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
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Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration
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Overview Service Integration Knowledge Representation Conclusion & Future
signeachtime. Note factoringa quarticinto two realquadraticsis differentthantrying to find four complexroots.Definition: A function f is analytic on an opensubsetR ⊂ C if f is complexdifferentiableeverywhereonR; f is entire if it is analyticonall of C.
2 Proof of the FundamentalTheoremvia Liouville
Theorem 2.1 (Liouville). If f(z) is analyticandboundedin thecomplex plane,thenf(z) is constant.
Wenow prove
Theorem 2.2 (Fundamental Theorem of Algebra). Let p(z) be a polynomialwith complex coefficientsof degreen. Thenp(z) hasn roots.
Proof. It is sufficient to show any p(z) hasoneroot, for by division we canthenwrite p(z) = (z − z0)g(z), with g of lowerdegree.
Notethatif
p(z) = anzn + an−1z
n−1 + · · ·+ a0, (2)
thenas|z| → ∞, |p(z)| → ∞. This followsas
p(z) = zn ·∣∣∣an +
an−1
z+ · · ·+ a0
zn
∣∣∣ . (3)
Assumep(z) is non-zeroeverywhere.Then 1p(z)
is boundedwhen |z| ≥ R.
Also, p(z) 6= 0, so 1p(z)
is boundedfor |z| ≤ R by continuity. Thus, 1p(z)
isa bounded,entire function, which must be constant. Thus, p(z) is constant,acontradictionwhich impliesp(z) musthave azero(ourassumption).
[Lev]
2
Formalized = Computerized
Semiformal – a pragmatic and practical compromiseanything informal that is intended to or could in principle beformalizedcombinations of informal and formal for both human andmachine audience
Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 3
Overview Service Integration Knowledge Representation Conclusion & Future
Looking up Related Information“What can I reuse—what is that good for —where/how is it applied?”
As of September 2010
MusicBrainz
(zitgist)
P20
YAGO
World Fact-book (FUB)
WordNet (W3C)
WordNet(VUA)
VIVO UFVIVO
Indiana
VIVO Cornell
VIAF
URIBurner
Sussex Reading
Lists
Plymouth Reading
Lists
UMBEL
UK Post-codes
legislation.gov.uk
Uberblic
UB Mann-heim
TWC LOGD
Twarql
transportdata.gov
.uk
totl.net
Tele-graphis
TCMGeneDIT
TaxonConcept
The Open Library (Talis)
t4gm
Surge Radio
STW
RAMEAU SH
statisticsdata.gov
.uk
St. Andrews Resource
Lists
ECS South-ampton EPrints
Semantic CrunchBase
semanticweb.org
SemanticXBRL
SWDog Food
rdfabout US SEC
Wiki
UN/LOCODE
Ulm
ECS (RKB
Explorer)
Roma
RISKS
RESEX
RAE2001
Pisa
OS
OAI
NSF
New-castle
LAAS
KISTIJISC
IRIT
IEEE
IBM
Eurécom
ERA
ePrints
dotAC
DEPLOY
DBLP (RKB
Explorer)
Course-ware
CORDIS
CiteSeer
Budapest
ACM
riese
Revyu
researchdata.gov
.uk
referencedata.gov
.uk
Recht-spraak.
nl
RDFohloh
Last.FM (rdfize)
RDF Book
Mashup
PSH
ProductDB
PBAC
Poké-pédia
Ord-nance Survey
Openly Local
The Open Library
OpenCyc
OpenCalais
OpenEI
New York
Times
NTU Resource
Lists
NDL subjects
MARC Codes List
Man-chesterReading
Lists
Lotico
The London Gazette
LOIUS
lobidResources
lobidOrgani-sations
LinkedMDB
LinkedLCCN
LinkedGeoData
LinkedCT
Linked Open
Numbers
lingvoj
LIBRIS
Lexvo
LCSH
DBLP (L3S)
Linked Sensor Data (Kno.e.sis)
Good-win
Family
Jamendo
iServe
NSZL Catalog
GovTrack
GESIS
GeoSpecies
GeoNames
GeoLinkedData(es)
GTAA
STITCHSIDER
Project Guten-berg (FUB)
MediCare
Euro-stat
(FUB)
DrugBank
Disea-some
DBLP (FU
Berlin)
DailyMed
Freebase
flickr wrappr
Fishes of Texas
FanHubz
Event-Media
EUTC Produc-
tions
Eurostat
EUNIS
ESD stan-dards
Popula-tion (En-AKTing)
NHS (EnAKTing)
Mortality (En-
AKTing)Energy
(En-AKTing)
CO2(En-
AKTing)
educationdata.gov
.uk
ECS South-ampton
Gem. Norm-datei
datadcs
MySpace(DBTune)
MusicBrainz
(DBTune)
Magna-tune
John Peel(DB
Tune)
classical(DB
Tune)
Audio-scrobbler (DBTune)
Last.fmArtists
(DBTune)
DBTropes
dbpedia lite
DBpedia
Pokedex
Airports
NASA (Data Incu-bator)
MusicBrainz(Data
Incubator)
Moseley Folk
Discogs(Data In-cubator)
Climbing
Linked Data for Intervals
Cornetto
Chronic-ling
America
Chem2Bio2RDF
biz.data.
gov.uk
UniSTS
UniRef
UniPath-way
UniParc
Taxo-nomy
UniProt
SGD
Reactome
PubMed
PubChem
PRO-SITE
ProDom
Pfam PDB
OMIM
OBO
MGI
KEGG Reaction
KEGG Pathway
KEGG Glycan
KEGG Enzyme
KEGG Drug
KEGG Cpd
InterPro
HomoloGene
HGNC
Gene Ontology
GeneID
GenBank
ChEBI
CAS
Affy-metrix
BibBaseBBC
Wildlife Finder
BBC Program
mesBBC
Music
rdfaboutUS Census
e-science data – with opaque mathematical modelsstatistical datasets – without mathematical derivation rulespublication databases – without mathematical contentChristoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 8
Overview Service Integration Knowledge Representation Conclusion & Future
Collaboration 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 documentspublishing in textbook style – could it bemore comprehensible?adapting notation (e.g. ⋅↝ ×, (nk)↝ C
kn) – not quite on demand
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 & Future
How to Enable Collaboration?
