Uncertainty and
Semantic web
Jennifer Sleeman
Agenda
 Define
uncertainty
 Provide background
 Show areas of research
 Highlight various approaches
 Provide a demonstration of Pronto
Definition - Uncertainty
 Knowledge
can be inaccurate or
incomplete
 Knowledge can be imprecise or “fuzzy”
….leads to uncertainty…
Definition - Uncertainty



Machine-readable information
Applications that work with random information (image
processing, geospatial, information retrieval, etc.)
Ontology concept definitions





Vague concepts:
Tall, Small, Big, ….
Green, Blue, ….
Few, Many, ….
Semantic web services
….work with uncertainty…
Background – Description Logic
Naming Conventions
Taken from Wikipedia [12].
Is representing uncertainty
necessary?
 Tim


Berner-Lee rejection of uncertainty
Not necessary [7]
Scalability issues [7]
Can you describe knowledge using a
“monotonic bivalent language”[7]?
What about grey?
Uncertainty
Is it necessary?
Taken from [5] presented at the URSW 2008.
General Approaches to Uncertainty
and Semantic Web

Incomplete/Distorted knowledge [1]
• Possibility degrees alternatives

Inability to define concepts precisely [1]
• Degree of truth

Conflicting alternatives [1]
• Degree of probability
According to [1], since how we solve uncertainty
problems depends upon the domain, it is hard
to define a single language extension.
Areas of Research
(based upon 2007/2008 URSW Conference agendas)
 Extending
Semantic Web to support
uncertainty
 Fuzzy theory
 Probability theory
 Uncertainty and Ontologies
 Uncertainty and Web Services
Extending the Semantic Web
 Extend
Semantic Web languages to
support probabilistic, possibilistic, and
fuzzy reasoning
 Can be at the ontology layer or the rules
layer
 Within the ontology layer proposals for:


Syntax and Semantics
Logical Formalisms
Fuzzy Theory
“…In classical set theory, the membership of
elements in a set is assessed in binary
terms according to a bivalent condition —
an element either belongs or does not
belong to the set. By contrast, fuzzy set
theory permits the gradual assessment of
the membership of elements in a set; this
is described with the aid of a membership
function valued in the real unit interval
[0, 1]…”[10]
Fuzzy Approaches
 Extending
languages such as OWL with
fuzzy extensions
 Extending Description Logic with fuzzy
extensions
 If a language is extended, one must
provide a way to support reasoning of the
language with the fuzzy extension
Rules and Uncertainty

Rules Interchange Format
 Rules Markup Language

For representing/interchanging rules

Attempt to provide ways to represent various
types of uncertainty [1]
 Not as much recent attention as ontology layer
 fuzzy RuleML defines way to specify
membership degree [1]

Example:
Taken from [1].
Fuzzy RDF

Extends syntax and semantics of RDF
 Triple extended to support real number on the
interval [0,1]


n: s p o [13]
Interpretation


Subject, object has degree of membership to
extension of predicate [13]
Satisfies statement if
• Membership degree of {subject, object} to the extension of
the predicate is >= to n [13]
Fuzzy RDF
 RDFS


extended
“Class extensions are fuzzy sets of domain
elements” [13]
Domains are fuzzy and their assignment to
properties can also be fuzzy [13]
 Inference
engines can be extended to
support such fuzziness
Fuzzy Description Logic
 Fuzzy

One such proposal
 Solve
problem of representing and
reasoning of fuzzy concepts
 With
concrete domains –
reasoning using concrete data types
 With fuzzy version domains are fuzzy
 Modifiers are supported (very, slightly, etc.)
[12]
Fuzzy Description Logic
Non-fuzzy Concrete Domain:
Concrete Fuzzy Domain:
Taken from [12].
Fuzzy Description Logic

Interpretations are fuzzy

From satisfied/unsatisfied to a degree of truth [0,1]

