[PPT] - Knowledge Base Exchange Marcelo Arenas 1 Elena Botoeva 2 Diego PowerPoint Presentation

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Knowledge Base Exchange

Marcelo Arenas1 Elena Botoeva2 Diego Calvanese2

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

Description Logics Workshop 14 July 2011, Barcelona

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Outline

1

Knowledge Base Exchange

2

Techniques for Deciding Knowledge Base Exchange

3

Conclusions

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Outline

1

Knowledge Base Exchange

2

Techniques for Deciding Knowledge Base Exchange

3

Conclusions

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Knowledge Base Exchange

Σ1 source signature Σ2 target signature

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Knowledge Base Exchange

Σ1 source signature Σ2 target signature M

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Knowledge Base Exchange

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

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Knowledge Base Exchange

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

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Mapping

A mapping specifies how a source KB should be translated into a target KB.

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Mapping

A mapping specifies how a source KB should be translated into a target KB. A mapping is a tuple M = (Σ1, Σ2, T12), where

◮ Σ1, Σ2 are disjoint signatures and ◮ T12 is a TBox with assertions of the form

C1 ⊑ C2, where C1 is a concept over Σ1, C2 is a concept over Σ2, R1 ⊑ R2, where R1 is a role over Σ1, R2 is a role over Σ2.

Arenas, Botoeva, Calvanese Knowledge Base Exchange 5/27

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Mapping

A mapping specifies how a source KB should be translated into a target KB. A mapping is a tuple M = (Σ1, Σ2, T12), where

◮ Σ1, Σ2 are disjoint signatures and ◮ T12 is a TBox with assertions of the form

C1 ⊑ C2, where C1 is a concept over Σ1, C2 is a concept over Σ2, R1 ⊑ R2, where R1 is a role over Σ1, R2 is a role over Σ2.

Let I be an interpretation of Σ1 and J an interpretation of Σ2. Then (I, J ) satisfies M, denoted (I, J ) | = M if

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Mapping

A mapping specifies how a source KB should be translated into a target KB. A mapping is a tuple M = (Σ1, Σ2, T12), where

◮ Σ1, Σ2 are disjoint signatures and ◮ T12 is a TBox with assertions of the form

C1 ⊑ C2, where C1 is a concept over Σ1, C2 is a concept over Σ2, R1 ⊑ R2, where R1 is a role over Σ1, R2 is a role over Σ2.

Let I be an interpretation of Σ1 and J an interpretation of Σ2. Then (I, J ) satisfies M, denoted (I, J ) | = M if

◮ C1

I ⊆ C2 J , for each C1 ⊑ C2 ∈ M, and

Arenas, Botoeva, Calvanese Knowledge Base Exchange 5/27

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Mapping

A mapping specifies how a source KB should be translated into a target KB. A mapping is a tuple M = (Σ1, Σ2, T12), where

◮ Σ1, Σ2 are disjoint signatures and ◮ T12 is a TBox with assertions of the form

C1 ⊑ C2, where C1 is a concept over Σ1, C2 is a concept over Σ2, R1 ⊑ R2, where R1 is a role over Σ1, R2 is a role over Σ2.

Let I be an interpretation of Σ1 and J an interpretation of Σ2. Then (I, J ) satisfies M, denoted (I, J ) | = M if

◮ C1

I ⊆ C2 J , for each C1 ⊑ C2 ∈ M, and

◮ R1

I ⊆ R2 J , for each R1 ⊑ R2 ∈ M.

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Solutions for Knowledge Base Exchange

Given an interpretation I of Σ1 and a set X of interpretations of Σ1, let SatM(I) = {J | (I, J ) | = M}, SatM(X) =

I∈X SatM(I).

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Solutions for Knowledge Base Exchange

Given an interpretation I of Σ1 and a set X of interpretations of Σ1, let SatM(I) = {J | (I, J ) | = M}, SatM(X) =

I∈X SatM(I).

Definition

Let M be a mapping, K1 a KB over Σ1, and K2 a KB over Σ2. K2 is a solution for K1 under M if: Mod(K2) ⊆ SatM(Mod(K1)).

