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# Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration

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Christoph Lange's Ph.D. defense (2011-03-11)

Christoph Lange's Ph.D. defense (2011-03-11)

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## Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web IntegrationPresentation Transcript

• Overview Service Integration Knowledge Representation Conclusion & Future Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration Christoph Lange Jacobs University, Bremen, Germany KWARC – Knowledge Adaptation and Reasoning for Content 2011-03-11 Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 1
• Overview Service Integration Knowledge Representation Conclusion & FutureWhy Mathematics? Mathematics ubiquitous foundation of science, technology, and engineering these have in common: rigorous style of argumentation symbolic formula language similar process of understanding results Mathematical Knowledge complex structures . . . that have been well studied and understood Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 2
• Overviewtime. Note factoring a quartic into two real quadratics is different than sign each Service Integration Knowledge Representation Conclusion & Future trying to ﬁnd four complex roots. Deﬁnition: A function f is analytic on an open subset R ⊂ C if f is complexSemiformal Mathematical Knowledge differentiable everywhere on R; f is entire if it is analytic on all of C.2 Proof of the Fundamental Theorem via Liouville Theorem 2.1 (Liouville). If f (z) is analytic and bounded in the complex plane, then f (z) is constant.Informal We now prove Theorem 2.2 (Fundamental Theorem of Algebra). Let p(z) be a polynomial Formalized = Computerized with complex coefﬁcients of degree n. Then p(z) has n roots. Proof. It is sufﬁcient to show any p(z) has one root, for by division we can then write p(z) = (z − z0 )g(z), with g of lower degree. Note that if p(z) = an z n + an−1 z n−1 + · · · + a0 , (2) then as |z| → ∞, |p(z)| → ∞. This follows as an−1 a0 p(z) = z n · an + + ··· + n . (3) z z 1 Assume p(z) is non-zero everywhere. Then p(z) is bounded when |z| ≥ R. 1 1 Also, p(z) = 0, so p(z) is bounded for |z| ≤ R by continuity. Thus, p(z) is a bounded, entire function, which must be constant. Thus, p(z) is constant, a contradiction which implies p(z) must have a zero (our assumption). [Lev] Semiformal – a pragmatic and practical compromise 2 anything informal that is intended to or could in principle be formalized combinations of informal and formal for both human and machine audience Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 3
• Overview Service Integration Knowledge Representation Conclusion & FutureCollaboration in Mathematics History of collaboration in the small: Hardy/Littlewood in the large: hundreds of mathematicians classifying the finite simple groups “industrialization” of research Utilizing the Social Web research blogs: Baez, Gowers, Tao Polymath: collaborative proofs Collaboration = creation, formalization, organization, understanding, reuse, application Polymath wiki/blog: P ≠ NP proof Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 4
• Overview Service Integration Knowledge Representation Conclusion & FutureAn Integrated View on a Collaboration WorkflowThe author(s): The reader(s): The reviewer(s): 0 original idea (in one’s “What does that 1 read paper (← ) mind) mean?”: missing 2 verify claims background, 1 formalize into used to different 3 point out problems structured document notation with the paper and 2 search existing its formal concepts “How does that knowledge to build work?” on “What is that good 3 validate formal for?” structure look up background 4 present in a information in cited comprehensible way publications 5 submit for review Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 5
• Overview Service Integration Knowledge Representation Conclusion & FutureLooking up Background Knowledge “What does that mean?” Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 6
• Overview Service Integration Knowledge Representation Conclusion & FutureAdapting the Presentation to FamiliarTerminology “What does that mean?” – here: unfamiliar unit system (imperial vs. metric) Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 7
• Overview Service Integration Knowledge Representation Conclusion & FuturePointing out and Discussing Problems Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 9
