Meeting on Mathematical Notations, Open University, 2007

1. Motivation State of the Art MathUI Knowledge Representation Services Conclusion
Editing and Using Mathematics in a Semantic Wiki
Meeting on Mathematical Notations 2007
Christoph Lange
Jacobs University, Bremen, Germany
(formerly International University Bremen)
KWARC – Knowledge Adaptation and Reasoning for Content
This work was supported by JEM-Thematic-Network ECP-038208.
October 23, 2007
Ch. Lange (Jacobs University Bremen) Editing and Using Mathematics in a Semantic Wiki October 23, 2007 1

2. Motivation State of the Art MathUI Knowledge Representation Services Conclusion
Abstract
SWiM is a semantic wiki for collaboratively . . .
building
editing
browsing
. . . mathematical knowledge represented in OMDoc [Kohlhase06].
Target audience:
mathematicians developing new theories
scholars learning mathematics
Outline:
Current SWiM prototype: user interface and architecture
Plans for further development: platform for semantic scientific services
Ch. Lange (Jacobs University Bremen) Editing and Using Mathematics in a Semantic Wiki October 23, 2007 2

3. Motivation State of the Art MathUI Knowledge Representation Services Conclusion
Collaborative Mathematics on the Web
Current projects: Wikipedia, PlanetMath, . . .
Easy to create and link knowledge items
Scientific services currently not available
E. g. a search for:
sqrt{a^2 + b^2} = c, or: x^2 + y^2 = z^2
All theorems about triangles for which a proof
exists
Neither system relates mathematical symbols to
their (formal) definitions or allows for formally
structured theories.
Ch. Lange (Jacobs University Bremen) Editing and Using Mathematics in a Semantic Wiki October 23, 2007 3

4. Motivation State of the Art MathUI Knowledge Representation Services Conclusion
Semantic Wikis and Semantic Markup
Semantic wikis understand the knowledge graph they contain
usually: 1 page = 1 real-world concept
pages and links typed with terms from ontologies
Structural semantic markup: In math, there are OMDoc-based
services for
learning assistance (→ ActiveMath [Melis+ 06])
semantic search (→ MathWebSearch [KohlhaseŞucan07])
publishing
theory management
proof verification
The challenge of joining both: Motivating the authors to structure
their knowledge!
Share the benefits of structural semantics with the author
Offer added-value services for the situation the author is in
Instant gratification: show up-to-date links to related pages
Ch. Lange (Jacobs University Bremen) Editing and Using Mathematics in a Semantic Wiki October 23, 2007 4

5. Motivation State of the Art MathUI Knowledge Representation Services Conclusion
The SWiM Prototype
SWiM prototype: IkeWiki [Schaffert06] with OMDoc
(http://kwarc.info/projects/swim/)
editing, presentation, navigation
Ch. Lange (Jacobs University Bremen) Editing and Using Mathematics in a Semantic Wiki October 23, 2007 5

6. Motivation State of the Art MathUI Knowledge Representation Services Conclusion
Demo (A mathematical document)
Ch. Lange (Jacobs University Bremen) Editing and Using Mathematics in a Semantic Wiki October 23, 2007 6

7. Motivation State of the Art MathUI Knowledge Representation Services Conclusion
Demo (Editing formulae)
Ch. Lange (Jacobs University Bremen) Editing and Using Mathematics in a Semantic Wiki October 23, 2007 7

8. Motivation State of the Art MathUI Knowledge Representation Services Conclusion
Demo (Editing notation definitions)
Ch. Lange (Jacobs University Bremen) Editing and Using Mathematics in a Semantic Wiki October 23, 2007 8

9. Motivation State of the Art MathUI Knowledge Representation Services Conclusion
Demo (An OpenMath content dictionary; old SWiM)
Ch. Lange (Jacobs University Bremen) Editing and Using Mathematics in a Semantic Wiki October 23, 2007 9

10. Motivation State of the Art MathUI Knowledge Representation Services Conclusion
Editing and Presenting OMDoc
Adapting OMDoc’s structures to a wiki:
Small, reusable pages: allow theory-constitutive statements to appear
in their own document
Simple link syntax: theory expands to theory.omdoc#theory.
Symbols and their notation defined in the wiki itself – requires flexible
presentation mechanism
Working on integration of mmlproc (MathML 3) [Müller07]
http://kwarc.info/projects/mmlkit/mmlproc/
Exploit structural knowledge: navigate through RDF knowledge graph
extracted from markup
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11. Motivation State of the Art MathUI Knowledge Representation Services Conclusion
Extracting Knowledge from OMDoc using an Ontology
Semantic web tools need explicit knowledge representation
Extract RDF from OMDoc markup, using document ontology as
vocabulary
Three layers of knowledge: objects, statements, theories
Formally modeled in OWL-DL (description logic)
Example
A wiki page: Extracted RDF triples (= graph):
proves
<omdoc> Proof Theorem
<proof id="pyth-proof" type type
for="pythagoras"> proves
... pyth-proof pythagoras
</proof>
</omdoc>
<pyth-proof, rdf:type, omdoc:Proof>
<pyth-proof, omdoc:proves, pythagoras>
(omdoc:* → OMDoc document ontology)
Ch. Lange (Jacobs University Bremen) Editing and Using Mathematics in a Semantic Wiki October 23, 2007 11

12. Motivation State of the Art MathUI Knowledge Representation Services Conclusion
Services for Science
Case studies in science and education planned
Hypothesis: other sciences structure knowledge similarly to
mathematics
Need to support tools scientists use, e. g. Mathematica
Integrate available tools, e. g. the MathWebSearch engine
Connecting to the semantic web: allow external services to operate on
the knowledge from the wiki
Ch. Lange (Jacobs University Bremen) Editing and Using Mathematics in a Semantic Wiki October 23, 2007 12

13. Motivation State of the Art MathUI Knowledge Representation Services Conclusion
Conclusion
SWiM supports the basic wiki features for OMDoc content.
On top of that base system, several services for science and education
will be prototyped and evaluated in case studies.
Easy Editing
Search
Proof for … 1
1 z −2 ?
2
pythagoras ∫−∞ e d ?
partial-diff-eqn ¿
proton 1.-------
---------
2.-------
---------
SWiM+ Publishing
Learning
Assistance
Change
Management
<xml>
...
</xml>
Ch. Lange (Jacobs University Bremen) Editing and Using Mathematics in a Semantic Wiki October 23, 2007 13