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Description Logic Knowledge Base Exchange
Elena Botoevasupervisor: Diego Calvanese
PhD Final ExaminationApril 10, 2014
Bolzano
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 1/33
Outline
1 Introduction
2 Summary of Work
3 Results
4 Technical DevelopmentUniversal SolutionsUniversal UCQ-solutionsUCQ-representations
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 2/33
Outline
1 Introduction
2 Summary of Work
3 Results
4 Technical DevelopmentUniversal SolutionsUniversal UCQ-solutionsUCQ-representations
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 3/33
Knowledge Base Exchange: a Simple Scenario
T1
A1
Category
BItem
BFootWear
?OpenShoes
?Strappy
?Plateau
?HighHeel
. . .
BApparel
. . .
Choose size5Size
Choose color5Color
Choose brand5Brand
Open Shoes: (3 items found)
brown sandStrappyeXX
wedge sandPlateaueXX
heel sandHighHeeleXX
Viewed as a knowledge base:
Item
FootWear
OpenShoes
Strappy
Plateau
HighHeel
. . .
Apparel
. . .
brown sand
Strappy
wedge sand
Plateau
heel sand
HighHeel
The website after restructuring:
Category
BProduct
BShoes
?Sandals
?Classic
?Platform
?Heeled
. . .
BClothing
. . .
Choose size5Size
Choose color5Color
Choose brand5Brand
Sandals: (3 products found)
brown sandClassiceXX
wedge sandPlatformeXX
heel sandHeeledeXX
A1
T1
?
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 4/33
Knowledge Base Exchange: a Simple Scenario
T1
A1
Category
BItem
BFootWear
?OpenShoes
?Strappy
?Plateau
?HighHeel
. . .
BApparel
. . .
Choose size5Size
Choose color5Color
Choose brand5Brand
Open Shoes: (3 items found)
brown sandStrappyeXX
wedge sandPlateaueXX
heel sandHighHeeleXX
Viewed as a knowledge base:
Item
FootWear
OpenShoes
Strappy
Plateau
HighHeel
. . .
Apparel
. . .
brown sand
Strappy
wedge sand
Plateau
heel sand
HighHeel
The website after restructuring:
Category
BProduct
BShoes
?Sandals
?Classic
?Platform
?Heeled
. . .
BClothing
. . .
Choose size5Size
Choose color5Color
Choose brand5Brand
Sandals: (3 products found)
brown sandClassiceXX
wedge sandPlatformeXX
heel sandHeeledeXX
A1
T1
?
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 4/33
Knowledge Base Exchange: a Simple Scenario
T1
A1
Category
BItem
BFootWear
?OpenShoes
?Strappy
?Plateau
?HighHeel
. . .
BApparel
. . .
Choose size5Size
Choose color5Color
Choose brand5Brand
Open Shoes: (3 items found)
brown sandStrappyeXX
wedge sandPlateaueXX
heel sandHighHeeleXX
Viewed as a knowledge base:
Item
FootWear
OpenShoes
Strappy
Plateau
HighHeel
. . .
Apparel
. . .
brown sand
Strappy
wedge sand
Plateau
heel sand
HighHeel
The website after restructuring:
Category
BProduct
BShoes
?Sandals
?Classic
?Platform
?Heeled
. . .
BClothing
. . .
Choose size5Size
Choose color5Color
Choose brand5Brand
Sandals: (3 products found)
brown sandClassiceXX
wedge sandPlatformeXX
heel sandHeeledeXX
A1
T1
?
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 4/33
Knowledge Base Exchange: a Simple Scenario
T1
A1
Category
BItem
BFootWear
?OpenShoes
?Strappy
?Plateau
?HighHeel
. . .
BApparel
. . .
Choose size5Size
Choose color5Color
Choose brand5Brand
Open Shoes: (3 items found)
brown sandStrappyeXX
wedge sandPlateaueXX
heel sandHighHeeleXX
Viewed as a knowledge base:
Item
FootWear
OpenShoes
Strappy
Plateau
HighHeel
. . .
Apparel
. . .
brown sand
Strappy
wedge sand
Plateau
heel sand
HighHeel
The website after restructuring:
Category
BProduct
BShoes
?Sandals
?Classic
?Platform
?Heeled
. . .
BClothing
. . .
Choose size5Size
Choose color5Color
Choose brand5Brand
Sandals: (3 products found)
brown sandClassiceXX
wedge sandPlatformeXX
heel sandHeeledeXX
A1
T1
?
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 4/33
Data Exchange [Fagin et al., 2003]
Σ1Σ1
source schema
Σ2Σ2
target schema
mappingM
I1
source instance
I2
target instance
best solution
Mapping M is a set of inclusions of conjunctive queries (CQs)
∀x , y(q1(x , y)→ ∃z q2(x , z)
).
I1 is a complete database instance.I2 is an incomplete database instance.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 5/33
Data Exchange [Fagin et al., 2003]
Σ1Σ1
source schema
Σ2Σ2
target schema
mappingM
I1
source instance
I2
target instance
best solution
Mapping M is a set of inclusions of conjunctive queries (CQs)
∀x , y(q1(x , y)→ ∃z q2(x , z)
).
I1 is a complete database instance.I2 is an incomplete database instance.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 5/33
Data Exchange [Fagin et al., 2003]
Σ1Σ1
source schema
Σ2Σ2
target schema
mappingM
I1
source instance
I2
target instance
best solution
Mapping M is a set of inclusions of conjunctive queries (CQs)
∀x , y(q1(x , y)→ ∃z q2(x , z)
).
I1 is a complete database instance.I2 is an incomplete database instance.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 5/33
Data Exchange Example
∀a, t.( AuthorOf (a, t) → ∃g .BookInfo(t, a, g) )M :
nabokov lolita
tolkien lotr
AuthorOfI1 :
nabokovlolita tragicomedy
tolkienlotr fantasy
BookInfoI2 :
nabokovlolita null1
tolkienlotr null2
BookInfoI ′2 :
I2 is a solution for I1 under M.I ′2 is a universal solution for I1 under M.
⇒ there is a homomorphism from I ′2 to I2.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 6/33
Data Exchange Example
∀a, t.( AuthorOf (a, t) → ∃g .BookInfo(t, a, g) )M :
nabokov lolita
tolkien lotr
AuthorOfI1 :
nabokovlolita tragicomedy
tolkienlotr fantasy
BookInfoI2 :
nabokovlolita null1
tolkienlotr null2
BookInfoI ′2 :
I2 is a solution for I1 under M.
I ′2 is a universal solution for I1 under M.⇒ there is a homomorphism from I ′2 to I2.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 6/33
Data Exchange Example
∀a, t.( AuthorOf (a, t) → ∃g .BookInfo(t, a, g) )M :
nabokov lolita
tolkien lotr
AuthorOfI1 :
nabokovlolita tragicomedy
tolkienlotr fantasy
BookInfoI2 :
nabokovlolita null1
tolkienlotr null2
BookInfoI ′2 :
I2 is a solution for I1 under M.I ′2 is a universal solution for I1 under M.
⇒ there is a homomorphism from I ′2 to I2.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 6/33
Data Exchange Example
∀a, t.( AuthorOf (a, t) → ∃g .BookInfo(t, a, g) )M :
nabokov lolita
tolkien lotr
AuthorOfI1 :
nabokovlolita tragicomedy
tolkienlotr fantasy
BookInfoI2 :
nabokovlolita null1
tolkienlotr null2
BookInfoI ′2 :
I2 is a solution for I1 under M.I ′2 is a universal solution for I1 under M.
⇒ there is a homomorphism from I ′2 to I2.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 6/33
Incomplete Data and Knowledge Exchange
A framework for Data Exchange with incomplete data was proposed by Arenas et al.[2011].
Σ1Σ1
source signature
Σ2Σ2
target signature
I1
incomplete source instance
I2
mappingM
solution
logical theory T1 logical theory T2
source KB target KB
solution
We specialize this framework to Description Logics, and in particular to DL-LiteR.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 7/33
Incomplete Data and Knowledge Exchange
A framework for Data Exchange with incomplete data was proposed by Arenas et al.[2011].
Σ1Σ1
source signature
Σ2Σ2
target signature
I1
incomplete source instance
I2
mappingM
solution
logical theory T1 logical theory T2
source KB target KB
solution
We specialize this framework to Description Logics, and in particular to DL-LiteR.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 7/33
Incomplete Data and Knowledge Exchange
A framework for Data Exchange with incomplete data was proposed by Arenas et al.[2011].
Σ1Σ1
source signature
Σ2Σ2
target signature
I1
incomplete source instance
I2
mappingM
solution
logical theory T1 logical theory T2
source KB target KB
solution
We specialize this framework to Description Logics, and in particular to DL-LiteR.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 7/33
Description Logic DL-LiteR
Description Logics (DLs) are decidable fragments of First-Order Logic,used as Knowledge Representation formalisms.
DLs talk about the domain of interest by means of
• concepts (unary predicates): Author, Book, A, B
• roles (binary predicates): AuthorOf, WrittenBy, P, R
DL-LiteR is a light-weight DL that asserts
• concept inclusions and disjointness of atomic concepts A,the domain ∃P and the range ∃P− of atomic roles P,
Book v ∃WrittenBy ,
• role inclusions and disjointness of atomic roles P andtheir inverses P−, AuthorOf vWrittenBy−,
• ground facts Author(nabokov), AuthorOf(nabokov,lolita),A(a), P(a, b).
TBox T
ABox A
Satisfiability check over a DL-LiteR KB K = 〈T ,A〉 can be donein polynomial time (in fact, in NLogSpace).
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 8/33
Description Logic DL-LiteR
Description Logics (DLs) are decidable fragments of First-Order Logic,used as Knowledge Representation formalisms.
