Publishing Math Lecture Notes as Linked Data

Publishing Math Lecture Notes as Linked Data
Catalin David1, Michael Kohlhase1, Christoph Lange1, Florian Rabe1, Nikita
Zhiltsov2 and Vyacheslav Zholudev1
1 Computer Science, Jacobs University Bremen, Germany
{c.david,m.kohlhase,ch.lange,f.rabe,v.zholudev}@jacobs-university.de
2 Mathematics, Kazan State University, Russia, nikita.zhiltsov@gmail.com
Abstract. We mark up a corpus of LATEX lecture notes semantically and
expose them as Linked Data in XHTML+MathML+RDFa. Our applica-
tion makes the resulting documents interactively browsable for students.
Our ontology helps to answer queries from students and lecturers, and
paves the path towards an integration of our corpus with external sites.
1 Application: Computer Science Lecture Notes
Over the last seven years, the second author has accumulated a large corpus
of teaching materials, comprising more than 2,000 slides, about 1,000 home-
work problems, and hundreds of pages of course notes, all written in LATEX. The
material covers a general first-year introduction to computer science, graduate
lectures on logics, and research talks on mathematical knowledge management.
This situation is typical for educators and researchers and represents the state
of the art in mathematics, physics, computer science, and engineering: LATEX
has proven suitable for writing high-quality lecture notes and publishing them
as PDF. However, in our educational setting, we would like to benefit from the
much larger degree of interactivity that screen reading and e-books support. For
example, while reading notes students want to directly look up the meaning of a
symbol (e. g. �) in a formula, or examples for a difficult concept (e. g. structural
induction). They may want to select advanced material for self-study from the
whole body of lecture notes, based on the topics covered in the lecture. They
want to use a search engine to find related material in other universities’ online
course notes, on mathematical web sites, or Wikipedia. Lecturers want to query
their repository for document parts reusable in an upcoming lecture, given the
prerequisites students are expected to meet and the material that has already
been covered. In a course for a special audience, e. g. mathematics for physicists,
they want to draw examples from that domain even though they are less familiar
with it. They also want to locate didactic gaps, such as concepts without ex-
amples, or unjustified proof steps. These services require semantic annotations
in the lecture notes that are understandable for external search engines. Plain
LATEX is barely usable for anything beyond on-screen reading and printing. Even
simple semantic annotations are uncommon, rare exceptions are the \title
command making its meaning explicit or \frac{a}{b} focusing on functional
structure instead of visual layout. This is especially problematic for symbols in
The final publication is available at www.springerlink.com
L. Aroyo et al. (Eds.): ESWC 2010, LNCS 6089, pp. 370–374, 2010.
c© Springer-Verlag Berlin Heidelberg 2010
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Publishing Math Lecture Notes as Linked Data 371
formulæ, which are often overloaded with multiple definitions or presentable in
different notations.
(n
k
)
can be a vector or a binomial coefficient, and a French or
Russian would write the latter as Ckn. Therefore, we have developed a semantic
representation of mathematical knowledge in LATEX and a presentation process
that preserves these semantic structures as Linked Data in the output, exposing
them to mashups for interactive exploration, as well as semantic searching and
querying. These are based on an ontology for mathematical knowledge so that
mathematical content can be linked across different repositories.
2 Research Background and Related Work
LATEX’s importance in scientific authoring and its extensibility by macros have
led to semantic extensions enabling modern publishing workflows. SALT (se-
mantically annotated LATEX [8]) marks up rhetorical structures and fine-grained
citations in scientific documents. Its markup is not sufficiently fine-grained for
formulæ, and its vocabulary is limited to rhetorics and citations and not exten-
sible. Our own sTEX offers macros for introducing new mathematical symbols
and using arbitrary metadata vocabularies. Some math e-learning systems, such
as ActiveMath [1] or MathDox [17], use semantic representations of formulæ and
higher-level structures, e. g. proof steps or course module dependencies, in the
standard XML languages OpenMath [5] and OMDoc [11]. They utilize seman-
tic structures but do not publish them in a standard representation like RDF,
which would promote general-purpose queries beyond the built-in services and
integration with other systems on the web. The Linking Open Data movement
promotes best practices for publishing data on the web [9], as standalone RDF
or embedded into HTML documents as RDFa [2]. Applications include Sindice,
an engine that crawls and indexes Linked Data [19], and the Sparks O3 Browser,
a mashup that utilizes RDFa annotations in HTML for interactive browsing [20].
Our interactive documents work similarly but additionally support annotations
in MathML formulæ. MathML has pioneered embedded annotations long before
RDFa, albeit with a more limited scope. Its parallel markup interlinks the ren-
dered appearance and the semantic structure of mathematical expressions; the
meaning of mathematical symbols is usually defined in lightweight ontologies
called OpenMath content dictionaries [4]. HELM (Hypertext Electronic Library
of Mathematics [3]) pioneered representing structures of mathematical knowl-
edge in RDF, e. g. what mathematical theory introduces a symbol, what of its
properties have been declared or asserted, and how the latter are proved. The
HELM ontology has not gained wide acceptance, though. At the time of its de-
velopment, there was no RDFa-like way of embedding RDF into web documents.
3 Architecture and Demo
Our architecture publishes semantically enriched LATEX lecture notes as XHTML+
MathML+RDFa Linked Data. We kept LATEX as an input language, as it is fa-
miliar to authors and well supported by editors, and as high-quality PDF can

