**Next message:**Bob McElrath: "Re: [Axiom-developer] Re: FeynCalc -> MAXIMA"**Previous message:**Camm Maguire: "Re: [Axiom-developer] Re: FeynCalc -> MAXIMA"**In reply to:**Tim Daly: "FeynCalc -> MAXIMA"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Mail actions:**[ respond to this message ] [ mail a new topic ]

Greetings! Just my 0.02 here --

I think the concept of literate programming axiom has adopted is

marvelous. The need for providing a ladder whereby a human brain can

easily expand the realm of its comprehension into a new area of

interest without expensive infrastructure and in a reasonable period

of time exceeds the need for a black box to spit out an answer in

assembly line fashion, IMHO, though both needs obviously exist and

will continue to do so. As long as we're talking 30 years, I think

the real coup would be to provide a framework which smooths the path

stretching from one end of this continuum to the other, and

concommitantly, provides utility and access to as wide a group of

interested volunteers as possible, especially in this frenetic age

when one's available time is likely to come in erratic and

unpredicatable short bursts.

Take care,

Tim Daly <daly@rio.sci.ccny.cuny.edu> writes:

*> I know this is a research problem though hardly one that merits papers
*

*> on the subject, I guess.
*

*>
*

*> My goal isn't to solve physics/math problems. My goal is to build a system
*

*> that will be used by computational mathematicians 30 years from now. Once
*

*> this is the stated goal several things become clear.
*

*>
*

*> One clear problem that every system suffers from is that the research
*

*> papers are disconnected from the code. Mathematicians do the research
*

*> and programmers do the code. Usually it is the same person with two
*

*> mindsets. So the math mindset writes the theory with theorems and
*

*> proofs then publishes it, possibly making claims (with no way to
*

*> verify the claims by others). The programmer mindset writes the code
*

*> which hopefully correctly implements the theory but never publishes it.
*

*> Or publishes it as a "contribution" to some system.
*

*>
*

*> Consider the issues this raises for computational mathematicians.
*

*>
*

*> First, claims are made which cannot be reproduced. Citing results of
*

*> the program runs without presenting the programs is equivalent to
*

*> citing theorems without providing proofs. How can a referee properly
*

*> review such work? Physics and chemistry require reproduced results
*

*> before claims are accepted.
*

*>
*

*> Second, the programs are either not available or published as
*

*> contributions. In the first case who is to know if the actual reason
*

*> for an algorithmic speedup turns out to be a compiler switch rather
*

*> than some theoretical reason like term ordering in a groebner basis
*

*> computation? Since it is unpublished the code is likely to die thus
*

*> undermining both the basis for the claim and the possibility that
*

*> other researchers can build on the work.
*

*>
*

*> The second case is even worse in some sense. I have 1100 domains
*

*> in Axiom (some of which I wrote) and 100+ algorithms in Magnus
*

*> with no theoretical documentation; indeed most have no documentation
*

*> at all. In the 30 year view how is the next generation supposed to
*

*> build upon the work we've done so far? How can they see the evolution
*

*> of algorithms? How can they maintain the code without the theory?
*

*>
*

*> Axiom represents over 30 years and over 300 man-years of research.
*

*> I don't believe that there will be funding to build systems that are
*

*> this large and this general. Even if one funded such an effort we
*

*> end up with a lot of rework that virtually no-one wants to do.
*

*>
*

*> So I'm proposing a goal for the 30 year horizon. We need to make an
*

*> effort to collect the theory and the code and reunite the two. I
*

*> realize that there are issues.
*

*>
*

*> One issue is, as you point out, that code has to deal with grubby
*

*> details which the theory can skip. But real design choices are made
*

*> when reducing theory to practice and these design choices greatly
*

