Parsing MathML

A.1 Use of MathML as Well-Formed XML

A MathML document must be a well-formed XML document using elements in the MathML namespace as defined by this specification, however it is not required that the document refer to any specific Document Type Definition (DTD) or schema that specifies MathML. It is sometimes advantageous not to specify such a language definition as these files are large, often much larger than the MathML expression and unless they have been previously cached by the MathML application, the time taken to fetch the DTD or schema may have an appreciable effect on the processing of the MathML document.

Note that if no DTD is specified with a DOCTYPE declaration, that entity references (for example to refer to MathML characters by name) may not be used. The document should be encoded in an encoding (for example UTF-8) in which all needed characters may be encoded as character data, or characters may be referenced using numeric character references, for example ∫ rather than ∫

If a MathML fragment is parsed without a DTD, in other words as a well-formed XML fragment, it is the responsibility of the processing application to treat the white space characters occurring outside of token elements as not significant.

However, in many circumstances, especially while producing or editing MathML, it is useful to use a language definition to constrain the editing process or to check the correctness of generated files. The following section, Section A.2 Using the RelaxNG Schema for MathML3, discusses the RelaxNG Schema for MathML3 [RELAX-NG], which forms a normative part of the specification. Following that, Section A.4 Using the MathML XML Schema, and Section A.3 Using the MathML DTD discuss alternative languages definition using the document type definitions (DTD) and the W3C XML schema language, [XMLSchemas], both of which are derived from the normative RelaxNG schema automatically. One should note that the schema definitions of the language is currently stricter than the DTD version. That is, a schema validating processor will declare invalid documents that are declared valid by a (DTD) validating XML parser. This is partly due to the fact that the XML schema language may express additional constraints not expressable in the DTD, and partly due to the fact that for reasons of compatibility with earlier releases, the DTD is intentionally forgiving in some places and does not enforce constraints that are specified in the text of this specification.

A.2 Using the RelaxNG Schema for MathML3

MathML documents should be validated using the RelaxNG Schema for MathML, either in the XML encoding (http://www.w3.org/Math/RelaxNG/mathml3/mathml3.rng) or in compact notation (http://www.w3.org/Math/RelaxNG/mathml3/mathml3.rnc) which is also shown below.

In contrast to DTDs there is no in-document method to associate a RelaxNG schema with a document.

We provide five RelaxNG schema for MathML3 in five parts:

The grammar for full MathML
The grammar for elements common to Content and Presentation
The grammar for Presentation MathML
The grammar for Strict Content MathML
The grammar for Content MathML3

A.2.1 Full MathML

The RelaxNG schema for full MathML builds on the schema describing the various parts of the language which are given in the following sections. It can be found at http://www.w3.org/Math/RelaxNG/mathml3/mathml3.rnc.

#     This is the Mathematical Markup Language (MathML) 3.0, an XML
#     application for describing mathematical notation and capturing
#     both its structure and content.
#
#     Copyright 1998-2010 W3C (MIT, ERCIM, Keio)
# 
#     Use and distribution of this code are permitted under the terms
#     W3C Software Notice and License
#     http://www.w3.org/Consortium/Legal/2002/copyright-software-20021231


default namespace m = "http://www.w3.org/1998/Math/MathML"

## Content  MathML
include "mathml3-content.rnc" 

## Presentation MathML
include "mathml3-presentation.rnc"

## math and semantics common to both Content and Presentation
include "mathml3-common.rnc"

A.2.2 Elements Common to Presentation and Content MathML

#     This is the Mathematical Markup Language (MathML) 3.0, an XML
#     application for describing mathematical notation and capturing
#     both its structure and content.
#
#     Copyright 1998-2010 W3C (MIT, ERCIM, Keio)
# 
#     Use and distribution of this code are permitted under the terms
#     W3C Software Notice and License
#     http://www.w3.org/Consortium/Legal/2002/copyright-software-20021231


default namespace m = "http://www.w3.org/1998/Math/MathML"
namespace local = ""

start = math

math = element math {math.attributes,MathExpression*}
MathExpression = semantics

NonMathMLAtt = attribute (* - (local:*|m:*)) {xsd:string} 

CommonDeprecatedAtt = attribute other {text}?

CommonAtt = attribute id {xsd:ID}?,
            attribute xref {text}?,
            attribute class {xsd:NMTOKENS}?,
            attribute style {xsd:string}?,
            attribute href {xsd:anyURI}?,
            CommonDeprecatedAtt,
            NonMathMLAtt*


math.attributes = CommonAtt,
               attribute display {"block" | "inline"}?,
               attribute maxwidth {length}?,
               attribute overflow {"linebreak" | "scroll" | "elide" | "truncate" | "scale"}?,
               attribute altimg {xsd:anyURI}?,
               attribute altimg-width {length}?,
               attribute altimg-height {length}?,
               attribute altimg-valign {length | "top" | "middle" | "bottom"}?,
               attribute alttext {text}?,
               attribute cdgroup {xsd:anyURI}?,
               math.deprecatedattributes

# the mathml3-presentation schema  adds additional attributes
# to the math element, all those valid on mstyle

math.deprecatedattributes = attribute mode {xsd:string}?,
                            attribute macros {xsd:string}?


name = attribute name {xsd:NCName}
cd = attribute cd {xsd:NCName}

src = attribute src {xsd:anyURI}?

annotation = element annotation {annotation.attributes,text}
                     
annotation-xml.model = (MathExpression|anyElement)*

anyElement =  element (* - m:*) {(attribute * {text}|text| anyElement)*}

annotation-xml = element annotation-xml {annotation.attributes,
                                         annotation-xml.model}
annotation.attributes = CommonAtt,
	                cd?,
                        name?,
                        DefEncAtt,
                        src?

