(3) Ontology-Based Data Access

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(1)Using Ontology-Based Data Access to Enable Context Recognition in the Presence of Incomplete Information Verteidigung der Dissertation Veronika Thost. June 19, 2017. 1 / 17.

(2) Context Recognition Example. Example context User Bob watches a video, but then starts working with a text editor and the video window is not visible anymore Possible system optimization: Save resources by decreasing quality parameters of the video. 2 / 17.

(3) Ontology-Based Data Access. Components in user focus?. ID APP TYPE w1 a1 mov w4 a2 text Window. SENSOR TYPE USER ITEM TIME s3 cam ann book5 20:10 s1 cam bob w1 20:13 Observation. 3 / 17.

(4) Ontology-Based Data Access. Components in user focus? SELECT ID FROM Win WHERE Win.ID=Obs.ITEM & Obs.TYPE=cam. ID APP TYPE w1 a1 mov w4 a2 text Window. SENSOR TYPE USER ITEM TIME s3 cam ann book5 20:10 s1 cam bob w1 20:13 Observation. 3 / 17.

(5) Ontology-Based Data Access Ontology: Domain Terminology. Components in user focus?. VideoPlayer. Application. EnergyIntensive. SystemCritical Window. Component. NotVisible. LooksAt. FocusesOn. User. ID APP TYPE w1 a1 mov w4 a2 text Window. SENSOR TYPE USER ITEM TIME s3 cam ann book5 20:10 s1 cam bob w1 20:13 Observation. 3 / 17.

(6) Ontology-Based Data Access Ontology: Domain Terminology. Components in user focus?. VideoPlayer. Application. EnergyIntensive. SystemCritical Window. Component. NotVisible. LooksAt. FocusesOn. User. TYPE = mov. ID APP TYPE w1 a1 mov w4 a2 text Window. SENSOR TYPE USER ITEM TIME s3 cam ann book5 20:10 s1 cam bob w1 20:13 Observation. 3 / 17.

(7) Ontology-Based Data Access Ontology: Domain Terminology. Components in user focus?. VideoPlayer. Application. EnergyIntensive. SystemCritical Window. Component. NotVisible. LooksAt. FocusesOn. User. ID APP TYPE w1 a1 mov w4 a2 text Window. SENSOR TYPE USER ITEM TIME s3 cam ann book5 20:10 s1 cam bob w1 20:13 Observation. 3 / 17.

(8) Ontology-Based Data Access Ontology: Domain Terminology. Components in user focus?. VideoPlayer. Application. EnergyIntensive. SystemCritical Window. Component. NotVisible. LooksAt. FocusesOn. User. User(bob) ID APP TYPE VideoPlayer(a1) w1 a1 mov Window(w1) w4 a2 text HasPart(a1, w1) Window. SENSOR TYPE USER ITEM TIME s3 cam ann book5 20:10 s1 cam bob w1 20:13 LooksAt(bob, w1) NotVisible(w1) Observation. 3 / 17.

(9) Ontology-Based Data Access Ontology: Domain Terminology. Components in user focus? ∃y .User(y ) ∧ FocusesOn(y , x) ∧ Component(x) VideoPlayer. Application. Answer: x. EnergyIntensive. SystemCritical Window. Component. NotVisible. LooksAt. FocusesOn. User. User(bob) ID APP TYPE VideoPlayer(a1) w1 a1 mov Window(w1) w4 a2 text HasPart(a1, w1) Window. SENSOR TYPE USER ITEM TIME s3 cam ann book5 20:10 s1 cam bob w1 20:13 LooksAt(bob, w1) NotVisible(w1) Observation. 3 / 17.

(10) Ontology-Based Data Access Ontology: Domain Terminology. Components in user focus? ∃y .User(y ) ∧ FocusesOn(y , x) ∧ Component(x) VideoPlayer v Application. Answer: x. EnergyIntensive. SystemCritical Window v Component LooksAt v FocusesOn. NotVisible User. User(bob) VideoPlayer(a1) Window(w1) HasPart(a1, w1) Window. LooksAt(bob, w1) Observation. NotVisible(w1). 3 / 17.

(11) Ontology-Based Data Access Ontology: Domain Terminology. Components in user focus? ∃y .User(y ) ∧ FocusesOn(y , x) ∧ Component(x). Answer: x. VideoPlayer v Application u EnergyIntensive u ¬SystemCritical Window v Component LooksAt v FocusesOn. NotVisible User. User(bob) VideoPlayer(a1) Window(w1) HasPart(a1, w1) Window. LooksAt(bob, w1) Observation. NotVisible(w1). 3 / 17.

(12) Ontology-Based Data Access Query Answering. Components in user focus? ∃y .User(y ) ∧ FocusesOn(y , x) ∧ Component(x). Answer: x = w1. VideoPlayer v Application u EnergyIntensive u ¬SystemCritical Window v Component LooksAt v FocusesOn. NotVisible User. User(bob) VideoPlayer(a1) Window(w1) HasPart(a1, w1) Window. LooksAt(bob, w1) Observation. NotVisible(w1). 3 / 17.

(13) Ontology-Based Data Access Query Answering. Components in user focus in the past, and not visible anymore (now)?. VideoPlayer v Application u EnergyIntensive u ¬SystemCritical Window v Component LooksAt v FocusesOn. NotVisible User. User(bob) VideoPlayer(a1) Window(w1) HasPart(a1, w1) Window. LooksAt(bob, w1) Observation. NotVisible(w1). 3 / 17.

(14) Ontology-Based Data Access Query Answering. Components in user focus in the past, and not visible anymore (now)?. VideoPlayer v Application u EnergyIntensive u ¬SystemCritical Window v Component. NotVisible. LooksAt v FocusesOn. User(bob). 20 : 11. User 20 : 13. 20 : 15. VideoPlayer(a1) Window(w1) HasPart(a1, w1) Window. LooksAt(bob, w1) Observation. NotVisible(w1). 3 / 17.

(15) Ontology-Based Data Access Query Answering. Components in user focus in the past, and not visible anymore (now)? (3P ∃y .User(y ) ∧ FocusesOn(y , x) ∧ Component(x)) ∧ NotVisible(x) VideoPlayer v Application u EnergyIntensive u ¬SystemCritical Window v Component. NotVisible. LooksAt v FocusesOn. User(bob). 20 : 11. User 20 : 13. 20 : 15. VideoPlayer(a1) Window(w1) HasPart(a1, w1) Window. LooksAt(bob, w1) Observation. NotVisible(w1). 3 / 17.

(16) Ontology-Based Data Access Query Answering with Rigid Names. Components in user focus in the past, and not visible anymore (now)? (3P ∃y .User(y ) ∧ FocusesOn(y , x) ∧ Component(x)) ∧ NotVisible(x) VideoPlayer v Application u EnergyIntensive u ¬SystemCritical Window v Component. NotVisible. LooksAt v FocusesOn. User(bob). 20 : 11. User 20 : 13. 20 : 15. VideoPlayer(a1) Window(w1) HasPart(a1, w1) Window. LooksAt(bob, w1) Observation. NotVisible(w1). 3 / 17.

