% language=us runpath=texruns:manuals/ontarget % \setupbackend % [format=pdf/ua-2] % \setuptagging % [state=start] \startcomponent ontarget-mathml \environment ontarget-style % \setupbodyfont[plex] % for the maps % \setupbodyfont[bonum] % for the maps %definefontfeature[fixedwidth][fixedwidth=0.30] %definefontfeature[fixedwidth][fixedwidth=0.28] \definefontfeature[fixedwidth][fixedwidth=0.27] \definefontfallback [MonoMathMath] [concrete-math.otf*fixedwidth] [0x0100-0x1FFFF] \definefontsynonym [MonoMathBase] [lmmono10-regular.otf*fixedwidth] [fallbacks=MonoMathMath] \definefontfallback [MonoTextMath] [concrete-math.otf] [0x0100-0x1FFFF] \definefontsynonym [MonoTextBase] [lmmono10-regular.otf] [fallbacks=MonoTextMath] \definefontsynonym [MonoMathBaseBold] [lmmonolt10-bold.otf*fixedwidth] [fallbacks=MonoMathMath] \definefont[MyMathFont] [MonoMathBase] \definefont[MyMathFontBold][MonoMathBaseBold] \definefont[MyTextFont] [MonoTextBase] \setuptyping [style=MyMathFont] \setupinteraction [state=start, color=, contrastcolor=, style=] \ifdefined\closure \else \definemathtopaccent [top:stretch] [closure] ["0305] \fi \startchapter[title=Meaningful math] \startlines Hans Hagen Mikael Sundqvist \stoplines \startsection[title={Introduction}] In this article we're going to discuss math in the perspective of accessibility. Although \CONTEXT\ already supports tagging in \PDF\ for quite a while, that specific kind of accessibility never took off, if only because very few viewers did anything useful with it. However, with universities introducing (whatever) validating features for documents pushed into the systems used for teaching, there was no way around picking up this thread. We start with some reflections about how we got here and then move on to some examples of how we deal with this in \LMTX. This project is part of a larger effort to get even better typeset math out of \TEX\ so we could benefit from some new features in the engine, even if they were never added with accessibility in mind. \stopsection \startsection[title={How about \MATHML}] Hans still remembers one of the first \TEX\ conferences that he attended where some evangelist from a large, in the meantime extinct, hardware company tried to convince the audience that something great had surfaced: \XML. In its wake came \MATHML, as far as we understood endorsed by large mainstream scientific publishers, some of which also disappeared in the meantime. He also remembers a user group meeting where, when showing some \MATHML\ processing, responses were of the kind \quotation {Why do you use \TEX\ at all if you want to use \MATHML\ (or \XML)} or even more curious \quotation {Why do you use \TEX\ if isn't for math?} Sometimes such comments are a way to put \CONTEXT\ (users) in their own category. Remarks of a similar kind are \quotation {One should not use \CONTEXT\ because it depends on some scripts}. The fact that scripts are used to (for instance) sort an index, process a bibliography or just manage the multiple run process (also think of \METAPOST\ graphics and position tracking in \DVI\ in former times) was seen as bad. Of course in the meantime many such features have proven to be useful, which is illustrated by the fact that other macro packages also have to deal with \MATHML\ and seem to use scripts. We mention this because decades of writing \CONTEXT\ code has taught us not to bother too much about what is said or happens outside our scope. Because we were actually confronted by \XML\ (or its precursor \SGML) it is no surprise that in \CONTEXT\ already pretty soon the angle bracket dominated format was supported and then of course \MATHML\ support is kind of easy to add to the portfolio. Although the way we deal with \XML\ is more sophisticated in \MKIV, and therefore the same in \MKXL\ aka \LMTX, the kind of things we do are the same. \stopsection \startsection[title=Practical use] At \PRAGMA\ we were never that thrilled by presentation \MATHML\ but content \MATHML\ has a certain charm and was actually more reliable when it came to rendering, if only because soon the usage pattern of the presentation variant became a wildcard. Where dealing with content \MATHML\ is kind of fun, handling presentation \MATHML\ is less so. For decades we've been involved in a project where the whole spectrum of Dutch secondary math education is covered. The content is mostly developed on a voluntary basis so for us it was mainly a way to experiment with large scale rendering of \XML\ mixed with math. The target was \PDF\ as well as the web, but we were not responsible for that part (although we can generate a web site if needed). However, as we often found out, the (limitations of) the web (\HTML) influenced the \XML\ coding in a negative way but we leave that aside. It just makes the source code less attractive over time. If we use that project as example, we can only conclude that \MATHML\ often looks like it failed to mature. We're talking about 30.000 files with over 400.000 math formulas but it being secondary school math they differ in complexity. The books are generated on demand and can differ in content and detail. There are of course also lots of images involved. A single run can involve thousands of files that are