thanks for all the references :)
I'm not sure if I'm going to repeat comments I made already somewhere.
First, has Dan Zwillinger weighed in? I think that it would be useful
to see what he has done.
Next, there are ambiguities among CAS and even within a single CAS.
For example, in Macsyma/ Maxima there is generally no semantics
associated with "=" or ">". But in some contexts, there is some meaning.
x>2*y
is a tree expression. It is not associated with x or with y.
assume(x>2*y) does mean something ... it puts info in a database.
Somehow encoding the method to extract this information into SEALATEX
(SeLaTeX?) in a CAS-independent way -- that's quite a task. In
particular, it would seem to require an understanding of what assume()
does in Maxima, and what is() does also.
x and not x has no particular meaning, but if x is explicitly true or
false,
Maxima simplifies it to false. If SEALATEX has a semantics -- are you
defining yet another CAS? Or perhaps you should link it 100% to Axiom's
semantics, which you presumably know about and can modify.
As far as recording stuff in DLMF -- there are presumably scope issues
("in this chapter n,m are natural numbers....") and maybe even a need
to make value assignments.
I think you need to model these in SEALATEX too.
Just musing about where you are heading.
RJF
On 8/18/2016 11:45 AM, Tim Daly wrote:
Fateman [0] raised a set of issues with the OpenMath
approach. We are not trying to be cross-platform in this
effort. Axiom does provide an algebraic scaffold so it is
possible that the selatex markup might be useful elsewhere
but that is not a design criterion.
Fateman[1] also raises some difficult cross-platform issues
that are not part of this design.
Fateman[2] shows that parsing tex with only syntactic markup
succeeded on only 43% of 10740 inputs. It ought to be posible
to increase this percentage given proper semantic markup.
(Perhaps there should be a competition similar to the deep
learning groups? PhDs have been awarded on incremental
improvements of the percentage)
This is a design-by-crawl approach to the semantic markup
idea. The hope is to get something running this week that
'works' but giving due consideration to global and long-term
issues. A first glance at CRC/NIST raises more questions
than answers as is usual with any research.
It IS a design goal to support a Computer Algebra Test Suite
(http://axiom-developer.org/axiom-website/CATS). It is very
tedious to hand construct test suites. It will be even more
tedious to construct them "second-level" by doing semantic
markup and then trying to use them as input, but the hope is
that eventually the CRC/NIST/G&R, etc will eventually be
published with semantics so computational mathematics can
stop working from syntax.
===========
Consideration 4: I/O transparency
Assume for the moment that we take a latex file containing
only formulas. We would like to be able to read this file so
it has computational mathematics (CM) semantics.
It is clear that there needs to be semantic tags that carry the
information but these tags have to be carefully designed NOT
to change the syntactic display. They may, as noted before,
require multiple semantic versions for a single syntax.
It is also clear that we would like to be able to output formulas
with CM semantics where currently we only output syntax.
===========
Consideration 5: I/O isomorphism
An important property of selatex is an isomorphism with
input/output. Axiom allows output forms to be defined for a
variety of targets so this does not seem to be a problem. For
input, however, this means that the reader has to know how
to expand \INT{3} into the correct domain. This could be done
with a stand-alone pre-processor from selatex->inputform.
It should be possible to read-then-write an selatex formula,
or write-then-read an selatex formula with identical semantics.
That might not mean that the I/O is identical though due to
things like variable ordering, etc.
===========
Consideration 6: Latex semantic macros
Semantic markup would be greatly simplified if selatex provided
a mechanism similar to Axiom's ability to define types "on the fly"
using either assignment
TYP:=FRAC(POLY(INT))
or macro form
TYP ==> FRAC(POLY(INT))
Latex is capable of doing this and selatex should probably include
a set of pre-defined common markups, such as
\FRINT ==> \FRAC\INT
===========
Consideration 7: selatex \begin{semantic} environment?
Currently Axiom provides a 'chunk' environment which surrounds
source code. The chunks are named so they can be extracted
individually or in groups
\begin{chunk}{a name for the chunk}
anything
\end{chunk}
We could provide a similar environment for semantics such as
\begin{semantics}{a name for the block}
\end{semantics}
which would provide a way to encapsulate markup and also allow
a particular block to be extracted in literate programming style.
===========
Consideration 8: Latex-time processing
Axiom currently creates specific files using \write to create
intermediate files (e.g. for tables). This technique can be used
to enhance latex-time debugging (where did it fail?).
It can be used to create Axiom files which pre-construct domains
needed when the input file with semantic markup is read.
This would help a stand-alone selatex->inputform preprocessor.
===========
Consideration 9: Design sketches
It is all well-and-good to hand-wave at this idea but a large
amount of this machinery already exists.
It would seem useful to develop an incremental test suite that
starts with "primitive" domains (e.g. INT), creating selatex I/O.
Once these are in place we could work on "type tower" markup
such as \FRAC\INT or \POLY\COMPLEX\FLOAT.
Following that might be pre-existing latex functions like \int, \sum,
\cos, etc.
To validate these ideas Axiom will include an selatex.sty file and
some unit tests files on primitive domain markup. That should be
enough to start the bikeshed discussions.
Ideas? Considerations? Suggestions?
Tim
[0] Fateman, Richard J.
"A Critique of OpenMath and Thoughts on
Encoding Mathematics, January, 2001"
https://people.eecs.berkeley.edu/~fateman/papers/openmathcrit.pdf
<https://people.eecs.berkeley.edu/%7Efateman/papers/openmathcrit.pdf>
[1] Fateman, Richard J.
"Verbs, Nouns, and Computer Algebra, or What's Grammar Got to
do with Math? ", December 18, 2008
https://people.eecs.berkeley.edu/~fateman/papers/nounverbmac.pdf
<https://people.eecs.berkeley.edu/%7Efateman/papers/nounverbmac.pdf>
[2] Fateman, Richard J.
"Parsing TeX into Mathematics",
https://people.eecs.berkeley.edu/~fateman/papers/parsing_tex.pdf
<https://people.eecs.berkeley.edu/%7Efateman/papers/parsing_tex.pdf>
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