Satisfiability of fuzzy axiom given fuzzy
interpretation [12]
 “Fuzzy axiom a logical consequence of a
knowledge base iff every model in the
knowledge base satisfies the fuzzy axiom” [12]
 Reasoning a problem

Computationally no calculus exists to check for
satisfiability of a fuzzy knowledge model [12]
Fuzzy OWL
 Extension
of OWL
 Example (describing the safety of a
location):


Without fuzzy, the location is either safe or not
safe
With fuzzy, the location is safe to a degree
 Classes
and properties are ‘fuzzy’
 A class is considered a fuzzy set [1]
 A property is a fuzzy relation over a set [1]
Fuzzy OWL
 Requires
extension of
to map OWL
entailment to
satisfiability [4]
 Reasoning changes in that when concepts
are represented as nodes in forest-like
representations, a “membership degree” is
associated with each node indicating it
belongs to a concept [4]
 Degrees added to OWL facts
Fuzzy OWL
Taken from [4].
Probability Theory
“..the central objects of probability theory are
random variables, stochastic processes,
and events: mathematical abstractions of
non-deterministic events or measured
quantities that may either be single
occurrences or evolve over time in an
apparently random fashion…” [11]
PR-OWL

Developed as an extension to OWL (basically an upper
ontology)


Uses MEBN logic rather than extending OWL



Represent conditional probability distribution [21]
MFrags organized into MEBN Theories (MTheories) [21]


A first order Bayesian logic [21]
Consists of entities and attributes
Attributes about entities and relationships to each other –
MEBN fragments (MFrag) [21]


Represents complex Bayesian models [21]
Collectively satisfy consistency constraints [21]
Goal

Provide a way to support Bayesian models
PR-OWL
Taken from [21].
BayesOWL










Express OWL ontologies as Bayesian networks by means of rules
For each node, a conditional probability table (CPT) is constructed [15]
All subject and object classes translated into concept nodes [15]
Arc drawn between 2 concept nodes if the 2 classes are related by
predicate [15]
Direction based on class hierarchy
L-Nodes generated during translation to represent OWL logical operators
True/false value for each node indicates whether the instance belongs to
the concept
CPTs are approximated using the “iterative proportional fitting procedure
(IPFP)” [15]
Restricted currently to OWL-DL taxonomies [15]
Goals


Support ontology reasoning using probabilistic approach
Support ontology mapping
BayesOWL
rdfs:subClassOf
owl:intersectionOf
owl:unionOf
owl:complementOf owl:equivalentClass owl:disjointWith
Taken from [15].
BayesOWL
•DAG constructed
•CPTs for LNodes specified
•Concept nodes
approximated
using D-IPFP
Taken from [15].
BayesOWL
 Reasoning



Support [15]
Concept satisfiability
Concept overlapping
Concept subsumption
 Extensions
to OWL to support probabilistic
representation [15]


PriorProb
CondProb
 Concept
Mapping [15]
BayesOWL
Extensions to OWL
Taken from [15].
Pronto




Non-monotonic probabilistic DL reasoner
Built on top of Pellet
Uses P-SHIQ(D) formalism [8]
Expressing uncertain axioms


Probabilistic Reasoning






Syntax based upon Lukasiewicz’s conditional constraints [8]
Lehmann’s lexicographic entailment [8]
Represents uncertain ontological knowledge and reasoning [8]
Capable of representing uncertainty in both ABox and TBox axioms
[8]
“All inferences are done in a totally ‘logical’ way” (no translation) [8]
Uses “OWL 1.1 axiom annotations to associate probability intervals
with uncertain OWL axioms” [8]
Doesn’t scale beyond “15 generic (TBox) conditional constraints” [9]
Pronto

Conditional constraints




(D|C)[l,u]
C and D concepts in P-SHIQ(D)
[l,u] closed interval within [0,1]
Supports overriding