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Solutions for Knowledge Base Exchange

Given an interpretation I of Σ1 and a set X of interpretations of Σ1, let SatM(I) = {J | (I, J ) | = M}, SatM(X) =

I∈X SatM(I).

Definition

Let M be a mapping, K1 a KB over Σ1, and K2 a KB over Σ2. K2 is a solution for K1 under M if: Mod(K2) ⊆ SatM(Mod(K1)). K2 is a universal solution for K1 under M if: Mod(K2) = SatM(Mod(K1)).

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Solutions for Knowledge Base Exchange: Example

Example

Let

K1 = T1, A1

T1 : B1 ⊑ A1 A1 : B1(b) and

M :

A1 ⊑ A2 B1 ⊑ B2

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Solutions for Knowledge Base Exchange: Example

Example

Let

K1 = T1, A1

T1 : B1 ⊑ A1 A1 : B1(b) and

M :

A1 ⊑ A2 B1 ⊑ B2 Then, K2 and K′

2 are solutions for K1 under M

K2 = T2, A2

T2 : ∅ A2 : B2(b), A2(b) and

K′

2 = T ′ 2, A′ 2

T ′

2 :

B2 ⊑ A2 A′

2 :

B2(b)

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Solutions for Knowledge Base Exchange: Example

Example

Let

K1 = T1, A1

T1 : B1 ⊑ A1 A1 : B1(b) and

M :

A1 ⊑ A2 B1 ⊑ B2 Then, K2 and K′

2 are solutions for K1 under M

K2 = T2, A2

T2 : ∅ A2 : B2(b), A2(b) and

K′

2 = T ′ 2, A′ 2

T ′

2 :

B2 ⊑ A2 A′

2 :

B2(b) Moreover, K2 is a universal solution for K1 under M, while K′

2 is not.

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

CQ-Solutions for Knowledge Base Exchange

We might want to relax the condition on solutions. If the main reasoning task performed over target KBs is CQ answering, then we can resort to a weaker notion of solution.

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

CQ-Solutions for Knowledge Base Exchange

We might want to relax the condition on solutions. If the main reasoning task performed over target KBs is CQ answering, then we can resort to a weaker notion of solution.

Definition

Let M be a mapping, K1 = T1, A1 a KB over Σ1, and K2 a KB over Σ2. K2 is a CQ-solution for K1 under M if for each CQ q over Σ2, cert(q, T1 ∪ M, A1) ⊆ cert(q, K2).

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

CQ-Solutions for Knowledge Base Exchange

We might want to relax the condition on solutions. If the main reasoning task performed over target KBs is CQ answering, then we can resort to a weaker notion of solution.

Definition

Let M be a mapping, K1 = T1, A1 a KB over Σ1, and K2 a KB over Σ2. K2 is a CQ-solution for K1 under M if for each CQ q over Σ2, cert(q, T1 ∪ M, A1) ⊆ cert(q, K2). K2 is a universal CQ-solution for K1 under M if for each CQ q over Σ2, cert(q, T1 ∪ M, A1) = cert(q, K2).

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

CQ-Solutions for Knowledge Base Exchange: Example

Example

Let

K1 = T1, A1

T1 : B1 ⊑ A1 A1 : B1(b) and

M :

A1 ⊑ A2 B1 ⊑ B2

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

CQ-Solutions for Knowledge Base Exchange: Example

Example

Let

K1 = T1, A1

T1 : B1 ⊑ A1 A1 : B1(b) and

M :

A1 ⊑ A2 B1 ⊑ B2 Then, K2 and K′

2 are universal CQ-solutions for K1 under M.

