B Content Markup Validation Grammar

This presents an informal EBNF grammar that can be used to validate the structure of Content Markup.

• It defines the valid expression trees in content markup. It does not define the rules for attribute validation. That must be done separately.
• The non-terminal Presentation_tags is a placeholder for a valid presentation element start tag or end tag.
• The string #PCDATA denotes XML parsed character data.
• Symbols beginning with '_' (for example _mmlarg) are internal symbols. A recursive grammar is usually required for their recognition.
• Symbols which are all in lowercase symbols (for example 'ci') are terminal symbols representing MathML content elements.
• Symbols beginning with Uppercase letters are terminals representating other tokens.

whitespace definitions including Presentation_tags

[1]	Presentation_tags	::=	"presentation"	/* placeholder */
[2]	Space	::=	#x09 | #x0A | #x0D | #x20	/* tab, lf, cr, space characters */
[3]	S	::=	(Space | Presentation_tags")*	/* treat presentation as space */

Characters, only for content validation characters

[4]

Char

::=

#x9 | #xA | #xD | [#x20-#xD7FF] | [#xE000-#xFFFD] | [#x10000-#x10FFFF]

/* valid XML chars */

• start(\%x) returns a valid start tag for the element \%x
• end(\%x) returns a valid end tag for the element \%x
• empty(\%x) returns a valid empty tag for the element \%x

 start(ci)    ::= "<ci>"
 end(cn)      ::= "</cn>"
 empty(plus)  ::= "<plus/>"

The reason for doing this is to avoid writing a grammar for all the attributes. The model below is not complete for all possible attribute values.

start and end tag functions

[5]	_start(\%x)	::=	"<\%x" (Char - '>')* ">"	/* returns a valid start tag for the element \%x */
[6]	_end(\%x)	::=	"<\%x" Space* ">"	/* returns a valid end tag for the element \%x */
[7]	_empty(\%x)	::=	"<\%x" (Char - '>')* "/>"	/* returns a valid empty tag for the element \%x */
[8]	_sg(\%x)	::=	S _start(\%x)	/* start tag preceded by optional whitespace */
[9]	_eg(\%x)	::=	_end(\%x) S	/* end tag followed by optional whitespace */
[10]	_ey(\%x)	::=	S _empty(\%x) S	/* empty tag preceded and followed by optional whitespace */

semantics, annotation, etc.

[11]	semantics	::=	_sg(semantics) _mmlarg _annot* _eg(semantics)
[12]	annotation	::=	_sg(annotation) #PCDATA _eg(annotation)
[13]	annotation-xml	::=	_sg(annotation-xml) _ANY _eg(annotation-xml)
[14]	_ANY	::=	"AnyXML"	/* placeholder for wellformed XML Fragment (not Mixed Content) */
[15]	_annot	::=	annotation | annotation-xml

mathml content constructs

[16]	_mmlarg	::=	_container | _token | _operator | _relation
[17]	_container	::=	_special | _constructor
[18]	_token	::=	ci | cn | csymbol | _constantsym
[19]	_special	::=	apply | lambda | reln | fn | semantics
[20]	_constructor	::=	interval | list | matrix | matrixrow | set | vector | piecewise | piece | otherwise
[21]	_qualifier	::=	lowlimit | uplimit | degree | logbase | domainofapplication | momentabout | condition	/* interval is both a qualifier and a constructor */
[22]	_constantsym	::=	integers | rationals | reals | naturalnumbers | complexes | primes | exponentiale | imaginaryi | notanumber | true | false | pi | eulergamma | infinity

relations

[23]	_relation	::=	_genrel | _setrel | _seqrel2ary
[24]	_genrel	::=	_genrel2ary | _genrelnary
[25]	_genrel2ary	::=	ne
[26]	_genrelnary	::=	eq | leq | lt | geq | gt
[27]	_setrel	::=	_seqrel2ary | _setrelnary
[28]	_setrel2ary	::=	in | notin | notsubset | notprsubset
[29]	_setrelnary	::=	subset | prsubset
[30]	_seqrel2ary	::=	tendsto

operators

[31]

_operator

::=

_funcop | _arithop | _calcop | _vcalcop | _seqop | _trigop | _classop | _statop | _lalgop | _logicop | _setop

functional operators

[32]	_funcop	::=	_funcop1ary | _funcopnary
[33]	_funcop1ary	::=	inverse | ident | domain | codomain | image
[34]	_funcopnary	::=	fn| compose	/* general user-defined function is n-ary */

