Description Logics Propositional Description Logics Enrico Franconi franconi@cs.man.ac.uk http://www.cs.man.ac.uk/˜franconi Department of Computer Science, University of Manchester (1/59)
Summary: where we stand • Description Logics as a formalization of O-O languages • Description Logics as a predicate level language • Concepts • Roles • Reasoning in Description Logics • Subsumption • FL − : the simplest structural description logic (2/59)
Why Description Logics? If predicate logic is directly used without some kind of restriction, then • the structure of the knowledge/information is lost (no variables, concepts as classes, and roles as properties), • the expressive power is too high for having good computational properties and efficient procedures. (3/59)
Axioms, Disjunctions and Negations Teaching - Assistant ⊑ ¬ Undergrad ⊔ Professor ∀ x . Teaching - Assistant ( x ) → ¬ Undergrad ( x ) ∨ Professor ( x ) A necessary condition in order to be a teaching assistant is to be either not undergraduated or a professor. Clearly, a graduated student being a teaching assistant is not necessarily a professor; moreover, it may be the case that some professor is not graduated. Teaching - Assistant . = ¬ Undergrad ⊔ Professor ∀ x . Teaching - Assistant ( x ) ↔ ¬ Undergrad ( x ) ∨ Professor ( x ) When the left-han side is an atomic concept, the “ ⊑ ” symbol introduces a primitive definition – giving only necessary conditions – while the “ . = ” symbol introduces a real definition – with necessary and sufficient conditions. In general, it is possible to have complex concept expressions at the left-hand side as well. (4/59)
ALC : the simplest propositional DL A I ⊆ ∆ I A primitive concept R I ⊆ ∆ I × ∆ I R primitive role ∆ I ⊤ top ⊥ ∅ bottom ∆ I \ C I ¬ C complement C I ∩ D I C ⊓ D conjunction C I ∪ D I C ⊔ D disjunction { x | ∀ y . R I ( x, y ) → C I ( y ) } ∀ R.C universal quant. { x | ∃ y . R I ( x, y ) ∧ C I ( y ) } ∃ R.C existential quant. (5/59)
Closed Propositional Language • Conjunction is interpreted as intersection of sets of individuals. • Disjunction is interpreted as union of sets of individuals. • Negation is interpreted as complement of sets of individuals. • ∃ R. ⊤ ⇐ ⇒ ∃ R. • ¬ ( C ⊓ D ) ⇐ ⇒ ¬ C ⊔ ¬ D • ¬ ( C ⊔ D ) ⇐ ⇒ ¬ C ⊓ ¬ D • ¬ ( ∀ R.C ) ⇐ ⇒ ∃ R. ¬ C • ¬ ( ∃ R.C ) ⇐ ⇒ ∀ R. ¬ C (6/59)
Negating Universal formulæ • ¬ ( ∀ R.C ) = ⇒ ∃ R. ¬ C • ¬ ( ∃ R.C ) = ⇒ ∀ R. ¬ C ∆ (Compare with FL − expressivity) (7/59)
Formal Semantics An interpretation I = (∆ I , · I ) consists of: • a nonempty set ∆ I (the domain ) • a function · I (the interpretation function ) that maps • every concept to a subset of ∆ I • every role to a subset of ∆ I × ∆ I • every individual to an element of ∆ I An interpretation function · I is an extension function if and only if it satisfies the semantic definitions of the language. (8/59)
Knowledge Bases Σ = � TBox , Abox � • Terminological Axioms: C ⊑ D , C . = D • Student . = Person ⊓ ∃ NAME . String ⊓ ∃ ADDRESS . String ⊓ ∃ ENROLLED . Course • Student ⊑ ∃ ENROLLED . Course • ∃ TEACHES . Course ⊑ ¬ Undergrad ⊔ Professor • Membership statements: C ( a ) , R ( a, b ) • Student ( john ) • ENROLLED ( john , cs415 ) • ( Student ⊔ Professor )( paul ) (9/59)
TBox: descriptive semantics Different semantics have been proposed for the TBox, depending on the fact whether cyclic statements are allowed or not. We consider now the descriptive semantics, based on classical logics. ⊑ D if C I ⊆ D I . • An interpretation I satisfies the statement C ˙ • An interpretation I satisfies the statement C . = D if C I = D I . An interpretation I is a model for a TBox T if I satisfies all the statements in T . (10/59)
ABox If I = (∆ I , · I ) is an interpretation, • C ( a ) is satisfied by I if a I ∈ C I . • R ( a, b ) is satisfied by I if ( a I , b I ) ∈ R I . A set A of assertions is called an ABox. An interpretation I is said to be a model of the ABox A if every assertion of A is satisfied by I . The ABox A is said to be satisfiable if it admits a model. An interpretation I = (∆ I , · I ) is said to be a model of a knowledge base Σ if every axiom of Σ is satisfied by I . A knowledge base Σ is said to be satisfiable if it admits a model. (11/59)
