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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krtzsch Knowledge-Based Systems Group at The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krtzsch

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

The Combined Approach to Query Answering in Horn-ALCHOIQ

David Carral, Irina Dragoste, Markus Krötzsch Knowledge-Based Systems Group at

/15 1

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 2

TBox

The Combined Approach

ABox Conjunctive query

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 2

TBox

The Combined Approach

  • 1. Materialisation Step

ABox Conjunctive query

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 2

TBox

The Combined Approach

Datalog Rule Set

  • 1. Materialisation Step

ABox Conjunctive query

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 2

TBox

The Combined Approach

Datalog Rule Engine Datalog Rule Set

  • 1. Materialisation Step

ABox Conjunctive query

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 2

TBox

The Combined Approach

Datalog Rule Engine Datalog Rule Set

  • 1. Materialisation Step

ABox Conjunctive query

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 2

TBox

The Combined Approach

Datalog Rule Engine Datalog Rule Set

  • 1. Materialisation Step

* Complete for CQA * May not be a model * Sound (and complete) for instance queries

ABox Conjunctive query

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 2

TBox

The Combined Approach

Datalog Rule Engine Datalog Rule Set

  • 2. Filtration Step
  • 1. Materialisation Step

* Complete for CQA * May not be a model * Sound (and complete) for instance queries

ABox Conjunctive query

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

  • Ans. 1
  • Ans. 2
  • Ans. 3
  • Ans. 4

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TBox

The Combined Approach

Datalog Rule Engine Datalog Rule Set

  • 2. Filtration Step
  • 1. Materialisation Step

* Complete for CQA * May not be a model * Sound (and complete) for instance queries

ABox Conjunctive query

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

  • Ans. 1
  • Ans. 2
  • Ans. 3
  • Ans. 4

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TBox

The Combined Approach

Datalog Rule Engine Datalog Rule Set

  • 2. Filtration Step
  • 1. Materialisation Step

* Complete for CQA * May not be a model * Sound (and complete) for instance queries

ABox Conjunctive query

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 3

C1 ⊓ … ⊓ Cn ⊑ D R ⊑ S ∃R . C ⊑ D R1 ⊓ … ⊓ Rn ⊑ ⊥ C ⊑ ∃R . D R(a, b) C ⊑ ≤ 1R . D A(a) C ⊑ {a}

The Combined Approach for Horn-ALCHOIQ

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 4

The Materialisation Step

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 5

C1 ⊓ … ⊓ Cn ⊑ D ↦ C1(x) ∧ … ∧ Cn(x) → D(x) ∃R . C ⊑ D ↦ R(x, y) ∧ C(y) → D(x) C ⊑ ∃R . D ↦ C(x) → R(x, tD) ∧ D(tD) R ⊑ S ↦ R(x, y) → S(x, y) C(a) ↦ C(a) R(a, b) ↦ R(a, b)

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 5

C1 ⊓ … ⊓ Cn ⊑ D ↦ C1(x) ∧ … ∧ Cn(x) → D(x) ∃R . C ⊑ D ↦ R(x, y) ∧ C(y) → D(x) C ⊑ ∃R . D ↦ C(x) → R(x, tD) ∧ D(tD) R ⊑ S ↦ R(x, y) → S(x, y) C(a) ↦ C(a) R(a, b) ↦ R(a, b)

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 6

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 6

a : C

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 6

a : C R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

d1 : D

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 6

a : C R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S d1 : D d2 : C

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 6

a : C d1 : D, E R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S d1 : D d2 : C

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 6

a : C d1 : D, E R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S a : C, F d1 : D d2 : C

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 6

a : C d1 : D, E R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R a : C, F d1 : D d2 : C d3 : D

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 6

a : C d1 : D, E R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S a : C, F d1 : D d2 : C d3 : D d4 : C

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

d3 : D, E

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a : C d1 : D, E R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S a : C, F d1 : D d2 : C d3 : D d4 : C

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

d3 : D, E d2 : C, F

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a : C d1 : D, E R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S a : C, F d1 : D d2 : C d3 : D d4 : C

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

d3 : D, E d2 : C, F

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a : C d1 : D, E R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S R a : C, F d1 : D d2 : C d3 : D d4 : C d5 : D

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

d6 : C d3 : D, E d2 : C, F

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a : C d1 : D, E R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S R S a : C, F d1 : D d2 : C d3 : D d4 : C d5 : D

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

d6 : C d5 : D, E d3 : D, E d2 : C, F

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a : C d1 : D, E R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S R S a : C, F d1 : D d2 : C d3 : D d4 : C d5 : D