K2 = T2, A2

T2 : ∅ A2 : B2(b), A2(b) and

K′

2 = T ′ 2, A′ 2

T ′

2 :

B2 ⊑ A2 A′

2 :

B2(b)

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

A New Problem: Representability

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

Arenas, Botoeva, Calvanese Knowledge Base Exchange 10/27

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

A New Problem: Representability

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

Arenas, Botoeva, Calvanese Knowledge Base Exchange 10/27

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

A New Problem: Representability

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

Arenas, Botoeva, Calvanese Knowledge Base Exchange 10/27

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

A New Problem: Representability

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

Arenas, Botoeva, Calvanese Knowledge Base Exchange 10/27

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

A New Problem: Representability

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

universal CQ-solution

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

A New Problem: Representability contd

Σ1 source signature Σ2 target signature M∗, T1 ∪ M | = M∗ M source TBox T1 A1 B1 C1 D1 target TBox T2 A2 B2 C2

representation

If such a T2 exists, we say that T1 is representable in M. T2 is called a representation of T1 in M.

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Representability: Example

Example

Let

T1 :

B1 ⊑ A1 and

M :

A1 ⊑ A2 B1 ⊑ B2 Then, T1 is representable in M and

T2 :

B2 ⊑ A2 is a representation of T1 in M. In this example (and later for the DL-Lite setting) we exploit that certain answers are characterised in terms of chase.

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Representability: Example contd

bla A1 B1 bla A2 B2 M T1 B1 ⊑ A1 T2 B2 ⊑ A2

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Representability: Example contd

bla A1 B1 bla A2 B2 M T1 B1 ⊑ A1 T2 B2 ⊑ A2 A1 b1 B1 a1 A1

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Representability: Example contd

bla A1 B1 bla A2 B2 M T1 B1 ⊑ A1 T2 B2 ⊑ A2 A1 b1 B1 a1 A1

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Representability: Example contd

bla A1 B1 bla A2 B2 M T1 B1 ⊑ A1 T2 B2 ⊑ A2 A1 b1 B1 a1 A1 chaseM,Σ2(A1) b1 B2 a1 A2 M

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Representability: Example contd

bla A1 B1 bla A2 B2 M T1 B1 ⊑ A1 T2 B2 ⊑ A2 A1 b1 B1 a1 A1 chaseT2(chaseM,Σ2(A1)) b1 A2 chaseM,Σ2(A1) b1 B2 a1 A2 M

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Representability: Example contd

bla A1 B1 bla A2 B2 M T1 B1 ⊑ A1 T2 B2 ⊑ A2 chaseT1(A1) b1 A1 A1 b1 B1 a1 A1 chaseT2(chaseM,Σ2(A1)) b1 A2 chaseM,Σ2(A1) b1 B2 a1 A2 M

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Representability: Example contd

bla A1 B1 bla A2 B2 M T1 B1 ⊑ A1 T2 B2 ⊑ A2 chaseT1(A1) b1 A1 b1 B1 a1 A1 chaseT2(chaseM,Σ2(A1)) b1 A2 b1 B2 a1 A2

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Representability: Example contd

bla A1 B1 bla A2 B2 M T1 B1 ⊑ A1 T2 B2 ⊑ A2 chaseT1(A1) b1 A1 b1 B1 a1 A1 chaseT2(chaseM,Σ2(A1)) b1 A2 b1 B2 a1 A2 chaseM,Σ2(chaseT1(A1)) b1 B2 a1 A2 b1 A2

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Representability: Example contd

bla A1 B1 bla A2 B2 M T1 B1 ⊑ A1 T2 B2 ⊑ A2 chaseT2(chaseM,Σ2(A1)) b1 A2 b1 B2 a1 A2 chaseM,Σ2(chaseT1(A1)) b1 B2 a1 A2 b1 A2

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Our Setting: Definite Inclusions

In this paper we tackle the problems for definite inclusions. A DL-LiteR inclusion is called definite if its right-hand side is an atomic concept or an atomic role. A DL-LiteR TBox is said to be definite if it consists of definite inclusions.

Arenas, Botoeva, Calvanese Knowledge Base Exchange 14/27

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Our Setting: Definite Inclusions

In this paper we tackle the problems for definite inclusions. A DL-LiteR inclusion is called definite if its right-hand side is an atomic concept or an atomic role. A DL-LiteR TBox is said to be definite if it consists of definite inclusions. Specifically, we consider definite mappings and DL-LiteRDFS KBs.