• Overview Service Integration Knowledge Representation Conclusion & FutureCollaboration Still has to be Enabled! Many collaboration tasks not currently well supported by machines For other tasks there is (limited) support creating and formalizing documents – semiformal!? search existing knowledge to build on – semiformal!? computation (recall unit conversion) – but not inside documents publishing in textbook style – could it be more comprehensible? adapting notation (e.g. ⋅ ×, n k Cn ) – not quite on demand k Existing machine services only focus on primitive tasks Can’t simply be put together, as they . . . . . . speak different languages . . . take different perspectives on knowledge Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 10
• Overview Service Integration Knowledge Representation Conclusion & FutureDocument Perspective: XML Markup XHTML+MathML(+OpenMath) ... is [itex] <mn>9144</mn> <mo>&InvisibleTimes;</mo> <mo>m</mo> [/itex] from city ... Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 11
• Overview Service Integration Knowledge Representation Conclusion & FutureHow to Enable Collaboration? Integrate a wide range of different services As they currently speak different languages, . . . first create a unified interoperability layer for knowledge representations (document vs. network perspective) then translate between different representations Tool: semantic web technology Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 13
• Overview Service Integration Knowledge Representation Conclusion & FutureContribution Building a collaboration environment is not trivial Collection of foundational, enabling technologies OMDoc+RDF(a), a unified interoperability layer for representing semiformal mathematical knowledge (document and network perspective) Design patterns for integrating services interactive assistance in published documents translations inside knowledge bases Evaluation of how effectively an integrated environment built that way (a semantic wiki for mathematics) supports practical workflows Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 14
• Overview Service Integration Knowledge Representation Conclusion & FutureSWiM, an Integrated Collaboration Environment Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 15
• Overview Service Integration Knowledge Representation Conclusion & FutureSWiM, an Integrated Collaboration Environment Semantic wiki, combining knowledge production and consumptionEditor for documents, Graph-based Localized discussionformulæ, metadata navigation forums Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 16
• Overview Service Integration Knowledge Representation Conclusion & FutureUsability Evaluation of the SWiM Prototype Integration is feasible, but is the result usable? learnable? effective? useful? satisfying to use? Can we effectively support maintenance workflows (on the OpenMath CDs)? Quick local fixing of minor errors (in text, formalization, or presentation) Peer review, and preparing major revisions by discussion In general: What particular challenges to usability does the integration of heterogenenous services entail? Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 17
• Overview Service Integration Knowledge Representation Conclusion & FutureFeedback Statements from Test Users positive statement successful 93 action 95 understood concept 36 18 not understood concept 18 unexpected bug negative 61 statement 43 dissatisfaction 52 44 51 confusion/uncertainty not understood expectation what to do not met Understanding only seems marginal, but had a high impact on successfully accomplishing tasks! Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 18
• Overview Service Integration Knowledge Representation Conclusion & FutureInterpretation and Consequences Usability hypotheses largely hold, but: Users with previous knowledge of related knowledge models or UIs had advantages Less experienced users frequently taken in by misconceptions; requested better explanations Users expected a more coherent integration User interfaces need Semantic Transparency (for learnability): self-explaining user interfaces familiar and consistent terminology (despite XML/RDF heterogeneity under the hood!) The SWiM user interface is not yet self-explaining Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 19
• Overview Service Integration Knowledge Representation Conclusion & FutureSelf-explaining Publicationsand Assistive Services Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 20
• Overview Service Integration Knowledge Representation Conclusion & FutureStructures of Mathematical Knowledge (MK) Goal: design unified interoperability layer for all relevant aspects of MK Different degrees of formality: informal, formalized, semiformal Classification of structural dimensions: logical/functional: symbols, objects, statements, theories rhetorical/document: from chapters down to phrases presentation: e.g. notation of symbols metadata: general administrative ones; applications/projects/people discussions about MK (e.g. about problems) Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 21