Logical Implication Σ | = ϕ if every model of Σ is a model of ϕ Example: TBox: ∃ TEACHES . Course ⊑ ¬ Undergrad ⊔ Professor ABox: TEACHES ( john , cs415 ) , Course ( cs415 ) , Undergrad ( john ) Σ | = Professor ( john ) (12/59)
Logical Implication What if: TBox: ∃ TEACHES . Course ⊑ Undergrad ⊔ Professor ABox: TEACHES ( john , cs415 ) , Course ( cs415 ) , Undergrad ( john ) ? Σ | = Professor ( john ) ? Σ | = ¬ Professor ( john ) (13/59)
Reasoning Services • Concept Satisfiability Σ �| = C ≡ ⊥ Student ⊓ ¬ Person the problem of checking whether C is satisfiable w.r.t. Σ , i.e. whether there exists a model I of Σ such that C I � = ∅ • Subsumption Σ | = C ⊑ D Student ⊑ Person the problem of checking whether C is subsumed by D w.r.t. Σ , i.e. whether C I ⊆ D I in every model I of Σ • Satisfiability Student . Σ �| = = ¬ Person the problem of checking whether Σ is satisfiable, i.e. whether it has a model • Instance Checking Σ | = C ( a ) Professor ( john ) the problem of checking whether the assertion C ( a ) is satisfied in every model of Σ (14/59)
Reasoning Services (cont.) • Retrieval { a | Σ | = C ( a ) } Professor ⇒ john • Realization { C | Σ | = C ( a ) } john ⇒ Professor (15/59)
� � � � � � � � � � � � � � � � � � � � � � ✁ � � � � � � � � � � � ✁ ✁ ✁ � � � � � � � � � � � � � � � � � � � � � � � � � � � ✁ � � � � � � � � � � � � � � � � � � � � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ✁ Reduction to satisfiability • Concept Satisfiability Σ �| = C ≡ ⊥ ↔ exists x s.t. Σ ∪ { C ( x ) } has a model • Subsumption Σ | = C ⊑ D ↔ Σ ∪ { ( C ⊓ ¬ D )( x ) } has no models D C ¬ D • Instance Checking Σ | = C ( a ) ↔ Σ ∪ {¬ C ( a ) } has no models (16/59)
The Taxonomy TOP ✚ ❃ ⑥ ❩ ✚ ❩ INANIMATE ANIMATE ✻ ✻ COURSE PERSON ✒ � ❅ ■ � ❅ STUDENT PROFESSOR ❅ ■ � ✒ ❅ � WORKING-STUDENT • Subsumption is a partial ordering relation in the space of concepts. • If we consider only named concepts, subsumption induces a taxonomy where only direct subsumptions are explicitly drawn. • A taxonomy is the minimal relation in the space of named concepts such that its rlflexive-transitive closure is the subsumption relation. (17/59)
The Taxonomy N . = ANIMATE ⊓ ( STUDENT ⊔ PROFESSOR ) TOP ❃ ✚ ❩ ⑥ ✚ ❩ INANIMATE ANIMATE ✻ ✻ ■ ❅ ❅ N COURSE PERSON ✘✘✘✘ ✿✻ ✒ � ❅ ■ � ❅ STUDENT PROFESSOR ■ ❅ � ✒ ❅ � WORKING-STUDENT • Subsumption is a partial ordering relation in the space of concepts. • If we consider only named concepts, subsumption induces a taxonomy where only direct subsumptions are explicitly drawn. • A taxonomy is the minimal relation in the space of named concepts such that its rlflexive-transitive closure is the subsumption relation. (17/59)
Classification • Given a concept C and a TBox T , for all concepts D of T determine whether D subsumes C , or D is subsumed by C . • Intuitively, this amounts to finding the “right place” for C in the taxonomy implicitly present in T . • Classification is the task of inserting new concepts in a taxonomy. It is sorting in partial orders. (18/59)
Reasoning procedures • Terminating, efficient and complete algorithms for deciding satisfiability – and all the other reasoning services – are available. • Algorithms are based on tableaux-calculi techniques. • Completeness is important for the usability of description logics in real applications. • Such algorithms are efficient for both average and real knowledge bases, even if the problem in the corresponding logic is in PSPACE or EXPTIME. (19/59)
Tableaux Calculus The Tableaux Calculus is a decision procedure solving the problem of satisfiability. If a formula is satisfiable, the procedure will constructively exhibit a model of the formula. The basic idea is to incrementally build the model by looking at the formula, by decomposing it in a top/down fashion. The procedure exhaustively looks at all the possibilities, so that it can eventually prove that no model could be found for unsatisfiable formulas. (20/59)
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