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Our Setting: Definite Inclusions

In this paper we tackle the problems for definite inclusions. A DL-LiteR inclusion is called definite if its right-hand side is an atomic concept or an atomic role. A DL-LiteR TBox is said to be definite if it consists of definite inclusions. Specifically, we consider definite mappings

◮ A mapping M is said to be definite if it is a definite TBox.

blabla blabla M and DL-LiteRDFS KBs.

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Our Setting: Definite Inclusions

In this paper we tackle the problems for definite inclusions. A DL-LiteR inclusion is called definite if its right-hand side is an atomic concept or an atomic role. A DL-LiteR TBox is said to be definite if it consists of definite inclusions. Specifically, we consider definite mappings

◮ A mapping M is said to be definite if it is a definite TBox.

blabla blabla M A1 A2 and DL-LiteRDFS KBs.

Arenas, Botoeva, Calvanese Knowledge Base Exchange 14/27

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Our Setting: Definite Inclusions

In this paper we tackle the problems for definite inclusions. A DL-LiteR inclusion is called definite if its right-hand side is an atomic concept or an atomic role. A DL-LiteR TBox is said to be definite if it consists of definite inclusions. Specifically, we consider definite mappings

◮ A mapping M is said to be definite if it is a definite TBox.

blabla blabla M A1 A2 ∃R1 A2 and DL-LiteRDFS KBs.

Arenas, Botoeva, Calvanese Knowledge Base Exchange 14/27

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Our Setting: Definite Inclusions

In this paper we tackle the problems for definite inclusions. A DL-LiteR inclusion is called definite if its right-hand side is an atomic concept or an atomic role. A DL-LiteR TBox is said to be definite if it consists of definite inclusions. Specifically, we consider definite mappings

◮ A mapping M is said to be definite if it is a definite TBox.

blabla blabla M A1 A2 ∃R1 A2 R1 R2 and DL-LiteRDFS KBs.

Arenas, Botoeva, Calvanese Knowledge Base Exchange 14/27

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Our Setting: Definite Inclusions

In this paper we tackle the problems for definite inclusions. A DL-LiteR inclusion is called definite if its right-hand side is an atomic concept or an atomic role. A DL-LiteR TBox is said to be definite if it consists of definite inclusions. Specifically, we consider definite mappings

◮ A mapping M is said to be definite if it is a definite TBox.

blabla blabla M A1 A2 ∃R1 A2 R1 R2 and DL-LiteRDFS KBs.

◮ We call DL-LiteRDFS the fragment of DL-LiteR obtained by

considering only definite DL-LiteR TBoxes.

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Outline

1

Knowledge Base Exchange

2

Techniques for Deciding Knowledge Base Exchange

3

Conclusions

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Computing (Universal) (CQ-)Solutions

Proposition

Let M be a definite mapping and K1 = T1, A1 a DL-LiteRDFS KB over Σ1. Then ∅, chaseM,Σ2(chaseT1(A1)) is a universal solution for K1 under M.

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Computing (Universal) (CQ-)Solutions

Proposition

Let M be a definite mapping and K1 = T1, A1 a DL-LiteRDFS KB over Σ1. Then ∅, chaseM,Σ2(chaseT1(A1)) is a universal solution for K1 under M. Note: in DL-LiteRDFS, the chase is always finite.

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Computing (Universal) (CQ-)Solutions

Proposition

Let M be a definite mapping and K1 = T1, A1 a DL-LiteRDFS KB over Σ1. Then ∅, chaseM,Σ2(chaseT1(A1)) is a universal solution for K1 under M. Note: in DL-LiteRDFS, the chase is always finite.

Theorem

For definite mappings and DL-LiteRDFS KBs, the problems of computing (universal) (CQ-)solutions can be solved in polynomial time.

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Checking Representability

Let us consider the checking problem associated with representability.

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Checking Representability

Let us consider the checking problem associated with representability.

Checking Representation

Input: a definite mapping M, a DL-LiteRDFS TBox T1 over Σ1, a DL-LiteRDFS TBox T2 over Σ2. Output: Yes, if T2 is a representation of T1 in M, NO, otherwise.