• Overview Service Integration Knowledge Representation Conclusion & FutureRequirements for Representing MK Structure Coverage Formal. Linking Comprh. satisfies Requirementa S.L.* S.R S.N S.M S.D F.R F.C L.A L.→ L.← C.S C.H O S T MathML 3 ++ – – – – – – ++ + + + + – + OpenMath 2 Objects ++ – – – – – – + – + – OpenMath 2 CDs ++b – – – – – – – OMDoc 1.2/S EX T ++b ++ + ++ + – ++ + – + – – MathLang ++ ++ – – – – – ++ + – – – DocBook 5 ++b – – – – +d – – – – + + – – TEI P5 ++c – – – – – ++ + + – + – – DITA 1.1 ++c – – – – +d – – – + + + – – EPUB 2.0.1/DTBook 3 ++c – – – – + – – – – – + – – CNXML 0.7/CollXML/mdml ++b + – – – – – – – – + – – Formalized languages ++ ++ – + – – – – – + – RDF(a) 1.1 (depends on vocabulary) + ++ ++ ++ + OMDoc 1.3/1.6 ++b ++ ++ ++ ++ ++e – ++ + ++e ++e ++e e +e Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 22
• Overview Service Integration Knowledge Representation Conclusion & FutureOMDoc+RDF(a) as an Interoperability Layer forExchanging and Reusing MK 1 Translate OMDoc to RDF formalize conceptual model as an ontology reused existing ontologies for rhetorics, metadata, etc. specified an XML→RDF translation for identifiers and structures 2 Embed RDFa into OMDoc extend OMDoc beyond mathematics embed arbitrary metadata into mathematical documents Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 23
• Overview Service Integration Knowledge Representation Conclusion & FutureCreating an RDF Resource from an XML Node<theory name="group"> <http://ma.th/group> <symbol name="op"> rdf:type omdoc:Theory ; Theory symbol <type> omdoc:homeTheoryOf M×M→M <http://ma.th/group#symbol> . rdf:type </type> <http://ma.th/group#symbol> rdf:type homeTheoryOf </symbol> rdf:type omdoc:Symbol ; . . . /group . . . /group#op</theory> omdoc:declaredType ... . Algorithm: Require: b, p, u, T , P ∈ U, n is an XML node, T is the URI of an ontology class or empty, P is the URI of an ontology property or empty Ensure: R ∈ U × U × (U ∪ L) is an RDF graph R← if u = ε then {if no explicit URI is defined by the rule, . . . } u ← mint(b, n) {. . . try to mint one, using built-in or custom minting functions (configurable per extraction module)} end if if u ≠ ε then {if we got a URI, . . . } if T ≠ ε then R ← R ∪ { u, rdf type, T } {make this resource an instance of the given class} end if if P ≠ ε then R ← R ∪ add_uri_property( , p, P, u) {create a link (e.g. of a type like hasPart) from the parent subject to this resource} end if for all c ∈ π NS (\$n ∗ \$n @∗) do {from each element and attribute child node (determined using an XPath evaluation function returning a nodeset) . . . } R ← R ∪ extract(b, c, u) {. . . recursively extract RDF, using the newly created resource as a parent subject} end for{i.e. the recursion terminates for nodes without children} end if return R Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 24
• Overview Service Integration Knowledge Representation Conclusion & FutureThe OMDoc Ontology (simplified) dependsOn, MathKnowledgeItem hasPart, subClassOf verbalizes Type other properties Theory Statement homeTheory imports From e imports, NonConstitutive p hasTy Import metaTheory Statement Notation Constitutive Definition Statement Proof Example Assertion proves Symbol hasDefinition Axiom exemplifies Definition bol Sym ers rend Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 25
• Overview Service Integration Knowledge Representation Conclusion & FutureMulti-step Dependencies Logical well-formedness: o hasDefinition ○ o usesSymbol o hasOccurrenceOfInDefinition o wellFormedNessDependsOn o dependsOn Validity of a proof: o hasStep ○ o stepJustifiedBy o validityDependsOn o dependsOn Dependency of published documents on notation definitions: o usesSymbol ○ o hasNotationDefinition o possiblyUsesNotationDefinition o presentationDependsOn o dependsOn . . . and their transitive closures Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 26
• Knowledge Representation Service IntegrationConclusion and Future Work Contribution of this thesis: building blocks for managing mathematical knowledge knowledge representation interoperability layer collaborative services methods and techniques for integrating them Planetary: e-Math on the Web supporting scientists in collaboratively gaining new knowledge contributing legacy MK collections to the Web of Data Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 27