DefEncAtt = attribute encoding {xsd:string}?,
            attribute definitionURL {xsd:anyURI}?

semantics = element semantics {semantics.attributes,
                               MathExpression, 
                              (annotation|annotation-xml)*}
semantics.attributes = CommonAtt,DefEncAtt,cd?,name?



length
# wrapped for display
 = xsd:string {
  pattern = '\s*((-?[0-9]*(\.[0-9]*)?(e[mx]|in|cm|mm|p[xtc]|%)?)|(negative)?((very){0,2}thi(n|
         ck)|medium)mathspace)\s*' 
}

A.2.3 The Grammar for Presentation MathML

#     This is the Mathematical Markup Language (MathML) 3.0, an XML
#     application for describing mathematical notation and capturing
#     both its structure and content.
#
#     Copyright 1998-2010 W3C (MIT, ERCIM, Keio)
# 
#     Use and distribution of this code are permitted under the terms
#     W3C Software Notice and License
#     http://www.w3.org/Consortium/Legal/2002/copyright-software-20021231

default namespace m = "http://www.w3.org/1998/Math/MathML"

MathExpression |= PresentationExpression

ImpliedMrow = MathExpression*

TableRowExpression = mtr|mlabeledtr

TableCellExpression = mtd

MstackExpression = MathExpression|mscarries|msline|msrow|msgroup

MsrowExpression = MathExpression|none

MultiScriptExpression = (MathExpression|none),(MathExpression|none)

mpadded-length
# wrapped for display
 = xsd:string {
  pattern = '\s*([\+\-]?[0-9]*(\.[0-9]*)?\s*((%?\s*(height|depth|width)?)|e[mx]|in|
         cm|mm|p[xtc]|((negative)?((very){0,2}thi(n|ck)|medium)mathspace)))\s*' }

linestyle = "none" | "solid" | "dashed"

verticalalign =
      "top" |
      "bottom" |
      "center" |
      "baseline" |
      "axis"

columnalignstyle = "left" | "center" | "right"

notationstyle =
     "longdiv" |
     "actuarial" |
     "radical" |
     "box" |
     "roundedbox" |
     "circle" |
     "left" |
     "right" |
     "top" |
     "bottom" |
     "updiagonalstrike" |
     "downdiagonalstrike" |
     "verticalstrike" |
     "horizontalstrike" |
     "madruwb"

idref = text
unsigned-integer = xsd:unsignedLong
integer = xsd:integer
number = xsd:decimal

character = xsd:string {
  pattern = '\s*\S\s*'}

color
# wrapped for display
 =  xsd:string {
  pattern = '\s*((#[0-9a-fA-F]{3}([0-9a-fA-F]{3})?)|[aA][qQ][uU][aA]|[bB][lL][aA][cC][kK]|
         [bB][lL][uU][eE]|[fF][uU][cC][hH][sS][iI][aA]|[gG][rR][aA][yY]|[gG][rR][eE][eE][nN]|
         [lL][iI][mM][eE]|[mM][aA][rR][oO][oO][nN]|[nN][aA][vV][yY]|[oO][lL][iI][vV][eE]|[pP][uU][rR][pP][lL][eE]|
         [rR][eE][dD]|[sS][iI][lL][vV][eE][rR]|[tT][eE][aA][lL]|[wW][hH][iI][tT][eE]|[yY][eE][lL][lL][oO][wW])\s*'}


group-alignment = "left" | "center" | "right" | "decimalpoint"
group-alignment-list = list {group-alignment+}
group-alignment-list-list = xsd:string {
  pattern = '(\s*\{\s*(left|center|right|decimalpoint)(\s+(left|center|right|decimalpoint))*\})*\s*' }
positive-integer = xsd:positiveInteger


TokenExpression = mi|mn|mo|mtext|mspace|ms

token.content = mglyph|malignmark|text

mi = element mi {mi.attributes, token.content*}
mi.attributes = 
  CommonAtt,
  CommonPresAtt,
  TokenAtt


mn = element mn {mn.attributes, token.content*}
mn.attributes = 
  CommonAtt,
  CommonPresAtt,
  TokenAtt


mo = element mo {mo.attributes, token.content*}
mo.attributes = 
  CommonAtt,
  CommonPresAtt,
  TokenAtt,
  attribute form {"prefix" | "infix" | "postfix"}?,
  attribute fence {"true" | "false"}?,
  attribute separator {"true" | "false"}?,
  attribute lspace {length}?,
  attribute rspace {length}?,
  attribute stretchy {"true" | "false"}?,
  attribute symmetric {"true" | "false"}?,
  attribute maxsize {length | "infinity"}?,
  attribute minsize {length}?,
  attribute largeop {"true" | "false"}?,
  attribute movablelimits {"true" | "false"}?,
  attribute accent {"true" | "false"}?,
  attribute linebreak {"auto" | "newline" | "nobreak" | "goodbreak" | "badbreak"}?,
  attribute lineleading {length}?,
  attribute linebreakstyle {"before" | "after" | "duplicate" | "infixlinebreakstyle"}?,
  attribute linebreakmultchar {text}?,
  attribute indentalign {"left" | "center" | "right" | "auto" | "id"}?,
  attribute indentshift {length}?,
  attribute indenttarget {idref}?,
  attribute indentalignfirst {"left" | "center" | "right" | "auto" | "id" | "indentalign"}?,
  attribute indentshiftfirst {length | "indentshift"}?,
  attribute indentalignlast {"left" | "center" | "right" | "auto" | "id" | "indentalign"}?,
  attribute indentshiftlast {length | "indentshift"}?


mtext = element mtext {mtext.attributes, token.content*}
mtext.attributes = 
  CommonAtt,
  CommonPresAtt,
  TokenAtt


mspace = element mspace {mspace.attributes, empty}
mspace.attributes = 
  CommonAtt,
  CommonPresAtt,
  TokenAtt,
  attribute width {length}?,
  attribute height {length}?,
  attribute depth {length}?,
  attribute linebreak {"auto" | "newline" | "nobreak" | "goodbreak" | "badbreak" | "indentingnewline"}?


ms = element ms {ms.attributes, token.content*}
ms.attributes = 
  CommonAtt,
  CommonPresAtt,
  TokenAtt,
  attribute lquote {text}?,
  attribute rquote {text}?


