The Complexity Landscape of Temporal Conjunctive Queriesstefborg/Talks/HiDeSt2018_slides.pdf · High-LevelModels Abstractentities&relations Discretizedinputsignals Ontology TheComplexityLandscapeofTemporalConjunctiveQueries

Stefan Borgwardt

The Complexity Landscape of TemporalConjunctive QueriesWorkshop on High-Level Declarative Stream Processing // Berlin, 25.09.2018

Declarative Stream Processing

The Complexity Landscape of Temporal Conjunctive Queries

© Stefan Borgwardt

Workshop on High-Level Declarative Stream Processing // Berlin, 25.09.2018

Slide 1 of 19

Declarative Stream Processing


© Stefan Borgwardt


Slide 1 of 19

High-Level Models

Abstract entities & relations

Discretized input signals

Ontology


© Stefan Borgwardt


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High-Level Models



Ontology


© Stefan Borgwardt


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High-Level Models



Ontology


© Stefan Borgwardt


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In This Talk� Description Logic / OWL Ontologies

� Temporal Logic Queries

� Computational Complexity

AC0 ⊆ NC1 ⊆ P ⊆ NP ⊆ PSPACE ⊆ EXP ⊆ NEXP ⊆ 2EXP


© Stefan Borgwardt


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© Stefan Borgwardt


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© Stefan Borgwardt


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Description Logic OntologiesConcept names Drone, Component, SensorRole names hasModel, isPartOfIndividual names drone-AB100,myDrone

Concepts Droneu ∃hasSensor.InfraredSensorAssertions drone-AB100 : DeliveryDrone,

(drone-AB100, ErnstReuterPlatz) : hasLocationConcept inclusions DeliveryDrone v Drone,InfraredSensor v ∃measures.Heat

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of

isPartOf(transitive)

hasPart(transitive)

Subclass of

isLandingSystemOf

hasLandingSystemhasSensor

isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel

isModelOfisFuselageOf

hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel

SpatialThing(external)

Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2


© Stefan Borgwardt


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Description Logic OntologiesConcept names Drone, Component, SensorRole names hasModel, isPartOfIndividual names drone-AB100,myDroneConcepts Droneu ∃hasSensor.InfraredSensor

Assertions drone-AB100 : DeliveryDrone,(drone-AB100, ErnstReuterPlatz) : hasLocation

Concept inclusions DeliveryDrone v Drone,InfraredSensor v ∃measures.Heat

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2


© Stefan Borgwardt


Slide 4 of 19

Description Logic OntologiesConcept names Drone, Component, SensorRole names hasModel, isPartOfIndividual names drone-AB100,myDroneConcepts Droneu ∃hasSensor.InfraredSensorAssertions drone-AB100 : DeliveryDrone,

(drone-AB100, ErnstReuterPlatz) : hasLocation

Concept inclusions DeliveryDrone v Drone,InfraredSensor v ∃measures.Heat

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2


© Stefan Borgwardt


Slide 4 of 19



Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2


© Stefan Borgwardt


Slide 4 of 19



Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2


© Stefan Borgwardt


Slide 4 of 19

Temporal Ontologies

A0 A1 . . . AnAssertions

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

Ontology O

?Query Q


© Stefan Borgwardt


Slide 5 of 19

Temporal Ontologies


Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

Ontology O

?Query Q


© Stefan Borgwardt


Slide 5 of 19

Temporal Ontologies


Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

Ontology O

?Query Q


© Stefan Borgwardt


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Temporal Conjunctive Queries (TCQs)Conjunctive queries, a subclass of SQL:

q(d) : ∃s, p. Drone(d)∧ hasSensor(d, s)∧measures(s, p)∧ HighAcceleration(p)

. . . combined via linear temporal operators (plus ∧, ∨, ¬):

Q(d) : Drone(d) ∧ ∃e. hasLocation(d, e)∧ ErnstReuterPlatz(e) ∧

♦− ∃f . hasLocation(d, f )∧ Hauptbahnhof(f )

(Baader, Borgwardt, and Lippmann 2013; Baader, Ghilardi, and Lutz 2012)


