Tagging Math Expressions in PDF documents with Lualatex

mathmltagger.lua

-- element tree, http://etree.luaforge.net/ (is a bit buggy for the {decl = false} option)
local el = require "etree"
-- nodelist visualization (optional), https://gist.github.com/556247
-- local viz = require "viznodelist"


function convertToMathML(head)
	return {tag="not implemented"}
end

-- create content MathML for every math formula
luatexbase.add_to_callback('mlist_to_hlist',
function(head, display, penalty)
	texio.write_nl('NEW mathlist')

	result = convertToMathML(head)
	if result ~= nil then
		et = etree.ElementTree({tag = "math", result}, {decl = false})
		local pdf = node.new("whatsit", "pdf_annot")
		local buffer = etree.StringBuffer()
		et:write(buffer)
		pdf.data = '/Subtype /MathML /Contents (' .. tostring(buffer) .. ')'
		head = node.insert_before(head, head, pdf)
	end
	return node.mlist_to_hlist(head, display, penalty)
end,
	"content MathML generator")

-- add content MathML as a PDF annotation
local whatsit   = node.id('whatsit')
local hlist     = node.id('hlist')
local vlist     = node.id('vlist')
local math_node = node.id('math')

local function add_size_to_annot(head, hbox)
	while head do
		typ = head.id
		if typ == vlist then
			add_size_to_annot(head.head, hbox)
		elseif typ == hlist then
			add_size_to_annot(head.head, {width=head.width,height=head.height,depth=head.depth})
		elseif typ == whatsit and head.subtype == 15 and
			string.sub(head.data, 1, 16) == '/Subtype /MathML' then
			if head.prev ~= nil and head.prev.id == math_node and head.prev.subtype == 0 then
				tail = head
				for test_node in node.traverse_id(math_node, head.next) do
					if test_node.subtype == 1 then
						tail = test_node
						break
					end
				end
				w, h, d = node.dimensions(head.prev, tail)
				hbox = {width=w,height=h,depth=d}
			end
			--texio.write_nl(string.format("add height %gpt, width %gpt, depth %gpt",hbox.height / 2^16, hbox.width / 2^16, hbox.depth / 2^16))
			head.width  = hbox.width
			head.height = hbox.height
			head.depth  = hbox.depth
		else
			-- texio.write_nl('found node '..node.type(head.id))
		end
		head = head.next
	end
end

local vpack_counter = 1

luatexbase.add_to_callback('pre_output_filter',
function(head)
	add_size_to_annot(head,{width=0,height=0,depth=0})
	-- viz.nodelist_visualize(head, "vpack"..vpack_counter..".gv")
	vpack_counter = vpack_counter + 1
	return head
end
	,"find math bounding box")

mathmltest.tex

\pdfcompresslevel=0 % to make everything visible in the pdf
\documentclass{article}
\usepackage{amssymb}
\usepackage{luacode}

\directlua{dofile("mathmltagger.lua")}

%% a collection of Hans Hagens MathML examples, and some additions
\begin{document}
$$b \equiv b$$
$$\sqrt[1+n]{64}$$
$$1 + x \over 1 - x$$
$$-1$$
$$a+b+c$$
$$x \ge 4$$
$$a b$$
$$ x\in\mathbb{N}$$
$$ 1A2C_{16} + 0101_{16} = 1B2D_{16}$$
$$ 2+5i\in\mathbb{C}$$
%% eq, neq, gt, lt, geq, leq
$$ a\le b\le c$$
%% equivalent, approx, implies
$$ a+b \equiv b+a $$
$$ 3.14159 \approx \pi $$
%%  minus, plus
$$37 -x$$
$$-37$$
$$-x+37$$
$$a+x$$
%% times
$$3p$$
%% divide
$$1-{1 \over 3}+{1\over 5}-{1\over 7}+\ldots = \frac{\pi}{4}$$
$${-b - \sqrt{a} \over (b-b) -\sqrt{a}}$$
%%$${-b - -b - \sqrt{a} \over (b-b)- -b -\sqrt{a}}$$
%% power
$$x^2 + \sin^2 x$$
%% root, degree
$$\sqrt[3]{64} = 4$$
%% sin, cos, tan, cot, scs, sec, ..
$$\sin(x+y)=\sin x \cos y + \cos x \sin y$$
$$\cos\pi = -1$$
%% log, ln, exp
$$\ln(e+2)\approx 1.55$$
$$e^2=7.3890560989307$$ %% is false!
%% quotient, rem
$$ \lfloor a/b \rfloor $$
%% factorial
$$ n! = n\times(n-1)\times(n-2)\times\cdots\times 1$$
%% min, max, gcd, lcm
$$z=\min\left\{(x+y),2x,{1\over y}\right\}$$
%% and, or, xor, not
$$1001_2 0101_2=0001$$
%% set, bvar
$$ \left\{1,4,8\right\}\neq$$
$$ \left\{x | 2<x<\right\}$$
%% list
$$[1,1,3]$$
$$D_{1,1,3}f$$
%% union, intersect, ...
$$U\cup V$$
$$U\cap V$$
$$v\in V$$
$$u\notin V$$
%% interval
$$(a,b]$$
$$(a,x)$$

%% inverse
$$ \sin^{-1}x$$

%% sum, product, limit, lowlimit, uplimit, bvar
$$ \sum_{i=1}^{n} {1 \over x} $$
$$ \prod_{i} {1 \over x}$$
$$ \prod_{x\in\mathbb{R}}f(x)$$
$$ \lim_{x\rightarrow 0}\sin x$$
%% int, diff, partialdiff, bvar, degree
$${d \left(\int_p^q f(x,a)dx \right) \over da}$$
$$f^\prime$$
$${d^2f(x) \over dx^2}$$
$${d^4f \over x df^2}$$
$${d^kf(x,y) \over x df(x,y)^m}$$
$${d^{m+n}f(x,y) \over x df(x,y)^m}$$
%% fn
\end{document}

This example demonstrates how Lua(La)Tex could be used to create math expression annotations in PDF documents. Here, the purpose is to tag math expressions with a bounding box. Obviously, it would be quite valuable to annotate the (La)Tex math expressions with the proper Content MathML, however this would go far beyond the simple program snippets presented here.

My own experience is that for simple LaTex math formulas it is quite easy to generate the proper Content MathML equivalents. However the approach using Context Free Grammar parsers (i.e. lpeg) doesn't apply well to the breadth of LaTex documents for which the meaning of math expressions is rather context sensitive!