mglyph = element mglyph {mglyph.attributes,mglyph.deprecatedattributes,empty}
mglyph.attributes = 
  CommonAtt,  CommonPresAtt,
  attribute src {xsd:anyURI}?,
  attribute width {length}?,
  attribute height {length}?,
  attribute valign {length}?,
  attribute alt {text}?
mglyph.deprecatedattributes =
  attribute index {integer}?,
  attribute mathvariant {"normal" | "bold" | "italic" | "bold-italic" | "double-struck" | "bold-fraktur" |
          "script" | "bold-script" | "fraktur" | "sans-serif" | "bold-sans-serif" | "sans-serif-italic" |
          "sans-serif-bold-italic" | "monospace" | "initial" | "tailed" | "looped" | "stretched"}?,
  attribute mathsize {"small" | "normal" | "big" | length}?,
  DeprecatedTokenAtt

msline = element msline {msline.attributes,empty}
msline.attributes = 
  CommonAtt,  CommonPresAtt,
  attribute position {integer}?,
  attribute length {unsigned-integer}?,
  attribute leftoverhang {length}?,
  attribute rightoverhang {length}?,
  attribute mslinethickness {length | "thin" | "medium" | "thick"}?

none = element none {none.attributes,empty}
none.attributes = 
  CommonAtt,
  CommonPresAtt

mprescripts = element mprescripts {mprescripts.attributes,empty}
mprescripts.attributes = 
  CommonAtt,
  CommonPresAtt


CommonPresAtt = 
  attribute mathcolor {color}?,
  attribute mathbackground {color | "transparent"}?

TokenAtt = 
  attribute mathvariant {"normal" | "bold" | "italic" | "bold-italic" | "double-struck" | "bold-fraktur" |
          "script" | "bold-script" | "fraktur" | "sans-serif" | "bold-sans-serif" | "sans-serif-italic" |
          "sans-serif-bold-italic" | "monospace" | "initial" | "tailed" | "looped" | "stretched"}?,
  attribute mathsize {"small" | "normal" | "big" | length}?,
  attribute dir {"ltr" | "rtl"}?,
  DeprecatedTokenAtt

DeprecatedTokenAtt = 
  attribute fontfamily {text}?,
  attribute fontweight {"normal" | "bold"}?,
  attribute fontstyle {"normal" | "italic"}?,
  attribute fontsize {length}?,
  attribute color {color}?,
  attribute background {color | "transparent"}?

MalignExpression = maligngroup|malignmark

malignmark = element malignmark {malignmark.attributes, empty}
malignmark.attributes = 
  CommonAtt, CommonPresAtt,
  attribute edge {"left" | "right"}?


maligngroup = element maligngroup {maligngroup.attributes, empty}
maligngroup.attributes = 
  CommonAtt, CommonPresAtt,
  attribute groupalign {"left" | "center" | "right" | "decimalpoint"}?


PresentationExpression = TokenExpression|MalignExpression|
                         mrow|mfrac|msqrt|mroot|mstyle|merror|mpadded|mphantom|
                         mfenced|menclose|msub|msup|msubsup|munder|mover|munderover|
                         mmultiscripts|mtable|mstack|mlongdiv|maction



mrow = element mrow {mrow.attributes, MathExpression*}
mrow.attributes = 
  CommonAtt, CommonPresAtt,
  attribute dir {"ltr" | "rtl"}?


mfrac = element mfrac {mfrac.attributes, MathExpression, MathExpression}
mfrac.attributes = 
  CommonAtt, CommonPresAtt,
  attribute linethickness {length | "thin" | "medium" | "thick"}?,
  attribute numalign {"left" | "center" | "right"}?,
  attribute denomalign {"left" | "center" | "right"}?,
  attribute bevelled {"true" | "false"}?


msqrt = element msqrt {msqrt.attributes, ImpliedMrow}
msqrt.attributes = 
  CommonAtt, CommonPresAtt


mroot = element mroot {mroot.attributes, MathExpression, MathExpression}
mroot.attributes = 
  CommonAtt, CommonPresAtt


mstyle = element mstyle {mstyle.attributes, ImpliedMrow}
mstyle.attributes = 
  CommonAtt, CommonPresAtt,
  mstyle.specificattributes,
  mstyle.generalattributes,
  mstyle.deprecatedattributes

mstyle.specificattributes =
  attribute scriptlevel {integer}?,
  attribute displaystyle {"true" | "false"}?,
  attribute scriptsizemultiplier {number}?,
  attribute scriptminsize {length}?,
  attribute infixlinebreakstyle {"before" | "after" | "duplicate"}?,
  attribute decimalpoint {character}?

