Re: Integers and Mathematical Correctness

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

On Mon, 29 Sep 2014 07:58:56 -0700 (PDT), Adam Beneschan wrote:

> On Sunday, September 28, 2014 12:47:18 AM UTC-7, Dmitry A. Kazakov wrote:
> 
>>> Therefore, any 
>>> operation that returns a modular type will automatically convert a
>>> negative value to the value modulo the modulus.
>> 
>> No. This is a description of some possible implementation of the operation
>> based on an integer type. This is not the only one implementation, e.g.
>> there exist machine instructions directly implementing modular operations
>> for moduli 2**16, 2**32 ...
> 
> I'm not sure we're on the same page... I'm talking about modular types in
> Ada, and my statement is taken directly from RM 3.5.4(19).

Should probably be rewritten, because it looks in contradiction with
4.5(10) that claims:

"The predefined operations on integer types either yield the mathematically
correct result or raise the exception Constraint_Error."

The mathematically correct result of any modulo K arithmetic operation is
an element of the ring {0, 1, ..., K-1}

   10 + 10 = 20  is mathematically incorrect for modulo 16

The mathematically correct one is:

   10 + 10 = 4

Though, of course

   10 + 10 ≡ 4  (mod 16)
   10 + 10 ≡ 20 (mod 16)
   10 + 10 ≡ 36 (mod 16)
   10 + 10 ≡ 52 (mod 16)
   ...

[ Elements of the ring can be considered ether numbers

   { 0, 1, ..., K-1 }

or sets of equivalence

  {{ 0, K, 2K, ... }, { 1, 1+K, 1+2K, ...}, ... , { K-1, 2K-1, 3K-1, ...}}

In ether case there is only one correct result, which cannot be "outside
the base range." ]

> (Of course, it's always possible for a clever optimizing compiler to
> generate code that skips a step.)

I bet that is the case for Unsigned_64, Unsigned_32 etc.

>> I also do not understand how all this might be related to rational numbers,
>> which are neither integer nor modular. 
> 
> There's no relation.

OK

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