Re: Friendly amendment to Motion 25
Dominique Lohez wrote:
Nate, p1788,
This short comment on a hurry, before the end of the discussion period
I greatly appreciate your effort to take into account the variable
dependence
I am very skeptical if it could work, but if it really does i would be
very glad
At the present time I am unable to decide if will vote for or against
this motion.
IMHO too many information are missing
I would like to find
1) The list of the so-called "native" decorations for a single
variable or when all the variable are independent
I'm assuming that bare intervals are promoted to the "best" decoration by
linear quality order (2). So any non-empty bare interval X would promote to
(X,D3), and any empty interval would promote to (Empty,D4). Users could of
course override the default promotion, if they wish, by providing explicity
the required decoration.
2) A formal definition of a dependent variable. What dependence
are allowed? What dependence are not worked out?
3) A definition of the domain of a function with dependent
variables
4) A definition of C D+ and D- with independent variable and
with dependent variables.
For an interval expression like:
sqrt( X ) + X
then X is the only independent variable of the entire expression; even
though sqrt(X) may be considered as an independent variable of the addition
operation, it is a dependent variable of the expression.
Generally speaking, the decoration given by S is for the individual
(current) operation, so its arguments are treated as independent variables
relative to the operation.
On the other hand, the decoration given by T is the historical (tracking)
decoration of the expression relative to the current evaluation point of the
expression, so it produces a decoration for a dependent variable relative to
the expression.
Evaluating the expression above, e.g.:
sqrt( X ) + X
= sqrt( (X,dd0) ) + (X,dd0) // promote X to decorated interval
= (sqrt(X),inf(S(sqrt,X),dd0)) + (X,dd0)
= (sqrt(X),inf(dd1,dd0)) + (X,dd0)
= (sqrt(X),dd2) + (X,dd0)
= (sqrt(X)+X,inf(S(+,sqrt(X),X),dd2,dd0))
= (sqrt(X)+X,inf(dd3,dd2,dd0))
= (sqrt(X)+X,dd4)
where dd0 is the "native" decoration for X, and dd1, ... dd4 are computed
decorations of intermediate steps of computation; then X is the independent
variable of the entire expression, because the decoration dd0 of X is not
given by T. The decoration dd1 of sqrt(X) is given by T, as is the
decoration dd3 of the addition operation. So neither of those results are
independent variables of the expression. The decoration dd4 of the entire
expression is also given by T, and is an dependent variable.
5) A completely worked out simple example ( f(x,y) = x*y
with y = x is enough )
Does the example above do better?
Nate