HTML-Math proposal

The next attachment to this letter is an HTML file
which is the first draft of our current proposal
for HTML-Math. The best way to read it is to put it
in a file and get a browser to view the file.
(It works with Netscape 2.0 on a Mac, at least.)
(Sorry I did not have time yet to install it at a
public URL at Wolfram.)

I'll be on the Monday conference call and I'll be eager
to hear any questions or comments any of you have by then.
(Though I realize it's Friday and the proposal is quite long,
and thus it might not be feasible to digest it in time for that.)

- Bruce

HTML-Math proposal summary

This letter summarizes the current proposal for HTML-Math from Wolfram
Research to the W3C HTML-Math Editorial Review Board. The current
proposal benefits greatly from prior suggestions from and discussions
with the other members of the board, for which we're very thankful.

This letter is by no means a complete, precise description of our
proposal -- many details are left out (most importantly, the complete
proposed standard character and operator dictionary and the precise set
of transformation rules for expanding the standard linear syntax
macros). These details will be supplied later if the general direction
of this proposal is accepted. I think enough of our proposal is
explained to give a good idea of its flavor and to serve as the basis
for further discussions.

Some important aspects remain to be discussed further by the group
before they are well enough understood to be part of a formal proposal,
notably how best to allow author extensions of the built-in character
and operator dictionary and transformation rules; these aspects are
left to be specified in future amendments to this proposal.

Note that this letter supersedes all prior proposals from Wolfram
Research, including the "position papers" (which were in general
more precise than I am trying to be in this summary, though they were
at a less concrete level). Note also that this letter is not an
"official document" but rather is part of our ongoing dialogue
with the HTML-Math ERB.


[description to be added later -- they're essentially the same ones
listed on the ERB's home page and shared by the group members]

<h2>Overall Architecture</h2>

HTML-Math can be embedded in an HTML document within SGML-style MATH
elements; i.e., it is preceded by a &lt;MATH> begin tag (possibly with
attributes) and followed by a &lt;/MATH> end tag. The contents of the MATH
element are called "HTML-Math source text".

HTML-Math is designed to be interpretable either by code in an HTML
browser, by a specialized "browser plugin" program, or by a standalone
program via the "foreign notation mechanism" of a general SGML
processing program. In order that HTML-Math is compatible between any
of these implementation modes, the first step in processing HTML-Math
source text is always the simultaneous parsing of SGML entities (used
to represent extended characters by name) and embedded SGML markup
(begin and end tags used to represent hierarchical structure and to
provide a place for adding attributes to subelements of a document).

HTML-Math is always processed by the following sequence of steps.
(Several of these steps make use of built-in information, consisting of
a dictionary of character and operator properties, and a set of
transformation rules; in a future amendment to this proposal, this
information will be author-extensible for all or part of a document.)

1. Parsing of SGML-style entities (which represent extended characters
by name) and markup tags.

2. Tokenization ("lexical analysis") of the non-markup source characters
(including those represented by SGML entities parsed in step 1). (Each
markup tag is treated as a single token; thus the output from this step
is a single linear sequence of tokens.)

3. Operator-precedence-based parsing of the resulting token sequence,
to generate an "expression tree". (When this letter needs to give
examples of such an expression tree in a way distinct from the source
notation, it will use the "display list representation", to be described.)

4. Application of transformation rules to the expression tree,
to generate another expression tree, called the final display list.

5. Rendering of the final display list, in the medium and style
chosen by the user of the rendering software.

Each of these steps is explained in more detail below. (This letter does
not attempt to specify every detail, however; this will be done by
subsequent addendums to this letter, if what is described here is accepted.)

But first I will show how the above steps unfold for a simple example,
as a general orientation to this proposal.

<h3>Example, and overview of processing steps</h3>

Here is a piece of text with one piece of embedded math and one
display equation:

	The solutions to the general quadratic equation
	&lt;math mode=inline>
	are given by
	x = {-b &amp;PlusMinus; &amp;root;{b^2-4ac}} &amp;over; 2a

If this example was rendered into ASCII it might look something like
	The solutions to the general quadratic equation ax  + bx + c = 0
	are given by:
		            / 2
		    -b +- \/ b  - 4ac
		x = -----------------

Here is a brief description of each of the steps in the parsing of this

<h4>step 0: find MATH elements</h4>

Each piece of source text between the begin and end tags of a MATH element
is parsed separately by HTML-Math. The first MATH element has the attribute
mode=inline, causing it to be displayed in-line within the surrounding text;
the second one has no mode attribute, so it uses the default value mode=display
and is shown as an unnumbered display equation. The rest of this description
concerns only the second MATH element.

<h4>steps 1 and 2: process extended characters (and markup); tokenize</h4>

Tokenization of the second MATH element results in the following token
sequence; each token is a list of a token-type and a string literal
(possibly containing extended characters) made from the token's source
characters, and some attributes (not shown):

	(mi "x")
	(mo "=")
	(mb "{")
	(mo "-")
	(mi "b")
	(mo "&amp;PlusMinus;")
	(mo "&amp;root;")
	(mb "{")
	(mi "b")
	(mo "^")
	(mn "2")
	(mo "-")
	(mn "4")
	(mi "a")
	(mi "c")
	(me "}")
	(me "}")
	(mo "&amp;over;")
	(mn "2")
	(mi "a")

The division of source characters into tokens, and the token types, are
determined from the dictionary of character and operator properties.
Each token may also contain a list of attributes and values which are
also defined by the dictionary, such as precedence for operator tokens,
but these are not shown above, for the sake of clarity.

[The ways of "escaping" characters which would otherwise affect the tokenizer
(like the double quote which delimits string literals (described below))
will be specified later. This can't be done with extended character notation
in a straightforward way, since step 1 is free to replace it with the
actual characters it represents; this is a necessary feature of an
architecture in which an SGML tool might preprocess HTML-Math.]

The full details of tokenization are given below, including a way of
representing a multi-character identifier. It is also possible to give
any token directly using SGML markup, e.g. <mn>2,3</mn> for a number
literal containing a comma; this is useful for representing individual
tokens which would not be tokenized in the desired way by the built-in
dictionary, or which would not be given the desired attributes.