loaded, combined, filtered and typeset and in spite of what one might expect processing speed is quite comfortable (not that much different from the average \CONTEXT\ run). Static and (screen optimized) interactive \PDF\ files, are path of the package. Some schools even generate their own variants, given that they are willing to install th e production system and invest some time in learning how to mess with the content and assembly files. The disappointment with \MATHML\ has a reason. We started very optimistic with content \MATHML\ but because that demanded transformations for the \HTML\ variant we moved to presentation \MATHML. Then there was some involvement from \OPENMATH\ participants and all was moved to that format because a related output demanded it. At that time \CONTEXT\ was able to process all three variants so for book production that was not really a problem. Nevertheless another change was needed, this time because browser support was (to put it mildly) fragile and with again a flirt with presentation \MATHML\ we finally ended up with \ASCIIMATH\ which has its own peculiarities but at least was supported via \MATHJAX. So that's where we are now: of the over 400.000 formulas 52.000 are presentation \MATHML\ and 360.000 are \ASCIIMATH. \stopsection \startsection[title=It comes and goes] Quite often when it comes to typesetting, the typewriter approach is taken as reference. As a consequence plenty of simple source tagging methods and fast rendering tools have showed up but it is only a matter of time before one hits limitations and coding becomes more complex due to handling border cases and exceptions, and rendering becomes slower or maybe even cumbersome because a complex layout is well, just complex to realize due to conflicting demands. This is why \TEX\ macro packages are large and complex too. So, when discussing for instance \PDF\ documents (other than straightforward texts like novels or simple reports) in the perspective of the internet, one tends to forget that \HTML\ is (at least initially) meant for viewing in browsers where it can reflow and adapt. The more one tends to markup there, the less easy it becomes and one only has to look at the source and style sheets to see that in the end it all converges to the same level of complexity. What matters here is not so much the rendering backend, but more the availability of a clean source, one that could be used to satisfy different needs, but quite often that is not the case: there is no way to control users. If \quote {simple} is the starting point, then \quote {complex} can be pretty ugly. The same is true for math: the more complex it gets, the more likely it is that the source is a bit of a mess. It is for that reason that in \CONTEXT\ we paid a lot of attention to help users keep the source clean. It is also why we spent years on upgrading the core \TEX\ math engine. Anyway, in the meantime \MATHML\ went from version 1 to version 2 and 3 and is now a hybrid of presentation, content and \OPENMATH\ with version 4 around the corner. Then there's also a light version showing up: \MATHML\ core. We can probably safely conclude that in the meantime \MATHJAX\ is mostly used in browsers and it's a combination of \JAVASCRIPT, \CSS, \TEX\ fonts that handles presentation \MATHML and \ASCIIMATH\ (or maybe even to some extent \LATEX\ math) either by element or by parsing the content of elements. One observation is that recent versions of \MATHML\ add lots of attributes to get things right. A with \SVG\ as soon as markup and styling enters the otherwise reasonable structured input it might get worse in the end. Since we delegate a lot to the engine we don't need that level of detail so for now we just ignore most of it. Some browsers did momentarily support \MATHML\ directly, just in order to drop it overnight or change the coverage. As we're talking of decades it clearly shows that it's a mess. It is therefore surprising (and impressive) that one can actually render math in web pages as for instance \WIKIPEDIA\ demonstrates. One can wonder what \MATHML\ core brings to this, especially as it expects browser support, while browser reputation is not that good. There is no reason to assume that the ever changing web landscape with a small number of implementations of supposedly generic formats will be better in the future. Maybe the problem is just complex. Anyway, it made me somewhat indifferent to the matter: we dealt with in the past, probably can deal with it in the future (if only by ignoring what never will be generally supported anyway). What is worrying, is that the existing already rather limited presentation \MATHML\ standard dropped (according to the Mozilla documentation) \type {mfenced}, \type {menclose} and some more, and on the other hand kept the redundant \type {sqrt}, didn't add some missing elements (there's more than operators, identifiers and numbers), introduced some weird replacements for the enclosed content (with school math as excuse -- well there's more to it than that). \stopsection \startsection[title=Messing up \PDF] A secondary element in this game is \PDF\ that has some \MATHML\ related tagging on board. There we have the fences so that can hardly be dropped from the