Re: Surprise in array concatenation

["Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de>]

On Wed, 07 Sep 2005 20:20:56 +0200, Georg Bauhaus wrote:

> Dmitry A. Kazakov wrote:
>> On Tue, 06 Sep 2005 17:51:11 +0200, Georg Bauhaus wrote:
> 
> Almost no real number can be represented in a computer.

OK, it is beaten to death. See Richard's paradox. Its free interpretation,
I quote J. E. Littlewood from memory, is: let there be integer numbers
which cannot be defined by sentences shorter than 12 words. "This set is
bounded and thus contains the lowest number N." This sentence defines N and
amazingly contains only 11 words.

>>>>Do you really believe
>>>>that 1 is *the* 1? Did you see a label "made in Heaven" on it? (:-))
>>>
>>>Type Positive starts at 1 and ends with Positive'Last.
> 
>> ... and it does this not symbolically, but physically
> 
> yes.
> 
>> by attaching a tiny
>> rubber thread to the Platonic number 1. Right? (:-))
> 
> no.

How else you could check if 1 really refers to the physical 1! Maybe some
scoundrel has reconnected all 1s in your computer to the physical 946.23,
while you slept! (:-))

>> You are trying to formulate some generic algorithm which does not work for
>> all index types.
> 
> My algorithm should work for arrays, using properties of arrays.
> A'First .. A'Last, where A'First > A'Last works well in loops.
> 
> When are you interested in computing the value half-way between
> the ends of a (non-numeric) enumeration type, attempting to use
> A'Length / 2?

Iteration algorithms in numeric analysis. Of course Length/2 is nonsense,
but (Xn+Xm)/2 or Xn+(Xm-Xn)/2 or its generalization G*(Xn+Xm) is used quite
often. For example, Xi can be from a vector space. G can be a matrix which
norm is 0.5.

>> This is why it is so important to have a consistent model
>> of indexing and arrays.
> 
> It is important to be consistently using the proper type of arrays
> in algorithms made for this type of arrays.
> 
>> For an unordered [index] type X it is wrong to
>> assume that:
> 
> Can there be an index type in Ada that isn't ordered?

Sure:

   type Unordered_Index is (A, B, C, D);
   function "<" (Left, Right : Unordered_Index)
      return Boolean is abstract;
   type Unordered_Array is array (Unordered_Index) of Integer;

>> 1. If L in X and U in X then there is always M in X between L and U
>> 2. That M can be computed as (L+U)/2
> 
> You can't usually compute M when X is an enumeration type.
> Otherwise, if A'Length <= 1, then (L+U)/2 = 0 = M.
> Thus, M can be computed but is not necessarily between L and U.

1. This is plain wrong. If L=U then (L+U)/2=L. In general, it is always so
that min(L,U) <= (L+U)/2 <= max(L,U), when ordered numbers are considered.
(For generalizations on unordered cases see "convex hull")

2. M could not be computed just because "+", "/", 2 aren't defined. Then of
course there could be no order;

3. Note that (1) is wrong for modular types. Not surprisingly, they aren't
numbers, but classes of equivalence (sets of numbers, infinite sets, BTW,
yet representable in the computer, how strange (:-)). As such they don't
posses the order of numbers. See "rings of congruence classes." Alas but,
here (modular types) Ada is again not very inconsistent.

-- 
Regards,
Dmitry A. Kazakov
http://www.dmitry-kazakov.de