Aspects of Mathematical Knowledge on the SemanticWeb

Summary of my work at DERI

Christoph Lange

Jacobs University, Bremen, GermanyKWARC – Knowledge Adaptation and Reasoning for Content

October 2, 2008

Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 1

Introduction

Ph.D. student, 2 of 3 years (+ n months) doneAt home in a group doing “Mathematical Knowledge Management”

dealing with mathematical knowledge, formality ranges fromhuman-friendly to computer-verifiablemy project: collaboration on semiformal knowledge (using a semanticwiki)

What I wanted to learn about the semantic web:engineering ontologies for scientific documentsuser interfaces for annotating and browsingrelation of social interaction to knowledge. . . and much more

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What I had

SWiM, a semantic wiki for mathematicsDocuments in OpenMath and OMDoc (think: “Semantic MathMLand more”)Practical approach:

1 analyse use cases (here: lightweight ontology engineering, collaborativeproof formalisation and documentation, lecture note authoring)

2 figure out conceptual model3 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 ofcollaboratively structuring mathematical knowledge, what additionalknowledge can be inferred from users’ contributions, and how can thisagain be utilised in order to improve collaboration?”

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Ontologies for Scientific Documents

Initial problem:I had a basic ontology that modelled structures of mathematicalknowledge; mainly statements (definition, theorem, proof, examples).Semi-formal knowledge often comes in documents that also containtextThere is a document structure (chapter, section, cross-reference), anda rhetorical structure, both of which can be independent from themathematical structure.

Collaboration with Tudor Groza: getting the model right (do it asSALT does with document ontology ↔ annotation ontology ↔rhetorical ontology)Expansion of the ontology, and rules for extracting these conceptsfrom OMDoc documents to RDF, supported by intern GordanRistovski

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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 easyselection of types of mathematical knowledge, fromphrase to theory level; deleting annotations;auto-completion of link targets (prepared)

Jana Giceva improved and extended syntax for annotating SALT-likerhetorical structures in OMDoc (i. e. using the SALTontology within the host language OMDoc, not LATEX)first ideas for a editing interface; visualisation ofrhetorical relations and blocks implemented (→ activedocuments)

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Argumentation about Mathematical Knowledge

Initial motivation: structured wiki discussions, well-defined workflowfor solving problems with knowledge in a wikiIn my case, a wiki page is an item of mathematical knowledge, e. g. atheoremissues discussed will be quite specific: e. g. “This theorem is hard tounderstand” (or wrong, or inadequately presented, . . . )Note: there is also argumentation within artifacts of scientificknowledge, but I’m more interested in argumentation about them

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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 wikipage, 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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Domain-Specific Argumentation

Assumptions:Possible problems depend on the type of knowledge itemPossible solutions depend on the type of knowledge item and the typeof problemStandard problems have standard solutions, with which software canassist

Survey (still running, tinyurl.com/5qdetd)Common issues: wrong, incomprehensible, uncommon style,underspecified, redundant, truth uncertainCommon solutions: directly improve affected knowledge item, split itWhen issues remain unresolved, it’s mostly due to insufficientrestructuring support

Ideas contributed by Tuukka Hastrup (how to scale this to the web;SIOC = Semantically Interlinked Online Communities) and StéphaneCorlosquet (making it work in the CMS Drupal)

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Domain-Specific Argumentation (Example)

discussion page

knowledgeitems

(OMDoc ontology)on wiki pages

theoremforum1

examplepost1: Issue

(Incomprehensible)

post6: Decision

post3: Agree

post2: Idea(ProvideExample)

post4: Disagree

post5: Agree

exemplifies

hasDiscussion(IkeWiki ontology)

has_container

has_replyresponseTo

resolvesInto

positionOn

onIdea

withPositionsonIssue

physical structure(SIOC)

argumentativestructure

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General Argumentation on Social Media Sites

Developing an argumentation module for SIOCJoint work with Uldis Bojars and Tudor GrozaUse cases, model, guidelines for usage; implementation to be done

Issue Idea Elaboration

Statement

Position

subClassOf

proposes_solution_for elaborates_on

Argument

Example Evaluation Justification

subClassOf

supports/challenges

agrees_with/disagrees_with/neutral_towards

Decision

decides

supported_by

arises_from

refers_to

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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 toannotate!)argumentation

small group of knowledge engineers (concessions required: integrationwith legacy Subversion repository)specialised editors: structured definitions, formulas, metadataevaluation needed

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Integrating OMDoc with the Semantic Web (Future)

Initial motivation: extensible metadata by importing vocabularies (sofar: hard-coded Dublin Core support)Model dependencies in a mathematical way: not just reference bywriting down a URI, but model RDF(S), OWL, and then semanticweb ontologies as mathematical theoriesFor semantic-web-like reasoning, export RDF(S), OWL from OMDocYet 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 insteadof rdfs:labels of type rdf:XMLLiteral)RDFa? Yes, but as an alternative syntax, not as a replacement. Wecan generate XHTML from OMDoc, and extract RDF from OMDoc(except formulas), so combine it. And provide a way back.

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Summary

What I hope(d) to learn here – to use it for mathematical knowledgemanagement:

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 ofsemantic wikis (and the semantic web)a case study of an ontology for a complex document format, withdocument structure, mathematical structure, and rhetorical structuredomain-specific argumentation in a semantic wikia more mathematical approach to ontology engineering

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