Can handle certain probabilistic conflicts
Flying birds/penguin problem
• Pronto allows “more specific constraints to override more
generic ones” [9]
• “if Pronto knows that Tweety is a Penguin and Penguin is a
subclass-of Bird, it will override the constraint
(FlyingObject|Bird)[0.9;1.0] by
(FlyingObject|Penguin)[0.0;0.05] and correctly entail
Tweety:(FlyingObject|owl:Thing)[0.0;0.05]. “ [9]
Uncertainty and Ontologies Mapping

Mapping a problem


Existing approaches - combination of syntactic and
semantic measures [18], use machine learning, or
linguistics and natural language processing [15]
Quality varies depending upon domain [18]

Wang argues without use of a thesaurus,
inaccuracies will occur [22]
 Problem:

When mapping a concept from ontology A to ontology
B there isn’t always a single concept match but rather
a number of concepts that match to some degree
Uncertainty and Ontologies Mapping

A proposed truth theory solution based on the
following [18]:

Dempster-Shafer, uncertain reasoning over potential
mappings
• Evidence Theory





Similarity matrix comparing all concepts/properties
Similarity measure of a concept between O1 and O2
DS combines evidence learned to form new belief
Promising approach
Multi-agent ontology mapping framework [18]


Not domain dependent
Doesn’t require large amounts of training data
Uncertainty and Ontologies Mapping
 A proposed




solution by Wang [22]:
ACAOM
Uses WordNet to calculate similarities for
node names
Name based mapping
Instance strategy
• More semantics more feasible to match
• Documents assigned to nodes

Uses vector space models to rank matches
Uncertainty and Ontologies Mapping

BayesOWL [15] also proposed a solution

Argue that existing similarity approaches will not work
• If degree of similarity is not present in both concepts being
matched [15]
• If concept itself is fuzzy [15]



Uses BayesOWL and belief propagation between
BNs [15]
Ontologies are first translated into BNs [15]
Use probabilistic evidence reasoning to determine
match [15]
Uncertainty and Ontologies – An
Ontology of Uncertainty

Proposed by the W3C UR3W-XG group
 Provides a vocabulary for representing different
types of uncertainty
 Was a good start but refinement needed [20]
 Strategy to use such an ontology as a way to
drive a reasoner


Open issue: coordination of reasoning of different
uncertainty models in knowledge base [19]
Uses SWRL rules to assign uncertainty to each
relation [19]
Uncertainty and Ontologies – An
Ontology of Uncertainty
Taken from [20].
Uncertainty and Web Services




Service discovery – what is best service for request?
Matching goal to service
Brokers used for filtering
Semantic Web Service Framework



Semantic Web Service Language – concepts/descriptions [17]
Semantic Web Service Ontology – conceptual model [17]
It is argued that current frameworks use first order and
description logics and “goal capabilities” are “based on
subsumption checking or query-answering”[16]
 Proposed approach uses Incident Calculus [16]
Demo - Pronto

Pronto Example: Breast Cancer Risk Models
 Models 2 types of risks – absolute and relative
 Combining risk factors to determine likelihood of
breast cancer for a woman [8]


Distinction between known and inferred
Pronto uses an ontology for knowledge
 Uses probabilistic statements to enable
computable inferencing [8]
 The probabilistic statements complement the
OWL syntax
Demo - Pronto
Risk factors relevant to breast cancer are subclasses of ‘RiskFactor’
 Categories of women that have certain risk factors are subclasses of
‘WomanWithRiskFactors’
 Women with risk of developing cancer subclass ‘WomanUnderBRCRisk’
 The goal:



“Compute the probability that a certain woman is an instance of some
WomanUnderBRCRisk subclass given that she is an instance of some
WomanWithRiskFactors subclass” [8]
“Infer generic probabilistic subsumption between classes under
WomanUnderBRCRisk and under WomanWithRiskFactors” [8]
Conditional constraints are used to represent ‘uncertain background
knowledge’ using the OWL 1.1 axiom annotations [8]
 The demo defines constraints to “express how risk factors influence the risk
of developing cancer” [8]
 Pronto combines the factors and computes the probability that a woman is
an instance of a subclass of ‘WomanUnderBRCRisk’

