Nate's objections to Motion 42
Nate, P1788
On 22 Jan 2013, at 21:08, Nathan T. Hayes wrote:
> I vote NO on Motion 42.
>
> There are several reasons I vote NO:
>
> -- Against the intent of Motion 8, the Motion 42 doesn't provide
> decorated interpretations of Empty such as (Empty,DEF), (Empty,DAC).
Because it has a rather different foundational principle from your scheme:
"it must tell the truth about some conceivable evaluation of a function
over a box", start of 8.8.4.
> For
> example, section 8.8.7 says decorated intersection operation may provide a
> decoration min(dx,dy), where dx and dy are the decorations of the input
> operands. So this gives:
>
> ([1,2],dac) intersect ([3,4],dac)
> = ([1,2] intersect [3,4],min(dac,dac))
> = (Empty,dac)
>
> But by section 8.8.4 the empty set is not permitted to be decorated with
> decoration dac, so the specification allows implementations that give
> contradictory results.
You are quite correct. This shows the danger of specifications that are
ad-hoc instead of derived from the mathematics. Back to the drawing board
with these (optional) versions of intersection & convexHull.
I regard these as of secondary importance; unless someone comes up with a
proven consistent spec of them, I'll drop them from the next draft.
> -- I believe everyone agrees Empty is a set. But if Empty is also an
> interval, then to call (Empty,ILL) a "decorated interval" on the one hand
> and "not an interval" on the other hand doesn't make any sense to me (it is
> a contradiction);
Your intersect bug is an issue of substance. This complaint is just
about use of language. One could rename (Empty,ILL) as BadInterval. My
reply to Guillaume gives (Empty,ILL) a theoretical backup, but I think
its main justification is pragmatic (8.8.3): a failed constructor call
needs to return *something* at level 2, and this is a good candidate.
> -- The ILL decoration may require strong Computer Algebra System
> (CAS) to prove; and even if such CAS is available, the ILL decoration may
> not always be provable. I think this decoration is unnecessary, too complex
> and should be dropped;
This has been answered by me and others.
> -- Since Motion 42 does not allow Empty to be decorated with
> decorations such as DEF and DAC, the motion must define these bare
> decorations as the compressed decorated intervals (Entire,DEF) and
> (Entire,DAC) respectively...
Why? Decorations, decorated intervals and compressed decorated intervals
are 3 quite different types (or families of types at Level 2 in the case
of the latter two) of objects. Your example in the following paragraphs
makes no sense to me. It seems to follow an error (IMO) in Motion 8,
which was written at a time when we didn't understand the need to treat
them as separate types.
>
> -- I understand compressed interval arithmetic was removed from
> Motion 42, but it appears these problems will remain when we do get to it.
No. It will follow Arnold's original "worst case scenario" system, which
cannot give contradictions.
> -- Decoration system with EIN that gives containment order
> EIN \subseteq DAC \subseteq DEF \subseteq GAP \supseteq NDF\supseteq EIN
There are *many* decoration schemes that give consistent and useful
results when correctly implemented and used. Motion 42's scheme and
yours are both in that category, I believe, but I prefer a scheme where
the reduced-graph of the containment order is a tree, not a DAG that
"joins up":
Motion 42 Kaucher group
trv GAP
/ \ / \
emp def NDF DEF
/ \ | \
ill dac | DAC
\ \ /
com EIN
I think the tree model will prove easier for users to understand.
Regards
John Pryce