Re: [Axiom-developer] BINGO, Curiosities with Axiom mathematical structures

[Ralf Hemmecke]

On 03/09/2006 08:27 PM, William Sit wrote:
> Martin:
>
> Good work (I'm impressed :-).

Sorry, but I am not so much. I've already pointed out one problem in my 
last mail and Bill saw another.

> I think your solution is in fact the way
> mathematicians build up algebraic structures: first define the underlying set
> and the basic operations, then prove that the axioms of certain algebraic
> structure are satisfied. Here you declare (rather than prove) these using
> Aldor's 'extend' mechanism.

OK, that is why "extend" is so great. But look at

define MyMonoid(T: Type, m: (T, T) -> T): Category == with {
    square: T-> T;
    default {square(t: T): T == m(t, t)}
}

(Didn't I write that?)

If you define

extend DomA: MyMonoid(DomA, f) == add;

then f must already be an operation in the exports of DomA. Well, we can 
do this.

define CatA: Category == with {
  f: (%, %) -> %;
}
DomA: CatA == add {f(x: %, y: %): % == ...}

Now we can extend DomA by MyMonoid(DomA, f$DomA).

Then DomA has the exports
  f: (%, %) -> %;
  square: DomA -> DomA;
That is a bit funny, because the type of DomA involves DomA.

> The beauty of your solution is that the notation (naming of the operations) is
> defined in the domain itself and then the algebraic properties declared by
> extension.

> The slight disadvantage is that these structural operations must be listed with
> each domain, and there is no default way (yet). So perhaps in MyMonoid, with
> parameters, there can be a default syntax, such as:

>   MyMonoid(T:Type, default{*:(T,T)->T)}):Category

Well, did you forget "== with;" here?

>   MyDualMonoid(T:Type, default{*:(T,T)->T, o:(T,T)->T}):Category ==
>     with{MyMonoid(T); MyMonoid(T, o)};

That does not look bad, but it will not work as expected if you used 
"Category". If you later declare something of type MyDualMonoid(..) it 
will only export two more symbols that can be checked through "has", 
namely "MyDualMonoid" and "MyMonoid". But see my last BINGO mail.
There must be another mechanism to achieve the desired behaviour. Via 
categories that does not work.

>   MyAbelianMonoid(T:Type, default{+:(T,T)->T}):Category ==
>     with{commutative(+); MyMonoid(T, +), ...};

Oh, "commutative(+)" is not declared. Aldor does not allow such 
attributes. "commutative(+)" must be a category. And since one cannot 
specify + without giving it a type, one would have to write something 
like "commutative(T, +)".

>   MyRing(T:Type, default{*:(T,T)->T, +:(T,T)->T}:Category ==
>     with{MyAbelianMonoid(T), MyMonoid(T), ...};
>   MyCommutativeRing(T:Type, default{*:(T,T)->T, +:(T,T)->T}:Category ==
>     with{MyAbelianMonoid(T), MyAbelianMonoid(T,*), ...};

> (the above omits the units for the operations, which could be added easily) so
> that 
>
>   extend MyInteger: MyRing(MyInteger) == add;
>
> would make sense.

Anyway, to me this all looks like specifying properties through tags.
We should, maybe rather write

  MyMonoid(T:Type, default{*:(T,T)->T)}): Property == properties;
   MyDualMonoid(T:Type, default{*:(T,T)->T, o:(T,T)->T}):Property ==
     properties{MyMonoid(T); MyMonoid(T, o)};

where I have just replaced Category by Property and "with" by "properties".

But all this is unsatisfactory, since if Aldor should allow properties, 
I would rather prefer to write some predicate logic formula than just 
declaring names for the properties.

Maybe adding the ability to Aldor to express properties by logical 
formulas that could also be used by proof checkers, that would be some 
step forward.

Ralf