Description Logics—Basics, Applications, and More Ian Horrocks Information Management Group University of Manchester, UK Ulrike Sattler Institut f¨ ur Theoretische Informatik TU Dresden, Germany TU Dresden 1 Germany
Overview of the Tutorial • History and Basics: Syntax, Semantics, ABoxes, Tboxes, Inference Problems and their interrelationship, and Relationship with other (logical) formalisms • Applications of DLs: ER-diagrams with i.com demo, ontologies, etc. including system demonstration • Reasoning Procedures: simple tableaux and why they work • Reasoning Procedures II: more complex tableaux, non-standard inference prob- lems • Complexity issues • Implementing/Optimising DL systems TU Dresden 2 Germany
Description Logics • family of logic-based knowledge representation formalisms well-suited for the representation of and reasoning about ➠ terminological knowledge ➠ configurations ➠ ontologies ➠ database schemata – schema design, evolution, and query optimisation – source integration in heterogeneous databases/data warehouses – conceptual modelling of multidimensional aggregation ➠ . . . • descendents of semantics networks, frame-based systems, and KL-ONE • aka terminological KR systems, concept languages, etc. TU Dresden 3 Germany
Architecture of a Standard DL System I Knowledge Base N F Terminology E I R N Father = Man ⊓ ∃ has child. ⊤ ... E T Human = Mammal ⊓ Biped . N E . . C R Description E F Logic A Concrete Situation S C Y E John : Human ⊓ Father S John has child Bill T . . . E M TU Dresden 4 Germany
✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✂ ✁ ✁ ✁ ✁ ✄ ✁ ✁ ✁ ✂ ✂ ✁ ✂ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✁ ✁ ✄ � � � � � � � � � � � � � � ✄ ✄ ✄ ✄ � � ✁ � ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � � � � � � � � � � ✄ Introduction to DL I A Description Logic - mainly characterised by a set of constructors that allow to build complex concepts and roles from atomic ones, concepts correspond to classes / are interpreted as sets of objects, roles correspond to relations / are interpreted as binary relations on objects, Example: Happy Father in the DL ALC Man ⊓ ( ∃ has-child . Blue ) ⊓ ( ∃ has-child . Green ) ⊓ ( ∀ has-child . Happy ⊔ Rich ) TU Dresden 5 Germany
Introduction to DL: Syntax and Semantics of ALC Semantics given by means of an interpretation I = (∆ I , · I ) : Constructor Syntax Example Semantics A I ⊆ ∆ I atomic concept A Human R I ⊆ ∆ I × ∆ I atomic role likes R For C, D concepts and R a role name C I ∩ D I C ⊓ D Human ⊓ Male conjunction C I ∪ D I C ⊔ D Nice ⊔ Rich disjunction ∆ I \ C I ¬ C ¬ Meat negation ∃ R.C ∃ has-child.Human { x | ∃ y. � x, y � ∈ R I ∧ y ∈ C I } exists restrict. ∀ has-child.Blond { x | ∀ y. � x, y � ∈ R I ⇒ y ∈ C I } ∀ R.C value restrict. TU Dresden 6 Germany
Introduction to DL: Other DL Constructors Constructor Syntax Example Semantics { x | |{ y. � x, y � ∈ R I }| ≥ n } ( ≥ n R ) ( ≥ 7 has-child) number restriction { x | |{ y. � x, y � ∈ R I }| ≤ n } ( ≤ n R ) ( ≤ 1 has-mother) R − has-child − {� x, y � | � y, x � ∈ R I } inverse role R ∗ has-child ∗ ( R I ) ∗ trans. role { x | � u I 1 , . . . , u I u 1 , . . . , u n .P h-father · age, age. > n � ∈ P } concrete domain etc. Many different DLs/DL constructors have been investigated TU Dresden 7 Germany
Introduction to DL: Knowledge Bases: TBoxes For terminological knowledge: TBox contains Concept definitions = C ˙ ( A a concept name, C a complex concept) A = Man ⊓ ∃ has-child.Human Father ˙ = Mammal ⊓ ∀ has-child − . Human Human ˙ ❀ introduce macros/names for concepts, can be (a)cyclic C 1 ⊑ C 2 Axioms ( C i complex concepts) ∃ favourite . Brewery ⊑ ∃ drinks . Beer ❀ restrict your models An interpretation I satisfies A . iff A I = C I a concept definition = C C 1 ⊑ C 2 iff C I 1 ⊆ C I an axiom 2 T iff I satisfies all definitions and axioms in T a TBox ❀ I is a model of T TU Dresden 8 Germany
Introduction to DL: Knowledge Bases: ABoxes For assertional knowledge: ABox contains Concept assertions a : C ( a an individual name, C a complex concept) John : Man ⊓ ∀ has-child. ( Male ⊓ Happy ) � a 1 , a 2 � : R Role assertions ( a i individual names, R a role) � John , Bill � : has-child An interpretation I satisfies iff a I ∈ C I a concept assertion a : C � a 1 , a 2 � : R iff � a I 1 , a I 2 � ∈ R I a role assertion A iff I satisfies all assertions in A an ABox ❀ I is a model of A TU Dresden 9 Germany