© Stefan Borgwardt


Slide 6 of 19

Temporal Conjunctive Queries (TCQs)Conjunctive queries, a subclass of SQL:

q(d) : ∃s, p. Drone(d)∧ hasSensor(d, s)∧measures(s, p)∧ HighAcceleration(p)

. . . combined via linear temporal operators (plus ∧, ∨, ¬):

Q(d) : Drone(d) ∧ ∃e. hasLocation(d, e)∧ ErnstReuterPlatz(e) ∧

♦− ∃f . hasLocation(d, f )∧ Hauptbahnhof(f )

(Baader, Borgwardt, and Lippmann 2013; Baader, Ghilardi, and Lutz 2012)


© Stefan Borgwardt


Slide 6 of 19

Temporal Concepts� Rigid names have a fixed interpretation: Drone, hasSensor

� Flexible names are allowed to change: hasLocation� No other form of temporal concepts, e.g.

Drone v ♦∃hasLocation.MaintenanceYard

More expressive and expensive

Undecidable in combination with rigid roles

No tools to develop temporal DL ontologies exist


© Stefan Borgwardt


Slide 7 of 19

Temporal Concepts� Rigid names have a fixed interpretation: Drone, hasSensor� Flexible names are allowed to change: hasLocation

� No other form of temporal concepts, e.g.






© Stefan Borgwardt


Slide 7 of 19

Temporal Concepts� Rigid names have a fixed interpretation: Drone, hasSensor� Flexible names are allowed to change: hasLocation� No other form of temporal concepts, e.g.






© Stefan Borgwardt


Slide 7 of 19

Temporal Concepts� Rigid names have a fixed interpretation: Drone, hasSensor� Flexible names are allowed to change: hasLocation� No other form of temporal concepts, e.g.





(Gabbay, Kurucz, Wolter, and Zakharyaschev 2003)


© Stefan Borgwardt


Slide 7 of 19

Answering TCQsSubclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

?

Does the query Q follow from the ontology O and observations A0, . . . ,An,taking the rigid names into account?

Do Q, O, and A0, . . . ,An have a common model,respecting the rigid names?


© Stefan Borgwardt


Slide 8 of 19

Answering TCQsSubclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

?

Does the query Q follow from the ontology O and observations A0, . . . ,An,taking the rigid names into account?

Do Q, O, and A0, . . . ,An have a common model,respecting the rigid names?


© Stefan Borgwardt


Slide 8 of 19

Complexity LandscapeDescription

logic

Data complexity Combined complexity

0 1 2 3 0 1 2 3

DL-Lite AC0

NC1

NP PSPACE

EL P CONP NP PSPACE CONEXP

ALC CONP EXP CONEXP 2EXP

SHIQ CONP ≤EXP 2EXP

SHO[Q|I] CONP ≤ ? ≤ EXP 2EXP

ALCHOIQ CONP ≤ ? ≤ dec. CO2NEXP ≤ ? ≤ dec.SROIQ CONP ≤ ? CO2NEXP ≤ ?

0 atemporal (CQs) 1 no rigid names 2 rigid concept names 3 rigid role names


© Stefan Borgwardt


Slide 9 of 19


logic


0 1 2 3 0 1 2 3

DL-Lite AC0

NC1

NP PSPACE








© Stefan Borgwardt


Slide 9 of 19


logic


0 1 2 3 0 1 2 3

DL-Lite AC0

NC1

NP PSPACE








© Stefan Borgwardt


Slide 9 of 19


logic


0 1 2 3 0 1 2 3

DL-Lite AC0

NC1

NP PSPACE








© Stefan Borgwardt


Slide 9 of 19


logic


0 1 2 3 0 1 2 3

DL-Lite AC0

NC1

NP PSPACE








© Stefan Borgwardt


Slide 9 of 19


logic


0 1 2 3 0 1 2 3

DL-Lite AC0

NC1

NP PSPACE





ALCHOIQ CONP ≤ ? ≤ dec. CO2NEXP ≤ ? ≤ dec.

SROIQ CONP ≤ ? CO2NEXP ≤ ?



