Re: [Axiom-developer] Design Thoughts on Semantic Latex (SELATEX)

[Tim Daly]

My initial approach was too heavy-handed and Axiom specific.

It seems the semantic markup task can be viewed as an editor

task. Editors don't care about semantics, they just work on text.

Viewed this way the markup's only function is decoration for

post-processing. It is (mostly) system independent.

The edit tasks seem to be {delete, insert, replace} and some

post-markup hints {function, type}

The delete markup tells weaver to remove the latex completely.

This is useful for things like {dx}

The insert markup adds missing semantic text, such as {*} which

is needed to indicate multiplication.

The replace markup gives alternate text for weaver, for things where

the function name might differ, e.g. \int -> integrate

The function markup names a function that weaver should call

which allows special handling in post-processing. It can be any

s-_expression_ (the weaver implementation is currently lisp-based).

The type markup passes type information to weaver so Axiom

knows the target type, useful for things like matrix.

The macros are

\newcommand{\AD}[1]{#1}% delete

\newcommand{\AI}[1]{}% insert

\newcommand{\AR}[2]{#1}% replace

\newcommand{\AF}[2]{#1}% function

\newcommand{\AT}[2]{#1}% type

Note that \AI outputs nothing, so 3\AI{*}x == 3x to latex.

\AD tells weaver to delete the text, e.g. {\AD ~dx} == ~dx to latex

\AR tells weaver to replace the text e.g. {\AR \pi}{\%pi}

\AF tells weaver to call a function, e.g. one that knows how to

rewrite the input in a special way, or does tracing, etc.

\AT adds target type information for Axiom

e.g. \AT{3x+6}{POLY(INT)} == 3x+6 to latex but passes it as

a Polynomial(Integer) to Axiom

For example,

$\int{\frac{dx}{ax*b}}$

becomes

$\AT{\AR{\int}{integrate}{\frac{\AD{dx}}{a\AI{*)x+b}}}{EXPR}{INT}}$

telling weaver that the target type (AT) is EXPR(INT),

the \int is really integrate

the dx is to be ignored and

the ax+b should read a*x+b

There is an obvious tradeoff of markup vs weaver.
For example. \int might be known to weaver.

Or expressions might call an equation rewriter to add {*}

The markup could vary from almost nothing to massive detail

depending on the downstream cleverness.

This initial markup set seems sufficient to handle every task

that requires semantics markup so far. The overhead seems

small and the gain seems large.

Now the only problem is post-processing the latex. Sigh.

There is no such thing as a simple job.

Tim

On Tue, Aug 23, 2016 at 7:27 PM, Tim Daly <address@hidden> wrote:

For those of you at home wishing to play along, there is a

selatex.test1 file at

http://axiom-developer.org/axiom-website/selatex.test1

containing 620 integrals.

Each line is a call of the form

  weaver(latex-string,axiom-string)

The goal is to transform the latex into Axiom.

Implicit is the idea that weaver will use the selatex tokens

to disambiguate the input. The current file has no selatex

tokens. They will be added as needed. The idea is to keep

the problem simple by adding print-invisible sematics to the

latex-string. In the ideal case the weaver program is trivial,

as is the markup. Any tradeoff should prioritize simplicity.

Another priority is to align the semantic markup with

Axiom domains in order to ground the semantics with code.

Once all of these calls translate correctly the Axiom output

routines need to output the latex-string with the added

semantic markup so the mapping is bi-directional.

The current file only looks at integration as I already have

the latex->axiom text available. Future test files will look

at other areas of interest. The long term goal is to parse

NIST/CRC/etc formulas.

Tim

On Sun, Aug 21, 2016 at 11:02 PM, Tim Daly <address@hidden> wrote:

Dan,

While paging through the CRC 31st Standard Mathematical

Tables I landed on page 219, section 3.4.1.2 Graph Invariants.

It would be a vast improvement if there were algorithms

associated with these invariants. Clearly they exist somewhere.

To "cross the gap" between tables and computational mathematics

it would be valuable to include implementations of these invariants.

It is hard to walk away from that page. An Axiom implementation

would be fun to write, especially given the next section that lists

different kinds of graphs which, presumably, would all have the

invariants. Even better, the graph algorithms are likely good

candidates for proof technology (ACL2 if done in Lisp, COQ if

done in Spad). Lisp has the advantage of an ANSI standard.