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

d6 : C d5 : D, E d4 : C, F d3 : D, E d2 : C, F

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a : C d1 : D, E R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S R S a : C, F d1 : D d2 : C d3 : D d4 : C d5 : D

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

d7 : D d6 : C d5 : D, E d4 : C, F d3 : D, E d2 : C, F

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a : C d1 : D, E R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S R S R a : C, F d1 : D d2 : C d3 : D d4 : C d5 : D

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

d7 : D d6 : C d5 : D, E d4 : C, F d3 : D, E d2 : C, F

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a : C d1 : D, E R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S R S R a : C, F d1 : D d2 : C d3 : D d4 : C d5 : D S d8 : C

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

d7 : D d6 : C d5 : D, E d4 : C, F d3 : D, E d2 : C, F

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a : C d1 : D, E R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S R S R a : C, F d1 : D d2 : C d3 : D d4 : C d5 : D S d8 : C

C(x) → R(x, tD) ∧ D(tD) D(x) → S(x, tC) ∧ C(tC) S(x, y) ∧ C(y) → E(x) R(x, y) ∧ E(y) → F(x)

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

d7 : D d6 : C d5 : D, E d4 : C, F d3 : D, E d2 : C, F

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a : C d1 : D, E R a : C

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S R S R a : C, F d1 : D d2 : C d3 : D d4 : C d5 : D S d8 : C

C(x) → R(x, tD) ∧ D(tD) D(x) → S(x, tC) ∧ C(tC) S(x, y) ∧ C(y) → E(x) R(x, y) ∧ E(y) → F(x)

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

d7 : D d6 : C d5 : D, E d4 : C, F d3 : D, E d2 : C, F

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a : C d1 : D, E R a : C R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S R S R a : C, F d1 : D d2 : C d3 : D d4 : C d5 : D S d8 : C

C(x) → R(x, tD) ∧ D(tD) D(x) → S(x, tC) ∧ C(tC) S(x, y) ∧ C(y) → E(x) R(x, y) ∧ E(y) → F(x)

tD : D

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

d7 : D d6 : C d5 : D, E d4 : C, F d3 : D, E d2 : C, F

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a : C d1 : D, E R S tC : C a : C R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S R S R a : C, F d1 : D d2 : C d3 : D d4 : C d5 : D S d8 : C

C(x) → R(x, tD) ∧ D(tD) D(x) → S(x, tC) ∧ C(tC) S(x, y) ∧ C(y) → E(x) R(x, y) ∧ E(y) → F(x)

tD : D

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

d7 : D d6 : C d5 : D, E d4 : C, F d3 : D, E d2 : C, F

/15 6

a : C d1 : D, E R R S tC : C a : C R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S R S R a : C, F d1 : D d2 : C d3 : D d4 : C d5 : D S d8 : C

C(x) → R(x, tD) ∧ D(tD) D(x) → S(x, tC) ∧ C(tC) S(x, y) ∧ C(y) → E(x) R(x, y) ∧ E(y) → F(x)

tD : D

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

tD : D, E d7 : D d6 : C d5 : D, E d4 : C, F d3 : D, E d2 : C, F

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a : C d1 : D, E R R S tC : C a : C R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S R S R a : C, F d1 : D d2 : C d3 : D d4 : C d5 : D S d8 : C

C(x) → R(x, tD) ∧ D(tD) D(x) → S(x, tC) ∧ C(tC) S(x, y) ∧ C(y) → E(x) R(x, y) ∧ E(y) → F(x)

tD : D

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

tD : D, E d7 : D d6 : C d5 : D, E d4 : C, F d3 : D, E d2 : C, F

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a : C d1 : D, E R R S tC : C a : C R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S R S R a : C, F d1 : D d2 : C d3 : D d4 : C d5 : D S d8 : C

C(x) → R(x, tD) ∧ D(tD) D(x) → S(x, tC) ∧ C(tC) S(x, y) ∧ C(y) → E(x) R(x, y) ∧ E(y) → F(x)

tC : C, F tD : D

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

tD : D, E d7 : D d6 : C d5 : D, E d4 : C, F d3 : D, E d2 : C, F

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a : C d1 : D, E R R S tC : C a : C R

C ⊑ ∃R . D D ⊑ ∃S . C ∃S . C ⊑ E ∃R . E ⊑ F C(a)

