Hybrid Unification in the Description Logic EL Franz Baader, Oliver Fern´ andez Gil and Barbara Morawska 9th International Symposium, FroCoS Nancy, September 20th, 2013 1/23
Outline 1 The Description Logic EL . 2 Unification in EL . 3 Hybrid Unification in EL . 4 Hybrid Unification is NP-complete. 5 Goal-Oriented Unification Algorithm. 6 Conclusions. 2/23
The Description Logic EL . Syntax and Semantics. EL concept descriptions built from finite sets: using concept constructors: N C := { Head injury , Severe } ⊤ , ⊓ , ∃ r . C N R := { status } Head injury ⊓ ∃ status . Severe Semantics An interpretation I is a pair ( △ I , . I ) that assigns: • subsets C I of △ I to concept names C and, • binary relations r I on △ I to role names r . Semantics of the constructors: ⊤ I := △ I ( C ⊓ D ) I := C I ∩ D I ( ∃ r . C ) I := { x | ∃ y : ( x , y ) ∈ r I ∧ y ∈ C I } 3/23
The Description Logic EL . Syntax and Semantics. EL concept descriptions built from finite sets: using concept constructors: N C := { Head injury , Severe } ⊤ , ⊓ , ∃ r . C N R := { status } Head injury ⊓ ∃ status . Severe Semantics An interpretation I is a pair ( △ I , . I ) that assigns: • subsets C I of △ I to concept names C and, • binary relations r I on △ I to role names r . Semantics of the constructors: ⊤ I := △ I ( C ⊓ D ) I := C I ∩ D I ( ∃ r . C ) I := { x | ∃ y : ( x , y ) ∈ r I ∧ y ∈ C I } 3/23
The Description Logic EL . Terminological Axioms. Concept definitions A ≡ C for A ∈ N C and a concept description C . A TBox T is a finite set of concept definitions: . . . Severe injury ≡ Injury ⊓ ∃ status . Severe . . . cyclic TBox: T = { A ≡ . . . ⊓ ∃ r . B ⊓ . . . , B ≡ . . . ⊓ ∃ r . A ⊓ . . . } General Concept Inclusions (GCIs) C ⊑ D for concept descriptions C , D . An ontology O is a finite set of GCIs: . . . Severe injury ⊑ Injury ⊓ ∃ status . Severe Injury ⊓ ∃ status . Severe ⊑ Severe injury . . . 4/23
The Description Logic EL . Terminological Axioms. Concept definitions A ≡ C for A ∈ N C and a concept description C . A TBox T is a finite set of concept definitions: . . . Severe injury ≡ Injury ⊓ ∃ status . Severe . . . cyclic TBox: T = { A ≡ . . . ⊓ ∃ r . B ⊓ . . . , B ≡ . . . ⊓ ∃ r . A ⊓ . . . } General Concept Inclusions (GCIs) C ⊑ D for concept descriptions C , D . An ontology O is a finite set of GCIs: . . . Severe injury ⊑ Injury ⊓ ∃ status . Severe Injury ⊓ ∃ status . Severe ⊑ Severe injury . . . 4/23
The Description Logic EL . Reasoning. Subsumption problem C ⊑ O D iff C I ⊆ D I for all models I of O . Equivalence problem C ≡ O D iff C ⊑ O D and D ⊑ O C . Both problems are polynomial for EL . 5/23
Unification in EL . Example. Unification can be used to detect redundancies in ontologies. Two concept descriptions that are meant to represent the concept of a patient with severe head injury : Patient ⊓ ∃ finding . ( Head injury ⊓ ∃ status . Severe ) Patient ⊓ ∃ finding . ( Severe injury ⊓ ∃ location . Head ) They are not equivalent, but can be made equivalent w.r.t. the following TBox: � Severe injury ≡ Injury ⊓ ∃ status . Severe � Head injury ≡ Injury ⊓ ∃ location . Head 6/23
Unification in EL . The decision problem. Unification problem N C is partitioned into a set of defined concepts N def and a set of primitive concepts N prim . Instance: An ontology O . A finite set of subsumptions Γ = { C 1 ⊑ ? D 1 , . . . , C n ⊑ ? D n } . Question: Is there an acyclic TBox T such that: C 1 ⊑ O∪T D 1 , . . . , C n ⊑ O∪T D n Observations O contains only concept names from N prim . The solution TBox T provides definitions for the concept names in N def . 7/23
Unification in EL . The decision problem. Unification problem N C is partitioned into a set of defined concepts N def and a set of primitive concepts N prim . Instance: An ontology O . A finite set of subsumptions Γ = { C 1 ⊑ ? D 1 , . . . , C n ⊑ ? D n } . Question: Is there an acyclic TBox T such that: C 1 ⊑ O∪T D 1 , . . . , C n ⊑ O∪T D n Observations O contains only concept names from N prim . The solution TBox T provides definitions for the concept names in N def . 7/23
Unification in EL . Previous Results. [Baader and Morawska 2009] Unification in EL without background ontology is NP-complete An EL -unification problem that is solvable w.r.t. the empty ontology has a solution that is a local acyclic TBox. Locality Atoms of Γ ( At ): C , ∃ r . D Non-variable atoms ( At nv ): At \ N def A TBox T is local if each concept definition in T is of the following form: X ≡ D 1 ⊓ . . . ⊓ D n where D i ∈ At nv for all i , 1 ≤ i ≤ D n . NP-decision procedure Guesses a local acyclic TBox and then checks whether it is a unifier. 8/23