DLs talk about the domain of interest by means of
• concepts (unary predicates): Author, Book, A, B
• roles (binary predicates): AuthorOf, WrittenBy, P, R
DL-LiteR is a light-weight DL that asserts
• concept inclusions and disjointness of atomic concepts A,the domain ∃P and the range ∃P− of atomic roles P,
Book v ∃WrittenBy ,
• role inclusions and disjointness of atomic roles P andtheir inverses P−, AuthorOf vWrittenBy−,
• ground facts Author(nabokov), AuthorOf(nabokov,lolita),A(a), P(a, b).
TBox T
ABox A
Satisfiability check over a DL-LiteR KB K = 〈T ,A〉 can be donein polynomial time (in fact, in NLogSpace).
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 8/33
Description Logic DL-LiteR
Description Logics (DLs) are decidable fragments of First-Order Logic,used as Knowledge Representation formalisms.
DLs talk about the domain of interest by means of
• concepts (unary predicates): Author, Book, A, B
• roles (binary predicates): AuthorOf, WrittenBy, P, R
DL-LiteR is a light-weight DL that asserts
• concept inclusions and disjointness of atomic concepts A,the domain ∃P and the range ∃P− of atomic roles P,
Book v ∃WrittenBy ,
• role inclusions and disjointness of atomic roles P andtheir inverses P−, AuthorOf vWrittenBy−,
• ground facts Author(nabokov), AuthorOf(nabokov,lolita),A(a), P(a, b).
TBox T
ABox A
Satisfiability check over a DL-LiteR KB K = 〈T ,A〉 can be donein polynomial time (in fact, in NLogSpace).
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 8/33
Outline
1 Introduction
2 Summary of Work
3 Results
4 Technical DevelopmentUniversal SolutionsUniversal UCQ-solutionsUCQ-representations
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 9/33
The Summary of my PhD Work
In this thesis, we
1 Propose a general framework for exchanging Description LogicKnowledge Bases.
2 Define and analyse relevant reasoning problems in this setting.
3 Develop reasoning techniques and characterize the computationalcomplexity of the problems.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 10/33
Outline
1 Introduction
2 Summary of Work
3 Results
4 Technical DevelopmentUniversal SolutionsUniversal UCQ-solutionsUCQ-representations
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 11/33
1. Knowledge Base Exchange Framework
source KB
target KB
solution
mappingM
T1
T2
Σ1Σ1
source signature
A1
A1
B1 C1
D1
Σ2Σ2
target signature
A2
A2
B2 C2
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 12/33
1. Knowledge Base Exchange Framework
source KB target KB
solution
mappingM
T1 T2
Σ1Σ1
source signature
A1
A1
B1 C1
D1
Σ2Σ2
target signature
A2
A2
B2 C2
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 12/33
2. Reasoning Problems
Solution
ABox
Decision problem
universal solution
preserves all models
universal UCQ-solution
preserves all answers toUnions of Conjunctive Queries
UCQ-representation
preserves all answers to UCQs,independently of the ABox
simpleABoxes
A(a
),∃R
(a),R
(a, b
)
extended
ABoxes
cont
ain
labe
led
nulls
membershipIs K2 (resp. T2) a solutionfor K1 (resp. T1) under M?
non-emptinessIs there any solutionfor K1 (resp. T1) under M?
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 13/33
2. Reasoning Problems
Solution
ABox
Decision problem
universal solution
preserves all models
universal UCQ-solution
preserves all answers toUnions of Conjunctive Queries
UCQ-representation
preserves all answers to UCQs,independently of the ABox
simpleABoxes
A(a
),∃R
(a),R
(a, b
)
extended
ABoxes
cont
ain
labe
led
nulls
membershipIs K2 (resp. T2) a solutionfor K1 (resp. T1) under M?
non-emptinessIs there any solutionfor K1 (resp. T1) under M?
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 13/33
2. Reasoning Problems
Solution
ABox
Decision problem
universal solution
preserves all models
universal UCQ-solution
preserves all answers toUnions of Conjunctive Queries
UCQ-representation
preserves all answers to UCQs,independently of the ABoxsimpleABoxes
A(a
),∃R
(a),R
(a, b
)
extended
ABoxes
cont
ain
labe
led
nulls
membershipIs K2 (resp. T2) a solutionfor K1 (resp. T1) under M?
non-emptinessIs there any solutionfor K1 (resp. T1) under M?
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 13/33
2. Reasoning Problems
Solution
ABox
Decision problem
universal solution
preserves all models
universal UCQ-solution
preserves all answers toUnions of Conjunctive Queries
UCQ-representation
preserves all answers to UCQs,independently of the ABoxsimpleABoxes
A(a
),∃R
(a),R
(a, b
)
extended
ABoxes
cont
ain
labe
led
nulls
membershipIs K2 (resp. T2) a solutionfor K1 (resp. T1) under M?
non-emptinessIs there any solutionfor K1 (resp. T1) under M?
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 13/33
3. Results
Universal solutions simple ABoxes extended ABoxesMembership PTime-complete NP-completeNon-emptiness PTime-complete PSpace-hard, in ExpTime
Universal UCQ-solutions simple ABoxes extended ABoxesMembership PSpace-hard in ExpTimeNon-emptiness in ExpTime PSpace-hard
UCQ-representations ComplexityMembership NLogSpace-completeNon-emptiness NLogSpace-complete
1 games
1 e4 e5 2 Nf3 Nc6 3 Bb5 3. . . a6 3. . . Nf6 4 Ba4
8 3VrzZ4Wb5™Xq1Tk4˜Wb2Un3—Vr7 zZ6Yp6šYp6YpzZ6Yp6šYp6Yp6 6YpzZ2UnzZzzZzzZ5 zZzzZz6šYpzzZz4 4WBzZzzZ6YPzZzzZ3 zZzzZzzZ2UNzZz2 6YP6šYP6YP6šYPz6šYP6YP6šYP1 3—VR2UN4˜WB5XQ1•TKzzZ3VR
a b c d e f g h
ad-hoc 2 automata
circuit value problem 3-colorability validity of QBF
3 game-theoretic
1 e4 e5 2 Nf3 Nc6 3 Bb5 3. . . a6 3. . . Nf6 4 Ba4
8 3VrzZ4Wb5™Xq1Tk4˜Wb2Un3—Vr7 zZ6Yp6šYp6YpzZ6Yp6šYp6Yp6 6YpzZ2UnzZzzZzzZ5 zZzzZz6šYpzzZz4 4WBzZzzZ6YPzZzzZ3 zZzzZzzZ2UNzZz2 6YP6šYP6YP6šYPz6šYP6YP6šYP1 3—VR2UN4˜WB5XQ1•TKzzZ3VR
a b c d e f g h
validity of QBF
4 5 graph-theoretic
reachability in directed graphs
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 14/33
3. Results
Universal solutions simple ABoxes extended ABoxesMembership PTime-complete NP-completeNon-emptiness PTime-complete PSpace-hard, in ExpTime
Universal UCQ-solutions simple ABoxes extended ABoxesMembership PSpace-hard in ExpTimeNon-emptiness in ExpTime PSpace-hard
UCQ-representations ComplexityMembership NLogSpace-completeNon-emptiness NLogSpace-complete
1 games
1 e4 e5 2 Nf3 Nc6 3 Bb5 3. . . a6 3. . . Nf6 4 Ba4
8 3VrzZ4Wb5™Xq1Tk4˜Wb2Un3—Vr7 zZ6Yp6šYp6YpzZ6Yp6šYp6Yp6 6YpzZ2UnzZzzZzzZ5 zZzzZz6šYpzzZz4 4WBzZzzZ6YPzZzzZ3 zZzzZzzZ2UNzZz2 6YP6šYP6YP6šYPz6šYP6YP6šYP1 3—VR2UN4˜WB5XQ1•TKzzZ3VR
a b c d e f g h
ad-hoc 2 automata
circuit value problem 3-colorability validity of QBF
3 game-theoretic
1 e4 e5 2 Nf3 Nc6 3 Bb5 3. . . a6 3. . . Nf6 4 Ba4
8 3VrzZ4Wb5™Xq1Tk4˜Wb2Un3—Vr7 zZ6Yp6šYp6YpzZ6Yp6šYp6Yp6 6YpzZ2UnzZzzZzzZ5 zZzzZz6šYpzzZz4 4WBzZzzZ6YPzZzzZ3 zZzzZzzZ2UNzZz2 6YP6šYP6YP6šYPz6šYP6YP6šYP1 3—VR2UN4˜WB5XQ1•TKzzZ3VR
a b c d e f g h
validity of QBF
4 5 graph-theoretic
reachability in directed graphs
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 14/33
3. Results
Universal solutions simple ABoxes extended ABoxesMembership PTime-complete NP-completeNon-emptiness PTime-complete PSpace-hard, in ExpTime
Universal UCQ-solutions simple ABoxes extended ABoxesMembership PSpace-hard in ExpTimeNon-emptiness in ExpTime PSpace-hard
UCQ-representations ComplexityMembership NLogSpace-completeNon-emptiness NLogSpace-complete
1 games
1 e4 e5 2 Nf3 Nc6 3 Bb5 3. . . a6 3. . . Nf6 4 Ba4
8 3VrzZ4Wb5™Xq1Tk4˜Wb2Un3—Vr7 zZ6Yp6šYp6YpzZ6Yp6šYp6Yp6 6YpzZ2UnzZzzZzzZ5 zZzzZz6šYpzzZz4 4WBzZzzZ6YPzZzzZ3 zZzzZzzZ2UNzZz2 6YP6šYP6YP6šYPz6šYP6YP6šYP1 3—VR2UN4˜WB5XQ1•TKzzZ3VR
a b c d e f g h
ad-hoc 2 automata
circuit value problem 3-colorability validity of QBF
3 game-theoretic
1 e4 e5 2 Nf3 Nc6 3 Bb5 3. . . a6 3. . . Nf6 4 Ba4
