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 < g...@integrable-solutions.net> 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 <axiom...@gmail.com> 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 >> >> >> >> >> _______________________________________________ >> Axiom-developer mailing list >> Axiom-developer@nongnu.org >> https://lists.nongnu.org/mailman/listinfo/axiom-developer >> >> >
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