Description Logics. an introduction into its basic ideas

Transcription

1 Description Logics an introduction into its basic ideas A. Heußner WS 2003/2004

2 Preview: Basic Idea: from Network Based Structures to DL AL : Syntax / Semantics Enhancements of AL Terminologies (TBox) Assertions (ABox) Inferences Rules and other Language Extensions from SHIQ to Semantic Web... Alexander Heußner: Description Logics 1

3 Knowledge Modeling: simple Example Try to model a very simple sentence: Calypso loves Ulysses Alexander Heußner: Description Logics 2

4 We said too much!!! Who said, Calypso is a woman and Ulysses a man? Why is Calypso wearing a skirt? What has a heart to do with love? What we know is that: there is something called Calypso there is something called Ulysses between them there is some thing called Love going on Calypso Love Ulysses Alexander Heußner: Description Logics 3

5 another simple Example: Grandparents love Children! relation between two sets not just about two individuals (Calypso, Ulysses,... ) in term of sets: Grandparents is a subset of the set of things who love Children? How do we construct, given the set of children, the set of all those who love them?? What is meant by those who love children? Do they love only one child or all children or a subset...? Def. think of concepts: Love.Children Love.Children Alexander Heußner: Description Logics 4

6 Definitions back to our example: Grandparent Love.Children if we can express inclusion, we can also express equivalence, so we can write definitions: Grandparent. = Human HasChildren. HasChildren.Human Love.Children a set of definitions forms a terminology (TBox) express facts (what is really going on) with assertions (ABox): Ulysses:Human Ulysses: Grandparent Alexander Heußner: Description Logics 5

7 DL - System Description Logics ( DL ) is the most recent name for a family of knowledge representation (KR) formalisms that represent the knowledge of an application domain (the world) by first defining the relevant concepts of the domain (its terminology), and then using these concepts to specify properties of objects and individuals occurring in the domain (the world description). [Baader/Nutt] Alexander Heußner: Description Logics 6

8 AL a simple DL atomic concepts, roles (binary relations of concepts) add concept constructors ( different languages!) syntax: let A be an atomic concept, C, D concepts and R a role C, D A ( atomic concept ) ( universal concept ) ( bottom concept ) A ( atomic(!) negation ) C D ( intersection ) R.C ( value restriction ) R. ( limited existential quantification ) Alexander Heußner: Description Logics 7

9 semantics: let I = ( I, I) be an interpretation constructor syntax semantics (set-theoretic) atomic concept A A I I atomic role R R I I I universal concept I = I bottom concept I = atomic negation A ( A) I = I A I intersection C D (C D) I = C I D I value restr. R.C {a I b.(a, b) R I b C I } lim. ex. quant. R. {a I b.(a, b) R I } union (U) C D C I D I full (E) R.C {a I b.(a, b) R I b C I } number restr. (N ) ( nr) {a I {b (a, b) R I } n} negation (C) C I C I Alexander Heußner: Description Logics 8

10 family of AL-languages: AL[U][E][N ][C] remember: C D ( C D) and R.C R. C therefore: AL N ALUE... Alexander Heußner: Description Logics 9

11 connection to FOL concept names A unary predicates P A concept roles R binary predicates P R concepts formulae with one free variable φ x (A) = P A (x) φ x ( C) = φ x (C) φ x (C D) = φ x (C) φ x (D) φ x (C D) = φ x (C) φ x (D) φ x ( R.C) = y.p R (x, y) φ y (C) φ x ( R.C) = y.p r (x, y) φ y (C) φ y is symmetric with x and y exchanged! two variables suffices (no =, no constants, no fct. symbols)! not all DL are purely first order (transitive closure, etc.)! ALC is decidable fragment of FOL with 2 variables (L) Alexander Heußner: Description Logics 10

12 Terminologies TBox concept definitions: A. = C (A is individual name, C is cmplx. concept) introduce macros/names for concepts can be (a)cyclic (need fixpoint semantics) axioms: C D (C, D cmplx. concept) restrict models Def. an interpretation I satisfies... a concept definition A =. C iff A I = C I an axiom C D iff C I D I a TBox T iff I satisfies all definitions and axioms in T I is a model of T Alexander Heußner: Description Logics 11

