Aspects of Mathematical Knowledge on the Semantic
                      Web
                             Summary of my wor...
Introduction


    Ph. D. student, 2 of 3 years (+ n months) done
    At home in a group doing “Mathematical Knowledge Man...
What I had

    SWiM, a semantic wiki for mathematics
    Documents in OpenMath and OMDoc (think: “Semantic MathML
    and...
Ontologies for Scientific Documents

   Initial problem:
          I had a basic ontology that modelled structures of mathe...
User Interfaces for Annotating and Browsing


   Implementation done by interns:
   Gordan Ristovski more and easier annot...
Argumentation about Mathematical Knowledge



   Initial motivation: structured wiki discussions, well-defined workflow
   f...
Resource       Edit      Discussion      History

Issue               Alice              2008–05–30         [Idea][Argumen...
Domain-Specific Argumentation

   Assumptions:
          Possible problems depend on the type of knowledge item
          P...
Domain-Specific Argumentation (Example)



                                                                                ...
General Argumentation on Social Media Sites

   Developing an argumentation module for SIOC
   Joint work with Uldis Boj¯r...
Other things I have done



   OpenMath case study: lightweight mathematical ontology engineering
   (http://wiki.openmath...
Integrating OMDoc with the Semantic Web (Future)

   Initial motivation: extensible metadata by importing vocabularies (so...
Summary

What I hope(d) to learn here – to use it for mathematical knowledge
management:
    engineering ontologies for sc...
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Aspects of Mathematical Knowledge on the Semantic Web

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Aspects of Mathematical Knowledge on the Semantic Web

  1. 1. Aspects of Mathematical Knowledge on the Semantic Web Summary of my work at DERI Christoph Lange Jacobs University, Bremen, Germany KWARC – Knowledge Adaptation and Reasoning for Content October 2, 2008 Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 1
  2. 2. Introduction Ph. D. student, 2 of 3 years (+ n months) done At home in a group doing “Mathematical Knowledge Management” dealing with mathematical knowledge, formality ranges from human-friendly to computer-verifiable my project: collaboration on semiformal knowledge (using a semantic wiki) What I wanted to learn about the semantic web: engineering ontologies for scientific documents user interfaces for annotating and browsing relation of social interaction to knowledge . . . and much more Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 2
  3. 3. What I had SWiM, a semantic wiki for mathematics Documents in OpenMath and OMDoc (think: “Semantic MathML and more”) Practical approach: 1 analyse use cases (here: lightweight ontology engineering, collaborative proof formalisation and documentation, lecture note authoring) 2 figure out conceptual model 3 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 of collaboratively structuring mathematical knowledge, what additional knowledge can be inferred from users’ contributions, and how can this again be utilised in order to improve collaboration?” Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 3
  4. 4. Ontologies for Scientific Documents Initial problem: I had a basic ontology that modelled structures of mathematical knowledge; mainly statements (definition, theorem, proof, examples). Semi-formal knowledge often comes in documents that also contain text There is a document structure (chapter, section, cross-reference), and a rhetorical structure, both of which can be independent from the mathematical structure. Collaboration with Tudor Groza: getting the model right (do it as SALT does with document ontology ↔ annotation ontology ↔ rhetorical ontology) Expansion of the ontology, and rules for extracting these concepts from OMDoc documents to RDF, supported by intern Gordan Ristovski Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 4
  5. 5. 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 easy selection of types of mathematical knowledge, from phrase to theory level; deleting annotations; auto-completion of link targets (prepared) Jana Giceva improved and extended syntax for annotating SALT-like rhetorical structures in OMDoc (i. e. using the SALT ontology within the host language OMDoc, not LTEX) A first ideas for a editing interface; visualisation of rhetorical relations and blocks implemented (→ active documents) Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 5
  6. 6. Argumentation about Mathematical Knowledge Initial motivation: structured wiki discussions, well-defined workflow for solving problems with knowledge in a wiki In my case, a wiki page is an item of mathematical knowledge, e. g. a theorem issues discussed will be quite specific: e. g. “This theorem is hard to understand” (or wrong, or inadequately presented, . . . ) Note: there is also argumentation within artifacts of scientific knowledge, but I’m more interested in argumentation about them Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 6
  7. 7. 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 wiki page, 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) Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 7
  8. 8. Domain-Specific Argumentation Assumptions: Possible problems depend on the type of knowledge item Possible solutions depend on the type of knowledge item and the type of problem Standard problems have standard solutions, with which software can assist Survey (still running, tinyurl.com/5qdetd) Common issues: wrong, incomprehensible, uncommon style, underspecified, redundant, truth uncertain Common solutions: directly improve affected knowledge item, split it When issues remain unresolved, it’s mostly due to insufficient restructuring support Ideas contributed by Tuukka Hastrup (how to scale this to the web; SIOC = Semantically Interlinked Online Communities) and Stéphane Corlosquet (making it work in the CMS Drupal) Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 8
  9. 9. Domain-Specific Argumentation (Example) hasDiscussion forum1 theorem (IkeWiki ontology) has_container exemplifies post1: Issue (Incomprehensible) example responseTo has_reply resolvesInto post2: Idea (ProvideExample) positionOn knowledge post3: Agree items (OMDoc ontology) on wiki pages post4: Disagree onIdea post5: Agree onIssue withPositions post6: Decision physical structure argumentative (SIOC) discussion page structure Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 9
  10. 10. General Argumentation on Social Media Sites Developing an argumentation module for SIOC Joint work with Uldis Boj¯rs and Tudor Groza a Use cases, model, guidelines for usage; implementation to be done refers_to supports/ Statement Argument challenges agrees_with/ subClassOf disagrees_with/ subClassOf neutral_towards arises_from Issue Idea Elaboration Example Evaluation Justification proposes_solution_for elaborates_on decides Position supported_by Decision Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 10
  11. 11. 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 to annotate!) argumentation small group of knowledge engineers (concessions required: integration with legacy Subversion repository) specialised editors: structured definitions, formulas, metadata evaluation needed Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 11
  12. 12. Integrating OMDoc with the Semantic Web (Future) Initial motivation: extensible metadata by importing vocabularies (so far: hard-coded Dublin Core support) Model dependencies in a mathematical way: not just reference by writing down a URI, but model RDF(S), OWL, and then semantic web ontologies as mathematical theories For semantic-web-like reasoning, export RDF(S), OWL from OMDoc Yet 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 instead of rdfs:labels of type rdf:XMLLiteral) RDFa? Yes, but as an alternative syntax, not as a replacement. We can generate XHTML from OMDoc, and extract RDF from OMDoc (except formulas), so combine it. And provide a way back. Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 12
  13. 13. Summary What I hope(d) to learn here – to use it for mathematical knowledge management: 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 of semantic wikis (and the semantic web) a case study of an ontology for a complex document format, with document structure, mathematical structure, and rhetorical structure domain-specific argumentation in a semantic wiki a more mathematical approach to ontology engineering Lange (Jacobs University) Aspects of Mathematical Knowledge on the Semantic Web October 2, 2008 13

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