(17) Ontology-Based Data Access Query Answering with Rigid Names. Components in user focus in the past, and not visible anymore (now)? (3P ∃y .User(y ) ∧ FocusesOn(y , x) ∧ Component(x)) ∧ NotVisible(x) VideoPlayer v Application u EnergyIntensive u ¬SystemCritical Window v Component. NotVisible. LooksAt v FocusesOn. User(bob). 20 : 11. User(bob). User 20 : 13. User(bob). 20 : 15. VideoPlayer(a1) Window(w1) HasPart(a1, w1) Window. Component(w1). Component(w1). LooksAt(bob, w1) Observation. NotVisible(w1). 3 / 17.

(18) Ontology-Based Data Access Query Answering with Rigid Names. Components in user focus in the past, and not visible anymore (now)? (3P ∃y .User(y ) ∧ FocusesOn(y , x) ∧ Component(x)) ∧ NotVisible(x) VideoPlayer v Application u EnergyIntensive u ¬SystemCritical Window v Component. NotVisible. LooksAt v FocusesOn. User(bob). 20 : 11. User(bob). User 20 : 13. User(bob). 20 : 15. VideoPlayer(a1) Window(w1) HasPart(a1, w1) Window. Component(w1). Component(w1). LooksAt(bob, w1) Observation. NotVisible(w1). 3 / 17.

(19) Outline • Temporal data: sequence of fact bases • Ontology: lightweight description logics (DLs) • Temporal queries: linear temporal logic (LTL) + conjunctive queries (CQs). 4 / 17.

(20) Outline • Temporal data: sequence of fact bases • Ontology: lightweight description logics (DLs) • Temporal queries: linear temporal logic (LTL) + conjunctive queries (CQs). I Problem: Results: Application:. Temporal query satisfiability Computational complexity Choose languages according to available resources (time and memory). 4 / 17.

(21) Outline • Temporal data: sequence of fact bases • Ontology: lightweight description logics (DLs) • Temporal queries: linear temporal logic (LTL) + conjunctive queries (CQs). Problem: Results: Application:. I. II. Temporal query satisfiability Computational complexity Choose languages according to available resources (time and memory). Temporal query answering Rewritability Hints for implementation (use existing tools). 4 / 17.

(22) Outline • Temporal data: sequence of fact bases • Ontology: lightweight description logics (DLs) • Temporal queries: linear temporal logic (LTL) + conjunctive queries (CQs). Why . . . • no temporal ontology language? expensive. 3P User v User. • DLs? user-friendly, well investigated, basis for W3C OWL standard • lightweight DLs? allow for efficient atemporal reasoning • CQs? describe complex networks. 4 / 17.

(23) Lightweight Description Logics Symbols • Individual names:. ann, bob, w1, . . . • Concept names:. Component, User, Window, . . . • Role names:. LooksAt, FocusesOn, HasPart, . . .. 5 / 17.

(24) Lightweight Description Logics Symbols. Semantics: I = (∆I , ·I ). • Individual names:. Interpretation domain ∆I and function ·I :. ann, bob, w1, . . . • Concept names:. Component, User, Window, . . . • Role names:. LooksAt, FocusesOn, HasPart, . . .. 5 / 17.

(25) Lightweight Description Logics Symbols. Semantics: I = (∆I , ·I ). • Individual names:. Interpretation domain ∆I and function ·I :. ann, bob, w1, . . . • Concept names:. Component, User, Window, . . . • Role names:. LooksAt, FocusesOn, HasPart, . . .. a1I. w1I. bobI. annI. 5 / 17.

(26) Lightweight Description Logics Symbols. Semantics: I = (∆I , ·I ). • Individual names:. Interpretation domain ∆I and function ·I :. ann, bob, w1, . . . • Concept names:. Component, User, Window, . . . • Role names:. LooksAt, FocusesOn, HasPart, . . .. a1I. w1I ComponentI. bobI. annI UserI. 5 / 17.

(27) Lightweight Description Logics Symbols. Semantics: I = (∆I , ·I ). • Individual names:. Interpretation domain ∆I and function ·I :. ann, bob, w1, . . . • Concept names:. Component, User, Window, . . . • Role names:. LooksAt, FocusesOn, HasPart, . . .. ApplicationI a1I WindowI I. w1. ComponentI. bobI. annI UserI. 5 / 17.

(28) Lightweight Description Logics Symbols. Semantics: I = (∆I , ·I ). • Individual names:. Interpretation domain ∆I and function ·I :. ann, bob, w1, . . . • Concept names:. Component, User, Window, . . . • Role names:. LooksAt, FocusesOn, HasPart, . . .. ApplicationI a1I WindowI I. w1. ComponentI. bobI. annI UserI. LooksAtI HasPartI 5 / 17.

(29) Lightweight Description Logics Symbols. Semantics: I = (∆I , ·I ). • Individual names:. Interpretation domain ∆I and function ·I :. ann, bob, w1, . . . • Concept names:. Component, User, Window, . . . • Role names:. LooksAt, FocusesOn, HasPart, . . .. ApplicationI a1I WindowI I. w1 Fact base F User(bob) LooksAt(bob, w1). ComponentI. bobI. annI UserI. LooksAtI HasPartI 5 / 17.

(30) Lightweight Description Logics Symbols. Semantics: I = (∆I , ·I ). • Individual names:. Interpretation domain ∆I and function ·I :. ann, bob, w1, . . . • Concept names:. Component, User, Window, . . . • Role names:. LooksAt, FocusesOn, HasPart, . . .. ApplicationI a1I WindowI I. w1 Fact base F User(bob) LooksAt(bob, w1). I |= F bobI ∈ UserI (bobI , w1I ) ∈ LooksAtI. ComponentI. bobI. annI UserI. LooksAtI HasPartI 5 / 17.

(31) Lightweight Description Logics Basic concepts. Semantics: I = (∆I , ·I ). • DL-Lite: User, ∃HasPart, ∃HasPart−. Interpretation domain ∆I and function ·I :. • EL: User, ∃HasPart.Window. ApplicationI a1I WindowI I. w1. ComponentI. bobI. annI UserI. LooksAtI HasPartI. 5 / 17.

(32) Lightweight Description Logics Basic concepts. Semantics: I = (∆I , ·I ). • DL-Lite: User, ∃HasPart, ∃HasPart−. Interpretation domain ∆I and function ·I :. • EL: User, ∃HasPart.Window. ApplicationI a1I WindowI I. w1. ComponentI. bobI. annI UserI. LooksAtI HasPartI. 5 / 17.

(33) Lightweight Description Logics Basic concepts • DL-Lite: User, ∃HasPart, ∃HasPart− • EL: User, ∃HasPart.Window. Ontology O Concept inclusions Window v Component VideoPlayer v ¬SystemCritical Role inclusions (·H ) LooksAt v FocusesOn. 5 / 17.

