Download - Revisiting defaults in description logics – and their role in aligning ontologies. (JIST 2014)
IntroductionContributionFree defaultsConclusion
Revisiting default description logics – and theirrole in aligning ontologies
Kunal Sengupta 1 Pascal Hitzler 1 Krzysztof Janowicz 2
1Wright State University, Dayton OH 45435, USA
2University of California, Santa Barbara, USA
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
1 IntroductionMotivationExamplesProposed Solution
2 Contribution
3 Free defaultsSemanticsDecidability
4 Conclusion
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
MotivationExamplesProposed Solution
1 IntroductionMotivationExamplesProposed Solution
2 Contribution
3 Free defaultsSemanticsDecidability
4 Conclusion
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
MotivationExamplesProposed Solution
Motivation
Heterogeneity is everywhere: Linked Data, Ontologies, Worldwide web
How to have ontology mappings that respect heterogeneity?
OWL (Description logics) is not suitable for ontology mapping
Why, you ask?
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
MotivationExamplesProposed Solution
Motivation
Heterogeneity is everywhere: Linked Data, Ontologies, Worldwide web
How to have ontology mappings that respect heterogeneity?
OWL (Description logics) is not suitable for ontology mapping
Why, you ask?
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
MotivationExamplesProposed Solution
Motivation
Heterogeneity is everywhere: Linked Data, Ontologies, Worldwide web
How to have ontology mappings that respect heterogeneity?
OWL (Description logics) is not suitable for ontology mapping
Why, you ask?
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
MotivationExamplesProposed Solution
Motivation
Heterogeneity is everywhere: Linked Data, Ontologies, Worldwide web
How to have ontology mappings that respect heterogeneity?
OWL (Description logics) is not suitable for ontology mapping
Why, you ask?
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
MotivationExamplesProposed Solution
An example!
Ontology A
a:hasWife v a:hasSpouse
symmetric(a:hasSpouse)
∃a:hasSpouse.a:Female v a:Male
∃a:hasSpouse.a:Male v a:Female
a:hasWife(a:john, a:mary)
a:Male(a:john)
a:Female(a:mary)
a:Male u a:Female v ⊥
Mappings
a:hasSpouse ≡ b:hasSpouse
a:Male ≡ b:Male
a:Female ≡ b:Female
Ontology B
symmetric(b:hasSpouse)
b:hasSpouse(b:mike, b:david)
b:Male(b:david)
b:Male(b:mike)
b:Female(b:anna)
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
MotivationExamplesProposed Solution
An example!
Ontology A
a:hasWife v a:hasSpouse
symmetric(a:hasSpouse)
∃a:hasSpouse.a:Female v a:Male
∃a:hasSpouse.a:Male v a:Female
a:hasWife(a:john, a:mary)
a:Male(a:john)
a:Female(a:mary)
a:Male u a:Female v ⊥
Mappings
a:hasSpouse ≡ b:hasSpouse
a:Male ≡ b:Male
a:Female ≡ b:Female
Ontology B
symmetric(b:hasSpouse)
b:hasSpouse(b:mike, b:david)
b:Male(b:david)
b:Male(b:mike)
b:Female(b:anna)
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
MotivationExamplesProposed Solution
An example!
Ontology A
a:hasWife v a:hasSpouse
symmetric(a:hasSpouse)
∃a:hasSpouse.a:Female v a:Male
∃a:hasSpouse.a:Male v a:Female
a:hasWife(a:john, a:mary)
a:Male(a:john)
a:Female(a:mary)
a:Male u a:Female v ⊥
Mappings
a:hasSpouse ≡ b:hasSpouse
a:Male ≡ b:Male
a:Female ≡ b:Female
Ontology B
symmetric(b:hasSpouse)
b:hasSpouse(b:mike, b:david)
b:Male(b:david)
b:Male(b:mike)
b:Female(b:anna)
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
MotivationExamplesProposed Solution
owl:sameAs example
Ontology A
a:Airport(a:kennedy)
a:Airport v a:Place
Ontology B
b:President(b:kennedy)
b:President v b:Person
b:Place u b:Person v ⊥
Mappings
a:Place ≡ b:Place
owl:sameAs(a:kennedy,b:kennedy)
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
MotivationExamplesProposed Solution
owl:sameAs example
Ontology A
a:Airport(a:kennedy)
a:Airport v a:Place
Ontology B
b:President(b:kennedy)
b:President v b:Person
b:Place u b:Person v ⊥
Mappings
a:Place ≡ b:Place
owl:sameAs(a:kennedy,b:kennedy)
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
MotivationExamplesProposed Solution
Using defaults for ontology mapping
Use statements like b:hasSpouse vd a:hasSpouse to denotemappings
For each pair that satisfies b:hasSpouse assume it satisfiesa:hasSpouse unless it causes an inconsistency
Assuming a:hasSpouse(b:mike, b:david) causes aninconsistency
The pair (b:mike, b:david) is treated as an exception to thestatement b:hasSpouse vd b:hasSpouse
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
MotivationExamplesProposed Solution
Solution
Use defaults to denote mappings, such that exceptions areallowed.
