Query-Based Entailment and Inseparability for ALC Ontologies Elena Botoeva Faculty of Computer Science, Free University of Bozen-Bolzano, Italy joint work with Carsten Lutz, Vladislav Ryzhikov, Frank Wolter and Michael Zakharyaschev Elena Botoeva(FUB) Query-Based Entailment and Inseparability for ALC Ontologies 1/7
Query Inseparability for Ontologies By an ontology O we mean • a knowledge base K = ( T , A ) , or • a TBox T . Query answering over ontologies is an important reasoning task. Elena Botoeva(FUB) Query-Based Entailment and Inseparability for ALC Ontologies 2/7
Query Inseparability for Ontologies By an ontology O we mean • a knowledge base K = ( T , A ) , or • a TBox T . Query answering over ontologies is an important reasoning task. Ontologies O 1 and O 2 are query-inseparable when we cannot distinguish between them by means of queries. Elena Botoeva(FUB) Query-Based Entailment and Inseparability for ALC Ontologies 2/7
Query Inseparability for Ontologies By an ontology O we mean • a knowledge base K = ( T , A ) , or • a TBox T . Query answering over ontologies is an important reasoning task. Ontologies O 1 and O 2 are query-inseparable when we cannot distinguish between them by means of queries. Applications • extracting modules • comparing versions of an ontology • forgetting some symbols from an ontology • exchanging knowledge Elena Botoeva(FUB) Query-Based Entailment and Inseparability for ALC Ontologies 2/7
Query Inseparability for Knowledge Bases Consider a class of queries Q ∈ { CQ , UCQ } , and a signature Σ of concept and role names. KBs K 1 = ( T 1 , A 1 ) and K 2 = ( T 2 , A 2 ) are Σ - Q inseparable , K 1 ≡ Q Σ K 2 , if K 1 | = q ( a ) ⇐ ⇒ K 2 | = q ( a ) for all Σ -queries q ∈ Q and all individuals a in K 1 and K 2 . Elena Botoeva(FUB) Query-Based Entailment and Inseparability for ALC Ontologies 3/7
Query Inseparability for Knowledge Bases Consider a class of queries Q ∈ { CQ , UCQ } , and a signature Σ of concept and role names. KBs K 1 = ( T 1 , A 1 ) and K 2 = ( T 2 , A 2 ) are Σ - Q inseparable , K 1 ≡ Q Σ K 2 , if K 1 | = q ( a ) ⇐ ⇒ K 2 | = q ( a ) for all Σ -queries q ∈ Q and all individuals a in K 1 and K 2 . Σ = { spots } Elena Botoeva(FUB) Query-Based Entailment and Inseparability for ALC Ontologies 3/7
Examples Query inseparability is different from logical equivalence: K 1 = ( { A ⊑ B } , { A ( a ) } ) and K 2 = ( ∅ , { A ( a ) , B ( a ) } ) K 1 ≡ (U)CQ K 1 �≡ K 2 but { A , B } K 2 Elena Botoeva(FUB) Query-Based Entailment and Inseparability for ALC Ontologies 4/7
Examples Query inseparability is different from logical equivalence: K 1 = ( { A ⊑ B } , { A ( a ) } ) and K 2 = ( ∅ , { A ( a ) , B ( a ) } ) K 1 ≡ (U)CQ K 1 �≡ K 2 but { A , B } K 2 Elena Botoeva(FUB) Query-Based Entailment and Inseparability for ALC Ontologies 4/7
Examples Query inseparability is different from logical equivalence: K 1 = ( { A ⊑ B } , { A ( a ) } ) and K 2 = ( ∅ , { A ( a ) , B ( a ) } ) K 1 ≡ (U)CQ K 1 �≡ K 2 but { A , B } K 2 UCQ-inseparability and CQ-inseparability are distinct: K 1 = ( { A ⊑ B ⊔ C } , { A ( a ) } ) and K 2 = ( ∅ , { A ( a ) } ) Elena Botoeva(FUB) Query-Based Entailment and Inseparability for ALC Ontologies 4/7
Examples Query inseparability is different from logical equivalence: K 1 = ( { A ⊑ B } , { A ( a ) } ) and K 2 = ( ∅ , { A ( a ) , B ( a ) } ) K 1 ≡ (U)CQ K 1 �≡ K 2 but { A , B } K 2 UCQ-inseparability and CQ-inseparability are distinct: K 1 = ( { A ⊑ B ⊔ C } , { A ( a ) } ) and K 2 = ( ∅ , { A ( a ) } ) K 1 �≡ UCQ K 1 ≡ CQ { A , B , C } K 2 but { A , B , C } K 2 Elena Botoeva(FUB) Query-Based Entailment and Inseparability for ALC Ontologies 4/7
Examples Query inseparability is different from logical equivalence: K 1 = ( { A ⊑ B } , { A ( a ) } ) and K 2 = ( ∅ , { A ( a ) , B ( a ) } ) K 1 ≡ (U)CQ K 1 �≡ K 2 but { A , B } K 2 UCQ-inseparability and CQ-inseparability are distinct: K 1 = ( { A ⊑ B ⊔ C } , { A ( a ) } ) and K 2 = ( ∅ , { A ( a ) } ) K 1 �≡ UCQ K 1 ≡ CQ { A , B , C } K 2 but { A , B , C } K 2 Signature makes a difference: K 1 ≡ UCQ { A } K 2 Elena Botoeva(FUB) Query-Based Entailment and Inseparability for ALC Ontologies 4/7
Query Inseparability for TBoxes Consider signatures: Σ 1 for ABoxes and Σ 2 for queries . TBoxes T 1 and T 2 are (Σ 1 , Σ 2 ) -(U)CQ inseparable , T 1 ≡ (U)CQ (Σ 1 , Σ 2 ) T 2 , if ( T 1 , A ) ≡ (U)CQ ( T 2 , A ) Σ 2 for all Σ 1 -ABoxes A . Elena Botoeva(FUB) Query-Based Entailment and Inseparability for ALC Ontologies 5/7
Main Results KBs : • (rooted) CQ-inseparability undecidable for ALC . • (rooted) UCQ-inseparability 2ExpTime-complete . TBoxes : • (rooted) CQ-inseparability undecidable for ALC . • CQ/UCQ-inseparability 2ExpTime-complete for Horn- ALC . • rooted CQ/UCQ-inseparability ExpTime-complete for Horn- ALC . Elena Botoeva(FUB) Query-Based Entailment and Inseparability for ALC Ontologies 6/7
See you at the poster! Elena Botoeva(FUB) Query-Based Entailment and Inseparability for ALC Ontologies 7/7
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