Representability in DL-Lite R Knowledge Base Exchange Marcelo Arenas - - PowerPoint PPT Presentation

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Representability in DL-Lite R Knowledge Base Exchange Marcelo Arenas - - PowerPoint PPT Presentation

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Knowledge Base Exchange Representability in DL - Lite R Conclusions Representability in DL-Lite R Knowledge Base Exchange Marcelo Arenas 1 Elena Botoeva 2 Diego Calvanese 2 Vladislav Ryzhikov 2 Evgeny Sherkhonov 3 1 Dept. of Computer Science, PUC

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SLIDE 1

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Representability in DL-LiteR Knowledge Base Exchange

Marcelo Arenas1 Elena Botoeva2 Diego Calvanese2 Vladislav Ryzhikov2 Evgeny Sherkhonov3

1 Dept. of Computer Science, PUC Chile

marenas@ing.puc.cl

2 KRDB Research Centre, Free Univ. of Bozen-Bolzano, Italy

lastname@inf.unibz.it ISLA, University of Amsterdam, Netherlands e.sherkhonov@uva.nl

Description Logics Workshop 7 June 2012, Rome

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 1/20

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SLIDE 2

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Outline

1 Knowledge Base Exchange 2 Representability in DL-LiteR 3 Conclusions

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 2/20

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SLIDE 3

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Outline

1 Knowledge Base Exchange 2 Representability in DL-LiteR 3 Conclusions

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 3/20

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SLIDE 4

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Knowledge Base Exchange

Problem

given a mapping M and a source knowledge base (KB) K1, compute a target KB K2 that is a solution for K1 under M.

Σ1 Σ1 source signature Σ2 Σ2 target signature

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 4/20

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SLIDE 5

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Knowledge Base Exchange

Problem

given a mapping M and a source knowledge base (KB) K1, compute a target KB K2 that is a solution for K1 under M.

Σ1 Σ1 source signature Σ2 Σ2 target signature M

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 4/20

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SLIDE 6

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Knowledge Base Exchange

Problem

given a mapping M and a source knowledge base (KB) K1, compute a target KB K2 that is a solution for K1 under M.

Σ1 Σ1 source signature Σ2 Σ2 target signature M source KB K1 T1 A1 B1 C1 D1 A1

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 4/20

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SLIDE 7

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Knowledge Base Exchange

Problem

given a mapping M and a source knowledge base (KB) K1, compute a target KB K2 that is a solution for K1 under M.

Σ1 Σ1 source signature Σ2 Σ2 target signature M source KB K1 T1 A1 B1 C1 D1 A1 target KB K2 T2 A2 B2 C2 A2

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 4/20

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SLIDE 8

Knowledge Base Exchange: Example

T1 : Item FootWear OpenShoes Strappy Plateau HighHeel . . . Apparel Gown LightGown FormalGown CocktailGown EveningGown . . .

hasMatchingOpenShoes matches

A1 :

brown_sand Strappy wedge_sand Plateau heel_sand HighHeel turq_gown LightGown

hasMatchingOpenShoes

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SLIDE 9

Knowledge Base Exchange: Example

T1 : Item FootWear OpenShoes Strappy Plateau HighHeel . . . Apparel Gown LightGown FormalGown CocktailGown EveningGown . . .

hasMatchingOpenShoes matches

Product Shoes Sandals Classic Platform Heeled Clothing Dress SummerDress WorkDress EveningDress

fitsWithSandals fitsWith

M : A1 :

brown_sand Strappy wedge_sand Plateau heel_sand HighHeel turq_gown LightGown

hasMatchingOpenShoes

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SLIDE 10

Example: Universal Solution

Definition

K2 is a universal solution for K1 under M if Mod(K2) = Mod(K1 ∪ M)|Σ2.

Theorem

If chaseT1(A1) is finite, then chaseT1∪M,Σ2(A1) is a universal solution for T1, A1 under M.

T1 ∪ M : Item FootWear OpenShoes Strappy Plateau HighHeel Apparel Gown LightGown . . . Product Shoes Sandals Classic Platform Heeled Clothing Dress SummerDress A1 : Strappy Plateau HighHeel LightGown Auniv

2

:

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SLIDE 11

Example: Universal Solution

Definition

K2 is a universal solution for K1 under M if Mod(K2) = Mod(K1 ∪ M)|Σ2.