Demo - Pronto
<owl:ObjectProperty rdf:about="#hasRiskFactor">
<rdfs:domain rdf:resource="#Person"/>
<rdfs:range rdf:resource="#RiskFactor"/>
</owl:ObjectProperty>
<owl:Class rdf:about="#WomanTakingEstrogen">
<owl:equivalentClass>
<owl:Restriction>
<owl:onProperty rdf:resource="#hasRiskFactor"/>
<owl:someValuesFrom rdf:resource="#Estrogen"/>
</owl:Restriction>
</owl:equivalentClass>
<rdfs:subClassOf rdf:resource="#Woman"/>
</owl:Class>
Taken from http://clarkparsia.com/pronto/cancer_ra.owl
Demo - Pronto
<owl:Class rdf:about="#WomanWithRiskFactors">
<owl:equivalentClass>
<owl:Class>
<owl:intersectionOf rdf:parseType="Collection">
<rdf:Description rdf:about="#Woman"/>
<owl:Restriction>
<owl:onProperty rdf:resource="#hasRiskFactor"/>
<owl:someValuesFrom rdf:resource="#RiskFactor"/>
</owl:Restriction>
</owl:intersectionOf>
</owl:Class>
</owl:equivalentClass>
<rdfs:subClassOf rdf:resource="#Woman"/>
</owl:Class>
Taken from http://clarkparsia.com/pronto/cancer_ra.owl
Demo - Pronto
<owl:Class rdf:about="#WomanAgedUnder50">
<owl:equivalentClass>
<owl:Class>
<owl:intersectionOf rdf:parseType="Collection">
<rdf:Description rdf:about="#Woman"/>
<owl:Restriction>
<owl:onProperty rdf:resource="#hasAge"/>
<owl:someValuesFrom rdf:resource="#AgeUnder50"/>
</owl:Restriction>
</owl:intersectionOf>
</owl:Class>
</owl:equivalentClass>
<rdfs:subClassOf rdf:resource="#WomanWithRiskFactors"/>
</owl:Class>
Taken from http://clarkparsia.com/pronto/cancer_ra.owl
Demo - Pronto
<owl:Class rdf:about="#WomanUnderAbsoluteBRCRisk">
<owl:equivalentClass>
<owl:Class>
<owl:intersectionOf rdf:parseType="Collection">
<rdf:Description rdf:about="#Woman"/>
<owl:Restriction>
<owl:onProperty rdf:resource="#hasRisk"/>
<owl:someValuesFrom rdf:resource="#AbsoluteBRCRisk"/>
</owl:Restriction>
</owl:intersectionOf>
</owl:Class>
</owl:equivalentClass>
</owl:Class>
Taken from http://clarkparsia.com/pronto/cancer_ra.owl
Demo - Pronto
<owl:Class rdf:about="#WomanUnderBRCRisk">
<owl:equivalentClass>
<owl:Class>
<owl:intersectionOf rdf:parseType="Collection">
<rdf:Description rdf:about="#Woman"/>
<owl:Restriction>
<owl:onProperty rdf:resource="#hasRisk"/>
<owl:someValuesFrom rdf:resource="#BRCRisk"/>
</owl:Restriction>
</owl:intersectionOf>
</owl:Class>
</owl:equivalentClass>
</owl:Class>
Taken from http://clarkparsia.com/pronto/cancer_ra.owl
Demo - Pronto
<owl:Class rdf:about="#WomanUnderIncreasedBRCRisk">
<owl:equivalentClass>
<owl:Class>