(34) Lightweight Description Logics Basic concepts • DL-Lite: User, ∃HasPart, ∃HasPart− • EL: User, ∃HasPart.Window. Ontology O Concept inclusions Window v Component VideoPlayer v ¬SystemCritical Role inclusions (·H ) LooksAt v FocusesOn. I |= O WindowI ⊆ ComponentI VideoPlayerI ∩ SystemCriticalI = ∅ LooksAtI ⊆ FocusesOnI. 5 / 17.

(35) Lightweight Description Logics Basic concepts • DL-Lite: User, ∃HasPart, ∃HasPart− • EL: User, ∃HasPart.Window. Ontology O Concept inclusions Window v Component VideoPlayer v ¬SystemCritical Role inclusions (·H ) LooksAt v FocusesOn. I |= O WindowI ⊆ ComponentI VideoPlayerI ∩ SystemCriticalI = ∅ LooksAtI ⊆ FocusesOnI. H H H DLs we focus on: DL-Lite H core , DL-Lite horn , DL-Lite krom , DL-Lite bool , EL. 5 / 17.

(36) Lightweight Description Logics Basic concepts • DL-Lite: User, ∃HasPart, ∃HasPart− • EL: User, ∃HasPart.Window. Ontology O Concept inclusions Window v Component VideoPlayer v ¬SystemCritical Role inclusions (·H ) LooksAt v FocusesOn. I |= O WindowI ⊆ ComponentI VideoPlayerI ∩ SystemCriticalI = ∅ LooksAtI ⊆ FocusesOnI. H H H DLs we focus on: DL-Lite H core , DL-Lite horn , DL-Lite krom , DL-Lite bool , EL. Temporal knowledge base (TKB). Semantics: I = (Ii )i≥0. K = hO, (Fi )0≤i≤n i. 5 / 17.

(37) Lightweight Description Logics Basic concepts • DL-Lite: User, ∃HasPart, ∃HasPart− • EL: User, ∃HasPart.Window. Ontology O Concept inclusions Window v Component VideoPlayer v ¬SystemCritical Role inclusions (·H ) LooksAt v FocusesOn. I |= O WindowI ⊆ ComponentI VideoPlayerI ∩ SystemCriticalI = ∅ LooksAtI ⊆ FocusesOnI. H H H DLs we focus on: DL-Lite H core , DL-Lite horn , DL-Lite krom , DL-Lite bool , EL. Temporal knowledge base (TKB) K = hO, (Fi )0≤i≤n i. Semantics: I = (Ii )i≥0. I |= K. Ii |= O for all i ≥ 0, Ii |= Fi for all i ∈ [0, n], and I respects individual and rigid names 5 / 17.

(38) Temporal Conjunctive Queries (TCQs). Components in user focus in the past, and not visible anymore (now)? ΦEx (x) := 3P ∃y .User(y )∧FocusesOn(y , x)∧Component(x) ∧NotVisible(x). 6 / 17.

(39) Temporal Conjunctive Queries (TCQs). Components in user focus in the past, and not visible anymore (now)? ΦEx (x) := 3P ∃y .User(y )∧FocusesOn(y , x)∧Component(x) ∧NotVisible(x) TCQ Φ, Ψ := CQ ϕ | ¬Φ | Φ ∧ Ψ | Φ ∨ Ψ |. 6 / 17.

(40) Temporal Conjunctive Queries (TCQs). Components in user focus in the past, and not visible anymore (now)? ΦEx (x) := 3P ∃y .User(y )∧FocusesOn(y , x)∧Component(x) ∧NotVisible(x) TCQ Φ, Ψ := CQ ϕ | ¬Φ | Φ ∧ Ψ | Φ ∨ Ψ | #F Φ (next) | #P Φ (previous) |. 6 / 17.

(41) Temporal Conjunctive Queries (TCQs). Components in user focus in the past, and not visible anymore (now)? ΦEx (x) := 3P ∃y .User(y )∧FocusesOn(y , x)∧Component(x) ∧NotVisible(x) TCQ Φ, Ψ := CQ ϕ | ¬Φ | Φ ∧ Ψ | Φ ∨ Ψ | #F Φ (next) | #P Φ (previous) | Φ U Ψ (until) | Φ S Ψ (since). 6 / 17.

(42) Temporal Conjunctive Queries (TCQs). Components in user focus in the past, and not visible anymore (now)? ΦEx (x) := 3P ∃y .User(y )∧FocusesOn(y , x)∧Component(x) ∧NotVisible(x) TCQ Φ, Ψ := CQ ϕ | ¬Φ | Φ ∧ Ψ | Φ ∨ Ψ | #F Φ (next) | #P Φ (previous) | Φ U Ψ (until) | Φ S Ψ (since). → 3P ϕ := true S ϕ (some time in the past). 6 / 17.

(43) Temporal Conjunctive Queries (TCQs). Components in user focus in the past, and not visible anymore (now)? ΦEx := 3P ∃y .User(y )∧FocusesOn(y , w1)∧Component(w1) ∧NotVisible(w1) TCQ Φ, Ψ := CQ ϕ | ¬Φ | Φ ∧ Ψ | Φ ∨ Ψ | #F Φ (next) | #P Φ (previous) | Φ U Ψ (until) | Φ S Ψ (since). → 3P ϕ := true S ϕ (some time in the past). Semantics: sequences I = (Ii )i≥0 of interpretations, Boolean queries. 6 / 17.

(44) Temporal Conjunctive Queries (TCQs). Components in user focus in the past, and not visible anymore (now)? ΦEx := 3P ∃y .User(y )∧FocusesOn(y , w1)∧Component(w1) ∧NotVisible(w1) TCQ Φ, Ψ := CQ ϕ | ¬Φ | Φ ∧ Ψ | Φ ∨ Ψ | #F Φ (next) | #P Φ (previous) | Φ U Ψ (until) | Φ S Ψ (since). → 3P ϕ := true S ϕ (some time in the past). Semantics: sequences I = (Ii )i≥0 of interpretations, Boolean queries. Example: I, 2 |= ΦEx if • I2 |= NotVisible(w1). 6 / 17.

(45) Temporal Conjunctive Queries (TCQs). Components in user focus in the past, and not visible anymore (now)? ΦEx := 3P ∃y .User(y )∧FocusesOn(y , w1)∧Component(w1) ∧NotVisible(w1) TCQ Φ, Ψ := CQ ϕ | ¬Φ | Φ ∧ Ψ | Φ ∨ Ψ | #F Φ (next) | #P Φ (previous) | Φ U Ψ (until) | Φ S Ψ (since). → 3P ϕ := true S ϕ (some time in the past). ϕ. true. 3P ϕ. Semantics: sequences I = (Ii )i≥0 of interpretations, Boolean queries. Example: I, 2 |= ΦEx if • I2 |= NotVisible(w1) • there is an i ∈ [0, 2] such that. Ii |= ∃y .User(y ) ∧ FocusesOn(y , w1) ∧ Component(w1) 6 / 17.

