
From:  Tim Daly 
Subject:  Re: [Axiomdeveloper] Design Thoughts on Semantic Latex (SELATEX) 
Date:  Sun, 21 Aug 2016 16:17:01 0400 
All
I began reading this topic's emails when they first appeared, and then fell behind.
Sorry for my late comments.
I admire your efforts.
They may be somewhat related to what I have done in the past.
My experience is as follows:
(1) CRC SMTF (Standard Mathematical Tables and Formula)
I put the ~700 integrals in CRC's SMTF into a format from which
(A) they could be typeset in LaTeX
(B) they could be converted into Mathematica (which either did a symbolic or numeric computation)  and this was done
I let Richard Fateman use them for his experiments.
(2) Elsevier's GR (Gradshteyn and Ryzhik's "Table of Integrals, Series, and Products")
I put the ~12,000 (if my memory is correct) integrals into a format from which
(A) they could be beautifully typeset in LaTeX
(B) they could be converted into Mathematica  and this was NOT done
Enclosed is a PDF file describing the work and the resulting format.
A different format was used for SMTF and GR.
While the SMTF work was not too arduous, the GR work was more than I had anticipated.
The input (the previous version of GR) had little syntactic structure and it took much effort to get it into shape.
I used (many different) regular expressions (in perl) to translate the bulk of the book, and then lots of hand tuning.
While I think you are well beyond my thinking on these topics, please let me know if I can help.
I am friends with Howard Cohl at NIST, who may be the current lead for DLMF ("Digital Library of Mathematical Functions" at dlmf.nist.gov).
Let me know if you need an introduction.
Dan Zwillinger address@hidden 6173882382On 8/20/2016 11:30 PM, Tim Daly wrote:
$\INTEG{3x~dx}{x}$\VARIABLE to get a correct parse so the _expression_ could bemarkup can exist. Axiom, for example, would not need the \INT orparsing possible. Depending on the system, more or less parsingThe semantics markup makes the display pretty and the semantic_expression_ into an inputform for the CAS. In Axiom that would beA 'weaver' program would see the integration _expression_ asmore clever but that's not the point being made here.These trivial macros can be made less verbose and certainlybut notice that the variable of integration is in the semantic markup.(integralsign) 3x dxas $\int{3x~dx}$, that is, anAn integer 3 can be wrapped as \INT{3} but will still display as 3.without showing up in the output. This allows the semantics to carryThis allows the variable of integration to be passed in the semanticsreplaced by the first argument.delimites the characters that will occur during expansion with the #1defines the number of expected arguments. The brace argumentThis defines 4 new latex markups. The number in the square brackets\newcommand{\INTEG}[2]{\int{#\newcommand{\POLY}[1]{#1}\newcommand{\VARIABLE}[1]{#1}\newcommand{\INT}[1]{#1}Some simple tests show that this works. Suppose selatex.sty contains:The game is to define latex markup that is transparent to the syntaxbut adds semantics for post processing.
1}}
(As an aside, INT, VARIABLE, and POLY just happen to be valid
Axiom domain abbreviations, hence the name choice. This choice
of names gives grounding to the semantics.)
Notice that \INTEG takes two arguments but will display only one.
additional, nondisplay information needed by the CAS.
Some examples follow.
A variable x can be wrapped as \VARIABLE{x}, displayed as x.
$\POLY{\INT{3}\VARIABLE{x}}$ will display as 3*x
$\INTEG{\POLY{\INT{3}\VARIABLE{x}~dx}}{x} will be the same result
$\INTEG{\POLY{\INT{3}\VARIABLE{x}~dx}}{x}$
with all of the semantic tags. The weaver's job is to rewrite this
integrate(3*x,x)
This validates the fundamental idea.
The next step is to write a simple weaver program. The clever path
would be to embed a declarative form of the parser syntax (BNF?)
as comments in selatex.sty. That way the latex semantics and the
weaver syntax are kept in sync.
Weaver would read the BNF comments from selatex.sty and
the formula with semantic markup as input and parse the semantic
markup into inputforms. (Wish I thought of this homework problem
when I taught the compiler course :) ).
Note that, depending on the BNF, weaver could be used to
generate output for Maxima's treebased representation.
An alternative next step is to look at a CRC book, recreate the
syntactic latex and then create the selatex.sty entries necessary
to generate weaver input.
Infinitesimal progress, but progress nontheless.
Tim
On Fri, Aug 19, 2016 at 12:45 AM, Tim Daly <address@hidden> wrote:
Suite" (CATS). Albert Rich has done this with RUBI and integration. That
One of the Axiom project goals is to develop a "Computer Algebra Test
work is already partially in the test suite and work has been done on the
pattern matching. Large datasets (like Kamke) are always welcome. Some,
such as Schaums were handdeveloped. This is tedious.
As Axiom develops more explanations and documentation it would be
useful to execute the formulas directly so there is a local incentive to beclear about semantics.
In the long term the hope is that we can just grab formulas directly from
their sources (ala literate programming). Your work makes it plain that raw
latex does not carry sufficient semantics. There is a global question of
how to make this work. Unfortunately a general crossplatform solution
is difficult (cite Dewar/Davenport/et al. for OpenMath).
Since Axiom is literate and extracting formulas is trivial it seems that
literate markup is a natural goal. Since Axiom uses abstract algebra
as a scaffold the type tower already has a lot of axiomatic semantics.