Briefly, the token types mi, mn, and mo represent tokens which will be
parsed as identifiers, numbers, and operators respectively; mb and me
represent begin and end tags, in this case "invisible grouping"
characters. (At this stage, the mo tokens which will typically be
rendered as "linear" operators in a 2-dimensional graphic medium (i.e.
shown between their operands in a horizontal row) are not distinguished
from the ones which will never be rendered directly since the
expressions containing them will be transformed before rendering. The
mi and mn tokens will all be rendered directly by default. The desire
to support other rendering media, and (eventually) both author- and
user- defined transformation rules in addition to the built-in rules,
is one of the reasons for not distinguishing these kinds of operators
at this stage.)

The SGML entities (e.g. &PlusMinus;) each represent extended characters.
They are treated the same as ordinary characters, in that their
tokenization and subsequent parsing is determined entirely by their
entries in the character and operator dictionary. The ones shown above
happen to be single-character operators, but others, e.g. "&alpha;",
are letters which would be tokenized as identifiers. (This proposal
will be accompanied later by a complete list of extended characters and
their properties, comprising the ones in standard character sets like
ISOtech and many new ones (and new names for old ones). All old names
will be case-sensitive, but all new names consisting of concatenated
words (as most will) will be allowed with the contained words
capitalized or not. These characters are used not only to represent
hundreds of renderable special characters which can appear in typeset
mathematics (not all of which are part of Unicode), but also some
nonrendering operators and identifiers used to generate certain layout
schemas or for "semantic disambiguation". Characters of all of these types
are used in various examples throughout this letter.)

<h4>step 3: parse according to operator attributes</h4>

The token sequence is then parsed according to the attributes of the
operator tokens (precedence, associativity, whether an operator acts as
a left or right bracket, whether it can embellish other operators).
<lit>{</lit> and <lit>}</lit> are grouping characters whose only effect
is to ensure that their contents are grouped into one subexpression by
the parser.

The parser decides as it forms each subexpression whether it is a term
or an operator (and thus how it is used during further parsing). In
most cases it is a term, but when operators are "embellished" (e.g.
subscripted) the resulting expressions remain operators, and retain the
precedence and other attributes of the base operator. (This doesn't
occur in the present example.)

The parser also introduces new tokens where necessary to represent
"missing terms" and "missing infix operators", and decides whether
missing operators should be parsed as "multiplication" (as in the above
example) or "named function application". (It's unfortunate that this
decision can't be deferred until the transformation rule stage, but
these two invisible operators have different precedences. Authors are
free to insert them explicitly instead of letting the parser choose
one. Once the present proposal is amended to allow author extensions,
authors will also be free to add transformation rules which further
transform expressions containing these invisible operators, or even
entirely new invisible operators.)

The token inserted in place of a missing term is (mi "&MissingTerm;").
The tokens inserted in place of missing infix operators are one of (mo
"&InvisibleTimes") or (mo "&FunctionApplication;"), depending on how
the parser interprets this invisible operator. The rule for deciding
which one is inserted is precisely this: an invisible function
application operator is inserted if and only if its left operand would
be an identifier or a scripted identifier, and the token to its right
is a left bracket operator (such as a left parentheses). By a scripted
identifier is meant any "left-nesting" of any number and type of
scripting schemas (subscripts, superscripts, prescripts, under or
overscripts) and non-directly-rendering schemas (e.g. font changes)
around an identifier token -- that is, the parser descends into the
base (first) argument of any such scripts (zero or more), then checks
whether it has reached an identifier.

These inserted tokens will typically render invisibly; the reasons they
are inserted explicitly by the parser are to allow them to be inserted
instead by the author with identical effects, and to simplify the later
use of transformation rules.

The expression tree generated by the parser can be represented in "display list
format" as follows (though it won't be suitable for display until some
transformation rules are applied); the "leaf nodes" are tokens as
described above, and the subexpressions grouped by the parser are
lists headed by "mterm" (as in this example) or "moperator":

	(mi "x")
	(mo "=")
				(mo "-")
				(mi "b")
			(mo "&amp;PlusMinus;")
				(mo "&amp;root;")
						(mi "b")
						(mo "^")
						(mn "2")
					(mo "-")
						(mn "4")
						(mo "&amp;InvisibleTimes;")
						(mi "a")
						(mo "&amp;InvisibleTimes;")
						(mi "c")
		(mo "&amp;over;")
			(mn "2")
			(mo "&amp;InvisibleTimes;")
			(mi "a")

<h4>step 4: transform to display list</h4>

The next stage is applying transformation rules to the parse tree to
generate the "display list". These rules expand the "linear syntax"
abbreviations for layout schemas (only some of which are used in this
example) into a more general form (which can also be given directly
using SGML markup). In a future version of this proposal, there will
also be provisions for author-defined rules for use in expanding
abbreviations or new constructs (perhaps with semantic connotations),
and users (of renderers) will be able to add or override rules for
expanding new constructs.

The display list is a representation of a single "displayable (or
renderable) object", which typically contains other displayable objects
as components. Each sublist is headed by the name of a "layout schema",
which can be thought of (in the terminology of object-oriented
programming) as a "class" of displayable objects. The layout schema
include the token types which can be rendered directly, as well as a
small list of compound forms corresponding to the "expression
constructors" used in most present typeset mathematics.

The complete list of layout schemas are given below in a separate
section, including for
each one the transformation rules used to interpret its linear syntax
form, and an SGML markup form in which it can be given in full
generality and with attributes. (Any HTML-Math expression can be given
in full SGML markup form, so that every subexpression is a separate
SGML element; or these forms can be mixed with the ordinary linear
syntax forms used in this example.)

The present example is transformed by the built-in rules to
give the following display list:

	(mi "x")
	(mo "=")
				(mo "-")
				(mi "b")
			(mo "&PlusMinus;")
						(mi "b")
						(mn "2")
					(mo "-")
						(mn "4")
						(mo "&InvisibleTimes;")
						(mi "a")
						(mo "&InvisibleTimes;")
						(mi "c")
			(mn "2")
			(mo "&InvisibleTimes;")
			(mi "a")

At the risk of excessive repetition:
each list in a display list like the above comes in one of the forms


where the layout-schema-name (e.g. mfraction, mrow, mroot) is one of the short
fixed list of layout schemas (given below), or

	(token-type-name "token-character-string")

where the token-type-name (e.g. mi, mn, mo) is one of the short fixed list
of token types (given below).

The process by which parsing and transformation generates the above
display list is not given for this example, but should be clear from
the descriptions below of the general rules for each step and from
the specific descriptions of the layout schemas involved.

<h4>step 5: rendering</h4>

The display list is suitable for more or less direct rendering, since it
is made of layout schema which are conceived as constructors for
expression renderings. For each layout schema in the complete list
given below, typical conventional renderings for 2-dimensional graphic
media are described.