(46) I Solving Satisfiability • Given: Boolean TCQ Φ + TKB K = hO, (Fi )0≤i≤n i • Sequences I = (Ii )i≥0 of interpretations • Complexity of TCQ entailment: I, n |= Φ for all I such that I |= K? • Solve TCQ satisfiability: Is there an I such that I |= K and I, n |= ¬Φ?. 7 / 17.

(47) I Solving Satisfiability . . . good complexities for lightweight DLs and TCQs? • Given: Boolean TCQ Φ + TKB K = hO, (Fi )0≤i≤n i • Sequences I = (Ii )i≥0 of interpretations • Complexity of TCQ entailment: I, n |= Φ for all I such that I |= K? • Solve TCQ satisfiability: Is there an I such that I |= K and I, n |= ¬Φ?. 7 / 17.

(48) I Solving Satisfiability . . . good complexities for lightweight DLs and TCQs? • Given: Boolean TCQ Φ + TKB K = hO, (Fi )0≤i≤n i • Sequences I = (Ii )i≥0 of interpretations • Complexity of TCQ entailment: I, n |= Φ for all I such that I |= K? • Solve TCQ satisfiability: Is there an I such that I |= K and I, n |= ¬Φ?. (i) [ |H]. Combined Complexity (ii). (iii). (i). Data Complexity (ii) (iii). DL-Lite [core|horn]. ≥PSpace. ?. ?. ?. ?. EL. ≥PSpace. ?. ?. ≥P. ?. ?. DL-Lite [krom|bool]. ≥PSpace. ?. ≥co-NP. ?. ≤ExpTime. ALCHQ1. ExpTime. co-NExpTime. co-NP. co-NP. ≤ExpTime. [ |H]. ≤2-ExpTime 2-ExpTime. ?. (i) no rigid names (ii) rigid concept names (iii) rigid role names (and rigid concept names) 1. (Baader et al. 2015) 7 / 17.

(49) I Solving Satisfiability . . . good complexities for lightweight DLs and TCQs? • Given: Boolean TCQ Φ + TKB K = hO, (Fi )0≤i≤n i • Sequences I = (Ii )i≥0 of interpretations • Complexity of TCQ entailment: I, n |= Φ for all I such that I |= K? • Solve TCQ satisfiability: Is there an I such that I |= K and I, n |= ¬Φ?. (i) [ |H]. Combined Complexity (ii). (iii). (i). Data Complexity (ii) (iii). DL-Lite [core|horn]. ≥PSpace. ?. ?. ?. ?. EL. ≥PSpace. ?. ?. ≥P. ?. ?. DL-Lite [krom|bool]. ≥PSpace. ?. ≥co-NP. ?. ≤ExpTime. ALCHQ1. ExpTime. co-NExpTime. co-NP. co-NP. ≤ExpTime. [ |H]. PSpace? 2-ExpTime. ?. (i) no rigid names (ii) rigid concept names (iii) rigid role names (and rigid concept names) 1. (Baader et al. 2015) 7 / 17.

(50) I Solving Satisfiability . . . good complexities for lightweight DLs and TCQs? • Given: Boolean TCQ Φ + TKB K = hO, (Fi )0≤i≤n i • Sequences I = (Ii )i≥0 of interpretations • Complexity of TCQ entailment: I, n |= Φ for all I such that I |= K? • Solve TCQ satisfiability: Is there an I such that I |= K and I, n |= ¬Φ?. (i) [ |H]. Combined Complexity (ii). (iii). (i). Data Complexity (ii) (iii). DL-Lite [core|horn]. ≥PSpace. ?. ?. ?. ?. EL. ≥PSpace. ?. ?. ≥P. ?. ?. DL-Lite [krom|bool]. ≥PSpace. ?. ≥co-NP. ?. ≤ExpTime. ALCHQ1. ExpTime. co-NExpTime. co-NP. co-NP. ≤ExpTime. [ |H]. PSpace? 2-ExpTime. FO rewritable?. (i) no rigid names (ii) rigid concept names (iii) rigid role names (and rigid concept names) 1. (Baader et al. 2015) 7 / 17.

(51) I Solving Satisfiability . . . good complexities for lightweight DLs and TCQs? • Given: Boolean TCQ Φ + TKB K = hO, (Fi )0≤i≤n i • Sequences I = (Ii )i≥0 of interpretations • Complexity of TCQ entailment: I, n |= Φ for all I such that I |= K? • Solve TCQ satisfiability: Is there an I such that I |= K and I, n |= ¬Φ?. (i) [ |H]. Combined Complexity (ii). (iii). (i). Data Complexity (ii) (iii). DL-Lite [core|horn]. ≥PSpace. ?. ?. ?. ?. EL. ≥PSpace. ?. ?. ≥P. ?. DL-Lite [krom|bool]. ≥PSpace. ?. ≥co-NP. ?. ≤ExpTime. ALCHQ1. ExpTime. co-NExpTime. co-NP. co-NP. ≤ExpTime. [ |H]. PSpace? 2-ExpTime. FO rewritable? Tractable?. (i) no rigid names (ii) rigid concept names (iii) rigid role names (and rigid concept names) 1. (Baader et al. 2015) 7 / 17.

(52) I Solving Satisfiability . . . good complexities for lightweight DLs and TCQs? • Given: Boolean TCQ Φ + TKB K = hO, (Fi )0≤i≤n i • Sequences I = (Ii )i≥0 of interpretations • Complexity of TCQ entailment: I, n |= Φ for all I such that I |= K? • Solve TCQ satisfiability: Is there an I such that I |= K and I, n |= ¬Φ?. (i) [ |H]. Combined Complexity (ii). (iii). (i). Data Complexity (ii) (iii). DL-Lite [core|horn]. ≥PSpace. ?. ?. ?. ?. FO rewritable?. EL. ≥PSpace. ?. ?. ≥P. ?. Tractable?. DL-Lite [krom|bool]. ≥PSpace. ?. ≥co-NP. ?. ALCHQ1. ExpTime. co-NExpTime. co-NP. co-NP. [ |H]. PSpace? 2-ExpTime. co-NP? ≤ExpTime. (i) no rigid names (ii) rigid concept names (iii) rigid role names (and rigid concept names) 1. (Baader et al. 2015) 7 / 17.

(53) I Solving Satisfiability: A General Algorithm (Baader et al. 2012, 2015) Satisfiability of ¬Φ w.r.t. hO, (Fi )0≤i≤n i → (Ii )i≥0 ? ΦEx = (3P ϕ1 ) ∧ ϕ2 ϕ1 := ∃y.User(y) ∧ FocusesOn(y, w1) ∧ Component(w1) ϕ2 := NotVisible(w1). ¬ΦEx = (¬3P ϕ1 ) ∨ ¬ϕ2. 8 / 17.