8 3VrzZ4Wb5™Xq1Tk4˜Wb2Un3—Vr7 zZ6Yp6šYp6YpzZ6Yp6šYp6Yp6 6YpzZ2UnzZzzZzzZ5 zZzzZz6šYpzzZz4 4WBzZzzZ6YPzZzzZ3 zZzzZzzZ2UNzZz2 6YP6šYP6YP6šYPz6šYP6YP6šYP1 3—VR2UN4˜WB5XQ1•TKzzZ3VR
a b c d e f g h
validity of QBF
4 5 graph-theoretic
reachability in directed graphs
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 14/33
3. Results
Universal solutions simple ABoxes extended ABoxesMembership PTime-complete NP-completeNon-emptiness PTime-complete PSpace-hard, in ExpTime
Universal UCQ-solutions simple ABoxes extended ABoxesMembership PSpace-hard in ExpTimeNon-emptiness in ExpTime PSpace-hard
UCQ-representations ComplexityMembership NLogSpace-completeNon-emptiness NLogSpace-complete
1 games
1 e4 e5 2 Nf3 Nc6 3 Bb5 3. . . a6 3. . . Nf6 4 Ba4
8 3VrzZ4Wb5™Xq1Tk4˜Wb2Un3—Vr7 zZ6Yp6šYp6YpzZ6Yp6šYp6Yp6 6YpzZ2UnzZzzZzzZ5 zZzzZz6šYpzzZz4 4WBzZzzZ6YPzZzzZ3 zZzzZzzZ2UNzZz2 6YP6šYP6YP6šYPz6šYP6YP6šYP1 3—VR2UN4˜WB5XQ1•TKzzZ3VR
a b c d e f g h
ad-hoc 2 automata
circuit value problem 3-colorability validity of QBF
3 game-theoretic
1 e4 e5 2 Nf3 Nc6 3 Bb5 3. . . a6 3. . . Nf6 4 Ba4
8 3VrzZ4Wb5™Xq1Tk4˜Wb2Un3—Vr7 zZ6Yp6šYp6YpzZ6Yp6šYp6Yp6 6YpzZ2UnzZzzZzzZ5 zZzzZz6šYpzzZz4 4WBzZzzZ6YPzZzzZ3 zZzzZzzZ2UNzZz2 6YP6šYP6YP6šYPz6šYP6YP6šYP1 3—VR2UN4˜WB5XQ1•TKzzZ3VR
a b c d e f g h
validity of QBF
4 5 graph-theoretic
reachability in directed graphs
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 14/33
3. Results
Universal solutions simple ABoxes extended ABoxesMembership PTime-complete NP-completeNon-emptiness PTime-complete PSpace-hard, in ExpTime
Universal UCQ-solutions simple ABoxes extended ABoxesMembership PSpace-hard in ExpTimeNon-emptiness in ExpTime PSpace-hard
UCQ-representations ComplexityMembership NLogSpace-completeNon-emptiness NLogSpace-complete
1 games
1 e4 e5 2 Nf3 Nc6 3 Bb5 3. . . a6 3. . . Nf6 4 Ba4
8 3VrzZ4Wb5™Xq1Tk4˜Wb2Un3—Vr7 zZ6Yp6šYp6YpzZ6Yp6šYp6Yp6 6YpzZ2UnzZzzZzzZ5 zZzzZz6šYpzzZz4 4WBzZzzZ6YPzZzzZ3 zZzzZzzZ2UNzZz2 6YP6šYP6YP6šYPz6šYP6YP6šYP1 3—VR2UN4˜WB5XQ1•TKzzZ3VR
a b c d e f g h
ad-hoc 2 automata
circuit value problem 3-colorability validity of QBF
3 game-theoretic
1 e4 e5 2 Nf3 Nc6 3 Bb5 3. . . a6 3. . . Nf6 4 Ba4
8 3VrzZ4Wb5™Xq1Tk4˜Wb2Un3—Vr7 zZ6Yp6šYp6YpzZ6Yp6šYp6Yp6 6YpzZ2UnzZzzZzzZ5 zZzzZz6šYpzzZz4 4WBzZzzZ6YPzZzzZ3 zZzzZzzZ2UNzZz2 6YP6šYP6YP6šYPz6šYP6YP6šYP1 3—VR2UN4˜WB5XQ1•TKzzZ3VR
a b c d e f g h
validity of QBF
4 5 graph-theoretic
reachability in directed graphs
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 14/33
Outline
1 Introduction
2 Summary of Work
3 Results
4 Technical DevelopmentUniversal SolutionsUniversal UCQ-solutionsUCQ-representations
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 15/33
The Essence of Knowledge Base ExchangeA mapping is a triple M = (Σ1,Σ2, T12),
where T12 is a set of DL-LiteR inclusions from Σ1 to Σ2
〈T1,A1〉 is a DL-LiteR knowledge base over Σ1 (source KB)〈T2,A2〉 is a DL-LiteR knowledge base over Σ2 (target KB)
source KB target KB
solution
mapping M
T1 T2
Σ1Σ1
source signature
A1
A1
B1 C1
D1
Σ2Σ2
target signature
A2
A2
B2 C2
For a KB K, we denote by UK the canonical model of K (chase in databases).
• 〈T2,A2〉 is a universal solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iff?
T2 = ∅ and UA2 is Σ2-homomorphically equivalent to U〈T1∪T12,A1〉.
• 〈T2,A2〉 is a universal UCQ-solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iffU〈T2,A2〉 is finitely Σ2-homomorphically equivalent to U〈T1∪T12,A1〉.
• T2 is a UCQ-representation for T1 under M = (Σ1,Σ2, T12) iff??
for each source ABox A1, U〈T2∪T12,A1〉 is Σ2-homomorphically equivalent to U〈T1∪T12,A1〉.
We present our 5 techniques that check the existence of the homomorphisms.
? plus one more condition with little technical impact
?? plus one more condition for checking consistency
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 16/33
The Essence of Knowledge Base ExchangeA mapping is a triple M = (Σ1,Σ2, T12),
where T12 is a set of DL-LiteR inclusions from Σ1 to Σ2
〈T1,A1〉 is a DL-LiteR knowledge base over Σ1 (source KB)〈T2,A2〉 is a DL-LiteR knowledge base over Σ2 (target KB)
source KB target KB
solution
mapping M
T1 T2
Σ1Σ1
source signature
A1
A1
B1 C1
D1
Σ2Σ2
target signature
A2
A2
B2 C2
For a KB K, we denote by UK the canonical model of K (chase in databases).
• 〈T2,A2〉 is a universal solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iff?
T2 = ∅ and UA2 is Σ2-homomorphically equivalent to U〈T1∪T12,A1〉.
• 〈T2,A2〉 is a universal UCQ-solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iffU〈T2,A2〉 is finitely Σ2-homomorphically equivalent to U〈T1∪T12,A1〉.
• T2 is a UCQ-representation for T1 under M = (Σ1,Σ2, T12) iff??
for each source ABox A1, U〈T2∪T12,A1〉 is Σ2-homomorphically equivalent to U〈T1∪T12,A1〉.
We present our 5 techniques that check the existence of the homomorphisms.
? plus one more condition with little technical impact
?? plus one more condition for checking consistency
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 16/33
The Essence of Knowledge Base ExchangeA mapping is a triple M = (Σ1,Σ2, T12),
where T12 is a set of DL-LiteR inclusions from Σ1 to Σ2
〈T1,A1〉 is a DL-LiteR knowledge base over Σ1 (source KB)〈T2,A2〉 is a DL-LiteR knowledge base over Σ2 (target KB)
source KB target KB
solution
mapping M
T1 T2
Σ1Σ1
source signature
A1
A1
B1 C1
D1
Σ2Σ2
target signature
A2
A2
B2 C2
For a KB K, we denote by UK the canonical model of K (chase in databases).
• 〈T2,A2〉 is a universal solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iff?
T2 = ∅ and UA2 is Σ2-homomorphically equivalent to U〈T1∪T12,A1〉.
• 〈T2,A2〉 is a universal UCQ-solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iffU〈T2,A2〉 is finitely Σ2-homomorphically equivalent to U〈T1∪T12,A1〉.
• T2 is a UCQ-representation for T1 under M = (Σ1,Σ2, T12) iff??
for each source ABox A1, U〈T2∪T12,A1〉 is Σ2-homomorphically equivalent to U〈T1∪T12,A1〉.
We present our 5 techniques that check the existence of the homomorphisms.
? plus one more condition with little technical impact
?? plus one more condition for checking consistency
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 16/33
The Canonical and Generating Models
Let K = 〈T ,A〉, where
A = {B(a)}T = {B v ∃R
∃R− v ∃S∃S− v ∃SR v Q}
Σ = {Q, S}
The canonical model UK
Σ
a
awR
awRwS
awRwSwS
B
···
R
,Q
S
S
The generating model GK
Σ
a
wR
wS
B
R
,Q
S
S
We call wR and wS witnesses of K, denoted Wit(K).Moreover, we write, e.g., a K wR , or wR K wS .
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 17/33
The Canonical and Generating Models
Let K = 〈T ,A〉, where
A = {B(a)}
T = {B v ∃R∃R− v ∃S∃S− v ∃SR v Q}
Σ = {Q, S}
The canonical model UK
Σ
a
awR
awRwS
awRwSwS
B
···
R
,Q
S
S
The generating model GK
Σ
a
wR
wS
B
R
,Q
S
S
We call wR and wS witnesses of K, denoted Wit(K).Moreover, we write, e.g., a K wR , or wR K wS .
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 17/33
The Canonical and Generating Models
Let K = 〈T ,A〉, where
A = {B(a)}T = {B v ∃R
∃R− v ∃S∃S− v ∃SR v Q}
Σ = {Q, S}
The canonical model UK
Σ
a
awR
awRwS
awRwSwS
B
···
R
,Q
S
S
The generating model GK
Σ
a
wR
wS
B
R
,Q
S
S
We call wR and wS witnesses of K, denoted Wit(K).Moreover, we write, e.g., a K wR , or wR K wS .
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 17/33
The Canonical and Generating Models
Let K = 〈T ,A〉, where
A = {B(a)}T = {B v ∃R
∃R− v ∃S
∃S− v ∃SR v Q}
Σ = {Q, S}
The canonical model UK
Σ
a
awR
awRwS
awRwSwS
B
···
R
,Q
S
S
The generating model GK
Σ
a
wR
wS
B
R
,Q
S
S
We call wR and wS witnesses of K, denoted Wit(K).Moreover, we write, e.g., a K wR , or wR K wS .