Theorem

If chaseT1(A1) is finite, then chaseT1∪M,Σ2(A1) is a universal solution for T1, A1 under M.

T1 ∪ M : Item FootWear OpenShoes Strappy Plateau HighHeel Apparel Gown LightGown . . . Product Shoes Sandals Classic Platform Heeled Clothing Dress SummerDress A1 : Strappy Plateau HighHeel LightGown Auniv

2

: Classic Platform Heeled SummerDress

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SLIDE 12

Example: Universal Solution

Definition

K2 is a universal solution for K1 under M if Mod(K2) = Mod(K1 ∪ M)|Σ2.

Theorem

If chaseT1(A1) is finite, then chaseT1∪M,Σ2(A1) is a universal solution for T1, A1 under M.

T1 ∪ M : Item FootWear OpenShoes Strappy Plateau HighHeel Apparel Gown LightGown . . . Product Shoes Sandals Classic Platform Heeled Clothing Dress SummerDress A1 : Strappy Plateau HighHeel LightGown Auniv

2

: Classic Platform Heeled SummerDress Sandals Dress

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SLIDE 13

Example: Universal Solution

Definition

K2 is a universal solution for K1 under M if Mod(K2) = Mod(K1 ∪ M)|Σ2.

Theorem

If chaseT1(A1) is finite, then chaseT1∪M,Σ2(A1) is a universal solution for T1, A1 under M.

T1 ∪ M : Item FootWear OpenShoes Strappy Plateau HighHeel Apparel Gown LightGown . . . Product Shoes Sandals Classic Platform Heeled Clothing Dress SummerDress A1 : Strappy Plateau HighHeel LightGown Auniv

2

: Classic Platform Heeled SummerDress Sandals Dress Shoes Clothing

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SLIDE 14

Example: Universal Solution

Definition

K2 is a universal solution for K1 under M if Mod(K2) = Mod(K1 ∪ M)|Σ2.

Theorem

If chaseT1(A1) is finite, then chaseT1∪M,Σ2(A1) is a universal solution for T1, A1 under M.

T1 ∪ M : Item FootWear OpenShoes Strappy Plateau HighHeel Apparel Gown LightGown . . . Product Shoes Sandals Classic Platform Heeled Clothing Dress SummerDress A1 : Strappy Plateau HighHeel LightGown Auniv

2

: Classic Platform Heeled SummerDress Sandals Dress Shoes Clothing Product

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SLIDE 15

Example: Universal Solution

Definition

K2 is a universal solution for K1 under M if Mod(K2) = Mod(K1 ∪ M)|Σ2.

Theorem

If chaseT1(A1) is finite, then chaseT1∪M,Σ2(A1) is a universal solution for T1, A1 under M.

T1 ∪ M : Item FootWear OpenShoes Strappy Plateau HighHeel Apparel Gown LightGown . . . Product Shoes Sandals Classic Platform Heeled Clothing Dress SummerDress hasMatchingOpenShoes matches fitsWithSandals fitsWith A1 : Strappy Plateau HighHeel LightGown

h a s M a t c h i n g O p e n S h

  • e

s

Auniv

2

: Classic Platform Heeled SummerDress Sandals Dress Shoes Clothing Product

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SLIDE 16

Example: Universal Solution

Definition

K2 is a universal solution for K1 under M if Mod(K2) = Mod(K1 ∪ M)|Σ2.

Theorem

If chaseT1(A1) is finite, then chaseT1∪M,Σ2(A1) is a universal solution for T1, A1 under M.

T1 ∪ M : Item FootWear OpenShoes Strappy Plateau HighHeel Apparel Gown LightGown . . . Product Shoes Sandals Classic Platform Heeled Clothing Dress SummerDress hasMatchingOpenShoes matches fitsWithSandals fitsWith A1 : Strappy Plateau HighHeel LightGown

h a s M a t c h i n g O p e n S h

  • e

s

Auniv

2

: Classic Platform Heeled SummerDress Sandals Dress Shoes Clothing Product fitsWithSandals

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SLIDE 17

Example: Universal Solution

Definition

K2 is a universal solution for K1 under M if Mod(K2) = Mod(K1 ∪ M)|Σ2.