(54) I Solving Satisfiability: A General Algorithm (Baader et al. 2012, 2015) Satisfiability of ¬Φ w.r.t. hO, (Fi )0≤i≤n i → (Ii )i≥0 ? ΦEx = (3P ϕ1 ) ∧ ϕ2 ϕ1 := ∃y.User(y) ∧ FocusesOn(y, w1) ∧ Component(w1) ϕ2 := NotVisible(w1). ¬ΦEx = (¬3P ϕ1 ) ∨ ¬ϕ2 1. Replace CQs ϕ1 , ϕ2 by propositional variables p1 , p2. 8 / 17.

(55) I Solving Satisfiability: A General Algorithm (Baader et al. 2012, 2015) Satisfiability of ¬Φ w.r.t. hO, (Fi )0≤i≤n i → (Ii )i≥0 ? ΦEx = (3P ϕ1 ) ∧ ϕ2 ϕ1 := ∃y.User(y) ∧ FocusesOn(y, w1) ∧ Component(w1) ϕ2 := NotVisible(w1). ¬ΦEx = (¬3P ϕ1 ) ∨ ¬ϕ2 1. Replace CQs ϕ1 , ϕ2 by propositional variables p1 , p2 (¬3P p1 ) ∨ ¬p2. 8 / 17.

(56) I Solving Satisfiability: A General Algorithm (Baader et al. 2012, 2015) Satisfiability of ¬Φ w.r.t. hO, (Fi )0≤i≤n i → (Ii )i≥0 ? ΦEx = (3P ϕ1 ) ∧ ϕ2 ϕ1 := ∃y.User(y) ∧ FocusesOn(y, w1) ∧ Component(w1) ϕ2 := NotVisible(w1). ¬ΦEx = (¬3P ϕ1 ) ∨ ¬ϕ2 1. Replace CQs ϕ1 , ϕ2 by propositional variables p1 , p2 (¬3P p1 ) ∨ ¬p2. 2. LTL satisfiability problem: Look for an LTL structure (wi )i≥0 that satisfies the formula at time point n wi : propositions true at i. 8 / 17.

(57) I Solving Satisfiability: A General Algorithm (Baader et al. 2012, 2015) Satisfiability of ¬Φ w.r.t. hO, (Fi )0≤i≤n i → (Ii )i≥0 ? ΦEx = (3P ϕ1 ) ∧ ϕ2 ϕ1 := ∃y.User(y) ∧ FocusesOn(y, w1) ∧ Component(w1) ϕ2 := NotVisible(w1). ¬ΦEx = (¬3P ϕ1 ) ∨ ¬ϕ2 1. Replace CQs ϕ1 , ϕ2 by propositional variables p1 , p2 (¬3P p1 ) ∨ ¬p2. 2. LTL satisfiability problem: Look for an LTL structure (wi )i≥0 that satisfies the formula at time point n wi : propositions true at i Possible LTL model: (wi )i≥0 = ∅, ∅, {p2 }, ∅ . . .. (for n = 2). 8 / 17.

(58) I Solving Satisfiability: A General Algorithm (Baader et al. 2012, 2015) Satisfiability of ¬Φ w.r.t. hO, (Fi )0≤i≤n i → (Ii )i≥0 ? ΦEx = (3P ϕ1 ) ∧ ϕ2 ϕ1 := ∃y.User(y) ∧ FocusesOn(y, w1) ∧ Component(w1) ϕ2 := NotVisible(w1). ¬ΦEx = (¬3P ϕ1 ) ∨ ¬ϕ2 1. Replace CQs ϕ1 , ϕ2 by propositional variables p1 , p2 (¬3P p1 ) ∨ ¬p2. 2. LTL satisfiability problem: Look for an LTL structure (wi )i≥0 that satisfies the formula at time point n wi : propositions true at i Possible LTL model: (wi )i≥0 = ∅, ∅, {p2 }, ∅ . . .. 3. (for n = 2). DL satisfiability problems (atemporal): Look for DL interpretations (Ii )i≥0 such that each Ii satisfies • hO, Fi i • the CQs according to wi : Ii |= ϕj iff pj ∈ wi 8 / 17.

(59) I Solving Satisfiability: A General Algorithm (Baader et al. 2012, 2015) • Ii |= hO, Fi i. ϕ1 := ∃y .User(y ) ∧ FocusesOn(y , w1) ∧ Component(w1) ϕ2 := NotVisible(w1) VideoPlayer v Application u EnergyIntensive u ¬SystemCritical Window v Component. NotVisible. LooksAt v FocusesOn. User(bob). F0. User F1. F2. VideoPlayer(a1) Window(w1) HasPart(a1, w1). LooksAt(bob, w1). NotVisible(w1). 9 / 17.

(60) I Solving Satisfiability: A General Algorithm (Baader et al. 2012, 2015) • Ii |= hO, Fi i. ϕ1 := ∃y .User(y ) ∧ FocusesOn(y , w1) ∧ Component(w1) ϕ2 := NotVisible(w1) VideoPlayer v Application u EnergyIntensive u ¬SystemCritical Window v Component. NotVisible. LooksAt v FocusesOn. User(bob). I0. User I1. I2. VideoPlayer(a1) Window(w1) HasPart(a1, w1). LooksAt(bob, w1). Component(w1). FocusesOn(bob, w1). NotVisible(w1). 9 / 17.

(61) I Solving Satisfiability: A General Algorithm (Baader et al. 2012, 2015) • Ii |= hO, Fi i • (wi )i≥0 = ∅, ∅, {p2 }, . . . : I0 , I1 must not satisfy ϕ1 , ϕ2. ϕ1 := ∃y .User(y ) ∧ FocusesOn(y , w1) ∧ Component(w1) ϕ2 := NotVisible(w1) VideoPlayer v Application u EnergyIntensive u ¬SystemCritical Window v Component. NotVisible. LooksAt v FocusesOn. User(bob). I0. User I1. I2. VideoPlayer(a1) Window(w1) HasPart(a1, w1). LooksAt(bob, w1). Component(w1). FocusesOn(bob, w1). NotVisible(w1). 9 / 17.

(62) I Solving Satisfiability: A General Algorithm (Baader et al. 2012, 2015) • Ii |= hO, Fi i • (wi )i≥0 = ∅, ∅, {p2 }, . . . : I0 , I1 must not satisfy ϕ1 , ϕ2. ϕ1 := ∃y .User(y ) ∧ FocusesOn(y , w1) ∧ Component(w1) ϕ2 := NotVisible(w1) VideoPlayer v Application u EnergyIntensive u ¬SystemCritical Window v Component. NotVisible. LooksAt v FocusesOn. User(bob). I0. User(bob). User I1. User(bob). I2. VideoPlayer(a1) Window(w1). Component(w1). Component(w1). HasPart(a1, w1). LooksAt(bob, w1). NotVisible(w1). Component(w1). FocusesOn(bob, w1) 9 / 17.

