Motivation State of the Art MathUI Knowledge Representation Services Conclusion

Editing and Using Mathematics in a Semantic WikiMeeting on Mathematical Notations 2007

Christoph Lange

Jacobs University, Bremen, Germany(formerly International University Bremen)

KWARC – Knowledge Adaptation and Reasoning for ContentThis work was supported by JEM-Thematic-Network ECP-038208.

October 23, 2007

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Motivation State of the Art MathUI Knowledge Representation Services Conclusion

Abstract

SWiM is a semantic wiki for collaboratively . . .buildingeditingbrowsing

. . . mathematical knowledge represented in OMDoc [Kohlhase06].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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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 itemsScientific services currently not availableE. g. a search for:

\sqrt{a^2 + b^2} = c, or: x^2 + y^2 = z^2All theorems about triangles for which a proofexists

Neither system relates mathematical symbols totheir (formal) definitions or allows for formallystructured theories.

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Motivation State of the Art MathUI Knowledge Representation Services Conclusion

Semantic Wikis and Semantic Markup

Semantic wikis understand the knowledge graph they containusually: 1 page =̂ 1 real-world conceptpages and links typed with terms from ontologies

Structural semantic markup: In math, there are OMDoc-basedservices for

learning assistance (→ ActiveMath [Melis+06])semantic search (→ MathWebSearch [KohlhaseŞucan07])publishingtheory managementproof verification

The challenge of joining both: Motivating the authors to structuretheir knowledge!

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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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

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Motivation State of the Art MathUI Knowledge Representation Services Conclusion

Demo (A mathematical document)

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Motivation State of the Art MathUI Knowledge Representation Services Conclusion

Demo (Editing formulae)

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Motivation State of the Art MathUI Knowledge Representation Services Conclusion

Demo (Editing notation definitions)

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Motivation State of the Art MathUI Knowledge Representation Services Conclusion

Demo (An OpenMath content dictionary; old SWiM)

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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 documentSimple link syntax: theory expands to theory.omdoc#theory.Symbols and their notation defined in the wiki itself – requires flexiblepresentation mechanism

Working on integration of mmlproc (MathML 3) [Müller07]http://kwarc.info/projects/mmlkit/mmlproc/

Exploit structural knowledge: navigate through RDF knowledge graphextracted from markup

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Motivation State of the Art MathUI Knowledge Representation Services Conclusion

Extracting Knowledge from OMDoc using an Ontology

Semantic web tools need explicit knowledge representationExtract RDF from OMDoc markup, using document ontology asvocabularyThree layers of knowledge: objects, statements, theoriesFormally modeled in OWL-DL (description logic)

ExampleA wiki page:<omdoc>

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

</proof></omdoc>

Extracted RDF triples (=̂ graph):

pyth-proof pythagoras

Proof Theoremtype type

proves

proves

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

(omdoc:* → OMDoc document ontology)

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Motivation State of the Art MathUI Knowledge Representation Services Conclusion

Services for Science

Case studies in science and education plannedHypothesis: other sciences structure knowledge similarly tomathematicsNeed to support tools scientists use, e. g. MathematicaIntegrate available tools, e. g. the MathWebSearch engineConnecting to the semantic web: allow external services to operate onthe knowledge from the wiki

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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 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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