This Note examines the treatment of bound variables in Content MathML. Bound variables are central representational primitives in mathematical languages. They allow one to express functions, quantification, and operators with qualifiers. The first edition of the MathML 2.0 Recommendation [MathML2] was somewhat vague about the identity conditions on bound variables, and as a consequence Content MathML applications were left to guess the exact meaning. This Note provides some of the rationale behind how this has been clarified in the second edition [MathML22e].
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This Note provides a self-contained explanation of bound variables in Content MathML. It contains non-normative interpretations of the MathML 2.0 Recommendation (Second Edition) [MathML22e] and it provides guidelines for representing mathematical objects in Content MathML. Please direct comments and report errors in this document to firstname.lastname@example.org.
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2 The problem
3 Background on bound variables in logics and mathematical foundations
5 Using definitionURL for Bound Variable Identification.
5.1 Why definitionURL?
5.2 Clarifying the Role of definitionURL in MathML
Bound variables are central representational primitives in
mathematical languages. They allow one to express functions, quantification,
and operators with qualifiers. The first edition of the MathML 2.0
Recommendation [MathML2] was vague about the identity conditions on bound
variables. For example, all examples in that Recommendation use simple
ci elements, for which there is never any doubt when
an instance of the bound variable is the same as the defining
occurrence. In fact, the question of identity only seriously arises
when one starts to use more elaborate presentations inside of the
elements. Content MathML
applications were left trying to decide
when two presentations are equal, possibly in the presence of
heavy use of
semantics tags and/or styling attributes.
How extensively could a
ci element be modified (even for
presentational purposes) without changing it mathematically?
The safe answer to this question was "not at all", but there were legitimate examples where at very least one of the presentations needed to be changed for expository purposes while retaining the mathematical identity. Furthermore, the first edition of the MathML 2.0 Recommendation [MathML2] did not elaborate on this point.
This Note investigates the recognition of instances of bound variables and shows how existing mechanisms for dealing with semantics can be used to make such identifications explicit without depending on the direct comparison of the presentational markup. This is the approach that has been used to clarify the notion of identity of bound variables in the second edition of the MathML 2.0 Recommendation [MathML22e].
When complex Presentation MathML annotations are used, the association of a particular instance of a ci with its defining occurrence in the bvar element can be difficult to establish, and the first edition of the MathML 2.0 Recommendation [MathML2] did not specify a way to do so. Note that a strict definition of identity based solely on the XML structure (possibly based on the XML Information Set) would be too restrictive.
In most purely Content MathML cases (including all the examples
in the the first edition of the MathML 2.0 Recommendation [MathML2])
there is no problem; the content of the bvar element
is a single ci,
whose content is just a simple character string. There is little doubt that
ci in the body of the apply element
that have the same name after whitespace normalizations is bound by the
<apply> <int/> <bvar><ci>x</ci></bvar> <interval><cn>0</cn><cn>1</cn></interval> <apply> </plus> <apply><power/><ci>x</ci><cn>2</cn></apply> <apply><times/><cn>2</cn><ci>x</ci></apply> <cn>5</cn> </apply> </apply>
However, the Recommendation does allow arbitrary Presentation MathML inside a
element and this feature is often used for variables with
sub/superscripts, or variables in different font families. In these cases,
it becomes less clear which variables are bound by the
bvar declaration. For example a simple color change might be used
to draw attention to one of the instances of a bound variable as in the
following variant of the the above example.
<apply> <int/> <bvar> <ci> <mstyle color="red"> <msub><mi>x</mi><mn>2</mn></msub> </mstyle> </ci> </bvar> <interval><cn>0</cn><cn>1</cn></interval> <apply> </plus> <apply><power/><ci><msub><mi>x</mi><mn>2</mn></msub></ci><cn>2</cn></apply> <apply><times/><cn>2</cn><ci><msub><mi>x</mi><mn>2</mn></msub></ci></apply> <cn>5</cn> </apply> </apply>
The goal is to clarify how to use presentational information on bound variables without losing track of their essential role as bound variables.
The problem of handling bound variables has been studied extensively in logic and the foundations of mathematics in the last one hundred years. In fact the handling of bound variables was the main contribution of Frege's treatment of first-order logic that is the basis for the set-theoretic foundation of mathematics. This section defines the terminology that will be used in the rest of this document.
A variable is called a [Definition: bound variable] in an expression of the form O x.B[x] , if O is a binding operator. We use B[x] for an expression (called the [Definition: body]) that may contain the identifier x. The occurrence of x directly after the operator O is called the [Definition: declaring occurence] and those in the body are called the [Definition: bound occurrences].
