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Re: [Axiom-developer] Design of Semantic Latex

From: Tim Daly
Subject: Re: [Axiom-developer] Design of Semantic Latex
Date: Thu, 1 Sep 2016 08:25:03 -0400

The weaver program can now process a latex document.
The end result is a tree structure of the same document.
There is still more to do, of course. Much more.

It is clear that the semantics of the markup tags are all in
the weaver program. This is obvious in hindsight since the
markup needs to be transparent to the print representation.

The parser needs to know the 'arity' of tags since \tag{a}{b}
would parse one way, \tag{a}, for a 1-arity tag and another
way \tag{a}{b} for a 2-arity tag. The code needs to be generalized
to parse given the arity.

The weaver program is structured so that the tree-parse output
is independent of Axiom. The Axiom rewrite will take the tree
as input and produce valid Axiom inputforms. This should make
it possible to target any CAS.

Onward and upward, as they say....


On Sat, Aug 27, 2016 at 1:28 PM, Tim Daly <address@hidden> wrote:

On Sat, Aug 27, 2016 at 12:14 PM, Richard Fateman <address@hidden> wrote:

Take up a book on complex analysis and see what problems you have
 as you try to encode the statements, or especially the homework
problems. I tried this decades ago with the text I used,
but probably any other text would do.

My last project at CMU (Tires) involved work on machine learning
using natural language (and Good-Old-Fashioned-AI (GOFAI)).
I'm not smart enough to make progress in natural language.


I think the emphasis on handbook or reference book representation
is natural, and I have certainly pursued this direction myself.  However
what you/we want to be able to encode is mathematical discourse. This
goes beyond "has the algorithm reproduced the reference value for an
integration."   Can you encode in semantic latex a description of the geometry
of the (perhaps infinitely layered) contour of a complex function?  You
might wonder if this is important, but then note that questions of this sort
appear in the problem section for chapter 1.

Like any research project, there has to be bounds on the ambition.

At this point, the goal is to modify the markup to disambiguate a latex
formula so the machine can import it. Axiom needs to import it to create
a test suite measuring progress against existing knowledge.

What you're describing seems to be a way to encode topological issues
dealing with the structure of the space underlying the formulas. I have no
idea how to encode the Bloch sphere or a torus or any other space except
by referencing an Axiom domain, which implicitly encodes it.

If the formula deals with quantum mechanics then the algorithms have an
implicit, mechanistic way of dealing with the Bloch sphere. So markup that
uses these function calls use this implicit grounding. Simllarly, markup that
uses a branch cut implicitly uses the implementation semantics.

Axiom and Mathematics have one set of branch cuts, Maple and Maxima
have another (at far as I can tell). So the markup decisions have to be
carefully chosen.

Here's the challenge then.  Take a mathematics book and "encode"
 it so that a program (hypothetically) could answer the problems at
the end of each chapter.

That's a much deeper can of worms than it appears. I spent a lot of
time in the question-answering literature. I have no idea how to make
progress in that area. The Tires project involved self-modifying lisp
based on natural language interaction with a human in the limited
domain of changing a car tire. See
(The grant ended before the projected ended. Sigh)


P.S. Tires is self-modifying lisp code. It "learns" by changing itself.
The initial code (the seed code) becomes "something else". One
interesting insight is that two versions of the seed code will diverge
based on "experience". That implies that you can't "teach by copy",
that is, you can't teach one system and then "just copy" it to another
existing system since their experiences (and the code structure)
will differ. Any system that "learns" will fail "teach by copy", I believe.
That means that AI will not have the exponential growth that everyone
seems to believe.

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