OpenMath content dictionaries are collections of mathematical symbols. Traditionally, content dictionaries are handcrafted by experts. The OpenMath speci cation requires a name and a textual description in English for each symbol in a dictionary. In our recently published MathML benchmark (MathMLBen), we represent mathematical formulae in Content MathML referring to Wikidata as the knowledge base for the grounding of the semantics. Based on this benchmark, we present an OpenMath content dictionary, which we generated automatically from Wikidata. Our Wikidata content dictionary consists of 330 entries. We used the 280 entries of the benchmark MathMLBen, as well as 50 entries that correspond to already existing items in the o cial OpenMath content dictionary entries. To create these items, we proposed the Wikidata property P5610. With this property, everyone can link OpenMath symbols and Wikidata items. By linking Wikidata and OpenMath data, the multilingual community maintained textual descriptions, references to Wikipedia articles, external links to other knowledge bases (such as the Wolfram Functions Site) are connected to the expert crafted OpenMath content dictionaries. Ultimately, these connections form a new content dictionary base. This provides multilingual background information for symbols in MathML formulae.
3. it is di cult to identify the reasons for the limited performance of a system.
As an alternative, we suggest breaking down complex MathIR tasks into several subtasks. The rst subtask is
to convert formulae from their source representation to a machine-readable format which describes the semantics.
With our MathMLBen project [11, 13], we created an open gold standard for this task. MathMLBen provides a
list of over 300 formulae and their textual contexts. The example formulae include formulae that were used in
the NTCIR search tasks [1, 2, 3], as well as formulae from the DLMF [
The
rst version of a Wikidata content dictionary As introduced earlier, a key feature of MathMLBen are special LaTeX(ML) macros [11]. Those macros link csymbol-elements in the MathML to entries in Wikidata. For example, one element of our gold standard includes the logistic function f (x) = 1+e kL(x x0) : We did not nd a symbol for the logistic function in the OpenMath content dictionaries1. However, articles regarding the logistic function can be found in many di erent languages in Wikipedia. Moreover, the Wikidata item Q1052379 connects all these Wikipedia articles. This item was manually improved by 19 users (excluding bots). Through these improvements, additional data such as properties, relations to other items and external identi ers were added. Figure 1 shows a screenshot of the item, including statements, external identi ers, and links to Wikipedia articles as well as other Wiki projects such as Wikisource and Wikiversity. In our gold standard, we used the semantic LaTeX macro2 \wf{Q1052379}{f} 1A list of all OpenMath symbols is available from https://www.openmath.org/symbols/.
2 The di erence between the macros nw and nwf is that nwf associates the role function with a symbol. For example, the rst invisible operator in f (a + b) is interpreted as function application rather than multiplication, if nwf is used [11]. to encode that LaTeXML should treat the symbol f as <csymbol cd="wikidata">Q1052379</csymbol> [11]. However, the content dictionary wikidata that the csymbol element refers to with its cd attribute did not exist.
Currently, there are more than 49 million items in Wikidata. Not all items are part of mathematical formulae. Thus, creating a Wikidata content dictionary based on all items would be impractical. In this paper, we present a Wikidata content dictionary that includes all the 280 entries used in MathMLBen. We call our content dictionary wikidata.ocd. It can be downloaded from https://cd.formulasearchengine.com/wikidata.ocd.
Listing 1 shows the content dictionary entry for the symbol logistic function (Q1052379). All 280 symbols follow the same pattern. This is because they generated them with MathTools [6] using the Wikidata item numbers given in the csymbol-elements in MathMLBen. The description includes the English label of the Wikidata entry (line 1366), the English description (not available in Listing 1), a link to the English Wikipedia article (line 1367) and nally a static link to the version of the Wikidata item that was used to create the content dictionary entry (line 1370). While this description is currently in English, the language is only a con guration parameter in our tool. Theoretically, any other language could be used. However, the success depends on the number of available community-maintained texts in that language (cf. Section 3).
Our content dictionary wikidata.ocd improves the standard compliance of the MathMLBen gold standard. It provides a content dictionary for third party MathML processing software that can read user-contributed content dictionaries without requiring a special implementation to fetch data from Wikidata. This improves the standard compliance the MathMLBen gold standard. 3
After having discussed how to automatically derive a content dictionary from a set of Wikidata items, we discuss how to create a cdbase that contains the standard OpenMath symbols from Wikidata in this chapter.
As Figure 2 shows, the nature of content dictionaries and Wikipedia pages (here in English) is di erent. While the CD description is brief in human-readable content, the Wikipedia page shows a lot of human-readable information. On the other hand, there are formal mathematical properties that are hard to extract from the Wikipedia page. Consequently, we analyse the strength and weaknesses of both approaches in the following. However, before doing so, we need to map entries in Wikidata to OpenMath. We therefore proposed a new property in Wikidata (P5610) of type external identi er which was approved on August 9th 2018. This identi er, labeled OpenMath ID, allows one to refer to OpenMath symbols from within Wikidata. That way, we connect both communities, Wikidata and OpenMath. Everyone (even without a Wikidata account) is now able to create new mappings.
The new property P5610 is of type string and has three constraints: a single-value, a unique value, and a
regexp lter ([a-z]+[
Listing 2 shows an entry in wikidata.ocd that corresponds to the Wikidata item plus. The description section contains more information than Listing 1. Line 269 is the English description from the item that represents the addition in Wikidata. Moreover, line 272 links to the de nition form of arith1#plus from the OpenMath content dictionary.
