Re: Revised Motion 26 decoration scheme
On Mon, July 18, 2011 17:27, Nate Hayes wrote:
> John Pryce wrote:
>> Note also that our text continues to include Nate's "bare object
>> arithmetic", whose importance for efficiency I accept.
>
> I'm glad to hear that, but I note we don't agree on how bare decorations
> are
> promoted to decorated intervals: Bare decorations should always be
> promoted
> to Empty decorated intervals, not Entire.
Motion 26 says that they promote (conceptually) to all possible intervals
(as the discarded interval could have been any). Entire plays no special
role.
> For example, consider the real function
> f(x,y) = y + 1 / g(x)
> and its interval extention over interval domain (X,Y) where both X and Y
> are
> nonempty. If we are computing with bare objects and g(X) is not
> continuous,
> then we have
> Y + 1 / def.
>
> Promoting the bare decoration def to (Empty,def) gives:
> Y + 1 / (Empty,def)
> = Y + (1/Empty,inf(ein,def))
> = Y + (Empty,def)
> = (Y+Empty,inf(dac,def))
> = (Empty,def)
> = def
> In other words, the exception "def" first occured in g(X) and this
> decoration is promoted all the way to the end of the computation. So the
> user knows in this case the reason for failure.
>
> On the other hand, promoting the bare decoration def to (Entire,def)
> gives:
> Y + 1 / (Entire,def)
> = Y + (1/Entire,inf(con,def))
> = Y + (Entire,con)
> = (Y+Entire,inf(dac,con))
> = (Entire,con)
> = con
> In this case, the exception "def" that occured in g(X) is lost by the
> subsequent computations: the user has less knowledge why the failure
> occurred.
If one has g(x)=sign(x) and X=[-1,1], the decorated evaluation would give
con and not def, and indeed, f is not everywhere defined. This property
must be preserved by the bare rules, since the use of the decoration might
depend on this.
Moreover, having a different semantics for the decorated and the bare case
(as proposed by you) would lead to confusion....
> I note motion 27 doesn't yet address arithmetic on bare objects. But my
> position is that bare decorations must promote to Empty, not Entire. This
> is
> because: once an exception occurs, the most important thing is to
> propagate
> the original cause or reason for the exception to the end of the lengthy
> computation. In other words, we no longer care about "containing"
> subsequent
> exceptions, since at that point all interval information is lost, anyways,
> and to do so leads trivially to "con" in almost all cases (and this is
> really uninformative).
We still must stick to the semantics of the decoration, which gives a
property of the evaluated functio n on the box in question. This
completely decides the rules.
Arnold Neumaier