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

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Jul 01, 2023 568 likes •748 views

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

SLIDE 1

Complexity of DLs

RWTH Aachen Germany 1

SLIDE 2

Complexity of DLs: Overview of the Complexity of Concept Consistency (co-)NP P PSpace ExpTime NExpTime

ALCreg ALCN O add regular roles add universal role ALCu ALCO ALC+ GCIs ALCIQR+ + QI still in ExpTime wrt general TBoxes Q Qual. NRs: (≥ n h-child Blond) ·¬,∩,∪ Boolean ops on roles O nominals: ”John” is a concept without ⊔ ALN subsumption of wrt acyc. TBoxes subsumption of FL0 ·R+ declare roles as transitive F feature chain (dis)agreement ⊓ and ∀ only wrt acyc. TBoxes ALCF ALCF ALCIO ALC¬ ALCIQO ALC¬,∩,∪ ALCN FL0 (co-NP) ALUN (NP) without ∃,

nly ¬A

ALE (co-NP)

nly ¬A

without ⊔ and NRs, add role hierarchies ALCHIQR+ I inverse roles: h-child− N NRs: (≥ n h-child)

(wrt acyc. TBoxes)

RWTH Aachen Germany 2

SLIDE 3

Complexity of DLs: Overview of the Complexity of Concept Consistency (co-)NP P PSpace ExpTime NExpTime

ALCreg ALCN O add regular roles add universal role ALCu ALCO ALC+ GCIs ALCIQR+ + QI still in ExpTime wrt general TBoxes Q Qual. NRs: (≥ n h-child Blond) I inverse roles: h-child− N NRs: (≥ n h-child) ·¬,∩,∪ Boolean ops on roles O nominals: ”John” is a concept without ⊔ ALN subsumption of wrt acyc. TBoxes subsumption of FL0 ·R+ declare roles as transitive F feature chain (dis)agreement ⊓ and ∀ only wrt acyc. TBoxes ALCF ALCF ALCIO ALC¬ ALCIQO ALC¬,∩,∪ ALCN FL0 (co-NP) ALUN (NP) ALE (co-NP)

nly ¬A

without ⊔ and NRs, add role hierarchies ALCHIQR+

(wrt acyc. TBoxes)

without ∃,

nly ¬A

RWTH Aachen Germany 3

SLIDE 4

Complexity of DLs: Overview of the Complexity of Concept Consistency (co-)NP P PSpace ExpTime NExpTime

ALCreg ALCN O add regular roles add universal role ALCu ALCO ALC+ GCIs ALCIQR+ + QI still in ExpTime wrt general TBoxes Q Qual. NRs: (≥ n h-child Blond) I inverse roles: h-child− ·¬,∩,∪ Boolean ops on roles O nominals: ”John” is a concept ALN subsumption of wrt acyc. TBoxes subsumption of FL0 ·R+ declare roles as transitive F feature chain (dis)agreement ⊓ and ∀ only wrt acyc. TBoxes ALCF ALCF ALCIO ALC¬ ALCIQO ALC¬,∩,∪ ALCN FL0 (co-NP) ALUN (NP) ALE (co-NP)

nly ¬A

without ⊔ and NRs, add role hierarchies ALCHIQR+ N NRs: (≥ n h-child)

(wrt acyc. TBoxes)

without ∃,

nly ¬A

without ⊔

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SLIDE 5

Complexity of DLs: Overview of the Complexity of Concept Consistency (co-)NP P PSpace ExpTime NExpTime

ALCreg ALCN O add regular roles add universal role ALCu ALCO ALC+ GCIs ALCIQR+ + QI still in ExpTime wrt general TBoxes Q Qual. NRs: (≥ n h-child Blond) I inverse roles: h-child− ·¬,∩,∪ Boolean ops on roles O nominals: ”John” is a concept ALN subsumption of wrt acyc. TBoxes subsumption of FL0 ·R+ declare roles as transitive F feature chain (dis)agreement ⊓ and ∀ only wrt acyc. TBoxes ALCF ALCF ALCIO ALC¬ ALCIQO ALC¬,∩,∪ ALCN FL0 (co-NP) ALUN (NP) ALE (co-NP) add role hierarchies ALCHIQR+ N NRs: (≥ n h-child)

(wrt acyc. TBoxes)

without ∃,

nly ¬A

without ⊔ and NRs,

nly ¬A

without ⊔

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SLIDE 6

Complexity of DLs: Overview of the Complexity of Concept Consistency (co-)NP P PSpace ExpTime NExpTime

ALCreg ALCN O add regular roles add universal role ALCu ALCO ALC+ GCIs ALCIQR+ + QI still in ExpTime wrt general TBoxes Q Qual. NRs: (≥ n h-child Blond) I inverse roles: h-child− ·¬,∩,∪ Boolean ops on roles O nominals: ”John” is a concept ALN subsumption of wrt acyc. TBoxes FL0 ·R+ declare roles as transitive F feature chain (dis)agreement wrt acyc. TBoxes ALCF ALCF ALCIO ALC¬ ALCIQO ALC¬,∩,∪ ALCN FL0 (co-NP) ALUN (NP) ALE (co-NP) add role hierarchies ALCHIQR+ N NRs: (≥ n h-child) ⊓ and ∀ only subsumption of

nly ¬A

without ⊔ and NRs, without ⊔ without ∃,

nly ¬A

(wrt acyc. TBoxes)