Theorem

If chaseT1(A1) is finite, then chaseT1∪M,Σ2(A1) is a universal solution for T1, A1 under M.

T1 ∪ M : Item FootWear OpenShoes Strappy Plateau HighHeel Apparel Gown LightGown . . . Product Shoes Sandals Classic Platform Heeled Clothing Dress SummerDress hasMatchingOpenShoes matches fitsWithSandals fitsWith A1 : Strappy Plateau HighHeel LightGown

h a s M a t c h i n g O p e n S h

  • e

s

Auniv

2

: Classic Platform Heeled SummerDress Sandals Dress Shoes Clothing Product fitsWithSandals fitsWith

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SLIDE 18

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Example: Universal UCQ-solution

Definition

K2 is a universal UCQ-solution for K1 under M if for each UCQ q over Σ2, cert(q, T1 ∪ M, A1) = cert(q, K2).

T2 : Product Shoes Sandals Classic Platform Heeled . . . Clothing Dress SummerDress WorkDress EveningDress . . .

fitsWithSandals fitsWith

A2 :

brown_sand Classic wedge_sand Platform heel_sand Heeled turq_gown SummerDress

fitsWithSandals Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 7/20

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SLIDE 19

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Outline

1 Knowledge Base Exchange 2 Representability in DL-LiteR 3 Conclusions

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 8/20

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SLIDE 20

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Representability Problem

Problem

given a mapping M and a source TBox T1, decide whether there exists a target TBox T2, such that for each ABox A1 over Σ1, T2, chaseM,Σ2(A1) is a universal UCQ-solution for T1, A1 under M.

Σ1 Σ1 source signature Σ2 Σ2 target signature M∗ s.t. M ⊆ M∗ and T1 ∪ M | = M∗ M T1 source TBox A1 B1 C1 D1

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 9/20

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SLIDE 21

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Representability Problem

Problem

given a mapping M and a source TBox T1, decide whether there exists a target TBox T2, such that for each ABox A1 over Σ1, T2, chaseM,Σ2(A1) is a universal UCQ-solution for T1, A1 under M.

Σ1 Σ1 source signature Σ2 Σ2 target signature M∗ s.t. M ⊆ M∗ and T1 ∪ M | = M∗ M T1 source TBox A1 B1 C1 D1 T2 target TBox A2 B2 C2

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 9/20

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SLIDE 22

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Representability Problem

Problem

given a mapping M and a source TBox T1, decide whether there exists a target TBox T2, such that for each ABox A1 over Σ1, T2, chaseM,Σ2(A1) is a universal UCQ-solution for T1, A1 under M.

Σ1 Σ1 source signature Σ2 Σ2 target signature M∗ s.t. M ⊆ M∗ and T1 ∪ M | = M∗ M T1 source TBox A1 B1 C1 D1 T2 target TBox A2 B2 C2 A1 for each

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 9/20

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SLIDE 23

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Representability Problem

Problem

given a mapping M and a source TBox T1, decide whether there exists a target TBox T2, such that for each ABox A1 over Σ1, T2, chaseM,Σ2(A1) is a universal UCQ-solution for T1, A1 under M.

Σ1 Σ1 source signature Σ2 Σ2 target signature M∗ s.t. M ⊆ M∗ and T1 ∪ M | = M∗ M T1 source TBox A1 B1 C1 D1 T2 target TBox A2 B2 C2 A1 for each chaseM,Σ2(A1) M

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 9/20

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SLIDE 24

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Representability Problem

Problem

given a mapping M and a source TBox T1, decide whether there exists a target TBox T2, such that for each ABox A1 over Σ1, T2, chaseM,Σ2(A1) is a universal UCQ-solution for T1, A1 under M.

