SWiM -- A Semantic Wiki for Mathematical Knowledge Management

SWiM – A Semantic Wiki for Mathematical
Knowledge Management
Christoph Lange
Computer Science, Jacobs University Bremen, ch.lange@jacobs-university.de
Abstract. SWiM is a semantic wiki for collaboratively building, edit-
ing and browsing mathematical knowledge represented in the domain-
specific structural semantic markup language OMDoc. It motivates users
to contribute to collections of mathematical knowledge by instantly shar-
ing the benefits of knowledge-powered services with them. SWiM is cur-
rently being used for authoring content dictionaries, i. e. collections of
uniquely identified mathematical symbols, and prepared for managing a
large-scale proof formalisation effort.
1 Research Background and Application Context:
Mathematical Knowledge Management
A great deal of scientific work consists of collaboratively authoring documents—
taking down first hypotheses, commenting on results of experiments, circulating
informal drafts inside a working group, and structuring, annotating, or reor-
ganising existing items of knowledge, finally leading to the publication of a
well-structured article or book. Here, we particularly focus on the domain of
mathematics and on tools that support collaborative authoring by utilising
the knowledge contained in the documents. In recent years, several semantic
markup languages have been developed to represent the clearly defined and hi-
erarchical structures of mathematics. The XML languages MathML [9], Open-
Math [11], and OMDoc [3] particularly aim at exchanging mathematical knowl-
edge on the web. OMDoc, employing Content MathML or OpenMath repre-
senting the functional structure of mathematical formulæ—as opposed to their
visual appearance—and adding support for mathematical statements (like sym-
bol declarations or axioms) and theories, has many applications in publishing,
education, research, and data exchange [3, chap. 26]. The main challenge is ac-
quiring a large collection of OMDoc-formalised knowledge that can power such
added-value services. In an open, collaborative environment, the workload can
be distributed among many authors, but as semantic markup makes fine-grained
structures explicit, it is tedious to author. As the community can only benefit
from added-value services after a substantial initial investment (writing, anno-
tating and linking) on the author’s part, we sought for motivating authors into
action by offering “elaborate [. . . ] services for the concrete situation” they are
in [2].
The final publication is available at www.springerlink.com
S. Bechhofer et al. (Eds.): ESWC 2008, LNCS 5021, pp. 832–837, 2008.
© Springer-Verlag Berlin Heidelberg 2008
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SWiM – A Semantic Wiki for Mathematical Knowledge Management 833
2 Key Technology: Semantic Wiki and Ontologies
Our research is motivated by the assumption that in this context a semantic
wiki comes in handy. OMDoc supports all levels of formalisation, from human-
readable texts to fully formal representations for automated theorem proving,
and semantic wikis have been found appropriate for collaboratively refining
knowledge models (cf. [13]). User motivation in semantic wikis by instant grat-
ification has been investigated in earlier works [1]. The ultimate goal of our
work is to achieve a feedback loop where users are supported to contribute well-
structured knowledge, which is then exploited to offer services, which in turn
facilitate editing and motivate new contributions [5].
<omdoc>
<proof id="pyth-proof"
for="pythagoras">
. . . </proof>
</omdoc>
extraction
RDF
pyth-proof pythagoras
Proof Theorem
type type
proves
proves
<pyth-proof, rdf:type, omdoc:Proof>
<pyth-proof, omdoc:proves, pythagoras>
Fig. 1. RDF extraction from OMDoc markup in a wiki page
Semantic markup has deep structures: an OMDoc document can contain the-
ories containing statements that contain formulæ referring to symbols defined
in other theories. This is uncommon for most semantic wikis, where the struc-
tures are rather flat and one aims at small pages to prevent editing conflicts and
to facilitate search and navigation. So to adapt OMDoc’s model of knowledge
to a semantic wiki, we had to choose an appropriate granularity of wiki pages
and arrived at one page holding one mathematical statement or one theory. To
make knowledge from OMDoc documents usable on the semantic web, informa-
tion about the resources represented by pages and their interrelations (e. g. “a
proof for the Pythagorean theorem”) are extracted to RDF. As a vocabulary
for this, we modeled OMDoc’s structures explicitly in a document ontology [5]
in OWL-DL. This ontology contains e. g. the information that both theorems
and proofs are specialisations of a general “mathematical statement”, and that
a proof can prove a theorem (Fig. 1). Moreover, generic transitive dependency
and containment relations have been modeled. For example, having one theory
import another theory (and reusing symbols defined there) establishes a depen-
dency. One theory logically contains its statements; similarly, statements can
contain sub-statements, as in the case of a proof that consists of multiple steps.
3 The SWiM 0.2 Prototype: IkeWiki + OMDoc
As a base system for the implementation, we chose IkeWiki [12]. Among the
systems evaluated, it offered the richest XML infrastructure—a key requirement