<owl:intersectionOf rdf:parseType="Collection">
<owl:Restriction>
<owl:onProperty rdf:resource="#hasRisk"/>
<owl:someValuesFrom rdf:resource="#IncreasedBRCRisk"/>
</owl:Restriction>
<rdf:Description rdf:about="#WomanUnderBRCRisk"/>
</owl:intersectionOf>
</owl:Class>
</owl:equivalentClass>
</owl:Class>
Taken from http://clarkparsia.com/pronto/cancer_ra.owl
Demo - Pronto
<owl:Class rdf:about="#WomanUnderLifetimeBRCRisk">
<owl:equivalentClass>
<owl:Class>
<owl:intersectionOf rdf:parseType="Collection">
<rdf:Description rdf:about="#Woman"/>
<owl:Restriction>
<owl:onProperty rdf:resource="#hasRisk"/>
<owl:someValuesFrom rdf:resource="#LifetimeBRCRisk"/>
</owl:Restriction>
</owl:intersectionOf>
</owl:Class>
</owl:equivalentClass>
<rdfs:subClassOf rdf:resource="#WomanUnderAbsoluteBRCRisk"/>
</owl:Class>
Taken from http://clarkparsia.com/pronto/cancer_ra.owl
Demo - Pronto
<owl:Class rdf:about="#WomanUnderModeratelyIncreasedBRCRisk">
<owl:equivalentClass>
<owl:Class>
<owl:intersectionOf rdf:parseType="Collection">
<rdf:Description rdf:about="#WomanUnderIncreasedBRCRisk"/>
<owl:Restriction>
<owl:onProperty rdf:resource="#hasRisk"/>
<owl:someValuesFrom rdf:resource="#ModeratelyIncreasedBRCRisk"/>
</owl:Restriction>
</owl:intersectionOf>
</owl:Class>
</owl:equivalentClass>
<rdfs:subClassOf rdf:resource="#WomanUnderIncreasedBRCRisk"/>
<owl:disjointWith rdf:resource="#WomanUnderStronglyIncreasedBRCRisk"/>
</owl:Class>
Taken from http://clarkparsia.com/pronto/cancer_ra.owl
Demo - Pronto
<owl:Class rdf:about="#WomanUnderModeratelyReducedBRCRisk">
<owl:equivalentClass>
<owl:Restriction>
<owl:onProperty rdf:resource="#hasRisk"/>
<owl:someValuesFrom
rdf:resource="#ModeratelyReducedBRCRisk"/>
</owl:Restriction>
</owl:equivalentClass>
<rdfs:subClassOf rdf:resource="#WomanUnderReducedBRCRisk"/>
<owl:disjointWith
rdf:resource="#WomanUnderStronglyReducedBRCRisk"/>
<owl:disjointWith
rdf:resource="#WomanUnderWeakelyReducedBRCRisk"/>
</owl:Class>
Taken from http://clarkparsia.com/pronto/cancer_ra.owl
Demo - Pronto
<!--Lifetime absolute risk-->
<!-- Any woman has a 12.3% risk of lifetime breast cancer -->
<owl11:Axiom>
<rdf:subject rdf:resource="#Woman"/>
<rdf:predicate rdf:resource="&rdfs;subClassOf"/>
<rdf:object rdf:resource="#WomanUnderLifetimeBRCRisk"/>
<pronto:certainty>0;0.123</pronto:certainty>
</owl11:Axiom>
<!-- If a woman has BRCA mutation, then the risk is beteen 30% and 85% -->
<owl11:Axiom>
