[Axiom-developer] [Complex Polynomial] (new)

[Bill Page]

Changes http://wiki.axiom-developer.org/ComplexPolynomial/diff
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*Andrea Bedini wrote:*

I'm just learning how to do some computations with Axiom,
but I've a problem with complex numbers.
 
Why this evalutate to zero ?
 
\begin{axiom}
A: Complex Polynomial Integer
A*conjugate A - A^2
\end{axiom}
 
Definitely wrong.  It looks like the source of the problem is:
\begin{axiom}
conjugate(a)
\end{axiom}

*Bill Page wrote:*

Of course I agree that it is wrong. But what would you expect
the answer to::

  conjugate(a)

to be? It could return unevaluated as just::

  conjugate(a)

but then what should the Type: be? Is such an expression
still necessarily a Complex Polynomial Integer?

This is dangerously close to the discussion that we had some
months ago about the meaning of "indeterminants", objects
which have a type but no specific value. For example, I can
say::

  A:Integer

but if I write::

  A+1

If you answered "yes" to the idea that 'conjugate(a)' is still
of type Complex Polynomial Integer, then surely 'A+1' is still
of type Integer is the same sense, but Axiom complains that
"A has not been given a value".

Perhaps evaluating the expression::

  conjugate(a)

should also complain about the lack of a value?

>From root Wed Dec 29 00:47:32 -0600 2004
From: root
Date: Wed, 29 Dec 2004 00:47:32 -0600
Subject: complex numbers
Message-ID: <address@hidden>
In-Reply-To: <address@hidden> (address@hidden)

seems to be a categorical error of some sort.

A: Complex Polynomial Integer

tells the system that 'A' is expected to have a value which is
Complex Polynomial Integer.

'conjugate' works on values, not potential values.

Thus, conjugate(A) has no meaning as 'A' has no value.
This should probably be an error.

If axiom could work with so that conjugate worked on the type
then axiom could work at some sort of an 'axiomatic' level
rather than a symbolic computation level. Perhaps when we join
forces with the ACL2 crowd we could state certain theorems and
have them applied in the absence of a value.

Tim


>From BillPage Wed Dec 29 01:02:59 -0600 2004
From: Bill Page
Date: Wed, 29 Dec 2004 01:02:59 -0600
Subject: complex numbers
Message-ID: <address@hidden>
In-Reply-To: <address@hidden>

Tim,


I know that this is opening up the whole big subject again,
but I do think that Axiom is already "two-faced" about this.

Consider for example that we can write:

(7) -> A: Complex Polynomial Integer
                                         Type: Void
(8) -> B: Complex Polynomial Integer
                                         Type: Void
(9) -> A+B

   (9)  B + A
                               Type: Complex Polynomial Integer

Neither A or B "has a value" but Axiom has no trouble agreeing
that A+B is still of type Complex Polynomial Integer.

I do not see any essential difference between this and

(10) -> A:Integer
                                         Type: Void
(11) -> B:Integer
                                         Type: Void
(12) -> A+B

   A is declared as being in Integer but has not been given a value.

Regards,
Bill Page.


>From root Wed Dec 29 01:36:08 -0600 2004
From: root
Date: Wed, 29 Dec 2004 01:36:08 -0600
Subject: complex numbers
Message-ID: <address@hidden>
In-Reply-To: <address@hidden> (address@hidden)


Clearly you're right and I agree with you.

The problem, as I see it, is that there are subtle degrees of meaning
that are easily stepped around when you work on paper but must be
clarified in computational mathematics.

A: Integer

might mean that 1) 'A' will hold a value which is an integer
under interpretation (1)

A+1 

is an error since 'A' has no value. But what does the type of
an unbound thing mean? Lisp assigns types to the values, not
the boxes. This is like labeling a box 'television'.

or perhaps that 2) 'A' is an indefinite object of type integer
under interpretation (2)

A+1

is another indeterminant integer, where '+' comes from Integer.

or perhaps that 3) 'A' obeys the laws applied to integers
under interpretation (3)

A+1

is a polynomial with 2 integers, representing a constant, where
'+' comes from Polynomial.

or perhaps that 4) 'A' is a symbol which hold integers
under interpretation (4)

A+1

is a polynomial in A with integer coefficients ignoring 'A's type
where '+' comes from Polynomial.

or perhaps that 5) 'A' is an element of a Ring and theorems can be applied
under interpretation (5)

A+1

is 'B', a new member of the Ring since '+' is a ring operator and both
'A' and '1' are ring elements. Ideally Axiom's types would be decorated
with axioms, like the ring axioms making reasoning about unbound but
typed objects possible. The '+' comes from the Category axioms of Integer.


