Complexity of DLs RWTH Aachen 1 Germany Complexity of DLs: - PowerPoint PPT Presentation

Complexity of DLs RWTH Aachen 1 Germany

Complexity of DLs: Overview of the Complexity of Concept Consistency P (co-)NP PSpace ExpTime NExpTime ALC reg ALUN (NP) ALCN add regular roles without ∃ , (wrt acyc. TBoxes) ALC u only ¬ A ALN + QI still in ExpTime add universal role without ⊔ ALE (co-NP) ALC + GCIs without ⊔ and NRs, ALCIQ R + wrt general TBoxes only ¬ A subsumption of subsumption of ALCN O ALCHIQ R + FL 0 FL 0 (co-NP) ALCO add role hierarchies ⊓ and ∀ only wrt acyc. TBoxes ALCIO I inverse roles: h-child − ALCIQO N NRs: ( ≥ n h-child ) Q Qual. NRs: ( ≥ n h-child Blond ) ALC ¬ ALC ¬ , ∩ , ∪ O nominals: ”John” is a concept ALCF F feature chain (dis)agreement · R + declare roles as transitive ALCF · ¬ , ∩ , ∪ Boolean ops on roles wrt acyc. TBoxes RWTH Aachen 2 Germany

Complexity of DLs: Overview of the Complexity of Concept Consistency P (co-)NP PSpace ExpTime NExpTime ALC reg ALUN (NP) ALCN add regular roles without ∃ , (wrt acyc. TBoxes) ALC u only ¬ A ALN + QI still in ExpTime add universal role without ⊔ ALE (co-NP) ALC + GCIs without ⊔ and NRs, ALCIQ R + wrt general TBoxes only ¬ A subsumption of subsumption of ALCN O ALCHIQ R + FL 0 FL 0 (co-NP) ALCO add role hierarchies ⊓ and ∀ only wrt acyc. TBoxes ALCIO I inverse roles: h-child − ALCIQO N NRs: ( ≥ n h-child ) Q Qual. NRs: ( ≥ n h-child Blond ) ALC ¬ ALC ¬ , ∩ , ∪ O nominals: ”John” is a concept ALCF F feature chain (dis)agreement · R + declare roles as transitive ALCF · ¬ , ∩ , ∪ Boolean ops on roles wrt acyc. TBoxes RWTH Aachen 3 Germany

Complexity of DLs: Overview of the Complexity of Concept Consistency P (co-)NP PSpace ExpTime NExpTime ALC reg ALUN (NP) ALCN add regular roles without ∃ , (wrt acyc. TBoxes) ALC u only ¬ A ALN + QI still in ExpTime add universal role without ⊔ ALE (co-NP) ALC + GCIs without ⊔ and NRs, ALCIQ R + wrt general TBoxes only ¬ A subsumption of subsumption of ALCN O ALCHIQ R + FL 0 FL 0 (co-NP) ALCO add role hierarchies ⊓ and ∀ only wrt acyc. TBoxes ALCIO I inverse roles: h-child − ALCIQO N NRs: ( ≥ n h-child ) Q Qual. NRs: ( ≥ n h-child Blond ) ALC ¬ ALC ¬ , ∩ , ∪ O nominals: ”John” is a concept ALCF F feature chain (dis)agreement · R + declare roles as transitive ALCF · ¬ , ∩ , ∪ Boolean ops on roles wrt acyc. TBoxes RWTH Aachen 4 Germany

Complexity of DLs: Overview of the Complexity of Concept Consistency P (co-)NP PSpace ExpTime NExpTime ALC reg ALUN (NP) ALCN add regular roles without ∃ , (wrt acyc. TBoxes) ALC u only ¬ A ALN + QI still in ExpTime add universal role without ⊔ ALE (co-NP) ALC + GCIs without ⊔ and NRs, ALCIQ R + wrt general TBoxes only ¬ A subsumption of subsumption of ALCN O ALCHIQ R + FL 0 FL 0 (co-NP) ALCO add role hierarchies ⊓ and ∀ only wrt acyc. TBoxes ALCIO I inverse roles: h-child − ALCIQO N NRs: ( ≥ n h-child ) Q Qual. NRs: ( ≥ n h-child Blond ) ALC ¬ ALC ¬ , ∩ , ∪ O nominals: ”John” is a concept ALCF F feature chain (dis)agreement · R + declare roles as transitive ALCF · ¬ , ∩ , ∪ Boolean ops on roles wrt acyc. TBoxes RWTH Aachen 5 Germany