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SLIDE 7

Complexity of DLs: Overview of the Complexity of Concept Consistency (co-)NP P PSpace ExpTime NExpTime

ALCreg ALCN O add regular roles add universal role ALCu ALCO ALC+ GCIs ALCIQR+ + QI still in ExpTime wrt general TBoxes Q Qual. NRs: (≥ n h-child Blond) I inverse roles: h-child− ·¬,∩,∪ Boolean ops on roles O nominals: ”John” is a concept ALN FL0 ·R+ declare roles as transitive F feature chain (dis)agreement wrt acyc. TBoxes ALCF ALCF ALCIO ALC¬ ALCIQO ALC¬,∩,∪ ALCN FL0 (co-NP) ALUN (NP) ALE (co-NP) add role hierarchies ALCHIQR+ N NRs: (≥ n h-child) wrt acyc. TBoxes subsumption of subsumption of ⊓ and ∀ only without ⊔ without ⊔ and NRs,

nly ¬A
nly ¬A

without ∃,

(wrt acyc. TBoxes)

RWTH Aachen Germany 7

SLIDE 8

Complexity of DLs: Overview of the Complexity of Concept Consistency (co-)NP P PSpace ExpTime NExpTime

ALCreg ALCN O add regular roles add universal role ALCu ALCO ALC+ GCIs ALCIQR+ + QI still in ExpTime wrt general TBoxes Q Qual. NRs: (≥ n h-child Blond) I inverse roles: h-child− ·¬,∩,∪ Boolean ops on roles O nominals: ”John” is a concept ALN FL0 ·R+ declare roles as transitive ALCF ALCF ALCIO ALC¬ ALCIQO ALC¬,∩,∪ ALCN FL0 (co-NP) ALUN (NP) ALE (co-NP) add role hierarchies ALCHIQR+ N NRs: (≥ n h-child) F feature chain (dis)agreement subsumption of without ⊔ without ∃,

nly ¬A

without ⊔ and NRs,

nly ¬A

subsumption of ⊓ and ∀ only wrt acyc. TBoxes

(wrt acyc. TBoxes)

wrt acyc. TBoxes

RWTH Aachen Germany 8

SLIDE 9

Complexity of DLs: Overview of the Complexity of Concept Consistency (co-)NP P PSpace ExpTime NExpTime

ALCreg ALCN O add regular roles add universal role ALCu ALCO ALC+ GCIs + QI still in ExpTime wrt general TBoxes Q Qual. NRs: (≥ n h-child Blond) I inverse roles: h-child− ·¬,∩,∪ Boolean ops on roles O nominals: ”John” is a concept ALN FL0 ALCF ALCF ALCIO ALC¬ ALCIQO ALC¬,∩,∪ ALCN FL0 (co-NP) ALUN (NP) ALE (co-NP) ALCHIQR+ N NRs: (≥ n h-child) F feature chain (dis)agreement ALCIQR+ ·R+ declare roles as transitive ⊓ and ∀ only subsumption of without ⊔ without ∃,

nly ¬A

without ⊔ and NRs,

nly ¬A

subsumption of wrt acyc. TBoxes

(wrt acyc. TBoxes)

add role hierarchies wrt acyc. TBoxes

RWTH Aachen Germany 9

SLIDE 10

Complexity of DLs: Overview of the Complexity of Concept Consistency (co-)NP P PSpace ExpTime NExpTime

ALCreg ALCN O ALCu ALCO + QI still in ExpTime Q Qual. NRs: (≥ n h-child Blond) I inverse roles: h-child− ·¬,∩,∪ Boolean ops on roles O nominals: ”John” is a concept ALN FL0 ·R+ declare roles as transitive ALCF ALCF ALCIO ALC¬ ALCIQO ALC¬,∩,∪ ALCN FL0 (co-NP) ALUN (NP) ALE (co-NP) ALCHIQR+ N NRs: (≥ n h-child) F feature chain (dis)agreement ALC without ⊔ subsumption of ⊓ and ∀ only wrt acyc. TBoxes subsumption of

nly ¬A

without ⊔ and NRs,

nly ¬A

without ∃,

(wrt acyc. TBoxes)

add role hierarchies wrt general TBoxes add universal role ALCIQR+ add regular roles wrt acyc. TBoxes

RWTH Aachen Germany 10

SLIDE 11

Complexity of DLs: Overview of the Complexity of Concept Consistency (co-)NP P PSpace ExpTime NExpTime

