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

[Tim Daly]

Indeed. Apparently the bleeding edge is actually Homotopy Type Theory.

I'm taking a course from Robert Harper on the subject now. I'm meeting

with a couple profs over spring break (this week), hoping to interest them

in Axiom. Unfortunately there is no grant funding behind Axiom so I'm

not sure if anything will come of it.

I'm also taking a course with Jeremy Avigad on the details of Lean

and Interactive Theorem Proving. I've been reading the source code

of the Lean kernel to try to understand how it works.

So much to know, so little time.

It is clear, however, that there is a nice mapping between these ideas

and Axiom's ideas. The goal is to get a "thin thread" through the system

that involves a proof of the GCD but to do it in such a way that proofs

become first-class objects, implemented in the Domain and propositions
are types defined in the Category, requiring proof in the Domain.

The idea of 'typeclass' matches the idea of Category (Axiom) nicely.

The idea of 'instance' matches the idea of Domain nicely.

Tim

On Sun, Mar 12, 2017 at 3:00 AM, Gabriel Dos Reis <address@hidden> wrote:

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:

Jeremy,

I read Wadler's paper. I find it amusing because these ideas were

implemented in Axiom long before the paper was published.
(http://202.3.77.10/users/karkare/courses/2010/cs653/Papers/ad-hoc-polymorphism.pdf)

One advantage, though, is their formalization. In the computer

algebra world there was work by Santas "A Type System for

Computer Algebra (http://daly.axiom-developer.org/Sant95.pdf)

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.

Tim

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