In this paper, we describe how to represent mathematical formulae in Content MathML referring to the open knowledge-base Wikidata for the grounding of the semantics. By doing so, we link identi ers and symbols in MathML to Wikidata items to annotate mathematical identi ers or operators. In contrast to other mathematical knowledgebases, which de ne symbols in a deductive fashion, the terms in Wikidata emerged inductively from Wikipedia articles in di erent languages. In this context, we discuss the term of a mathematical formula content and its relation to the data representation in Content MathML and Wikidata in detail.
We are confronted with a constantly increasing rate of published documents from
the disciplines of Science Technology, Mathematics and Engineering (STEM)
that contain a large amount of mathematical formulae [
In this chapter, we will review the currently available encoding standards for the
semantics of mathematical formulae to lay the foundations for our contribution
of a new semantic annotation in Content MathML linking to Wikidata items.
(La)TeX LaTeX (shortening of Lamport TeX named after its author) is a
software package for the typesetting system TeX that was rst released in 1983 [
OpenMath \OpenMath is an extensible standard for representing the
semantics of mathematical objects" [
MathML The Mathematical Markup Language (MathML), is \an XML
application for encoding mathematics on the Web" [
We de ne a mathematical formula content as the composition of its constituting
features, the properties or attributes that are relevant to identify the content.
There are mainly three types of formula constituents: numbers, identi ers, and
operators. To uniquely de ne the formula content, additional information on
the relations between these constituents is required. The Content MathML tree
can be visualized by VMEXT, as illustrated in [
This can be illustrated examining r = px2 + y2. This formula contains one number (2), three identi ers (r; x; y) and three operators (=; p; +, ^), if we classify the equality sign = as operator. In functional notation, it can be written as a nested function:
=(r,\sqrt(+(^(x,2),^(y,2))))
Having included the set of numbers, identi ers, and operators as well as their functional relations in our de nition of a formula content, we still need a disambiguation of the identi ers (and operators - although they are much less ambiguous) in the sense of de ning the mathematical data type and/or physical units. In our example, the identi ers r; x and y could be any variables, complex numbers, real numbers or represent radius and coordinates. Besides, also a number could need clari cation of its meaning. E.g., 3:14 could be an arbitrary value or a rounded representation of the irrational transcendent number . So a clari cation of the meaning is the last ingredient to grasp the formula content.
In short, we summarize our de nition of a formula content as: { The set of numbers, identi ers and operators (including equality = or inequality signs <; >) the formula contains. { The (nested) functional relations of the identi ers and numbers, with the operators being the functions and numbers or identi ers being the variables. { The individual meanings (disambiguation) of the numbers, identi ers, and operators. or more formally: A formula content F
F = fN ; I; O; Rg is given by the sets of the numbers N , identi ers I and operators O the formula contains and the set of functional relations R. In the rst three sets, each element is a (symbol, meaning)-tupel while the last set contains the nested functionality of the formula.
The example formula r = px2 + y2 is formalized as:
Fr = fN = f(2, natural number)g; I = f(r, radius), (x, dimension), (y, dimension)g; O = f(=, equality sign),(\sqrt(), square root); (^(), power operator)g; R = f=(r, \sqrt(+(^(x,2),^(y,2))))gg
The disambiguation of 2 as natural number, = as equality sign, \sqrt() as square root and ^() as power operator could arguably be omitted, but for completeness, we ll all tuples (number, disambiguation), (identi er, disambiguation) and (operator, disambiguation). 3.2
Our de nition of a formula content can be mapped into Content MathML with Wikidata annotation, i.e., using Wikidata as an OpenMath Content Dictionary. Figure 2 shows an example markup of the formula E = mc2 from physics with its identi ers linked to Wikidata items referenced by their QID, e.g., Q11379 for \energy".
Since the semantics of the identi ers or operators is taken care of by the Wikidata Content Dictionary, we claim that our de nition of a formula content can fully be covered by a description in Content MathML with Wikidata markup. In the following, we illustrate how the functionality of a formula can be mapped to Strict Content MathML, an XML encoding of OpenMath objects with Content Dictionaries. Table 1 contains the most important tags in Strict Content MathML and OpenMath representation, which will be described below3.
The tags are intended to use as follows: { The <cn> (\content number") element is built to contain numbers such as integers, real and (double) oating point numbers. Additionally, an e-notation as well as complex-cartesian and complex-polar notations are supported. In the case of complex numbers, the real and imaginary part can be separated using the <sep> tag (possibly be rewritten using the <apply> element). { The <csymbol> (\content symbol") element is used in <annotation-xml> encoding to refer to a Content Dictionary by the cd attribute. Identi ers or operators can be cross-referenced by an id and xref attribute respectively. { The <ci> (\content identi er") element is a markup for mathematical variables who - in contrast to symbols - do not have a xed value. All variables with the same name in the same scope (see bindings <bind> and bound variables <bvar> below) are considered equal or identical. 3 The description follows the W3C Recommendation for Content Markup available online at https://www.w3.org/TR/MathML3/chapter4.html#contm.cds. { The <cs> (\content string") element encodes string literals in the form of text. { The <apply> element is used to fundamentally build compound objects by recursively applying a function or an operator to some arguments (numbers, symbols or variables). E.g. x + 3 can be implemented as <apply><csymbol cd="arith1">plus</csymbol> <ci>x</ci><cn>3</cn></apply>. { The <bind> element is indented to build mathematical expressions where a variable is bound in the scope of a function, operator or quanti er. The latter can be an integral, a sum, product or logical quanti er such as 8 (for all ) or 9 (there exists) while the former is the bound dummy variable (renaming it does not change the meaning of the expression). { The <bvar> element denotes the individual bound variables that occur as children in a nested binding expression. { The <share> can be used to avoid duplicate passages by allowing for copies that are pasted by reference. This can be done using the href and id attributes. { The <semantics>, <annotation>, and <annotation-xml> element wrap content elements that provide additional semantic markup or annotations via a Content Dictionary. The tags are regarded as part of both Presentation MathML and Content MathML.