The natural join of literate latex and abstract algebra is clearly
semantic markup, aka selatex.
===========
Consideration 10: semantic>inputform translation (weaver? :) )
>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.
I am NOT defining another CAS. The goal is a "welldesigned hack" using
universally understood latex, a latex package, and a translation program.
The selatex idea is only partially Axiom specific. \INT, for instance, seems
pretty generic. However, if the idea is to read formulas and disambiguate
a=b (boolean) vs a=b (equation) then the markup needs to be grounded
to have meaning. Axiom's domains (BOOLEAN) and (EQ) as the ground
\BOOLEAN(a=b)
\EQ(a=b)
are unambiguous relative to each other in Axiom. I don't know enough
about Maxima to understand how this might translate.
The extracted formulas with the decorated semantics still needs a
semantics>inputform (weaver) preprocessor which could be Maxima
specific. This would lead to debate about what "equality" means, of course.
Axiom has tried to create a firstorder "rosetta stone" to translate between
systems (rosetta.pdf [1]) but it is too shallow to consider providing
crossplatform semantics.
=============
Consideration 11: \scope in selatex
>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.
(See Consideration 6)
Clearly there are scoping issues. My current thinking is to create a
\scope markup that would manage the environment(s). This is not
a new issue (see "Lisp in Small Pieces" [0])
There seem to be three concerns.
First is the scope name, with something like 'global' as a keyword.Second is the "closure chain" of other scopes.
Third is the symbol being scoped.
\scope{name}{chain}{symbol}
The weaver program would walk this chain to create the proper
file syntax for system input.
============
Consideration 12: System specific commands \axiom
Along with the formulas it is clear that some system specific
input may be required, such as loading files, clearing workspaces,
etc. Some of these may be done in the weaver program, such as
between formulas. Others may need to be added to the semantics
block. So a markup that provides verbatim quoting per system
might be defined, e.g.
\axiom{)clear all} %clear the workspace
which would simply quote an input line.
==============
Note that so far all that is being suggested is transparent formula
markups which do not impact the presentation, some special tags
(\scope, \axiom,...) and a weaver program, along with the ability to
read the latex and extract named formulas (aka a literate program,
which Axiom already can do).
It ought to be possible (by design) to create a semantic version of
CRC that any system could import, assuming a "sufficiently clever
weaver".
On a more ambitious note, I am trying to find a way to keep the selatex
markup "hidden" in a pdf and use it as the clipboard paste when the
formula is selected. Anyone with a clue, please help.
===============
[0] Queinnec, Christopher, "Lisp in Small Pieces" ISBN 9780521545662
(2003)
[1] Wester, Michael J. and Daly, TImothy "Rosetta"
On Thu, Aug 18, 2016 at 5:30 PM, Richard Fateman <address@hidden> wrote:
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 CASindependent 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:
[1] Fateman, Richard J.some unit tests files on primitive domain markup. That should beTo validate these ideas Axiom will include an selatex.sty file and\cos, etc.Following that might be preexisting latex functions like \int, \sum,such as \FRAC\INT or \POLY\COMPLEX\FLOAT.starts with "primitive" domains (e.g. INT), creating selatex I/O.with a standalone preprocessor from selatex>inputform.to expand \INT{3} into the correct domain. This could be doneinput, however, this means that the reader has to know howvariety of targets so this does not seem to be a problem. Forinput/output. Axiom allows output forms to be defined for awith CM semantics where currently we only output syntax.to change the syntactic display. They may, as noted before,information but these tags have to be carefully designed NOTIt is clear that there needs to be semantic tags that carry theit has computational mathematics (CM) semantics.Fateman [0] raised a set of issues with the OpenMath
approach. We are not trying to be crossplatform 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 crossplatform 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 designbycrawl approach to the semantic markup
idea. The hope is to get something running this week that
'works' but giving due consideration to global and longterm
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://axiomdeveloper.org/axiomwebsite/CATS ). It is very
tedious to hand construct test suites. It will be even more
tedious to construct them "secondlevel" 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.
only formulas. We would like to be able to read this file so
===========
Consideration 4: I/O transparency
Assume for the moment that we take a latex file containing
require multiple semantic versions for a single syntax.
It is also clear that we would like to be able to output formulas
===========
Consideration 5: I/O isomorphism
An important property of selatex is an isomorphism with
It should be possible to readthenwrite an selatex formula,
or writethenread 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 predefined 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: Latextime processing
Axiom currently creates specific files using \write to create
intermediate files (e.g. for tables). This technique can be used
to enhance latextime debugging (where did it fail?).
It can be used to create Axiom files which preconstruct domains
needed when the input file with semantic markup is read.
This would help a standalone selatex>inputform preprocessor.
===========
Consideration 9: Design sketches
It is all wellandgood to handwave at this idea but a large
amount of this machinery already exists.
It would seem useful to develop an incremental test suite that
Once these are in place we could work on "type tower" markup
enough to start the bikeshed discussions.
Ideas? Considerations? Suggestions?
Tim[0] Fateman, Richard J.
"A Critique of OpenMath and Thoughts onEncoding Mathematics, January, 2001"https://people.eecs.berkeley.edu/~fateman/papers/openmathcri t.pdf
"Verbs, Nouns, and Computer Algebra, or What's Grammar Got to[2] Fateman, Richard J.
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