Σ1 Σ1 source signature Σ2 Σ2 target signature M∗ s.t. M ⊆ M∗ and T1 ∪ M | = M∗ M T1 source TBox A1 B1 C1 D1 T2 target TBox A2 B2 C2 A1 for each chaseM,Σ2(A1) M K1 K2

universal UCQ-solution Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 9/20

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SLIDE 25

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Representability: Example

T1 : M : FootWear OpenShoes Shoes Sandals

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 10/20

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SLIDE 26

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Representability: Example

T1 : M : FootWear OpenShoes T2 : Shoes Sandals

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 10/20

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SLIDE 27

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Representability: Example

T1 : M : FootWear OpenShoes T2 : Shoes Sandals It is easy to see that T2 is a UCQ-representation of T1 under M: for each ABox A1 of the form {OpenShoes(s)} or {FootWear(s)}, chaseT1∪M,Σ2(A1) = chaseM∪T2,Σ2(A1), hence T2, chaseM,Σ2(A1) is a universal UCQ-solution for T1, A1 under M.

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 10/20

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SLIDE 28

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Representability: Progress

  • In DL11, we solved the representability problem for DL-LiteRDFS, i.e.,

TBox inclusions of the form A1 ⊑ A2 ∃R1 ⊑ A2 R1 ⊑ R2

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 11/20

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SLIDE 29

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Representability: Progress

  • In DL11, we solved the representability problem for DL-LiteRDFS, i.e.,

TBox inclusions of the form A1 ⊑ A2 ∃R1 ⊑ A2 R1 ⊑ R2

  • In this work we address inclusions of the form

A1 ⊑ ∃R2

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 11/20

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SLIDE 30

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Representability: Progress

  • In DL11, we solved the representability problem for DL-LiteRDFS, i.e.,

TBox inclusions of the form A1 ⊑ A2 ∃R1 ⊑ A2 R1 ⊑ R2

  • In this work we address inclusions of the form

A1 ⊑ ∃R2 Thus, we solve the representability problem for DL-Litepos

R ,

the fragment of DL-LiteR without disjointness assertions.

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 11/20

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SLIDE 31

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Checking Representability

  • We first show how to check whether a given TBox T2 is

a representation of T1 under M.

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 12/20

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SLIDE 32

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Checking Representability

  • We first show how to check whether a given TBox T2 is

a representation of T1 under M.

◮ We provide polynomial time conditions to verify that. Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 12/20

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SLIDE 33

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Checking Representability

  • We first show how to check whether a given TBox T2 is

a representation of T1 under M.

◮ We provide polynomial time conditions to verify that. ◮ These conditions ensure that for each ABox A1 over Σ1,

chaseT1∪M,Σ2(A1) is homomorphically equivalent to chaseM∪T2,Σ2(A1).

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 12/20

slide-34
SLIDE 34

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Checking Representability

  • We first show how to check whether a given TBox T2 is

a representation of T1 under M.

◮ We provide polynomial time conditions to verify that. ◮ These conditions ensure that for each ABox A1 over Σ1,

chaseT1∪M,Σ2(A1) is homomorphically equivalent to chaseM∪T2,Σ2(A1).

◮ We also show that these conditions are necessary. Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 12/20

slide-35
SLIDE 35

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Checking Representability

  • We first show how to check whether a given TBox T2 is

a representation of T1 under M.

◮ We provide polynomial time conditions to verify that. ◮ These conditions ensure that for each ABox A1 over Σ1,

chaseT1∪M,Σ2(A1) is homomorphically equivalent to chaseM∪T2,Σ2(A1).

◮ We also show that these conditions are necessary.

In what follows, I will show how to ensure that for each ABox A1 over Σ1 chaseT1∪M,Σ2(A1) → chaseM∪T2,Σ2(A1).

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 12/20

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SLIDE 36

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Inclusion A1 ⊑ ∃R2

T1 ∪ M : FootWear OpenShoes Gown LightGown

hasMatchingOpenShoes

Shoes Sandals Dress SummerDress

fitsWithSandals

A1 :

turq_gown LightGown Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 13/20

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SLIDE 37

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Inclusion A1 ⊑ ∃R2

T1 ∪ M : FootWear OpenShoes Gown LightGown

hasMatchingOpenShoes

Shoes Sandals Dress SummerDress

fitsWithSandals

A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress Dress n Sandals Shoes

fitsWithSandals Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 13/20

slide-38
SLIDE 38

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Inclusion A1 ⊑ ∃R2

T1 ∪ M : FootWear OpenShoes Gown LightGown

hasMatchingOpenShoes

Shoes Sandals Dress SummerDress

fitsWithSandals

A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress Dress n Sandals Shoes

fitsWithSandals

chaseM

,Σ2(A1) :