standard. Then there is a reasonable addition in the sense that one can embed the whole formula instead. But all this is done in the perspective of accessibility and getting that from typeset text is a bit of a challenge unless one renders a dedicated version suitable for reading out loud, and there is no reason not to have two variants of the same document: it might even make more sense than cooking up methods to use the (always suboptimal) tagged content. Within reasonable bounds we always supported tagging but we have to admit that we never really got much feed back on that and the reason is obvious: there has hardly been any support from viewers. It is all one big gamble and likely older tagged documents are not that useable anyway because one had to gamble about how to tag best. Also, validating applications (part of school platforms) don't know that well how to check and report on it anyway. That leaves us with guesswork and just trying to do the best we can: using common sense works out best. One \quote {problem} is that \CONTEXT\ users have never been involved in \MATHML\ discussions, \PDF\ tagging and (if there ever were) discussions about math in the \TEX\ engine, simply because early in \TEX's history \LATEX\ became the de facto standard of operating. In that perspective, looking from the \CONTEXT\ end, it's not much different from looking at how \MICROSOFT\ word or other ways of coding documents influence these standards. We just need to either adapt or ignore what happens and the latter is often the best (given the mentioned instable situation). And so, we just focus on the rendering, not caring much if it can be resembled in for instance \MATHML\ or tags. We're pretty sure that hardly any user wants to cut and paste a formula from a \PDF\ document and accessibility can often be made better by better and|/|or different rendering (maybe combine with some robust tagging). The best we can do is to produce a clean output and make sure that enough information is in the file to (maybe) help someone who doesn't have access to the source. \stopsection \startsection[title=What do we want?] When we talk about accessibility for visually disabled readers, there is not much we can say about it simply because we cannot experience the same. The closest we can come to it is by listening to a document being read in (say) a podcast. One doesn't see the rendered output and depends on some software reader. Now, if accessibility has been taken into account, of course a dedicated \PDF\ file with a layout that has no fancy rendering is best but what publisher is willing to provide two versions for that purpose? In some way we have a binary situation: if you can see the document, even when it has to be enlarged, there is little need for listening, and otherwise there is need for a nice layout. But in the end the single document demand will drive the curious mix of fancy layout and straightforward reading. But we fear that as long as big business, politics and money are involved we're unlikely to get anywhere close to a nice solution. That said, the route we've chosen is serialization. When a user is forced to validate accessibility in some application and the verdict is negative, the first question is: \quotation {So, how should it be done then?} and of course there is seldom an answer to this. Adding tags to a \PDF\ document is often enough to get into the green, but as said that tells nothing about the useability. We can embed the \MATHML\ representation and that actually satisfies some checkers. If we then add the serialized (read: spoken) math as actual text property the score can become positive there too. Serialization started as a gimmick when we tried to satisfy demands and at first we actually produced a document where all formulas could be replaced by their \quote {spoken} variant. But then we realized that if someone has a visual handicap that might work out well but that it still is not much different from a typeset formula. Real accessibility would be better served with a more dedicated rendering of the whole document, not some partial serialization. We therefore decided to stick to the combination of embedded \MATHML\ and serialization in actual text annotations and just wait till viewers start supporting that. We do the best we can at our (rendering) end and leave it at that. It must be said that when we started with this side track it was actually kind of fun. Compared to all we did with math (fonts, engine, additional and upgraded macros) this was a relatively simple activity, given that \CONTEXT\ already had most on board and we could benefit from recently added features to the engine. It also made for a nice experiment with math dictionaries, something we had (and have) on the agenda anyway. Also, as with many mechanisms it might evolve over time, for instance because languages have different demands and currently we only cover English and Swedish. The tests are done with some real math books and the upcoming math manual. We're talking of thousands of inline and display formulas of different complexity here. For testing we have various tracing options which of course adds to the runtime, but generally speaking the overhead is quite acceptable, especially given the task at hand. We started with some reflections on \MATHML\ as input and ended with \MATHML\ as output. We already had that for quite a while and could even embed it for tracing but now embedding is more integrated. The serialization is a nice addition to that so let's now explain a bit more about that. \stopsection \startsection[title={What does it mean?