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 require multiple semantic versions for a single syntax. =========== Consideration 5: I/O isomorphism 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. 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 on Encoding Mathematics, January, 2001" https://people.eecs.berkeley.edu/~fateman/papers/openmathcrit.pdf"Verbs, Nouns, and Computer Algebra, or What's Grammar Got to [2] Fateman, Richard J. _______________________________________________ Axiom-developer mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-developer |
I agree that this is
doable and would be useful; but I would include a built-in (or
separate) lint that gives a context for troubleshooting when the
57% (or whatever) occurs. Ray On 08/18/2016 02:45 PM, Tim Daly wrote:
-- Two views on life: life is an art not to be learned by observation. George Santayana:Interpretations of Poetry and Religion It's kinda nice to participate in your life Raymond Rogers _______________________________________________ Axiom-developer mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-developer |
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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:
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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 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" On Thu, Aug 18, 2016 at 5:30 PM, Richard Fateman <[hidden email]> wrote:
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The game is to define latex markup that is transparent to the syntax but adds semantics for post processing. (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.) additional, non-display information needed by the CAS. 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 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 <[hidden email]> wrote:
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All Dan Zwillinger [hidden email] 617-388-2382 On 8/20/2016 11:30 PM, Tim Daly wrote:
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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 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. Axiom bugs and misprints in published texts. 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 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 <[hidden email]> wrote:
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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 "Consideration 12: System Specific Commands"... which implies that the latex environment and quoting macros have to be implemented. Sigh. On Sun, Aug 21, 2016 at 4:17 PM, Tim Daly <[hidden email]> wrote:
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Dan, While paging through the CRC 31st Standard MathematicalTo "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 <[hidden email]> wrote:
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For those of you at home wishing to play along, there is a selatex.test1 file atweaver(latex-string,axiom-stri 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. NIST/CRC/etc formulas. On Sun, Aug 21, 2016 at 11:02 PM, Tim Daly <[hidden email]> wrote:
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My initial approach was too heavy-handed and Axiom specific. It seems the semantic markup task can be viewed as an editor 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 <[hidden email]> wrote:
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Hi Tim:
Would it be simpler to only add semantic markups to algorithmic descriptions in papers? Authors can be asked to provide a separate chunk with [Axiom] semantic markups (in essence, a skeleton implementation or pseudo-code of the algorithm involved---skeleton because the data structures of mathematical objects are usually ignored in a math paper). This would avoid having to mess with the latex source (already hard to read sometimes) or to "weave" to remove the semantic markups to recapture the latex: all that is needed would be to ignore the semantic chunk). Put another way, the semantic chunk is a direct (by hand or automatic) translation of the latex version of an algorithm chunk.
Also, what would be the scope of the added semantic macros in LaTeX (like \AD, \INT)? Can their scope be limited only to semantic chunks?
William
William Sit
Professor Emeritus Department of Mathematics The City College of The City University of New York New York, NY 10031
homepage: wsit.ccny.cuny.edu
From: Axiom-developer <axiom-developer-bounces+wyscc=[hidden email]> on behalf of Tim Daly <[hidden email]>
Sent: Thursday, August 25, 2016 6:17 AM To: Dan Zwillinger Cc: Richard Fateman; James Davenport; [hidden email]; Mike Dewar; axiom-dev; [hidden email] Subject: Re: [Axiom-developer] Design Thoughts on Semantic Latex (SELATEX) My initial approach was too heavy-handed and Axiom specific.
It seems the semantic markup task can be viewed as an editor 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
<[hidden email]> wrote:
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In reply to this post by Tim Daly
Hi Tim:
A question: how would you handle overloading of operators like * ("multiplication") in a semantic mark-up? Need the markup be as detailed as the compiler requires or just sloppy enough that the interpreter can figure out the correct semantic?
William
William Sit
Professor Emeritus Department of Mathematics The City College of The City University of New York New York, NY 10031
homepage: wsit.ccny.cuny.edu
From: Axiom-developer <axiom-developer-bounces+wyscc=[hidden email]> on behalf of Tim Daly <[hidden email]>
Sent: Thursday, August 25, 2016 6:17 AM To: Dan Zwillinger Cc: Richard Fateman; James Davenport; [hidden email]; Mike Dewar; axiom-dev; [hidden email] Subject: Re: [Axiom-developer] Design Thoughts on Semantic Latex (SELATEX) My initial approach was too heavy-handed and Axiom specific.
