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Re: [Axiom-developer] Ad-hoc polymorphism paper

From: Gabriel Dos Reis
Subject: Re: [Axiom-developer] Ad-hoc polymorphism paper
Date: Sat, 11 Mar 2017 23:00:41 -0800

Reimplementing AXIOM systems with a Hindley-Milner style polymorphism will take the computer algebra community at least three or four decades back -- with no improvement.

-- Gaby

On Fri, Mar 3, 2017 at 4:21 PM, Tim Daly <address@hidden> wrote:

I read Wadler's paper. I find it amusing because these ideas were
implemented in Axiom long before the paper was published.

One advantage, though, is their formalization. In the computer
algebra world there was work by Santas "A Type System for
Computer Algebra (
who visited the Axiom group at IBM Research.

Axiom introduces the idea of "conditional categories" which does
not seem to be anywhere else. (see Santas paper) You can say

C0: Category == with (if % has Ring then Ring)

so you can assert Ring in the current Domain (called %)
if the Category chain includes Ring. Knowing that, the
Domain (Instance in Lean) can conditionally add functions

D0 : C0 ==

  if % has Ring then
      foo: % -> %

Axiom was far ahead of its time and once it has a proof mechanism
it will be far ahead of all other computer algebra systems.

I'd really like to replace the current type-resolution machinery in Axiom
with a more formal approach. The compiler requires explicit types
everywhere. The interpreter does type inference. It would be interesting
to re-implement it using a Hindley/Milner style machine. I suspect that
would raise some interesting research questions as Axiom implements
a very general coerce/convert mechanism. Even so, there are special
cases hard-coded into the interpreter.

A theory-based machine would be much easier to prove correct.


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