}] In \TEX\ we often focus mainly on how to typeset math formulas and not so much about the meaning of those formulas. It should be clear that \im{\sqrt{x}} stands for the square root of \im{x}, but many other symbols are used with more than one meaning. When we see a formula, we can often guess the intended meaning from the context, in particular if the author has used standard notation, introduced new notation, not used the same notation to mean several things, and kept the notation as simple as possible. There are, however, ambiguous cases. We give one example where the situation might not be completely clear. If, in a manuscript on complex analysis, we meet the formula \im{\bar{z} \in \widebar{U}}, we will likely interpret the first bar as the complex conjugate of the complex number \im{z}. But the meaning of \im{\widebar{U}} is perhaps less clear. The \im{\in} hints that it should be a set. One standard notation implies that it denotes the closure of the set \im{U}. But it could also, in principle, mean the set of complex conjugate of the elements in the set \im{U}. Even if the bars over these characters look the same in the pdf file, it would be good if it was possible also to carry the meaning somehow. If somebody copies the formula from the pdf they should get something sensible out of it when pasting it elsewhere. It is therefore important to work with the symbols predefined in Unicode math, and not to come up with their own weird symbols by clipping, rotating, or in other problematic ways combining symbols and perhaps also rules. Unicode math contains a lot of symbols. Many of them are described with a name that rather says something about the shape than about how they are supposed to be used. Given that we are free to use whatever symbol to denote anything, this is perhaps natural. But it is also problematic. Take \im{\char"2297} (its official name is CIRCLED TIMES), for example. It comes with two synonyms that tell a bit how it can be used as \quotation{tensor product} and as \quotation{vector pointing into page}. For the first usage the macro name \tex{otimes} has traditionally in \TEX\ been attached to the symbol. But, as the synonym says, sometimes it also denotes a vector pointing into the page, and then one can question both the macro name and the binary operation class that is attached to it. If one wants to use this symbol in the latter meaning it is natural to define a new macro that typesets the symbol, with a matching class. Observe, however, that such a construct will not change the meaning for someone copying the symbol from the pdf. It will still be CIRCLED TIMES. % This looks like a \im{\char"2BBE} (CIRCLED X), with a hat on top. There is also % \im{\char"2BBF} (CIRCLED BOLD X). % \bgroup % \switchtobodyfont[newcomputermodern] % \startformula % \char"2A36 \neq \hat{\char"2297} % \stopformula % \egroup \stopsection \startsection[title={Accessibility, tagging}] When it comes to accessibility, the situation becomes even more interesting. How shall a screen reader read the symbol \im{\char"2297}? As \quotation{CIRCLED TIMES}, as \quotation{tensor product} or as \quotation{points into the page}? Or perhaps as \quotation{otimes}? \CONTEXT\ has for a long time been able to tag documents that include mathematics and also export them to MathML. As of now, the formulas are transformed into some core MathML and included as attachments in the pdf file. Meaning easily gets lost in this conversion, so one can question how accessible the result actually is for a person who needs to have it read aloud. Moreover, the MathML itself, or the flavor of it, sometimes changes. For example, the \typ{mfenced} element recently got deprecated, leading to compatibility issues for a lot of existing documents. This lack of stability makes it both difficult and demotivating to support tagging. It can be useful to have the MathML if one wants to export and show the output on a web site. One shall remember, though, that the typeset math from \CONTEXT\ that one gets in a pdf file is not in general equivalent (features differ) to the MathML produced, so it will not be perfect. The example \im{\bar{z} \in \widebar{U}}, discussed in the introduction comes out like this (we have replaced the math italic z and U so that they show below since they are not present in the monotype font we use) \startbuffer z ̅ ∈ U ̅ \stopbuffer \typebuffer Let us also mention that it is not easy to verify that the tagging actually works. At Lund university, where Mikael is working, the tool Ally (as a plugin to the Canvas LMS) is used to check the tagged pdf files, and it does usually mark tagged pdf files from \CONTEXT\ as being perfectly tagged. But even here, things do change. At some point it was changed so that formulas was seen as attached images, and then it complained about lacking alternative texts. It is also an