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Checking Representability

Let us consider the checking problem associated with representability.

Checking Representation

Input: a definite mapping M, a DL-LiteRDFS TBox T1 over Σ1, a DL-LiteRDFS TBox T2 over Σ2. Output: Yes, if T2 is a representation of T1 in M, i.e., for each A1, T2, chaseM,Σ2(A1) is a universal CQ-solution for T1, A1 under M. NO, otherwise.

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Checking Representability

Let us consider the checking problem associated with representability.

Checking Representation

Input: a definite mapping M, a DL-LiteRDFS TBox T1 over Σ1, a DL-LiteRDFS TBox T2 over Σ2. Output: Yes, if T2 is a representation of T1 in M, i.e., for each A1, T2, chaseM,Σ2(A1) is a universal CQ-solution for T1, A1 under M. NO, otherwise. We base our technique on the notion of the translation set M(α, µ).

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Translation Set M(α, µ)

Let α be a DL-LiteRDFS inclusion over Σ1, and µ ∈ M. Then M(α, µ), is the set of DL-LiteRDFS inclusions over Σ2 such that, if there exists an inclusion ν ∈ M as in the table, then β ∈ M(α, µ).

α µ ν β E1 ⊑ A1 A1 ⊑ A2 E1 ⊑ E2 E2 ⊑ A2 ∃R1 ⊑ A1 A1 ⊑ A2 ∃R1 ⊑ E2 E2 ⊑ A2 R1 ⊑ R2 ∃R2 ⊑ A2 R1 ⊑ S1 S1 ⊑ S2 R1 ⊑ R2 R2 ⊑ S2 ∃S1 ⊑ A2 ∃R1 ⊑ E2 E2 ⊑ A2 R1 ⊑ R2 ∃R2 ⊑ A2 ∃S1

− ⊑ A2

∃R1

− ⊑ E2

E2 ⊑ A2 R1 ⊑ R2 ∃R2

− ⊑ A2 Arenas, Botoeva, Calvanese Knowledge Base Exchange 18/27

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Translation Set M(α, µ)

Let α be a DL-LiteRDFS inclusion over Σ1, and µ ∈ M. Then M(α, µ), is the set of DL-LiteRDFS inclusions over Σ2 such that, if there exists an inclusion ν ∈ M as in the table, then β ∈ M(α, µ).

α µ ν β E1 ⊑ A1 A1 ⊑ A2

T1 E1 A1 α T2 A2 µ

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Translation Set M(α, µ)

Let α be a DL-LiteRDFS inclusion over Σ1, and µ ∈ M. Then M(α, µ), is the set of DL-LiteRDFS inclusions over Σ2 such that, if there exists an inclusion ν ∈ M as in the table, then β ∈ M(α, µ).

α µ ν β E1 ⊑ A1 A1 ⊑ A2 E1 ⊑ E2

T1 E1 A1 α T2 A2 µ E2 E ′

2

ν ν′

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Translation Set M(α, µ)

Let α be a DL-LiteRDFS inclusion over Σ1, and µ ∈ M. Then M(α, µ), is the set of DL-LiteRDFS inclusions over Σ2 such that, if there exists an inclusion ν ∈ M as in the table, then β ∈ M(α, µ).

α µ ν β E1 ⊑ A1 A1 ⊑ A2 E1 ⊑ E2 E2 ⊑ A2

T1 E1 A1 α T2 A2 µ E2 E ′

2

ν ν′

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Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions

Translation Set M(α, µ)

Let α be a DL-LiteRDFS inclusion over Σ1, and µ ∈ M. Then M(α, µ), is the set of DL-LiteRDFS inclusions over Σ2 such that, if there exists an inclusion ν ∈ M as in the table, then β ∈ M(α, µ).

