MathML

Version 3 of the MathML specification was released as a W3C recommendation on 20 October 2010. A recommendation of A MathML for CSS Profile was later released on 7 June 2011;[8] this is a subset of MathML suitable for CSS formatting. Another subset, Strict Content MathML, provides a subset of content MathML with a uniform structure and is designed to be compatible with OpenMath. Other content elements are defined in terms of a transformation to the strict subset. New content elements include <bind> which associates bound variables (<bvar>) to expressions, for example a summation index. The new <share> element allows structure sharing.[9]

The development of MathML 3.0 went through a number of stages. In June 2006, the W3C rechartered the MathML Working Group to produce a MathML 3 Recommendation until February 2008, and in November 2008 extended the charter to April 2010. A sixth Working Draft of the MathML 3 revision was published in June 2009. On 10 August 2010 version 3 graduated to become a "Proposed Recommendation" rather than a draft.[9] An implementation of MathML 2 landed in WebKit around this same time,[10] with a Chromium implementation following a couple of years later,[11] although that implementation was removed from Chromium after less than a year.[12]

The Second Edition of MathML 3.0 was published as a W3C Recommendation on 10 April 2014.[2] The specification was approved as an ISO/IEC international standard 40314:2015 on 23 June 2015.[13] Also in 2015, the MathML Association was founded to support the adoption of the MathML standard.[14] At that time, according to a member of the MathJax team, none of the major browser makers paid any of their developers for any MathML-rendering work; whatever support existed was overwhelmingly the result of unpaid volunteer time/work.[15]

In August 2021, a new specification called MathML Core was published, described as the “core subset of Mathematical Markup Language, or MathML, that is suitable for browser implementation.”[16] MathML Core set itself apart from MathML 3.0 by including detailed rendering rules and integration with CSS, automated browser support testing resources, and focusing on a fundamental subset of MathML. An implementation was added to Chromium at the beginning of 2023.[17]

Presentation MathML focuses on the display of an equation, and has about 30 elements. The elements' names all begin with m. A Presentation MathML expression is built up out of tokens that are combined using higher-level elements, which control their layout. Finer details of presentation are affected by close to 50 attributes.

Token elements generally only contain characters (not other elements). They include:

• <mi>x</mi> – identifiers;
• <mo>+</mo> – operators;
• <mn>2</mn> – numbers;
• <mtext>such that</mtext> – text.

Note, however, that these token elements may be used as extension points, allowing markup in host languages. MathML in HTML5 allows most inline HTML markup in mtext, and <mtext><b>non</b> zero</mtext> is conforming, with the HTML markup being used within the MathML to mark up the embedded text (making the first word bold in this example).

These are combined using layout elements, that generally contain only elements. They include:

• <mrow> – a horizontal row of items;
• <msup>, <munderover>, and others – superscripts, limits over and under operators like sums, etc.;
• <mfrac> – fractions;
• <msqrt> and <mroot> – roots;
• <mfenced> – surrounding content with fences, such as parentheses.

As usual in HTML and XML, many entities are available for specifying special symbols by name, such as π and →. An interesting feature of MathML is that entities also exist to express normally-invisible operators, such as ⁢ (or the shorthand ⁢) for implicit multiplication. They are:

• U+2061 FUNCTION APPLICATION;
• U+2062 INVISIBLE TIMES;
• U+2063 INVISIBLE SEPARATOR;
• U+2064 INVISIBLE PLUS.

The full specification of MathML entities[22] is closely coordinated with the corresponding specifications for use with HTML and XML in general.[23]

Thus, the expression ${\displaystyle ax^{2}+bx+c}$ requires two layout elements: one to create the overall horizontal row and one for the superscripted exponent. However, the individual tokens also have to be identified as identifiers (<mi>), operators (<mo>), or numbers (<mn>). Adding the token markup, the full form ends up as

<mrow>
	<mi>a</mi> <mo>&InvisibleTimes;</mo> <msup><mi>x</mi><mn>2</mn></msup>
	<mo>+</mo><mi>b</mi><mo>&InvisibleTimes;</mo><mi>x</mi>
	<mo>+</mo><mi>c</mi>
</mrow>

A complete document that consists of just the MathML example above, is shown here:

<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE math PUBLIC "-//W3C//DTD MathML 2.0//EN" "http://www.w3.org/Math/DTD/mathml2/mathml2.dtd">
<math xmlns="http://www.w3.org/1998/Math/MathML">
	<mrow>
		<mi>a</mi> <mo>&InvisibleTimes;</mo> <msup><mi>x</mi><mn>2</mn></msup>
		<mo>+</mo><mi>b</mi><mo>&InvisibleTimes;</mo><mi>x</mi>
		<mo>+</mo><mi>c</mi>
	</mrow>
</math>

Content MathML focuses on the semantics, or meaning, of the expression rather than its layout. Central to Content MathML is the <apply> element that represents function application. The function being applied is the first child element under <apply>, and its operands or parameters are the remaining child elements. Content MathML uses only a few attributes.

Tokens such as identifiers and numbers are individually marked up, much as for Presentation MathML, but with elements such as <ci> and <cn>. Rather than being merely another type of token, operators are represented by specific elements, whose mathematical semantics are known to MathML: <times>, <power>, etc. There are over a hundred different elements for different functions and operators.[24]

For example, <apply><sin/><ci>x</ci></apply> represents ${\displaystyle \sin(x)}$ and <apply><plus/><ci>x</ci><cn>5</cn></apply> represents ${\displaystyle x+5}$. The elements representing operators and functions are empty elements, because their operands are the other elements under the containing <apply>.

The expression ${\displaystyle ax^{2}+bx+c}$ could be represented as

<math>
	<apply>
		<plus/>
		<apply>
			<times/>
			<ci>a</ci>
			<apply>
				<power/>
				<ci>x</ci>
				<cn>2</cn>
			</apply>
		</apply>
		<apply>
			<times/>
			<ci>b</ci>
			<ci>x</ci>
		</apply>
		<ci>c</ci>
	</apply>
</math>

Content MathML is nearly isomorphic to expressions in a functional language such as Scheme and other dialects of Lisp. <apply>...</apply> amounts to Scheme's (...), and the many operator and function elements amount to Scheme functions. With this trivial literal transformation, plus un-tagging the individual tokens, the example above becomes:

(plus
  (times a (power x 2))
  (times b x)
  c)

This reflects the long-known close relationship between XML element structures, and LISP or Scheme S-expressions.[25][26]

According to the OM Society,[27] OpenMath Content Dictionaries can be employed as collections of symbols and identifiers with declarations of their semantics – names, descriptions and rules. A 2018 paper presented at the SIGIR conference[28] proposed that the semantic knowledge base Wikidata could be used as an OpenMath Content Dictionary to link semantic elements of a mathematical formula to unique and language-independent Wikidata items.

• W3C Recommendation: Mathematical Markup Language (MathML) 1.01 Specification
• W3C Recommendation: Mathematical Markup Language (MathML) Version 2.0 (Second Edition)
• W3C Recommendation: Mathematical Markup Language (MathML) Version 3.0 (Third Edition)