In first-order logic (and all other foundations of mathematics), bound variables have two crucial properties:
[Definition: Bound Variable Identity ] Bound variables (BVI) are declared by the quantifier (or generally the binding operator such as a sum, product, limit,...) and all other occurrences of the bound variable are just references to this "declaring" occurrence. Outside the scope of the binding operators, bound variables are invisible.
[Definition: Bound Variable Names ] A bound variable name (BVN) is provided for the traditional symbolic presentation. This allows for a linearized (formula-like) representation of the functional dependency in formula-trees, as in
For all x. x > 1 → x2 > x
To be consistent with the logical requirement that names are essentially irrelevant outside their scope, the rule of alphabetic renaming allows one to systematically rename bound variables. In particular, the expression above is mathematically equivalent to
For all y. y > 1 → y2 > y
Bound variable identity provides the semantical essence of bound variables. There are formulations of mathematics without (explicit) bound variables (Category theory, combinatory logic, De Bruijn Indices), but they are usually hard for humans to understand.
Bound variable names help humans understand the formulae involved. In fact, un-intuitive renaming can render complex formulae unintelligible. This is also the main reason why the variable-free logics mentioned above have remained theoretical tools rather than communication devices.
In the first edition of MathML 2.0 [MathML2]
Content MathML only explicitly supported a
representation of bound variables (only the Presentation MathML
content could be used for identification). As a consequence, it was
difficult to decide the identity question, a resolution of which
was needed for formal treatment of Content MathML formulae/objects.
This is not a problem in most cases, where names are simple. However, when names involve complex (Presentation MathML) objects, the BVN-based specification potentially leaves open the issue of when a bound occurrence refers to a declaring occurrence -- the central question to be resolved in order to decode the meaning of the expression.
There are really only a few sensible choices for identifying a bound variable with its defining occurrence. They include:
XML-Infoset-equality: two Presentation MathML names are equal, if and only if their infosets are. This criterion is probably best supported by XML. (See appendix A and a general discussion of equality)
Explicit links: introducing some specific link to the declaring occurrence or the binding operator (such as used in the HELM project).
The XML-Infoset-equality approach is impractical for complicated presentations where the whole issue of equivalence of presentations becomes as complicated as the issue of equivalence of images. This BVN-based identification schema leads at best to computationally expensive equality notions.
However, explicit links back to the defining occurrence of a bound variable can provide an optional alternative identification schema especially for use in the more difficult cases. These links make the bound variable identification (BVI) explicit and the implemenation makes natural and intuitive use of existing machinery in MathML. The advantages of such an approach include:
By design, presentation is permitted in Content MathML in only two rather
controlled ways. It may occur (much like an image) inside a
ci element, or it may be tied in via the
element. The troublesome case is the content of
The view taken when this was introduced was that such
presentation was analogous to embedding an image.
It was to be an indecomposable token. Anything more complicated
was to be done using semantics tags. By analogy with images,
ci elements are equal just if they are essentially
identical which may formally defined by equality of the
ci elements with two completely different
presentation contents (albeit visually indistinguishable) to be
equivalent would be the same as expecting images in two different
files and/or formats to be identical. Short of
"decomposing" the objects and doing some potentially
complicated analysis, equivalence could not possibly be
When there is a need to transform Content MathML representation into other content formats, the binding relation of the bound variables must be identified. In a BVN-based approach, this entails the need to compute the equivalence of variable names as Presentation MathML representations. The sophisticated equality tests that may be needed here are not very well-supported by XML tools - some sort of specialized XML diff program appears to be implied. Thus, there are too many factors such This is a serious problem the adoption of Content MathML in more formal settings.
Compatibility with existing applications is important. Thus the solution must augment the existing BVN-based treatment of bound variables with a BVI-based treatment that allows the user to make the identity condition of bound variables clear, where it is not obvious from the name alone.
<lambda> <bvar> <ci id="the-boundvar">complex presentation</ci> </bvar> <apply> <plus/> <ci definitionURL="#the-boundvar">complex presentation</ci> <ci definitionURL="#the-boundvar">complex presentation</ci> </apply> </lambda>
Instead of computing whether "complex presentation" is
identical in each occurrence, it is only necessary to check whether the
definitionURL point to the same declared
content of the
definitionURL is a URI, which has well-understood
identity conditions (URIs are the basis of the Web).
Of course, use of this BVI-based approach to bound variables is a strictly optional way of clarifying bound variable semantics.