For the remainder of the section, we discuss the di erences between traditional OpenMath symbol de nition entries and Wikidata generated symbol de nitions. Our Wikidata content dictionary contains 330 symbols in a single content dictionary. In contrast, there are 289 o cial OpenMath symbols divided into 38 content dictionaries. 247 OpenMath symbols have a role attribute (application 193, constant 39, binder 3, semanticattribution 2, error 3). While we did research on identifying Wikidata items as numerical constants [14], this information is not included in the current version of wikidata.ocd. Moreover, the o cial OpenMath content dictionaries contain 149 examples, 180 formal mathematical properties (FMP), and 179 commented mathematical properties (CMP). Currently, wikidata.ocd does not contain any of the aforementioned features. Due to the lack of time, the MathMLBen data has not been converted to the OpenMath XML format, which would be required to create examples. Deriving reasonable CMP or FMP from Wikidata requires semantic enhancement of the de ning formula statement which are currently only available in presentation form. The description eld in the o cial OpenMath content dictionaries is on average 131 words, as compared to 212 words (including 14 words for the reference to the source) in wikidata.ocd. While Wikidata items are not divided into a structure comparable to content dictionaries, they have hierarchical relations such as the instance of (P31) relation. As displayed in Table 1, the instance of relation is not modelled consistently. We hypothesise that this is typical for corpora which emerged from community interactions. Finally, the symbol names in the standard OpenMath dictionaries are easier to remember for English speakers. Therefore, using IDE or smart editors is a prerequisite to work with Wikidata items conveniently. Otherwise, the long numeric item identi ers are hard to read and to remember. To support this purpose, we released the node module codemirror-wikidata [12], which provides autocompletion based on the description rather than on the numeric values. 4
In this paper, we released a rst version of the Wikidata content dictionary wikidata.ocd to the public. It contains all the symbols used in the MathMLBen open gold standard. Moreover, it contains 50 entities that correspond to standard OpenMath symbols. Furthermore, we introduced the new Wikidata property P5610 and described how it can be used to create an alternative cdbase. Also, we compared symbol descriptions that were generated automatically from Wikidata to the manually crafted OpenMath symbol de nitions. While multilingualism and links to Wikipedia might be considered as an advantage of the Wikidata cdbase, many other formal aspects such as structure and type information are currently better modeled in the traditional OpenMath content dictionaries. On the other hand, Wikidata has far more items that could be possibly used as symbol de nitions.
Future research should investigate how the missing formal information in wikidata.ocd can be automatically extracted. If there was a mechanism to generate content dictionaries from Wikidata that have the same formal quality as the current OpenMath content dictionaries, a good foundation for CD extension, based on Wikidata, would ease the expansion of the OpenMath standard. Another promising research direction is to better understand how information given in distributed data sources can be connected using alignments [7, 9, 10]. Acknowledgments We thank the Wikimedia Foundation and Wikimedia Deutschland for providing cloud computing facilities and for providing o ce space for us. 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). The author would like to thank Howard Cohl for constructive criticism of the manuscript.
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OpenMath ID Instance of
arith1#abs piecewise function, even function, idempotent function arith1#divide binary operation
arith1#gcd function arith1#minus binary operation, operation arith1#plus binary operation arith1#power operation arith1#root type of mathematical function, algebraic function arith1#sum mathematical expression arith1#times binary operation arith1#unary minus calculus1#diff unary operation, mathematical concept
fns1#lambda Wikimedia disambiguation page fns1#left compose operator, operation hypergeo0#gamma function integer1#factorial function interval1#interval oo part limit1#limit mathematical concept limit1#null integer, Fibonacci number, triangular number, automorphic number, even number, non-negative integer, 0 number class, non-positive integer linalg1#determinant invariant
linalg2#matrix array data structure, tensor linalg2#matrixrow row and column vectors linalg2#vector tensor list1#list creative work logic1#and logical connective, boolean function logic1#equivalent transitive relation, symmetric relation, re exive relation nums1#e real number, transcendental number, irrational number nums1#i square root, mathematical constant, Gaussian integer,
imaginary number nums1#infinity mathematical concept nums1#pi real number, transcendental number, mathematical con
stant, irrational number relation1#approx relation, estimation relation1#eq equivalence relation, partial order relation1#geq inequation relation1#gt inequation relation1#leq inequation relation1#lt inequation relation1#neq inequality sign
set1#in binary relation, subclass set1#intersect binary operation, set operation set1#set Wikidata metaclass, Wikidata metaclass, Wikidata
metaclass, class (set theory), formalization, collection transc1#arccos inverse trigonometric function, decreasing function transc1#arctan inverse trigonometric function, increasing function transc1#cos trigonometric function, even function transc1#cosh hyperbolic function, even function transc1#exp exponential function, type of mathematical function transc1#ln transc1#log transc1#sin transc1#sinh transc1#tan transc1#tanh type of mathematical function, logarithm type of mathematical function, type of mathematical function, multivalued function, elementary transcendental function trigonometric function, odd function hyperbolic function, odd function trigonometric function hyperbolic function [12] [13] [14]
Moritz Schubotz. \VMEXT2: A Visual Wikidata aware Content MathML Editor". In: Joint Proceedings of the CME-EI, FMM, CAAT, FVPS, M3SRD, OpenMath Workshops, Doctoral Program and Work in Progress at the Conference on Intelligent Computer Mathematics 2018 co-located with the 11th Conference on Intelligent Computer Mathematics (CICM 2018). Ed. by Osman Hasan et al. 2018.
Moritz Schubotz et al. \Improving the Representation and Conversion of Mathematical Formulae by Considering their Textual Context". In: Proceedings of the 18th ACM/IEEE on Joint Conference on Digital Libraries, JCDL 2018, Fort Worth, TX, USA, June 03-07, 2018. Ed. by Jiangping Chen et al. ACM, 2018.