Christoph Lange is a PhD student studying mathematical knowledge management using semantic web technologies. He developed a semantic wiki called SWiM for collaboratively structuring mathematical knowledge. His work included developing ontologies for modeling the structure of mathematical documents, creating user interfaces for annotating and browsing knowledge, and implementing domain-specific argumentation in wikis. He hopes his research on applying these techniques to the complex domain of mathematics can help advance semantic web technologies and knowledge management.

1. Aspects of Mathematical Knowledge on the Semantic
Web
Summary of my work at DERI
Christoph Lange
Jacobs University, Bremen, Germany
KWARC – Knowledge Adaptation and Reasoning for Content
October 2, 2008
Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 1

2. Introduction
Ph. D. student, 2 of 3 years (+ n months) done
At home in a group doing “Mathematical Knowledge Management”
dealing with mathematical knowledge, formality ranges from
human-friendly to computer-verifiable
my project: collaboration on semiformal knowledge (using a semantic
wiki)
What I wanted to learn about the semantic web:
engineering ontologies for scientific documents
user interfaces for annotating and browsing
relation of social interaction to knowledge
. . . and much more
Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 2

3. What I had
SWiM, a semantic wiki for mathematics
Documents in OpenMath and OMDoc (think: “Semantic MathML
and more”)
Practical approach:
1 analyse use cases (here: lightweight ontology engineering, collaborative
proof formalisation and documentation, lecture note authoring)
2 figure out conceptual model
3 implement it in a prototype (within the SWiM sandbox)
4 evaluate how people use it
Leitmotiv of my research
“How can users be motivated and supported to make the effort of
collaboratively structuring mathematical knowledge, what additional
knowledge can be inferred from users’ contributions, and how can this
again be utilised in order to improve collaboration?”
Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 3

4. Ontologies for Scientific Documents
Initial problem:
I had a basic ontology that modelled structures of mathematical
knowledge; mainly statements (definition, theorem, proof, examples).
Semi-formal knowledge often comes in documents that also contain
text
There is a document structure (chapter, section, cross-reference), and
a rhetorical structure, both of which can be independent from the
mathematical structure.
Collaboration with Tudor Groza: getting the model right (do it as
SALT does with document ontology ↔ annotation ontology ↔
rhetorical ontology)
Expansion of the ontology, and rules for extracting these concepts
from OMDoc documents to RDF, supported by intern Gordan
Ristovski
Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 4