372 David, Kohlhase, Lange, Rabe, Zhiltsov, Zholudev
be obtained from it. With sTEX (semantically enhanced TEX), we have intro-
duced LATEX macros for marking up the semantic structure of formulæ and doc-
uments [12] and manually annotated our complete corpus using the sTEX plugin
for the Emacs editor. One can, e. g., declare a symbol union, formally define it,
and make its semantic representation \union{A,B,C} expand to A\cup B\cup C
for human-readable rendering. There are environments for mathematical state-
ments and theories, e. g. \begin{example}[for=union]. LATEXML transforms
this into a semantically equivalent intermediate XML representation, using the
standard XML languages OpenMath for formulæ [5] and OMDoc for higher-level
structures [11]. Finally, our JOMDoc rendering library [10] generates human-
readable output from this XML – an output that still contains the full semantic
structure as annotations. A custom Java implementation renders formulæ as
parallel markup of Presentation MathML annotated with OpenMath3; render-
ing higher-level structures as XHTML+RDFa [2] is implemented in XSLT. RDF
is extracted from XML by our Krextor XML→RDF library [15], which generates
URIs for all mathematical objects in a document. It uses our OMDoc ontology
(cf. [14]) as a vocabulary for representing all mathematical structures (e. g. “d
is a definition, e is an example for d”) plus full text, inspired by HELM and
designed as a more expressive counterpart of the OMDoc XML schema.
The whole transformation process is integrated into our versioned XML
database TNTBase [22]; see http://kwarc.info/LinkedLectures. TNTBase
has a Subversion-compatible interface making it suitable as a lecture notes repos-
itory. The TEX→XML and XML→RDF transformations are automatically trig-
gered by a hook upon committing a new revision of an sTEX lecture module. If
the generated OMDoc+OpenMath is not schema-valid, the commit is rejected.
On the other hand, it follows Linked Data best practices and, depending on
the MIME type an HTTP client requests, serves a document as OMDoc, as
RDF (only a structural outline, not the full text and formulæ), or as XHTML+
MathML+RDFa. The latter contains JavaScript code from our JOBAD library
for interactive documents [13,7], which operationalizes the annotations – Linked
Data and other – in the rendered documents. JOBAD’s definition lookup deter-
mines the OpenMath annotation of the Presentation MathML symbol the user
clicked on, from that obtains the URI of the symbol, and then requests XHTML
from that URI (resulting in the symbol’s declaration and definition), which is
then displayed in a popup. The RDFa annotations are used for making parts of a
document (e. g. steps of a structured proof) foldable, and for displaying the local
neighborhood in the RDF graph (e. g. related examples) in popups; this is im-
plemented using the rdfQuery library [18], relying on the Linked Data structure
in the latter case. Further third-party services can be integrated in a mashup
style; we have demonstrated this for a unit conversion service [13,7]. Besides en-
abling JOBAD’s services, we have implemented machinery to load the extracted
3 A proposal for fully representing formulæ in RDF [16] has not gained wide accep-
tance. RDF-based reasoners are often limited to decidable first order logic subsets,
which is insufficient for mathematical applications, and XML has a straightforward
notion of order (e. g. of the arguments of an operator or of a set constructor).

Publishing Math Lecture Notes as Linked Data 373
RDF into a triple store and query it using SPARQL. We also provide a widget
for formulating queries without knowing SPARQL and the OMDoc ontology. It
allows to ask some non-trivial queries, e. g. “find examples for all concepts from
graph theory (about which I’m planning a lecture), assuming as prerequisites
the concepts from formal languages (and their prerequisites)”. This would yield
the parse tree of a context-free language as an example for the concept “tree” –
as operating systems were not among the prerequisites.
Our demo shows the complete pipeline in action: (i) annotating a document
with our sTEX Emacs mode, (ii) committing it to TNTBase, (iii) automatic
translation to OMDoc, schema validation, and RDF extraction,(iv) loading the
extracted RDF data into a triple store, (v) retrieving the document in different
representations, (vi) browsing the XHTML+MathML+RDFa rendering, (vii) in-
teracting with the Linked Data in it, (viii) and querying a triple store. Addition-
ally, we will demonstrate the generation of PDF from the sTEX sources.
4 Conclusion and Outlook
Our architecture makes legacy LATEX lecture notes available as Linked Data.
We expose these data to external clients but have also implemented services for
interactively exploring the XHTML+MathML+RDFa presentation of our data.
We are also working on preserving some of the semantics in the PDF output, as
SALT does. Evaluation of our enriched lecture notes by the student end users
is planned for the next semester. To the best of our knowledge, we are the first
provider of RDF-based Linked Data in the domain of mathematics and among
the first to operationalize the Linked Data structures of formula markup. Having
successfully transformed more than 300,000 normal, non-semantic LATEX docu-
ments from arxiv.org to XHTML+Presentation MathML [21] and working on
machinery for automatically annotating them using natural language process-
ing, we will soon be able to expose even more mathematical knowledge as Linked
Open Data; however, due to the inherent complexity of mathematical knowledge,
with a less formal semantics than manually annotated documents. Our lecture

374 David, Kohlhase, Lange, Rabe, Zhiltsov, Zholudev
notes are self-contained so far, but we are now starting to reap the benefits
of Linked Data by linking them to other data sets, e. g. DBpedia [6], whose
mathematical knowledge does not have a semantics as strong as ours, but which
provides abundant informal background knowledge, e. g. about the originators
of mathematical theories. On the other hand, hardly any well-known mathemat-
ical site (e. g. planetmath.org and mathworld.wolfram.com) currently exposes
machine-understandable metadata. We promote our technology, starting with
lightweight RDFa annotation using the OMDoc ontology, as a migration path
towards their integration into a true mathematical Semantic Web.
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