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RE: [Axiom-developer] Re: BINGO, Curiosities with Axiom mathematical str
From: |
Bill Page |
Subject: |
RE: [Axiom-developer] Re: BINGO, Curiosities with Axiom mathematical structures |
Date: |
Sat, 11 Mar 2006 01:21:17 -0500 |
On March 10, 2006 7:30 AM Ralf Hemmecke wrote:
> ...
> extend MyInteger: MyMonoid(MyInteger, *$MyInteger) == add;
> ...
> stdout << "1: " << (Z has MyMonoid(Z, *$Z)) << newline;
> stdout << "2: " << (Z has MyMonoid(Z, +$Z)) << newline;
> stdout << "3: " << (M has MyMonoid(M, *$M)) << newline;
> stdout << "4: " << (M has MyMonoid(M, +$M)) << newline;
> ...
> and try to explain the output.
>
> 1: T
> 2: T
> 3: T
> 4: T
>
> ---------------------------------------------------
>
> OK, and now the explanation. One would like the output
> (T, F, T, F), rigth?
>
> Now look at MyInteger. It has type
>
> with {
> ArithmeticType; -- includes +, *, 0, 1
> MyMonoid(MyInteger, *$MyInteger);
> }
>
> Well, but MyMonoid is a **function**. So the type is (nearly)
> with {
> ArithmeticType;
> square: MyInteger -> MyInteger;
> default {square(t: MyInteger): T == (*$MyInteger)(t, t)}
> }
>
> You see, the * is hidden behind the == sign. The actual exports
> are that of ArithmeticType together with "square". It should be
> clear that then 2 and 4 from above also return true.
>
Your explanation looks good but then I thought: What about those
categories in Axiom that take a member of some domain as a parameter.
If this parameter does not appear in any export of the category
then surely this same "error" would occur right?
Consider for example 'DirectProductCategory'
http://wiki.axiom-developer.org/axiom--test--1/src/algebra/VectorSpad/tangle
?submit=tangle&chunk=category+DIRPCAT+DirectProductCategory
It's paramters are:
DirectProductCategory(dim:NonNegativeInteger, R:Type)
but 'dim' does not appear in the signatures of the list of exports
as one can verify with:
')sh DIRPCAT
In fact 'dim' only to the right of the == some default functions
such as 'dimension' and 'size'.
One might predict using your analysis above that 'DIRPCAT(2,INT)'
and 'DIRPCAT(3,INT)' would be treated identically. But try this:
(1) -> DIRPROD(2,INT) has DIRPCAT(2,INT)
(1) true
Type: Boolean
(2) -> DIRPROD(2,INT) has DIRPCAT(3,INT)
(2) false
Type: Boolean
-------
How does Axiom know that (2) is false?
Could this have something to do with the way SPAD compiles categories
compared to Aldor?
Regards,
Bill Page.
- RE: [Axiom-developer] BINGO, Curiosities with Axiom mathematical structures, (continued)
- RE: [Axiom-developer] BINGO, Curiosities with Axiom mathematical structures, Bill Page, 2006/03/09
- Re: [Axiom-developer] BINGO,Curiosities with Axiom mathematical structures, William Sit, 2006/03/10
- Re: [Axiom-developer] BINGO, Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
- [Axiom-developer] Re: BINGO, Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/10
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/09
- RE: [Axiom-developer] Curiosities with Axiom mathematical structures, Bill Page, 2006/03/10
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/14
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, William Sit, 2006/03/10
- [Axiom-developer] Re: BINGO, Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/10
- RE: [Axiom-developer] Re: BINGO, Curiosities with Axiom mathematical structures,
Bill Page <=
- Re: [Axiom-developer] Re: BINGO, Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/11
- Re: [Axiom-developer] Re: BINGO, Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
- Re: [Axiom-developer] Re: BINGO,Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/13
- RE: [Axiom-developer] Re: BINGO, Curiosities with Axiom mathematical structures, Bill Page, 2006/03/13
- Re: [Axiom-developer] Re: BINGO,Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/14
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/10
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/13
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/13
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/13
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, William Sit, 2006/03/13