mstyle.generalattributes =
  attribute accent {"true" | "false"}?,
  attribute accentunder {"true" | "false"}?,
  attribute align {"left" | "right" | "center"}?,
  attribute alignmentscope {list {("true" | "false") +}}?,
  attribute bevelled {"true" | "false"}?,
  attribute charalign {"left" | "center" | "right"}?,
  attribute charspacing {length | "loose" | "medium" | "tight"}?,
  attribute close {text}?,
  attribute columnalign {list {columnalignstyle+} }?,
  attribute columnlines {list {linestyle +}}?,
  attribute columnspacing {list {(length) +}}?,
  attribute columnspan {positive-integer}?,
  attribute columnwidth {list {("auto" | length | "fit") +}}?,
  attribute crossout {list {("none" | "updiagonalstrike" | "downdiagonalstrike" | "verticalstrike" | "horizontalstrike")*}}?,
  attribute denomalign {"left" | "center" | "right"}?,
  attribute depth {length}?,
  attribute dir {"ltr" | "rtl"}?,
  attribute edge {"left" | "right"}?,
  attribute equalcolumns {"true" | "false"}?,
  attribute equalrows {"true" | "false"}?,
  attribute fence {"true" | "false"}?,
  attribute form {"prefix" | "infix" | "postfix"}?,
  attribute frame {linestyle}?,
  attribute framespacing {list {length, length}}?,
  attribute groupalign {group-alignment-list-list}?,
  attribute height {length}?,
  attribute indentalign {"left" | "center" | "right" | "auto" | "id"}?,
  attribute indentalignfirst {"left" | "center" | "right" | "auto" | "id" | "indentalign"}?,
  attribute indentalignlast {"left" | "center" | "right" | "auto" | "id" | "indentalign"}?,
  attribute indentshift {length}?,
  attribute indentshiftfirst {length | "indentshift"}?,
  attribute indentshiftlast {length | "indentshift"}?,
  attribute indenttarget {idref}?,
  attribute largeop {"true" | "false"}?,
  attribute leftoverhang {length}?,
  attribute length {unsigned-integer}?,
  attribute linebreak {"auto" | "newline" | "nobreak" | "goodbreak" | "badbreak"}?,
  attribute linebreakmultchar {text}?,
  attribute linebreakstyle {"before" | "after" | "duplicate" | "infixlinebreakstyle"}?,
  attribute lineleading {length}?,
  attribute linethickness {length | "thin" | "medium" | "thick"}?,
  attribute location {"w" | "nw" | "n" | "ne" | "e" | "se" | "s" | "sw"}?,
  attribute longdivstyle {"lefttop" | "stackedrightright" | "mediumstackedrightright" | "shortstackedrightright" |
          "righttop" | "left/\right" | "left)(right" | ":right=right" | "stackedleftleft" |
          "stackedleftlinetop"}?,
  attribute lquote {text}?,
  attribute lspace {length}?,
  attribute mathsize {"small" | "normal" | "big" | length}?,
  attribute mathvariant {"normal" | "bold" | "italic" | "bold-italic" | "double-struck" | "bold-fraktur" |
          "script" | "bold-script" | "fraktur" | "sans-serif" | "bold-sans-serif" | "sans-serif-italic" |
          "sans-serif-bold-italic" | "monospace" | "initial" | "tailed" | "looped" | "stretched"}?,
  attribute maxsize {length | "infinity"}?,
  attribute minlabelspacing {length}?,
  attribute minsize {length}?,
  attribute movablelimits {"true" | "false"}?,
  attribute mslinethickness {length | "thin" | "medium" | "thick"}?,
  attribute notation {text}?,
  attribute numalign {"left" | "center" | "right"}?,
  attribute open {text}?,
  attribute position {integer}?,
  attribute rightoverhang {length}?,
  attribute rowalign {list {verticalalign+} }?,
  attribute rowlines {list {linestyle +}}?,
  attribute rowspacing {list {(length) +}}?,
  attribute rowspan {positive-integer}?,
  attribute rquote {text}?,
  attribute rspace {length}?,
  attribute selection {positive-integer}?,
  attribute separator {"true" | "false"}?,
  attribute separators {text}?,
  attribute shift {integer}?,
  attribute side {"left" | "right" | "leftoverlap" | "rightoverlap"}?,
  attribute stackalign {"left" | "center" | "right" | "decimalpoint"}?,
  attribute stretchy {"true" | "false"}?,
  attribute subscriptshift {length}?,
  attribute superscriptshift {length}?,
  attribute symmetric {"true" | "false"}?,
  attribute valign {length}?,
  attribute width {length}?

mstyle.deprecatedattributes =
  DeprecatedTokenAtt,
  attribute veryverythinmathspace {length}?,
  attribute verythinmathspace {length}?,
  attribute thinmathspace {length}?,
  attribute mediummathspace {length}?,
  attribute thickmathspace {length}?,
  attribute verythickmathspace {length}?,
  attribute veryverythickmathspace {length}?

math.attributes &= CommonPresAtt
math.attributes &= mstyle.specificattributes
math.attributes &= mstyle.generalattributes




merror = element merror {merror.attributes, ImpliedMrow}
merror.attributes = 
  CommonAtt, CommonPresAtt


mpadded = element mpadded {mpadded.attributes, ImpliedMrow}
mpadded.attributes = 
  CommonAtt, CommonPresAtt,
  attribute height {mpadded-length}?,
  attribute depth {mpadded-length}?,
  attribute width {mpadded-length}?,
  attribute lspace {mpadded-length}?,
  attribute voffset {mpadded-length}?


mphantom = element mphantom {mphantom.attributes, ImpliedMrow}
mphantom.attributes = 
  CommonAtt, CommonPresAtt


mfenced = element mfenced {mfenced.attributes, MathExpression*}
mfenced.attributes = 
  CommonAtt, CommonPresAtt,
  attribute open {text}?,
  attribute close {text}?,
  attribute separators {text}?


menclose = element menclose {menclose.attributes, ImpliedMrow}
menclose.attributes = 
  CommonAtt, CommonPresAtt,
  attribute notation {text}?


msub = element msub {msub.attributes, MathExpression, MathExpression}
msub.attributes = 
  CommonAtt, CommonPresAtt,
  attribute subscriptshift {length}?


msup = element msup {msup.attributes, MathExpression, MathExpression}
msup.attributes = 
  CommonAtt, CommonPresAtt,
  attribute superscriptshift {length}?


msubsup = element msubsup {msubsup.attributes, MathExpression, MathExpression, MathExpression}
msubsup.attributes = 
  CommonAtt, CommonPresAtt,
  attribute subscriptshift {length}?,
  attribute superscriptshift {length}?


munder = element munder {munder.attributes, MathExpression, MathExpression}
munder.attributes = 
  CommonAtt, CommonPresAtt,
  attribute accentunder {"true" | "false"}?,
  attribute align {"left" | "right" | "center"}?


mover = element mover {mover.attributes, MathExpression, MathExpression}
mover.attributes = 
  CommonAtt, CommonPresAtt,
  attribute accent {"true" | "false"}?,
  attribute align {"left" | "right" | "center"}?


munderover = element munderover {munderover.attributes, MathExpression, MathExpression, MathExpression}
munderover.attributes = 
  CommonAtt, CommonPresAtt,
  attribute accent {"true" | "false"}?,
  attribute accentunder {"true" | "false"}?,
  attribute align {"left" | "right" | "center"}?


mmultiscripts = element mmultiscripts {mmultiscripts.attributes, MathExpression,MultiScriptExpression*,(mprescripts,MultiScriptExpression*)?}
mmultiscripts.attributes = 
  msubsup.attributes


mtable = element mtable {mtable.attributes, TableRowExpression*}
mtable.attributes = 
  CommonAtt, CommonPresAtt,
  attribute align {xsd:string {
    pattern ='\s*(top|bottom|center|baseline|axis)\s*[0-9]*'}}?,
  attribute rowalign {list {verticalalign+} }?,
  attribute columnalign {list {columnalignstyle+} }?,
  attribute groupalign {group-alignment-list-list}?,
  attribute alignmentscope {list {("true" | "false") +}}?,
  attribute columnwidth {list {("auto" | length | "fit") +}}?,
  attribute width {"auto" | length}?,
  attribute rowspacing {list {(length) +}}?,
  attribute columnspacing {list {(length) +}}?,
  attribute rowlines {list {linestyle +}}?,
  attribute columnlines {list {linestyle +}}?,
  attribute frame {linestyle}?,
  attribute framespacing {list {length, length}}?,
  attribute equalrows {"true" | "false"}?,
  attribute equalcolumns {"true" | "false"}?,
  attribute displaystyle {"true" | "false"}?,
  attribute side {"left" | "right" | "leftoverlap" | "rightoverlap"}?,
  attribute minlabelspacing {length}?