<rdf:subject rdf:resource="#WomanWithBRCAMutation"/>
<rdf:predicate rdf:resource="&rdfs;subClassOf"/>
<rdf:object rdf:resource="#WomanUnderLifetimeBRCRisk"/>
<pronto:certainty>0.3;0.85</pronto:certainty>
</owl11:Axiom>
<!-- If it's BRCA1 mutation, then the lifetime risk is between 60% and 80% -->
<owl11:Axiom>
<rdf:subject rdf:resource="#WomanWithBRCA1Mutation"/>
<rdf:predicate rdf:resource="&rdfs;subClassOf"/>
<rdf:object rdf:resource="#WomanUnderLifetimeBRCRisk"/>
<pronto:certainty>0.6;0.8</pronto:certainty>
</owl11:Axiom>
Taken from http://clarkparsia.com/pronto/cancer_cc.owl
Demo - Pronto
<!-- Age-related risk-->
<owl11:Axiom>
<rdf:subject rdf:resource="#WomanAgedUnder20"/>
<rdf:predicate rdf:resource="&rdfs;subClassOf"/>
<rdf:object rdf:resource="#WomanUnderShortTermBRCRisk"/>
<pronto:certainty>0;0.0005</pronto:certainty>
</owl11:Axiom>
<owl11:Axiom>
<rdf:subject rdf:resource="#WomanAged2030"/>
<rdf:predicate rdf:resource="&rdfs;subClassOf"/>
<rdf:object rdf:resource="#WomanUnderShortTermBRCRisk"/>
<pronto:certainty>0;0.004</pronto:certainty>
</owl11:Axiom>
<owl11:Axiom>
<rdf:subject rdf:resource="#WomanAged3040"/>
<rdf:predicate rdf:resource="&rdfs;subClassOf"/>
<rdf:object rdf:resource="#WomanUnderShortTermBRCRisk"/>
<pronto:certainty>0;0.014</pronto:certainty>
</owl11:Axiom>
Taken from http://clarkparsia.com/pronto/cancer_cc.owl
Demo - Pronto
<owl11:Axiom>
<rdf:subject rdf:resource="#WomanAged4050"/>
<rdf:predicate rdf:resource="&rdfs;subClassOf"/>
<rdf:object rdf:resource="#WomanUnderShortTermBRCRisk"/>
<pronto:certainty>0;0.025</pronto:certainty>
</owl11:Axiom>
<owl11:Axiom>
<rdf:subject rdf:resource="#WomanAged5060"/>
<rdf:predicate rdf:resource="&rdfs;subClassOf"/>
<rdf:object rdf:resource="#WomanUnderShortTermBRCRisk"/>
<pronto:certainty>0;0.035</pronto:certainty>
</owl11:Axiom>
<owl11:Axiom>
<rdf:subject rdf:resource="#WomanAged6070"/>
<rdf:predicate rdf:resource="&rdfs;subClassOf"/>
<rdf:object rdf:resource="#WomanUnderShortTermBRCRisk"/>
<pronto:certainty>0;0.039</pronto:certainty>
</owl11:Axiom>
Taken from http://clarkparsia.com/pronto/cancer_cc.owl
Demo - Pronto
<!--owl11:Axiom>
<rdf:subject rdf:resource="#Julie"/>
<rdf:predicate rdf:resource="&rdf;type"/>
<rdf:object rdf:resource="#WomanAged3040"/>
<pronto:certainty>1;1</pronto:certainty>
</owl11:Axiom>
<owl11:Axiom>
<rdf:subject rdf:resource="#Mary"/>
<rdf:predicate rdf:resource="&rdf;type"/>