(63) I Solving Satisfiability: A General Algorithm (Baader et al. 2012, 2015) • Ii |= hO, Fi i • (wi )i≥0 = ∅, ∅, {p2 }, . . . : I0 , I1 must not satisfy ϕ1 , ϕ2. ϕ1 := ∃y .User(y ) ∧ FocusesOn(y , w1) ∧ Component(w1) ϕ2 := NotVisible(w1) VideoPlayer v Application u EnergyIntensive u ¬SystemCritical Window v Component. NotVisible. LooksAt v FocusesOn. User(bob). I0. User(bob). User I1. User(bob). I2. VideoPlayer(a1) Window(w1). Component(w1). Component(w1). HasPart(a1, w1). LooksAt(bob, w1). NotVisible(w1). Component(w1). FocusesOn(bob, w1) 9 / 17.

(64) I Solving Satisfiability: First Results for TCQ Entailment. (i) [ |H]. Combined Complexity (ii). (iii). (i). Data Complexity (ii) (iii). DL-Lite [core|horn]. ≥PSpace. ?. ≤co-NExpTime. ?. ?. EL. ≥PSpace. ?. ≤co-NExpTime. ≥P. ?. ≤co-NP. DL-Lite [krom|bool]. ≥PSpace. ?. ≤2-ExpTime. ≥co-NP. ?. ≤ExpTime. ALCHQ1. ExpTime. co-NExpTime. 2-ExpTime. co-NP. co-NP. ≤ExpTime. [ |H]. ≤co-NP. (i) no rigid concept or role names (ii) rigid concept names (iii) rigid role names (and rigid concept names). 1. (Baader et al. 2015) 10 / 17.

(65) I Solving Satisfiability: PSpace Combined Complexity LTL satisfiability algorithm (Sistla and Clarke 1985): If LTL model exists, then there is a periodic one. 11 / 17.

(66) I Solving Satisfiability: PSpace Combined Complexity LTL satisfiability algorithm Model (wi )i≥0 for (¬3P p1 ) ∨ ¬p2 ? (Sistla and Clarke 1985): If LTL model exists, then there is a periodic one. 11 / 17.

(67) I Solving Satisfiability: PSpace Combined Complexity LTL satisfiability algorithm Model (wi )i≥0 for (¬3P p1 ) ∨ ¬p2 ? (Sistla and Clarke 1985): If LTL model exists, then there is a periodic one. • Guess start s and end e of the period • Memory: LTL formula sets Wi−1 , Wi , Ws representing wi−1 , wi , ws. 11 / 17.

(68) I Solving Satisfiability: PSpace Combined Complexity LTL satisfiability algorithm Model (wi )i≥0 for (¬3P p1 ) ∨ ¬p2 ? (Sistla and Clarke 1985): If LTL model exists, then there is a periodic one. • Guess start s and end e of the period • Memory: LTL formula sets Wi−1 , Wi , Ws representing wi−1 , wi , ws • Iterate over time t and always • •. • •. Wi−1 := Wi Wi := guess a set of subformulas Check if Wi may follow after Wi−1. p1 ∈ Wi−1 ⇒ 3P p1 ∈ Wi. At s: Ws := Wi At e: check if Ws may follow after Wi 11 / 17.

(69) I Solving Satisfiability: PSpace Combined Complexity LTL satisfiability algorithm Model (wi )i≥0 for (¬3P p1 ) ∨ ¬p2 ? → (It )t≥0 ? (Sistla and Clarke 1985): If LTL model exists, then there is a periodic one. • Guess start s and end e of the period • Memory: LTL formula sets Wi−1 , Wi , Ws representing wi−1 , wi , ws • Iterate over time t and always • •. • •. Wi−1 := Wi Wi := guess a set of subformulas Check if Wi may follow after Wi−1. p1 ∈ Wi−1 ⇒ 3P p1 ∈ Wi. At s: Ws := Wi At e: check if Ws may follow after Wi 11 / 17.

(70) I Solving Satisfiability: PSpace Combined Complexity LTL satisfiability algorithm Model (wi )i≥0 for (¬3P p1 ) ∨ ¬p2 ? → (It )t≥0 ? (Sistla and Clarke 1985): If LTL model exists, then there is a periodic one. • Guess start s and end e of the period • Memory: LTL formula sets Wi−1 , Wi , Ws representing wi−1 , wi , ws • Iterate over time t and always • • •. • •. Wi−1 := Wi Wi := guess a set of subformulas Check if Wi may follow after Wi−1 DL satisfiability testing on the fly: Look for It such that • It |= hO, Ft i • It |= ϕj iff pj ∈ wt (given by Wi ). p1 ∈ Wi−1 ⇒ 3P p1 ∈ Wi. At s: Ws := Wi At e: check if Ws may follow after Wi 11 / 17.

(71) I Solving Satisfiability: PSpace Combined Complexity with Rigid Names LTL satisfiability algorithm Model (wi )i≥0 for (¬3P p1 ) ∨ ¬p2 ? → (It )t≥0 ? (Sistla and Clarke 1985): If LTL model exists, then there is a periodic one. • Guess start s and end e of the period • Memory: LTL formula sets Wi−1 , Wi , Ws representing wi−1 , wi , ws • Iterate over time t and always • • •. • •. Wi−1 := Wi Wi := guess a set of subformulas Check if Wi may follow after Wi−1 DL satisfiability testing on the fly: Look for It such that • It |= hO, Ft i • It |= ϕj iff pj ∈ wt (given by Wi ). p1 ∈ Wi−1 ⇒ 3P p1 ∈ Wi. At s: Ws := Wi At e: check if Ws may follow after Wi 11 / 17.

(72) I Solving Satisfiability: PSpace Combined Complexity with Rigid Names LTL satisfiability algorithm Model (wi )i≥0 for (¬3P p1 ) ∨ ¬p2 ? → (It )t≥0 ? (Sistla and Clarke 1985): If LTL model exists, then there is a periodic one • Guess a polynomial amount of data D. User(bob),. . .. • Guess start s and end e of the period • Memory: LTL formula sets Wi−1 , Wi , Ws representing wi−1 , wi , ws • Iterate over time t and always • • •. • •. Wi−1 := Wi Wi := guess a set of subformulas Check if Wi may follow after Wi−1 DL satisfiability testing on the fly: Look for It such that • It |= hO, Ft i • It |= ϕj iff pj ∈ wt (given by Wi ) + additional tests w.r.t. D At s: Ws := Wi At e: check if Ws may follow after Wi. p1 ∈ Wi−1 ⇒ 3P p1 ∈ Wi. Ii |= User(bob) . . .?. 11 / 17.

