**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
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*> that will be used by computational mathematicians 30 years from now. Once
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*> this is the stated goal several things become clear.
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*>
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*> One clear problem that every system suffers from is that the research
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*> papers are disconnected from the code. Mathematicians do the research
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*> and programmers do the code. Usually it is the same person with two
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*> mindsets. So the math mindset writes the theory with theorems and
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*> proofs then publishes it, possibly making claims (with no way to
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*> verify the claims by others). The programmer mindset writes the code
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*> which hopefully correctly implements the theory but never publishes it.
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*> Or publishes it as a "contribution" to some system.
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*>
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*> Consider the issues this raises for computational mathematicians.
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*>
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*> First, claims are made which cannot be reproduced. Citing results of
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*> the program runs without presenting the programs is equivalent to
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*> citing theorems without providing proofs. How can a referee properly
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*> review such work? Physics and chemistry require reproduced results
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*> before claims are accepted.
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*>
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*> Second, the programs are either not available or published as
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*> contributions. In the first case who is to know if the actual reason
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*> for an algorithmic speedup turns out to be a compiler switch rather
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*> than some theoretical reason like term ordering in a groebner basis
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*> computation? Since it is unpublished the code is likely to die thus
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*> undermining both the basis for the claim and the possibility that
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*> other researchers can build on the work.
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*>
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*> The second case is even worse in some sense. I have 1100 domains
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*> in Axiom (some of which I wrote) and 100+ algorithms in Magnus
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*> with no theoretical documentation; indeed most have no documentation
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*> at all. In the 30 year view how is the next generation supposed to
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*> build upon the work we've done so far? How can they see the evolution
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*> of algorithms? How can they maintain the code without the theory?
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*>
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*> Axiom represents over 30 years and over 300 man-years of research.
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*> I don't believe that there will be funding to build systems that are
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*> this large and this general. Even if one funded such an effort we
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*> end up with a lot of rework that virtually no-one wants to do.
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*>
*

*> So I'm proposing a goal for the 30 year horizon. We need to make an
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*> effort to collect the theory and the code and reunite the two. I
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*> realize that there are issues.
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*>
*

*> One issue is, as you point out, that code has to deal with grubby
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*> details which the theory can skip. But real design choices are made
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*> when reducing theory to practice and these design choices greatly
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*> affect the results. We need to encourage the practice of explaining
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*> these design decisions. For example, how are infinite objects (like
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*> groups) represented? We have learned that in simple domains like
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*> polynomials there are a wide range of design choices (dense, sparse,
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*> recursive, etc) that are appropriate for different problems.
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*>
*

*> Another issue is that current systems don't "reach up" close enough to
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*> the theory. The gap between the theory and the implementation (I call
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*> it the impedance mismatch) is too large for most systems. For
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*> instance, Magnus is implemented in C++ which is WAY too close to the
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*> machine and very, very far away from Infinite Group Theory (the Magnus
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*> domain). Thus the burden of crossing this gap falls on the
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*> programmer. Systems like Axiom are much closer to the mathematics. But
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*> not close enough. We need systems that span this gap in carefully
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*> structured ways so we can be efficient without being obscure.
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*> This is one of the root causes of your comment that "the
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*> practical implementation of the algorithm is often connected to the
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*> published algorithm in complicated ways". The implemented algorithm
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*> should not be much longer than the published one.
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*>
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*> If we look at the 30 year horizon it is clear that all papers in
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*> computational mathematics will be online. We must set standards
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*> now, or at least strive for good examples, that make it possible
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*> to use the research effectively. In today's terms we should be
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*> able to "drag and drop" a computational mathematics paper onto
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*> a system like Axiom and have it immediately available. (In 30
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*> year terms Axiom should know the "intentional stance" of the
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*> researcher and automatically incorporate the algorithms).
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*>
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*> One of the key problems is that "Computational Mathematics" is
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*> like "Computer Science" was 30 years ago. Comp Sci was a branch
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*> of the Math dept (numerical analysis), Engineering (circuit
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*> minimization), or Business (spreadsheets). It was not recognized
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*> as its own subject with yet.
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*>
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*> Today Computational Mathematics is growing out of Math (research
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*> papers with no code), Comp Sci (research papers in polynomial
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*> representation), Physics (clifford algebras, hopf algebras), or
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*> Engineering (matrix methods), etc. It is not recognized as its
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*> own subject yet (at least not everywhere. Risc-Linz, UWO, Waterloo,
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*> and a few other places seem to have done so).
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*>
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*> My current religious zealotism and wild-eyed, irrational planning
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*> (I admit it's over-the-top-painful) claim is that we need to start
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*> with an old idea "Literate Programming" and evolve it to suit the
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*> needs of the next generation Computational Mathematician. Thus
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*> all of Axiom (and soon Magnus) has been rewritten into TeX documents.
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*> There are no C, Lisp, Spad, Makefile, etc files. Now I'm trying to
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*> ensure that new code added to the system includes the theory (or
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*> at least permission to use the paper so I can write the literate
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*> document).
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*>
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*> Thus I would really like to see the papers that provide the theory for
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*> FeynCalc as well as the code. If I can write one of the algorithms in
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*> Axiom in a few dozen lines that would be much clearer than a few
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*> thousand lines of C and I'd have the research paper attached.
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*>
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*> It's a hard problem but we have 30 years to solve it.
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*>
*

*> Tim "the 30 year horizon" Daly
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*> axiom@tenkan.org
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*> daly@idsi.net
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*>
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*>
*

*>
*

*>
*

*>
*

-- 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 ]

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