However, HTML-Math does not specify or require any particular rendering
behavior. This is because it is intended to represent expressions in a
way that allows them to be rendered to various quite different media
(including, for example, interactive speech), and even within one
medium, to be rendered according to the style preferences of an
individual user, and in a way which suitably fits the context provided
by the surrounding document.

On the other hand, HTML-Math <em>does</em> specify some contextual
information which must be available for the rendering of any
subexpression. This information includes certain attributes from
surrounding HTML-Math or HTML elements (or, in the future, attributes
specified by author- or user- specified rendering rules), and also
certain attributes inherited from the location of a MATH element in a
surrounding document (such as the text font, fontsize, and baseline
position), which may ultimately be determined by non-math browser code
either from the document itself or from something about its display
environment. If an HTML browser supports HTML-Math embedded in an HTML
document by means of an external program (e.g. a "plugin" or "helper
application"), it must supply these attributes to that external program
in order to allow rendered expressions to reasonably fit with their

(A complete list of rendering attributes is not given in this letter.
The mode attribute (which can be display or inline) has been mentioned
already; among the attributes not mentioned so far are whether
subscripts and superscripts should be positioned as is conventional for
math or for chemical formulas. These attributes can be given on any
HTML-Math element (when it's expressed as SGML markup) and apply to it
and to all enclosed elements.)

HTML-Math also specifies a few semantic conventions which the layout
primitives are intended to convey, when this might be necessary for
correct rendering; for example, 2-dimensional renderers may render
fractions with horizontal fraction bars or infix slashes according to
the width of the fraction elements and the available width of the
display, but this would not be correct for "columns" or "vertical
vectors" as opposed to fractions.

<h4>copy commands, rendering, and computer algebra systems</h4>

When HTML-Math expressions are rendered into a potentially interactive
medium (e.g. a window on a computer screen, which is also a potential
acceptor of gestures with a mouse), a renderer may provide various
"copy commands" which can be used to copy entire expressions or
subexpressions, in various formats, into other documents or programs.
There may be separate commands for copying either the HTML-Math source
text associated with a subexpression, or the rendered output in various

Note that there are two distinct ways in which source text for an
expression might be copied -- either with or without the
document-supplied contextual information which modifies its
interpretation or rendering in the present environment. (Even entire
expressions may be affected by contextual information in larger parts
of the surrounding HTML document.) It is suggested that both kinds of
commands be provided. The present standard is intended to make it clear
exactly which such information needs to be copied and how it can be
represented in the copied source text.

Some computer algebra systems should be able to accept HTML-Math input
directly. For the sake of others, some renderers may provide copy
commands which translate HTML-Math into the native input form for those
systems. Such commands can be considered to be doing rendering into a
special medium, which is intended to be displayed to a program rather
than to a human being. When renderers allow their users to specify
additional transformation rules for rendering into various media and/or
in various styles, it is suggested that the list of supported media and
styles (as well as the rendering rules for each one) be
user-extensible, and that "copy rendered form" commands be provided for
each medium and style for which the user has provided any rendering
rules. This will allow the creators of computer algebra systems to
publish lists of suggested "rendering rules" for translating HTML-Math
expressions into the input formats of their systems, which can be
easily installed by users for use in their renderers, and further
modified or extended by users when desired.

<h4>rendering of syntax errors</h4>

It's possible for HTML-Math source text to have "syntax errors", though
the language is specified so as to make this rare. (In the case of
mismatched begin and end tags this may or may not prevent
HTML-Math-specific code from ever seeing the incorrect source text,
depending on the characteristics of the browser code for HTML as a
whole.) In all cases in which HTML-Math-specific code detects a syntax
error in HTML-Math source text, HTML-Math specifies only that the
renderer should (1) not crash or otherwise generate any runtime error,
(2) render correct subexpressions in the standard way (to the extent
that parsing can separate them from the erroneous ones), and (3) render
incorrect subexpressions in a way which makes it obvious to a human
viewer that they are incorrect (e.g. as a "visual error message" of
some kind). HTML-Math encourages renderers to make it as easy as
possible for human viewers to learn the nature of an error from the
rendering of an erroneous subexpression (even when they have minimal
knowledge of HTML-Math), but there are no formal specifications about
how this should be done.

The purpose of the requirement that errors be obvious in the rendering
is to ensure that authors can use any HTML-Math browser for "testing"
their HTML-Math source text, and can be sure that if it "appears to
work right" in their test browser, that it is correct standard
HTML-Math and therefore can be expected to "work right" in other

<h4>upward-compatible extensions</h4>

There is one exception to the above rule about the rendering of
erroneous expressions, since it would otherwise
disallow upward-compatible extensions added by creators of specific
browsers to the source language accepted by HTML-Math (when these would
be syntax errors in the standard source language), such as adding new
extended characters or layout schemas: such extensions to a renderer
are allowed, provided that the renderer can be run in a mode which
disables all nonstandard extensions and treats them as syntax errors as
described above. Renderers with nonstandard extensions should make it
easy for users to discover the existence of this "strict standard mode"
and how to use it.

<h2>Example 2</h2>

As another example, the indefinite integral of dx over x can be
represented as
	&amp;int; &amp;dd; x &amp;over; x
where the extended characters used represent (respectively) the integral
sign (a large operator with precedence somewhat higher than +), the
"differential d" (a high-precedence prefix operator), and an infix
operator for forming fractions with horizontal bars (with precedence
near that of division).

(The integral sign character can also be called &amp;integral; or
&amp;Integral;. The name &amp;int; is provided since it is already part of
the ISOtech character set.)

This source text is parsed into the form
	(mo "&amp;int;")
			(mo "&amp;dd;")
			(mi "x")
		(mo "&amp;over;")
		(mi "x")
and then transformed into the form for rendering
	(mo "&amp;int;")
			(mo "&amp;dd;")
			(mi "x")
		(mi "x")

When the result is rendered, the integral sign (being a large operator)
is rendered in a larger font size.

If a definite integral was desired, this would be represented
by embellishing the &amp;int; operator with a subscript and superscript,
which could be done using their linear syntax forms by (for example)
	&amp;int;_1%2 &amp;dd; x &amp;over; x

The details of the features introduced with this example
are given below.

