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Re: Revised Motion 26 decoration scheme

Arnold Neumaier wrote:

On Wed, July 20, 2011 16:43, Nate Hayes wrote:

Arnold Neumaier wrote:

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....



This is already a problem for Motion 26, since for g(x)=floor(x) and
X=[4,8]
the full decorated interval computation gives
    def
but when evaluated using bare objects (by Motion 26) it gives
    con


This is not a problem, since in both cases, the answer says something
correct about the function.
Only by coincidence, since the decoration provided is trivially correct for any computation. 



Whereas in your example, the bare answer makes a wrong statement about
the function.
It states the evalutation of f(X,Y) is not possible because one of the sub-expressions was not continuous. This is an entirely correct statement about the function. 

Nate