13 Assertions ABox concept assertions: role assertion: < a 1, a 2 >: R a : C (a is individual name) Def. an interpretation I satisfies (with respect to T )... a concept assertion a : C iff a I C I a role assertion < a 1, a 2 > : R iff < a I, b I > R I a ABox A iff I satisfies all assertions in A I is a model of A!!! Open-World-Assumption model of A and T is an abstraction of a concrete world Alexander Heußner: Description Logics 12

14 Tasks of Inference in TBox: satisfiability: a concept C is satisfiable with respect to T iff there exists a model I of T such that C I subsumption: C T D (or T = C D) iff C I D I equivalence: C. = T D iff C I = D I (for every I of T ) disjointness: C T D iff C I D I = Alexander Heußner: Description Logics 13

15 in ABox: consistency of ABox A with T : there exists interpretation I which is a model of both A and T instantiation of assertion α by A: A = α iff every interpret. I which satisfies A also satisfies α retrieval / realization: find mostspecific concept for given individuals / find individual to given concepts Alexander Heußner: Description Logics 14

16 use (logical) Deduction: usage: make implicit knowledge explicit! Prop. all above tasks can be reduced to subsumption / satisfiability all above tasks decidable in DL (without proof ;-) need only one reasoning algorithm use standard reasoning algorithms (highly improved tableaux-algorithms like [Fact], [Racer],... ) Alexander Heußner: Description Logics 15

17 Rules extend KnowledgeBase (T,A) with rules: C D iff every instance individual of C is instance of D add trigger rules to Knowledge Base extend semantics...! attention: C D not equals D C Alexander Heußner: Description Logics 16

18 Language Extensions use usual binary relation operators as role constructors (R S, R, R S, R + (closure), R (inverse),... ) expressive number restriction: qualified restriction: 2 haschild.male number variables: Person α haschild.male α haschild.female set-/ one-of -relation: set of individual names {a 1,..., a n } with {a 1...} I = {a I 1...} introduce individual names (nominals) also in DL can describe finite sets: {a 1 } is singleton, {a 1 } {a 2 }... {a n } possible loss of properties like finite-models, decidability,... Alexander Heußner: Description Logics 17

19 DL and Ontologies use DL to build up ontologies from bottom-up instead of manually create a hierarchy and then assign properties to the concepts: first assign to each concept a logic definition which is then used to infer a classification advantages: hierarchy of ontology evolves with added concepts algorithm for classification is proved to be correct Alexander Heußner: Description Logics 18

20 from DL to Semantic Web web ontology languages OIL and OWL are based on DL : S = ALC with transitive roles SH = S with role Hierarchies SHI = SH with Inverse roles (R ) SHIQ = SHI with number restriction semantic foundation of ontology via semantics of SHIQ Alexander Heußner: Description Logics 19

21 References [Areces] Carlos Areces: Ontologies and Description Logics, presentation from May 2003, (link to PDF-version) [Baader/Nutt] Franz Baader, Werner Nutt: Basic Description Logics in Baader(ed.) et.al. Description Logics Handbook, Cambridge 2002 [Brachman/Nardi] Danielle Nardi, Ronald J. Brachman: An Introduction to Description Logics in Baader(ed.) et.al. Description Logics Handbook, Cambridge 2002 [Description Logics] Description Logics Website at [weblinks tested: ] Alexander Heußner: Description Logics 20

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23 Thanks to Carlos Areces whose eidetic introduction this presentation tried to track... the DL -community at Manchester, especially Ulrike Sattler, from whom I lent some foils ( to be honest, this presentation is a melange of the ones above ) this presentation was designed and made with the help of vim, (pdf)latex, P 4, xfig, and gv under OpenBSD; many thanks to the community for the possibility to use FREE Software «A. Heußner December 2003 back to start