
From:  Tim Daly 
Subject:  Re: [Axiomdeveloper] Design of Semantic Latex 
Date:  Thu, 1 Sep 2016 08:25:03 0400 
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,
https://www.amazon.com/FunctionsComplexVariableTechnique Mathematics/dp/0898715954
but probably any other text would do.My last project at CMU (Tires) involved work on machine learning
using natural language (and GoodOldFashionedAI (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 latexformula 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 issuesdealing with the structure of the space underlying the formulas. I have noidea how to encode the Bloch sphere or a torus or any other space exceptby referencing an Axiom domain, which implicitly encodes it.If the formula deals with quantum mechanics then the algorithms have animplicit, mechanistic way of dealing with the Bloch sphere. So markup thatuses these function calls use this implicit grounding. Simllarly, markup thatuses a branch cut implicitly uses the implementation semantics.Axiom and Mathematics have one set of branch cuts, Maple and Maximahave another (at far as I can tell). So the markup decisions have to becarefully 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 oftime in the questionanswering literature. I have no idea how to makeprogress in that area. The Tires project involved selfmodifying lispbased on natural language interaction with a human in the limiteddomain of changing a car tire. See(The grant ended before the projected ended. Sigh)TimP.S. Tires is selfmodifying lisp code. It "learns" by changing itself.The initial code (the seed code) becomes "something else". Oneinteresting insight is that two versions of the seed code will divergebased 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 anotherexisting 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 everyoneseems to believe.
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