Introduzione alle logiche descrittive

Transcription

1 Introduzione alle logiche descrittive I principali formalismi di KR Reti semantiche Sistemi a Frame Sistemi di Produzione FOL (First Order Logic) e vari altri linguaggi logici Logiche descrittive

2 Le logiche descrittive Originariamente nate per dare una semantica formale ai sistemi a frame e alle reti semantiche Sono frammenti decidibili di FOL che supportano ragionamento tassonomico efficiente, ora anche per varianti piuttosto espressive. Alcune applicazioni recenti Costruzione e validazione di ontologie Ragionamento su modelli concettuali (per database) Description Logics Main characteristics: Description logics describe domain in terms of concepts, roles and individuals Concepts correspond to classes / are interpreted as sets of objects, roles correspond to relations / are interpreted as binary relations on objects Typical inferences Is a description satisfiable (non contradictory)? Is a description more general than another one? Can we derive that an individual is an instance of a certain concept?

3 DL Architecture Knowledge Base Tbox (schema) Man Human u Male Happy-Father Man u has-child Female u Abox (data) John : Happy-Father hjohn, Maryi : has-child Inference System Interface Description Logic Family Technically, DL are characterised by: A set of constructors for building complex concepts and roles from simpler ones A set of axioms for asserting facts about concepts, roles and individuals Example: the concept describing happy fathers could be written: Man È haschild.female È haschild.male È haschild.(rich Ë Happy)

4 DL Semantics DL semantics defined by interpretations: I = ( I, I),where I is the domain (a non-empty set) I is an interpretation function that maps: Concept (class) name A subset A I of I Role (property) name R binary relation R I over I Individual name i element i I of I ALC ALC is the smallest DL that is propositionally closed Constructors include booleans (and, or, not), and Restrictions on role successors (e.g., haschild.female and haschild.(rich Ë Happy )

5 ALC Semantics DL Concept and Role Constructors Different DLs are characterized by supporting specific constructors. Examples: Number restrictions (cardinality constraints) on roles, e.g., 3 haschild, 1 hasmother Qualified number restrictions, e.g., 2 haschild.female, 1 hasparent.male Nominals (singleton concepts), e.g., {Italy} Concrete domains (datatypes), e.g., hasage.( 21), Inverse roles, e.g., haschild (hasparent) Role Inclusion, e.g., hasdaughter (included in haschild) Transitive roles, e.g., haschild* (descendant) Role composition, e.g., hasparent o hasbrother (uncle)

6 Semantics of qualified number restrictions DL Knowledge Base Normally separated into 2 parts: TBox and ABox T (Tbox) is a set of axioms of the form: C v D (concept inclusion) C D (concept equivalence) R v S (role inclusion) R S (role equivalence) R + v R (role transitivity) The axioms describe structure of domain (i.e., a conceptual schema), e.g.: HappyFather Man È haschild.female È Elephant Ç Animal È Large È Grey ancestor + v ancestor

7 DL Knowledge Base A (Abox) is a set of axioms of the form x D (concept instantiation) hx,yi R (role instantiation) The axioms describe a concrete situation (data), e.g.: John HappyFather <John,Mary> haschild Separation of TBox and ABox is conceptually and practically convenient Note: The ABox has not a closed-world assumption DL Knowledge Base Two sorts of Tbox axioms often distinguished Definitions C v D or C D where C is a concept name (atomic concept) General Concept Inclusion axioms (GCIs) C v D where C in an arbitrary concept

8 Knowledge Base Semantics An interpretation I satisfies (models) an axiom A (I ² A): I ² C v D iff C I D I I ² C D iff C I = D I I ² R v S iff R I S I I ² R S iff R I = S I I ² R + v R iff (R I ) + R I I ² x D iff x I D I I ² hx,yi R iff (x I,y I ) R I Knowledge Base Semantics I satisfies a Tbox T (I ² T ) iff I satisfies every axiom A in T I satisfies an Abox A (I ² A) iff I satisfies every axiom A in A I satisfies a KB K (I ² K) iff I satisfies both T and A I is called a model of T, A, K, respectively.

9 Inference Tasks Knowledge is correct (captures intuitions) C subsumes D w.r.t. K iff for every model I of K, C I D I Knowledge is minimally redundant (no unintended synonyms) C is equivalent to D w.r.t. K iff for every model I of K, C I = D I Knowledge is meaningful (classes can have instances) C is satisfiable w.r.t. K iff there exists some model I of K s.t. C I Querying knowledge Inference Tasks x is an instance of C w.r.t. K iff for every model I of K, x I C I hx,yi is an instance of R w.r.t. K iff for, every model I of K, (x I,y I ) R I Knowledge base consistency A KB K is consistent iff there exists some model I of K

10 Short History of Reasoning Mechanisms for Description Logics Phase 1: Incomplete systems (Back, Classic, Loom,... ) Based on structural algorithms Phase 2: Development of tableau algorithms and complexity results Tableau-based systems for Pspace logics (e.g., Kris, Crack) Investigation of optimization techniques Phase 3: Tableau algorithms for very expressive DLs Highly optimized tableau systems for ExpTime logics (e.g., FaCT, DLP, Racer) Relationship to modal logic and decidable fragments of FOL Phase 4: Mature implementations Mainstream applications Databases Consistency of conceptual schemata (EER, UML etc.) Schema integration Query subsumption (w.r.t. a conceptual schema) Ontologies and Semantic Web Ontology engineering (design, maintenance, integration) Reasoning with ontology-based markup (meta-data) Service description and discovery

11 Advances in DL automatic reasoning References Basic Description Logics, F.Baader, W. Nutt, in Handbook of Description Logics Acknowledgement These slides contain substantial material from slides by Ian Horrocks, University of Manchester.