Re: Axiom musings...

[Tim Daly]

Geometric algebra also affects another "in-process" goal.

I have BLAS and LAPACK in the Axiom sources (volume 10.5).

I've spent some time on the question of changing BLAS to use
John Gustafson's UNUM representation, which eliminates a lot
of code because various "standard errors" cannot occur. See
his book "The End Of Error"
https://www.amazon.com/End-Error-Computing-Chapman-Computational/dp/1482239868

But since Geometric algebra is coordinate free, many of the
computations can be done symbolically and then evalulated
in final form.

BLAS re-caste in Geometric Algebra means that some of the
errors, such as roundoff, cannot occur in the symbolic form.

This has the potential to make Axiom's BLAS and LAPACK
computations faster and more accurate.

Tim


On 9/5/20, Tim Daly <axiomcas@gmail.com> wrote:
> I'm in the process of re-architecting Axiom, of course.
>
> The primary research effort, as you know, is incorporating
> proof technology.
>
> But in the process of re-architecting there are more things
> to consider. Two of them are "front and center" at the moment.
>
> One concern is "Geometric Algebra". See
> http://geometricalgebra.net/
> https://www.youtube.com/watch?v=0fF2xToQmgs&list=PLsSPBzvBkYjzcQ4eCVAntETNNVD2d5S79
>
> Geometric algebra unifies a lot of mathematics. In particular,
> it "cleans up" linear algebra, creating a "coordinate-free"
> representation. This greatly simplifies and unifies a lot of
> mathematics.
>
> So the question becomes, can this be used to "re-represent"
> Axiom mathematics dependent on linear algebra? I don't
> know but the idea has a lot of potential for simplification.
>
>
> The second concern is "Category Theory". This theory
> provides a simplification and a generalization of various
> ideas in Axiom. It also puts constraints on things like an
> Axiom "category" to Axiom "category" functors so that the
> conversion preserves the mathematical "Category"
> structure and properties.
>
> MIT has a "course" on "Programming with Categories"
> https://www.youtube.com/playlist?list=PLhgq-BqyZ7i7MTGhUROZy3BOICnVixETS
> which makes things rather more understandable.
>
> So one question is how to re-represent Axiom's type
> structure so that it has a correct mathematical "Category"
> structure. This, of course, raises the question of Group
> Theory with Type Theory with Proof Theory with Category
> Theory.
>
> Getting all of this "aligned" (and hopefully reasonably
> correct) will give Axiom a solid mathematical foundation.
>
> Mathematics has changed a lot since Axiom was created
> and many of those changes have shown that we need a
> much stronger basis for ad-hoc things like coercion, etc.
>
>
> Tim
>
>
> On 8/21/20, Tim Daly <axiomcas@gmail.com> wrote:
>> A briliant essay:
>>
>> In exactly the same way a small change in axioms
>> (of which we cannot be completely sure) is capable,
>> generally speaking, of leading to completely different
>> conclusions than those that are obtained from theorems
>> which have been deduced from the accepted axioms.
>> The longer and fancier is the chain of deductions
>> ("proofs"), the less reliable is the final result.
>>
>> https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
>>
>>
>> On 8/8/20, Tim Daly <axiomcas@gmail.com> wrote:
>>> Mark,
>>>
>>> You're right, of course. The problem is too large.
>>> So what. is the plan to achieve a research result?
>>>
>>> There are 3 major restrictions on the effort (so far).
>>>
>>> First, the focus is on the GCD in NonNegativeInteger.
>>> Volume 13 is basically a collection of published thoughts
>>> by various authors on the GCD, a background literature
>>> search. Build a limited system with essentially one user
>>> visible function (the NNI GCD) and implement all of the
>>> ideas there. This demonstrates inheritance of axioms,
>>> specification of functions, pre- and post-conditions,
>>> proof integration, provisos, the new compiler, etc.
>>>
>>> Second, make the SANE GCD work in the current Axiom
>>> system by generating compatible code. This gives a
>>> stepping-stone approach where things can be grounded.
>>> Obviously none of the new proof ideas will be expected
>>> to work in the current system but it "gives a place to stand".
>>>
>>> Third, develop a lattice of functions. The idea is to attack the
>>> functions that  depend on almost nothing, prove them correct,
>>> and use them to prove functions that only depend on the
>>> prior layer. I did this with the category structure when I first
>>> got the system since it was necessary to bootstrap Axiom
>>> without a running system (something that was not possible
>>> with the IBM/NAG version). That effort took several months
>>> so I expect that function-lattice to take about the same time.
>>>
>>> This makes the research "incremental" so that a result can
>>> be achieved in one lifetime. Like a PhD thesis, it is initially
>>> intended as a small step forward but still be a valid instance
>>> of "computational mathematics", deeply combining proof and
>>> computer algebra.
>>>
>>> Tim
>>>
>>
>