SWiM – A Semantic Wiki for Mathematical Knowledge Management

Motivation State of the Art OMDoc+Wiki Usability MathUI Knowledge Representation Services Conclusion

SWiM – A Semantic Wiki for Mathematical Knowledge
Management
Workshop on Mathematical User Interfaces at MKM 2007

Christoph Lange

Jacobs University, Bremen, Germany
(formerly International University Bremen)
KWARC – Knowledge Adaptation and Reasoning for Content

June 27, 2007

Ch. Lange (Jacobs University Bremen) SWiM – A Semantic Wiki for MKM June 27, 2007 1


Abstract

SWiM is a semantic wiki for collaboratively . . .
building
editing
browsing
. . . mathematical knowledge represented in OMDoc.
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



Collaborative Mathematics on the Web

Current collaborative projects for maths:
Wikipedia – wiki encyclopedia covering (almost)
anything
PlanetMath – wiki-like system dedicated to
mathematics
Connexions – courseware repository and content
management system
...
Easy to create and link knowledge items
⇒ wikis also suitable for non-public research
projects
Scientiﬁc services currently not available



Deﬁciencies of Wikipedia etc. in Knowledge Management
Imagine a scholar searching for:
sqrt{a^2 + b^2} = c, or:
x^2 + y^2 = z^2
Impossible in Wikipedia and
PlanetMath, which use
A
presentational LTEX
In principle possible (but not
implemented) in Connexions,
thanks to Content MathML
All theorems about triangles for
which a proof exists
Possible in semantic wikis
Neither system relates mathematical
symbols to their (formal) deﬁnitions
or allows for formally structured
theories.


Semantic Wikis

Conventional wikis do not understand the knowledge graph they
contain
Semantic wikis address this problem:
usually: 1 page = 1 real-world concept
pages and links typed with terms from ontologies
Examples: Semantic MediaWiki, IkeWiki, . . .



Structural Semantic Markup

Structural semantic markup: common way to represent scientiﬁc
knowledge
For OMDoc, e. g., there are services for
learning assistance (→ ActiveMath)
semantic search (→ MathWebSearch)
publishing
theory management
proof veriﬁcation



OMDoc in a Semantic Wiki – Meeting Users’ Needs

Semantic wikis are appropriate base system for community-authored
knowledge models:
support stepwise formalisation workflow
offer enhanced navigation and search
Who is willing to author OMDoc and OpenMath?
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



The SWiM Prototype
SWiM prototype: IkeWiki+OMDoc
(http://kwarc.info/projects/swim/)
editing, presentation, navigation



Editing OMDoc

So far, only raw OMDoc XML editing supported
Minimum feedback when XML is invalid
Adapting OMDoc 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.



Presenting OMDoc
Adapted OMDoc XSLTs (→ XHTML+MathML), with caching
Symbols in formulae point to their <symbol> declarations.
Hyperlinked source code view
doc.omdoc
A−tmpl.xsl A.omdoc
uses symbols from compile XSL
documents A, B, C XSL templates for templates defines symbols
symbols from A from present− and their presentation
ation elements
collect documents (expres.xsl)
whose symbols
are used
(exincl.xsl)
doc−incl.xsl omdoc2pmml.xsl
omdoc2pmml.xsl
omdoc2pmml.xsl
include −tmpl style generic templates
generic templates
sheets of documents forgeneric templates
OMDoc to MathML
for OMDoc to MathML
whose symbols are used transformation to MathML
for OMDoc
transformation
transformation

(... and for other output formats)

doc2pmml.xsl
transform
to MathML bind doc−incl.xsl
and omdoc2pmml.xsl
XSL transformation
xsl:include
doc.xml italic generated file
MathML output
document



Exploiting Knowledge

Semantic wiki: One page = one concept
SWiM: One page = one statement or theory
(small, reusable pages, but not too small)
Note: presentation process allows for includes!
Relations between theories and statements made explicit as RDF
subject–predicate–object triples
Incoming and outgoing links displayed in a box, grouped by type
(→ instant gratiﬁcation)



An Ontology of OMDoc
OMDoc allows for modeling three layers of mathematical knowledge:
objects, statements, and theories.
Make these layers explicit – formalise OMDoc’s structures in an
ontology (OWL-DL; description logic)
Partly done for OMDoc; further steps: extending, generalising
imports
Theory

lives in contains-1 Concept depends on

Statement
Symbol
deﬁned by
Deﬁnition


Extracting Knowledge from OMDoc

Pages are in OMDoc – knowledge buried in the markup
Semantic Web tools need more explicit representation of the
knowledge (RDF)
Extract RDF from markup (so far only on page level)

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)



Extension Towards Sciences

We are concerned with a technology transfer of MKM applications to
general scientiﬁc knowledge management.
OMDoc has already been successfully extended towards physics; other
sciences will follow
Need an “upper” ontology of scientiﬁc markup languages with . . .
objects/statements/theories
containment, dependency
. . . as an abstraction layer for services exploiting the knowledge.
Need to support tools scientists use
Editing wiki pages with a local text editor . . .
. . . or a specialised application like Mathematica.
I. e. allow for internal and external services operating on the data in
the wiki or on knowledge extracted from it.



Planned Services

Goal: support the way scientists work and collaborate!
Dependency maintenance (lightweight in SWiM, or enhance backend
by change manager like locutor )
Learning assistance by suggesting annotated or inferred prerequisites
(in SWiM, or using ActiveMath’s web services)
Editing aids like ontology-based auto-completion
Search using MathWebSearch in the backend and oﬀering interface in
the UI
Community-aware reading and browsing: lectora case study
Feed lectora with log of users’ interactions
Adapt e. g. presentation to preferences of a user or his community
Case study in education from September 2007



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>


SWiM – A Semantic Wiki for Mathematical Knowledge Management

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