<h2>Layout schemas</h2>

Every expression in HTML-Math ultimately specifies the relative
sizes and arrangement of a collection of symbols layed out in a
"logically 2-dimensional" manner. This structure is specified not
as coordinates, but in terms of a small set of "perceptual primitives"
or "layout schemas" which are sufficient to describe almost all of
the commonly used notations in existing typeset mathematics. This
choice of level of representation is both as general and as abstract
as possible while still being based on the structure of the notation
for an expression, rather than purely on its semantic structure or

Although the layout schemas and the typical notations rendered with them
are described in this letter in 2-dimensional terms since they are most
commonly understood that way, they can also be considered as abstract
expression constructors, so that HTML-Math notations are not inherently
tied to physically 2-dimensional media, but can equally well be
rendered into other media such as interactive speech or computer
algebra systems.

Primitive token types such as "variable" or "number" are also
considered a form of layout schema, though they have no substructure,
because each of these token types is conventionally rendered
differently. In the terminology of object-oriented programming,
the layout schemas can be considered the subclasses of the class
of renderable objects.

Each layout schema has a name beginning with "m" (for "math"). This name
is used in the "display list representation" (as shown in the example
given above) as the head of the display list for an instance of a given
schema, and is also the SGML element name for the SGML form of each
schema (i.e. the name used in the begin and end tags). (The initial "m"
is partly to avoid collisions with other HTML tag names, and to make it
easy for a reader to tell which tags are specific to HTML-Math. Note
that certain non-math-specific HTML tags may be embedded in HTML-Math
expressions, e.g. links, anchors, or font changes.)

The tag names with just one letter after the "m" are token types; the
others are layout schemas, or (in the case of mterm and moperator,
which are not strictly layout schemas but are included in the following
list anyway) part of the expression tree generated by the parser. (The
names are not usually related to the linear syntax forms in which some
instances of a layout schema can be given.)

The following list of token types and layout schemas includes for each
one a description of its intended purpose, conventional rendering (not
a formal part of the standard), and semantic connotations (if any). The
SGML markup form and the linear syntax form is also given; for the
token types, the tokenization rules are discussed.

There are also some more HTML-Math examples showing the processing steps
for schemas not covered in the example discussed earlier.

<h3>Token type schemas</h3>

Here is a short summary of the token type schemas described here:

Name    Represents                    Some examples in HTML-Math source

mi      variable or identifier        a       \sin     &lt;mi>num-trees&lt;/mi>
mn      number literal                3.1              &lt;mn>3.1e10&lt;/mn>
mt      text string                   "such that"      &lt;mt>such that&lt;/mt>
mo      operator (rendered or not)    +                &lt;mo
mb      begin tag or {                {       &lt;mterm>  &lt;mfraction>
me      end tag	or }                  }       &lt;/mterm> &lt;/mfraction>

<h4>mi: variable or identifier</h4>

The mi token type represents "identifiers", named elements typically
used as variable names. Characters defined as letter-like by the
dictionary are tokenized into individual identifiers (even if they are
not separated by whitespace). (Rationale: single-letter variable names
are much more common in math than multi-character variable names.)

A backslash followed by one or more letterlike characters or digits is
tokenized into a single identifier (even if it starts with a digit);
thus \sin and \3d are both single identifiers. The backslash is not
part of the identifier name -- thus \x\y and xy are turned into the
same pair of identifier tokens.

Note that the sequence of letterlike characters forming a single
token after \ can't include whitespace. If it contains extended
characters given in SGML entity notation, these must be letterlike,
and the entity names should be terminated with ";" rather than with
whitespace (except perhaps for the last one).

E.g., to specify a single identifier which looks like "cos" except
that the middle letter is a hypothetical extended character, one
might use
(followed by some non-letterlike non-digit character).

Any character sequence may be designated as an identifier by enclosing
it within &lt;mi>...&lt;/mi>.

Identifiers are typically rendered (in a 2-dimensional graphic medium)
by displaying the characters of the name in a
closely-proportionally-spaced horizontal row, with single-character
identifiers rendered in italic (except for certain characters such as
double-struck capital letters like the Z often used to represent the
set of integers).

In future amendments to this proposal, it will be possible to associate
a semantic type and a locality of reference to a given identifier in a
given scope of a source document, and to specify a instance of an
identifier as a "defining instance" in some scope, but the issues
involved are not discussed in this letter.

<h4>mn: number literal</h4>

The mn token type represents "numeric literals", sequences of digits and
decimal points typically used to represent numbers directly. The
precise rules for tokenization of number literals are: one number
literal token is formed from every maximal-length sequence of one or
more digits mixed with zero or more decimal points, with no two decimal
points adjacent. (If two such sequences overlap; the first one is used.
However, this case is impossible, so this rule is never used.)

(The character dictionary determines which characters count as "digits"
and as "decimal points"; in the standard dictionary these are
"0123456789" and "." respectively. Note that the standard dictionary
also declares "." as an operator, but its use as a decimal point in a
legal number literal overrides its use as an operator.)

Neither commas nor minus signs (nor any form of "scientific notation")
are automatically treated as parts of number literals. (E.g., -3 is
parsed as a unary negation operator applied to 3.)

However, any character sequence may be designated as a number literal
by enclosing it within &lt;mn>...&lt;/mn>.

Number literals are typically rendered as a closely spaced row of their
constituent characters, not in italics.

<h4>mt: text string</h4>

The mt token type represents text strings. These can be given as "string
literals", i.e. as character sequences between double quotation marks
(") (which are not part of the strings), making use of precisely the
same character escape sequences [to be specified later]
or extended character names as text
outside of string literals can use, or they can be given as arbitrary
character sequences between &lt;mt>...&lt;/mt> tags.

String literals are typically rendered the same way as text which surrounds
the MATH element. When exported into a computer algebra system, they should
typically be represented as string literals in the format of that system.

<h4>mo: operator</h4>

The mo token type represents an operator token, whose properties
(listed below) will affect the grouping of subexpressions by the

The dictionary of character and operator properties defines certain
characters as operator characters, and certain sequences of these
characters as operators, with specific values of the properties listed
below. The tokenizer turns maximal sequences of operator characters
into operator tokens, and gives them attributes corresponding to the
properties in the dictionary. When several potential operator tokens
overlap, the leftmost one is chosen.

(When the dictionary is made author extensible in a future version of
this proposal, it may be possible for authors to declare character
sequences as operators which would otherwise be tokenized as
identifiers, but this will never be done in the standard dictionary.
Rationale: authors must be able to write any sequence of letter-like
characters without worrying that it will be tokenized as an operator by
default in some future versions of HTML-Math.)

Any character sequence can be specified as an operator by enclosing
it within &lt;mo>...&lt;/mo> tags; the properties given below will have
default values (to be specified later along with the full standard
operator dictionary) unless specific values are specified using
attributes within the begin tag, e.g. &lt;mo prefix=true prec=400>++&lt;/mo>.

