A Similarity Measure for the ALN Description Logic Nicola Fanizzi, - PowerPoint PPT Presentation

A Similarity Measure for the ALN Description Logic Nicola Fanizzi, Claudia d’Amato Dipartimento di Informatica • Universit` a degli Studi di Bari Campus Universitario, Via Orabona 4, 70125 Bari, Italy CILC 2006 ⋄ Bari

Introduction & Motivation The Reference Representation Language A Similarity Measure for ALN Conclusions and Further Developments Contents Introduction & Motivation 1 Motivations Objectives 2 The Reference Representation Language Knowledge Base & Subsumption Normal Form A Similarity Measure for ALN 3 Definition Similarity Measure: example Measure Involving Individuals Discussion 4 Conclusions and Further Developments Conclusions Future Work N. Fanizzi, C. d’Amato A Similarity Measure

Introduction & Motivation The Reference Representation Language Motivations A Similarity Measure for ALN Objectives Conclusions and Further Developments Motivations Ontological knowledge Result of a complex process of knowledge acquisition Plays a key role for interoperability in the Semantic Web perspective Is expressed by standard ontology mark-up languages which are supported by well-founded semantics of Description Logics (DLs) Need of services able to build knowledge bases automatically or semi-automatically This can be done by the use of inductive inference services N. Fanizzi, C. d’Amato A Similarity Measure

Introduction & Motivation The Reference Representation Language Motivations A Similarity Measure for ALN Objectives Conclusions and Further Developments Objectives Induction of structural knowledge is known is ML (concept formation). This is generally applied on zero-order representations. our Goal → to make clusters of concepts or individuals asserted by mean ontological knowledge Problem → to define a similarity/dissimilarity measure applicable to ontology languages N. Fanizzi, C. d’Amato A Similarity Measure

Introduction & Motivation The Reference Representation Language Knowledge Base & Subsumption A Similarity Measure for ALN Normal Form Conclusions and Further Developments Why ALN Logic Knowledge representation by mean Description Logic ( ALN ) Description Logic is the counterpart framework of OWL language standard de facto for the knowledge representation in the Semantic Web N. Fanizzi, C. d’Amato A Similarity Measure

Introduction & Motivation The Reference Representation Language Knowledge Base & Subsumption A Similarity Measure for ALN Normal Form Conclusions and Further Developments The Representation Language Primitive concepts N C = { C , D , . . . } : subsets of a domain Primitive roles N R = { R , S , . . . } : binary relations on the domain Interpretation I = (∆ I , · I ) where ∆ I : domain of the interpretation and · I : interpretation function : Name Syntax Semantics ⊤ ∆ I top concept ⊥ ∅ bottom concept A I ⊆ ∆ I primitive concept A ∆ I \ A I ¬ A primitive negation concept conjunction C 1 ⊓ C 2 C I 1 ∩ C I 2 { x ∈ ∆ I | ∀ y ∈ ∆ I (( x , y ) ∈ R I → y ∈ C I ) } universal restriction ∀ R . C { x ∈ ∆ I | |{ y ∈ ∆ I | ( x , y ) ∈ R I } |≤ n } at-most restriction ≤ n . R { x ∈ ∆ I | |{ y ∈ ∆ I | ( x , y ) ∈ R I } |≥ n } at-least restriction ≥ n . R N. Fanizzi, C. d’Amato A Similarity Measure

Introduction & Motivation The Reference Representation Language Knowledge Base & Subsumption A Similarity Measure for ALN Normal Form Conclusions and Further Developments Knowledge Base & Subsumption K = �T , A� T-box T is a set of definitions C ≡ D , meaning C I = D I , where C is the concept name and D is a description A-box A contains extensional assertions on concepts and roles e.g. C ( a ) and R ( a , b ), meaning, resp., that a I ∈ C I and ( a I , b I ) ∈ R I . Subsumption Given two concept descriptions C and D , C subsumes D , denoted by C ⊒ D , iff for every interpretation I , it holds that C I ⊇ D I N. Fanizzi, C. d’Amato A Similarity Measure

Introduction & Motivation The Reference Representation Language Knowledge Base & Subsumption A Similarity Measure for ALN Normal Form Conclusions and Further Developments Examples Instances of concept definitions: Single ≡ Person ⊓ ≤ 0 . isMarriedTo Polygamist ≡ Person ⊓ ∀ isMarriedTo . Person ⊓ ≥ 2 . isMarriedTo Bigamist ≡ Person ⊓ ∀ isMarriedTo . Person ⊓ = 2 . isMarriedTo MalePolygamist ≡ Male ⊓ Person ⊓ ∀ isMarriedTo . Person ⊓ ≥ 2 . isMarriedTo The following are instances of simple assertions: Male(Bob), Person(Mary), Single(Jhon), isMarriedTo(Bob , Mary) It is easy to see that the following relationship holds: Poligamist ⊒ MalePolygamist. N. Fanizzi, C. d’Amato A Similarity Measure

