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 KnowledgeManagement

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

http://kwarc.info

http://www.jacobs-university.de


http://kwarc.info


Abstract

SWiM is a semantic wiki for collaboratively . . .buildingeditingbrowsing

. . . mathematical knowledge represented in OMDoc.Target audience:

mathematicians developing new theoriesscholars learning mathematics

Outline:Current SWiM prototype: user interface and architecturePlans for further development: platform for semantic scientific services


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Collaborative Mathematics on the Web

Current collaborative projects for maths:Wikipedia – wiki encyclopedia covering (almost)anythingPlanetMath – wiki-like system dedicated tomathematicsConnexions – courseware repository and contentmanagement system. . .

Easy to create and link knowledge items⇒ wikis also suitable for non-public researchprojects

Scientific services currently not available


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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 andPlanetMath, which usepresentational LATEXIn principle possible (but notimplemented) in Connexions,thanks to Content MathML

All theorems about triangles forwhich a proof exists

Possible in semantic wikis

Neither system relates mathematicalsymbols to their (formal) definitionsor allows for formally structuredtheories.


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Semantic Wikis

Conventional wikis do not understand the knowledge graph theycontainSemantic wikis address this problem:

usually: 1 page =̂ 1 real-world conceptpages and links typed with terms from ontologies

Examples: Semantic MediaWiki, IkeWiki, . . .


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Structural Semantic Markup

Structural semantic markup: common way to represent scientificknowledgeFor OMDoc, e. g., there are services for

learning assistance (→ ActiveMath)semantic search (→ MathWebSearch)publishingtheory managementproof verification


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OMDoc in a Semantic Wiki – Meeting Users’ Needs

Semantic wikis are appropriate base system for community-authoredknowledge models:

support stepwise formalisation workflowoffer enhanced navigation and search

Who is willing to author OMDoc and OpenMath?Share the benefits of structural semantics with the authorOffer added-value services for the situation the author is inInstant gratification: show up-to-date links to related pages


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The SWiM Prototype

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


http://kwarc.info


http://kwarc.info/projects/swim/


Editing OMDoc

So far, only raw OMDoc XML editing supportedMinimum feedback when XML is invalidAdapting OMDoc to a wiki:Small, reusable pages: allow theory-constitutive statements to appear

in their own documentSimple link syntax: theory expands to theory.omdoc#theory.


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Presenting OMDoc

Adapted OMDoc XSLTs (→ XHTML+MathML), with cachingSymbols in formulae point to their <symbol> declarations.Hyperlinked source code view

XSL transformation

xsl:include

generated fileitalic

transformto MathML

doc.xml

MathML outputdocument

omdoc2pmml.xsl

generic templatesfor OMDoc to MathMLtransformation

omdoc2pmml.xsl


omdoc2pmml.xsl


A.omdoc

and their presentationdefines symbols

compile XSLtemplates

from present−ation elements(expres.xsl)

A−tmpl.xsl

XSL templates forsymbols from A

doc.omdoc

uses symbols fromdocuments A, B, C

collect documentswhose symbols

are used(exincl.xsl)

doc2pmml.xsl

bind doc−incl.xsland omdoc2pmml.xsl

doc−incl.xsl

include −tmpl stylesheets of documentswhose symbols are used

(... and for other output formats)


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Exploiting Knowledge

Semantic wiki: One page = one conceptSWiM: 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 RDFsubject–predicate–object triplesIncoming and outgoing links displayed in a box, grouped by type(→ instant gratification)


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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 anontology (OWL-DL; description logic)Partly done for OMDoc; further steps: extending, generalising

Theory

Concept

Statement

Symbol

Definition

vimports

contains-1 depends on

defined by

lives in

v

v

v


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Extracting Knowledge from OMDoc

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

ExampleA wiki page:

<omdoc><proof id="pyth-proof"for="pythagoras">...

</proof></omdoc>

Extracted RDF triples (=̂ graph):

pyth-proof pythagoras

Proof Theorem

type typeproves

proves

<pyth-proof, rdf:type, omdoc:Proof><pyth-proof, omdoc:proves, pythagoras>

(omdoc:* → OMDoc document ontology)


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Extension Towards Sciences

We are concerned with a technology transfer of MKM applications togeneral scientific knowledge management.OMDoc has already been successfully extended towards physics; othersciences will followNeed an “upper” ontology of scientific markup languages with . . .

objects/statements/theoriescontainment, 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 inthe wiki or on knowledge extracted from it.


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Planned Services

Goal: support the way scientists work and collaborate!Dependency maintenance (lightweight in SWiM, or enhance backendby 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-completionSearch using MathWebSearch in the backend and offering interface inthe UICommunity-aware reading and browsing: lectora case study

Feed lectora with log of users’ interactionsAdapt e. g. presentation to preferences of a user or his communityCase study in education from September 2007


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Conclusion

SWiM supports the basic wiki features for OMDoc content.On top of that base system, several services for science and educationwill be prototyped and evaluated in case studies.

SWiM+

1

¿∫

−∞

ze−12?2

d ?

1.----------------2.----------------

<xml> ...</xml>

ChangeManagement

Publishing

Search

LearningAssistance

pythagoraspartial-diff-eqnproton

Proof for …

Easy Editing


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swim – a semantic wiki for mathematical knowledge management

Technology