interesting fact that exporting a \PDF\ file, verified as perfectly tagged, into sound (also possible in Canvas LMS), it does not read the formulas correctly, if at all. \stopsection \startsection[title={Dictionaries}] With the right mark up and choice of notation from the writer, it should be possible to have it read different things, depending on the context. We therefore came up with dictionaries. They make it possible for symbols to carry a meaning and context, in addition to the atom class. In fact, we shall think of it as something that is independent of the atom class. A Unicode character can thus have several instances, where different instances might belong to different groups. Of course the math class can also vary. Thus, for \im{\char"2297} it could be like this: \starttextdisplay \setupalign[middle]\dontleavehmode \starttabulate[||||||] \NC Symbol \NC Macro \NC Class \NC Group \NC Meaning \NC \NR \NC \im{\char"2297} \NC \tex{tensorproduct} \NC binary operator \NC binary operator \NC tensor product \NC \NR \NC \im{\char"2297} \NC \tex{pointsintopage} \NC ordinary \NC unary arithmetic \NC points into the page \NC \NR \stoptabulate \stoptextdisplay The idea is then that the user can specify the groups used in a document. If one typesets a document on mathematical logic, one can load the groups that are attached to logic, and one will have these macros predefined, likely with the intended meaning. One can of course add or change the setup as well. \stopsection \startsection[title={Formulas converted into text}] One reason to introduce dictionaries with groups, in addition to atom classes, is that we can now use the label system in \CONTEXT\ to attach to each symbol also a label that tells how it could be read out. The same has been done for various macros, and as a result we can convert formulas into \quotation{spoken mathematics}, something that will be easily read by screen readers, since it is only text. Of course, given the amount of symbols and macros, we are not complete. In fact, we do not want to be complete either, and the reason is simple: We cannot know how various authors want their formulas to be spoken. So, what we have is merely a proof of concept, with a set of interpretations that covers many basic usages of commonly used symbols. \setuptagging[state=on] \enabletrackers[math.textblobs] \setupnote [mathnote] [location=margin] % \enabledirectives[structures.tags.math.standalone] To get a hold of it, let us look at a few simple examples, where after each formula we show how it is interpreted in text. \startbuffer \startformula 1 + 2 = 3 \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula 3^2 + 4^2 = 5^2 \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula \frac{3}{6} = \frac{1}{2} = 1/2 \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula \sqrt{9} = 3 \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula \sin \left(\frac{\pi}{6}\right) = \frac{1}{2} \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula \conjugate{1 + 2\ii} = 1 - 2\ii \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula \frac{1 + 2}{3 + 4} = \frac{3}{7} \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \disabletrackers[math.textblobs] \setuptagging[state=off] \stopsection \startsection[title={Some difficulties}] The process has really been trial and error. for sure there is space for improvements and variations, but we believe that the main structure is there. Different areas of mathematics come with different notations and different ways to interpret. So, if for example a logician wants to take this up, there is for sure some basic tuning needed before it works as expected. One of the difficulties we encountered along the way was how to work with parentheses. When we write \m {a(b + c)} we likely read it as \quotation {\m {a} times \m {b} plus \m {c}}. But we cannot read it like that, since that could equally well be interpreted as \m {ab + c}. We need the parentheses to be interpreted as some group: \setuptagging[state=on] \enabletrackers[math.textblobs] \startbuffer \startformula a(b + c) \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \disabletrackers[math.textblobs] \setuptagging[state=off] On the other hand, when we write \m {f(x)} it is likely that it shall be interpreted as \quotation{\m{f} of \m {x}} rather than \quotation {\m {f} times \m {x}}. \setuptagging[state=on] \enabletrackers[math.textblobs] \startbuffer \startformula f(x) \neq f\of(x) \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] In addition to the \tex {of} to handle this case, we also introduced the possibility to declare glyphs as being functions. So, it is possible to do \starttyping \registermathfunction[𝑓] \stoptyping and then leave out the \tex {of}. In fact, one of the main difficulties has been to control when the explicit \quotation {times} shall be there and when it shall not. There are several special cases; we have likely missed a few. It is also possible to declare whole alphabets as being for example vectors or matrices. We can do \startbuffer \registermathsymbol[default][en][lowercasebold][the vector] \stopbuffer \typebuffer \getbuffer and then use them as follows: \startbuffer \startformula (\alpha + \beta) \mathbf{u} = \alpha \mathbf{u} + \beta \mathbf{u} \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \stopsection \startsection[title={A few more examples}] We give a few more examples for you to ponder. \startbuffer \startformula a__1(1 + x) + (1 + y)b__1 - a_2(1 + z) - (1 + u)b_2 \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula a__{0}.a__{1}\notimes a__{2} \ldots a__{n} \ldots \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula h'\of(x) \neq h'(x) \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula s\of(1) = s\of(\set{0}) = \set{0} \cup \set{\set{0}} \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula a\sqrt{x} = ax^{1/2} \neq ax^{1/3} = a\root[3]{x} \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula \rationals = \set{\frac{p}{q} \fence p,q \in \integers \land q \neq 0} \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula f \mapsas x \mapsto x + \exp(x) \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula \lim_{k \tendsto +\infty} \frac{A__k}{B__k} \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula \Gamma__1^^2__3^^4 \neq \Gamma__1^^2^^{}__3^^4 \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula \int_{a}^{b} f'(x) \dd x = f(b) - f(a) \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula \int_{\Omega} f \dd \mu = 0 \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula \sigma \of (A \transpose{A}) \setminus \set{0} = \sigma \of (\transpose{A} A) \setminus \set{0} \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \startbuffer \startformula \frac{\partial^3 u}{\partial x^2 \partial y} \stopformula \stopbuffer \typebuffer \getbuffer \placenotes[mathnote] \disabletrackers[math.textblobs] \setuptagging[state=off] \stopsection \startsection[title=Development] Getting all these meanings right was sort of trivial with all the already present tools in \CONTEXT. We will not go into details but it basically boils down to the following: \startitemize \startitem A user keys in a formula, preferably using the proper, structure related commands, but this comes natural. \stopitem \startitem A raw formula (noad list) enters a chain of \LUA\ processing, this has always been the approach in \MKIV, so nothing new here. \stopitem \startitem Although there is some tagging happening at the \TEX\ end, we also intercept some (yet) missing cases in \LUA, again that already happened. \stopitem \startitem When a formula is typeset, the result is turned into an \MATHML\ representation, and here we just use the existing export facilities of \CONTEXT. \stopitem \startitem The now \XML\ encoded formula is then used for generating the meaning, which is the new thing in \LMTX. \stopitem \stopitemize If tagging is also needed we embed the \MATHML\ as well as the meaning, so contrary to \MKIV, we no longer tag the individual components but treat the formula as a whole. After all, this sequential tagging made little sense anyway. When we go from \MATHML\ to meaning, we use information about characters that is part of a dictionary subsystem: we put symbols into categories which permits us to render with different spacing etc.\ based on the domain we're dealing with but it also permits assigning s different meaning. This math sub-project gave us a good reason to finally add this dictionary information to the character database (which of course gave us some good laughs about how weird math sometimes get coded, also in the perspective of \UNICODE). Right from the start we decided that for real useful accessibility we needed to support multiple languages, which is why we started with simultaneous English and Swedish. We will add support for Dutch and depending on needs and input other languages. Such additions are relatively easy and fit well into the way \CONTEXT\ is set up. In \in {figure} [fig:meaningexamples:42] \in {and} [fig:meaningexamples:10] you see a few pages from a large document with examples that we used for development. This document will be in the \CONTEXT\ distribution and will be extended when there is need for more. It can also help translators. The document can show more details, like dictionary information, if needed. \startplacefigure[title={The file with examples that we test.},reference=fig:meaningexamples:42] \externalfigure[ontarget-mathml-meanings][page=42,width=\textwidth] \stopplacefigure \startplacefigure[title={The file with examples that we test.},reference=fig:meaningexamples:10] \externalfigure[ontarget-mathml-meanings][page=10,width=\textwidth] \stopplacefigure \startsection[title={A longer example, revisited}] We show below again the example from the introduction, this time with the math interpretations written out. We show this on separate pages in because here we have a somewhat complex mixed tracing situation (here we use the \STIX\ font). \bgroup \page[yes] \setuptagging[state=on] \enabletrackers[math.textblobs] % \enabledirectives[structures.tags.math.standalone] \setupnote [mathnote] [location=page] \registermathfunction[𝑓] \registermathfunction[𝑔] \registermathfunction[φ] \switchtobodyfont[stixtwo,9pt] \samplefile{ontarget-mathml-example.tex} \page[yes] \egroup % \disabledirectives[structures.tags.math.standalone] \disabletrackers[math.textblobs] \setuptagging[state=off] \stopsection \stopchapter \stopcomponent