S R S R S R a : C, F d1 : D d2 : C d3 : D d4 : C d5 : D S d8 : C

C(x) → R(x, tD) ∧ D(tD) D(x) → S(x, tC) ∧ C(tC) S(x, y) ∧ C(y) → E(x) R(x, y) ∧ E(y) → F(x)

a : C, F tC : C, F tD : D

Materialisation Step: EL

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 7

Materialisation Step: (Horn-)ELI

C1 ⊓ … ⊓ Cn ⊑ D ↦ C1(x) ∧ … ∧ Cn(x) → D(x) ∃R . C ⊑ D ↦ R(x, y) ∧ C(y) → D(x), C(x) ∧ R−(x, t𝔽) → R−(x, t𝔽⊓D) ∧ ⋀E∈𝔽⊓DE(t𝔽⊓D) for every conjunction 𝔽 of concept names C ⊑ ∃R . D ↦ C(x) → R(x, tD) ∧ D(tD) R ⊑ S ↦ R(x, y) → S(x, y), R−(x, y) → S−(x, y)

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 7

Materialisation Step: (Horn-)ELI

C1 ⊓ … ⊓ Cn ⊑ D ↦ C1(x) ∧ … ∧ Cn(x) → D(x) ∃R . C ⊑ D ↦ R(x, y) ∧ C(y) → D(x), C(x) ∧ R−(x, t𝔽) → R−(x, t𝔽⊓D) ∧ ⋀E∈𝔽⊓DE(t𝔽⊓D) for every conjunction 𝔽 of concept names C ⊑ ∃R . D ↦ C(x) → R(x, tD) ∧ D(tD) R ⊑ S ↦ R(x, y) → S(x, y), R−(x, y) → S−(x, y)

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 8

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a)

Materialisation Step: Horn-ELI

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a)

Materialisation Step: Horn-ELI

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R a : C,E b : C

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R a : C,E b : C D : tD R

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R a : C,E b : C D : tD R

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

R

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

S

/15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R a : C,E b : C D : tD R tC : C

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

R

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

S

/15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R a : C,E b : C D : tD R tC : C R

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

R

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

S

/15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R a : C,E b : C D, F : tD⊓F R D : tD R tC : C R

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

R

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

a : C,E,G S

/15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R a : C,E b : C D, F : tD⊓F R D : tD R tC : C R

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

R

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

a : C,E,G S S

/15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R a : C,E b : C D, F : tD⊓F R D : tD R tC : C R

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

R

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

a : C,E,G S S S

/15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R a : C,E b : C D, F : tD⊓F R D : tD R tC : C R

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

R tC⊓E : C,E

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

a : C,E,G R S S S

/15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R a : C,E b : C D, F : tD⊓F R D : tD R tC : C R

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

R tC⊓E : C,E

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

a : C,E,G R S S R S

/15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R a : C,E b : C D, F : tD⊓F R D : tD R tC : C R

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

R tC⊓E : C,E

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

a : C,E,G tC⊓E : C,E,G R S S R S

/15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R a : C,E b : C D, F : tD⊓F R D : tD R tC : C R

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

R tC⊓E : C,E

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

a : C,E,G tC⊓E : C,E,G R S S R S

/15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R a : C,E b : C D, F : tD⊓F R D : tD R tC : C R

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

R tC⊓E : C,E

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

a : C,E,G tC⊓E : C,E,G R S S R S

/15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R a : C,E b : C D, F : tD⊓F R D : tD R tC : C R

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

R tC⊓E : C,E

q1 = ∃y, w . C(a) ∧ R(a, y) ∧ S(y, w) ∧ G(w)

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

a : C,E,G tC⊓E : C,E,G R S S R S

/15 8

a : C,E,G d1 : D,F d2 : C,E,G d3 : D,F R S R b : C e1 : D e2 : C e3 : D R S R a : C,E b : C D, F : tD⊓F R D : tD R tC : C R

∃R− . E ⊑ F ∃S− . F ⊑ E C ⊑ ∃R . D ∃R . F ⊑ G D ⊑ ∃S . C C(a) C(b) E(a) C(x) → R(x, tD) ∧ D(tD)

Materialisation Step: Horn-ELI

D(x) → S(x, tC) ∧ C(tC) E(x) ∧ R(x, tD) → R(x, tD⊓F) ∧ D(tD⊓F) ∧ F(tD⊓F) F(x) ∧ S(x, tC) → S(x, tC⊓E) ∧ C(tC⊓E) ∧ E(tC⊓E) R(x, y) ∧ F(y) → G(x)

R tC⊓E : C,E

q2 = ∃y, w . C(a) ∧ R(a, y) ∧ S(y, w) ∧ R(w, y) q1 = ∃y, w . C(a) ∧ R(a, y) ∧ S(y, w) ∧ G(w)