(note minus is both 1ary and 2ary)

arithmetic operators

[35]	_arithop	::=	_arithop1ary | _arithop2ary | _arithopnary | root
[36]	_arithop1ary	::=	abs | conjugate | factorial | minus | arg | real | imaginary | floor | ceiling
[37]	_arithop2ary	::=	quotient | divide | minus | power | rem
[38]	_arithopnary	::=	plus | times | max | min | gcd | lcm

calculus and vector calculus

[39]	_calcop	::=	int | diff | partialdiff
[40]	_vcalcop	::=	divergence | grad | curl | laplacian

sequences and series

[41]

_seqop

::=

sum | product | limit

elementary classical functions and trigonometry

[42]	_classop	::=	exp | ln | log
[43]	_trigop	::=	sin | cos | tan | sec | csc | cot | sinh | cosh | tanh | sech | csch | coth | arcsin | arccos | arctan

statistics operators

[44]	_statop	::=	_statopnary | moment
[45]	_statopnary	::=	mean | sdev | variance | median | mode

linear algebra operators

[46]	_lalgop	::=	_lalgop1ary |_lalgop2ary | _lalgopnary
[47]	_lalgop1ary	::=	determinant | transpose
[48]	_lalgop2ary	::=	vectorproduct | scalarproduct | outerproduct
[49]	_lalgopnary	::=	selector

logical operators

[50]	_logicop	::=	_logicop1ary | _logicopnary | _logicop2ary | _logicopquant
[51]	_logicop1ary	::=	not
[52]	_logicop2ary	::=	implies | equivalent | approx | factorof
[53]	_logicopnary	::=	and | or | xor
[54]	_logicopquant	::=	forall | exists

set theoretic operators

[55]	_setop	::=	_setop1ary |_setop2ary | _setopnary
[56]	_setop1ary	::=	card
[57]	_setop2ary	::=	setdiff
[58]	_setopnary	::=	union | intersect | cartesianproduct

operator groups

[59]	_unaryop	::=	_funcop1ary | _arithop1ary | _trigop | _classop | _calcop | _vcalcop | _logicop1ary | _lalgop1ary | _setop1ary
[60]	_binaryop	::=	_arithop2ary | _setop2ary | _logicop2ary | _lalgop2ary
[61]	_naryop	::=	_arithopnary | _statopnary | _logicopnary | _lalgopnary | _setopnary | _funcopnary
[62]	_specialop	::=	_special | ci | csymbol
[63]	_ispop	::=	int | sum | product
[64]	_diffop	::=	diff | partialdiff
[65]	_binaryrel	::=	_genrel2ary | _setrel2ary | _seqrel2ary
[66]	_naryrel	::=	_genrelnary | _setrelnary

separator

[67]

sep

::=

_ey(sep)

leaf tokens and data content of leaf elements

[68]	_mdatai	::=	(#PCDATA | Presentation_tags)*	/* note _mdata includes Presentation constructs here. */
[69]	_mdatan	::=	(#PCDATA | sep | Presentation_tags)*	/* note _mdata includes Presentation constructs here. */
[70]	ci	::=	_sg(ci) _mdatai _eg(ci)
[71]	cn	::=	_sg(cn) _mdatan _eg(cn)
[72]	csymbol	::=	_sg(csymbol) _mdatai _eg(csymbol)

condition - constraints. constraints contains either a single reln (relation), or an apply holding a logical combination of relations, or a set (over which the operator should be applied).

condition

[73]

condition

::=

_sg(condition) reln | apply | set _eg(condition)

domains for integral, sum , product, and specials

[74]	_domainofapp	::=	domainofapplication | _domainabbrev
[75]	_domainabbrev	::=	(lowlimit uplimit?) | uplimit | interval | condition

Note that apply is used in place of the deprecated reln in MathML2.0 for relational operators as well as arithmetic, algebraic etc.