ALCreg ALCN O ALCu ALCO + QI still in ExpTime Q Qual. NRs: (≥ n h-child Blond) I inverse roles: h-child− ·¬,∩,∪ Boolean ops on roles O nominals: ”John” is a concept ALN FL0 ·R+ declare roles as transitive ALCF ALCF ALCIO ALC¬ ALCIQO ALC¬,∩,∪ ALCN FL0 (co-NP) ALUN (NP) ALE (co-NP) ALCHIQR+ N NRs: (≥ n h-child) F feature chain (dis)agreement ALCIQR+ ALC without ⊔ subsumption of ⊓ and ∀ only wrt acyc. TBoxes subsumption of without ⊔ and NRs,

nly ¬A
nly ¬A

without ∃,

(wrt acyc. TBoxes)

add regular roles add universal role wrt general TBoxes add role hierarchies wrt acyc. TBoxes

RWTH Aachen Germany 11

SLIDE 12

Complexity of DLs: Overview of the Complexity of Concept Consistency (co-)NP P PSpace ExpTime NExpTime

ALCreg ALCN O ALCu ALCO + QI still in ExpTime Q Qual. NRs: (≥ n h-child Blond) I inverse roles: h-child− ·¬,∩,∪ Boolean ops on roles O nominals: ”John” is a concept ALN FL0 ·R+ declare roles as transitive ALCF ALCF ALCIO ALC¬ ALCIQO ALC¬,∩,∪ ALCN FL0 (co-NP) ALUN (NP) ALE (co-NP) ALCHIQR+ N NRs: (≥ n h-child) F feature chain (dis)agreement ALCIQR+ ALC without ⊔ subsumption of ⊓ and ∀ only wrt acyc. TBoxes subsumption of

nly ¬A

without ⊔ and NRs,

nly ¬A

without ∃,

(wrt acyc. TBoxes)

add regular roles add universal role wrt general TBoxes add role hierarchies wrt acyc. TBoxes

RWTH Aachen Germany 12

SLIDE 13

Complexity of DLs: Overview of the Complexity of Concept Consistency (co-)NP P PSpace ExpTime NExpTime

ALCreg ALCN O ALCu ALCO + QI still in ExpTime Q Qual. NRs: (≥ n h-child Blond) I inverse roles: h-child− ·¬,∩,∪ Boolean ops on roles O nominals: ”John” is a concept ALN FL0 ALCF ALCF ALCIO ALC¬ ALCIQO ALC¬,∩,∪ ALCN FL0 (co-NP) ALUN (NP) ALE (co-NP) ALCHIQR+ N NRs: (≥ n h-child) F feature chain (dis)agreement ALCIQR+ ALC ·R+ declare roles as transitive without ⊔ subsumption of ⊓ and ∀ only wrt acyc. TBoxes subsumption of

nly ¬A

without ⊔ and NRs,

nly ¬A

without ∃,

(wrt acyc. TBoxes)

add regular roles add universal role wrt general TBoxes wrt acyc. TBoxes add role hierarchies

RWTH Aachen Germany 13

SLIDE 14

Complexity of DLs: Overview of the Complexity of Concept Consistency (co-)NP P PSpace ExpTime NExpTime

ALCreg ALCN O ALCu ALCO + QI still in ExpTime Q Qual. NRs: (≥ n h-child Blond) I inverse roles: h-child− O nominals: ”John” is a concept ALN FL0 ALCF ALCF ALCIO ALC¬ ALCIQO ALC¬,∩,∪ ALCN FL0 (co-NP) ALUN (NP) ALE (co-NP) ALCHIQR+ N NRs: (≥ n h-child) F feature chain (dis)agreement ALCIQR+ ALC ·R+ declare roles as transitive ·¬,∩,∪ Boolean ops on roles without ⊔ subsumption of ⊓ and ∀ only wrt acyc. TBoxes subsumption of

nly ¬A

without ⊔ and NRs,

nly ¬A

without ∃,

(wrt acyc. TBoxes)

add regular roles add universal role wrt general TBoxes add role hierarchies wrt acyc. TBoxes

RWTH Aachen Germany 14

SLIDE 15

Complexity of DLs: Overview of the Complexity of Concept Consistency (co-)NP P PSpace ExpTime NExpTime

ALCreg ALCN O ALCu ALCO + QI still in ExpTime Q Qual. NRs: (≥ n h-child Blond) I inverse roles: h-child− O nominals: ”John” is a concept ALN FL0 ALCF ALCF ALCIO ALC¬ ALCIQO ALC¬,∩,∪ ALCN FL0 (co-NP) ALUN (NP) ALE (co-NP) ALCHIQR+ N NRs: (≥ n h-child) F feature chain (dis)agreement ALCIQR+ ALC ·R+ declare roles as transitive ·¬,∩,∪ Boolean ops on roles without ⊔ subsumption of ⊓ and ∀ only wrt acyc. TBoxes subsumption of

nly ¬A

without ⊔ and NRs,

nly ¬A

without ∃,

(wrt acyc. TBoxes)

add regular roles add universal role wrt general TBoxes add role hierarchies wrt acyc. TBoxes

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SLIDE 16

Complexity of DLs: What was left out We left out a variety of complexity results for ➫ concept consistency of other DLs (e.g., those with “concrete domains”) ➫ other standard inferences (e.g., deciding consistency of ABoxes w.r.t. TBoxes) ➫ “non-standard” inferences such as – matching and unification of concepts – rewriting concepts – least common subsumer (of a set of concepts) – most specific concept (of an ABox individual)

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