mlabeledtr = element mlabeledtr {mlabeledtr.attributes, TableCellExpression+}
mlabeledtr.attributes = 
  mtr.attributes


mtr = element mtr {mtr.attributes, TableCellExpression*}
mtr.attributes = 
  CommonAtt, CommonPresAtt,
  attribute rowalign {"top" | "bottom" | "center" | "baseline" | "axis"}?,
  attribute columnalign {list {columnalignstyle+} }?,
  attribute groupalign {group-alignment-list-list}?


mtd = element mtd {mtd.attributes, ImpliedMrow}
mtd.attributes = 
  CommonAtt, CommonPresAtt,
  attribute rowspan {positive-integer}?,
  attribute columnspan {positive-integer}?,
  attribute rowalign {"top" | "bottom" | "center" | "baseline" | "axis"}?,
  attribute columnalign {columnalignstyle}?,
  attribute groupalign {group-alignment-list}?


mstack = element mstack {mstack.attributes, MstackExpression*}
mstack.attributes = 
  CommonAtt, CommonPresAtt,
  attribute align {xsd:string {
    pattern ='\s*(top|bottom|center|baseline|axis)\s*[0-9]*'}}?,
  attribute stackalign {"left" | "center" | "right" | "decimalpoint"}?,
  attribute charalign {"left" | "center" | "right"}?,
  attribute charspacing {length | "loose" | "medium" | "tight"}?


mlongdiv = element mlongdiv {mlongdiv.attributes, MstackExpression,MstackExpression,MstackExpression+}
mlongdiv.attributes = 
  msgroup.attributes,
  attribute longdivstyle {"lefttop" | "stackedrightright" | "mediumstackedrightright" | "shortstackedrightright" |
          "righttop" | "left/\right" | "left)(right" | ":right=right" | "stackedleftleft" |
          "stackedleftlinetop"}?


msgroup = element msgroup {msgroup.attributes, MstackExpression*}
msgroup.attributes = 
  CommonAtt, CommonPresAtt,
  attribute position {integer}?,
  attribute shift {integer}?


msrow = element msrow {msrow.attributes, MsrowExpression*}
msrow.attributes = 
  CommonAtt, CommonPresAtt,
  attribute position {integer}?


mscarries = element mscarries {mscarries.attributes, (MsrowExpression|mscarry)*}
mscarries.attributes = 
  CommonAtt, CommonPresAtt,
  attribute position {integer}?,
  attribute location {"w" | "nw" | "n" | "ne" | "e" | "se" | "s" | "sw"}?,
  attribute crossout {list {("none" | "updiagonalstrike" | "downdiagonalstrike" | "verticalstrike" | "horizontalstrike")*}}?,
  attribute scriptsizemultiplier {number}?


mscarry = element mscarry {mscarry.attributes, MsrowExpression*}
mscarry.attributes = 
  CommonAtt, CommonPresAtt,
  attribute location {"w" | "nw" | "n" | "ne" | "e" | "se" | "s" | "sw"}?,
  attribute crossout {list {("none" | "updiagonalstrike" | "downdiagonalstrike" | "verticalstrike" | "horizontalstrike")*}}?


maction = element maction {maction.attributes, MathExpression+}
maction.attributes = 
  CommonAtt, CommonPresAtt,
  attribute actiontype {text}?,
  attribute selection {positive-integer}?

A.2.4 The Grammar for Strict Content MathML3

The grammar for Strict Content MathML3 can be found at http://www.w3.org/Math/RelaxNG/mathml3/mathml3-strict-content.rnc.

#     This is the Mathematical Markup Language (MathML) 3.0, an XML
#     application for describing mathematical notation and capturing
#     both its structure and content.
#
#     Copyright 1998-2010 W3C (MIT, ERCIM, Keio)
# 
#     Use and distribution of this code are permitted under the terms
#     W3C Software Notice and License
#     http://www.w3.org/Consortium/Legal/2002/copyright-software-20021231


default namespace m = "http://www.w3.org/1998/Math/MathML"

ContExp = semantics-contexp | cn | ci | csymbol | apply | bind | share | cerror | cbytes | cs

cn = element cn {cn.attributes,cn.content}
cn.content = text
cn.attributes = attribute type {"integer" | "real" | "double" | "hexdouble"}

semantics-ci = element semantics {semantics.attributes,(ci|semantics-ci), 
  (annotation|annotation-xml)*}

semantics-contexp = element semantics {semantics.attributes,ContExp, 
  (annotation|annotation-xml)*}

ci = element ci {ci.attributes, ci.content}
ci.attributes = CommonAtt, ci.type?
ci.type = attribute type {"integer" | "rational" | "real" | "complex" | "complex-polar" | "complex-cartesian" |
          "constant" | "function" | "vector" | "list" | "set" | "matrix"}
ci.content = text


csymbol = element csymbol {csymbol.attributes,csymbol.content}

SymbolName = xsd:NCName
csymbol.attributes = CommonAtt, cd
csymbol.content = SymbolName

BvarQ = bvar*
bvar = element bvar { ci | semantics-ci}

apply = element apply {CommonAtt,apply.content}
apply.content = ContExp+


bind = element bind {CommonAtt,bind.content}
bind.content = ContExp,bvar*,ContExp

share = element share {CommonAtt, src, empty}

cerror = element cerror {cerror.attributes, csymbol, ContExp*}
cerror.attributes = CommonAtt

cbytes = element cbytes {cbytes.attributes, base64}
cbytes.attributes = CommonAtt
base64 = xsd:base64Binary

cs = element cs {cs.attributes, text}
cs.attributes = CommonAtt

MathExpression |= ContExp

A.2.5 The Grammar for Content MathML

The grammar for Content MathML3 builds on the grammar for the Strict Content MathML subset, and can be found at http://www.w3.org/Math/RelaxNG/mathml3/mathml3-content.rnc.