The properties of any operator token include:


<li>Whether the operator can be prefix, infix, or postfix (more than one
form can be allowed). (Prefix operators have no left operand and
postfix operators have no right operand.)</li>

<li>A left and right precedence (which can be used to define
associativity and bracketing behaviors, as described below). Left
precedences are only meaningful when left operands are allowed, so they
are optional otherwise; similarly with right precedences. In fact, a
separate right precedence can be given for use in the infix and prefix
cases (as is done with a minus sign) or in the large-operator case;
this is not allowed with the left precedence (since that would make it
impossible in some cases to parse embellished operators

<li>Whether the operator can be used to embellish other operators.
Embellishing an operator means using it within some compound layout
schema (e.g. giving it a subscript) so that the resulting expression
has the same parsing properties as the bare operator (and in
particular, so that the operator's original precedence is used to parse
the terms surrounding the expression for the entire embellished
operator). (The parser will copy all attributes of the original operator
to the expression it generates to represent the embellished operator,
which is headed by "moperator" (see below).)</li>

<li>Whether this operator is always a "large operator" (such as the
integral or summation signs). This affects parsing in a way which
supersedes some other attributes (e.g., large operators have only right
operands). It is also possible to turn ordinary operators into large
ones (see the mlargeop layout schema, below).</li>

<li>Whether this operator is "stretchy". This is typically used for
brackets. This affects only rendering, and typically means the
operator's vertical size depends on that of its operands (for brackets,
operands are what is contained between matching pairs).</li>

The attribute names and values corresponding to these properties are:
prefix=true (means this form is allowed, not required)<br>
embellisher=true (means use to embellish other operators is allowed, not
large=true (means always parsed and rendered as a large operator)<br>

The numbers used as precedences must be integers (positive or negative).
Higher numbers mean higher precedences, i.e., stronger binding.

The parser groups a term with the adjacent operator which has the higher
precedence (assuming it is being used in a form which takes an operand
on that side). If these precedences are equal, it groups the term with
<em>both</em> operators; this feature is used to define "bracketing
operators" such as parentheses so that the token sequence parsed from
	(mo "(")
	(mi "x")
	(mo ")")
groups into the single expression tree
		(mo "(")
		(mi "x")
		(mo ")")

In the standard dictionary, all kinds of left brackets are prefix
operators with the same right precedence (which happens to be 0), and
all right brackets are postfix operators with that same value of right
precedence (that is, also 0), which means that even brackets of
different kinds can group together, e.g. in expressions like

(Individual brackets can be prevented from grouping at all by enclosing
them in &lt;mterm>...&lt;/mterm> (see below). Note that the invisible
grouping characters { and } are not operators at all, and can't be
prevented from grouping (nor, of course, do they render as curly braces);
extended characters are provided which do parse as regular brackets
and render as curly braces.)

The same feature of grouping a term with both adjacent operators is used
to allow certain operators to have "flat" or "n-ary" associativity,
e.g. + and &amp;InvisibleTimes;. This is what causes the source text "4ac"
(in the example given far above) to parse to a single (mterm ...)
subexpression containing three subterms (which are mn and mi tokens for
4, a, and c) separated by two (invisible) operator tokens.

Some operators are intended instead to be left or right associative; for
example, the superscripting operator ^ (see the mscripts layout schema
below) is right associative, meaning that a^b^c parses in the same way
as a^{b^c}. This is achieved (in the standard dictionary) by giving ^ a
slightly higher left precedence than right precedence. Similarly, left
associative operators have a slightly higher right precedence than
their left precedence.

Sometimes, more than one operator has the same left and right
precedence; this is true, for example, of relational operators, so that
sequences of inequalities turn into single subexpressions even when
(e.g.) both < and <= (or &amp;LessEqual;) are used in the same sequence.

Note that even infix + and infix - are flat-associative with the same
precedences; this means that the source text "a - b + c" parses into a
single subexpression of five tokens, which is appropriate for rendering
even though it is not the most convenient structure for some other
purposes such as evaluation (though it is not in any sense inconsistent
with the semantic meaning of the expression).

The properties described above are sufficient to generate all possible
behaviors of any operator in HTML-Math. For convenience, alternative
attributes are provided which set the above properties in typical ways.
(These can presently be used only in &lt;mo> tags, but in the future
will be most commonly used when authors can add new operators to the
dictionary.) These attributes have the default value "unused" so that
they will have no effect unless set explicitly. These attributes are:

<li>prec=number (sets both left and right prec to this number,
perhaps modified by the assoc attribute)</li>

<li>assoc=right, left, flat
(determines slight increment or decrement
of a left or right prec set by the prec attribute)</li>

<li>bracket=left, right
(sets the apropriate values of prefix, infix (false), postfix, all precs)</li>

After the parser chooses between alternative forms of an operator token,
it generates a modified token with only the appropriate attributes set,
for passing to subsequent stages (transformation rules and rendering).
This may be important if those stages use the attribute values in some
way; e.g., the renderer may wish to add a different amount of spacing
to the left of a prefix operator (which has no left operand) or an
infix operator (which has one), or to make the spacing depend on the
absolute precedence, or a user-specified transformation rule may depend
on whether an operator was used in prefix form.

Some operators are normally never rendered directly; instead they are
treated as "macros" for expressing other forms (like layout schemas) in
an abbreviated way. For example, this happens to all the operators in
the linear syntax forms of the layout schemas. This is implemented by
built-in transformation rules.

Other operators will be rendered as "themselves". Typically they are
rendered as if they were the same text characters (possibly extended
characters) used to name them, with surrounding spacing adjusted by the
renderer to best convey the structure of the expression.

Large operators are typically rendered specially: in a larger than normal
font size, and with any embellishing scripts placed in different positions
depending on whether the expression mode is inline or display. Stretchy
operators are also typically rendered specially (as described earlier).

<h4>mb: begin tag or {<br>me: end tag or }</h4>

The mb and me token types are used by the tokenizer to represent begin
and end tags (respectively) of HTML-Math-specific SGML elements which can
contain subexpressions (but not elements like mn which contain only
character sequences and are turned into single tokens). They are also
used to represent the left and right invisible grouping characters, {
and }, respectively.

These tokens are never rendered directly; what they each mean is
described under the element name. It is an error for these begin and
end tokens not to match exactly. (HTML-Math does not allow end tags to
be left out or to be given in abbreviated forms.)