α µ ν β R1 ⊑ S1 ∃S1 ⊑ A2

T1 ∃R1 ∃R−

1

R1 ∃S1 ∃S−

1

S1 α T2 A2 µ

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Translation Set M(α, µ)

Let α be a DL-LiteRDFS inclusion over Σ1, and µ ∈ M. Then M(α, µ), is the set of DL-LiteRDFS inclusions over Σ2 such that, if there exists an inclusion ν ∈ M as in the table, then β ∈ M(α, µ).

α µ ν β R1 ⊑ S1 ∃S1 ⊑ A2 ∃R1 ⊑ E2 R1 ⊑ R2

T1 ∃R1 ∃R−

1

R1 ∃S1 ∃S−

1

S1 α T2 A2 µ E2 ∃R2 ∃R−

2

R2 ν ν′

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Translation Set M(α, µ)

Let α be a DL-LiteRDFS inclusion over Σ1, and µ ∈ M. Then M(α, µ), is the set of DL-LiteRDFS inclusions over Σ2 such that, if there exists an inclusion ν ∈ M as in the table, then β ∈ M(α, µ).

α µ ν β R1 ⊑ S1 ∃S1 ⊑ A2 ∃R1 ⊑ E2 E2 ⊑ A2 R1 ⊑ R2 ∃R2 ⊑ A2

T1 ∃R1 ∃R−

1

R1 ∃S1 ∃S−

1

S1 α T2 A2 µ E2 ∃R2 ∃R−

2

R2 ν ν′

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Reverse Translation Set M−(β, ν)

Let β be a DL-LiteRDFS inclusion over Σ2, and ν ∈ M. Then M−(β, ν), is the set of DL-LiteRDFS inclusions over Σ1 such that, if there exists an inclusion µ ∈ M as in the table, then α ∈ M−(β, ν).

α µ ν β E1 ⊑ A1 A1 ⊑ A2 E1 ⊑ E2 E2 ⊑ A2 ∃R1 ⊑ A1 A1 ⊑ A2 ∃R1 ⊑ E2 R1 ⊑ S1 ∃S1 ⊑ A2 R1 ⊑ S1 ∃S1

− ⊑ A2

∃R1

− ⊑ E2

∃R1 ⊑ A1 A1 ⊑ A2 R1 ⊑ R2 ∃R2 ⊑ A2 R1 ⊑ S1 ∃S1 ⊑ A2 R1 ⊑ S1 ∃S1

− ⊑ A2

R1 ⊑ R2 ∃R2

− ⊑ A2

R1 ⊑ S1 S1 ⊑ S2 R1 ⊑ R2 R2 ⊑ S2

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Checking Representability

We get the following characterisation of representations.

Proposition

Let M be a definite mapping, T1 a DL-LiteRDFS TBox over Σ1, and T2 a DL-LiteRDFS TBox over Σ2. Then T2 is a representation of T1 in M if and only if for each inclusion α, s.t. T1 | = α, and for each inclusion µ ∈ M left-compatible with rhs(α), there exists β ∈ M(α, µ), s.t. T2 | = β, and for each inclusion β, s.t. T2 | = β, and for each inclusion ν ∈ M right-compatible with lhs(β), there exists α ∈ M−(β, ν), s.t. T1 | = α.

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Deciding Representability

Theorem

Let M be a definite mapping and T1 a DL-LiteRDFS TBox over Σ1. Then we can check whether T1 is representable in M in polynomial time.

Proof.

1 Take M(T1, M) = M(α, µ), where the union ranges over all α,

s.t. T1 | = α, and µ ∈ M is left-compatible with rhs(α);

2 Remove from M(T1, M) every β s.t. there exists an inclusion

ν ∈ M right-compatible with lhs(β) and for each α ∈ M−(β, ν), T1 | = α. Let the resulting TBox be denoted with T2 = Rep(T1, M).