It seems worthwhile to take sections like this, expand them

across computational and proof tools, and publish them in a

form that is generally useful. It is "nice to know" that a graph

has a radius but it would be even better if I could "just point and

click" to import the algorithm.

Axiom has been pushing literate programming for years. The

tools exist to "make it so", as the saying goes.

Tim

On Sun, Aug 21, 2016 at 10:40 PM, Tim Daly <address@hidden> wrote:

Like any research problem it is a struggle to get a useful grip on this.

Looking at G&R (I just ordered the latest, mine is 4th edition), the

task quickly gets out of hand.

The CATS tests in the past were created by reading the printed latex

in various volumes and hand-translating them to Axiom input.

It is not difficult to re-create the latex input for these examples.

Doing so and combining the results gives a set of examples with

matching input latex and output Axiom. The homework problem is

to write the necessary markup and weaver.

It is immediately obvious that this is more challenging than it seems.

For example, when writing y'(x)=0, Axiom needs y:=operator 'y

so it knows about the symbol as an operator. This falls under

"Consideration 12: System Specific Commands"... which implies

that the latex environment and quoting macros have to be

implemented. Sigh.

There is no such thing as a simple job.

Anyway, at least there is a way to make a proof of concept

prototype that reproduces existing results.

Tim

On Sun, Aug 21, 2016 at 4:17 PM, Tim Daly <address@hidden> wrote:

Dan,

Welcome.

Re: Howard Cohl. Yes, I'd like an introduction. It seems important to

make DLMF, CRC, and other sources contain enough semantics that

they can be read by a computer algebra system. There are an

enormous number of issues, such as what to do with functions

unknown to the CAS, which need to be thought through.

I believe that NIST/CRC/G&R collections with semantic markup will

have a great normalizing effect on CAS development since it will raise

cross-platform questions like "What percent of G&R do you handle?".

Albert Rich (RUBI)[0] has been doing this for integration using patterns.

This can only benefit computational mathematics in the long term.

I've also campaigned for associating algorithms with published tables.

It is important in the long term to have reference versions. The ACM

used to do this years ago. I'd like to see a Gruntz algorithm for limits

where it can be applied, for instance. It would also provide a focus on

"missing algorithms" or edge cases. Davenport/Trager/Bronstein

algorithms promise a decision procedure but there are no existing

complete implementations. The tables could highlight missing cases,

giving focus to efforts to complete the procedure.

It will also put back-pressure on the tables to define different versions

of the same formulas based on domains (C, R, etc).

"The GR work was more than I had anticipated"... wins the award for

understatement of the decade.

The goal of this effort is to make it possible to read those
formulas directly into a CAS. Axiom is my primary target but
it should be done in a somewhat system agnostic form.

I've spent well over a year creating the computer algebra test suite.

It would be so much easier and more useful if the original sources

could be read directly.

I read your paper. There is an interesting mix of syntax and semantics.

I guess the difference in this effort is that the semantic markup is

intended to be transparent and grounded. The transparent aspect

keeps the tables unchanged in print form. The grounded aspect keeps

the semantics based on some algorithmic base. This implies, of

course, that there IS an algorithmic base which does not yet exist

for some of the work.

As you can see from the CATS work I've been trying to validate

Axiom's results with respect to published texts. This has found both

Axiom bugs and misprints in published texts.

The Kamke[1] suite was the first effort for differential equations.

The Spiegel[2] chapter 14 on indefinite integrals for integration.

The von Seggern[3] book on curves and surfaces for graphics.

The Legendre and Grazini[4] on Pasta by Design for 3D graphics.

The RUBI work on integration.

and, currently I'm re-creating the numerics that were lost when NAG

released the open source version, leaving me swimming through

Luke's[5] Algorithms book.

which, to quote a famous phrase "was more than I had anticipated".

Your Handbook of Integration[6] has a section on various known

"Caveats, How an integration result may be incorrect". This raises the

wonderful topic of branch cuts yet again. I did some testing and it

seems that Axiom and Mathematica share one set while Maple and

Maxima share another.

All of which leads to a need to create better reference materials that

are generally available (unlike the ACM algorithms for non-paying

customers) and directly useful for computational mathematics.