The invisible grouping characters are equivalent to the tags &lt;mg>
and &lt;/mg>. Their behavior is described under the tag name mg.

<h3>schemas for internal use</h3>

Some schemas are generated by the parser, but intended to be used up by
transformation rules rather than being rendered directly -- thus they
are not truly layout schemas, so I refer to them only as "schemas".
They can be the heads of subexpressions in display lists generated by
the parser, but are not normally present in display lists presented to
the renderer.

(When authors can modify the built-in transformation rules, it may
become possible for these schemas to be presented to the renderer,
which the renderer should treat as an error, as described in the
earlier section on rendering erroneous expressions.)

These schemas can be given directly in source text using SGML begin and
end tags of the same name. In this case, the tokenizer produces mb and
me tokens (described above) for the begin and end tags respectively,
which are then parsed by the parser as if they were a special kind of
brackets (different from ordinary brackets since they must always
match, and in some cases don't prevent ordinary brackets from matching
"across" them), producing a schema named with the tag name.

The { and } invisible grouping characters are treated by the tokenizer
precisely the same as the &lt;mg> and &lt;/mg> tags, respectively. (In the
main example I showed them as generating the tokens (mb "{") and (me
"}"), which was for the clarity of that description, but they act just
as if they generated (mb "mg") and (me "mg") respectively. Of course,
this fact has no visible effect once the parser has properly matched
them; the actual internal representation used is of course not
specified by HTML-Math (for these or any other data structures)
provided the behavior is as specified here.)

The complete set of such schemas is:

Name         Purpose                           Example use

mg           invisible grouping (aka { })      {1-x} &amp;over; {1+x}
mterm        term-like expression sequence     a+b
moperator    embellished operator              +_2
mlargeop     makes regular operators large

<h4>mg: invisible grouping</h4>

The effect of surrounding some HTML-Math source text with { and } or
&lt;mg> and &lt;/mg> is to force the parser to parse it separately and
group it into a single subexpression.

In some cases this forces bracket operators not to match anything
outside, but this explicitly does <em>not</em> happen to a bracket
operator, or to an embellished bracket operator, which by itself
constitutes the entire renderable contents of the {...} form.

In no case does use of invisible grouping, by itself,
force an operator to be treated as a term.

<h4>mterm: represent or force term-like expression sequence</h4>

Most subexpressions generated by the parser should be treated as terms
if they are part of larger expressions. The sequence of component
expressions which form them is grouped by the parser into an mterm
schema. (This will later be transformed into some layout schema for
rendering.) For example, a+b is parsed (before transformation
rules are applied) as (mterm (mi "a") (mo "+") (mi "b")).
(This will then be transformed by standard rules into the layout schema
(mrow (mi "a") (mo "+") (mi "b")), since (mo "+") is not used as the
linear syntax for some other layout schema.)

Source text enclosed in &lt;mterm>...&lt;/mterm> tags is parsed
normally, but then explicitly "forced" to be treated as a single term
(for the purposes of further parsing)
even if it would otherwise not be.

For example, &lt;mterm>+&lt;/mterm> is parsed (before transformation
rules are applied) as (mterm (mo "+")) and &lt;mterm>+_2&lt;/mterm> is
parsed (also before transformation rules) as (mterm (moperator (mo "+")
(mo "_") (mn "2"))).

The explicitly added mterms will be removed by transformation rules
before rendering, so their only effect on rendering is the indirect one
of producing layout schemas which have operators in positions that
would normally be used for terms; this may, for example, affect the
spacing around those expressions, depending on the spacing rules used
in the renderer.

<h4>moperator: embellished operators</h4>

When an operator is directly followed by another (postfix or infix)
operator with the attribute embellisher=true, the first operator is
"embellished" by the second one (and by its right operand, if any). The
parser generates an internal expression headed by moperator (before
applying transformation rules), and treats it as if it was itself an
operator with the attributes of its "base" (the embellished operator),
i.e. uses them for parsing the surrounding source text.

For example, in the expression
a +_2 b
the + is embellished by the _ (the infix operator for subscript) and the
2. Thus the parser generates (before transformation rules are used)
(mterm (mi "a") (moperator (mo "+") (mo "_") (mn "2")) (mi "b")).

<h4>mlargeop: makes regular operators large</h4>

The standard dictionary provides a small number of operators with the
large=true attribute (such as the integral and summation signs), but a
large number of operators may be used this way in mathematical
notation. HTML-Math provides for this by allowing any operator to be
surrounded by the &lt;mlargeop>...&lt;/mlargeop> tags, thereby turning
it into a large operator in this instance. For example,
&lt;mlargeop>&amp;Union;&lt;/mo> is a large "set union" operator made
from &amp;Union; which is an ordinary set union operator. An expression
representing the union of all sets in the set S might be represented as
	&lt;mlargeop>&amp;Union;&lt;/mo>_{s&amp;Element;S} s
where the extended character &amp;Element; is the set-membership
operator (which looks something like a small &amp;epsilon;).

The precedence of a large operator generated by mlargeop
is determined by the first successful method from among:
<li>the precedence given by attributes in the &lt;mlargeop> begin tag;</li>
<li>the largeprec attribute of the original operator, if it has one;</li>
<li>just higher than the ordinary precedence of the original operator.</li>

<h3>"Expression constructor" layout schemas</h3>

All renderable expressions with subexpressions are represented by one of the
following layout schemas:

Name            Represents              Some examples in HTML-Math source

mrow            horizontal sequence     a+b     [0,1)   &amp;int;
&amp;ee;^-x^2 &amp;dd; x
mfraction       fraction                2 &amp;over; 3      {1-x}
&amp;over; {1+x}
mroot           radical (nth root)      &amp;root; 2        &amp;root; 2 % n
mscripts        subscript or superscript or aligned pair
                                        a_1     x^2     &amp;Sum;_{x=1}%n
munderscript    underscript             &amp;RightArrow;__"word"
moverscript     overscript              x^^&amp;Cap;
mprescripts     presubscript or presuperscript or aligned pair
                                        F___0   F^^^1   F___0%%%1
mbox            hides all internal structure from renderer

All of these except mrow are typically rendered in a "2-dimensional" form
when rendering into 2-dimensional graphical media.

<h4>mrow: horizontal sequence</h4>

Most operators are (in 2-dimensional media) rendered in a horizontal row
between or next to their operands. The layout schema used in this case
is mrow.