(73) I Solving Satisfiability: Results for TCQ Entailment Combined Complexity (i). (ii). [ |H] DL-Lite [core|horn]. PSpace. PSpace. EL. PSpace. PSpace. (iii) PSpace ≥co-NExpTime. DL-Lite [krom|bool]. ≥PSpace. ?. ≤2-ExpTime. DL-Lite H [krom|bool]. ≥PSpace. ?. ≤2-ExpTime. ALCHQ1. ExpTime. co-NExpTime. 2-ExpTime. (i) no rigid names (ii) rigid concept names (iii) rigid role names (and rigid concept names). 1. (Baader et al. 2015) 12 / 17.

(74) I Solving Satisfiability: Results for TCQ Entailment Combined Complexity (i). (ii). [ |H] DL-Lite [core|horn]. PSpace. PSpace. EL. PSpace. PSpace. (iii) PSpace co-NExpTime. DL-Lite [krom|bool]. ≥PSpace. ?. ≤2-ExpTime. DL-Lite H [krom|bool]. ≥PSpace. ?. ≤2-ExpTime. ALCHQ1. ExpTime. co-NExpTime. 2-ExpTime. (i) no rigid names (ii) rigid concept names (iii) rigid role names (and rigid concept names). PSpace: rigid roles critical if DL powerful enough. 1. (Baader et al. 2015) 12 / 17.

(75) I Solving Satisfiability: Results for TCQ Entailment Combined Complexity (i). (ii). [ |H] DL-Lite [core|horn]. PSpace. PSpace. EL. PSpace. PSpace. (iii) PSpace co-NExpTime. DL-Lite [krom|bool]. ExpTime. ?. ≤2-ExpTime. DL-Lite H [krom|bool]. 2-ExpTime. 2-ExpTime. 2-ExpTime. ExpTime. co-NExpTime. 2-ExpTime. ALCHQ. 1. (i) no rigid names (ii) rigid concept names (iii) rigid role names (and rigid concept names). PSpace: rigid roles critical if DL powerful enough TCQ satisfiability in DL-Litebool reducible to DL-Lite krom User v Male t Female CIs > v Male t Male, Male v ¬Male, . . . TCQ ¬∃x.User(x) ∧ Male(x) ∧ Female(x) 1. (Baader et al. 2015) 12 / 17.

(76) I Solving Satisfiability: Results for TCQ Entailment Combined Complexity (i). (ii). [ |H] DL-Lite [core|horn]. PSpace. PSpace. EL. PSpace. PSpace. DL-Lite [krom|bool]. ExpTime. co-NExpTime. 2-ExpTime. DL-Lite H [krom|bool]. 2-ExpTime. 2-ExpTime. 2-ExpTime. ExpTime. co-NExpTime. 2-ExpTime. ALCHQ1. (iii) PSpace co-NExpTime. (i) no rigid names (ii) rigid concept names (iii) rigid role names (and rigid concept names). PSpace: rigid roles critical if DL powerful enough TCQ satisfiability in DL-Litebool reducible to DL-Lite krom User v Male t Female CIs > v Male t Male, Male v ¬Male, . . . TCQ ¬∃x.User(x) ∧ Male(x) ∧ Female(x) 1. (Baader et al. 2015) 12 / 17.

(77) I Solving Satisfiability: Reduction of SAT (¬x ∨ ¬y ∨ z) ∧ . . . satisfiable iff Φ is satisfiable w.r.t. hO, (Fi )0≤i≤n i. Select a literal in the TCQ, and ensure valid assignments: A(x) iff ¬A(x) 2P C(c) → L(c) ∨ #F L(c) ∨ #F #F L(c). Transfer choice of TCQ to literal individuals. A to express assignment ∃R.L v A Represent formula in the fact bases, three per clause C(c) R(x, c). F0. F1 R(y , c). F2 R(z, c). 13 / 17.



(80) I Solving Satisfiability: Reduction of SAT (¬x ∨ ¬y ∨ z) ∧ . . . satisfiable iff Φ is satisfiable w.r.t. hO, (Fi )0≤i≤n i. Select a literal in the TCQ, and ensure valid assignments: A(x) iff ¬A(x) 2P C(c) → L(c) ∨ #F L(c) ∨ #F #F L(c) ∧ 2P ¬∃u, v .S(u, v ) ∧ A(u) ∧ A(v ). Transfer choice of TCQ to literal individuals. A to express assignment ∃R.L v A Represent formula in the fact bases, three per clause C(c). F0. F1. F2. R(x, c). R(y , c). R(z, c). S(x, x). S(y , y ). S(z, z) 13 / 17.

(81) I Solving Satisfiability: Results for TCQ Entailment Data Complexity. [ |H]. DL-Lite [core|horn]. (i). (ii). (iii). ALogTime. ALogTime. ALogTime. ≥P. co-NP. co-NP. ≥co-NP. ?. ≤ExpTime. co-NP. co-NP. ≤ExpTime. EL [ |H] DL-Lite [krom|bool]. ALCHQ. 1. (i) no rigid names (ii) rigid concept names (iii) rigid role names (and rigid concept names). DL-Lite: no FO rewritability ALogTime: efficient parallel algorithms exist!. 1. (Baader et al. 2015) 14 / 17.

(82) I Solving Satisfiability: Results for TCQ Entailment Data Complexity. [ |H]. DL-Lite [core|horn]. (i). (ii). (iii). ALogTime. ALogTime. ALogTime. P. co-NP. co-NP. ≥co-NP. ?. ≤ExpTime. co-NP. co-NP. ≤ExpTime. EL [ |H] DL-Lite [krom|bool]. ALCHQ. 1. (i) no rigid names (ii) rigid concept names (iii) rigid role names (and rigid concept names). DL-Lite: no FO rewritability ALogTime: efficient parallel algorithms exist! EL: best result possible if no rigid symbols, but already rigid concepts critical. 1. (Baader et al. 2015) 14 / 17.

(83) I Solving Satisfiability: Results for TCQ Entailment Data Complexity. [ |H]. DL-Lite [core|horn]. (i). (ii). (iii). ALogTime. ALogTime. ALogTime. P. co-NP. co-NP. co-NP. co-NP. ≤ExpTime. co-NP. co-NP. ≤ExpTime. EL [ |H] DL-Lite [krom|bool]. ALCHQ. 1. (i) no rigid names (ii) rigid concept names (iii) rigid role names (and rigid concept names). DL-Lite: no FO rewritability ALogTime: efficient parallel algorithms exist! EL: best result possible if no rigid symbols, but already rigid concepts critical Upper bounds: apply general approach 1. (Baader et al. 2015) 14 / 17.

(84) II Rewritability of Temporal Query Answering • Positive Temporal QL queries: LTL without negation + QL queries • Temporal KB with ontology in some lightweight logic L • QL and L must satisfy certain requirements. → Rewritability of QL queries w.r.t. KBs in L. 15 / 17.

(85) II Rewritability of Temporal Query Answering • Positive Temporal QL queries: LTL without negation + QL queries • Temporal KB with ontology in some lightweight logic L • QL and L must satisfy certain requirements. → Rewritability of QL queries w.r.t. KBs in L. rewrite. + QL Query Φ. Ontology O in L. QL’ Query Φ0. Answers to Φ0 over F = Answers to Φ w.r.t. hO, F i. 15 / 17.

