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)


1 IntroductionMotivationExamplesProposed Solution

2 Contribution

3 Free defaultsSemanticsDecidability

4 Conclusion



MotivationExamplesProposed Solution


2 Contribution


4 Conclusion




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?




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)




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)




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




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?




2 Contribution


4 Conclusion



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.



SemanticsDecidability


2 Contribution


4 Conclusion




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




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




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




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




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 |




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




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.




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.




Decidability




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




Some notations




IndKB = {a, b}P(IndKB) = {{a}, {b}, {a, b}}

Pn(IndKB) ={({a}, {a}), ({a}, {b}), ({b}, {a}), . . . , ({a, b}, {a, b})}, for| δ |= 2





Some notations




IndKB = {a, b}P(IndKB) = {{a}, {b}, {a, b}}Pn(IndKB) ={({a}, {a}), ({a}, {b}), ({b}, {a}), . . . , ({a, b}, {a, b})}, for| δ |= 2





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






1 Xi ≡ (Ci u Di u {a1, . . . , ak}) t (¬Ci u {a1, . . . , ak})

2 Ci u ¬{a1, . . . , ak} v Di

Intuition








1 Xi ≡ (Ci u Di u {a1, . . . , ak}) t (¬Ci u {a1, . . . , ak})2 Ci u ¬{a1, . . . , ak} v Di

Intuition






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.




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




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




Role defaults

Note: Similar construction can be used to show decidability of roledefaults under the semantics of free-defaults




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




2 Contribution


4 Conclusion



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



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 ...


Revisiting defaults in description logics – and their role in aligning ontologies. (JIST 2014)

Technology

Transcript of Revisiting defaults in description logics – and their role in aligning ontologies. (JIST 2014)