An operator like + has no special transformation rule to specify
its layout, so a "default" rule is used which turns any (mterm ...)
schema which remains after other rules have been tried
into an (mrow ...) layout schema with the same arguments.
For example, a+b will be parsed into (mterm (mi "a") (mo "+") (mi "b"))
and then transformed by this rule into the renderable form
(mrow (mi "a") (mo "+") (mi "b")).

Renderers typically use spacing rules within an mrow
which are sensitive to whether the constituents are
terms or operators (including embellished operators),
to the type of operator (e.g. prefix or infix),
and sometimes to the relative precedence of nested operators.

An mrow can be specified directly as an SGML element by source text
which looks like
	&lt;mrow> arg1 &lt;mc> arg2 &lt;mc> ... &lt;mc> argn &lt;/mrow>
where "argi" means the source text for the ith argument, and
&lt;mc> ("c" stands for "comma") is a special HTML-Math empty element
used only to separate multiple arguments of the SGML forms of schemas.
(It can be used in any schema which allows more than one argument.)

This form can be used to specify any mrow with one or more arguments.

A missing argument (e.g. in &lt;mrow>&lt;/mrow> or between two &lt;mc>s)
is replaced by a nested empty mrow (i.e. one with no arguments), which
is neither an operator nor a term, and (typically) renders invisibly
with zero width. There is no way to specify an empty mrow "by itself"
in source text. (Empty mrows are used internally to represent certain
other missing constructs, e.g. in the mscripts layout schema. For this
use, it is important that an empty mrow is not equivalent to the
missing terms or operators sometimes inserted by the parser, and that
it can't be represented except as an argument to the SGML form of a


A fraction with a horizontal bar is usually specified using the
linear syntax form
	numerator &amp;over; denominator
making use of the extended character &amp;over; which is an infix operator
with precedence near that of division (the / infix operator).

It can also be specified in the SGML form
	&lt;mfraction> numerator &lt;mc> denominator &lt;/mfraction>
It's an error if there are other than two arguments given in this form
(i.e. other than one &lt;mc>).
[In SGML form, certain rendering attributes which will be described
later can be added to the begin tag, e.g. to modify the appearance of
the horizontal bar.]

This layout schema carries the semantic connotation of a fraction,
i.e. something which is semantically equivalent to division.
This is important, because it means
renderers are allowed to render it as if it was an mrow containing
the / operator, e.g. if this is necessary due to the display width being
too small
(or whenever their user prefers it that way).

<h4>mroot: radical (nth root)</h4>

A radical sign (typically used to represent square or nth roots)
can be specified by using the linear syntax
	&amp;root; x
	&amp;root; x % n
In the first case there is typically no "n" shown (in the place of the "nth

The semantic connotations include: n is equivalent to 2 if it's missing,
and this expression can be rendered instead as a 1/nth power if necessary
or desired.

The SGML form can be either of
	&lt;mroot> x &lt;/mroot>
	&lt;mroot> x &lt;mc> n &lt;/mroot>
It's an error if there are other than one or two arguments given in this form
(i.e. other than zero or one &lt;mc>).

[In SGML form, certain rendering attributes which will be described later
can be added to the begin tag, e.g. to modify the appearance
of the horizontal bar above the expression whose root is being extracted.]

<h4>mscripts: subscript or superscript, or a vertically aligned pair of

The mscripts layout schema represents an expression with one or both of a
subscript and superscript, which are intended to be "vertically aligned"
if both are present. (Note that this "vertical alignment" is logically
meaningful even in non-2-dimensional media, e.g. in tensor notations.)

The following linear syntax forms generate the same thing as the following
SGML forms (which all correspond to an expression tree (in display list format)
of (mscripts arg1 arg2) for some arguments):

Form        SGML form                                        Explanation

x_a         &lt;mscripts> x &lt;mc> a &lt;mc>   &lt;/mscripts>
x^b         &lt;mscripts> x &lt;mc>   &lt;mc> b &lt;/mscripts>
x_a%b       &lt;mscripts> x &lt;mc> a &lt;mc> b &lt;/mscripts>
aligned pair
x^b%a       &lt;mscripts> x &lt;mc> a &lt;mc> b &lt;/mscripts>
aligned pair

It's an error if the mscript element has other than three arguments
(separated by &lt;mc>s).

The _ and ^ operators are each right-associative. [Full details of their
precedences, and those of all other linear syntax operators related to
scripts of all kinds, including the % used above, will be described
later with the full table of operators and precedences.]

How the _ ^ and % operators interact to add multiple "scripts"
to one "base" is described in a separate section below.

<h4>mscripts used to represent tensors</h4>

Additional infix operators %_ and %^ are provided which are left-associative
but which also generate subscripts and superscripts. These are for use
in representing tensor notations in the following way.

"Left-nested" mscript layout schemas (i.e. where the first argument of
each one is the next one, from outermost to innermost in a chain) are
interpreted as representing vertically aligned pairs of tensor indices
(from farthest away to closest to the base expression). (This means
that the source text will contain the indices from left to right.)

It is acceptable for some indices to be missing, and these will be
rendered invisibly.

(This special interpretation of left-nested mscripts schemas also
extends through left-nested mprescripts schemas in the same nested
chain, and through any schemas which have no effect on rendering (such
as font changing schemas), but <em>not</em> through any other schemas
(in particular, not through moverscript, munderscript, or mbox
schemas). The order of adding a new layer of scripts and a new layer of
prescripts doesn't matter.

A typical rendering algorithm for this case (ignoring the possibility of
unusually tall subscripts or superscripts) would determine the
horizontal positions of the script arguments of an mscripts layout
schema normally (i.e. based on the horizontal position and width of the
entire base argument), but to determine their vertical positions would
"burrow down" into the base (through any left-nested mscripts,
mprescripts, and schemas with no effect on rendering) and depend only
on the vertical position and height of whatever it found inside. A more
careful algorithm might make use of a general grid or table layout
facility to position all the scripts at once. Effectively, all the
layout schemas in one of the left-nested chains being considered here
form a single renderable object. (The reason they are not represented
as a single level in one layout schema object is to make the
transformation rules which form them from the linear syntax operators
much simpler to express than would be possible that way, especially
when scripts are mixed with prescripts.)

<h4>digression: format of transformation rules</h4>

The present proposal makes use of built-in transformation rules
to turn linear syntax forms of some operators into internal forms.

Although the present proposal defines no "official" appearance or format
for those rules (nor even their properties in general),
there is some use for such a format in order to document the built-in
rules (and, perhaps, to allow them in practice to be read from a file
rather than hardwired into the rendering code, if desired).
(Furthermore, it is expected that a future amendment to this proposal
will provide a way for authors to add such rules themselves,
for which a format will be needed.)

