The DL-Lite Family of Languages A FO Perspective Alessandro Artale KRDB Research Centre – Free University of Bozen-Bolzano Joint work with D. Calvanese, R. Kontchakov, M. Zakharyaschev TU Dresden. December 14–15, 2011
Recommended Readings [1] A. Artale, D. Calvanese, R. Kontchakov and M. Zakharyaschev. The DL-Lite family and relations . JAIR, 36:1–69, 2009. [2] D. Calvanese, G. De Giacomo, D. Lembo, M. Lenzerini, and R. Rosati. DL-Lite: Tractable description logics for ontologies . Proceedings of AAAI 2005. [3] D. Calvanese, G. De Giacomo, D. Lembo, M. Lenzerini, R. Rosati. Tractable reasoning and efficient query answering in DLs: The DL-Lite family . Journal of Automated Reasoning, 39:385–429, 2007. Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 1/40
Outline 1. Ontology based data access 2. The DL-Lite -family of ontology languages: DL-Lite bool , DL-Lite horn , DL-Lite core , DL-Lite krom 3. Translation to the one-variable fragment of First-Order Logic 4. Answering UCQ 5. Conclusions Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 2/40
Ontologies in Computer Science • Ontologies are formal specifications of a particular domain • Used to represent information at the conceptual level in terms of classes/concepts/entities and relationships between them • Typically expressed in logic: – First Order Logic – Description Logics: a specialized formalism (typically a fragment of FOL) for expressing knowledge in terms of classes and relationships Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 3/40
Ontologies in Computer Science • Ontologies are formal specifications of a particular domain • Used to represent information at the conceptual level in terms of classes/concepts/entities and relationships between them • Typically expressed in logic: – First Order Logic – Description Logics: a specialized formalism (typically a fragment of FOL) for expressing knowledge in terms of classes and relationships • Share strong similarities with other representation formalisms in Computer Science – Frame systems in Artificial Intelligence – ER diagrams in databases and information systems – UML class diagrams in software engineering – Constraints over a relational schema (inclusion and key dependencies) Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 3-a/40
Ontology based data access Desiderata: achieve logical transparency in access to data: • Hide to the user where and how data are stored • Present to the user a conceptual view of the data • Query the data sources through the conceptual model Query over conceptual layer Conceptual Layer Ontology Data Layer As in Data Integration, but with a rich conceptual description as the global view Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 4/40
Description Logics: The DL-Lite family The DL-Lite DLs provide an answer to our basic question: For which ontology languages can we answer queries over an ontology efficiently (in data complexity)? • DL-Lite is a family of DLs optimized according to the tradeoff between expressive power and data complexity • The DL-Lite family establishes the maximal subset of DLs constructs for which data complexity of query answering is L OG S PACE – Query answering techniques leverage on RDBMS technology (i.e. SQL) Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 5/40
Objectives of the Lecture 1. To show how the basic DL-Lite in [CDLLR,AAAI05; CDLLR,KR06] can be extended with full Booleans, cardinalities and role inclusion axioms obtaining the logic DL-Lite bool and three sublanguages: DL-Lite krom , DL-Lite core and DL-Lite horn 2. To characterize the first-order logic nature of class of DL-Lite DLs 3. To provide tight combined complexity results for reasoning in the new languages showing that: • Cardinalities are harmless; • Role inclusions, in most cases, destroy the nice computational behavior of DL-Lite ! 4. To show the L OG S PACE data complexity result of answering positive existential queries in DL-Lite horn . Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 6/40