Integrate a wide range of different services
As they currently speak different languages, . . .first create a unified interoperability layer for knowledgerepresentations (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 & Future
Contribution
Building a collaboration environment is not trivial
Collection of foundational, enabling technologiesOMDoc+RDF(a), a unified interoperability layer for representingsemiformal mathematical knowledge (document and networkperspective)Design patterns for integrating services
interactive assistance in published documentstranslations inside knowledge bases
Evaluation of how effectively an integrated environment builtthat way (a semanticwiki for mathematics) supports practicalworkflows
Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 14
Overview Service Integration Knowledge Representation Conclusion & Future
SWiM, an Integrated Collaboration Environment
Developed formaintainingmathematical knowledge collectionsChristoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 15
Overview Service Integration Knowledge Representation Conclusion & Future
Feedback Statements from Test Users
36understoodconcept
93
positivestatement
95
successfulaction
61negativestatement
52
confusion/uncertainty51
expectationnot met
44
not understoodwhat to do
43dissatisfaction
18 unexpected bug18 not understood concept
Understanding only seemsmarginal, but had a high impact onsuccessfully accomplishing tasks!Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 18
Overview Service Integration Knowledge Representation Conclusion & Future
Interpretation and Consequences
Usability hypotheses largely hold, but:
Users with previous knowledge of related knowledge models orUIs had advantagesLess 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 interfacesfamiliar and consistent terminology (despite XML/RDFheterogeneity 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 & Future
Structures of Mathematical Knowledge (MK)
Goal: design unified interoperability layer for all relevant aspects ofMK
Different degrees of formality: informal, formalized, semiformal
Classification of structural dimensions:
logical/functional: symbols, objects, statements, theoriesrhetorical/document: from chapters down to phrasespresentation: e.g. notation of symbolsmetadata: general administrative ones;applications/projects/peoplediscussions 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 & Future
OMDoc+RDF(a) as an Interoperability Layer forExchanging and Reusing MK
1 Translate OMDoc to RDF
formalize conceptual model as an ontologyreused existing ontologies for rhetorics, metadata, etc.specified an XML→RDF translation for identifiers and structures
2 Embed RDFa into OMDoc
extend OMDoc beyond mathematicsembed arbitrary metadata into mathematical documents
Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 23
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 graphR← ∅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 customminting functions (configurable per extraction module)}end ifif u ≠ ε then {if we got a URI, . . . }
if T ≠ ε thenR← R ∪ {⟨u, rdf ∶type, T⟩} {make this resource an instance of the given class}
end ifif 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 iffor all c ∈ πNS($n/ ∗ ∣$n/@∗) do {from each element and attribute child node (determined using an XPath evaluation functionreturning 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 ifreturn R
Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 24
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
a proofs onlyb contribution of this thesisc intentionally delegated to SALTd intentionally delegated to DCMI Terms, ccREL, the OpenMath CD ontology,and other vocabularies
e contribution of this thesis: a modernized ontology, which I have developedfor the purpose of maintaining OpenMath CDs
f contribution of this thesisChristoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 29
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
extend OMDoc beyond mathematics:coherently express all mathematical and related knowledge inthe same languageembed arbitrary metadata into mathematical documentslink mathematical documents to related external resources
Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 31
Write Expressive RDF Vocabularies in OMDocImplementation and alignment of this structural ontology require:
selectively use more expressivity“just” OWL DL does not capture all concepts of interestmetadata inheritance, applicability of problem/solution types toprimary knowledge, etc. require first- or second-order logicOMDoc supports heterogeneous formalization!
comprehensive and comprehensible documentationfor developers and end usersreuse 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 ontologyengineering
Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 32
quickly fixing minor errorsfixing and verifying notationspeer review and preparing major revisions by discussionserving information needs of learners and instructorsmanaging a project
Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 35
An existing presentation markup (HTML) editor, extended into a versatile reusableediting component for logical and document structures, formulæ, symbolnotation definitions, metadataChristoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 36
Knowledge in foreign repositories represented in differentlanguages
Even services operating on the same repository may speakdifferent languages
e.g. semantic XML markup for authoring and publishing . . .. . . and RDF graphs for retrieval and linking
Transparently translate between them!XML to RDFRestricted language (e.g. OWL) to richer language (e.g. OMDoc)Different granularities (e.g. file system vs. fine-grainedknowledge base)
Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 44
SWiM, an Integrated Collaboration EnvironmentUse cases specifically considered:
Quickly Fixing Minor ErrorsFixing and Verifying NotationsPeer Review and Preparing Major Revisions by DiscussionManaging a Project
Features:Client for versioned repositories (legacy content)Utilizing dependencies, e.g. for publishingLocal access to the editorArgumentative discussions
Conclusion:Integrating heterogeneous services is feasibleImprovement over wiki state of the artIncubator for new services and system components
Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 46
Content analysis of discussion posts created by domainexperts
Do the user interface and knowledge model allow for exactassociation of problem reports to knowledge items?Does the knowledge model capture common argumentationprimitives?
Community survey: Are the services useful for the OpenMathcommunity?
Supervised experiments with test users:Are the knowledge model and the user interface learnable?Do they effectively support the three workflows?
Christoph Lange Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration 2011-03-11 47