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 9

ℛEq = {x ≈ y → y ≈ x, x ≈ y ∧ y ≈ z → x ≈ z} ∪ {C(x) ∧ x ≈ y → C(y) ∣ C ∈ N+

C} ∪

{ℝ(x, y) ∧ x ≈ z → ℝ(z, y), ℝ(x, y) ∧ y ≈ z → ℝ(x, z) ∣ ℝ ∈ N⊓

R}

Materialisation Step: Horn-ALCHOIQ

ℛTop = {C(x) → ⊤ (x) ∣ C ∈ NC} ∪ {ℝ(x, y) → ⊤ (x) ∧ ⊤ (y) ∣ ℝ ∈ N⊓

R}

ℛRole = {ℝ(x, y) ∧ N(y) → ℝ−(x, y) ∣ ℝ ∈ N⊓

R} ∪

{ℝ(x, y) → R(x, y) ∣ ℝ ∈ N⊓

R, R ∈ NR}

ℛNm = {N(a), ⊤ (a) ∣ a ∈ NI}

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SLIDE 60

The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 10

Materialisation Step: Horn-ALCHOIQ

C1 ⊓ … ⊓ Cn ⊑ D ↦ C1(x) ∧ … ∧ Cn(x) → D(x) C ⊑ {a} ↦ C(x) → x ≈ a ∃R . C ⊑ D ↦ R(x, y) ∧ C(y) → D(x), C(x) ∧ ℝ−(x, t𝔽) → ℝ−(x, t𝔽⊓D) ∧ ⋀E∈𝔽⊓DE(t𝔽⊓D) for every ℝ ∈ N⊓

R with R ∈ ℝ and 𝔽 ∈ N⊓ C

C ⊑ ∃R . D ↦ C(x) → R(x, tD) ∧ D(tD) R ⊑ S ↦ ℝ(x, y) → (ℝ ⊓ S)(x, y), ℝ−(x, y) → (ℝ− ⊓ S−)(x, y) for every ℝ ∈ N⊓

R with R ∈ ℝ

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 11

Materialisation Step: Horn-ALCHOIQ

For all ℝ, 𝕋 ∈ N⊓

R with R ∈ ℝ and R ∈ 𝕋, and all 𝔽, 𝔾 ∈ N⊓ C

D(y) ∧ R−(y, x) ∧ C(x) ∧ R(x, z) ∧ D(z) ∧ N(z) → y ≈ z C(x) ∧ ℝ(x, t𝔽) ∧ D(t𝔽) ∧ 𝕋(x, t𝔾) ∧ F(t𝔾) → (ℝ ⊓ 𝕋)(x, t𝔽⊓𝔾) ∧ ⋀

X∈𝔽⊓𝔾

X(t𝔽⊓𝔾) D(y) ∧ ℝ−(y, x) ∧ C(x) ∧ 𝕋(x, t𝔽) ∧ D(t𝔽) → ⋀

X∈𝔽

X(y) ∧ (ℝ− ⊓ 𝕋−)(y, x) D(y) ∧ R−(y, x) ∧ C(x) ∧ N(x) → N(y) C ⊑ ≤ 1R . D ↦

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 11

Materialisation Step: Horn-ALCHOIQ

For all ℝ, 𝕋 ∈ N⊓

R with R ∈ ℝ and R ∈ 𝕋, and all 𝔽, 𝔾 ∈ N⊓ C

D(y) ∧ R−(y, x) ∧ C(x) ∧ R(x, z) ∧ D(z) ∧ N(z) → y ≈ z C(x) ∧ ℝ(x, t𝔽) ∧ D(t𝔽) ∧ 𝕋(x, t𝔾) ∧ F(t𝔾) → (ℝ ⊓ 𝕋)(x, t𝔽⊓𝔾) ∧ ⋀