© Stefan Borgwardt


Slide 9 of 19


logic


0 1 2 3 0 1 2 3

DL-Lite AC0

NC1

NP PSPACE








© Stefan Borgwardt


Slide 9 of 19

Propositional Abstraction� To decide satisfiability of Q over O and A0, . . . ,An,replace the inner CQs q1, . . . , qm by propositional variables p1, . . . , pm.

Qp : p1 ∧ p2 ∧ ♦−p3

� possible worldsW = {W1, . . . ,Wk} ⊆ 2{p1,...,pm}, e.g. {p1, p3}� mapping A : {0, . . . , n}→ {1, . . . , k}

WA(0) WA(1) . . . WA(n)

?Qp


© Stefan Borgwardt


Slide 10 of 19


Qp : p1 ∧ p2 ∧ ♦−p3

� possible worldsW = {W1, . . . ,Wk} ⊆ 2{p1,...,pm}, e.g. {p1, p3}

� mapping A : {0, . . . , n}→ {1, . . . , k}

WA(0) WA(1) . . . WA(n)

?Qp


© Stefan Borgwardt


Slide 10 of 19


Qp : p1 ∧ p2 ∧ ♦−p3

� possible worldsW = {W1, . . . ,Wk} ⊆ 2{p1,...,pm}, e.g. {p1, p3}� mapping A : {0, . . . , n}→ {1, . . . , k}

WA(0) WA(1) . . . WA(n)

?Qp


© Stefan Borgwardt


Slide 10 of 19

Main Approach

WA(0) WA(1) . . . WA(n) tn+1 . . . Qk

?Qp

T-satisfiability

LTL model t0, t1, . . . of Qp with� ti = WA(i) for 0 ≤ i ≤ n� ti ∈W for i > n

P / PSPACE

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

R-satisfiability

Models of

� QA(i),Ai and O for 0 ≤ i ≤ n� Qi and O for 1 ≤ i ≤ kthat respect the rigid names.

Qi :∧pj∈Wi

qj ∧∧pj /∈Wi

¬qj


© Stefan Borgwardt


Slide 11 of 19

Main Approach

QA(0),A0 QA(1),A1 . . . QA(n),An Q1 . . . Qk

?Qp

T-satisfiability

LTL model t0, t1, . . . of Qp with� ti = WA(i) for 0 ≤ i ≤ n� ti ∈W for i > n

P / PSPACE

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

R-satisfiability

Models of

� QA(i),Ai and O for 0 ≤ i ≤ n� Qi and O for 1 ≤ i ≤ kthat respect the rigid names.

Qi :∧pj∈Wi

qj ∧∧pj /∈Wi

¬qj


© Stefan Borgwardt


Slide 11 of 19

R-satisfiability in Case 1 (No Rigid Names)

QA(0),A0 QA(1),A1 . . . QA(n),An Q1 . . . Qk

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

independent atemporal satisfiability checks

“maximal” setW, mapping A



(Baader, Borgwardt, and Lippmann 2013; Borgwardt and Thost 2015b)


© Stefan Borgwardt


Slide 12 of 19


QA(0),A0 QA(1),A1 . . . QA(n),An Q1 . . . Qk

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2







© Stefan Borgwardt


Slide 12 of 19


QA(0),A0 QA(1),A1 . . . QA(n),An Q1 . . . Qk

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2







© Stefan Borgwardt


Slide 12 of 19

R-satisfiability in Case 3 (Rigid Role Names)

QA(0),A0 QA(1),A1 . . . QA(n),An Q1 . . . Qk

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

integrate satisfiability check by renaming the flexible names:

Q(0)A(0),A

(0)0, . . . ,Q(n)

A(n),A(n)n , Q(n+1)

1, . . . ,Q(n+k)

k , O(0), . . . ,O(n+k)





© Stefan Borgwardt


Slide 13 of 19


QA(0),A0 QA(1),A1 . . . QA(n),An Q1 . . . Qk

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2


Q(0)A(0),A

(0)0, . . . ,Q(n)

A(n),A(n)n , Q(n+1)

1, . . . ,Q(n+k)

k , O(0), . . . ,O(n+k)





© Stefan Borgwardt


Slide 13 of 19


QA(0),A0 QA(1),A1 . . . QA(n),An Q1 . . . Qk

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2


Q(0)A(0),A

(0)0, . . . ,Q(n)