b:hasSpouse : a:hasSpouse
a:hasSpouse
But DLs + Defaults = Undecidable logics [Baader, Hollunder95].
Workaround: Defaults apply only to named individuals[Baader, Hollunder 95].
What about un-named individuals?
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
MotivationExamplesProposed Solution
Solution
Use defaults to denote mappings, such that exceptions areallowed.
b:hasSpouse : a:hasSpouse
a:hasSpouse
But DLs + Defaults = Undecidable logics [Baader, Hollunder95].
Workaround: Defaults apply only to named individuals[Baader, Hollunder 95].
What about un-named individuals?
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
1 IntroductionMotivationExamplesProposed Solution
2 Contribution
3 Free defaultsSemanticsDecidability
4 Conclusion
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
Contribution
Free defaults
Application of defaults not limited to named individuals∗∗
Defaults with role inclusions are also decidable.
A new, more powerful language to define mapping betweenontologies.
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
Contribution
Free defaults
Application of defaults not limited to named individuals∗∗
Defaults with role inclusions are also decidable.
A new, more powerful language to define mapping betweenontologies.
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
Contribution
Free defaults
Application of defaults not limited to named individuals∗∗
Defaults with role inclusions are also decidable.
A new, more powerful language to define mapping betweenontologies.
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
1 IntroductionMotivationExamplesProposed Solution
2 Contribution
3 Free defaultsSemanticsDecidability
4 Conclusion
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Syntax
A new operator vd that represents default inclusions
C vd D is a default class inclusion
A default-knowledge-base is denoted as (KB, δ), where KB isa DL knowledge base and δ is a set of default axioms
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Semantics (Intuitive)
C vd D
Every named individual in C is also in D unless it causes aninconsistency
Every un-named individual in C is also in D (it behavesexactly same as v)
Exceptions occur only in named individuals
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Default satisfying individuals
A named individual a is said to satisfy a default C vd D if:
1 a ∈ CI ,DI , or
2 a ∈ (¬C )I
Key: We want to maximize the sets of individuals that satisfy eachdefault
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Default satisfying individuals
A named individual a is said to satisfy a default C vd D if:
1 a ∈ CI ,DI , or
2 a ∈ (¬C )I
Key: We want to maximize the sets of individuals that satisfy eachdefault
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Interpretation
An interpretation I for a default-knowledge-base (KB, δ)
1 (∆I , .I) as usual
2 Additionally, δI denotes the tuple (X I1 , . . . ,XIn )
3 X Ii is the set of interpreted named individuals satisfying thei th default Ci vd Di
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Preference relation >KB,δ
I, J are two interpretations of default-knowledge-base (KB, δ).Then I >KB,δ J if:
1 aI = aJ for all a ∈ KB
2 X Ii ⊇ XJi for all 1 ≤ i ≤| δ |3 X Ii ⊃ XJi for some 1 ≤ i ≤| δ |
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Example
Knowledge Base
Bird(a)
Bird(b)
Penguin(c)
Penguin v Bird
Penguin u Fly v ⊥δ = {Bird vd Fly}
Interpretation I
∆I = {a, b, c}BirdI = {a, b, c}
PenguinI = {c}FlyI = {a}δI = ({a})
Interpretation J
∆J = {a, b, c}BirdJ = {a, b, c}
PenguinJ = {c}FlyJ = {a, b}δJ = ({a, b})
J >KB,δ I
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Example
Knowledge Base
Bird(a)
Bird(b)
Penguin(c)
Penguin v Bird
Penguin u Fly v ⊥δ = {Bird vd Fly}
Interpretation I
∆I = {a, b, c}BirdI = {a, b, c}
PenguinI = {c}FlyI = {a}δI = ({a})
Interpretation J
∆J = {a, b, c}BirdJ = {a, b, c}
PenguinJ = {c}FlyJ = {a, b}δJ = ({a, b})
J >KB,δ I
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
d-model
An interpretation I of (KB, δ) is called a default model or d-modelif
I satisfies all axioms of KB
CIi ⊆ DIi , for all un-named individuals
I is maximal with respect to the preference relation >KB,δ
A default-knowledge-base that has a d-model is called d-satisfiable.Note: Reiter’s normal defaults are always satisfiable.