turq_gown SummerDress

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 13/20

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SLIDE 39

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Inclusion A1 ⊑ ∃R2

T1 ∪ M : FootWear OpenShoes Gown LightGown

hasMatchingOpenShoes

Shoes Sandals Dress SummerDress

fitsWithSandals

A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress Dress n Sandals Shoes

fitsWithSandals

chaseM

,Σ2(A1) :

turq_gown SummerDress

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 13/20

slide-40
SLIDE 40

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Inclusion A1 ⊑ ∃R2

T1 ∪ M : FootWear OpenShoes Gown LightGown

hasMatchingOpenShoes

Shoes Sandals Dress SummerDress

fitsWithSandals

A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress Dress n Sandals Shoes

fitsWithSandals

chaseM∪T2,Σ2(A1) : turq_gown SummerDress Dress · · · m Sandals Shoes · · ·

fitsWithSandals Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 13/20

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SLIDE 41

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Inclusion A1 ⊑ ∃R2

T1 ∪ M : FootWear OpenShoes Gown LightGown

hasMatchingOpenShoes

T2 : Shoes Sandals Dress SummerDress

fitsWithSandals

A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress Dress n Sandals Shoes

fitsWithSandals

chaseM∪T2,Σ2(A1) : turq_gown SummerDress Dress · · · m Sandals Shoes · · ·

fitsWithSandals Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 13/20

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SLIDE 42

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Inclusion A1 ⊑ ∃R2 cont.

T1 ∪ M : FootWear OpenShoes Gown LightGown

hasMatchingOpenShoes

Shoes Sandals Dress SummerDress A1 :

turq_gown LightGown Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 14/20

slide-43
SLIDE 43

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Inclusion A1 ⊑ ∃R2 cont.

T1 ∪ M : FootWear OpenShoes Gown LightGown

hasMatchingOpenShoes

Shoes Sandals Dress SummerDress A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress Dress n Sandals Shoes

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 14/20

slide-44
SLIDE 44

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Inclusion A1 ⊑ ∃R2 cont.

T1 ∪ M : FootWear OpenShoes Gown LightGown

hasMatchingOpenShoes

Shoes Sandals Dress SummerDress A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress Dress n Sandals Shoes chaseM

,Σ2(A1) :

turq_gown SummerDress

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 14/20

slide-45
SLIDE 45

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Inclusion A1 ⊑ ∃R2 cont.

T1 ∪ M : FootWear OpenShoes Gown LightGown

hasMatchingOpenShoes

Shoes Sandals Dress SummerDress A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress Dress n Sandals Shoes chaseM

,Σ2(A1) :

turq_gown SummerDress

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 14/20

slide-46
SLIDE 46

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Inclusion A1 ⊑ ∃R2 cont.

T1 ∪ M : FootWear OpenShoes Gown LightGown

hasMatchingOpenShoes

Shoes Sandals Dress SummerDress A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress Dress n Sandals Shoes chaseM∪T2,Σ2(A1) : turq_gown SummerDress Dress · · · Sandals Shoes · · ·

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 14/20

slide-47
SLIDE 47

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Inclusion A1 ⊑ ∃R2 cont.

T1 ∪ M : FootWear OpenShoes Gown LightGown

hasMatchingOpenShoes

T2 : Shoes Sandals Dress SummerDress A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress Dress n Sandals Shoes chaseM∪T2,Σ2(A1) : turq_gown SummerDress Dress · · · Sandals Shoes · · ·

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 14/20

slide-48
SLIDE 48

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Inclusion A1 ⊑ ∃R2 cont.

T1 ∪ M : FootWear OpenShoes Gown LightGown

hasMatchingOpenShoes

Shoes Sandals Dress SummerDress A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress Dress n Sandals Shoes chaseM∪T2,Σ2(A1) : turq_gown SummerDress Dress · · · m1 mk−1 mk Sandals Shoes · · ·

R1

2

· · · Rk

2

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 14/20

slide-49
SLIDE 49

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Inclusion A1 ⊑ ∃R2 cont.