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 17/33
The Canonical and Generating Models
Let K = 〈T ,A〉, where
A = {B(a)}T = {B v ∃R
∃R− v ∃S∃S− v ∃S
R v Q}
Σ = {Q, S}
The canonical model UK
Σ
a
awR
awRwS
awRwSwS
B
···
R
,Q
S
S
The generating model GK
Σ
a
wR
wS
B
R
,Q
S
S
We call wR and wS witnesses of K, denoted Wit(K).Moreover, we write, e.g., a K wR , or wR K wS .
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 17/33
The Canonical and Generating Models
Let K = 〈T ,A〉, where
A = {B(a)}T = {B v ∃R
∃R− v ∃S∃S− v ∃SR v Q}
Σ = {Q, S}
The canonical model UK
Σ
a
awR
awRwS
awRwSwS
B
···
R,Q
S
S
The generating model GK
Σ
a
wR
wS
B
R,Q
S
S
We call wR and wS witnesses of K, denoted Wit(K).Moreover, we write, e.g., a K wR , or wR K wS .
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 17/33
The Canonical and Generating Models
Let K = 〈T ,A〉, where
A = {B(a)}T = {B v ∃R
∃R− v ∃S∃S− v ∃SR v Q}
Σ = {Q, S}
The canonical model UKΣ
a
awR
awRwS
awRwSwS
B
···
R
,Q
S
S
The generating model GKΣ
a
wR
wS
B
R
,Q
S
S
We call wR and wS witnesses of K, denoted Wit(K).Moreover, we write, e.g., a K wR , or wR K wS .
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 17/33
Outline
1 Introduction
2 Summary of Work
3 Results
4 Technical DevelopmentUniversal SolutionsUniversal UCQ-solutionsUCQ-representations
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 18/33
Membership for Simple Universal Solutions is in PTime
A2 is a universal solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iff? there exist
• a Σ2-homomorphism from UA2 to U〈T1∪T12,A1〉,
EASY
• a Σ2-homomorphism from U〈T1∪T12,A1〉 to UA2 .
via Reachability Games on graphs
For a KB K, an ABox A, and a signature Σ,we construct a reachability game (G,F ) such that
there exists a Σ-homomorphism from UK to UA
iff
Duplicator has a strategy against Spoiler
to avoid F in G.
Decidable in PTime.
1
2 3
4 5 6 7
8 9 10
11
Duplicator wins from 1Spoiler wins from 1
G
F
? plus one more condition of no technical interest
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 19/33
Membership for Simple Universal Solutions is in PTime
A2 is a universal solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iff? there exist
• a Σ2-homomorphism from UA2 to U〈T1∪T12,A1〉, EASY
• a Σ2-homomorphism from U〈T1∪T12,A1〉 to UA2 .
via Reachability Games on graphs
For a KB K, an ABox A, and a signature Σ,we construct a reachability game (G,F ) such that
there exists a Σ-homomorphism from UK to UA
iff
Duplicator has a strategy against Spoiler
to avoid F in G.
Decidable in PTime.
1
2 3
4 5 6 7
8 9 10
11
Duplicator wins from 1Spoiler wins from 1
G
F
? plus one more condition of no technical interest
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 19/33
Membership for Simple Universal Solutions is in PTime
A2 is a universal solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iff? there exist
• a Σ2-homomorphism from UA2 to U〈T1∪T12,A1〉, EASY
• a Σ2-homomorphism from U〈T1∪T12,A1〉 to UA2 . via Reachability Games on graphs
For a KB K, an ABox A, and a signature Σ,we construct a reachability game (G,F ) such that
there exists a Σ-homomorphism from UK to UA
iff
Duplicator has a strategy against Spoiler
to avoid F in G.
Decidable in PTime.
1
2 3
4 5 6 7
8 9 10
11
Duplicator wins from 1Spoiler wins from 1
G
F
? plus one more condition of no technical interest
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 19/33
Membership for Simple Universal Solutions is in PTime
A2 is a universal solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iff? there exist
• a Σ2-homomorphism from UA2 to U〈T1∪T12,A1〉, EASY
• a Σ2-homomorphism from U〈T1∪T12,A1〉 to UA2 . via Reachability Games on graphs
For a KB K, an ABox A, and a signature Σ,we construct a reachability game (G,F ) such that
there exists a Σ-homomorphism from UK to UA
iff
Duplicator has a strategy against Spoiler
to avoid F in G.
Decidable in PTime.
1
2 3
4 5 6 7
8 9 10
11
Duplicator wins from 1Spoiler wins from 1
G
F
? plus one more condition of no technical interest
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 19/33
Membership for Simple Universal Solutions is in PTime
A2 is a universal solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iff? there exist
• a Σ2-homomorphism from UA2 to U〈T1∪T12,A1〉, EASY
• a Σ2-homomorphism from U〈T1∪T12,A1〉 to UA2 . via Reachability Games on graphs
For a KB K, an ABox A, and a signature Σ,we construct a reachability game (G,F ) such that
there exists a Σ-homomorphism from UK to UA
iff
Duplicator has a strategy against Spoiler
to avoid F in G.
Decidable in PTime.
1
2 3
4 5 6 7
8 9 10
11
Duplicator wins from 1
Spoiler wins from 1
G
F
? plus one more condition of no technical interest
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 19/33
Membership for Simple Universal Solutions is in PTime
A2 is a universal solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iff? there exist
• a Σ2-homomorphism from UA2 to U〈T1∪T12,A1〉, EASY
• a Σ2-homomorphism from U〈T1∪T12,A1〉 to UA2 . via Reachability Games on graphs
For a KB K, an ABox A, and a signature Σ,we construct a reachability game (G,F ) such that
there exists a Σ-homomorphism from UK to UA
iff
Duplicator has a strategy against Spoiler
to avoid F in G.
Decidable in PTime.
1
2 3
4 5
6
7
8
9
10
11
Duplicator wins from 1
Spoiler wins from 1
G
F
? plus one more condition of no technical interest
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 19/33
Membership for Simple Universal Solutions is in PTime
A2 is a universal solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iff? there exist
• a Σ2-homomorphism from UA2 to U〈T1∪T12,A1〉, EASY
• a Σ2-homomorphism from U〈T1∪T12,A1〉 to UA2 . via Reachability Games on graphs
For a KB K, an ABox A, and a signature Σ,we construct a reachability game (G,F ) such that
there exists a Σ-homomorphism from UK to UA
iff
Duplicator has a strategy against Spoiler
to avoid F in G. Decidable in PTime.
1
2 3
4 5
6
7
8
9
10
11
Duplicator wins from 1Spoiler wins from 1
G
F
? plus one more condition of no technical interest
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 19/33
1 Reachability game to check Σ-homomorphism from UK to UA
a 7→ a
a, a wR
wR 7→ a
a, wR wR
a, a wS
wS 7→ b
b, wS wQ
G
F
The reachability game GΣ(GK,UA) = (G,F )
C
C
C
C
C
C
C
of polynomialsize
a
awR
awRwR
awS
awSwQ···
R ′
R ′
S ′
Q ′
UΣK
a
b
R ′
S ′
UΣA
h
Duplicator has a strategy against Spoiler to avoid F in G from a 7→ a iff
UK is Σ-homomorphically embeddable into UA.
decidable in polynomial time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 20/33
1 Reachability game to check Σ-homomorphism from UK to UA
a 7→ a
a, a wR
wR 7→ a
a, wR wR
a, a wS
wS 7→ b
b, wS wQ
G
F
The reachability game GΣ(GK,UA) = (G,F )
C
C
C
C
C
C
C
of polynomialsize
a
awR
awRwR
awS
awSwQ···
R ′
R ′
S ′
Q ′
UΣK
a
b
R ′
S ′
UΣA
h
Duplicator has a strategy against Spoiler to avoid F in G from a 7→ a iff
UK is Σ-homomorphically embeddable into UA.
decidable in polynomial time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 20/33
1 Reachability game to check Σ-homomorphism from UK to UA
a 7→ a
a, a wR
wR 7→ a
a, wR wR
a, a wS
wS 7→ b
b, wS wQ
G
F
The reachability game GΣ(GK,UA) = (G,F )
C
C
C
C
C
C
C
of polynomialsize
a
awR
awRwR
awS
awSwQ···
R ′
R ′
S ′
Q ′
UΣK
a
b
R ′
S ′
UΣA
h
Duplicator has a strategy against Spoiler to avoid F in G from a 7→ a iff
UK is Σ-homomorphically embeddable into UA.
decidable in polynomial time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 20/33
1 Reachability game to check Σ-homomorphism from UK to UA
a 7→ a
a, a wR
wR 7→ a
a, wR wR
a, a wS
wS 7→ b
b, wS wQ
G
F
The reachability game GΣ(GK,UA) = (G,F )
C
C
C
C
C
C
C
of polynomialsize
a
awR
awRwR
awS
awSwQ···
R ′
R ′
S ′
Q ′
UΣK
a
b
R ′
S ′
UΣA
h
Duplicator has a strategy against Spoiler to avoid F in G from a 7→ a iff
UK is Σ-homomorphically embeddable into UA.
decidable in polynomial time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 20/33
1 Reachability game to check Σ-homomorphism from UK to UA
a 7→ a
a, a wR
wR 7→ a
a, wR wR
a, a wS
wS 7→ b
b, wS wQ
G
F
The reachability game GΣ(GK,UA) = (G,F )
C
C
C
C
C
C
C
of polynomialsize
a
awR
awRwR
awS
awSwQ···
R ′
R ′
S ′
Q ′
UΣK
a
b
R ′
S ′
UΣA
h
Duplicator has a strategy against Spoiler to avoid F in G from a 7→ a iff
UK is Σ-homomorphically embeddable into UA.