<rdf:object rdf:resource="#WomanWithBRCA1Mutation"/>
<pronto:certainty>1;1</pronto:certainty>
</owl11:Axiom>
<owl11:Axiom>
<rdf:subject rdf:resource="#Ann"/>
<rdf:predicate rdf:resource="&rdf;type"/>
<rdf:object rdf:resource="#WomanWithMotherBRCAffected"/>
<pronto:certainty>1;1</pronto:certainty>
</owl11:Axiom>
<owl11:Axiom>
<rdf:subject rdf:resource="#Ann"/>
<rdf:predicate rdf:resource="&rdf;type"/>
<rdf:object rdf:resource="#AshkenaziJewishWoman"/>
<pronto:certainty>0.9;0.95</pronto:certainty>
</owl11:Axiom-->
Taken from http://clarkparsia.com/pronto/cancer_cc.owl
Demo - Pronto
<owl11:Axiom>
<rdf:subject rdf:resource="#Helen"/>
<rdf:predicate rdf:resource="&rdf;type"/>
<rdf:object rdf:resource="#PostmenopausalWoman"/>
<pronto:certainty>1;1</pronto:certainty>
</owl11:Axiom>
<owl11:Axiom>
<rdf:subject rdf:resource="#Helen"/>
<rdf:predicate rdf:resource="&rdf;type"/>
<rdf:object rdf:resource="#WomanTakingEstrogen"/>
<pronto:certainty>1;1</pronto:certainty>
</owl11:Axiom>
<owl11:Axiom>
<rdf:subject rdf:resource="#Helen"/>
<rdf:predicate rdf:resource="&rdf;type"/>
<rdf:object rdf:resource="#WomanTakingProgestin"/>
<pronto:certainty>1;1</pronto:certainty>
</owl11:Axiom>
Taken from http://clarkparsia.com/pronto/cancer_cc.owl
Demo - Pronto
<owl11:Axiom>
<rdf:subject rdf:resource="#AshkenaziJewishWoman"/>
<rdf:predicate rdf:resource="&rdfs;subClassOf"/>
<rdf:object rdf:resource="#WomanWithBRCAMutation"/>
<pronto:certainty>0.025;0.025</pronto:certainty>
</owl11:Axiom>
<owl11:Axiom>
<rdf:subject rdf:resource="#WomanWithBRCAMutation"/>
<rdf:predicate rdf:resource="&rdfs;subClassOf"/>
<rdf:object rdf:resource="#WomanUnderLifetimeBRCRisk"/>
<pronto:certainty>0.3;0.85</pronto:certainty>
</owl11:Axiom>
Demo - Pronto
 Running
query (generic TBox conditional
constraint) (C|D)[l,u] [9]
entail
http://clarkparsia.com/pronto/cancer_ra.ow
l#AshkenaziJewishWoman
http://clarkparsia.com/pronto/cancer_ra.ow
l#WomanUnderLifetimeBRCRisk
Demo - Pronto
Query : entail
Result: 34:
(WomanUnderLifetimeBRCRisk|AshkenaziJewishWoman)[0.0075;0.123]
Explanation:
Explaining the generic constraint 34:
(WomanUnderLifetimeBRCRisk|AshkenaziJewish
Woman)[0.0075;0.123]:
Lower bound is because of:
[[8: (WomanWithBRCAMutation|AshkenaziJewishWoman)[0.025;0.025], 7:
(WomanUnderLi
fetimeBRCRisk|WomanWithBRCAMutation)[0.3;0.85]]]
Upper bound is because of:
[[10: (WomanUnderLifetimeBRCRisk|Woman)[0.0;0.123]]]
Result computed in 6266ms
Want to learn more?

