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Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for...
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
Tractable Query Answering for Expressive Ontologies and Rules*
David Carral, Irina Dragoste, Markus Krötzsch Knowledge-Based Systems Group at
/231
*Published at ISWC 2017
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /232
Disjunctive Existential Rules
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /233
Disjunctive Existential Rules
Facts
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /233
HasParent(x, y) ∧ HasSister(y, z) → HasAunt(x, z)Human(x) → ∃y .HasParent(x, y) ∧ Human(y)Animal(x) → Herbivore(x) ∨ Carnivore(x) ∨ Omnivore(x)P(x, a, y) ∧ R(y, w) ∧ S(w, x) → ∃v . (R(w, v) ∧ A(v)) ∨ D(x)
Disjunctive Existential Rules
Facts
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /233
HasParent(x, y) ∧ HasSister(y, z) → HasAunt(x, z)Human(x) → ∃y .HasParent(x, y) ∧ Human(y)Animal(x) → Herbivore(x) ∨ Carnivore(x) ∨ Omnivore(x)P(x, a, y) ∧ R(y, w) ∧ S(w, x) → ∃v . (R(w, v) ∧ A(v)) ∨ D(x)
Disjunctive Existential Rules
FactsHasFriend(stan, kyle)
Dead(kenny)P(a, c,d)
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /233
HasParent(x, y) ∧ HasSister(y, z) → HasAunt(x, z)Human(x) → ∃y .HasParent(x, y) ∧ Human(y)Animal(x) → Herbivore(x) ∨ Carnivore(x) ∨ Omnivore(x)P(x, a, y) ∧ R(y, w) ∧ S(w, x) → ∃v . (R(w, v) ∧ A(v)) ∨ D(x)
Disjunctive Existential Rules
FactsHasFriend(stan, kyle)
Dead(kenny)P(a, c,d)
Remark. If normalised, SROIQ ontologies can be translated into equivalent programs of disjunctive existential rules.
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The Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /235
Director(x) → ∃y .Directs(x, y) ∧ Film(y)
The Disjunctive Chase
Film(x) → ∃z . IsDirectedBy(x, z) ∧ Director(z)Film(ai)
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Director(x) → ∃y .Directs(x, y) ∧ Film(y)Film(x) → ∃z . IsDirectedBy(x, z) ∧ Director(z)Film(ai)
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /235
Director(x) → Directs(x, fy(x)) ∧ Film( fy(x))Film(x) → IsDirectedBy(x, fz(x)) ∧ Director( fz(x))Film(ai)
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /235
Film(ai)
Director(x) → Directs(x, y(x)) ∧ Film(y(x))Film(x) → IsDirectedBy(x, z(x)) ∧ Director(z(x))
The Skolem Disjunctive Chase
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Film(ai)
ai : Film
Director(x) → Directs(x, y(x)) ∧ Film(y(x))Film(x) → IsDirectedBy(x, z(x)) ∧ Director(z(x))
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /235
Film(ai)
ai : Film
z(ai) : Director
Director(x) → Directs(x, y(x)) ∧ Film(y(x))Film(x) → IsDirectedBy(x, z(x)) ∧ Director(z(x))
DirectedBy
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /235
Film(ai)
ai : Film
z(ai) : Director
Director(x) → Directs(x, y(x)) ∧ Film(y(x))Film(x) → IsDirectedBy(x, z(x)) ∧ Director(z(x))
DirectedBy
Directs
z(y(ai)) : Film
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /235
Film(ai)
ai : Film
z(ai) : Director
Director(x) → Directs(x, y(x)) ∧ Film(y(x))Film(x) → IsDirectedBy(x, z(x)) ∧ Director(z(x))
DirectedBy DirectedBy
Directs
z(y(ai)) : Film
y(z(y(ai))) : Director
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /235
Film(ai)
ai : Film
z(ai) : Director
Director(x) → Directs(x, y(x)) ∧ Film(y(x))Film(x) → IsDirectedBy(x, z(x)) ∧ Director(z(x))
DirectedBy DirectedBy
Directs
z(y(ai)) : Film
y(z(y(ai))) : Director
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /235
Film(ai)
ai : Film
z(ai) : Director
Director(x) → Directs(x, y(x)) ∧ Film(y(x))Film(x) → IsDirectedBy(x, z(x)) ∧ Director(z(x))
DirectedBy DirectedBy
Directs
z(y(ai)) : Film
y(z(y(ai))) : Director
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /236
Film(ai)Film(x) → IsDirectedBy(x, z(x)) ∧ Director(z(x))Director(x) → Directs(x, y(x)) ∧ Film(y(x))
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /236
Film(ai)Film(x) → IsDirectedBy(x, z(x)) ∧ Director(z(x))Director(x) → Directs(x, y(x)) ∧ Film(y(x))
ai : F
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /236
Film(ai)Film(x) → IsDirectedBy(x, z(x)) ∧ Director(z(x))Director(x) → Directs(x, y(x)) ∧ Film(y(x))
ai : F ai : F
y(ai) : F
DB
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /236
Film(ai)Film(x) → IsDirectedBy(x, z(x)) ∧ Director(z(x))Director(x) → Directs(x, y(x)) ∧ Film(y(x))
ai : F ai : F
y(ai) : F
DB
D
z(y(ai)) : F
ai : F
y(ai) : F
DB
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /236
