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

From: Ralf Hemmecke
Subject: Re: [Axiom-developer] BINGO, Curiosities with Axiom mathematical structures
Date: Fri, 10 Mar 2006 00:39:47 +0100
User-agent: Thunderbird 1.5 (X11/20051201)

On 03/09/2006 08:27 PM, William Sit wrote:

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

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
  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.


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