Motion P1788/M0013.02:ComparisonOperations : NO
My vote on Motion 13 is NO.
The P1788 review process is alive and well. I think Dominique Lohez and Vincent Lefevre have identified flaws in (the position paper of) Motion 13 that demand a significant revision before it can be accepted. Before Dominique's 20th April mail I hadn't properly noticed that it starts at Level 2 not Level 1.
In fact it's worse (though this is arguably its wording rather than its substance): it starts at Level 3, because Section 2 begins:
> If aa and bb are intervals of IF bar with bounds a1 <= a2 and b1 <= b2 respectively, then these 7 relations are defined by:
> aa=bb :<==> a1=b1 and a2=b2
etc. That is, the relations are stated to be *defined* in terms of the bounds. No wonder the empty set looks as if it is an afterthought.
In Motion 3 we agreed that
A. intervals are subsets of the reals R;
B. the empty set Empty is included;
C. unbounded intervals are included.
So far that we can treat interval comparisons in terms of set theory, thus far will the behaviour of Empty, and unbounded intervals, drop out automatically. It may not always be intuitive, but it will be provably consistent.
So what would make me vote YES is a recasting in terms of set theory. Below, is my attempt in this direction.
Taking Occam's Razor to the situation, I think all we need to assume about R, here, is that it is a nonempty linearly ordered set (O, <=) with no least or greatest element. The concept of a closed interval is well-defined in such an O. Since we only combine finitely many intervals, notions such as completeness are irrelevant.
Infinite intervals are defined as we do at present, by adjoining placeholders called -oo and +oo to O to get an extended set O*.
Consider the 7 proposed relations, using Dan Zuras' names for them (22 Apr 2010 03:46). I replace the level-3 definitions by ones that use just set concepts and <=.
Let aa=[a1,b1] and bb=[b1,b2] be nonempty.
> aa equals bb <==>
Standard set equality relation: (aa subset bb) and (bb subset aa)
> aa subset bb <==>
Standard subset relation: for all a in aa, a is in bb.
> aa lessEqual bb <==> a1 <= b1 && a2 <= b2
Vincent (23 April 2010 16:56) noted this has a set definition
for all b in bb, there exists a in aa such that a <= b
and
for all a in aa, there exists b in bb such that a <= b.
> aa precedeTouch bb <==> a2 <= b1
for all a in aa, for all b in bb, a <= b
The ones that involve < rather than <= are more doubtful:
> aa interior bb <==> b1 < a1 && a2 < b2
This is more subtle because "interior" is a topology concept, using the "natural order topology" on any linearly ordered set. The level 3 definition differs from the topological one, for infinite endpoints. E.g. (Entire interior Entire) is true topologically but not with the level 3 definition.
Query: Which do practitioners want or is the behaviour at infinite endpoints immaterial?
> aa strictLess bb <==> a1 < b1 && a2 < b2
I shall use the analogue of "lessEqual":
for all b in bb, there exists a in aa such that a < b
and
for all a in aa, there exists b in bb such that a < b.
This agrees with the level 3 definition for bounded intervals, I think. But not for unbounded ones, e.g.
([-oo,0] strictLess [-oo,1]), and (Entire strictLess Entire)
are true in my definition but not in the level 3 one.
Comments please.
> aa precede bb <==> a2 < b1
for all a in aa, for all b in bb, a < b
seems to agree with the level 3 definition for all nonempty aa and bb whether bounded or not.
In a following email, I aim to check these against Dan's list of the behaviour of Empty (email cited above).
There's something I don't understand in Dominique's email, see below.
Finally, the Vienna proposal has got there before us, as so often, see 5.4(2). I think Vincent's version of "lessEqual" can be written in the notation there, as
(aa <=SA bb) and (aa <=AS bb).
Regards
John
===========================================
On 30 Apr 2010, at 11:11, Dominique Lohez wrote:
> My vote is NO
>
> Rationales
> 1) The comparison operations SHOULD be defined at defined at level 1 and then developed at level 2 and further
> Such an approach would be consistent with the the methodology alway used in the working group up to now
>
> 2) The primitive operations provided by the standard MUST be be defined such that for any pair of intervals one and only one operation provided the true result.
I don't know what this sentence means.
> This is not the case for the primitive operations proposed in the motion 13
> I think that the relative position of intervals sharing a common bound should be carefully distinguished, since these operations will reveal of crucial for the discussion of reverse operators.
...
> However further work is needed to be able to handled both bounded and unbounded intervals
>
> 3) For the shake of convenience further very often used relations MAY be defined
Please suggest what you would include here.
On 30 Apr 2010, at 15:14, Vincent Lefevre wrote:
> I also vote NO.
>
> As it is now, this motion is wrong from a mathematical point of view.
>
> On 2010-04-30 12:11:25 +0200, Dominique Lohez wrote:
>> My vote is NO
>>
>> Rationales
>>
>> 1) The comparison operations SHOULD be defined at defined at level 1
>> and then developed at level 2 and further
>> Such an approach would be consistent with the the methodology
>> alway used in the working group up to now
>
> I agree, and moreover, comparisons must be completely defined, in
> particular concerning the empty set. I don't see how { IFF bar, <= }
> can be a lattice, as the empty set is taken into account.
>
> The motion says:
>
> The greatest lower bound and the least upper bound of an interval
> with the empty set are both the empty set.
>
> So, this would mean that X <= Empty and Empty <= X for any X, but
> with the consequence that <= would no longer be an order relation
> (since no antisymmetric).
>
> I think that X <= Empty and Empty <= X should be false for any
> non-empty X. If defined as operations of P1788, glb and lub
> should return NaI on (X,Empty).