Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration

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


Why 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


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 complex

Semiformal 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


Collaboration 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


An Integrated View on a Collaboration Workflow

The 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


Looking up Background Knowledge

“What does that mean?”



Adapting the Presentation to Familiar
Terminology

“What does that mean?” – here: unfamiliar unit system (imperial vs.
metric)



Looking up Related Information
“What can I reuse — what is that good for — where/how is it applied?”
Sussex St.
Reading Andrews NDL
Audio- Lists Resource subjects t4gm
MySpace scrobbler Lists
Moseley (DBTune) (DBTune) RAMEAU
Folk NTU SH lobid
GTAA Plymouth Resource
Lists
Organi-
Reading
Lists
sations
Music The Open ECS
Magna- Brainz Music
DB tune Library LCSH South-
(Data Brainz LIBRIS ampton
Tropes lobid Ulm
Incubator) (zitgist) Man- EPrints
Resources
chester
Surge Reading
biz. Music RISKS
Radio Lists The Open ECS
data. John Brainz
Discogs Library PSH Gem. UB South-
gov.uk Peel (DBTune)
FanHubz (Data In- (Talis) Norm- Mann- ampton
(DB cubator) Jamendo datei heim RESEX
Tune)
Popula- Poké- DEPLOY
Last.fm
tion (En- pédia
Artists Last.FM Linked RDF
AKTing) research EUTC (DBTune) (rdfize) LCCN VIAF Book Wiki
data.gov Produc- Pisa Eurécom
P20 Mashup semantic
NHS .uk tions classical web.org
(EnAKTing) Pokedex
(DB
Mortality Tune) PBAC ECS
(En-
AKTing)
BBC MARC (RKB Budapest
Program Codes Explorer)
Energy education OpenEI BBC List Semantic Lotico Revyu OAI
(En- CO2 data.gov mes Music Crunch SW
AKTing) (En- .uk Chronic- Linked Dog
NSZL Base
AKTing) ling Event- MDB RDF Food IRIT
America Media Catalog
ohloh
BBC DBLP ACM IBM
Good- BibBase
Ord- Wildlife (RKB
Openly Recht- win
nance Finder Explorer)
Local spraak. Family DBLP
legislation Survey Tele- New VIVO UF
.gov.uk nl graphis York flickr (L3S) New-
VIVO castle
Times URI wrappr OpenCal Indiana RAE2001
UK Post- Burner ais DBLP
codes statistics (FU
VIVO CiteSeer Roma
data.gov LOIUS Taxon iServe Berlin) IEEE
.uk Cornell
Concept Geo
World data
ESD Fact- OS dcs
Names book dotAC
stan- reference Project
Linked Data NASA (FUB) Freebase
dards data.gov Guten-
.uk
for Intervals (Data GESIS Course-
transport DBpedia berg STW ePrints CORDIS
Incu- ware
data.gov bator) (FUB)
Fishes ERA UN/
.uk
of Texas Geo LOCODE
Uberblic
Euro- Species
The stat dbpedia TCM SIDER Pub KISTI
(FUB) lite Gene STITCH Chem JISC
London Geo KEGG
DIT LAAS
Gazette TWC LOGD Linked Daily OBO Drug
Eurostat Data UMBEL lingvoj Med
(es) Disea-
YAGO Medi some
Care ChEBI KEGG NSF
Linked KEGG KEGG
Linked Drug Cpd
GovTrack rdfabout Glycan
Sensor Data CT Bank Pathway
US SEC Open Reactome
(Kno.e.sis) riese Uni
Cyc Lexvo Path-
totl.net way Pfam PDB
Semantic HGNC
XBRL
WordNet KEGG KEGG
(VUA) Linked Taxo- CAS Reaction
rdfabout Twarql UniProt Enzyme
EUNIS Open nomy
US Census Numbers PRO- ProDom
SITE Chem2
UniRef Bio2RDF
Climbing WordNet SGD Homolo
Linked (W3C) Affy- Gene
Cornetto
GeoData metrix PubMed Gene
UniParc
Ontology
GeneID
Airports
Product
DB UniSTS MGI
Gen
Bank OMIM InterPro
As of September 2010

e-science data – with opaque mathematical models
statistical datasets – without mathematical derivation rules
publication databases – without mathematical content


Pointing out and Discussing Problems



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 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


Document Perspective: XML Markup

XHTML+MathML(+OpenMath)
... is <math>
<mn>9144</mn>
<mo>⁢</mo>
<mo>m</mo>
</math> from city ...