*> affect the results. We need to encourage the practice of explaining
*

*> these design decisions. For example, how are infinite objects (like
*

*> groups) represented? We have learned that in simple domains like
*

*> polynomials there are a wide range of design choices (dense, sparse,
*

*> recursive, etc) that are appropriate for different problems.
*

*>
*

*> Another issue is that current systems don't "reach up" close enough to
*

*> the theory. The gap between the theory and the implementation (I call
*

*> it the impedance mismatch) is too large for most systems. For
*

*> instance, Magnus is implemented in C++ which is WAY too close to the
*

*> machine and very, very far away from Infinite Group Theory (the Magnus
*

*> domain). Thus the burden of crossing this gap falls on the
*

*> programmer. Systems like Axiom are much closer to the mathematics. But
*

*> not close enough. We need systems that span this gap in carefully
*

*> structured ways so we can be efficient without being obscure.
*

*> This is one of the root causes of your comment that "the
*

*> practical implementation of the algorithm is often connected to the
*

*> published algorithm in complicated ways". The implemented algorithm
*

*> should not be much longer than the published one.
*

*>
*

*> If we look at the 30 year horizon it is clear that all papers in
*

*> computational mathematics will be online. We must set standards
*

*> now, or at least strive for good examples, that make it possible
*

*> to use the research effectively. In today's terms we should be
*

*> able to "drag and drop" a computational mathematics paper onto
*

*> a system like Axiom and have it immediately available. (In 30
*

*> year terms Axiom should know the "intentional stance" of the
*

*> researcher and automatically incorporate the algorithms).
*

*>
*

*> One of the key problems is that "Computational Mathematics" is
*

*> like "Computer Science" was 30 years ago. Comp Sci was a branch
*

*> of the Math dept (numerical analysis), Engineering (circuit
*

*> minimization), or Business (spreadsheets). It was not recognized
*

*> as its own subject with yet.
*

*>
*

*> Today Computational Mathematics is growing out of Math (research
*

*> papers with no code), Comp Sci (research papers in polynomial
*

*> representation), Physics (clifford algebras, hopf algebras), or
*

*> Engineering (matrix methods), etc. It is not recognized as its
*

*> own subject yet (at least not everywhere. Risc-Linz, UWO, Waterloo,
*

*> and a few other places seem to have done so).
*

*>
*

*> My current religious zealotism and wild-eyed, irrational planning
*

*> (I admit it's over-the-top-painful) claim is that we need to start
*

*> with an old idea "Literate Programming" and evolve it to suit the
*

*> needs of the next generation Computational Mathematician. Thus
*

*> all of Axiom (and soon Magnus) has been rewritten into TeX documents.
*

*> There are no C, Lisp, Spad, Makefile, etc files. Now I'm trying to
*

*> ensure that new code added to the system includes the theory (or
*

*> at least permission to use the paper so I can write the literate
*

*> document).
*

*>
*

*> Thus I would really like to see the papers that provide the theory for
*

*> FeynCalc as well as the code. If I can write one of the algorithms in
*

*> Axiom in a few dozen lines that would be much clearer than a few
*

*> thousand lines of C and I'd have the research paper attached.
*

*>
*

*> It's a hard problem but we have 30 years to solve it.
*

*>
*

*> Tim "the 30 year horizon" Daly
*

*> axiom@tenkan.org
*

*> daly@idsi.net
*

*>
*

*>
*

*>
*

*>
*

*>
*

-- Camm Maguire camm@enhanced.com ========================================================================== "The earth is but one country, and mankind its citizens." -- Baha'u'llah

**Next message:**Bob McElrath: "Re: [Axiom-developer] Re: FeynCalc -> MAXIMA"**Previous message:**Camm Maguire: "Re: [Axiom-developer] Re: FeynCalc -> MAXIMA"**In reply to:**Tim Daly: "FeynCalc -> MAXIMA"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Mail actions:**[ respond to this message ] [ mail a new topic ]

*
This archive was generated by hypermail 2b29
: 03/25/17-12:40:03 PM Z CET
*