decidable in polynomial time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 20/33
1 Reachability game to check Σ-homomorphism from UK to UA
a 7→ a
a, a wR
wR 7→ a
a, wR wR
a, a wS
wS 7→ b
b, wS wQ
G
F
The reachability game GΣ(GK,UA) = (G,F )
C
C
C
C
C
C
C
of polynomialsize
a
awR
awRwR
awS
awSwQ···
R ′
R ′
S ′
Q ′
UΣK
a
b
R ′
S ′
UΣA
h
Duplicator has a strategy against Spoiler to avoid F in G from a 7→ a iff
UK is Σ-homomorphically embeddable into UA.
decidable in polynomial time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 20/33
1 Reachability game to check Σ-homomorphism from UK to UA
a 7→ a
a, a wR
wR 7→ a
a, wR wR
a, a wS
wS 7→ b
b, wS wQ
G
F
The reachability game GΣ(GK,UA) = (G,F )
C
C
C
C
C
C
C
of polynomialsize
a
awR
awRwR
awS
awSwQ···
R ′
R ′
S ′
Q ′
UΣK
a
b
R ′
S ′
UΣA
h
Duplicator has a strategy against Spoiler to avoid F in G from a 7→ a iff
UK is Σ-homomorphically embeddable into UA.
decidable in polynomial time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 20/33
1 Reachability game to check Σ-homomorphism from UK to UA
a 7→ a
a, a wR
wR 7→ a
a, wR wR
a, a wS
wS 7→ b
b, wS wQ
G
F
The reachability game GΣ(GK,UA) = (G,F )
C
C
C
C
C
C
C
of polynomialsize
a
awR
awRwR
awS
awSwQ···
R ′
R ′
S ′
Q ′
UΣK
a
b
R ′
S ′
UΣA
h
Duplicator has a strategy against Spoiler to avoid F in G from a 7→ a iff
UK is Σ-homomorphically embeddable into UA.
decidable in polynomial time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 20/33
1 Reachability game to check Σ-homomorphism from UK to UA
a 7→ a
a, a wR
wR 7→ a
a, wR wR
a, a wS
wS 7→ b
b, wS wQ
G
F
The reachability game GΣ(GK,UA) = (G,F )
C
C
C
C
C
C
C
of polynomialsize
a
awR
awRwR
awS
awSwQ···
R ′
R ′
S ′
Q ′
UΣK
a
b
R ′
S ′
UΣA
h
Duplicator has a strategy against Spoiler to avoid F in G from a 7→ a iff
UK is Σ-homomorphically embeddable into UA.
decidable in polynomial time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 20/33
1 Reachability game to check Σ-homomorphism from UK to UA
a 7→ a
a, a wR
wR 7→ a
a, wR wR
a, a wS
wS 7→ b
b, wS wQ
G
F
The reachability game GΣ(GK,UA) = (G,F )
C
C
C
C
C
C
C
of polynomialsize
a
awR
awRwR
awS
awSwQ···
R ′
R ′
S ′
Q ′
UΣK
a
b
R ′
S ′
UΣA
h
Duplicator has a strategy against Spoiler to avoid F in G from a 7→ a iff
UK is Σ-homomorphically embeddable into UA.
decidable in polynomial time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 20/33
1 Reachability game to check Σ-homomorphism from UK to UA
a 7→ a
a, a wR
wR 7→ a
a, wR wR
a, a wS
wS 7→ b
b, wS wQ
G
F
The reachability game GΣ(GK,UA) = (G,F )
C
C
C
C
C
C
C
of polynomialsize
a
awR
awRwR
awS
awSwQ···
R ′
R ′
S ′
Q ′
UΣK
a
b
R ′
S ′
UΣA
h
Duplicator has a strategy against Spoiler to avoid F in G from a 7→ a iff
UK is Σ-homomorphically embeddable into UA.
decidable in polynomial time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 20/33
Non-emptiness for Extended Universal Solutions is in ExpTime
There exists a universal solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iff?
U〈T1∪T12,A1〉 is Σ2-homomorphically embeddable to a finite subset of itself.
For KBs K1, K2, and a signature Σ, we construct a two-way alternating automaton(2ATA) A that accepts a tree encoding a finite subset C of UK2 such that
UK1 is Σ-homomorphically embeddable into C.
A accepts trees T of the form
T
εR
1a
1 · 1w
1 · 1 · 1S ···
for a ∈ Ind(K2), w ∈Wit(K2).
A “launches” two threads from ε:
• one thread verifies that T encodes a finitesubset C of UK2 (stops when sees an S).
• the other thread tries to find aΣ-homomorphism from UK1 to Cby traversing T down and up.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 21/33
Non-emptiness for Extended Universal Solutions is in ExpTime
There exists a universal solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iff?
U〈T1∪T12,A1〉 is Σ2-homomorphically embeddable to a finite subset of itself.
For KBs K1, K2, and a signature Σ, we construct a two-way alternating automaton(2ATA) A that accepts a tree encoding a finite subset C of UK2 such that
UK1 is Σ-homomorphically embeddable into C.
A accepts trees T of the form
T
εR
1a
1 · 1w
1 · 1 · 1S ···
for a ∈ Ind(K2), w ∈Wit(K2).
A “launches” two threads from ε:
• one thread verifies that T encodes a finitesubset C of UK2 (stops when sees an S).
• the other thread tries to find aΣ-homomorphism from UK1 to Cby traversing T down and up.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 21/33
Non-emptiness for Extended Universal Solutions is in ExpTime
There exists a universal solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iff?
U〈T1∪T12,A1〉 is Σ2-homomorphically embeddable to a finite subset of itself.
For KBs K1, K2, and a signature Σ, we construct a two-way alternating automaton(2ATA) A that accepts a tree encoding a finite subset C of UK2 such that
UK1 is Σ-homomorphically embeddable into C.
A accepts trees T of the form
T
εR
1a
1 · 1w
1 · 1 · 1S ···
for a ∈ Ind(K2), w ∈Wit(K2).
A “launches” two threads from ε:
• one thread verifies that T encodes a finitesubset C of UK2 (stops when sees an S).
• the other thread tries to find aΣ-homomorphism from UK1 to Cby traversing T down and up.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 21/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?
there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?
there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?
there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?
there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?
there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?
there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?
there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?
there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?
there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?
there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?
there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?
there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?
there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?
there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
2 2ATA A to check Σ-homomorphism from UK1 to a finite subset C of UK2
TC
εR
1a
1 · 1wQ
1 · 1 · 1S ···
εr
···
···
q0
qf
αa
ωQ
ωQ
qh
νb
κaP
νP
κaP
νP
C is a finite subsetof UK2
?there is a Σ-homomorphismfrom UK1
to C?
A run of A over TC
States occurringinfinitely often
b
bwP
bwPwP ···
P ′
P ′
P ′
UΣK1
a
awQ
P ′
P ′
awQwQ
P ′
···
CΣ
h
There exists an accepting run of the 2ATA A over TC iff
UK1 is Σ2-homomorphically embeddable into C, a finite subset of UK2 .
decidable in exponential time
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 22/33
Outline
1 Introduction
2 Summary of Work
3 Results
4 Technical DevelopmentUniversal SolutionsUniversal UCQ-solutionsUCQ-representations
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 23/33
Membership for Universal UCQ-Solutions is in ExpTime
〈T2,A2〉 is a universal UCQ-solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iffU〈T2,A2〉 is finitely Σ2-homomorphically equivalent to U〈T1∪T12,A1〉.
For KBs K1, K2, and a signature Σ, we use reachability games for checking whetherUK1 is finitely Σ-homomorphically embeddable into UK2 .
Now, UK2 is in general infinite.
• we cannot use the game GΣ(GK1 ,UK2 ) = (Gi ,Fi ), a straightforward extension ofGΣ(GK,UA), as Gi is in general infinite.
• so we define a game GΣ(GK1 ,GK2 ) = (Gf ,Ff ), where Gf is of exponential size andthe states have a more complicated structure involving
{u1, . . . , uk} 7→ w .
Hence, we obtain an ExpTime upper bound.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 24/33
Membership for Universal UCQ-Solutions is in ExpTime
〈T2,A2〉 is a universal UCQ-solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iffU〈T2,A2〉 is finitely Σ2-homomorphically equivalent to U〈T1∪T12,A1〉.
For KBs K1, K2, and a signature Σ, we use reachability games for checking whetherUK1 is finitely Σ-homomorphically embeddable into UK2 .
Now, UK2 is in general infinite.
• we cannot use the game GΣ(GK1 ,UK2 ) = (Gi ,Fi ), a straightforward extension ofGΣ(GK,UA), as Gi is in general infinite.
• so we define a game GΣ(GK1 ,GK2 ) = (Gf ,Ff ), where Gf is of exponential size andthe states have a more complicated structure involving
{u1, . . . , uk} 7→ w .
Hence, we obtain an ExpTime upper bound.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 24/33
Membership for Universal UCQ-Solutions is in ExpTime
〈T2,A2〉 is a universal UCQ-solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iffU〈T2,A2〉 is finitely Σ2-homomorphically equivalent to U〈T1∪T12,A1〉.
For KBs K1, K2, and a signature Σ, we use reachability games for checking whetherUK1 is finitely Σ-homomorphically embeddable into UK2 .
Now, UK2 is in general infinite.
• we cannot use the game GΣ(GK1 ,UK2 ) = (Gi ,Fi ), a straightforward extension ofGΣ(GK,UA), as Gi is in general infinite.
• so we define a game GΣ(GK1 ,GK2 ) = (Gf ,Ff ), where Gf is of exponential size andthe states have a more complicated structure involving
{u1, . . . , uk} 7→ w .
Hence, we obtain an ExpTime upper bound.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 24/33
Membership for Universal UCQ-Solutions is in ExpTime
〈T2,A2〉 is a universal UCQ-solution for 〈T1,A1〉 under M = (Σ1,Σ2, T12) iffU〈T2,A2〉 is finitely Σ2-homomorphically equivalent to U〈T1∪T12,A1〉.