3 Check whether T2 is a representation of T1 in M. ◮ If the check succeeds, then T1 is representable in M. ◮ Otherwise, T1 is not representable in M. Arenas, Botoeva, Calvanese Knowledge Base Exchange 21/27

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Constructing Representations: Example

Example

T1 : ∃R1 ⊑ A1 M : ∃R1 ⊑ B2 A1 ⊑ A2 B1 ⊑ B2 R1 ⊑ R2 B1 ∃R1 ∃R−

1

R1 A1 B2 ∃R2 ∃R−

2

R2 A2

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Constructing Representations: Example

Example

T1 : ∃R1 ⊑ A1 M : ∃R1 ⊑ B2 A1 ⊑ A2 B1 ⊑ B2 R1 ⊑ R2 B1 ∃R1 ∃R−

1

R1 A1 B2 ∃R2 ∃R−

2

R2 A2 M(T1, M) : ∃R2 ⊑ A2 B2 ⊑ A2

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Constructing Representations: Example

Example

T1 : ∃R1 ⊑ A1 M : ∃R1 ⊑ B2 A1 ⊑ A2 B1 ⊑ B2 R1 ⊑ R2 B1 ∃R1 ∃R−

1

R1 A1 B2 ∃R2 ∃R−

2

R2 A2 M(T1, M) : ∃R2 ⊑ A2 B2 ⊑ A2

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Constructing Representations: Example

Example

T1 : ∃R1 ⊑ A1 M : ∃R1 ⊑ B2 A1 ⊑ A2 B1 ⊑ B2 R1 ⊑ R2 B1 ∃R1 ∃R−

1

R1 A1 B2 ∃R2 ∃R−

2

R2 A2 M(T1, M) : ∃R2 ⊑ A2 B2 ⊑ A2

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Constructing Representations: Example

Example

T1 : ∃R1 ⊑ A1 M : ∃R1 ⊑ B2 A1 ⊑ A2 B1 ⊑ B2 R1 ⊑ R2 B1 ∃R1 ∃R−

1

R1 A1 B2 ∃R2 ∃R−

2

R2 A2 M(T1, M) : ∃R2 ⊑ A2 B2 ⊑ A2

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Constructing Representations: Example

Example

T1 : ∃R1 ⊑ A1 M : ∃R1 ⊑ B2 A1 ⊑ A2 B1 ⊑ B2 R1 ⊑ R2 B1 ∃R1 ∃R−

1

R1 A1 B2 ∃R2 ∃R−

2

R2 A2 M(T1, M) : ∃R2 ⊑ A2 B2 ⊑ A2 T2 : ∃R2 ⊑ A2

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Constructing Representations: Example

Example

T1 : ∃R1 ⊑ A1 M : ∃R1 ⊑ B2 A1 ⊑ A2 B1 ⊑ B2 R1 ⊑ R2 B1 ∃R1 ∃R−

1

R1 A1 B2 ∃R2 ∃R−

2

R2 A2 T1 is representable in M and T2 is a representation of T1 in M. M(T1, M) : ∃R2 ⊑ A2 B2 ⊑ A2 T2 : ∃R2 ⊑ A2

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A New Problem: Weak Representability

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

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A New Problem: Weak Representability

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

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A New Problem: Weak Representability

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

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A New Problem: Weak Representability

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

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A New Problem: Weak Representability

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

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A New Problem: Weak Representability

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

universal CQ-solution

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Deciding Weak Representability

Theorem

Let M be a definite mapping and T1 a DL-LiteRDFS TBox over Σ1. Then T1 is weakly representable in M.

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Outline

1

Knowledge Base Exchange

2

Techniques for Deciding Knowledge Base Exchange

3

Conclusions

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Conclusions and Future Work

We have specialised the framework for KB exchange to the case of DLs. We have defined new reasoning tasks: representability and weak representability of a TBox in a mapping. We have shown the following results for definite mappings and DL-LiteRDFS KBs:

◮ the problems of computing (universal) (CQ-)solutions can be solved

in polynomial time.

◮ the problem of representability of a TBox in a mapping is decidable in

polynomial time.

◮ every DL-LiteRDFS TBox is weakly representable in a definite mapping.

We plan to extend the results to the case of full DL-LiteR. The issues to explore:

◮ labelled nulls in the chase ◮ disjointness constraints Arenas, Botoeva, Calvanese Knowledge Base Exchange 26/27

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Thank you for your attention!

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