A(n),A(n)n , Q(n+1)

1, . . . ,Q(n+k)

k , O(0), . . . ,O(n+k)





© Stefan Borgwardt


Slide 13 of 19


logic


0 1 2 3 0 1 2 3

DL-Lite AC0

NC1

NP PSPACE







(Baader, Borgwardt, and Lippmann 2015a,b; Borgwardt and Thost 2015b; Thost 2017)

(Borgwardt and Thost 2015a)


© Stefan Borgwardt


Slide 14 of 19


logic


0 1 2 3 0 1 2 3

DL-Lite AC0

NC1

NP PSPACE







(Baader, Borgwardt, and Lippmann 2015a,b; Borgwardt and Thost 2015b; Thost 2017)

(Borgwardt and Thost 2015a)


© Stefan Borgwardt


Slide 14 of 19

DL-Lite AC0

NC1

NP PSPACE

� Guess additional information P� Do not constructW and A explicitly� Combine T-satisfiability and R-satisfiability checks

Q0,A0 Q1,A1 . . . Qn,An Qn+1 . . .

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

P P P P P P P


© Stefan Borgwardt


Slide 15 of 19

DL-Lite AC0

NC1

NP PSPACE


Q0,A0 Q1,A1 . . . Qn,An Qn+1 . . .

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

P P P P P P P


© Stefan Borgwardt


Slide 15 of 19

DL-Lite AC0

NC1

NP PSPACE


Q0,A0 Q1,A1 . . . Qn,An Qn+1 . . .

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

P P P P P P P


© Stefan Borgwardt


Slide 15 of 19

DL-Lite AC0

NC1

NP PSPACE


Q0,A0 Q1,A1 . . . Qn,An Qn+1 . . .

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

P P P P P P P


© Stefan Borgwardt


Slide 15 of 19

DL-Lite AC0

NC1

NP PSPACE


Q0,A0 Q1,A1 . . . Qn,An Qn+1 . . .

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

P P P P P P P


© Stefan Borgwardt


Slide 15 of 19

DL-Lite AC0

NC1

NP PSPACE


Q0,A0 Q1,A1 . . . Qn,An Qn+1 . . .

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

P P P P P P P


© Stefan Borgwardt


Slide 15 of 19

DL-Lite AC0

NC1

NP PSPACE


Q0,A0 Q1,A1 . . . Qn,An Qn+1 . . .

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

P P P P P P P


© Stefan Borgwardt


Slide 15 of 19

DL-Lite AC0

NC1

NP PSPACE


Q0,A0 Q1,A1 . . . Qn,An Qn+1 . . .

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

P P P P P P P


© Stefan Borgwardt


Slide 15 of 19

DL-Lite AC0

NC1

NP PSPACE

� Rewrite the R-satisfiability check for QA(i) into a first-order query QO,Pi

and answer it over the database Ai

� Use an ALOGTIME Turing machine for T-satisfiability

QO,P0

. . . . . . QO,Pn2

QO,Pn2+1 . . . . . . QO,P

n

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

PP P

P P P P


© Stefan Borgwardt


Slide 16 of 19

DL-Lite AC0

NC1

NP PSPACE


and answer it over the database Ai� Use an ALOGTIME Turing machine for T-satisfiability

QO,P0

. . . . . . QO,Pn2

QO,Pn2+1 . . . . . . QO,P

n

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

PP P

P P P P


© Stefan Borgwardt


Slide 16 of 19

DL-Lite AC0

NC1

NP PSPACE



QO,P0

. . . . . . QO,Pn2

QO,Pn2+1 . . . . . . QO,P

n

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

PP P

P P P P


© Stefan Borgwardt


Slide 16 of 19

DL-Lite AC0

NC1

NP PSPACE



QO,P0

. . . . . . QO,Pn2

QO,Pn2+1 . . . . . . QO,P

n

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

PP P

P P P P


© Stefan Borgwardt


Slide 16 of 19

DL-Lite AC0

NC1

NP PSPACE



QO,P0

. . . . . . QO,Pn2

QO,Pn2+1 . . . . . . QO,P

n

Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

PP P

P P P P


© Stefan Borgwardt


Slide 16 of 19

NegationDisallowing negation in TCQs:

Description

logic


0 1 2 3 0 1 2 3

DL-Lite AC0

NP

EL P NP

ALC CONP EXP

SHIQ CONP 2EXP

(Borgwardt, Lippmann, and Thost 2015; Bourgaux and Turhan 2017)

bounded history

encoding

Closed-world negation?