X∈𝔽⊓𝔾

X(t𝔽⊓𝔾) D(y) ∧ ℝ−(y, x) ∧ C(x) ∧ 𝕋(x, t𝔽) ∧ D(t𝔽) → ⋀

X∈𝔽

X(y) ∧ (ℝ− ⊓ 𝕋−)(y, x) D(y) ∧ R−(y, x) ∧ C(x) ∧ N(x) → N(y) C ⊑ ≤ 1R . D ↦

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 12

Materialisation Step: Horn-ALCHOIQ

For all ℝ, 𝕋 ∈ N⊓

R with R ∈ ℝ and R ∈ 𝕋, and all 𝔽, 𝔾 ∈ N⊓ C

C(x) ∧ ℝ(x, t𝔽) ∧ D(t𝔽) ∧ 𝕋(x, t𝔾) ∧ F(t𝔾) → (ℝ ⊓ 𝕋)(x, t𝔽⊓𝔾) ∧ ⋀

X∈𝔽⊓𝔾

X(t𝔽⊓𝔾) C ⊑ ≤ 1R . D ↦

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 12

Materialisation Step: Horn-ALCHOIQ

a : C tB⊓D : B, D R R⊓S tA : A, D

For all ℝ, 𝕋 ∈ N⊓

R with R ∈ ℝ and R ∈ 𝕋, and all 𝔽, 𝔾 ∈ N⊓ C

C(x) ∧ ℝ(x, t𝔽) ∧ D(t𝔽) ∧ 𝕋(x, t𝔾) ∧ F(t𝔾) → (ℝ ⊓ 𝕋)(x, t𝔽⊓𝔾) ∧ ⋀

X∈𝔽⊓𝔾

X(t𝔽⊓𝔾) C ⊑ ≤ 1R . D ↦

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 12

Materialisation Step: Horn-ALCHOIQ

a : C tB⊓D : B, D R R⊓S tA : A, D

For all ℝ, 𝕋 ∈ N⊓

R with R ∈ ℝ and R ∈ 𝕋, and all 𝔽, 𝔾 ∈ N⊓ C

C(x) ∧ ℝ(x, t𝔽) ∧ D(t𝔽) ∧ 𝕋(x, t𝔾) ∧ F(t𝔾) → (ℝ ⊓ 𝕋)(x, t𝔽⊓𝔾) ∧ ⋀

X∈𝔽⊓𝔾

X(t𝔽⊓𝔾) C ⊑ ≤ 1R . D ↦ C(x) ∧ (R ⊓ S)(x, tB⊓D) ∧ D(tB⊓D) ∧ R(x, tA) ∧ D(tA) → (R ⊓ S)(x, tA⊓B⊓D) ∧ A(tA⊓B⊓D) ∧ B(tA⊓B⊓D) ∧ D(tA⊓B⊓D)

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 12

Materialisation Step: Horn-ALCHOIQ

a : C tB⊓D : B, D R R⊓S tA : A, D tA⊓B⊓D : A, B, D R⊓S

For all ℝ, 𝕋 ∈ N⊓

R with R ∈ ℝ and R ∈ 𝕋, and all 𝔽, 𝔾 ∈ N⊓ C

C(x) ∧ ℝ(x, t𝔽) ∧ D(t𝔽) ∧ 𝕋(x, t𝔾) ∧ F(t𝔾) → (ℝ ⊓ 𝕋)(x, t𝔽⊓𝔾) ∧ ⋀

X∈𝔽⊓𝔾

X(t𝔽⊓𝔾) C ⊑ ≤ 1R . D ↦ C(x) ∧ (R ⊓ S)(x, tB⊓D) ∧ D(tB⊓D) ∧ R(x, tA) ∧ D(tA) → (R ⊓ S)(x, tA⊓B⊓D) ∧ A(tA⊓B⊓D) ∧ B(tA⊓B⊓D) ∧ D(tA⊓B⊓D)

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SLIDE 67

The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 13

Evaluation

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SLIDE 68

The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 14

Materialisation Step: Implementation and Evaluation

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SLIDE 69

The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 14

Materialisation Step: Implementation and Evaluation

Konclude Materialisation Step

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SLIDE 70

The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 14

Materialisation Step: Implementation and Evaluation

Konclude Materialisation Step

* Use RDFox as a Datalog engine and add rules on demand

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SLIDE 71

The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 14

Materialisation Step: Implementation and Evaluation

Konclude Materialisation Step

* Use RDFox as a Datalog engine and add rules on demand * Rule count per ontology: 108+6 (LUBM), 254+19 (UOBM), 481+14 (Reactome),317+59 (Uniprot)

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SLIDE 72

The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch /15 14

Materialisation Step: Implementation and Evaluation

Konclude Materialisation Step

* Use RDFox as a Datalog engine and add rules on demand * Rule count per ontology: 108+6 (LUBM), 254+19 (UOBM), 481+14 (Reactome),317+59 (Uniprot) * RDFox uses renaming to deal with equality.

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The Combined Approach to Query Answering in Horn-ALCHOIQ David Carral, Irina Dragoste, Markus Krötzsch

The Combined Approach to Query Answering in Horn-ALCHOIQ

David Carral, Irina Dragoste, Markus Krötzsch Knowledge-Based Systems Group at

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