apply construct

[76]	apply	::=	_sg(apply) _applybody | _relnbody _eg(apply)
[77]	_applybody	::=	( _unaryop _mmlarg )	/* 1-ary ops */
			| (_binaryop _mmlarg _mmlarg)	/* 2-ary ops */
			| (_naryop _mmlarg*)	/* n-ary ops, enumerated arguments */
			| (_naryop bvar* _domainofapp? _mmlarg)	/* n-ary ops, over domain of application */
			| (_specialop _mmlarg*)	/* special ops can be applied to anything */
			| (_specialop bvar* _domainofapp? _mmlarg)	/* special ops, over domain of application */
			| (_ispop bvar* _domainofapp? _mmlarg)	/* integral, sum, product */
			| (_diffop bvar* _mmlarg)	/* differential ops */
			| (log logbase? _mmlarg)	/* logs */
			| (moment degree? momentabout? _mmlarg*)	/* statistical moment */
			| (root degree? _mmlarg)	/* radicals - default is square-root */
			| (limit bvar* lowlimit? condition? _mmlarg)	/* limits */
			| (_logicopquant bvar* _domainofapp? _mmlarg)	/* quantifier with explicit bound variables */

Equations and relations - reln uses lisp-like syntax (like apply) the bvar and condition elements are used to construct a "such that" or "where" constraint on the relation. Note that reln is deprecated but still valid in MathML2.0.

equations and relations

[78]	reln	::=	_sg(reln) _relnbody _eg(reln)
[79]	_relnbody	::=	( _binaryrel bvar* condition? _mmlarg _mmlarg ) | ( _naryrel bvar* condition? _mmlarg* )

fn construct Note that fn is deprecated but still valid in MathML2.0

[80]	fn	::=	_sg(fn) _fnbody _eg(fn)
[81]	_fnbody	::=	Presentation_tags | _mmlarg

lambda construct

[82]	lambda	::=	_sg(lambda) _lambdabody _eg(lambda)
[83]	_lambdabody	::=	bvar* _domainofapp? _mmlarg	/* multivariate lambda calculus */

declare construct

[84]	declare	::=	_sg(declare) _declarebody _eg(declare)
[85]	_declarebody	::=	ci (fn | constructor)?

constructors

[86]	interval	::=	_sg(interval) _mmlarg _mmlarg _eg(interval)	/* start, end define interval */
[87]	set	::=	_sg(set) _lsbody _eg(set)
[88]	list	::=	_sg(list) _lsbody _eg(list)
[89]	_lsbody	::=	_mmlarg*	/* enumerated arguments */
			| (bvar* _domainofapp _mmlarg)	/* generated arguments */
[90]	matrix	::=	_sg(matrix) matrixrow* _eg(matrix)
			| _sg(matrix) bvar* _domainofapp? _mmlarg _eg(matrix)	/* vectors over domain of application */
[91]	matrixrow	::=	_sg(matrixrow) _mmlarg* _eg(matrixrow)	/* allows matrix of operators */
[92]	vector	::=	_sg(vector) _mmlarg* _eg(vector)
			| _sg(vector) bvar* _domainofapp? _mmlarg _eg(vector)	/* vectors over domain of application */
[93]	piecewise	::=	_sg(piecewise) piece* otherwise? _eg(piecewise)
[94]	piece	::=	_sg(piece) _mmlarg _mmlarg _eg(piece)	/* used by piecewise */
[95]	otherwise	::=	_sg(otherwise) _mmlarg _eg(otherwise)	/* used by piecewise */

bound variables

[96]	_cisemantics	::=	_sg(semantics) _citoken _annot* _eg(semantics)
[97]	_citoken	::=	ci | _cisemantics
[98]	bvar	::=	_sg(bvar) _citoken degree? _eg(bvar)
[99]	degree	::=	_sg(degree) _mmlarg _eg(degree)

other qualifiers - note the contained _mmlarg could be a reln

[100]	lowlimit	::=	_sg(lowlimit) _mmlarg _eg(lowlimit)
[101]	uplimit	::=	_sg(uplimit) _mmlarg _eg(uplimit)
[102]	logbase	::=	_sg(logbase) _mmlarg _eg(logbase)
[103]	domainofapplication	::=	_sg(domainofapplication) _mmlarg _eg(domainofapplication)
[104]	momentabout	::=	_sg(momentabout) _mmlarg _eg(momentabout)

The top level math element. Allow declare only at the head of a math element.

math

[105]

math

::=

_sg(math) declare* _mmlarg* _eg(math)