© Stefan Borgwardt


Slide 17 of 19


Description

logic


0 1 2 3 0 1 2 3

DL-Lite AC0

NP

EL P NP

ALC CONP EXP

SHIQ CONP 2EXP


bounded history

encoding



© Stefan Borgwardt


Slide 17 of 19


Description

logic


0 1 2 3 0 1 2 3

DL-Lite AC0

NP

EL P NP

ALC CONP EXP

SHIQ CONP 2EXP


bounded history

encoding



© Stefan Borgwardt


Slide 17 of 19

Metric Temporal Operators

(Baader, Borgwardt, Koopmann, Ozaki, and Thost 2017; Brandt, Kalaycı,

Kontchakov, Ryzhikov, Xiao, and Zakharyaschev 2017; Gutiérrez-Basulto, Jung, and Ozaki 2016; Thost 2018)

� Refer to specific time intervals,

e.g. “in the last 10 minutes”:

♦[−10,0] ∃f . hasLocation(d, f )∧ Hauptbahnhof(f )

� Often cause an exponential blow-up ...

� ... unless restricted to intervals [0, c], [c,∞), [−c, 0], (−∞,−c]


© Stefan Borgwardt


Slide 18 of 19






♦[−10,0] ∃f . hasLocation(d, f )∧ Hauptbahnhof(f )� Often cause an exponential blow-up ...



© Stefan Borgwardt


Slide 18 of 19






♦[−10,0] ∃f . hasLocation(d, f )∧ Hauptbahnhof(f )� Often cause an exponential blow-up ...



© Stefan Borgwardt


Slide 18 of 19

OutlookDescription

logic


0 1 2 3 0 1 2 3

DL-Lite AC0

NC1

NP PSPACE






Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

?

Thank you!


© Stefan Borgwardt


Slide 19 of 19

OutlookDescription

logic


0 1 2 3 0 1 2 3

DL-Lite AC0

NC1

NP PSPACE






Subclass of

Subclass of

Subclass of

Subclass of

Subclass of

hasCategory

Subclass of


hasPart(transitive)

Subclass of

isLandingSystemOf


isSensorOf

isDirectPartOf

successorOf

Subclass of

Subclass of

hasModel


hasFuselage

hasDirectPart

isVariantOf

Subclass of

predecessorOf

AirCraftComp...

Sensor3

LandingSystem

Fuselage

HardwareCo...

Component

ProductModel


Drone,UAS, UAV

LiftGenerator...

GroundStatio...

Category2

?

Thank you!The Complexity Landscape of Temporal Conjunctive Queries

© Stefan Borgwardt


Slide 19 of 19

References IBaader, Franz, Stefan Borgwardt, Patrick Koopmann, Ana Ozaki, and Veronika Thost (2017). “Metric Temporal

Description Logics with Interval-Rigid Names”. In: Proc. FroCoS’17. Volume 10483. LNCS, pages 60–76.Baader, Franz, Stefan Borgwardt, and Marcel Lippmann (2013). “Temporalizing Ontology-Based Data Access”.

In: Proc. CADE’13. Volume 7898. LNCS, pages 330–344.— (2015a). “Temporal Conjunctive Queries in Expressive Description Logics with Transitive Roles”. In: Proc.AI’15. Volume 9457. LNAI, pages 21–33.

— (2015b). “Temporal Query Entailment in the Description Logic SHQ”. In: J. Web Semant. 33, pages 71–93.Baader, Franz, Silvio Ghilardi, and Carsten Lutz (2012). “LTL over Description Logic Axioms”. In: ACM T.Comput. Log. 13.3, 21:1–21:32.