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
d-entailment
An axiom α is d-entailed by a default-knowledge-base if it istrue in all the d-models.
We follow skeptical reasoning.
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Decidability
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Some notations
IndKB is the set of all the named individuals occurring in KB.
P(IndKB) is the power set of IndKB
Pn(IndKB) = P(IndKB)× . . .n-1 times × P(IndKB), where n isthe cardinality of δ
IndKB = {a, b}
P(IndKB) = {{a}, {b}, {a, b}}Pn(IndKB) ={({a}, {a}), ({a}, {b}), ({b}, {a}), . . . , ({a, b}, {a, b})}, for| δ |= 2
Note: Pn(IndKB) is a set of n-tuples that contains all thepossible combinations for δI
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Some notations
IndKB is the set of all the named individuals occurring in KB.
P(IndKB) is the power set of IndKB
Pn(IndKB) = P(IndKB)× . . .n-1 times × P(IndKB), where n isthe cardinality of δ
IndKB = {a, b}P(IndKB) = {{a}, {b}, {a, b}}
Pn(IndKB) ={({a}, {a}), ({a}, {b}), ({b}, {a}), . . . , ({a, b}, {a, b})}, for| δ |= 2
Note: Pn(IndKB) is a set of n-tuples that contains all thepossible combinations for δI
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Some notations
IndKB is the set of all the named individuals occurring in KB.
P(IndKB) is the power set of IndKB
Pn(IndKB) = P(IndKB)× . . .n-1 times × P(IndKB), where n isthe cardinality of δ
IndKB = {a, b}P(IndKB) = {{a}, {b}, {a, b}}Pn(IndKB) ={({a}, {a}), ({a}, {b}), ({b}, {a}), . . . , ({a, b}, {a, b})}, for| δ |= 2
Note: Pn(IndKB) is a set of n-tuples that contains all thepossible combinations for δI
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Some notations
IndKB is the set of all the named individuals occurring in KB.
P(IndKB) is the power set of IndKB
Pn(IndKB) = P(IndKB)× . . .n-1 times × P(IndKB), where n isthe cardinality of δ
IndKB = {a, b}P(IndKB) = {{a}, {b}, {a, b}}Pn(IndKB) ={({a}, {a}), ({a}, {b}), ({b}, {a}), . . . , ({a, b}, {a, b})}, for| δ |= 2
Note: Pn(IndKB) is a set of n-tuples that contains all thepossible combinations for δI
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Knowledge base re-writing
Consider some P = (X1, . . . ,Xn) ∈ Pn(IndKB). Then KBP isobtained by adding the following axioms to KB, for each Ci vd Di
1 Xi ≡ (Ci u Di u {a1, . . . , ak}) t (¬Ci u {a1, . . . , ak})2 Ci u ¬{a1, . . . , ak} v Di
Intuition
Recall, an individual a satisfies a default C vd D ifa ∈ CI ,DI or a ∈ (¬C )I
Unknowns always satisfy the defaults
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Knowledge base re-writing
Consider some P = (X1, . . . ,Xn) ∈ Pn(IndKB). Then KBP isobtained by adding the following axioms to KB, for each Ci vd Di
1 Xi ≡ (Ci u Di u {a1, . . . , ak}) t (¬Ci u {a1, . . . , ak})
2 Ci u ¬{a1, . . . , ak} v Di
Intuition
Recall, an individual a satisfies a default C vd D ifa ∈ CI ,DI or a ∈ (¬C )I
Unknowns always satisfy the defaults
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Knowledge base re-writing
Consider some P = (X1, . . . ,Xn) ∈ Pn(IndKB). Then KBP isobtained by adding the following axioms to KB, for each Ci vd Di
1 Xi ≡ (Ci u Di u {a1, . . . , ak}) t (¬Ci u {a1, . . . , ak})2 Ci u ¬{a1, . . . , ak} v Di
Intuition
Recall, an individual a satisfies a default C vd D ifa ∈ CI ,DI or a ∈ (¬C )I
Unknowns always satisfy the defaults
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Decidability of d-satisfiability
Lemma
Let (KB, δ) be a default-knowledge-base, if KBP is classicallysatisfiable for some P ∈ Pn(IndKB), then (KB, δ) is d-satisfiable.