Attend the 2009 URSW Conference

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Visit W3C Uncertainty Reasoning for the World Wide
Web Incubator Group
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http://c4i.gmu.edu/ursw/2008/
Download Pronto
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http://www.w3.org/2005/Incubator/urw3/
Review presentations from last year’s conference
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http://c4i.gmu.edu/ursw/2009/
http://pellet.owldl.com/pronto/
Download FiRE
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http://www.image.ece.ntua.gr/~nsimou/FiRE/
References
[1] - Stoilos,Simou,Stamou,Kollias,“Uncertainty and the Semantic Web”, http://www.image.ece.ntua.gr/php/savepaper.php?id=445, 2006, IEEE
[2] – 2008 Conference, “Uncertainty Reasoning for the Semantic Web”, http://c4i.gmu.edu/ursw/2008/index.html
[3] - 2007 Conference, “Uncertainty Reasoning for the Semantic Web”, http://c4i.gmu.edu/ursw/2007/index.html
[4] - Stoilos,Stamou,Tzouvaras,Pan,Horrocks, “Fuzzy OWL: Uncertainty and the Semantic Web”, http://www.image.ntua.gr/papers/398.pdf
[5] - Lassila, “Some Personal Thoughts on Semantic Web and “Non-symbolic” AI”, http://c4i.gmu.edu/ursw/2008/talks/URSW2008_Keynote_Lassila.pdf, 2008,
ISWC
[6] – Williams,Bastin,Cornford,Ingram, “Describing and Communicating Uncertainty within the Semantic Web”,
http://c4i.gmu.edu/ursw/2008/papers/URSW2008_F3_WilliamsEtAl.pdf
[7] – Sanchez, “Fuzzy logic and semantic web”,
http://books.google.com/books?id=Cidej8b4ESIC&pg=PA4&lpg=PA4&dq=monotonic+bivalent+language&source=bl&ots=mtbZcZfaO7&sig=VtGqKXurrzl5HOw36UBTeTpdoE&hl=en&ei=sBIASpuJFonItgeKnpyTBw&sa=X&oi=book_result&ct=result&resnum=1#PPP1,M1
[8] – Klinov, Parsia, “Demonstrating Pronto: a Non-monotonic Probabilistic OWL Reasoner”,
http://www.webont.org/owled/2008dc/papers/owled2008dc_paper_2.pdf
[9] – Klinov, “Introducing Pronto: Probabilistic DL Reasoning in Pellet“, http://clarkparsia.com/weblog/2007/09/27/introducing-pronto/
[10] – Wikipedia Fuzzy Set theory, http://en.wikipedia.org/wiki/Fuzzy_set
[11] – Wikipedia Probability Theory, http://en.wikipedia.org/wiki/Probability_theory
[12] – Straccia, “A Fuzzy Description Logic for the Semantic Web”, http://www.win.tue.nl/~aserebre/ks/Lit/Straccia2006.pdf
[13] – Mazzieri, Dragoni, “A Fuzzy Semantics for Semantic Web Languages”, http://ftp.informatik.rwth-aachen.de/Publications/CEUR-WS/Vol-173/paper2.pdf
[14] – Wikipedia Description Logic, http://en.wikipedia.org/wiki/Description_logic
[15] – Ding, Peng, Pan, “BayesOWL: Uncertainty Modeling in Semantic Web Ontologies”, http://ebiquity.umbc.edu/_file_directory_/papers/217.pdf
[16] – Martin-recurerda1, Robertson2, “Discovery and Uncertainty in Semantic Web Services”, http://ftp.informatik.rwth-aachen.de/Publications/CEUR-WS/Vol173/paper4.pdf
[17] – “Semantic Web Services Framework (SWSF) Overview”, http://www.w3.org/Submission/SWSF/
[18] – Nagy,Vargas-Vera,Motta, “Uncertain Reasoning for Creating Ontology Mapping on the Semantic Web”,
http://c4i.gmu.edu/ursw/2007/files/papers/URSW2007_P2_NagyVeraMotta.pdf
[19] – Ceravolo, Damiani,Leida, “Which Role for an Ontology of Uncertainty?”, http://c4i.gmu.edu/ursw/2008/papers/URSW2008_P6_CeravoloEtAl.pdf
[20] – Laskey, Laskey, “Uncertainty Reasoning for the World Wide Web: Report on the URW3-XG Incubator Group”,
http://c4i.gmu.edu/ursw/2008/papers/URSW2008_FX_LaskeyLaskey.pdf
[21] – Costa, Laskey, “PR-OWL: A Framework for Probabilistic Ontologies”, http://volgenau.gmu.edu/~klaskey/papers/FOIS2006_CostaLaskey.pdf
[22] – Wang, “Integrating Uncertainty Into Ontology Mapping”, http://iswc2007.semanticweb.org/papers/955.pdf