Complexity of DLs: Overview of the Complexity of Concept Consistency P (co-)NP PSpace ExpTime NExpTime ALC reg ALUN (NP) ALCN add regular roles without ∃ , (wrt acyc. TBoxes) ALC u only ¬ A ALN + QI still in ExpTime add universal role without ⊔ ALE (co-NP) ALC + GCIs without ⊔ and NRs, ALCIQ R + wrt general TBoxes only ¬ A subsumption of subsumption of ALCN O ALCHIQ R + FL 0 FL 0 (co-NP) ALCO add role hierarchies ⊓ and ∀ only wrt acyc. TBoxes ALCIO I inverse roles: h-child − ALCIQO N NRs: ( ≥ n h-child ) Q Qual. NRs: ( ≥ n h-child Blond ) ALC ¬ ALC ¬ , ∩ , ∪ O nominals: ”John” is a concept ALCF F feature chain (dis)agreement · R + declare roles as transitive ALCF · ¬ , ∩ , ∪ Boolean ops on roles wrt acyc. TBoxes RWTH Aachen 6 Germany

Complexity of DLs: Overview of the Complexity of Concept Consistency P (co-)NP PSpace ExpTime NExpTime ALC reg ALUN (NP) ALCN add regular roles without ∃ , (wrt acyc. TBoxes) ALC u only ¬ A ALN + QI still in ExpTime add universal role without ⊔ ALE (co-NP) ALC + GCIs without ⊔ and NRs, ALCIQ R + wrt general TBoxes only ¬ A subsumption of subsumption of ALCN O ALCHIQ R + FL 0 FL 0 (co-NP) ALCO add role hierarchies ⊓ and ∀ only wrt acyc. TBoxes ALCIO I inverse roles: h-child − ALCIQO N NRs: ( ≥ n h-child ) Q Qual. NRs: ( ≥ n h-child Blond ) ALC ¬ ALC ¬ , ∩ , ∪ O nominals: ”John” is a concept ALCF F feature chain (dis)agreement · R + declare roles as transitive ALCF · ¬ , ∩ , ∪ Boolean ops on roles wrt acyc. TBoxes RWTH Aachen 7 Germany

Complexity of DLs: Overview of the Complexity of Concept Consistency P (co-)NP PSpace ExpTime NExpTime ALC reg ALUN (NP) ALCN add regular roles without ∃ , (wrt acyc. TBoxes) ALC u only ¬ A ALN + QI still in ExpTime add universal role without ⊔ ALE (co-NP) ALC + GCIs without ⊔ and NRs, ALCIQ R + wrt general TBoxes only ¬ A subsumption of subsumption of ALCN O ALCHIQ R + FL 0 FL 0 (co-NP) ALCO add role hierarchies ⊓ and ∀ only wrt acyc. TBoxes ALCIO I inverse roles: h-child − ALCIQO N NRs: ( ≥ n h-child ) Q Qual. NRs: ( ≥ n h-child Blond ) ALC ¬ ALC ¬ , ∩ , ∪ O nominals: ”John” is a concept ALCF F feature chain (dis)agreement · R + declare roles as transitive ALCF · ¬ , ∩ , ∪ Boolean ops on roles wrt acyc. TBoxes RWTH Aachen 8 Germany

Complexity of DLs: Overview of the Complexity of Concept Consistency P (co-)NP PSpace ExpTime NExpTime ALC reg ALUN (NP) ALCN add regular roles without ∃ , (wrt acyc. TBoxes) ALC u only ¬ A ALN + QI still in ExpTime add universal role without ⊔ ALE (co-NP) ALC + GCIs without ⊔ and NRs, ALCIQ R + wrt general TBoxes only ¬ A subsumption of subsumption of ALCN O ALCHIQ R + FL 0 FL 0 (co-NP) ALCO add role hierarchies ⊓ and ∀ only wrt acyc. TBoxes ALCIO I inverse roles: h-child − ALCIQO N NRs: ( ≥ n h-child ) Q Qual. NRs: ( ≥ n h-child Blond ) ALC ¬ ALC ¬ , ∩ , ∪ O nominals: ”John” is a concept ALCF F feature chain (dis)agreement · R + declare roles as transitive ALCF · ¬ , ∩ , ∪ Boolean ops on roles wrt acyc. TBoxes RWTH Aachen 9 Germany

