UncertaintyAndSW.ppt
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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
Visit W3C Uncertainty Reasoning for the World Wide
Web Incubator Group
http://c4i.gmu.edu/ursw/2008/
Download Pronto
http://www.w3.org/2005/Incubator/urw3/
Review presentations from last year’s conference
http://c4i.gmu.edu/ursw/2009/
http://pellet.owldl.com/pronto/
Download FiRE
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
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