The simplest DL-Lite Language: DL-Lite core DL-Lite core Ontology language: B 1 ⊑ B 2 , B 1 ⊑ ¬ B 2 with: • Concept Inclusions: − → A | ∃ R | ⊥ B P | P − − → R A ( c ) , ¬ A ( c ) P ( c, d ) , ¬ P ( c, d ) • ABox assertions: with c , d constants Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 7/40
The most expressive DL-Lite Language: DL-Lite R , N bool DL-Lite R , N bool Ontology language: C 1 ⊑ C 2 , with: • Concept Inclusions: − → B | ¬ C | C 1 ⊓ C 2 C − → A | ≥ q R | ⊥ B P | P − − → R R 1 ⊑ R 2 • Role Inclusions: • ABox assertions ( A ): A ( c ) , ¬ A ( c ) P ( c, d ) , ¬ P ( c, d ) , with c, d constants. Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 8/40
The most expressive DL-Lite Language: DL-Lite R , N bool DL-Lite R , N bool Ontology language: C 1 ⊑ C 2 , with: • Concept Inclusions: − → B | ¬ C | C 1 ⊓ C 2 C − → A | ≥ q R | ⊥ B P | P − − → R R 1 ⊑ R 2 • Role Inclusions: • ABox assertions ( A ): A ( c ) , ¬ A ( c ) P ( c, d ) , ¬ P ( c, d ) , with c, d constants. • A TBox, T , is a set of concept and role inclusions. A TBox, T , is what we call an Ontology, O . • A Knowledge Base is a pair K = ( T , A ) Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 8-a/40
core , DL-Lite R , N krom , DL-Lite R , N DL-Lite R , N horn DL-Lite R , N core Ontology language: B 1 ⊑ B 2 , B 1 ⊑ ¬ B 2 DL-Lite R , N krom Ontology language: B 1 ⊑ B 2 , B 1 ⊑ ¬ B 2 , ¬ B 1 ⊑ B 2 DL-Lite R , N k B k ⊑ B horn Ontology language: � Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 9/40
DL-Lite – Example ⊑ Manager Employee ⊑ AreaManager Manager ⊑ TopManager Manager AreaManager ⊓ TopManager ⊑ ⊥ ∃ WorksFor ⊑ Employee ∃ WorksFor − ⊑ Project ∃ WorksFor − ⊑ Project ≥ 2 Manages ⊑ ⊥ ≥ 2 Manages − ⊑ ⊥ . . . See [Artale et. al.,ER07] for more details on the correspondence between DL-Lite and conceptual data models Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 10/40
Semantics of DL-Lite Construct Syntax Example Semantics A I ⊆ ∆ I atomic concept A Doctor P I ⊆ ∆ I × ∆ I atomic role P child { ( d, e ) ∈ ∆ I × ∆ I | ( e, d ) ∈ P I } child − P − inverse role ⊥ ⊥ ∅ empty concept C I 1 ∩ C I C 1 ⊓ C 2 Doctor ⊓ Male conjunction 2 ∆ I \ C I ¬ C ¬ ( Doctor ⊓ Male ) negation { d ∈ ∆ I | ♯ { e ∈ ∆ I | ( d, e ) ∈ R I } ≥ n } ≥ 2 child − ≥ nR cardinalities Cl I ⊆ Cr I Cl ⊑ Cr Father ⊑ ≥ 1 child inclusion asser. a I ∈ A I memb. asser. A ( a ) Father ( bob ) ( a I , b I ) ∈ P I memb. asser. P ( a, b ) child ( bob , ann ) Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 11/40
Relevant reasoning tasks We are interested in: 1. Checking the consistency of the ontology ( Schema Consistency ) 2. Checking the consistency of single classes in the ontology ( Class Consistency ) 3. Checking whether new constraints hold in the ontology (e.g. discovering new I SA – Class Subsumption ) 4. Checking the consistency of the data wrt the ontology 5. Answering queries expressed over the ontology by means of the underlying data Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 12/40
FO Translation Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 13/40
DL-Lite N bool is NP-complete – Upper Bound • Class consistency for DL-Lite N bool can be reduced to formula satisfiability for the one-variable fragment QL 1 of first-order logic without equality and functions. • Formula satisfiability for the one-variable fragment QL 1 is known to be NP-complete [BGG:97]. • First we present a lengthy yet quite ‘natural’ and ‘transparent’ reduction · † ; • Then we shall see that this reduction can be substantially optimised to a L OG S PACE reduction · ‡ . Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 14/40
DL-Lite N bool is NP-complete – Upper Bound – Translating C • Inductive translation of Concepts, C ∗ : ( ⊥ ) ∗ = ⊥ ( A ) ∗ = A ( x ) ( ¬ C ) ∗ = ¬ C ∗ ( x ) ( C 1 ⊓ C 2 ) ∗ = C ∗ 1 ( x ) ∧ C ∗ 2 ( x ) ( ≥ q R ) ∗ = E q R ( x ) Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 15/40
DL-Lite N bool is NP-complete – Upper Bound – Translation K † • Translation of K =(TBox,ABox): The lengthy translation K † . T ∗ ∧ � A † ∧ � � �� � R ∈ role ± ( K ) R † � K † � ε ( R ) ∧ δ ( R ) ∧ = R ∈ role ± ( K ) Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 16/40
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