Re: A decorations question
P1788 and Nate
On 3 Mar 2010, at 18:11, Nate Hayes wrote:
> I think what I hear you saying is this is just another example why Bounded attribute is a thorn in the side, and why things get much simpler and cleaner if it goes away altogether, e.g., it circles back to some of Ian's comments. If so, I agree.
As part of writing v02 of the draft standard text I continue to ponder these things. I want to give "bounded" a fair crack of the whip, so I continue with my questions...
I think -- as soon as one has chosen a model where intervals are sets, as we have -- there is a "standard interpretation" (SI) of intervals. Like the Copenhagen Interpretation of quantum theory, it has difficulties and critics, but it seems the most cogent one around. To be more precise, it is an interpretation of the "meaning" of evaluating an explicit expression f(x_1,...,x_n) over intervals, i.e computing ff(xx_1,...,xx_n) where ff is the interval version of f.
- A newly created interval, say xx=[1,2], represents a number
x that is unknown, but is assumed to lie within [1,2],
AND in some sense "ranges over" the whole of [0,2].
The second part is rarely made explicit, and I want to know
if you agree with it.
- The interval yy = ff(xx_1,...,xx_n) is an attempt to get a
tight enclosure of range(f; xx_1,...,xx_n), which of course
is the set of all y=f(x_1,...,x_n) as all the x_i do their
"ranging" independently over their intervals xx_i.
This affects the rules for setting the "bounded" values: definitely bounded, possibly bounded, definitely unbounded.
Examples.
(a) After xx=[0,1], yy=1/xx we must have yy is definitely unbounded because our SI says it definitely contains 1/x values with x -> 0.
(b) Then zz=yy+yy =[0,oo] is definitely unbounded because yy is so, and zz "really" comprises values y+y with y in yy, and these include values y -> oo (this is independent of our knowledge that it "even more really" contains values 1/x + 1/x where x -> 0).
(c) But ww=yy-yy=[-oo,oo] is possibly bounded because it "really" comprises values y-y with y in yy, so it is "really" [0,0].
Underlying this, and part of my SI it seems, is that intervals like yy, zz, ww, that are computed from raw data such as literal intervals xx=[0,1], are NOT raw data. I.e. the unknown y,z or w that belongs to it does not range freely over its interval, but is a function of x and therefore only ranges "as x tells it to".
Do you agree? The other extreme, namely _each_ interval qq has its own variable q that is regarded as ranging over all of qq, makes "bounded" quite pointless. For then each interval just "is itself", and to say that [0,oo], say, is possibly bounded is a contradiction.
We can't define consistent semantics for "bounded" till we have sorted out this semi-philosophical question. And, IMO, we can't decide whether "bounded" should be kept or dumped, till we have tried to produce _at least one_ consistent (and useful) semantic model for it. If we can't find even one, we should definitely dump it!
Your views please.
Regards
John