T1 ∪ M : FootWear OpenShoes Gown LightGown

hasMatchingOpenShoes

T2 : Shoes Sandals Dress SummerDress A1

2

Ak−1

2

R

1 2

· · · R

k 2

A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress Dress n Sandals Shoes chaseM∪T2,Σ2(A1) : turq_gown SummerDress Dress · · · m1 mk−1 mk Sandals Shoes · · ·

R1

2

· · · Rk

2

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 14/20

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SLIDE 50

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Deciding Representability in DL-Litepos

R

Theorem

Let M be a DL-Litepos

R

mapping and T1 a DL-Litepos

R

TBox over Σ1. Then we can check whether T1 is representable under M in polynomial time. This algorithm is similar to the one for DL-LiteRDFS:

  • we construct the maximal possible TBox over Σ2 and check

whether it is a representation.

  • If the check fails, then T1 is not representable under M.

Otherwise, this TBox is a representation.

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 15/20

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SLIDE 51

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Weak Representability

Problem

given a mapping M and a source TBox T1, decide whether there exists a mapping M⋆ such that M ⊆ M⋆, T1 ∪ M | = M⋆, and T1 is UCQ-representable under M⋆.

Σ1 Σ1 source signature Σ2 Σ2 target signature M T1 source TBox A1 B1 C1 D1

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 16/20

slide-52
SLIDE 52

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Weak Representability

Problem

given a mapping M and a source TBox T1, decide whether there exists a mapping M⋆ such that M ⊆ M⋆, T1 ∪ M | = M⋆, and T1 is UCQ-representable under M⋆.

Σ1 Σ1 source signature Σ2 Σ2 target signature M M∗ s.t. M ⊆ M∗ and T1 ∪ M | = M∗ T1 source TBox A1 B1 C1 D1

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 16/20

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SLIDE 53

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Weak Representability

Problem

given a mapping M and a source TBox T1, decide whether there exists a mapping M⋆ such that M ⊆ M⋆, T1 ∪ M | = M⋆, and T1 is UCQ-representable under M⋆.

Σ1 Σ1 source signature Σ2 Σ2 target signature M M∗ s.t. M ⊆ M∗ and T1 ∪ M | = M∗ T1 source TBox A1 B1 C1 D1 T2 target TBox A2 B2 C2

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 16/20

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SLIDE 54

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Weak Representability

Problem

given a mapping M and a source TBox T1, decide whether there exists a mapping M⋆ such that M ⊆ M⋆, T1 ∪ M | = M⋆, and T1 is UCQ-representable under M⋆.

Σ1 Σ1 source signature Σ2 Σ2 target signature M M∗ s.t. M ⊆ M∗ and T1 ∪ M | = M∗ T1 source TBox A1 B1 C1 D1 T2 target TBox A2 B2 C2 A1 for each

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 16/20

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SLIDE 55

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Weak Representability

Problem

given a mapping M and a source TBox T1, decide whether there exists a mapping M⋆ such that M ⊆ M⋆, T1 ∪ M | = M⋆, and T1 is UCQ-representable under M⋆.

Σ1 Σ1 source signature Σ2 Σ2 target signature M M∗ s.t. M ⊆ M∗ and T1 ∪ M | = M∗ T1 source TBox A1 B1 C1 D1 T2 target TBox A2 B2 C2 A1 for each chaseM,Σ2(A1) M∗

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 16/20

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SLIDE 56

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Weak Representability

Problem

given a mapping M and a source TBox T1, decide whether there exists a mapping M⋆ such that M ⊆ M⋆, T1 ∪ M | = M⋆, and T1 is UCQ-representable under M⋆.

Σ1 Σ1 source signature Σ2 Σ2 target signature M M∗ s.t. M ⊆ M∗ and T1 ∪ M | = M∗ T1 source TBox A1 B1 C1 D1 T2 target TBox A2 B2 C2 A1 for each chaseM,Σ2(A1) M∗ K1 K2

universal CQ-solution Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 16/20

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SLIDE 57

Deciding Weak Representability

Recall that every DL-LiteRDFS TBox is weakly representable under every DL-LiteRDFS mapping (M⋆ is the maximal mapping s.t. T1 ∪ M | = M⋆). However, there exist a DL-Litepos

R

mapping M and TBox T1 s.t. T1 is not weakly representable under M.

slide-58
SLIDE 58

Deciding Weak Representability

Recall that every DL-LiteRDFS TBox is weakly representable under every DL-LiteRDFS mapping (M⋆ is the maximal mapping s.t. T1 ∪ M | = M⋆). However, there exist a DL-Litepos

R

mapping M and TBox T1 s.t. T1 is not weakly representable under M.