It seems the semantic markup task can be viewed as an editor 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
<[hidden email]> wrote:
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In reply to this post by William Sit-3
William, It is unlikely that authors will provide a special chunk for Axiom in papers.are quite interesting (see http://mathpix.com). Unfortunately, there isn't enough information in the latex. For instance, are your formulas given over the real or complex domain? In the longer term I am campaigning to bend these tomes toward a more computational mathematics basis. Instead of listing the names of 20 invariant graph algorithms we really need reference versions of the algorithms. And we need them in machine-readable form. And we need them now so a whole generation of computational mathematicians do not write yet-another-CAS from scratch. Tim On Thu, Aug 25, 2016 at 9:13 AM, William Sit <[hidden email]> wrote:
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In reply to this post by William Sit-3
Suppose the latex has MN. This might be a symbol, two integers or two matrices.\AT takes an expression and a type target. The weaver program sees that this is a multiplication from On Thu, Aug 25, 2016 at 9:22 AM, William Sit <[hidden email]> wrote:
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In reply to this post by Tim Daly
Tim:
Thanks for taking the time to address my ignorant questions. You wrote: "For instance, are your formulas given over the real or complex domain?"
This question is of course relevant in computation. Even in Axiom, which allows domains (belonging to a specific Axiom category) as parameters in function or package calls, the compiler needs to know the exact domain at compile time (and with some more effort,
delay this knowledge to run time). Your example with \AT for matrix multiplication MN illustrates that.
However, mathematics is different. We do NOT have to name any specific domain. We can say, the algorithm works for any field k. How do you turn that into computer code without making a choice for k? Or, we can say the algorithm works for any matrix in GL(n,
k), for any positive integer n over any field k.
Your answer to the question on overloading is of course the "middle way", but the problem above (unspecified domains in a category, or element in a domain) could cascade and so there has to be a non-specific translation (or way to mark-up), perhaps with
a "default" specification in case computation becomes necessary.
Somehow I got (perhaps incorrectly) the impression that your proposed target is weaver(latex paper, axiom paper)---of course, a paper is also a string.
For a limited application (formulas like integrals), such generality is perhaps not needed. For that purpose, I do not believe we need a new semantic mark-up layer---if I follow your progress correctly, you already have a direct [semi-automatic?] translation
program (or a bunch of macros) that inputs the latex source for a formula (or a scanned image with "mathematical OCR" software) and outputs the Axiom code (or better still, an Axiom package that allows [domain] parameters). As you acknowledged, the selatex
test file with weaver(latex string, axiom string) does not yet provide the semantic content (that's the semi-automatic part: choosing default domains). Why do we need to "unweave" an axiom string with semantic mark-up back to latex (with or without semantic)?
Is it to ensure that weave has an inverse? I don't see that to be the case, since we have to make choices for domains to give full computational semantics but don't for in the latex string, even including full mathematical semantics. I think weave is one to
many in general, but unweave can be one to one and thus possibly loses the generality of input latex string given to the weave routine.
William
William Sit
Professor Emeritus Department of Mathematics The City College of The City University of New York New York, NY 10031
homepage: wsit.ccny.cuny.edu
From: Tim Daly <[hidden email]>
Sent: Thursday, August 25, 2016 2:50 PM To: William Sit Cc: Dan Zwillinger; Richard Fateman; James Davenport; [hidden email]; Mike Dewar; axiom-dev; [hidden email] Subject: Re: [Axiom-developer] Design Thoughts on Semantic Latex (SELATEX) William,
It is unlikely that authors will provide a special chunk for Axiom in papers.are quite interesting (see http://mathpix.com). Unfortunately, there isn't
enough information in the latex. For instance, are your formulas given
over the real or complex domain?
In the longer term I am campaigning to bend these tomes toward a
more computational mathematics basis. Instead of listing the names of
20 invariant graph algorithms we really need reference versions of the
algorithms. And we need them in machine-readable form. And we need
them now so a whole generation of computational mathematicians do
not write yet-another-CAS from scratch.
Tim
On Thu, Aug 25, 2016 at 9:13 AM, William Sit
<[hidden email]> wrote:
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Extracting Mathematical Semantics from LATEX Documents Jurgen Stuber and Mark van den Brand, Abstract. We report on a project to use SGLR parsing and term rewriting with ELAN4 to extract the semantics of mathematical formulas from a LATEX document and representing them in MathML. The LATEX document we used is part of the Digital Library of Mathematical Functions (DLMF) project of the US National Institute of Standards and Technology (NIST) and obeys project-specific conventions, which contains macros for mathematical constructions, among them 200 predefined macros for special functions, the subject matter of the project. The SGLR parser can parse general context-free languages, which suffices to extract the structure of mathematical formulas from calculus that are written in the usual mathematical style, with most parentheses and multiplication signs omitted. The parse tree is then rewritten into a more concise and uniform internal syntax that is used as the base for extracting MathML or other semantical information. On Thu, Aug 25, 2016 at 8:05 PM, Tim Daly <[hidden email]> wrote:
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