Film(ai)
ai : F
y(ai) : F
DB
DBD
z(y(ai)) : F
y(z(y(ai))) : D
Film(x) → IsDirectedBy(x, z(x)) ∧ Director(z(x))Director(x) → Directs(x, y(x)) ∧ Film(y(x))
ai : F ai : F
y(ai) : F
DB
D
z(y(ai)) : F
ai : F
y(ai) : F
DB
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /236
Film(ai)
ai : F
y(ai) : F
DB
DBD
z(y(ai)) : F
y(z(y(ai))) : D
Film(x) → IsDirectedBy(x, z(x)) ∧ Director(z(x))Director(x) → Directs(x, y(x)) ∧ Film(y(x))
ai : F ai : F
y(ai) : F
DB
D
z(y(ai)) : F
ai : F
y(ai) : F
DB
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /237
Animal(a) Animal(b)Animal(x) → Vertebrate(x) ∨ Invertebrate(x)
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /237
Animal(a) Animal(b)Animal(x) → Vertebrate(x) ∨ Invertebrate(x)
A : a b : A
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /237
Animal(a) Animal(b)Animal(x) → Vertebrate(x) ∨ Invertebrate(x)
A : a b : A
V, A : a b : Ab : AI, A : a
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /237
Animal(a) Animal(b)Animal(x) → Vertebrate(x) ∨ Invertebrate(x)
A : a b : A
V, A : a b : Ab : AI, A : a
V, A : a b : A, VV, A : a b : A, I
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /237
Animal(a) Animal(b)Animal(x) → Vertebrate(x) ∨ Invertebrate(x)
A : a b : A
V, A : a b : Ab : AI, A : a
V, A : a b : A, VV, A : a b : A, I I, A : a b : A, V
I, A : a b : A, I
The Skolem Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /238
Ensuring Tractability of the Disjunctive Chase
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /239
Existential Dependency Graph
A(x) → ∃y . S(x, y) ∧ B(y)B(x) → ∃z .R(x, z) ∧ D(z)D(x) → E(x)E(x) → ∃w .R(x, w)
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
A(x) → S(x, y(x)) ∧ B(y(x))B(x) → R(x, z(x)) ∧ D(z(x))D(x) → E(x)E(x) → R(x, w(x))
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
/239
Existential Dependency Graph
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
A(x) → S(x, y(x)) ∧ B(y(x))B(x) → R(x, z(x)) ∧ D(z(x))D(x) → E(x)E(x) → R(x, w(x))
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
/239
Existential Dependency Graph
y
z
w
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
A(x) → S(x, y(x)) ∧ B(y(x))B(x) → R(x, z(x)) ∧ D(z(x))D(x) → E(x)E(x) → R(x, w(x))
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
/239
Existential Dependency Graph
y
z
w
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
A(x) → S(x, y(x)) ∧ B(y(x))B(x) → R(x, z(x)) ∧ D(z(x))D(x) → E(x)E(x) → R(x, w(x))
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
/239
Existential Dependency Graph
y
z
w
A(c)S(c, y(c)), B(y(c))R(y(c), z(y(c))), D(z(y(c)))
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
A(x) → S(x, y(x)) ∧ B(y(x))B(x) → R(x, z(x)) ∧ D(z(x))D(x) → E(x)E(x) → R(x, w(x))
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
/239
Existential Dependency Graph
y
z
w
A(c)S(c, y(c)), B(y(c))R(y(c), z(y(c))), D(z(y(c)))
z(y(c))
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
A(x) → S(x, y(x)) ∧ B(y(x))B(x) → R(x, z(x)) ∧ D(z(x))D(x) → E(x)E(x) → R(x, w(x))
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
/239
Existential Dependency Graph
y
z
w
z(y(c))
![Page 37: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/37.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
A(x) → S(x, y(x)) ∧ B(y(x))B(x) → R(x, z(x)) ∧ D(z(x))D(x) → E(x)E(x) → R(x, w(x))
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
/239
Existential Dependency Graph
y
z
w
z(y(c))
![Page 38: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/38.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
A(x) → S(x, y(x)) ∧ B(y(x))B(x) → R(x, z(x)) ∧ D(z(x))D(x) → E(x)E(x) → R(x, w(x))
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
/239
Existential Dependency Graph
y
z
w
z(y(c))B(c)R(c, z(c)), D(z(c)),E(z(c)),R(z(c), w(z(c)))
![Page 39: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/39.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
A(x) → S(x, y(x)) ∧ B(y(x))B(x) → R(x, z(x)) ∧ D(z(x))D(x) → E(x)E(x) → R(x, w(x))
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
/239
Existential Dependency Graph
y
z
w
z(y(c))
w(z(c))
B(c)R(c, z(c)), D(z(c)),E(z(c)),R(z(c), w(z(c)))
![Page 40: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/40.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
A(x) → S(x, y(x)) ∧ B(y(x))B(x) → R(x, z(x)) ∧ D(z(x))D(x) → E(x)E(x) → R(x, w(x))
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
/239
Existential Dependency Graph
y
z