To these ends, here is an example of some built-in rules
and a description of their format and operation.

The actions of the infix scripting operators can be described
(and are in fact implemented) using the following transformation rules:

$base _ $sub     ->  &lt;mscripts> $base &lt;mc> $sub &lt;mc>

$base ^ $super   ->  &lt;mscripts> $base &lt;mc>      &lt;mc> $super

$base %_ $sub    ->  $base _ $sub

$base %^ $super  ->  $base ^ $super

&lt;mscripts>   $base &lt;mc>      &lt;mc> $super &lt;/mscripts>  %  $sub  ->
  &lt;mscripts> $base &lt;mc> $sub &lt;mc> $super &lt;/mscripts>

&lt;mscripts>   $base &lt;mc> $sub &lt;mc>        &lt;/mscripts>  %  $super  ->
  &lt;mscripts> $base &lt;mc> $sub &lt;mc> $super &lt;/mscripts>

The format of these rules in general makes use of the following
two constructs:
Construct                     Purpose

template -> result            infix operator "->" for representing one rule

$name                         formal parameter or "pattern variable"

Such a rule is used by finding a subexpression which matches its template
(or pattern) (which generates a necessary set of bindings of the pattern
variables to subexpressions of the matched expression), and replacing that
subexpression with the "result" after substituting the same pattern
variable bindings in the result.

A list of such rules is used by using the first rule which matches,
and repeatedly transforming an expression until no rules match.
(But in the above example, the order of the rules doesn't matter.)

A list of rules should actually be repeatedly applied to all the subexpressions
of an expression tree, deepest first; and whenever a rule is used,
applied recursively and immediately to the result generated (after
substitution of bindings for pattern variables). This matters in the
present example (but explaning why it matters here is left as an exercise
for the reader).

How the rules of the example behave specifically (i.e. what they are for)
is described in the next section. [End of digression.]

<h4>how the infix operators for scripting work
to fill in the desired script positions on one base</h4>

To repeat:
The actions of the infix scripting operators can be described
(and are in fact implemented) using the transformation rules
given in the above explanation of the transformation rule format.

Here is how these rules actually work:

The first set of rules just say that the infix operators _ and ^
each make a new mscripts element from their arguments,
in each case leaving the unused script position empty:

$base _ $sub     ->  &lt;mscripts> $base &lt;mc> $sub &lt;mc>

$base ^ $super   ->  &lt;mscripts> $base &lt;mc>      &lt;mc> $super

The second set of rules simply say that the %_ and %^ operators
do precisely the same thing:

$base %_ $sub    ->  $base _ $sub

$base %^ $super  ->  $base ^ $super
(The differences between these operators and _ and ^
are entirely in their precedences and associativities.)

The final set of rules say what the % operator does:
it fills in an empty script position remaining in the outermost
mscripts element:

&lt;mscripts>   $base &lt;mc>      &lt;mc> $super &lt;/mscripts>  %  $sub  ->
  &lt;mscripts> $base &lt;mc> $sub &lt;mc> $super &lt;/mscripts>

&lt;mscripts>   $base &lt;mc> $sub &lt;mc>        &lt;/mscripts>  %  $super  ->
  &lt;mscripts> $base &lt;mc> $sub &lt;mc> $super &lt;/mscripts>

By the use of these operators, a piece of source text can add new
subscripts and superscripts to a given base from innermost to
outermost, alternating subscripts with superscripts as it pleases, but
once adding a script to a farther-right index position than before, can
never "go back" to an empty position farther to the left.

A typical pattern for entering a tensor would be: for each pair of
vertically aligned index positions, enter them in one of the forms
	$base %_ $sub
	$base %^ $super
	$base %_ $sub % $super
depending on which indices are present. Since the %_ and %^ operators
always skip to the next index position whereas % never does
(which is all evident from the above rules),
this pattern will always work (unless <em>both</em> of a pair of
vertically aligned index positions are empty, which is presumably
a very rare case!). (Authors who prefer entering superscripts first
can use the $base %^ $super % $sub form when both scripts are present.)

For example, the tensor which should render something like
(with four indices in three aligned columns)
could be entered as either
    x %^ a %^ b % c %_ d
    x %^ a %_ c % b %_ d
(using only left-associative operators).


[to be described later]


[to be described later]

<h4>mprescripts: presubscript or presuperscript or aligned pair of these</h4>

[to be described later]

<h4>mbox: hides internal structure from renderer</h4>

The mbox layout schema takes exactly one argument and has no effect on
parsing (it does not force the argument to be a term, etc, any more
than { } does). It tells a renderer to ignore any internal structure in
the argument expression which might otherwise alter its interpretation
of a layout schema -- only the overall "size and shape" of the argument
is allowed to affect rendering of layout schema containing the mbox.
(In non-graphical media, "size and shape" refers to whatever parameters
must be taken heed of in order to fit a rendered subexpression into a
rendered whole expression without undue "gaps" or "overlaps". E.g., in
speech, "size and shape" might refer to the time interval occupied by a

Other than that, mbox has no effect on rendering (it's an invisible
wrapper around its argument).

Its main use is to separate left-nested mscripts
or mprescripts layout schemas from being interpreted as specifying
scripts in successive tensor index positions on the same base;
i.e. it forces (e.g.) &lt;mbox>x^2&lt;/mbox>^2 to look more like
     2 2

However, since a renderer is allowed in principle to use arbitrary rules
to allow subexpression structure to affect rendering of a whole expression,
use of mbox may have other affects as well.

<h2>Additional Topics</h2>

The following additional topics
(besides the ones mentioned near the word "later" in the above letter)
will be dealt with later in more detailed versions of this proposal:

<li>how tables or grids or matrices are represented</li>
<li>special markup with optional semantic information
(such as information about identifier types and scopes)</li>
<li>equation numbering and related topics</li>
<li>special markup with optional hints to renderers,
such as linebreaking hints or hints about good choices of subexpressions
which can be collapsed or expanded interactively by the viewer</li>
<li>embedded html elements of various kinds (e.g. links)</li>
<li>relation to CSS</li>
<li>changes to font, bold or italic, font size</li>
<li>how diacritics are represented as overscripts</li>
<li>the "prime" postfix operator, which is really a superscript</li>
<li>SGML rules for terminator character of entities</li>
<li>extended chars included directly in source file also allowed</li>
<li>details of character properties defined in the dictionary</li>