T1 ∪ M : OpenShoes LightGown

hasMatchingOpenShoes

Sandals SummerDress A1 :

turq_gown LightGown

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SLIDE 59

Deciding Weak Representability

Recall that every DL-LiteRDFS TBox is weakly representable under every DL-LiteRDFS mapping (M⋆ is the maximal mapping s.t. T1 ∪ M | = M⋆). However, there exist a DL-Litepos

R

mapping M and TBox T1 s.t. T1 is not weakly representable under M.

T1 ∪ M : OpenShoes LightGown

hasMatchingOpenShoes

Sandals SummerDress A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress n Sandals

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SLIDE 60

Deciding Weak Representability

Recall that every DL-LiteRDFS TBox is weakly representable under every DL-LiteRDFS mapping (M⋆ is the maximal mapping s.t. T1 ∪ M | = M⋆). However, there exist a DL-Litepos

R

mapping M and TBox T1 s.t. T1 is not weakly representable under M.

T1 ∪ M : OpenShoes LightGown

hasMatchingOpenShoes

Sandals SummerDress T2 : A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress n Sandals

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SLIDE 61

Deciding Weak Representability

Recall that every DL-LiteRDFS TBox is weakly representable under every DL-LiteRDFS mapping (M⋆ is the maximal mapping s.t. T1 ∪ M | = M⋆). However, there exist a DL-Litepos

R

mapping M and TBox T1 s.t. T1 is not weakly representable under M.

T1 ∪ M : OpenShoes LightGown

hasMatchingOpenShoes

Sandals SummerDress T2 : A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress n Sandals chaseM∪T2,Σ2(A1) : turq_gown SummerDress Sandals

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SLIDE 62

Deciding Weak Representability

Recall that every DL-LiteRDFS TBox is weakly representable under every DL-LiteRDFS mapping (M⋆ is the maximal mapping s.t. T1 ∪ M | = M⋆). However, there exist a DL-Litepos

R

mapping M and TBox T1 s.t. T1 is not weakly representable under M.

T1 ∪ M : OpenShoes LightGown

hasMatchingOpenShoes

Sandals SummerDress T2 : A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress n Sandals chaseM∪T2,Σ2(A1) : turq_gown SummerDress Sandals

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SLIDE 63

Deciding Weak Representability

Recall that every DL-LiteRDFS TBox is weakly representable under every DL-LiteRDFS mapping (M⋆ is the maximal mapping s.t. T1 ∪ M | = M⋆). However, there exist a DL-Litepos

R

mapping M and TBox T1 s.t. T1 is not weakly representable under M.

T1 ∪ M : OpenShoes LightGown

hasMatchingOpenShoes

Sandals SummerDress T2 : A1 :

turq_gown LightGown

chaseT1∪M,Σ2(A1) : turq_gown SummerDress n Sandals chaseM∪T2,Σ2(A1) : turq_gown SummerDress Sandals

Theorem

Weak representability can be solved in polynomial time for DL-Litepos

R

mappings and TBoxes.

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SLIDE 64

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Outline

1 Knowledge Base Exchange 2 Representability in DL-LiteR 3 Conclusions

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 18/20

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SLIDE 65

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Conclusions and Future Work

  • We continued our work on KB exchange in DL-Lite.
  • We addressed the problem of UCQ-representability in DL-Litepos

R , the

fragment of DL-LiteR without disjointness assertions.

  • We showed that the problem of representability and weak

representability of a DL-Litepos

R

TBox under a DL-Litepos

R

mapping is decidable in polynomial time.

  • To solve representability in full DL-LiteR it remains to add disjointness

constraints.

  • There are also other open problems in the context of KB exchange, e.g.,

computing universal solutions, checking solution etc.

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SLIDE 66

Knowledge Base Exchange Representability in DL-LiteR Conclusions

Thank you for your attention!

Elena Botoeva Representability in DL-LiteR Knowledge Base Exchange 20/20