The exact interpretation chosen appears to be dictated by the 
underlying code and is not the same everywhere.

Axiom is the product of many people, some of whom have chosen
different interpretations. Indeed, some of the interpretations
didn't exist before the computational aspects of mathematics
came into play.

There are quite a few areas of research that could be followed.
Indeed, working out the implications of the several meanings of 
'A' is a PhD topic, and a rather hard one at that. The issue 
heads off into questions of provisos, questions of reasoning with
theorems and axioms, etc.

Axiom's main strength has always been as a research platform where
it is possible to work out these ideas and reduce them to practice.

Unfortunately, research funding seems nowhere to be found.

Tim


>From BobMcElrath Wed Dec 29 12:12:44 -0600 2004
From: Bob McElrath
Date: Wed, 29 Dec 2004 12:12:44 -0600
Subject: complex numbers
Message-ID: <address@hidden>
In-Reply-To: <address@hidden>


root address@hidden wrote:
> Clearly you're right and I agree with you.
> 
> The problem, as I see it, is that there are subtle degrees of meaning
> that are easily stepped around when you work on paper but must be
> clarified in computational mathematics.

Why cannot we allow *all*?  I expect I should be able to coerce 'A+1' to
an Integer, Polynomial Integer, Expression Integer, etc.  The default
seems to be Integer, which seems fine.

The error seems to be in the Polynomial domain:

    (16) -> c:Complex Integer           
                                                      Type: Void
    (17) -> conjugate(c)     
     
       c is declared as being in Complex Integer but has not been given a 
          value.
    (17) -> c:Complex Polynomial Integer
                                                      Type: Void
    (18) -> conjugate(c)

       (18)  c
                                    Type: Complex Polynomial Integer

Adding the Polynomial to its domain causes this to go wrong...

    (19) -> c:Polynomial Complex Integer
                                                                       Type: 
Void
    (20) -> conjugate(c)
       There are 4 exposed and 1 unexposed library operations named 
          conjugate having 1 argument(s) but none was determined to be 
        ...

I'm surprised this doesn't work.  A Polynomial on a Ring is still a
member of that ring and should inherit its functions.  (in this case,
conjugate)  It also strikes me that Polynomial Complex Integer is the
proper type here, not Complex Polynomial Integer...clearly they are
inequivalent.

> A: Integer
> 
> might mean that 1) 'A' will hold a value which is an integer
> under interpretation (1)
> 
> A+1 
> 
> is an error since 'A' has no value. But what does the type of
> an unbound thing mean? Lisp assigns types to the values, not
> the boxes. This is like labeling a box 'television'.

This is the approach taken by Integer, Complex, and Float it seems.

> or perhaps that 2) 'A' is an indefinite object of type integer
> under interpretation (2)
> 
> A+1
> 
> is another indeterminant integer, where '+' comes from Integer.
> 
> or perhaps that 3) 'A' obeys the laws applied to integers
> under interpretation (3)
> 
> A+1
> 
> is a polynomial with 2 integers, representing a constant, where
> '+' comes from Polynomial.

I cannot seem to construct an example of type (3).  Given: c:Integer and
I want to construct a Polynomial Integer containing 'c', how would I do
it?

> or perhaps that 4) 'A' is a symbol which hold integers
> under interpretation (4)
> 
> A+1
> 
> is a polynomial in A with integer coefficients ignoring 'A's type
> where '+' comes from Polynomial.

The type of 'A' cannot be ignored since 'A+1' must Polynomial must have
a Field such as Integer in its constructor.

> or perhaps that 5) 'A' is an element of a Ring and theorems can be applied
> under interpretation (5)
> 
> A+1
> 
> is 'B', a new member of the Ring since '+' is a ring operator and both
> 'A' and '1' are ring elements. Ideally Axiom's types would be decorated
> with axioms, like the ring axioms making reasoning about unbound but
> typed objects possible. The '+' comes from the Category axioms of Integer.

Now this truly is a research project.  ;)

> The exact interpretation chosen appears to be dictated by the 
> underlying code and is not the same everywhere.
> 
> Axiom is the product of many people, some of whom have chosen
> different interpretations. Indeed, some of the interpretations
> didn't exist before the computational aspects of mathematics
> came into play.

I think this *must* be clarified, and a unifying set of assumptions be
applied across all domains.

> There are quite a few areas of research that could be followed.
> Indeed, working out the implications of the several meanings of 
> 'A' is a PhD topic, and a rather hard one at that. The issue 
> heads off into questions of provisos, questions of reasoning with
> theorems and axioms, etc.

This does not seem so hard, but maybe I am being naive.

Cheers,
Bob McElrath [Univ. of California at Davis, Department of Physics]


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