Theorem
The task of determining d-satisfiability of default-knowledge-basesis decidable.
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Decidability of d-satisfiability
Lemma
Let (KB, δ) be a default-knowledge-base, if KBP is classicallysatisfiable for some P ∈ Pn(IndKB), then (KB, δ) is d-satisfiable.
Theorem
The task of determining d-satisfiability of default-knowledge-basesis decidable.
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Decidability of d-entailment tasks
For non-montonic logics such as free-defaults entailment tasksare not directly reducible to satisfiability check
We need to check all models for d-entailments
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Proof sketch
KBP is constructed using P ∈ Pn(IndKB)
Pn(IndKB) is finite
An order can be computed on Pn(IndKB) based on maximalityof its elements (finite time)
Construct a set MaxP consisting of maximal classicallysatisfiable KBPs
MaxP is generated in finite time due to the finiteness ofPn(IndKB)
Theorem
A DL axiom α is entailed by a default-knowledge-base (KB, δ) iff itis classically entailed by every KBP ∈ MaxP
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Proof sketch
KBP is constructed using P ∈ Pn(IndKB)
Pn(IndKB) is finite
An order can be computed on Pn(IndKB) based on maximalityof its elements (finite time)
Construct a set MaxP consisting of maximal classicallysatisfiable KBPs
MaxP is generated in finite time due to the finiteness ofPn(IndKB)
Theorem
A DL axiom α is entailed by a default-knowledge-base (KB, δ) iff itis classically entailed by every KBP ∈ MaxP
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Role defaults
Note: Similar construction can be used to show decidability of roledefaults under the semantics of free-defaults
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
SemanticsDecidability
Back to the example
Now, the mapping can be denoted asb:hasSpouse vd a:hasSpouse, a:hasSpouse vd b:hasSpouse
b:hasSpouseI = {(b:david, b:mike), (b:mike, b:david),(a:john, a:marry), (a:marry, a:john)}
a:hasSpouseI = {((a:john, a:marry), (a:marry, a:john)}
Notice that heterogeneity is still respected and at the sametime relations can be carried over
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
1 IntroductionMotivationExamplesProposed Solution
2 Contribution
3 Free defaultsSemanticsDecidability
4 Conclusion
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
Conclusion
Drawback
The free-defaults don’t work when un-named individuals areimplicit exceptions
Future work
Investigation of decidability of more general defaults in DLs
Effecient algorithmization of d-entailment tasks
Implementation
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
Conclusion
Drawback
The free-defaults don’t work when un-named individuals areimplicit exceptions
Future work
Investigation of decidability of more general defaults in DLs
Effecient algorithmization of d-entailment tasks
Implementation
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)
IntroductionContributionFree defaultsConclusion
Questions?
DaSe Lab for Data Semantics at Wright state university
Twitter : @DaSeLab, @sengupta kunal
Website : http://wright.edu/~sengupta.4
Topics : Semantic Web, Knowledge representation andreasoning, Description Logics, Rules, Ontology alignment,Applications ...
Kunal Sengupta , Pascal Hitzler , Krzysztof Janowicz Revisiting default description logics (JIST 2014)