Introduction & Motivation The Reference Representation Language Knowledge Base & Subsumption A Similarity Measure for ALN Normal Form Conclusions and Further Developments Other Inference Services instance checking decide whether an individual is an instance of a concept retrieval find all invididuals instance of a concept realization problem finding the concepts which an individual belongs to, especially the most specific one, if any: most specific concept Given an A-Box A and an individual a , the most specific concept of a w.r.t. A is the concept C , denoted MSC A ( a ), such that A | = C ( a ) and C ⊑ D , ∀ D such that A | = D ( a ). N. Fanizzi, C. d’Amato A Similarity Measure

Introduction & Motivation The Reference Representation Language Knowledge Base & Subsumption A Similarity Measure for ALN Normal Form Conclusions and Further Developments Normal Form C is in ALN normal form iff C ≡ ⊥ or C ≡ ⊤ or if � � P ⊓ ( ∀ R . C R ⊓ ≥ n . R ⊓ ≤ m . R ) C = R ∈ N R P ∈ prim( C ) where: C R = val R ( C ), n =min R ( C ) and m = max R ( C ) prim( C ) set of all (negated) atoms occurring at C ’s top-level val R ( C ) conjunction C 1 ⊓ · · · ⊓ C n in the value restriction on R , if any (o.w. val R ( C ) = ⊤ ); min R ( C ) = max { n ∈ N | C ⊑ ( ≥ n . R ) } (always finite number); max R ( C ) = min { n ∈ N | C ⊑ ( ≤ n . R ) } (if unlimited max R ( C ) = ∞ ) For any R , every sub-description in val R ( C ) is in normal form. N. Fanizzi, C. d’Amato A Similarity Measure

Introduction & Motivation Definition The Reference Representation Language Similarity Measure: example A Similarity Measure for ALN Measure Involving Individuals Conclusions and Further Developments Discussion A Similarity Measure for ALN : Definition / I L = ALN / ≡ the set of all concepts in ALN normal form I canonical interpretation of A A-Box s : L × L �→ [0 , 1] defined ∀ C , D ∈ L : s ( C , D ) := λ [ s P (prim( C ) , prim( D )) + 1 1 � + s (val R ( C ) , val R ( D )) + | N R | · | N R | R ∈ N R � · s N ((min R ( C ) , max R ( C )) , (min R ( D ) , max R ( D )))] R ∈ N R where λ ∈ ]0 , 1] (let λ = 1 / 3), N. Fanizzi, C. d’Amato A Similarity Measure

Introduction & Motivation Definition The Reference Representation Language Similarity Measure: example A Similarity Measure for ALN Measure Involving Individuals Conclusions and Further Developments Discussion A Similarity Measure for ALN : Defintion / II P C ∈ prim( C ) P I Q D ∈ prim( D ) Q I | � C ∩ � D | s P (prim( C ) , prim( D )) := P C ∈ prim( C ) P I Q D ∈ prim( D ) Q I | � C ∪ � D | s N (( m C , M C ) , ( m D , M D )) := min( M C , M D ) − max( m C , m D ) + 1 max( M C , M D ) − min( m C , m D ) + 1 s N (( m C , M C ) , ( m D , M D )) := 0 if min( M C , M D ) > max( m C , m D ) N. Fanizzi, C. d’Amato A Similarity Measure

Introduction & Motivation Definition The Reference Representation Language Similarity Measure: example A Similarity Measure for ALN Measure Involving Individuals Conclusions and Further Developments Discussion Similarity Measure: example... Let A be the considered ABox Person(Meg) , ¬ Male(Meg) , hasChild(Meg,Bob) , hasChild(Meg,Pat) , Person(Bob) , Male(Bob) , hasChild(Bob,Ann) , Person(Pat) , Male(Pat) , hasChild(Pat,Gwen) , Person(Gwen) , ¬ Male(Gwen) , Person(Ann) , ¬ Male(Ann) , hasChild(Ann,Sue) , marriedTo(Ann,Tom) , Person(Sue) , ¬ Male(Sue) , Person(Tom) , Male(Tom) and let C and D be two descriptions in ALN normal form: C ≡ Person ⊓ ∀ marriedTo . Person ⊓ ≤ 1 . hasChild ≡ Male ⊓ ∀ marriedTo . (Person ⊓ ¬ Male) ⊓ ≤ 2 . hasChild D N. Fanizzi, C. d’Amato A Similarity Measure

Introduction & Motivation Definition The Reference Representation Language Similarity Measure: example A Similarity Measure for ALN Measure Involving Individuals Conclusions and Further Developments Discussion ...Similarity Measure: example... In order to compute s ( C , D ) let us consider: Let be λ := 1 3 N R = { hasChild, marriedTo } → | N R | = 2  1  s P (prim( C ) , prim( D )) + 1 � s ( C , D ) := s (val R ( C ) , val R ( D )) + 3 2 R ∈ N R  1 � + s N ((min R ( C ) , max R ( C )) , (min R ( D ) , max R ( D )))  2 R ∈ N R N. Fanizzi, C. d’Amato A Similarity Measure