The definitionURL attribute is used in Content MathML to point to the (external) definition of an operator. The relevant part of the Recommendation is 18.104.22.168, where it says
definitionURL [...] points to an external definition of the
semantics of the symbol or construct being declared. The value is
a URL or URI that should point to some kind of definition. This
definition overrides the MathML default semantics. [...]
The main statement here is that the value of the
attribute is a URI points to a definition of the semantics of the
element. In the foundation of mathematics it is an accepted fact that
bound variables behave just like symbols/constants, only that the
scope (and definition) of bound variables are more local than
symbols/constants. In fact most foundational systems treat the
symbols/constants as bound at the theory level. There is a very clear
understanding that the declaring instance (in MathML the
element) behaves like the (local) definition of the symbol. In fact
many modern foundational systems make this a key design decision.
definitionURL is conformant to the current DTD and content
One sometimes sees Content MathML representations like the following one.
<declare definitionURL=".../formalpowerseries"><ci>F</ci></declare> <apply> <set/> <bvar><ci>F</ci></bvar> <condition> <apply> <in/> <ci>F</ci> <csymbol definitionURL="XXX">My Special Formal PSeries</csymbol> </apply> </condition> <ci>x</ci> </apply>
declare element is used to give the bound variable
ci element a
definitionURL to attribute a property to a
MathML element (in this case being a formal power series), possibly to allow
presentation algorithms may make use of this extra knowledge to display the
cis in a different manner.
This of course
conflicts with the use of
definitionURL as recommended above, since
then one cannot possibly use
definitionURL to refer back to the
defining instance of the bvar, and at the same time to the external definition
of the special formal power series.
This example is somewhat contrived in that the
is being used in the spirit of a type declaration.
In some sense, this example actually reflects a misunderstanding
definitionURL attribute. The
is supposedly reserved for "Definitions", i.e. for
statements that fix the meaning of an element (possibly up to isomorphism).
No matter how the points are argued, the example represents a legitimate authoring need and must be accommodated in some manner. The only conflict comes when one tries to use the BVI style of linking bound variables. Fortunately, in that case, it is a prime candidate for the typing through the use of the new general typing mechanism for Content MathML [MathMLTypes], so no functionality is lost: the above example can be re-represented explicitly by
<declare> <ci definitionURL="#the-bvar">F</ci> <semantics> <ci definitionURL="#the-bvar">F</ci> <annotation-xml definitionURL="types.html#type_of"> <csymbol definitionURL=".../formalpowerseries"/> </annotation-xml> </semantics> </declare> <apply> <set/> <bvar><ci id="the-bvar">F</ci></bvar> <condition> <apply> <in/> <ci definitionURL="#the-bvar">F</ci> <csymbol definitionURL="XXX">My Special Formal PSeries</csymbol> </apply> </condition> <ci>x</ci> </apply>
The declare merely associates the
element with an appropriate semantics tag which provides the type and
all is well. Note that in this case, the
ci element that is the
first child of the
declare element also links back to the
ci element in the
bvar element. Clearly, the
definitionURL attribute on the
ci element in the first
child of the
declare element restricts the substitution to those
ci that carry a matching
attribute. Without this, the substitution would have affected other
<ci> F </ci> elements outside the scope of the
apply element shown in the example, but inside the
element that governs the scope of the
The definition of equality of expressions for purposes of
declare is not discussed in version 2.0 of the
Recommendation beyond noting that the content of
elements is treated much like an image of the formula and
then systematically basing all examples on a very simple
notion of equality of the element content as ASCII strings.
While this is a very simple example of XML information set
equality, there are a number of different definitions that
strictly speaking could not be ruled out.
This could be one of many plausible criteria, including:
string equality: two Presentation MathML names are equal, iff their string values (just throw away all element contributions) are. This is the simplest criterion computationally, but it confuses
<math> <msup> <mi>x</mi> <mn>2</mn> </msup> </math>
<math> <msub> <mi>x</mi> <mn>2</mn> </msub> </math>
which both have the string value
XML-Infoset-equality: two Presentation MathML names are equal, if and only if their infosets are. This criterion is probably best supported by XML, among the non-trivial ones.
MathML-Infoset equality: two Presentation
MathML names are equal, if and only if their infosets are equal after
text node content. This is
the equality that is suggested for MathML token
observational equality: two Presentation MathML names are equal, if and only if their recommended rendering (according to the MathML specification) are equal. This is probably the most adequate one (after all BVN approaches are geared towards humans), but this is very hard to compute and even specify unambiguously.
For purposes of this Note, the notion of equality has generally been based on XML information sets after white space normalization. Roughly speaking, this means that if all attributes that are present have the same value as strings, and recursively, the element content is the same the same, then the objects are identical.