For KBs K1, K2, and a signature Σ, we use reachability games for checking whetherUK1 is finitely Σ-homomorphically embeddable into UK2 .
Now, UK2 is in general infinite.
• we cannot use the game GΣ(GK1 ,UK2 ) = (Gi ,Fi ), a straightforward extension ofGΣ(GK,UA), as Gi is in general infinite.
• so we define a game GΣ(GK1 ,GK2 ) = (Gf ,Ff ), where Gf is of exponential size andthe states have a more complicated structure involving
{u1, . . . , uk} 7→ w .
Hence, we obtain an ExpTime upper bound.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 24/33
Outline
1 Introduction
2 Summary of Work
3 Results
4 Technical DevelopmentUniversal SolutionsUniversal UCQ-solutionsUCQ-representations
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 25/33
UCQ-representability
mappingM:
T12
T1
T2
UCQ-representation
such that for each ABox A1 and each query q
A1
cert(q, 〈T1 ∪ T12,A1〉)
cert(q, 〈T2 ∪ T12,A1〉)T1
T12
T12
T2
Σ1Σ1
source signature
A1
B1 C1
D1
Σ2Σ2
target signature
A2
B2 C2
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 26/33
UCQ-representability
mappingM:
T12
T1 T2
UCQ-representation
such that for each ABox A1 and each query q
A1
cert(q, 〈T1 ∪ T12,A1〉)
cert(q, 〈T2 ∪ T12,A1〉)T1
T12
T12
T2
Σ1Σ1
source signature
A1
B1 C1
D1
Σ2Σ2
target signature
A2
B2 C2
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 26/33
UCQ-representability
mappingM:
T12
T1 T2
UCQ-representation
such that for each ABox A1 and each query q
A1
cert(q, 〈T1 ∪ T12,A1〉)
cert(q, 〈T2 ∪ T12,A1〉)T1
T12
T12
T2
Σ1Σ1
source signature
A1
B1 C1
D1
Σ2Σ2
target signature
A2
B2 C2
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 26/33
UCQ-representability
mappingM:
T12
T1 T2
UCQ-representation
such that for each ABox A1 and each query q
A1
cert(q, 〈T1 ∪ T12,A1〉)
cert(q, 〈T2 ∪ T12,A1〉)
T1
T12
T12
T2
Σ1Σ1
source signature
A1
B1 C1
D1
Σ2Σ2
target signature
A2
B2 C2
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 26/33
UCQ-representability
mappingM:
T12
T1 T2
UCQ-representation
such that for each ABox A1 and each query q
A1
cert(q, 〈T1 ∪ T12,A1〉)
cert(q, 〈T2 ∪ T12,A1〉)T1
T12
T12
T2
Σ1Σ1
source signature
A1
B1 C1
D1
Σ2Σ2
target signature
A2
B2 C2
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 26/33
UCQ-representability
mappingM:
T12
T1 T2
UCQ-representation
such that for each ABox A1 and each query q
A1
cert(q, 〈T1 ∪ T12,A1〉)
cert(q, 〈T2 ∪ T12,A1〉)T1
T12
T12
T2
Σ1Σ1
source signature
A1
B1 C1
D1
Σ2Σ2
target signature
A2
B2 C2
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 26/33
Membership for UCQ-representations is in NLogSpaceT2 is a UCQ-representation of T1 under M = (Σ1,Σ2, T12) iff:
Let A1 = {X (a)}.
1)
T1 ∪ T12 |= X v X ′
X ,X ′a
G〈T1∪T12,A1〉
iff
T2 ∪ T12 |= X v X ′
X ,X ′a
G〈T2∪T12,A1〉
2)
a 〈T1∪T12,A1〉 wR
Xa
BwR
S
G〈T1∪T12,A1〉
⇒
there exists y in G〈T2∪T12,A1〉:
X
,By =
a
By
Sy1 ···yn
By
G〈T2∪T12,A1〉
Let A1 = {X (a),Y (a)}
3) 〈T1 ∪ T12,A1〉 is inconsistent iff 〈T2 ∪ T12,A1〉 is inconsistent
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 27/33
Membership for UCQ-representations is in NLogSpaceT2 is a UCQ-representation of T1 under M = (Σ1,Σ2, T12) iff:
Let A1 = {X (a)}.
1)
T1 ∪ T12 |= X v X ′
X ,X ′a
G〈T1∪T12,A1〉
iff
T2 ∪ T12 |= X v X ′
X ,X ′a
G〈T2∪T12,A1〉
2)
a 〈T1∪T12,A1〉 wR
Xa
BwR
S
G〈T1∪T12,A1〉
⇒
there exists y in G〈T2∪T12,A1〉:
X
,By =
a
By
Sy1 ···yn
By
G〈T2∪T12,A1〉
Let A1 = {X (a),Y (a)}
3) 〈T1 ∪ T12,A1〉 is inconsistent iff 〈T2 ∪ T12,A1〉 is inconsistent
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 27/33
Membership for UCQ-representations is in NLogSpaceT2 is a UCQ-representation of T1 under M = (Σ1,Σ2, T12) iff:
Let A1 = {X (a)}.
1)
T1 ∪ T12 |= X v X ′
X ,X ′a
G〈T1∪T12,A1〉
iff
T2 ∪ T12 |= X v X ′
X ,X ′a
G〈T2∪T12,A1〉
2)
a 〈T1∪T12,A1〉 wR
Xa
BwR
S
G〈T1∪T12,A1〉
⇒
there exists y in G〈T2∪T12,A1〉:
X
,By =
a
By
S
y1 ···yn
By
G〈T2∪T12,A1〉
Let A1 = {X (a),Y (a)}
3) 〈T1 ∪ T12,A1〉 is inconsistent iff 〈T2 ∪ T12,A1〉 is inconsistent
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 27/33
Membership for UCQ-representations is in NLogSpaceT2 is a UCQ-representation of T1 under M = (Σ1,Σ2, T12) iff:
Let A1 = {X (a)}.
1)
T1 ∪ T12 |= X v X ′
X ,X ′a
G〈T1∪T12,A1〉
iff
T2 ∪ T12 |= X v X ′
X ,X ′a
G〈T2∪T12,A1〉
2)
a 〈T1∪T12,A1〉 wR
Xa
BwR
S
G〈T1∪T12,A1〉
⇒
there exists y in G〈T2∪T12,A1〉:
X ,By = a
By
Sy1 ···yn
By
G〈T2∪T12,A1〉
Let A1 = {X (a),Y (a)}
3) 〈T1 ∪ T12,A1〉 is inconsistent iff 〈T2 ∪ T12,A1〉 is inconsistent
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 27/33
Membership for UCQ-representations is in NLogSpaceT2 is a UCQ-representation of T1 under M = (Σ1,Σ2, T12) iff:
Let A1 = {X (a)}.
1)
T1 ∪ T12 |= X v X ′
X ,X ′a
G〈T1∪T12,A1〉
iff
T2 ∪ T12 |= X v X ′
X ,X ′a
G〈T2∪T12,A1〉
2)
a 〈T1∪T12,A1〉 wR
Xa
BwR
S
G〈T1∪T12,A1〉
⇒
there exists y in G〈T2∪T12,A1〉:
X
,By =
a
By
S
y1 ···yn
By
G〈T2∪T12,A1〉
Let A1 = {X (a),Y (a)}
3) 〈T1 ∪ T12,A1〉 is inconsistent iff 〈T2 ∪ T12,A1〉 is inconsistent
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 27/33
Membership for UCQ-representations is in NLogSpaceT2 is a UCQ-representation of T1 under M = (Σ1,Σ2, T12) iff:
Let A1 = {X (a)}.
1)
T1 ∪ T12 |= X v X ′
X ,X ′a
G〈T1∪T12,A1〉
iff
T2 ∪ T12 |= X v X ′
X ,X ′a
G〈T2∪T12,A1〉
2)
a 〈T1∪T12,A1〉 wR
Xa
BwR
S
G〈T1∪T12,A1〉
⇒
there exists y in G〈T2∪T12,A1〉:
X
,By =
a
By
S
y1 ···yn
By
G〈T2∪T12,A1〉
Let A1 = {X (a),Y (a)}
3) 〈T1 ∪ T12,A1〉 is inconsistent iff 〈T2 ∪ T12,A1〉 is inconsistent
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 27/33
Non-emptiness for UCQ-representations is in NLogSpace
Let Σ1 = {A(·),B(·),C(·)}, Σ2 = {A′(·),B ′(·),C ′(·)}, and T1 = {A v B}.
Is there T2, a UCQ-representation for T1 under M = (Σ1,Σ2, T12),where T12 is as follows (gray arrows):
A
C
B
A′
C ′
B ′
NO
A
C
B
A′
C ′
B ′
YES
A
C
B
A′
C ′
B ′
YES
A
C
B
A′
C ′
B ′
NO
A
C
B
A′
C ′
B ′
YES
We have that T1 ∪ T12 |= A v B ′.So T2 should be such that T2 ∪ T12 |= A v B ′.
If T2 ∪ T12 |= C v B ′, then it should be the case T1 ∪ T12 |= C v B ′.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 28/33
Non-emptiness for UCQ-representations is in NLogSpace
Let Σ1 = {A(·),B(·),C(·)}, Σ2 = {A′(·),B ′(·),C ′(·)}, and T1 = {A v B}.
Is there T2, a UCQ-representation for T1 under M = (Σ1,Σ2, T12),where T12 is as follows (gray arrows):
A
C
B
A′
C ′
B ′
NO
A
C
B
A′
C ′
B ′
YES
A
C
B
A′
C ′
B ′
YES
A
C
B
A′
C ′
B ′
NO
A
C
B
A′
C ′
B ′
YES
We have that T1 ∪ T12 |= A v B ′.So T2 should be such that T2 ∪ T12 |= A v B ′.
If T2 ∪ T12 |= C v B ′, then it should be the case T1 ∪ T12 |= C v B ′.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 28/33
Non-emptiness for UCQ-representations is in NLogSpace
Let Σ1 = {A(·),B(·),C(·)}, Σ2 = {A′(·),B ′(·),C ′(·)}, and T1 = {A v B}.