Concluding, we claim that the semantic level of our de ned formula content is covered by the <semantics>, <annotation> and <annotation-xml> markup while the structure and functional level can be described by utilizing the <apply> and <bind> environments.
The remainder of this paper describes the construction, maintenance and
quality assessment of a MathML benchmark dataset for the evaluation of
automated semantic Information Retrieval. We recently introduced it [
Expressions 1 to 100 are random samples taken from the \National
Institute of Informatics Testbeds and Community for Information access Research
Project" (NTCIR) 11 Math Wikipedia Task [
Expressions 101 to 200 are random samples taken from the \NIST
Digital Library of Mathematical Functions" (DLMF) [
Expressions 201 to 305 were selected from the NTCIR arXiv and
NTCIR12 Wikipedia dataset retrieval. 70 % of these formulae were taken from the
arXiv [
We created a Graphical User Interface (see Figure 3) as a web application
with a variety of input elds to easily maintain the gold standard. For each Gold
ID entry or formula, you can chose a Formula Name, specify a Formula Type
(de nition, equation, relation or general formula) and insert the Original Input
TeX and manually Corrected TeX together with a Hyperlink to the source.
The Semantic LaTeX Input eld is used for the semantic annotations,
providing the basis for the generation of Content MathML with Wikidata annotations
by LaTeXML [
As described in the gold-standard [
Finally, we will discuss in detail how our proposed usage of Wikidata both as
LaTeX markup and a MathML Content Dictionary will improve grasping the
semantics of formula content. We will compare our approach of implementing
Wikidata annotation in comparison to already existing DLMF LaTeX macros
and default OpenMath Content Dictionaries.
LaTeXML [
associated Legendre function of the second kind or { \JacobiP{\alpha}{\beta}{n}@{x}: Pn( ; )(x):
Jacobi polynomial.
However, the de ned macros are naturally not extensive, and expansion requires community consensus and implementation. In contrast, Wikidata items can be easily created at any time by anyone. This is why we propose using Wikidata markup as a supplement. We are aware of the fact that since the content of Wikidata items is monitored by a much larger community than the DLMF responsibles can be either bene cial or harmful to the quality of the semantic markup. 4.2
To assess the quality of Wikidata annotations at the level of MathML markup, we will now compare the Content Dictionaries (CDs) using the simple example of adding two numbers a + b.
Using the plus tag of the OpenMath CD arith1 for arithmetic operations4, Strict Content MathML reads <apply><csymbol cd="arith1">plus</csymbol><ci>a</ci><ci>b</ci></apply> <apply><csymbol cd="arith1">plus</csymbol><ci>b</ci><ci>a</ci></apply>. In contrast, a proposed Wikidata markup reads in LaTeX \w{Q12916}{a} \w{Q32043}{+} \w{Q12916}{b} and the Content MathML annotation generated by LaTeXML is While the arith1 -markup allows for a speci cation of the identi ers a and b as "integer", "rational", "real", "complex" etc., the Wikidata-markup assigns them to the Wikidata item real number (Q12916). Additionally it is possible to specify the meaning, e.g. annotate as a physical observable (Q845789). The plus-operation is assigned to the Wikidata item addition (Q32043).
The Wikidata markup is larger and maybe less easy to read, but as already discussed it can be extended at all time by creating or adjusting Wikidata items. Furthermore, it has the advantage that all items are linked to Wikipedia articles that provide extensive human-readable descriptions. Moreoover, MathML is seldomly edited manually. Therefore, we propose to use tools such as VMEXT, which uses human readable labels rather than the Wikidata QIDs in the GUI. 4 For the speci cation see http://www.openmath.org/cd/arith1#sum.
With our proposed Wikidata markup, we achieved advantages of both LaTeX and MathML markup at the same time. The Wikidata macros in LaTeX are easy to write and read and thus adequate for editors who are not familiar with the syntax and structure of MathML. It will be an important mission to convince researchers in the mathematical sciences of the bene ts of making their published content machine-readable. In their documents, they should mark as many entities as possible: - named entities, e.g. \w{Q210546}{Equivalence principle} - whole formulae, e.g. \w{Q210546}{$E=mc^2$} - parts of formulae, e.g. $\w{Q11379}{E} = \w{Q11423}{m} \w{Q2111}{c}^2$ We are planing to develop a GUI to facilitate the annotation process. This will produce labeled data to use for the future discovery and recognition of formula concepts, our long-term research goal.
Formula Concept Discovery (FCD) will be an approach to develop an understanding of the de nition of a mathematical concept by using labeled data from Wikipedia, the arXiv, other resources and self-annotated formulae. We strive to engineer an automatic retrieval of the de ning formula of a given mathematical concept within a document. Machine Learning methods such as formula clustering, decision trees, and feature analysis will help to approximate our ambition.
Formula Concept Recognition (FCR) will subsequently be applicable to spot the de ned formula concepts in given documents by identifying its semantic components (i.e., the constituting identi ers, operators, and numbers) with Wikidata items. If there is nally a su cient amount of labeled data, we can explore the possibility of using neural networks to recognize a given formula by matching its features to concept labels, starting with a small set of classi ers. Our future research will eventually aim at a largely representation-independent de nition and recognition of formula concepts. 6
We thank Wikimedia Foundation and Wikimedia Deutschland providing cloud computing facilities and inviting us for a research visit. This work was supported by the FITWeltweit program of the German Academic Exchange Service (DAAD) as well as the German Research Foundation (DFG grant GI-1259-1).