#     This is the Mathematical Markup Language (MathML) 3.0, an XML
#     application for describing mathematical notation and capturing
#     both its structure and content.
#
#     Copyright 1998-2010 W3C (MIT, ERCIM, Keio)
# 
#     Use and distribution of this code are permitted under the terms
#     W3C Software Notice and License
#     http://www.w3.org/Consortium/Legal/2002/copyright-software-20021231

include "mathml3-strict-content.rnc"{
  cn.content = (text | mglyph | sep | PresentationExpression)* 
  cn.attributes = CommonAtt, DefEncAtt, attribute type {text}?, base?

  ci.attributes = CommonAtt, DefEncAtt, ci.type?
  ci.type = attribute type {text}
  ci.content = (text | mglyph | PresentationExpression)* 

  csymbol.attributes = CommonAtt, DefEncAtt, attribute type {text}?,cd?
  csymbol.content = (text | mglyph | PresentationExpression)* 

  bvar = element bvar { (ci | semantics-ci) & degree?}

  cbytes.attributes = CommonAtt, DefEncAtt

  cs.attributes = CommonAtt, DefEncAtt

  apply.content = ContExp+ | (ContExp, BvarQ, Qualifier*, ContExp*)

  bind.content = apply.content
}

base = attribute base {text}


sep = element sep {empty}
PresentationExpression |= notAllowed


DomainQ = (domainofapplication|condition|interval|(lowlimit,uplimit?))*
domainofapplication = element domainofapplication {ContExp}
condition = element condition {ContExp}
uplimit = element uplimit {ContExp}
lowlimit = element lowlimit {ContExp}

Qualifier = DomainQ|degree|momentabout|logbase
degree = element degree {ContExp}
momentabout = element momentabout {ContExp}
logbase = element logbase {ContExp}

type = attribute type {text}
order = attribute order {"numeric" | "lexicographic"}
closure = attribute closure {text}


ContExp |= piecewise


piecewise = element piecewise {CommonAtt, DefEncAtt,(piece* & otherwise?)}

piece = element piece {CommonAtt, DefEncAtt, ContExp, ContExp}

otherwise = element otherwise {CommonAtt, DefEncAtt, ContExp}


DeprecatedContExp = reln | fn | declare
ContExp |= DeprecatedContExp

reln = element reln {ContExp*}
fn = element fn {ContExp}
declare = element declare {attribute type {xsd:string}?,
                           attribute scope {xsd:string}?,
                           attribute nargs {xsd:nonNegativeInteger}?,
                           attribute occurrence {"prefix"|"infix"|"function-model"}?,
                           DefEncAtt, 
                           ContExp+}


interval.class = interval
ContExp |= interval.class


interval = element interval { CommonAtt, DefEncAtt,closure?, ContExp,ContExp}

unary-functional.class = inverse | ident | domain | codomain | image | ln | log | moment
ContExp |= unary-functional.class


inverse = element inverse { CommonAtt, DefEncAtt, empty}
ident = element ident { CommonAtt, DefEncAtt, empty}
domain = element domain { CommonAtt, DefEncAtt, empty}
codomain = element codomain { CommonAtt, DefEncAtt, empty}
image = element image { CommonAtt, DefEncAtt, empty}
ln = element ln { CommonAtt, DefEncAtt, empty}
log = element log { CommonAtt, DefEncAtt, empty}
moment = element moment { CommonAtt, DefEncAtt, empty}

lambda.class = lambda
ContExp |= lambda.class


lambda = element lambda { CommonAtt, DefEncAtt, BvarQ, DomainQ, ContExp}

nary-functional.class = compose
ContExp |= nary-functional.class


compose = element compose { CommonAtt, DefEncAtt, empty}

binary-arith.class = quotient | divide | minus | power | rem | root
ContExp |= binary-arith.class


quotient = element quotient { CommonAtt, DefEncAtt, empty}
divide = element divide { CommonAtt, DefEncAtt, empty}
minus = element minus { CommonAtt, DefEncAtt, empty}
power = element power { CommonAtt, DefEncAtt, empty}
rem = element rem { CommonAtt, DefEncAtt, empty}
root = element root { CommonAtt, DefEncAtt, empty}

unary-arith.class = factorial | minus | root | abs | conjugate | arg | real | imaginary | floor | ceiling | exp
ContExp |= unary-arith.class


factorial = element factorial { CommonAtt, DefEncAtt, empty}
abs = element abs { CommonAtt, DefEncAtt, empty}
conjugate = element conjugate { CommonAtt, DefEncAtt, empty}
arg = element arg { CommonAtt, DefEncAtt, empty}
real = element real { CommonAtt, DefEncAtt, empty}
imaginary = element imaginary { CommonAtt, DefEncAtt, empty}
floor = element floor { CommonAtt, DefEncAtt, empty}
ceiling = element ceiling { CommonAtt, DefEncAtt, empty}
exp = element exp { CommonAtt, DefEncAtt, empty}

nary-minmax.class = max | min
ContExp |= nary-minmax.class


max = element max { CommonAtt, DefEncAtt, empty}
min = element min { CommonAtt, DefEncAtt, empty}

nary-arith.class = plus | times | gcd | lcm
ContExp |= nary-arith.class


plus = element plus { CommonAtt, DefEncAtt, empty}
times = element times { CommonAtt, DefEncAtt, empty}
gcd = element gcd { CommonAtt, DefEncAtt, empty}
lcm = element lcm { CommonAtt, DefEncAtt, empty}

nary-logical.class = and | or | xor
ContExp |= nary-logical.class


and = element and { CommonAtt, DefEncAtt, empty}
or = element or { CommonAtt, DefEncAtt, empty}
xor = element xor { CommonAtt, DefEncAtt, empty}

unary-logical.class = not
ContExp |= unary-logical.class


not = element not { CommonAtt, DefEncAtt, empty}

binary-logical.class = implies | equivalent
ContExp |= binary-logical.class


implies = element implies { CommonAtt, DefEncAtt, empty}
equivalent = element equivalent { CommonAtt, DefEncAtt, empty}

quantifier.class = forall | exists
ContExp |= quantifier.class


forall = element forall { CommonAtt, DefEncAtt, empty}
exists = element exists { CommonAtt, DefEncAtt, empty}

nary-reln.class = eq | gt | lt | geq | leq
ContExp |= nary-reln.class


eq = element eq { CommonAtt, DefEncAtt, empty}
gt = element gt { CommonAtt, DefEncAtt, empty}
lt = element lt { CommonAtt, DefEncAtt, empty}
geq = element geq { CommonAtt, DefEncAtt, empty}
leq = element leq { CommonAtt, DefEncAtt, empty}

