Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions Knowledge Base Exchange Marcelo Arenas 1 Elena Botoeva 2 Diego Calvanese 2 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 Arenas, Botoeva, Calvanese Knowledge Base Exchange 1/27
Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions Outline Knowledge Base Exchange 1 Techniques for Deciding Knowledge Base Exchange 2 Conclusions 3 Arenas, Botoeva, Calvanese Knowledge Base Exchange 2/27
Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions Outline Knowledge Base Exchange 1 Techniques for Deciding Knowledge Base Exchange 2 Conclusions 3 Arenas, Botoeva, Calvanese Knowledge Base Exchange 3/27
Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions Knowledge Base Exchange Σ 1 Σ 2 target signature source signature Arenas, Botoeva, Calvanese Knowledge Base Exchange 4/27
Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions Knowledge Base Exchange M Σ 1 Σ 2 target signature source signature Arenas, Botoeva, Calvanese Knowledge Base Exchange 4/27
Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions Knowledge Base Exchange M Σ 1 Σ 2 target signature source signature T 1 A 1 D 1 B 1 C 1 A 1 source KB K 1 Arenas, Botoeva, Calvanese Knowledge Base Exchange 4/27
Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions Knowledge Base Exchange M Σ 1 Σ 2 target signature source signature T 1 T 2 A 1 D 1 A 2 B 1 C 1 B 2 C 2 A 1 A 2 source KB K 1 target KB K 2 Arenas, Botoeva, Calvanese Knowledge Base Exchange 4/27
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. Arenas, Botoeva, Calvanese Knowledge Base Exchange 5/27
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 = (Σ 1 , Σ 2 , T 12 ), where ◮ Σ 1 , Σ 2 are disjoint signatures and ◮ T 12 is a TBox with assertions of the form C 1 ⊑ C 2 , where C 1 is a concept over Σ 1 , C 2 is a concept over Σ 2 , R 1 ⊑ R 2 , where R 1 is a role over Σ 1 , R 2 is a role over Σ 2 . Arenas, Botoeva, Calvanese Knowledge Base Exchange 5/27
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 = (Σ 1 , Σ 2 , T 12 ), where ◮ Σ 1 , Σ 2 are disjoint signatures and ◮ T 12 is a TBox with assertions of the form C 1 ⊑ C 2 , where C 1 is a concept over Σ 1 , C 2 is a concept over Σ 2 , R 1 ⊑ R 2 , where R 1 is a role over Σ 1 , R 2 is a role over Σ 2 . Let I be an interpretation of Σ 1 and J an interpretation of Σ 2 . Then ( I , J ) satisfies M , denoted ( I , J ) | = M if Arenas, Botoeva, Calvanese Knowledge Base Exchange 5/27
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 = (Σ 1 , Σ 2 , T 12 ), where ◮ Σ 1 , Σ 2 are disjoint signatures and ◮ T 12 is a TBox with assertions of the form C 1 ⊑ C 2 , where C 1 is a concept over Σ 1 , C 2 is a concept over Σ 2 , R 1 ⊑ R 2 , where R 1 is a role over Σ 1 , R 2 is a role over Σ 2 . Let I be an interpretation of Σ 1 and J an interpretation of Σ 2 . Then ( I , J ) satisfies M , denoted ( I , J ) | = M if I ⊆ C 2 J , for each C 1 ⊑ C 2 ∈ M , and ◮ C 1 Arenas, Botoeva, Calvanese Knowledge Base Exchange 5/27
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 = (Σ 1 , Σ 2 , T 12 ), where ◮ Σ 1 , Σ 2 are disjoint signatures and ◮ T 12 is a TBox with assertions of the form C 1 ⊑ C 2 , where C 1 is a concept over Σ 1 , C 2 is a concept over Σ 2 , R 1 ⊑ R 2 , where R 1 is a role over Σ 1 , R 2 is a role over Σ 2 . Let I be an interpretation of Σ 1 and J an interpretation of Σ 2 . Then ( I , J ) satisfies M , denoted ( I , J ) | = M if I ⊆ C 2 J , for each C 1 ⊑ C 2 ∈ M , and ◮ C 1 I ⊆ R 2 J , for each R 1 ⊑ R 2 ∈ M . ◮ R 1 Arenas, Botoeva, Calvanese Knowledge Base Exchange 5/27
Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions Solutions for Knowledge Base Exchange Given an interpretation I of Σ 1 and a set X of interpretations of Σ 1 , let Sat M ( I ) = {J | ( I , J ) | = M} , Sat M ( X ) = � I∈X Sat M ( I ) . Arenas, Botoeva, Calvanese Knowledge Base Exchange 6/27
Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions Solutions for Knowledge Base Exchange Given an interpretation I of Σ 1 and a set X of interpretations of Σ 1 , let Sat M ( I ) = {J | ( I , J ) | = M} , Sat M ( X ) = � I∈X Sat M ( I ) . Definition Let M be a mapping, K 1 a KB over Σ 1 , and K 2 a KB over Σ 2 . K 2 is a solution for K 1 under M if: Mod ( K 2 ) ⊆ Sat M ( Mod ( K 1 )) . Arenas, Botoeva, Calvanese Knowledge Base Exchange 6/27
Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions Solutions for Knowledge Base Exchange Given an interpretation I of Σ 1 and a set X of interpretations of Σ 1 , let Sat M ( I ) = {J | ( I , J ) | = M} , Sat M ( X ) = � I∈X Sat M ( I ) . Definition Let M be a mapping, K 1 a KB over Σ 1 , and K 2 a KB over Σ 2 . K 2 is a solution for K 1 under M if: Mod ( K 2 ) ⊆ Sat M ( Mod ( K 1 )) . K 2 is a universal solution for K 1 under M if: Mod ( K 2 ) = Sat M ( Mod ( K 1 )) . Arenas, Botoeva, Calvanese Knowledge Base Exchange 6/27
Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions Solutions for Knowledge Base Exchange: Example Example Let K 1 = �T 1 , A 1 � M : and A 1 ⊑ A 2 T 1 : B 1 ⊑ A 1 B 1 ⊑ B 2 A 1 : B 1 ( b ) Arenas, Botoeva, Calvanese Knowledge Base Exchange 7/27
Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions Solutions for Knowledge Base Exchange: Example Example Let K 1 = �T 1 , A 1 � M : and A 1 ⊑ A 2 T 1 : B 1 ⊑ A 1 B 1 ⊑ B 2 A 1 : B 1 ( b ) Then, K 2 and K ′ 2 are solutions for K 1 under M K ′ 2 = �T ′ 2 , A ′ 2 � K 2 = �T 2 , A 2 � and T 2 : ∅ T ′ B 2 ⊑ A 2 2 : A 2 : B 2 ( b ) , A 2 ( b ) A ′ 2 : B 2 ( b ) Arenas, Botoeva, Calvanese Knowledge Base Exchange 7/27
Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions Solutions for Knowledge Base Exchange: Example Example Let K 1 = �T 1 , A 1 � M : and A 1 ⊑ A 2 T 1 : B 1 ⊑ A 1 B 1 ⊑ B 2 A 1 : B 1 ( b ) Then, K 2 and K ′ 2 are solutions for K 1 under M K ′ 2 = �T ′ 2 , A ′ 2 � K 2 = �T 2 , A 2 � and T 2 : ∅ T ′ B 2 ⊑ A 2 2 : A 2 : B 2 ( b ) , A 2 ( b ) A ′ 2 : B 2 ( b ) Moreover, K 2 is a universal solution for K 1 under M , while K ′ 2 is not. Arenas, Botoeva, Calvanese Knowledge Base Exchange 7/27
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. Arenas, Botoeva, Calvanese Knowledge Base Exchange 8/27
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, K 1 = �T 1 , A 1 � a KB over Σ 1 , and K 2 a KB over Σ 2 . K 2 is a CQ-solution for K 1 under M if for each CQ q over Σ 2 , cert ( q , �T 1 ∪ M , A 1 � ) ⊆ cert ( q , K 2 ) . Arenas, Botoeva, Calvanese Knowledge Base Exchange 8/27
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, K 1 = �T 1 , A 1 � a KB over Σ 1 , and K 2 a KB over Σ 2 . K 2 is a CQ-solution for K 1 under M if for each CQ q over Σ 2 , cert ( q , �T 1 ∪ M , A 1 � ) ⊆ cert ( q , K 2 ) . K 2 is a universal CQ-solution for K 1 under M if for each CQ q over Σ 2 , cert ( q , �T 1 ∪ M , A 1 � ) = cert ( q , K 2 ) . Arenas, Botoeva, Calvanese Knowledge Base Exchange 8/27
Knowledge Base Exchange Techniques for Deciding Knowledge Base Exchange Conclusions CQ-Solutions for Knowledge Base Exchange: Example Example Let K 1 = �T 1 , A 1 � M : and A 1 ⊑ A 2 T 1 : B 1 ⊑ A 1 B 1 ⊑ B 2 A 1 : B 1 ( b ) Arenas, Botoeva, Calvanese Knowledge Base Exchange 9/27
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