Re: Csets, modals and Motion 3
On 25 Apr 2009 at 12:49, Nate Hayes wrote:
> Dear John, et. al.,
>
> Thank you for your very thoughtful comments on this topic.
>
>
> > Motion 3 says that 1788 should define an interval to be a closed
> > connected subset of the reals R. Clearly, this excludes both cset models,
> > and modal interval models, from being *directly* supported by the
> > standard.
> >
> > ...
> >
> > In view of the usefulness of these other systems, how should they be
> > supported *on top* of a system that is close to Vienna? This seems
> > quite easy for modals (as Nate has confirmed), and Vienna already
> > provides low-level functions to aid a modal implementation.
>
>
> I feel I need to make some clarifications, but then also let me make a few
> suggestions:
>
>
> I believe Motion 3 allows *unbounded* subsets of the reals R, e.g., the
> interval (-Inf,5] is the set of all real numbers { x | x <= 5 }. This is
> different than allowing extended-real subsets of R*, where intervals may
> contain the infinities, which are not real numbers.
>
> The results I show in Chapter 7 of my introductory paper conform to the
> former interpretation, not the latter. So I believe they are compatible with
> Motion 3. Note that the modal interval schema also aligns with Ulrich's
> position paper for classical interval arithmetic in the special cases when
> both modal interval operands are existential, and it satisfies Siegfried's
> suggested property that 0*X=0 for any interval X.
>
> I'm also working on a follow-up paper to show how the results in Chapter 7
> were originally obtained (many years ago) from the "Semantic Theorem for f*"
> of Gardenes, et. al., since some people have asked where do these results
> come from. For example, the modal interval operation
>
> (+Inf,3]+(-Inf,2]=[0,5].
>
> I don't suggest, mathematically, that Inf-Inf=0 (the infinities are not even
> members of the intervals), but that, by implication, we may use this rule
> inside a computer to compute the correct endpoints of such a modal interval
> result. The mathematical justification for the interval [0,5] is obtained
> from the "Semantic Theorem for f*" (and requires some involved explanation).
> It is just fortunate coincidence that the rule Inf-Inf=0 as a mnemonic
> device always provides the same answer. Same for Inf/Inf=1 and 0*Inf=0.
>
> Such a result doesn't follow c-set interpretation. However, I believe it
> does not preclude a c-set application from using multi-intervals or OPI
> datatypes (see, e.g., Section 8.1 of my introductory modal paper) as a
> mechanism for c-set compatibility, should 1788 decide to standardize the
> Kaucher intervals or Motion 3.
>
> There may also be as-of-yet undiscovered applications of modal intervals
> within the c-set paradigm, too. So this is an example of why it is not a
> good idea that 1788 should leave the meaning of nonstandard intervals
> "implementation defined." There can only be one meaning for nonstandard
> intervals in any given application, and if two are needed, this creates a
> conflict. I therefore see it is the responsibility of 1788 to standardize
> the meaning of nonstandard intervals to prevent such problems out in the
> field between various interval arithmetic applications and hardware.
>
> It is my wish to make 1788 useful to the widest possible audience (this
> includes c-sets, modals and possibly mid-rad arithmetic), and to do this
> while following George's admonitions to K.I.S.S.
>
> In my view, modals are directly compatible (or even coincide) with 1788
> proposals such as Vienna Proposal and Ulrich's position paper, and
> standardizing the Kaucher intervals will, at the same time, not preclude
> c-set compatibility. Modals are also compatible, I believe, with Motion 3.
>
> For this reason, and since the modals come basically for free, at no extra
> cost (in both software and hardware), and since they also significantly
> reduce the needed amounts of "implementation defined" and/or "undefined"
> behaviour in a standard like 1788, I believe these are all compelling
> reasons to standardize them, i.e., standardizing the Kaucher intervals is
> a solution to K.I.S.S., in my opinion.
>
> Sincerely,
>
> Nate Hayes
> Sunfish Studio, LLC
Let me add something about the "K.I.S.S. principle".
From mathematical point of view a definition of a mathematical object
(in our case "interval") is done by means of axioms. A definition is
simple if the corresponding system of axioms is simple and applicable.
Thus the definition of a real number is based on the algebraic systems of
a ring (when addition and multiplication are involved) and a vector space
(when addition and multiplication by scalar are involved). These two
systems proved to be the simplest and eventually were accepted worldwide.
Consequently zero and negative numbers were accepted (although
they are not easy to be understood).
Similarly, the simplest definition of an interval is based on the Kaucher
system of axioms (when addition, multiplication and inclusion are involved)
and the quasivector system (when addition, multiplication by scalar and
inclusion are involved). These systems involve improper intervals --
similarly to real number systems, which involve negative numbers. The
reason behind is that computing in a group is simpler than computing in
a semigroup.
However, the concept of a group derived from a semigroup by means of
embedding leads to improper elements. It is not easy for the human mind
to understand this. That is why it took more than 20 centuries for the
humanity before the concept of a real number was finally coined.
Most colleagues have specialized in techniques, using just proper
intervals. Certainly, they do not like Kaucher/modal intervals. It is not
wondering that these colleagues will vote against an implementation of
Kaucher/modal intervals in the standard. For them Kaucher intervals are
too "complex" and do not conform to "K.I.S.S. principle".
However, the truth is that Kaucher IA is simpler than the "standard" one.
Just because a group is simpler than a semigroup.
Svetoslav