Complexity of DLs: Overview of the Complexity of Concept Consistency P (co-)NP PSpace ExpTime NExpTime ALC reg ALUN (NP) ALCN add regular roles without ∃ , (wrt acyc. TBoxes) ALC u only ¬ A ALN + QI still in ExpTime add universal role without ⊔ ALE (co-NP) ALC without ⊔ and NRs, ALCIQ R + wrt general TBoxes only ¬ A subsumption of subsumption of ALCN O ALCHIQ R + FL 0 FL 0 (co-NP) ALCO add role hierarchies ⊓ and ∀ only wrt acyc. TBoxes ALCIO I inverse roles: h-child − ALCIQO N NRs: ( ≥ n h-child ) Q Qual. NRs: ( ≥ n h-child Blond ) ALC ¬ ALC ¬ , ∩ , ∪ O nominals: ”John” is a concept ALCF F feature chain (dis)agreement · R + declare roles as transitive ALCF · ¬ , ∩ , ∪ Boolean ops on roles wrt acyc. TBoxes RWTH Aachen 10 Germany

Complexity of DLs: Overview of the Complexity of Concept Consistency P (co-)NP PSpace ExpTime NExpTime ALC reg ALUN (NP) ALCN add regular roles without ∃ , (wrt acyc. TBoxes) ALC u only ¬ A ALN + QI still in ExpTime add universal role without ⊔ ALE (co-NP) ALC without ⊔ and NRs, ALCIQ R + wrt general TBoxes only ¬ A subsumption of subsumption of ALCN O ALCHIQ R + FL 0 FL 0 (co-NP) ALCO add role hierarchies ⊓ and ∀ only wrt acyc. TBoxes ALCIO I inverse roles: h-child − ALCIQO N NRs: ( ≥ n h-child ) Q Qual. NRs: ( ≥ n h-child Blond ) ALC ¬ ALC ¬ , ∩ , ∪ O nominals: ”John” is a concept ALCF F feature chain (dis)agreement · R + declare roles as transitive ALCF · ¬ , ∩ , ∪ Boolean ops on roles wrt acyc. TBoxes RWTH Aachen 11 Germany

Complexity of DLs: Overview of the Complexity of Concept Consistency P (co-)NP PSpace ExpTime NExpTime ALC reg ALUN (NP) ALCN add regular roles without ∃ , (wrt acyc. TBoxes) ALC u only ¬ A ALN + QI still in ExpTime add universal role without ⊔ ALE (co-NP) ALC without ⊔ and NRs, ALCIQ R + wrt general TBoxes only ¬ A subsumption of subsumption of ALCN O ALCHIQ R + FL 0 FL 0 (co-NP) ALCO add role hierarchies ⊓ and ∀ only wrt acyc. TBoxes ALCIO I inverse roles: h-child − ALCIQO N NRs: ( ≥ n h-child ) Q Qual. NRs: ( ≥ n h-child Blond ) ALC ¬ ALC ¬ , ∩ , ∪ O nominals: ”John” is a concept ALCF F feature chain (dis)agreement · R + declare roles as transitive ALCF · ¬ , ∩ , ∪ Boolean ops on roles wrt acyc. TBoxes RWTH Aachen 12 Germany

Complexity of DLs: Overview of the Complexity of Concept Consistency P (co-)NP PSpace ExpTime NExpTime ALC reg ALUN (NP) ALCN add regular roles without ∃ , (wrt acyc. TBoxes) ALC u only ¬ A ALN + QI still in ExpTime add universal role without ⊔ ALE (co-NP) ALC without ⊔ and NRs, ALCIQ R + wrt general TBoxes only ¬ A subsumption of subsumption of ALCN O ALCHIQ R + FL 0 FL 0 (co-NP) ALCO add role hierarchies ⊓ and ∀ only wrt acyc. TBoxes ALCIO I inverse roles: h-child − ALCIQO N NRs: ( ≥ n h-child ) Q Qual. NRs: ( ≥ n h-child Blond ) ALC ¬ ALC ¬ , ∩ , ∪ O nominals: ”John” is a concept ALCF F feature chain (dis)agreement · R + declare roles as transitive ALCF · ¬ , ∩ , ∪ Boolean ops on roles wrt acyc. TBoxes RWTH Aachen 13 Germany