(86) II Rewritability of Temporal Query Answering • Positive Temporal QL queries: LTL without negation + QL queries • Temporal KB with ontology in some lightweight logic L • QL and L must satisfy certain requirements. → Rewritability of QL queries w.r.t. KBs in L • Generic rewritability result for PTQ answering. rewrite. + Temporal QL Query Φ. Ontology O in L. Temporal QL’ Query Φ0. Answers to Φ0 over (F )0≤i≤n = Answers to Φ w.r.t. hO, (F )0≤i≤n i. 15 / 17.

(87) II Rewritability of Temporal Query Answering • Positive Temporal QL queries: LTL without negation + QL queries • Temporal KB with ontology in some lightweight logic L • QL and L must satisfy certain requirements. → Rewritability of QL queries w.r.t. KBs in L • Generic rewritability result for PTQ answering • Many formalisms satisfy our requirements. → Tools for answering QL queries often exist. rewrite. + Temporal QL Query Φ. Ontology O in L. Temporal QL’ Query Φ0. Answers to Φ0 over (F )0≤i≤n = Answers to Φ w.r.t. hO, (F )0≤i≤n i. 15 / 17.

(88) II Rewritability of Temporal Query Answering L. QL. QL0. EL++. subs.. subs.. DL-Lite R. CQ. UCQ. dr ELH⊥. CQ. FO=. DL-Lite N horn. CQ. FO=. DL-Lite R. UCQ. PEQ. DL-Lite. CQ. UCQ. ¬. CQ. Datalog. ELHI. DL-Lite R. CQ. UCQ. DL-Lite +. CQ. UCQ+. Horn-ALCHIQ. CQ. UCQ. IQ. IQ. SROEL(u, ×). IQ. IQ. ±. CQ. UCQ. +. LDL. Datalog. family. 16 / 17.

(89) Summary & Outlook • Ontology-based data access: common domain terminology and knowledge • We need extensions for recognizing complex contexts • Temporal query answering w.r.t. ontologies in lightweight logics. 17 / 17.

(90) Summary & Outlook • Ontology-based data access: common domain terminology and knowledge • We need extensions for recognizing complex contexts • Temporal query answering w.r.t. ontologies in lightweight logics. 17 / 17.

(91) Summary & Outlook • Ontology-based data access: common domain terminology and knowledge • We need extensions for recognizing complex contexts • Temporal query answering w.r.t. ontologies in lightweight logics. Combined and data complexity of TCQ satisfiability. Rewritability of TQ answering. 17 / 17.

(92) Summary & Outlook • Ontology-based data access: common domain terminology and knowledge • We need extensions for recognizing complex contexts • Temporal query answering w.r.t. ontologies in lightweight logics. Combined and data complexity of TCQ satisfiability • Description logics DL-Lite and EL • Solutions inherently exponential • New algorithms: PSpace combined complexity in many cases • Feasible data complexity for DL-LiteH horn • Similar results for TQs where QL = DL axioms (not in this talk). Rewritability of TQ answering. 17 / 17.

(93) Summary & Outlook • Ontology-based data access: common domain terminology and knowledge • We need extensions for recognizing complex contexts • Temporal query answering w.r.t. ontologies in lightweight logics. Combined and data complexity of TCQ satisfiability • Description logics DL-Lite and EL • Solutions inherently exponential • New algorithms: PSpace combined complexity in many cases • Feasible data complexity for DL-LiteH horn • Similar results for TQs where QL = DL axioms (not in this talk). Rewritability of TQ answering • Generic rewritability result for positive TQs • Conditions are satisfied by many existing formalisms • Hints at implementations. 17 / 17.

(94) Summary & Outlook • Ontology-based data access: common domain terminology and knowledge • We need extensions for recognizing complex contexts • Temporal query answering w.r.t. ontologies in lightweight logics. Metric temporal logic operators? Other DLs? Combined and data complexity of TCQ satisfiability • Description logics DL-Lite and EL • Solutions inherently exponential • New algorithms: PSpace combined complexity in many cases • Feasible data complexity for DL-LiteH horn. The co-NP/ExpTime gap?. • Similar results for TQs where QL = DL axioms (not in this talk). Rewritability of TQ answering. Implementations? Use cases?. • Generic rewritability result for positive TQs. Other restrictions?. • Conditions are satisfied by many existing formalisms • Hints at implementations. 17 / 17.

(95) • S. Borgwardt, M. Lippmann, and T.: Temporalizing Rewritable Query Languages over Knowledge Bases. Journal of Web Semantics, 2015. • S. Borgwardt and T.: Temporal Query Answering in DL-Lite with Negation. In Proc. of GCAI, EasyChair, 2015. Temporal Query Answering in the Description Logic EL. In Proc. of IJCAI, AAAI Press, 2015. • S. Borgwardt, M. Lippmann, and T.: Temporal Query Answering in the Description Logic DL-Lite. In Proc. of FroCoS, LNCS, 2013. • T. and E. Zenker: Temporal Query Answering in a Fuzzy World. In Proc. of the Posters and Demos Track of SEMANTiCS, CEUR WS, 2015. • S. Borgwardt and T.: Temporal Query Answering in the Description Logic EL (ext. abstract). In Proc. of DL, CEUR WS, 2015. • T., J. Holste, and Ö. Özçep: On Implementing Temporal Query Answering in DL-Lite (ext. abstract). In Proc. of DL, CEUR WS, 2015. • S. Borgwardt, M. Lippmann, and T.: Temporal Query Answering in DL-Lite (best student paper). In Proc. of DL, CEUR WS, 2013..

(96) • F. Baader, S. Borgwardt, P. Koopmann, A. Ozaki, and T.: Metric Temporal Description Logics with Interval-Rigid Names. In Proc. of FroCoS, LNCS, 2017. • F. Baader, S. Borgwardt, P. Koopmann, A. Ozaki, and T.: Metric Temporal Description Logics with Interval-Rigid Names (ext. abstract). In Proc. of DL, CEUR WS, 2017..

(97) Thank you! Stefan Borgwardt Franz Baader Marcel Lippmann Markus Krötzsch Anni-Yasmin Turhan Ana Ozaki Kerstin Achtruth Carsten Lutz My family and friends. .. HAEC.

(98) (Baader et al. 2015) F. Baader, S. Borgwardt, and M. Lippmann: Temporal Query Entailments in the Description Logic SHQ. Journal of Web Semantics, 2015. (Baader et al. 2012) F. Baader, S. Ghilardi, and C. Lutz: LTL over Description Logic Axioms. ACM Transactions on Computational Logic, 2012. (Sistla and Clarke 1985) A. P. Sistla and E. M. Clarke: The Complexity of Propositional Linear Temporal Logics, Journal of the ACM, 1985..

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