The current plan is to take some tables, find or re-create the latex,

invent a semantic markup, and then write the "weaver". At this point

the research is still at the "proof of concept" stage. (tex formula

sources are most welcome).

Ultimately I'd really like to see a book of formulas and algorithms

that I can just drag-and-drop into Axiom and be able to use them

without lifetimes of work.

Actually, that 's only the penultimate goal. I have augmented

Axiom to include proofs (ACL2,COQ) so I'd also like to see proofs,

(this IS mathematics, after all) but maybe we'll leave that for
next month :-)

Tim

[0] Rich, Albert "Rule-based Mathematics"
http://www.apmaths.uwo.ca/~arich/

[1] Kamke. E. "Differentialgleichungen Losungsmethoden und Losungen"

Chelsea Publishing Company, New York, 1959

[2] Spiegel, Murray R. "Mathematical Handbook", Schaum's Outline

Series; McGraw-Hill Book Company 1968

[3] von Seggern, David "CRC Standard Curves and Surfaces",

CRC Press, 1993 ISBN 0-8493-0196-3

[4] Legendre, George L. and Grazini, Stefano "Pasta by Design",

Thames and Hudson, 2001

[5] Luke, Yudell "Algorithms for the Computation of Mathematical

Functions", Academic Press, 1977 ISBN 0-12-459940-6

[6] Zwillinger, Daniel "Handbook of Integration" Jones and Bartlett,

London, 1992

 

On Sun, Aug 21, 2016 at 10:16 AM, Dan Zwillinger <address@hidden> wrote:

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
617-388-2382

On 8/20/2016 11:30 PM, Tim Daly wrote:

The game is to define latex markup that is transparent to the syntax

but adds semantics for post processing.

Some simple tests show that this works. Suppose selatex.sty contains:

\newcommand{\INT}[1]{#1}

\newcommand{\VARIABLE}[1]{#1}

\newcommand{\POLY}[1]{#1}

\newcommand{\INTEG}[2]{\int{#1}}

This defines 4 new latex markups. The number in the square brackets

defines the number of expected arguments. The brace argument

delimites the characters that will occur during expansion with the #1

replaced by the first argument.

(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.

This allows the variable of integration to be passed in the semantics

without showing up in the output. This allows the semantics to carry

additional, non-display information needed by the CAS.

Some examples follow.

An integer 3 can be wrapped as \INT{3} but will still display as 3.

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

as $\int{3x~dx}$, that is, an

  (integralsign) 3x dx

but notice that the variable of integration is in the semantic markup.

These trivial macros can be made less verbose and certainly

more clever but that's not the point being made here.

A 'weaver' program would see the integration _expression_ as

$\INTEG{\POLY{\INT{3}\VARIABLE{x}~dx}}{x}$

with all of the semantic tags. The weaver's job is to rewrite this

_expression_ into an inputform for the CAS. In Axiom that would be

integrate(3*x,x)

The semantics markup makes the display pretty and the semantic

parsing possible. Depending on the system, more or less parsing

markup can exist. Axiom, for example, would not need the \INT or

\VARIABLE to get a correct parse so the _expression_ could be

$\INTEG{3x~dx}{x}$

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 tree-based representation.

An alternative next step is to look at a CRC book, re-create the

syntactic latex and then create the selatex.sty entries necessary

to generate weaver input.

Infinitesimal progress, but progress non-the-less.

Tim

On Fri, Aug 19, 2016 at 12:45 AM, Tim Daly <address@hidden> wrote:

One of the Axiom project goals is to develop a "Computer Algebra Test

Suite" (CATS). Albert Rich has done this with RUBI and integration. That

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 hand-developed. 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 be

clear 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 cross-platform 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 "well-designed 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) pre-processor which could be Maxima
specific. This would lead to debate about what "equality" means, of course.

Axiom has tried to create a first-order "rosetta stone" to translate between

systems (rosetta.pdf [1]) but it is too shallow to consider providing
cross-platform 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 978-0521545662
(2003)

[1] Wester, Michael J. and Daly, TImothy "Rosetta"

http://axiom-developer.org/axiom-website/rosetta.pdf

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 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

[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

[2] Fateman, Richard J.

"Parsing TeX into Mathematics",
https://people.eecs.berkeley.edu/~fateman/papers/parsing_tex.pdf