• Knowledge Representation Service IntegrationSelf-explanation with System Ontologies System ontologies in Planetary: structural ontologies, workflow ontologies, argumentation ontology Customizable in the environment (= mathematical documents) “The ontology is the API” Self-explaining user interface via ontology documentation Theorem …… Example hasDiscussion ` SIOC forum1 definition argumentation subClassOf (IkeWiki ontology) module (partly shown) Position agrees_with/ agrees_with/ exemplifies disagrees_with disagrees_with post1: Issue Domain-specific Math. Know- (UnclearWh.Useful) example argumentation ledge Item classes (partly shown) supported_by has_reply elaborates_on subClassOf post2: Elaboration OMDoc ontology Ontology has_container agrees_with Decision Entity post3: Position resolvesInto decides decides proposes_ solution_for knowledge resolves_into post4: Idea items (ProvideExample) (OMDoc ontology) Issue Idea proposes_solution_for supports on wiki pages subClassOf subClassOf decides post5: Evaluation Wrong Inappropriate Incomprehensible Provide Keep as Delete agrees_with for Domain Example Bad Example post6: Position post7: Decision supported_by argumentative physical structure structure (SIOC Core) discussion page (SIOC Arg.) Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 28
• Knowledge Representation Service IntegrationStructural Coverage of RDF Vocabularies Structures Logical/functional Rhet. Notation Metadata Discussion Objects Stmts. Theories N3 Vocabularies + – – – – – OpenMath CD – – – HELM + + – – – – MoWGLI + ++ – – + – MathLang DRa – + – – – – – PML – ++a – – – – – SALT – – – ++ – – + OntoReST – – – ++ – – – DILIGENT – – – – – – + DCMI Terms – – – – – ++ – OMDocb ++ + –c + –d – OpenMath CDe – + – SIOC Argumentationf – – – – – – ++ a proofs only b contribution of this thesis c intentionally delegated to SALT d intentionally delegated to DCMI Terms, ccREL, the OpenMath CD ontology, and other vocabularies e contribution of this thesis: a modernized ontology, which I have developed Christoph the purpose of maintaining OpenMath CDs for Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 29
• Knowledge Representation Service IntegrationTranslate OMDoc to RDF formalized OMDoc’s conceptual model as an ontology abstracted from XML schema, generalized (e.g. dependencies) comprehensible for services (via RDF semantics) annotation vocabulary for XHTML+RDFa published from OMDoc reused existing ontologies for rhetorics, metadata, etc. specified an XML→RDF translation for: identifiers of structural concepts (peculiarities of URI formats) the structures and relations themselves Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 30
• Knowledge Representation Service IntegrationEmbed RDFa into OMDoc extend OMDoc beyond mathematics: coherently express all mathematical and related knowledge in the same language embed arbitrary metadata into mathematical documents link mathematical documents to related external resources Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 31
• Knowledge Representation Service IntegrationWrite Expressive RDF Vocabularies in OMDoc Implementation and alignment of this structural ontology require: selectively use more expressivity “just” OWL DL does not capture all concepts of interest metadata inheritance, applicability of problem/solution types to primary knowledge, etc. require first- or second-order logic OMDoc supports heterogeneous formalization! comprehensive and comprehensible documentation for developers and end users reuse existing ontologies, or adapt and integrate them (modularity!) Result: useful for our ontologies and metadata vocabularies, . . . . . . but also for other ontologies (reimplemented FOAF) existing MKM services become available for ontology engineering Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 32
• Knowledge Representation Service IntegrationMKM Services for Ontology Engineering Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 33
• Knowledge Representation Service IntegrationConcrete Workflows and Usage Scenarios Scenarios I studied: In the OpenMath Content Dictionaries: Quickly Fixing Minor Errors Fixing and Verifying Notations Peer Review and Preparing Major Revisions by Discussion In Michael Kohlhase’s computer science lecture notes: Serving Information Needs of Learners and Instructors In a collection of software engineering documents (contracts, requirements, manuals), and in the Flyspeck collaborative proof formalization effort: Managing a Project Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 34
• Knowledge Representation Service IntegrationPrimitive Services that we Need accom- Creating Formal./ Under- Reusing Applying plishes Organizing standing Editing Validating Publishing Inf. Retr. Arguing quickly fixing minor errors fixing and verifying notations peer review and preparing major revisions by discussion serving information needs of learners and instructors managing a project Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 35
• Knowledge Representation Service IntegrationA Unified Visual Editing Component An existing presentation markup (HTML) editor, extended into a versatile reusable editing component for logical and document structures, formulæ, symbol notation definitions, metadata Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 36