Unification in EL . Previous Results. [Baader and Morawska 2009] Unification in EL without background ontology is NP-complete An EL -unification problem that is solvable w.r.t. the empty ontology has a solution that is a local acyclic TBox. Locality Atoms of Γ ( At ): C , ∃ r . D Non-variable atoms ( At nv ): At \ N def A TBox T is local if each concept definition in T is of the following form: X ≡ D 1 ⊓ . . . ⊓ D n where D i ∈ At nv for all i , 1 ≤ i ≤ D n . NP-decision procedure Guesses a local acyclic TBox and then checks whether it is a unifier. 8/23
Unification in EL . Previous Results. [Baader and Morawska 2009] Unification in EL without background ontology is NP-complete An EL -unification problem that is solvable w.r.t. the empty ontology has a solution that is a local acyclic TBox. Locality Atoms of Γ ( At ): C , ∃ r . D Non-variable atoms ( At nv ): At \ N def A TBox T is local if each concept definition in T is of the following form: X ≡ D 1 ⊓ . . . ⊓ D n where D i ∈ At nv for all i , 1 ≤ i ≤ D n . NP-decision procedure Guesses a local acyclic TBox and then checks whether it is a unifier. 8/23
Unification in EL . Previous Results. Extending unification to non-empty ontologies • The notion of locality does not work. • Unification problems with solution, but no local unifiers. • The algorithm is complete only for cycle restricted ontologies [Baader, Borgwardt, Morawska 2011]. O �| = C ⊑ ∃ w . C A different approach Extend what is accepted as a solution to the unification problem: • Allow the TBox T to be cyclic. • Use greatest fixpoint semantics to interpret defined concepts in T . 9/23
Unification in EL . Previous Results. Extending unification to non-empty ontologies • The notion of locality does not work. • Unification problems with solution, but no local unifiers. • The algorithm is complete only for cycle restricted ontologies [Baader, Borgwardt, Morawska 2011]. O �| = C ⊑ ∃ w . C A different approach Extend what is accepted as a solution to the unification problem: • Allow the TBox T to be cyclic. • Use greatest fixpoint semantics to interpret defined concepts in T . 9/23
Hybrid Unification. Greatest fixpoint semantics [Nebel 1991]. Example • N prim := { Node } and N def := { INode } • T := { INode ≡ Node ⊓ ∃ edge . INode } • J is an interpretation of Node and edge : . . . m 1 m 2 m 3 n 1 • How to extend J to a model I of T ? classical models: { m 1 , m 2 , . . . } ∪ { n 1 } , { m 1 , m 2 , . . . } , { n 1 } or ∅ . gfp model: { m 1 , m 2 , . . . } ∪ { n 1 } . Formally, For two extensions I 1 , I 2 of J : I 1 ≺ J I 2 iff X I 1 ⊆ X I 2 for each X ∈ N def . The gfp model is the greatest element of ≺ J that models T . 10/23
Hybrid Unification. Greatest fixpoint semantics [Nebel 1991]. Example • N prim := { Node } and N def := { INode } • T := { INode ≡ Node ⊓ ∃ edge . INode } • J is an interpretation of Node and edge : . . . m 1 m 2 m 3 n 1 • How to extend J to a model I of T ? classical models: { m 1 , m 2 , . . . } ∪ { n 1 } , { m 1 , m 2 , . . . } , { n 1 } or ∅ . gfp model: { m 1 , m 2 , . . . } ∪ { n 1 } . Formally, For two extensions I 1 , I 2 of J : I 1 ≺ J I 2 iff X I 1 ⊆ X I 2 for each X ∈ N def . The gfp model is the greatest element of ≺ J that models T . 10/23
Hybrid Unification. Greatest fixpoint semantics [Nebel 1991]. Example • N prim := { Node } and N def := { INode } • T := { INode ≡ Node ⊓ ∃ edge . INode } • J is an interpretation of Node and edge : . . . m 1 m 2 m 3 n 1 • How to extend J to a model I of T ? classical models: { m 1 , m 2 , . . . } ∪ { n 1 } , { m 1 , m 2 , . . . } , { n 1 } or ∅ . gfp model: { m 1 , m 2 , . . . } ∪ { n 1 } . Formally, For two extensions I 1 , I 2 of J : I 1 ≺ J I 2 iff X I 1 ⊆ X I 2 for each X ∈ N def . The gfp model is the greatest element of ≺ J that models T . 10/23
Hybrid Unification. Hybrid EL -Ontologies. Hybrid EL -ontology [Brandt and Model 2005] A hybrid EL -ontology is a pair ( O , T ) where O is an EL -ontology and T is a (cyclic) TBox. An interpretation I is hybrid model of a hybrid ontology ( O , T ) iff: • I is an extension of a model J of O . • I is a gfp model of T . Subsumption w.r.t. hybrid ontologies C is subsumed by D w.r.t. ( O , T ) ( C ⊑ gfp , O∪T D ) iff: C I ⊆ D I , for every hybrid model I of ( O , T ) Subsumption is decidable in polynomial time. [Brandt and Model 2005, Novakovic 2007] 11/23
Recommend
More recommend
