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

[Tim Daly]

The design goal of semantic latex markup would be to decorate

existing formulas with type information. This would greatly benefit

computational mathematics. A single formula would be explicit

about its domain and range. Indeed, a single formula might have

more general form over C than over R, for example. Semantic

markup would result in multiple versions of the same formula.

Assumptions, such as branch cuts, could be made explicit and

available to the community. This might not set a standard but it might

encourage machinery to handle different decisions explicitly.

Provisos, such as limitations on the range (x in [1,2]), would be made

available for things like cylindrical algebraic decomposition (CAD)

algorithms.

The CRC and NIST collections with semantic markup would create

a "computer algebra test suite" (CATS) that all of the systems could

use as a reference.

Pattern matching systems, such as Albert Rich's RUBI matcher or

Richard Fateman's TILU would have much more information on

which to base patterns.

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Consideration 0: Where are semantics inserted?

A search of the TeX package name space shows that selatex is unused.

This allows \usepackage{selatex} without conflict.

Axiom has 1177 categories/domains/packages (hereafter just called 'types'

except when being more specific).

A zero-th attempt at selatex markup amounts to a file (selatex.sty) that

contains a \newcommand for each type, for instance,

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

which would allow 'raw' markup anywhere since latex simply ends up

removing all the type decorations, as in

\INT{3}  becomes just 3

============

Consideration 1: Where are semantics processed?

These markups are essentially invisible to tex so they could be inserted

anywhere. The associated semantics would be given by a program that

reads the tex. This makes it possible to mark up formulas from the NIST

and CRC books without any change to the result.

Axiom 'pamphlet' files are just raw latex files. There is no "weave" step

which Knuth used to extract latex. This is a feature we want to be

careful to preserve. Only a "tangle" program exists and it only extracts

a named chunk so it is trivial.

If a formula is in a named chunk it can be extracted as raw text so that

\begin{chunk}{formula1}

\INT{3}

\end{chunk}

when given the command "tangle formula.file formula1" we would see

\INT{3}

on standard output.

This implies that all semantic processing is done by an external program.

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Consideration 2: Some latex semantics might help

The INT type has only a single argument but many types have more than

one. We could modify selatex.sty to include multiple "required" arguments

where they are appropriate, as in,

\newcommand{\SQMATRIX}[2] ...

so that we could requires the correct number of arguments. This could

eliminate trivial errors.

============

Consideration 3: More latex semantics might help?

The first argument of SQMATRIX is a non-negative integer. The second

is anything of category Ring so we would see

SQMATRIX(3,INT)

SQMATRIX(3,COMPLEX(FLOAT))

It might be possible to let latex check that the second argument is in

the correct category. It is not clear how to encode such information

(or whether it is a good idea) in selatex. Some experiments need to

be done.

Ideas? Considerations? Suggestions?

Tim