a flavor?
P1788, Siegfried, Bill
My problem with viewing Rump interval arithmetic as a flavor was that I didn't see what its Level 1 is. But in fact the solution seems quite simple. If I'm right, it *is* a flavor. Please analyse the following and tell me if it works.
Rump arithmetic basics
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See
[1] "Interval Arithmetic over Finitely Many Endpoints", BIT Numerical Mathematics, 52(4):1059–1075, 2012.
R1. Siegfried defines IR to comprise all connected subsets of R. That is, Empty together with all closed, open and half-open nonempty intervals, possibly unbounded.
R2. Each instance of Rump arithmetic is defined by a finite subset B of IR whose members are pairwise disjoint, so they can be uniquely arranged b_1 < b_2 < ... < b_k where the "<" means "strictly to the left of" (our strictPrecedes relation of Table 10.4). He calls them *interval bounds*.
If inf b_1 = -oo and sup b_k = +oo then B is called *admissible*. Assume this henceforth.
R3. His set of intervals IB is defined as the set of ordered pairs (a,b) with a,b in B and a <= b, together with Empty. IMO it is better to see this as a *representation*, and identify (a,b) -- as Siegfried does most of the time -- with its "range", which is convexHull(a,b), an interval. Hence IB is a finite subset of IR.
R4. IB is closed under intersection and convexHull, so it is a complete lattice in the set-containment order, with Empty at the bottom and (as B is admissible) Entire at the top.
Hence each subset S of R has a well-defined IB-hull, namely the intersection of all IB-intervals that contain S.
R5. Siegfried defines (see [1], Def 2.11) the *natural interval extension* of a function f, for a particular IB, to be the interval function F that maps a box A to the IB-hull of the exact range, range(f; A).
Actually he uses "strict" evaluation so A must be contained in Domain(f), else a special value NaI is returned. But one can extend this to use the "loose" evaluation of the set-based standard.
And I find this rather perverse, mixing up Level 1 with Level 2. It seems to me, since IR itself is closed under arbitrary intersections, that one should define this in two stages:
- form a Level 1 natural interval extension (IR-interval extension),
which can act on arbitrary boxes, not just IB-boxes (boxes whose
components are IB-intervals);
- take the IB-hull of this to get the IB-interval extension.
This has the same effect.
Matching with flavors
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F1. The set of Level 1 intervals of the flavor is IR. The flavor's "contains" relation is normal set containment.
F2. Each B specifies an interval type of the flavor. Its set of intervals is defined to be IB, so I'll call the type IB also. From R3, IB is indeed a finite subset of the Level 1 intervals, as 1788 requires.
F3. IR clearly includes the 1788 common intervals as a subset.
F4. The IR-interval extension I propose in R5 above is a natural choice for "the Level 1 value" of function f evaluated over box A. A function G (see [1], Def 2.11 again) that returns *some* IB-interval enclosing this Level 1 value, is exactly what P1788 §7 regards as an IB version of the point function f.
For common evaluations, such a G always returns an interval, never NaI.
Each flavor can define its own special values such as NaI. I conclude that, subject to some changes that affect concept and presentation, but not implementation:
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Rump arithmetic is a flavor.
Any admissible B defines a type.
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The only thing missing is that it must implement a decoration system including at least "com". Of course, I would prefer it to use
- loose evaluation;
- the same decoration system as the set-based flavor;
- hence a different meaning of NaI.
But it is a flavor either way.
If this analysis is correct I look forward to an implementation of a Rump flavor containing his T and H (Tiny and Huge) elements of B.
Please pick holes.
John Pryce
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On 2013 Dec 1, at 16:57, John Pryce wrote:
> On 2013 Dec 1, at 16:10, G. William (Bill) Walster wrote:
>> Is Siegfried's proposed interval foundation yet another example of an system that would not be a P1788 standard conforming flavor?
>>
>> I found is paper very innovative and interesting.
>
> So did I. Unfortunately it seemed to me on first reading that (any nontrivial realisation of) it was incompatible with P1788 since early in the project's history. If I'm wrong, and it can indeed be a flavor, that would be great.
>
> At the time he made the "Rump system" public, I felt it was too late to discard what we had done and start again. So I think Siegfried is a bit unfair to the group in saying "the only comment I heard from the stds-1788 mail group was that there is no reference implementation".
>
> I hope someone implements the system, and I look forward to try it out.
>
> Regards
>
> John Pryce