binary-reln.class = neq | approx | factorof | tendsto
ContExp |= binary-reln.class


neq = element neq { CommonAtt, DefEncAtt, empty}
approx = element approx { CommonAtt, DefEncAtt, empty}
factorof = element factorof { CommonAtt, DefEncAtt, empty}
tendsto = element tendsto { CommonAtt, DefEncAtt, type?, empty}

int.class = int
ContExp |= int.class


int = element int { CommonAtt, DefEncAtt, empty}

Differential-Operator.class = diff
ContExp |= Differential-Operator.class


diff = element diff { CommonAtt, DefEncAtt, empty}

partialdiff.class = partialdiff
ContExp |= partialdiff.class


partialdiff = element partialdiff { CommonAtt, DefEncAtt, empty}

unary-veccalc.class = divergence | grad | curl | laplacian
ContExp |= unary-veccalc.class


divergence = element divergence { CommonAtt, DefEncAtt, empty}
grad = element grad { CommonAtt, DefEncAtt, empty}
curl = element curl { CommonAtt, DefEncAtt, empty}
laplacian = element laplacian { CommonAtt, DefEncAtt, empty}

nary-setlist-constructor.class = set | \list
ContExp |= nary-setlist-constructor.class


set = element set { CommonAtt, DefEncAtt, type?, BvarQ*, DomainQ*, ContExp*}
\list = element \list { CommonAtt, DefEncAtt, order?, BvarQ*, DomainQ*, ContExp*}

nary-set.class = union | intersect | cartesianproduct
ContExp |= nary-set.class


union = element union { CommonAtt, DefEncAtt, empty}
intersect = element intersect { CommonAtt, DefEncAtt, empty}
cartesianproduct = element cartesianproduct { CommonAtt, DefEncAtt, empty}

binary-set.class = in | notin | notsubset | notprsubset | setdiff
ContExp |= binary-set.class


in = element in { CommonAtt, DefEncAtt, empty}
notin = element notin { CommonAtt, DefEncAtt, empty}
notsubset = element notsubset { CommonAtt, DefEncAtt, empty}
notprsubset = element notprsubset { CommonAtt, DefEncAtt, empty}
setdiff = element setdiff { CommonAtt, DefEncAtt, empty}

nary-set-reln.class = subset | prsubset
ContExp |= nary-set-reln.class


subset = element subset { CommonAtt, DefEncAtt, empty}
prsubset = element prsubset { CommonAtt, DefEncAtt, empty}

unary-set.class = card
ContExp |= unary-set.class


card = element card { CommonAtt, DefEncAtt, empty}

sum.class = sum
ContExp |= sum.class


sum = element sum { CommonAtt, DefEncAtt, empty}

product.class = product
ContExp |= product.class


product = element product { CommonAtt, DefEncAtt, empty}

limit.class = limit
ContExp |= limit.class


limit = element limit { CommonAtt, DefEncAtt, empty}

unary-elementary.class = sin | cos | tan | sec | csc | cot | sinh | cosh | tanh | sech | csch | coth | arcsin | arccos | arctan | arccosh | arccot | arccoth | arccsc | arccsch | arcsec | arcsech | arcsinh | arctanh
ContExp |= unary-elementary.class


sin = element sin { CommonAtt, DefEncAtt, empty}
cos = element cos { CommonAtt, DefEncAtt, empty}
tan = element tan { CommonAtt, DefEncAtt, empty}
sec = element sec { CommonAtt, DefEncAtt, empty}
csc = element csc { CommonAtt, DefEncAtt, empty}
cot = element cot { CommonAtt, DefEncAtt, empty}
sinh = element sinh { CommonAtt, DefEncAtt, empty}
cosh = element cosh { CommonAtt, DefEncAtt, empty}
tanh = element tanh { CommonAtt, DefEncAtt, empty}
sech = element sech { CommonAtt, DefEncAtt, empty}
csch = element csch { CommonAtt, DefEncAtt, empty}
coth = element coth { CommonAtt, DefEncAtt, empty}
arcsin = element arcsin { CommonAtt, DefEncAtt, empty}
arccos = element arccos { CommonAtt, DefEncAtt, empty}
arctan = element arctan { CommonAtt, DefEncAtt, empty}
arccosh = element arccosh { CommonAtt, DefEncAtt, empty}
arccot = element arccot { CommonAtt, DefEncAtt, empty}
arccoth = element arccoth { CommonAtt, DefEncAtt, empty}
arccsc = element arccsc { CommonAtt, DefEncAtt, empty}
arccsch = element arccsch { CommonAtt, DefEncAtt, empty}
arcsec = element arcsec { CommonAtt, DefEncAtt, empty}
arcsech = element arcsech { CommonAtt, DefEncAtt, empty}
arcsinh = element arcsinh { CommonAtt, DefEncAtt, empty}
arctanh = element arctanh { CommonAtt, DefEncAtt, empty}

nary-stats.class = mean | sdev | variance | median | mode
ContExp |= nary-stats.class


mean = element mean { CommonAtt, DefEncAtt, empty}
sdev = element sdev { CommonAtt, DefEncAtt, empty}
variance = element variance { CommonAtt, DefEncAtt, empty}
median = element median { CommonAtt, DefEncAtt, empty}
mode = element mode { CommonAtt, DefEncAtt, empty}

nary-constructor.class = vector | matrix | matrixrow
ContExp |= nary-constructor.class


vector = element vector { CommonAtt, DefEncAtt, BvarQ, DomainQ, ContExp*}
matrix = element matrix { CommonAtt, DefEncAtt, BvarQ, DomainQ, ContExp*}
matrixrow = element matrixrow { CommonAtt, DefEncAtt, BvarQ, DomainQ, ContExp*}

unary-linalg.class = determinant | transpose
ContExp |= unary-linalg.class


determinant = element determinant { CommonAtt, DefEncAtt, empty}
transpose = element transpose { CommonAtt, DefEncAtt, empty}

nary-linalg.class = selector
ContExp |= nary-linalg.class


selector = element selector { CommonAtt, DefEncAtt, empty}

binary-linalg.class = vectorproduct | scalarproduct | outerproduct
ContExp |= binary-linalg.class