Is there T2, a UCQ-representation for T1 under M = (Σ1,Σ2, T12),where T12 is as follows (gray arrows):
A
C
B
A′
C ′
B ′
NO
A
C
B
A′
C ′
B ′
YES
A
C
B
A′
C ′
B ′
YES
A
C
B
A′
C ′
B ′
NO
A
C
B
A′
C ′
B ′
YES
We have that T1 ∪ T12 |= A v B ′.So T2 should be such that T2 ∪ T12 |= A v B ′.
If T2 ∪ T12 |= C v B ′, then it should be the case T1 ∪ T12 |= C v B ′.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 28/33
Non-emptiness for UCQ-representations is in NLogSpace
Let Σ1 = {A(·),B(·),C(·)}, Σ2 = {A′(·),B ′(·),C ′(·)}, and T1 = {A v B}.
Is there T2, a UCQ-representation for T1 under M = (Σ1,Σ2, T12),where T12 is as follows (gray arrows):
A
C
B
A′
C ′
B ′
NO
A
C
B
A′
C ′
B ′
YES
A
C
B
A′
C ′
B ′
YES
A
C
B
A′
C ′
B ′
NO
A
C
B
A′
C ′
B ′
YES
We have that T1 ∪ T12 |= A v B ′.So T2 should be such that T2 ∪ T12 |= A v B ′.
If T2 ∪ T12 |= C v B ′, then it should be the case T1 ∪ T12 |= C v B ′.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 28/33
Non-emptiness for UCQ-representations is in NLogSpace
Let Σ1 = {A(·),B(·),C(·)}, Σ2 = {A′(·),B ′(·),C ′(·)}, and T1 = {A v B}.
Is there T2, a UCQ-representation for T1 under M = (Σ1,Σ2, T12),where T12 is as follows (gray arrows):
A
C
B
A′
C ′
B ′
NO
A
C
B
A′
C ′
B ′
YES
A
C
B
A′
C ′
B ′
YES
A
C
B
A′
C ′
B ′
NO
A
C
B
A′
C ′
B ′
YES
We have that T1 ∪ T12 |= A v B ′.So T2 should be such that T2 ∪ T12 |= A v B ′.
If T2 ∪ T12 |= C v B ′, then it should be the case T1 ∪ T12 |= C v B ′.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 28/33
Non-emptiness for UCQ-representations is in NLogSpace
Let Σ1 = {A(·),B(·),C(·)}, Σ2 = {A′(·),B ′(·),C ′(·)}, and T1 = {A v B}.
Is there T2, a UCQ-representation for T1 under M = (Σ1,Σ2, T12),where T12 is as follows (gray arrows):
A
C
B
A′
C ′
B ′
NO
A
C
B
A′
C ′
B ′
YES
A
C
B
A′
C ′
B ′
YES
A
C
B
A′
C ′
B ′
NO
A
C
B
A′
C ′
B ′
YES
We have that T1 ∪ T12 |= A v B ′.So T2 should be such that T2 ∪ T12 |= A v B ′.
If T2 ∪ T12 |= C v B ′, then it should be the case T1 ∪ T12 |= C v B ′.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 28/33
Publications
Conference Publications
• E. Botoeva, R. Kontchakov, V. Ryzhikov, F. Wolter, and M. Zakharyaschev.Query inseparability for description logic knowledge bases.In Proc. of the 14th Int. Conf. on the Principles of Knowledge Representation and Reasoning (KR2014). To appear.
• M. Arenas, E. Botoeva, D. Calvanese, and V. Ryzhikov.Exchanging OWL 2 QL knowledge bases.In Proc. of the 23rd Int. Joint Conf. on Artificial Intelligence (IJCAI 2013), pages 703-710, 2013.
• M. Arenas, E. Botoeva, D. Calvanese, V. Ryzhikov, and E. Sherkhonov.Exchanging description logic knowledge bases.In Proc. of the 13th Int. Conf. on the Principles of Knowledge Representation and Reasoning (KR2012), pages 563-567.
Workshop Publications
• M. Arenas, E. Botoeva, D. Calvanese, and V. Ryzhikov.Computing solutions in OWL 2 QL knowledge exchange.In Proc. of the 26th Int. Workshop on Description Logic (DL 2013), volume 1014, pages 4-16, 2013.
• M. Arenas, E. Botoeva, D. Calvanese, V. Ryzhikov, and E. Sherkhonov.Representability in DL-Liter knowledge base exchange.In Proc. of the 25th Int. Workshop on Description Logic (DL 2012), volume 846, 2012.
• M. Arenas, E. Botoeva, and D. Calvanese.Knowledge base exchange.In Proc. of the 24th Int. Workshop on Description Logic (DL 2011), volume 745, 2011.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 29/33
Publications cont.
Technical Reports
• M. Arenas, E. Botoeva, D. Calvanese, and V. Ryzhikov.Exchanging OWL 2 QL knowledge bases (extended version).CoRR Technical Report arXiv:1304.5810, arXiv.org e-Print archive, 2013. Available athttp://arxiv.org/abs/1304.5810.
Under Submission
• M. Arenas, E. Botoeva, D. Calvanese, and V. Ryzhikov.Knowledge base exchange: The case of OWL 2 QL.Under submission to a journal.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 30/33
Thank you for your attention!
Universal solutions simple ABoxes extended ABoxesMembership PTime-complete NP-completeNon-emptiness PTime-complete PSpace-hard, in ExpTime
Universal UCQ-solutions simple ABoxes extended ABoxesMembership PSpace-hard in ExpTimeNon-emptiness in ExpTime PSpace-hard
UCQ-representations ComplexityMembership NLogSpace-completeNon-emptiness NLogSpace-complete
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 31/33
Marcelo Arenas, Jorge Perez, and Juan L. Reutter. Data exchange beyond completedata. pages 83–94, 2011.
Ronald Fagin, Phokion G. Kolaitis, Renee J. Miller, and Lucian Popa. Data exchange:Semantics and query answering. pages 207–224, 2003.
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 32/33
Knowledge Base Exchange: Example
∃AuthorOf − v ∃BookGenreAuthorOf − v WrittenByTaxNumber v SSN
M :
nabokov lolita
tolkien lotr
AuthorOfA1 :
∃AuthorOf v AuthorAuthor v ∃TaxNumber
T1 :
nabokovlolita
tolkienlotr
WrittenBy
nabokov null1
tolkien null2
SSNA2 :
lolita null3
lotr null4
BookGenre
nabokovlolita
tolkienlotr
WrittenByA2 :
∃WrittenBy− v ∃SSN∃WrittenBy v ∃BookGenreT2 :
〈T2,A2〉 is a universal-UCQ solution for 〈T1,A1〉 under M (with simple ABoxes).
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 32/33
Knowledge Base Exchange: Example
∃AuthorOf − v ∃BookGenreAuthorOf − v WrittenByTaxNumber v SSN
M :
nabokov lolita
tolkien lotr
AuthorOfA1 :
∃AuthorOf v AuthorAuthor v ∃TaxNumber
T1 :nabokovlolita
tolkienlotr
WrittenBy
nabokov null1
tolkien null2
SSNA2 :
lolita null3
lotr null4
BookGenre
nabokovlolita
tolkienlotr
WrittenByA2 :
∃WrittenBy− v ∃SSN∃WrittenBy v ∃BookGenreT2 :
A2 is a universal solution for 〈T1,A1〉 under M (with extended ABoxes).
〈T2,A2〉 is auniversal-UCQ solution for 〈T1,A1〉 under M (with simple ABoxes).