Borgwardt, Stefan, Marcel Lippmann, and Veronika Thost (2015). “Temporalizing Rewritable Query

Languages over Knowledge Bases”. In: J. Web Semant. 33, pages 50–70.Borgwardt, Stefan and Veronika Thost (2015a). “Temporal Query Answering in DL-Lite with Negation”. In: Proc.GCAI’15. Volume 36. EPiC, pages 51–65.

— (2015b). “Temporal Query Answering in the Description Logic EL”. In: Proc. IJCAI’15, pages 2819–2825.Bourgaux, Camille and Anni-Yasmin Turhan (2017). “Temporal Query Answering in DL-Lite over Inconsistent

Data”. In: Proc. ISWC’17. Volume 10587. LNCS, pages 121–137.


© Stefan Borgwardt


Slide 20 of 19

References IIBrandt, Sebastian, Elem Güzel Kalaycı, Roman Kontchakov, Vladislav Ryzhikov, Guohui Xiao, and

Michael Zakharyaschev (2017). “Ontology-Based Data Access with a Horn Fragment of Metric Temporal

Logic”. In: Proc. AAAI’17, pages 1070–1076.Gabbay, Dov M., Agi Kurucz, Frank Wolter, and Michael Zakharyaschev (2003). Many-Dimensional ModalLogics: Theory and Applications. Volume 148. Studies in Logic and the Foundations of Mathematics.

Gutiérrez-Basulto, Víctor, Jean Christoph Jung, and Ana Ozaki (2016). “On Metric Temporal Description

Logics”. In: Proc. ECAI’16. Volume 285. FAIA, pages 837–845.Thost, Veronika (2017). “News on Temporal Conjunctive Queries”. In: Joint Proc. of the Web Stream Processingworkshop (WSP’17) and the 2nd Int. Workshop on Ontology Modularity, Contextuality, and Evolution(WOMoCoE’17). Volume 1936. CEUR-WS, pages 1–16.

— (2018). “Metric Temporal Extensions of DL-Lite and Interval-Rigid Names”. In: Proc. KR’18. To appear.


© Stefan Borgwardt


Slide 21 of 19

PicturesErnst-Reuter-Platz by “Andreas Levers”, CC BY-NC 2.0, title slide

Taipeh Taiwan Street Head by “tingyaoh”, CC0 1.0, slides 1, 19

Drone Logistics Drone Package Drone by “geralt”, CC0 1.0, slides 1, 19

Space Center SpaceX Control Center by “SpaceX-Imagery”, CC0 1.0, slides 1, 19

Network Rectangle Rings Networking by “geralt”, CC0 1.0, slides 1, 2

Auto Sensor by “Frank Derks”, CC BY 2.0, slide 2

Artificial Neural Network Ann by “ahmedgad”, CC0 1.0, slide 2

Dronetology by “David Martín Lammerding”, CC BY 4.0, slide 4 and later

World Globe Earth Planet Blue by “qimono”, CC0 1.0, slide 17, adapted

Stopwatch Timer Clock Symbol Icon by “skeeze”, CC0 1.0, slide 18


© Stefan Borgwardt


Slide 22 of 19

https://www.flickr.com/photos/96dpi/4500935582/

https://creativecommons.org/licenses/by-nc/2.0/

https://pixabay.com/en/taipei-taiwan-street-head-2057819/

https://creativecommons.org/publicdomain/zero/1.0/

https://pixabay.com/en/drone-logistics-drone-package-drone-2816244/


https://pixabay.com/en/space-center-spacex-control-center-693251/


https://pixabay.com/en/network-rectangle-rings-networking-1989138/


https://www.flickr.com/photos/eurosporttuning/16442169022/

https://creativecommons.org/licenses/by/2.0/

https://pixabay.com/en/artificial-neural-network-ann-3501528/


http://www.dronetology.net/

https://creativecommons.org/licenses/by/4.0/

https://pixabay.com/en/world-globe-earth-planet-blue-1303628/


https://pixabay.com/en/stopwatch-timer-clock-symbol-icon-2624277/


Documents

The Complexity Landscape of Temporal Conjunctive Queriesstefborg/Talks/HiDeSt2018_slides.pdf · High-LevelModels Abstractentities&relations Discretizedinputsignals Ontology TheComplexityLandscapeofTemporalConjunctiveQueries