834 Christoph Lange
for adding OMDoc support—and was found to be most extensible [4]. Its backend
consists of a PostgreSQL database for the page contents, a Jena RDF store for
the RDF graph and the ontologies. Additional ontologies can easily be imported.
The frontend heavily relies on the Dojo Ajax toolkit.
Technically, the extension of IkeWiki to SWiM required supporting OMDoc
in addition to the HTML-like wiki page format. To foster stepwise formalisation
of informal text, we chose to mix OMDoc fragments with wiki markup. Thus
we could still rely on IkeWiki’s WYSIWYG HTML editor, which just had to be
enhanced by support for OMDoc XML elements. Moreover, this choice allowed
for an easier maintenance of the OMDoc-related enhancements to the SWiM code
base and avoided changes to the underlying database schema. The document
ontology is preloaded into the RDF store. RDF triples are extracted from the
OMDoc markup upon saving a page or importing an OMDoc file. Additional
XSLT template rules care for rendering embedded OMDoc fragments. In order
to render mathematical formulæ, there is a notation definition for every semantic
symbol. These notation definitions can be imported and edited right in the wiki,
as parts of OMDoc documents [6]. An efficient, specialised renderer supporting
the upcoming MathML 3 standard [10,9] applies them to the symbols in the
formulæ. In the editing view, statement- and theory-level structures of OMDoc
are made accessible as special HTML tables, whereas mathematical formulæ
given in semantic markup are made accessible in a simplified ASCII notation
of OpenMath. OMDoc documents are browsable via inline links manually set in
the informal parts, via links from occurrences of symbols in formulæ to the place
of their declaration, set by the formula renderer, and via RDF links, displayed in
a separate box by IkeWiki. The latter comprise those triples that are extracted
from the markup (cf. Fig. 1), as well as triples inferred by a reasoner1.
SWiM also relies on the ontology for reacting on changes to notation defini-
tions. When an author changes a notation definition n for a symbol s, exactly
those wiki pages that contain a formula using s or that include other pages
containing such formulæ need to be re-rendered. Looking up the symbol s ren-
dered by n, the formulæ fi using s, or pages (transitively) including the fi would
be clumsy in the OMDoc XML sources, but is easy in the RDF graph, as this
information is extracted from the documents and represented using ontology
properties such as NotationDefinition–renders–Symbol and Statement–contains–
Formula; Formula–uses–Symbol. This service allows for instant visual debugging
of notation definitions [6]. For upcoming releases, more ontology-powered ser-
vices are planned, including more general change management, learning assis-
tance, and editing facilitations like editing of subsections and auto-completion of
link targets [7]. There is some evidence that many services can be based on the
most generic relations of dependency and (physical or logical) containment [5].
1 The ontology is prepared for DL reasoning, but currently only the RDFS reasoner
built into Jena is used.

SWiM – A Semantic Wiki for Mathematical Knowledge Management 835
Fig. 2. A mathematical document in SWiM
With scientists and knowledge engineers in mind, we envisage SWiM as a devel-
opment environment that conveniently supports refactorings of knowledge2.
4 Use Cases and Applications
Now that viewing, browsing, editing, importing and exporting mathematical
documents basically works, we are evaluating SWiM in practical settings. The
Flyspeck project is about large-scale formalisation of a proof of the Kepler con-
jecture. We are starting to support this effort by “crowdsourcing” the knowledge
compiled so far (hundreds of proof sketches that are not yet machine-verifiable)
on a SWiM site [8]. The main challenge is giving an interested visitor an im-
pression of the extent of the project and, using appropriate SPARQL queries,
showing him where work needs to be done. Currently we are investigating how
the original LATEX sources can be utilised by automatically converting them to
HTML with MathML, then to informal OMDoc, breaking that into wiki pages,
and letting the users formalise them stepwisely. For the upcoming OpenMath 3
standard, SWiM is currently being extended to an editor for OpenMath Con-
tent Dictionaries [6], which could be regarded as flat OMDoc theories that just
define symbols and do not import anything. There, mainly editing Dublin Core
metadata and notation definitions is of interest.
2 This is common in mathematics, e. g. in algebra: If one just needs groups, they can
be defined by a theory with the four well-known axioms. For explicitly modeling
related structures as well, one would break this into smaller theories—semigroup
just defining an associative operation on a set, monoid importing this and extending
it by an identity element, and finally the refactored group, adding inverse elements.

836 Christoph Lange
5 Conclusion and Related Work
SWiM makes mathematical documents editable collaboratively and particularly
facilitates browsing them by exploiting the knowledge they contain. Domain-
specific services are powered by an ontology that models structures of documents—
an advantage over generic semantic wikis, which would not be able to offer addi-
tional services for mathematical knowledge. Competing non-semantic approaches
like the math encyclopædia PlanetMath (evaluated in [4]) are less flexible, as they
cannot exploit the structures of their presentation-oriented LATEX formulæ and
rely on a fixed set of metadata. Most services for editing and browsing need to be
hard-coded, which potentially restricts the scale of knowledge managment tasks
the systems can be applied to. The SWiM approach of integrating a semantic
markup language into a wiki by choosing an appropriate page granularity, mod-
eling a document ontology, and extracting relevant facts from the markup into
RDF has successfully been applied to OMDoc and the closely related but syn-
tactically different OpenMath [6] and is likely to be portable to other domains
as well, e. g. for the chemical markup language CML.
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