Singletons of T
> On 5 Mar 2012, at 02:10, Vincent Lefevre wrote:
> > Even though not all FP arithmetic are IEEE-754 conforming, the
> > fact that P1788 already requires +inf and -inf means that NaN
> > should be available in the FP implementation.
> >
> > BTW, I think that P1788 should allow non-FP formats (e.g.
> > double-double and rational arithmetics). Requirements could
> > be at least: the number of values is finite, the format is
> > symmetric (-x is exact), and +inf, -inf and NaN are available.
>
> I think this is an excellent suggestion. As we're seeing, the interface of intervals to floating point is necessary but contentious.
> Since, early on, we rejected tying P1788 exclusively to IEEE 754, we need some ground rules like this.
> Vincent's rules are *necessary* properties to my philosophy of how we create this interface, except for "-x is exact", which is sensible rather than necessary.
An intersting example of such non-FP format is a set of all singleton intervals ST of some interval type T, augmented with +oo and -oo .
(Singleton interval is an interval that contains exactly one real point).
Let us see examples:
1) T=infsup_F.
ST = { [u,u] | u \in F, u - is finite } U { -oo , +oo } isomorphic to F;
2) T=midrad_F.
ST = { <m,0> | m \in F, m - is finite } U { -oo, +oo } isomorphic to F;
3) Tripple interval type T. <r,i,s>=[r+i,r+s]
ST = { <r,z,z> | r \in F, z \in F, r and z - are finite } U { -oo, +oo } isomorphic to double-F numbers
{ r + z | r \in F, z \in F }
If T satisfies additional requirement: "Each nonempty interval from T contains at least one singleton interval from T".
then mid_ST(x) \in x for any nonempty interval from T.
We can consider function
mid_ST: T -> ST
as non-arithmetic function
center: T -> T.
It maps intervals from T into singlton intervals from T .
This function can be nicely used in to create center-forms.
-Dima