Introduction to DL: Basic Inference Problems Is C I ⊆ D I in all interpretations I ? Subsumption: C ⊑ D Is C I ⊆ D I in all models I of T ? w.r.t. TBox T : C ⊑ T D ❀ structure your knowledge, compute taxonomy Consistency: Is C consistent w.r.t. T ? Is there a model I of T with C I � = ∅ ? of ABox A : Is A consistent? Is there a model of A ? of KB ( T , A ): Is ( T , A ) consistent? Is there a model of both T and A ? Inference Problems are closely related: C ⊑ T D iff C ⊓ ¬ D is in consistent w.r.t. T , (no model of I has an instance of C ⊓ ¬ D ) C is consistent w.r.t. T iff not C ⊑ T A ⊓ ¬ A ❀ Decision Procdures for consistency (w.r.t. TBoxes) suffice TU Dresden 10 Germany
Introduction to DL: Basic Inference Problems II For most DLs, the basic inference problems are decidable , with complexities between P and ExpTime . Why is decidability important? Why does semi-decidability not suffice? If subsumption (and hence consistency) is undecidable, and ➠ subsumption is semi-decidable, then consistency is not semi-decidable ➠ consistency is semi-decidable, then subsumption is not semi-decidable ➠ Quest for a “highly expressive” DL with “practicable” inference problems where expressiveness depends on the application practicability changed over the time TU Dresden 11 Germany
Introduction to DL: History Complexity of Inferences provided by DL systems over the time Investigation of Complexity of Inference Problems/Algorithms starts Loom KL-ONE NIKL Undecidable Fact, DLP, Race ExpTime Crack, Kris PSpace NP PTime Classic (AT&T) late early mid late ’80s ’90s ’90s ’90s TU Dresden 12 Germany
Introduction to DL: State-of-the-implementation-art In the last 5 years, DL-based systems were built that ✔ can handle DLs far more expressive than ALC (close relatives of converse-DPDL) • Number restrictions: “people having at most 2 cats and exactly 1 dog” • Complex roles: inverse (“has-child” — “child-of”), transitive closure (“offspring” — “has-child”), role inclusion (“has-daughter” — “has-child”), etc. ✔ implement provably sound and complete inference algorithms (for ExpTime-complete problems) ✔ can handle large knowledge bases (e.g., Galen medical terminology ontology: 2,740 concepts, 413 roles, 1,214 axioms) ✔ are highly optimised versions of tableau-based algorithms ✔ perform (surprisingly well) on benchmarks for modal logic reasoners (Tableaux’98, Tableaux’99) TU Dresden 13 Germany
Relationship with Other Logical Formalisms: First Order Predicate Logic Most DLs are decidable fragments of FOL: Introduce a unary predicate A for a concept name A a binary relation R for a role name R Translate complex concepts C, D as follows: t x ( A ) = A( x ) , t y ( A ) = A( y ) , t x ( C ⊓ D ) = t x ( C ) ∧ t x ( D ) , t y ( C ⊓ D ) = t y ( C ) ∧ t y ( D ) , t x ( C ⊔ D ) = t x ( C ) ∨ t x ( D ) , t y ( C ⊔ D ) = t y ( C ) ∨ t y ( D ) , t x ( ∃ R.C ) = ∃ y. R( x, y ) ∧ t y ( C ) , t y ( ∃ R.C ) = ∃ x. R( y, x ) ∧ t x ( C ) , t x ( ∀ R.C ) = ∀ y. R( x, y ) ⇒ t y ( C ) , t y ( ∀ R.C ) = ∀ x. R( y, x ) ⇒ t x ( C ) . A TBox T = { C i . = D i } is translated as � Φ T = ∀ x. t x ( C i ) ⇔ t x ( D i ) 1 ≤ i ≤ n TU Dresden 14 Germany
Relationship with Other Logical Formalisms: First Order Predicate Logic II C is consistent iff its translation t x ( C ) is satisfiable, C is consistent w.r.t. T iff its translation t x ( C ) ∧ Φ T is satisfiable, C ⊑ D iff t x ( C ) ⇒ t x ( D ) is valid C ⊑ T D iff Φ t ⇒ ∀ x. ( t x ( C ) ⇒ t x ( D )) is valid. ❀ ALC is a fragment of FOL with 2 variables (L2), known to be decidable ❀ ALC with inverse roles and Boolean operators on roles is a fragment of L2 ❀ further adding number restrictions yields a fragment of C2 (L2 with “counting quantifiers”), known to be decidable ✦ in contrast to most DLs, adding transitive roles (binary relations/ transitive closure operator) to L2 leads to undecidability ✦ many DLs (like many modal logics) are fragments of the Guarded Fragment ✦ most DLs are less complex than L2: L2 is NExpTime-complete, most DLs are in ExpTime TU Dresden 15 Germany
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