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 32/33
Knowledge Base Exchange: Example
∃AuthorOf − v ∃BookGenreAuthorOf − v WrittenByTaxNumber v SSN
M :
nabokov lolita
tolkien lotr
AuthorOfA1 :
∃AuthorOf v AuthorAuthor v ∃TaxNumber
T1 :
nabokovlolita
tolkienlotr
WrittenBy
nabokov null1
tolkien null2
SSNA2 :
lolita null3
lotr null4
BookGenre
nabokovlolita
tolkienlotr
WrittenByA2 :
∃WrittenBy− v ∃SSN∃WrittenBy v ∃BookGenreT2 :
〈T2,A2〉 is a universal-UCQ solution for 〈T1,A1〉 under M (with simple ABoxes).Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 32/33
3 Reachability game to check Σ-homomorphism from UK1 to UK2
∅, {a,wR} 7→ a
{a,wR}, a, a wS
{a,wR}, {wS ,wR} 7→wT
{wS ,wR},wT , wS wS
{wS ,wR}, {wS ,wR} 7→wT
{a,wR}, a, wR wR
{a,wR}, {wR} 7→ a
{wR}, a, wR wR
{wR}, {wR} 7→ a
Gs
The reachability game G sΣ = (Gs ,Fs)
C
C
C
C
C
C
C
C
C
UK1 UK2
h
a
awS
awSwR
awSw2R
awSw3R
aw 2S
aw 2SwR
aw 2Sw
2R
aw 3S··· ···
···
S
S
R
R
R S
R
R
a
awT
awTwT
···
S ,R−
S ,R−
R
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 33/33
3 Reachability game to check Σ-homomorphism from UK1 to UK2
∅, {a,wR} 7→ a
{a,wR}, a, a wS
{a,wR}, {wS ,wR} 7→wT
{wS ,wR},wT , wS wS
{wS ,wR}, {wS ,wR} 7→wT
{a,wR}, a, wR wR
{a,wR}, {wR} 7→ a
{wR}, a, wR wR
{wR}, {wR} 7→ a
Gs
The reachability game G sΣ = (Gs ,Fs)
C
C
C
C
C
C
C
C
C
UK1 UK2
h
a
awS
awSwR
awSw2R
awSw3R
aw 2S
aw 2SwR
aw 2Sw
2R
aw 3S··· ···
···
S
S
R
R
R S
R
R
a
awT
awTwT
···
S ,R−
S ,R−
R
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 33/33
3 Reachability game to check Σ-homomorphism from UK1 to UK2
∅, {a,wR} 7→ a
{a,wR}, a, a wS
{a,wR}, {wS ,wR} 7→wT
{wS ,wR},wT , wS wS
{wS ,wR}, {wS ,wR} 7→wT
{a,wR}, a, wR wR
{a,wR}, {wR} 7→ a
{wR}, a, wR wR
{wR}, {wR} 7→ a
Gs
The reachability game G sΣ = (Gs ,Fs)
C
C
C
C
C
C
C
C
C
UK1 UK2
ha
awS
awSwR
awSw2R
awSw3R
aw 2S
aw 2SwR
aw 2Sw
2R
aw 3S··· ···
···
S
S
R
R
R S
R
R
a
awT
awTwT
···
S ,R−
S ,R−
R
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 33/33
3 Reachability game to check Σ-homomorphism from UK1 to UK2
∅, {a,wR} 7→ a
{a,wR}, a, a wS
{a,wR}, {wS ,wR} 7→wT
{wS ,wR},wT , wS wS
{wS ,wR}, {wS ,wR} 7→wT
{a,wR}, a, wR wR
{a,wR}, {wR} 7→ a
{wR}, a, wR wR
{wR}, {wR} 7→ a
Gs
The reachability game G sΣ = (Gs ,Fs)
C
C
C
C
C
C
C
C
C
UK1 UK2
ha
awS
awSwR
awSw2R
awSw3R
aw 2S
aw 2SwR
aw 2Sw
2R
aw 3S··· ···
···
S
S
R
R
R S
R
R
a
awT
awTwT
···
S ,R−
S ,R−
R
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 33/33
3 Reachability game to check Σ-homomorphism from UK1 to UK2
∅, {a,wR} 7→ a
{a,wR}, a, a wS
{a,wR}, {wS ,wR} 7→wT
{wS ,wR},wT , wS wS
{wS ,wR}, {wS ,wR} 7→wT
{a,wR}, a, wR wR
{a,wR}, {wR} 7→ a
{wR}, a, wR wR
{wR}, {wR} 7→ a
Gs
The reachability game G sΣ = (Gs ,Fs)
C
C
C
C
C
C
C
C
C
UK1 UK2
ha
awS
awSwR
awSw2R
awSw3R
aw 2S
aw 2SwR
aw 2Sw
2R
aw 3S··· ···
···
S
S
R
R
R S
R
R
a
awT
awTwT
···
S ,R−
S ,R−
R
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 33/33
3 Reachability game to check Σ-homomorphism from UK1 to UK2
∅, {a,wR} 7→ a
{a,wR}, a, a wS
{a,wR}, {wS ,wR} 7→wT
{wS ,wR},wT , wS wS
{wS ,wR}, {wS ,wR} 7→wT
{a,wR}, a, wR wR
{a,wR}, {wR} 7→ a
{wR}, a, wR wR
{wR}, {wR} 7→ a
Gs
The reachability game G sΣ = (Gs ,Fs)
C
C
C
C
C
C
C
C
C
UK1 UK2
ha
awS
awSwR
awSw2R
awSw3R
aw 2S
aw 2SwR
aw 2Sw
2R
aw 3S··· ···
···
S
S
R
R
R S
R
R
a
awT
awTwT
···
S ,R−
S ,R−
R
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 33/33
3 Reachability game to check Σ-homomorphism from UK1 to UK2
∅, {a,wR} 7→ a
{a,wR}, a, a wS
{a,wR}, {wS ,wR} 7→wT
{wS ,wR},wT , wS wS
{wS ,wR}, {wS ,wR} 7→wT
{a,wR}, a, wR wR
{a,wR}, {wR} 7→ a
{wR}, a, wR wR
{wR}, {wR} 7→ a
Gs
The reachability game G sΣ = (Gs ,Fs)
C
C
C
C
C
C
C
C
C
UK1 UK2
ha
awS
awSwR
awSw2R
awSw3R
aw 2S
aw 2SwR
aw 2Sw
2R
aw 3S··· ···
···
S
S
R
R
R S
R
R
a
awT
awTwT
···
S ,R−
S ,R−
R
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 33/33
3 Reachability game to check Σ-homomorphism from UK1 to UK2
∅, {a,wR} 7→ a
{a,wR}, a, a wS
{a,wR}, {wS ,wR} 7→wT
{wS ,wR},wT , wS wS
{wS ,wR}, {wS ,wR} 7→wT
{a,wR}, a, wR wR
{a,wR}, {wR} 7→ a
{wR}, a, wR wR
{wR}, {wR} 7→ a
Gs
The reachability game G sΣ = (Gs ,Fs)
C
C
C
C
C
C
C
C
C
UK1 UK2
ha
awS
awSwR
awSw2R
awSw3R
aw 2S
aw 2SwR
aw 2Sw
2R
aw 3S··· ···
···
S
S
R
R
R S
R
R
a
awT
awTwT
···
S ,R−
S ,R−
R
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 33/33
3 Reachability game to check Σ-homomorphism from UK1 to UK2
∅, {a,wR} 7→ a
{a,wR}, a, a wS
{a,wR}, {wS ,wR} 7→wT
{wS ,wR},wT , wS wS
{wS ,wR}, {wS ,wR} 7→wT
{a,wR}, a, wR wR
{a,wR}, {wR} 7→ a
{wR}, a, wR wR
{wR}, {wR} 7→ a
Gs
The reachability game G sΣ = (Gs ,Fs)
C
C
C
C
C
C
C
C
C
UK1 UK2
ha
awS
awSwR
awSw2R
awSw3R
aw 2S
aw 2SwR
aw 2Sw
2R
aw 3S··· ···
···
S
S
R
R
R S
R
R
a
awT
awTwT
···
S ,R−
S ,R−
R
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 33/33
3 Reachability game to check Σ-homomorphism from UK1 to UK2
∅, {a,wR} 7→ a
{a,wR}, a, a wS
{a,wR}, {wS ,wR} 7→wT
{wS ,wR},wT , wS wS
{wS ,wR}, {wS ,wR} 7→wT
{a,wR}, a, wR wR
{a,wR}, {wR} 7→ a
{wR}, a, wR wR
{wR}, {wR} 7→ a
Gs
The reachability game G sΣ = (Gs ,Fs)
C
C
C
C
C
C
C
C
C
UK1 UK2
ha
awS
awSwR
awSw2R
awSw3R
aw 2S
aw 2SwR
aw 2Sw
2R
aw 3S··· ···
···
S
S
R
R
R S
R
R
a
awT
awTwT
···
S ,R−
S ,R−
R
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 33/33
3 Reachability game to check Σ-homomorphism from UK1 to UK2
∅, {a,wR} 7→ a
{a,wR}, a, a wS
{a,wR}, {wS ,wR} 7→wT
{wS ,wR},wT , wS wS
{wS ,wR}, {wS ,wR} 7→wT
{a,wR}, a, wR wR
{a,wR}, {wR} 7→ a
{wR}, a, wR wR
{wR}, {wR} 7→ a
Gs
The reachability game G sΣ = (Gs ,Fs)
C
C
C
C
C
C
C
C
C
UK1 UK2
ha
awS
awSwR
awSw2R
awSw3R
aw 2S
aw 2SwR
aw 2Sw
2R
aw 3S··· ···
···
S
S
R
R
R S
R
R
a
awT
awTwT
···
S ,R−
S ,R−
R
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 33/33
3 Reachability game to check Σ-homomorphism from UK1 to UK2
∅, {a,wR} 7→ a
{a,wR}, a, a wS
{a,wR}, {wS ,wR} 7→wT
{wS ,wR},wT , wS wS
{wS ,wR}, {wS ,wR} 7→wT
{a,wR}, a, wR wR
{a,wR}, {wR} 7→ a
{wR}, a, wR wR
{wR}, {wR} 7→ a
Gs
The reachability game G sΣ = (Gs ,Fs)
C
C
C
C
C
C
C
C
C
UK1 UK2
ha
awS
awSwR
awSw2R
awSw3R
aw 2S
aw 2SwR
aw 2Sw
2R
aw 3S··· ···
···
S
S
R
R
R S
R
R
a
awT
awTwT
···
S ,R−
S ,R−
R
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 33/33
3 Reachability game to check Σ-homomorphism from UK1 to UK2
∅, {a,wR} 7→ a
{a,wR}, a, a wS
{a,wR}, {wS ,wR} 7→wT
{wS ,wR},wT , wS wS
{wS ,wR}, {wS ,wR} 7→wT
{a,wR}, a, wR wR
{a,wR}, {wR} 7→ a
{wR}, a, wR wR
{wR}, {wR} 7→ a
Gs
The reachability game G sΣ = (Gs ,Fs)
C
C
C
C
C
C
C
C
C
UK1 UK2
ha
awS
awSwR
awSw2R
awSw3R
aw 2S
aw 2SwR
aw 2Sw
2R
aw 3S··· ···
···
S
S
R
R
R S
R
R
a
awT
awTwT
···
S ,R−
S ,R−
R
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 33/33
3 Reachability game to check Σ-homomorphism from UK1 to UK2
∅, {a,wR} 7→ a
{a,wR}, a, a wS
{a,wR}, {wS ,wR} 7→wT
{wS ,wR},wT , wS wS
{wS ,wR}, {wS ,wR} 7→wT
{a,wR}, a, wR wR
{a,wR}, {wR} 7→ a
{wR}, a, wR wR
{wR}, {wR} 7→ a
Gs
The reachability game G sΣ = (Gs ,Fs)
C
C
C
C
C
C
C
C
C
UK1 UK2
ha
awS
awSwR
awSw2R
awSw3R
aw 2S
aw 2SwR
aw 2Sw
2R
aw 3S··· ···
···
S
S
R
R
R S
R
R
a
awT
awTwT
···
S ,R−
S ,R−
R
Elena Botoeva(FUB) Description Logic Knowledge Base Exchange 33/33