5. User Interfaces for Annotating and Browsing
Implementation done by interns:
Gordan Ristovski more and easier annotation support in the editor
(got his hands dirty in JavaScript): toolbars for easy
selection of types of mathematical knowledge, from
phrase to theory level; deleting annotations;
auto-completion of link targets (prepared)
Jana Giceva improved and extended syntax for annotating SALT-like
rhetorical structures in OMDoc (i. e. using the SALT
ontology within the host language OMDoc, not LTEX) A
first ideas for a editing interface; visualisation of
rhetorical relations and blocks implemented (→ active
documents)
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6. Argumentation about Mathematical Knowledge
Initial motivation: structured wiki discussions, well-defined workflow
for solving problems with knowledge in a wiki
In my case, a wiki page is an item of mathematical knowledge, e. g. a
theorem
issues discussed will be quite specific: e. g. “This theorem is hard to
understand” (or wrong, or inadequately presented, . . . )
Note: there is also argumentation within artifacts of scientific
knowledge, but I’m more interested in argumentation about them
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7. Resource Edit Discussion History
Issue Alice 2008–05–30 [Idea][Argument][Agree][Disagree][Decision]
It’s hard to find out how to improve content (= resources) in semantic wikis
Agree Bob 2008–05–31
Indeed, besides automated approaches it’s hard to get focused feedback from users.
Idea Claire 2008–06–01 [Argument][Agree][Disagree][Decision]
So let’s make wiki discussions semantic!
Argument Dave 2008–06–02 [Agree][Disagree]
We could take types from the DILIGENT argumentation ontology for the posts.
Argument Eric 2008–06–03 [Agree][Disagree]
And every discourse should be connected to resources corresponding to the wiki
page, and there should be domain-specific Idea and Issue subclasses.
Agree Anonymous 2008–06–04
That’s great, then the wiki could assist with the realisation of an approved idea.
Decision Christoph 2008–06–05
So let’s do it! (Available in the latest IkeWiki, domain-specific extension for math-
ematics in SWiM, IkeWiki’s math edition, is work in progress)
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8. Domain-Specific Argumentation
Assumptions:
Possible problems depend on the type of knowledge item
Possible solutions depend on the type of knowledge item and the type
of problem
Standard problems have standard solutions, with which software can
assist
Survey (still running, tinyurl.com/5qdetd)
Common issues: wrong, incomprehensible, uncommon style,
underspecified, redundant, truth uncertain
Common solutions: directly improve affected knowledge item, split it
When issues remain unresolved, it’s mostly due to insufficient
restructuring support
Ideas contributed by Tuukka Hastrup (how to scale this to the web;
SIOC = Semantically Interlinked Online Communities) and Stéphane
Corlosquet (making it work in the CMS Drupal)
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9. Domain-Specific Argumentation (Example)
hasDiscussion
forum1 theorem
(IkeWiki ontology)
has_container exemplifies
post1: Issue
(Incomprehensible) example
responseTo
has_reply resolvesInto
post2: Idea
(ProvideExample)
positionOn knowledge
post3: Agree items
(OMDoc ontology)
on wiki pages
post4: Disagree onIdea
post5: Agree onIssue
withPositions
post6: Decision
physical structure argumentative
(SIOC) discussion page structure
Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 9

10. General Argumentation on Social Media Sites
Developing an argumentation module for SIOC
Joint work with Uldis Boj¯rs and Tudor Groza
a
Use cases, model, guidelines for usage; implementation to be done
refers_to
supports/
Statement Argument
challenges
agrees_with/
subClassOf disagrees_with/ subClassOf
neutral_towards
arises_from
Issue Idea Elaboration Example Evaluation Justification
proposes_solution_for elaborates_on
decides Position
supported_by
Decision
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11. Other things I have done
OpenMath case study: lightweight mathematical ontology engineering
(http://wiki.openmath.org)
no rhetorical structures, no documents (but still a lot of structures to
annotate!)
argumentation
small group of knowledge engineers (concessions required: integration
with legacy Subversion repository)
specialised editors: structured definitions, formulas, metadata
evaluation needed
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12. Integrating OMDoc with the Semantic Web (Future)
Initial motivation: extensible metadata by importing vocabularies (so
far: hard-coded Dublin Core support)
Model dependencies in a mathematical way: not just reference by
writing down a URI, but model RDF(S), OWL, and then semantic
web ontologies as mathematical theories
For semantic-web-like reasoning, export RDF(S), OWL from OMDoc
Yet another ontology language?
No, we gain benefits: better validation, different reasoning services:
dependency graph checking, search for terms, . . .
Yes, but easy integration into documents (literal programming instead
of rdfs:labels of type rdf:XMLLiteral)
RDFa? Yes, but as an alternative syntax, not as a replacement. We
can generate XHTML from OMDoc, and extract RDF from OMDoc
(except formulas), so combine it. And provide a way back.
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13. Summary
What I hope(d) to learn here – to use it for mathematical knowledge
management:
engineering ontologies for scientific documents !
user interfaces for annotating and browsing !
relation of social interaction to knowledge !
What I hope to contribute to the semantic web:
mathematics as a quite complex use case pointing out some limits of
semantic wikis (and the semantic web)
a case study of an ontology for a complex document format, with
document structure, mathematical structure, and rhetorical structure
domain-specific argumentation in a semantic wiki
a more mathematical approach to ontology engineering
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