Network Perspective: RDF Graphs

Look up Related Information: Point out and Discuss Problems:
Sussex St.
hasDiscussion `
forum1 definition
Reading Andrews NDL
Audio- Lists Resource subjects

(IkeWiki ontology)
t4gm
MySpace scrobbler Lists
Moseley (DBTune) (DBTune) RAMEAU
Folk NTU SH lobid
GTAA Plymouth Resource
Lists
Organi-
Reading
sations

exemplifies
Lists
Music The Open ECS
Magna- Brainz Music
DB Library LCSH South-

post1: Issue
tune (Data Brainz LIBRIS
lobid ampton Ulm
Tropes Incubator) (zitgist) Man- Resources EPrints
chester
Surge Reading

(UnclearWh.Useful)
biz. Music RISKS
Radio Lists The Open ECS
data. Brainz

example
John Discogs Library PSH Gem. UB South-
gov.uk Peel (DBTune)
FanHubz (Data In- (Talis) Norm- Mann- ampton
(DB

has_reply
cubator) Jamendo datei heim RESEX

elaborates_on
Tune)
Popula- Poké- DEPLOY
Last.fm
tion (En- pédia
Artists Last.FM Linked RDF
AKTing) research EUTC (DBTune) (rdfize) LCCN VIAF Book Wiki
data.gov Produc- Pisa Eurécom
P20 Mashup semantic
NHS

post2: Elaboration
.uk tions classical
Pokedex web.org
(EnAKTing) (DB
Mortality Tune) PBAC ECS
(En-
AKTing)
BBC MARC (RKB Budapest
Codes

has_container
Program Explorer)
OpenEI BBC

agrees_with
Energy education Semantic Lotico Revyu OAI
List
(En- CO2 data.gov mes Music Crunch SW
AKTing) (En- .uk Chronic- Linked Dog
NSZL Base
AKTing) ling Event- MDB RDF Food IRIT
Catalog

resolvesInto
America Media ohloh
BBC DBLP ACM

post3: Position
Good- BibBase IBM
Ord- Wildlife (RKB
Openly Recht- win
nance Finder Explorer)
Local spraak. Family DBLP
legislation Survey Tele- New

proposes_
VIVO UF
.gov.uk nl graphis York flickr (L3S) New-
VIVO castle
Times URI wrappr OpenCal Indiana RAE2001
UK Post- Burner ais DBLP
codes statistics
data.gov
.uk
LOIUS Taxon
Concept Geo
World
iServe
VIVO
Cornell
(FU
Berlin)
data
IEEE
CiteSeer Roma
solution_for knowledge
post4: Idea items
ESD Fact- OS dcs
Names book dotAC
stan- reference Project
Linked Data NASA (FUB) Freebase
dards data.gov Guten-
.uk
for Intervals (Data GESIS Course-

(ProvideExample)
STW CORDIS

(OMDoc ontology)
transport Incu- DBpedia berg ePrints
(FUB) ware
data.gov bator) Fishes ERA UN/
.uk
of Texas Geo LOCODE
Uberblic

on wiki pages
Euro- Species

supports
The stat dbpedia TCM SIDER Pub KISTI
(FUB) lite Gene STITCH Chem JISC

decides
London Geo KEGG
DIT LAAS
Gazette TWC LOGD Linked Daily OBO Drug
Eurostat Data UMBEL lingvoj Med
Disea-

post5: Evaluation
(es)
YAGO Medi some
Care ChEBI KEGG NSF
Linked KEGG KEGG
Linked Drug Cpd
GovTrack rdfabout Glycan
Sensor Data CT Bank Pathway
Open

agrees_with
US SEC riese Reactome
(Kno.e.sis) Cyc
Uni
Lexvo Path-
totl.net way Pfam PDB
Semantic HGNC
XBRL
KEGG

post6: Position
WordNet KEGG
(VUA) Linked Taxo- CAS Reaction
rdfabout Twarql UniProt Enzyme
EUNIS Open nomy
US Census Numbers PRO- ProDom
SITE Chem2
UniRef Bio2RDF
Climbing WordNet SGD Homolo
Linked (W3C) Affy- Gene
Cornetto
GeoData metrix PubMed Gene

post7: Decision
UniParc
Ontology
GeneID
Airports
Product
DB UniSTS
Gen
MGI supported_by
Bank OMIM InterPro

argumentative
As of September 2010

physical structure structure
(SIOC Core) discussion page (SIOC Arg.)



How 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



Contribution

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



SWiM, an Integrated Collaboration Environment



SWiM, an Integrated Collaboration Environment

Semantic wiki, combining knowledge production and consumption

Editor for documents, Graph-based Localized discussion
formulæ, metadata navigation forums



Usability 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?



Feedback 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!


Interpretation 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



Self-explaining Publications
and Assistive Services



Structures 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)



Requirements 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


OMDoc+RDF(a) as an Interoperability Layer for
Exchanging 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



Creating 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


The 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


Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration

Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration

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