vectorproduct = element vectorproduct { CommonAtt, DefEncAtt, empty}
scalarproduct = element scalarproduct { CommonAtt, DefEncAtt, empty}
outerproduct = element outerproduct { CommonAtt, DefEncAtt, empty}

constant-set.class = integers | reals | rationals | naturalnumbers | complexes | primes | emptyset
ContExp |= constant-set.class


integers = element integers { CommonAtt, DefEncAtt, empty}
reals = element reals { CommonAtt, DefEncAtt, empty}
rationals = element rationals { CommonAtt, DefEncAtt, empty}
naturalnumbers = element naturalnumbers { CommonAtt, DefEncAtt, empty}
complexes = element complexes { CommonAtt, DefEncAtt, empty}
primes = element primes { CommonAtt, DefEncAtt, empty}
emptyset = element emptyset { CommonAtt, DefEncAtt, empty}

constant-arith.class = exponentiale | imaginaryi | notanumber | true | false | pi | eulergamma | infinity
ContExp |= constant-arith.class


exponentiale = element exponentiale { CommonAtt, DefEncAtt, empty}
imaginaryi = element imaginaryi { CommonAtt, DefEncAtt, empty}
notanumber = element notanumber { CommonAtt, DefEncAtt, empty}
true = element true { CommonAtt, DefEncAtt, empty}
false = element false { CommonAtt, DefEncAtt, empty}
pi = element pi { CommonAtt, DefEncAtt, empty}
eulergamma = element eulergamma { CommonAtt, DefEncAtt, empty}
infinity = element infinity { CommonAtt, DefEncAtt, empty}

A.2.6 MathML as a module in a RelaxNG Schema

Normally, a MathML expression does not constitute an entire XML document. MathML is designed to be used as the mathematics fragment of larger markup languages. In particular it is designed to be used as a module in documents marked up with the XHTML family of markup languages. As RelaxNG directly supports modular development, this is usually very easy: an XHTML+MathML schema can be specified as simply as

# A RelaxNG Schema for  XHTML+MathML
include "xhtml.rnc"
math = external "mathml3.rnc"
Inline.class |= math
Block.class |= math

assuming that we have access to a modular RelaxNG schema for XHTML that uses Inline.class and Block.class to collect the the content models for inline and block-level elements.

Customizing the MathML3 schema so that we can restrict the content of annotation-xml elements is similarly simple, for example:

# A RelaxNG Schema for MathML with OpenMath3 annotations
omobj = external "openmath3.rnc" 
include "mathml3.rnc" {anotation-xml.model = omobj}

The MathML3 schema is organized so that subsetting to one of the sublanguages specified here is easy. To include strict content MathML3 in a schema just include

include "mathml3-common.rnc"
include "mathml3-strict-content.rnc"

instead of include mathml3.rnc.

For details about RelaxNG grammars and modularization see [RELAX-NG] or [RelaxNGBook].

A.3 Using the MathML DTD

A.3.1 Document Validation Issues

The use of namespace prefixes creates an issue for DTD validation of documents embedding MathML. DTD validation requires knowing the literal (possibly prefixed) element names used in the document. However, the Namespaces in XML Recommendation [Namespaces] allows the prefix to be changed at arbitrary points in the document, since namespace prefixes may be declared on any element.

The MathML DTD uses the strategy outlined in [Modularization] of making every element name in the DTD be accessed by an entity reference. This means that by declaring a couple of entities to specify the prefix before the DTD is loaded, the prefix can be chosen by a document author, and compound DTDs that include several modules can, without changing the module DTDs, specify unique prefixes for each module to avoid clashes.

A.3.2 Attribute values in the MathML DTD

Note that the DTD is far more permissive than the Relax NG schema, many constraints on attribute values may not be enforced using DTD syntax. For example, many attributes expect numerical values or specific micro-syntax, but are declared as CDATA (arbitrary string) in the DTD. However, lack of enforcement of a requirement in the DTD does not imply that the requirement is not part of the MathML language itself, or that it will not be enforced by a particular MathML renderer. (See Section 2.3.2 Handling of Errors for a description of how MathML renderers should respond to MathML errors.)

Furthermore, the MathML DTD is provided for convenience; although it is intended to be fully compatible with the text of the specification, the text should be taken as definitive if there is a contradiction. (Any contradictions which may exist between various chapters of the text should be resolved by favoring Chapter 7 Characters, Entities and Fonts first, then Chapter 3 Presentation Markup, Chapter 4 Content Markup, then Section 2.1 MathML Syntax and Grammar, and then other parts of the text.)

A.3.3 DOCTYPE declaration for MathML

MathML documents should be validated using the XML DTD for MathML, http://www.w3.org/Math/DTD/mathml3/mathml3.dtd.

Documents using this DTD should contain a DOCTYPE declaration of the form:

<!DOCTYPE math 
    PUBLIC "-//W3C//DTD MathML 3.0//EN"
           "http://www.w3.org/Math/DTD/mathml3/mathml3.dtd">

The URI may be changed to that of a local copy of the DTD if required.

A.4 Using the MathML XML Schema

MathML fragments can be validated using the XML Schema for MathML, located at http://www.w3.org/Math/XMLSchema/mathml3/mathml3.xsd. The provided schema has been mechanically generated from the Relax NG schema, it omits some constraints that can not be enforced using XSD syntax.

A.4.1 Associating the MathML schema with MathML fragments

Similarly to the DOCTYPE declaration used in documents, it is possible to link a MathML fragment to the XML Schema, as shown below.

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
          xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
          xsi:schemaLocation="http://www.w3.org/1998/Math/MathML
            http://www.w3.org/Math/XMLSchema/mathml3/mathml3.xsd">
...
</mml:math>

The xsi:schemaLocation attribute belongs to the XML Schema instance namespace, defined in [XMLSchemas]. The value of the attribute is a pair of URIs: the first is the MathML namespace URI and the second is the location of the schema for that namespace. The second URI may be changed to that of a local copy if required.

As the XML Schema specification indicates, the use of the schemaLocation attribute is only a hint for schema processors: validation by a MathML-aware processor can be performed without requiring that the schemaLocation attribute be declared in the instance. Moreover, a processor can even ignore or override the specified schema URI, if it is directed to.

A Parsing MathML