w
z(y(c))
w(z(c))
![Page 41: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/41.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
A(x) → S(x, y(x)) ∧ B(y(x))B(x) → R(x, z(x)) ∧ D(z(x))D(x) → E(x)E(x) → R(x, w(x))
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
/239
Existential Dependency Graph
y
z
w
z(y(c))
w(z(c))
![Page 42: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/42.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
A(x) → S(x, y(x)) ∧ B(y(x))B(x) → R(x, z(x)) ∧ D(z(x))D(x) → E(x)E(x) → R(x, w(x))
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
/239
Existential Dependency Graph
y
z
w
z(y(c))
w(z(c))
A(c)S(c, y(c)),B(y(c)),C(y(c)),E(y(c)),R(y(c), w(y(c)))
![Page 43: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/43.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
A(x) → S(x, y(x)) ∧ B(y(x))B(x) → R(x, z(x)) ∧ D(z(x))D(x) → E(x)E(x) → R(x, w(x))
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
/239
Existential Dependency Graph
y
z
w
z(y(c))
w(z(c))
A(c)S(c, y(c)),B(y(c)),C(y(c)),E(y(c)),R(y(c), w(y(c)))
w(y(c))
![Page 44: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/44.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
A(x) → S(x, y(x)) ∧ B(y(x))B(x) → R(x, z(x)) ∧ D(z(x))D(x) → E(x)E(x) → R(x, w(x))
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
/239
Existential Dependency Graph
y
z
w
z(y(c))
w(z(c))
w(y(c))
![Page 45: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/45.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
A(x) → S(x, y(x)) ∧ B(y(x))B(x) → R(x, z(x)) ∧ D(z(x))D(x) → E(x)E(x) → R(x, w(x))
B(x) ∧ C(x) → E(x)S(x, y) → C(x)
/239
Existential Dependency Graph
y
z
w
![Page 46: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/46.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2310
(a) Acyclicity
y
zw
![Page 47: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/47.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2310
(a) Acyclicity
y
zwz(c)
![Page 48: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/48.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2310
(a) Acyclicity
y
zwz(c)
w(z(c))
![Page 49: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/49.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2310
(a) Acyclicity
y
zw y(w(z(c)))z(c)
w(z(c))
![Page 50: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/50.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2310
(a) Acyclicity
y
zw
z(y(w(z(c))))
y(w(z(c)))z(c)
w(z(c))
![Page 51: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/51.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2310
(a) Acyclicity
y
zw
z(y(w(z(c))))
y(w(z(c)))z(c)
w(z(c)) w(z(y(w(z(c)))))
![Page 52: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/52.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2310
(a) Acyclicity
y
zw
z(y(w(z(c))))
y(w(z(c)))z(c)
w(z(c)) w(z(y(w(z(c)))))
………
![Page 53: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/53.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2310
(a) Acyclicity
y
zw Remark. If the existential dependency graph of a given set of rules is acyclic, then the set of terms introduced during the computation of the chase is finite.
![Page 54: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/54.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2311
(f) Arity at Most 1Film(x) → ∃y . IsFilmDirectedBy(x, y) ∧ Director(y)
A(x) ∧ B(x, w) ∧ C(x, z) → ∃z .R(x, w, z)
![Page 55: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/55.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2311
(f) Arity at Most 1Film(x) → ∃y . IsFilmDirectedBy(x, y) ∧ Director(y)Film(x) → IsFilmDirectedBy(x, y(x)) ∧ Director(y(x))
A(x) ∧ B(x, y) ∧ C(x, z) → ∃z .R(x, y, z)A(x) ∧ B(x, w) ∧ C(x, z) → R(x, w, z(x, w))
![Page 56: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/56.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2311
(f) Arity at Most 1Film(x) → ∃y . IsFilmDirectedBy(x, y) ∧ Director(y)Film(x) → IsFilmDirectedBy(x, y(x)) ∧ Director(y(x))
A(x) ∧ B(x, y) ∧ C(x, z) → ∃z .R(x, y, z)A(x) ∧ B(x, w) ∧ C(x, z) → R(x, w, z(x, w))
Remark. If the arity of every function symbol in the skolemisation of a program is at most 1, then every term in the chase is of the form x1(…xn(c)…) with c constant.
![Page 57: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/57.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2312
(f) Arity at Most 1
Remark. If the arity of every function symbol in the skolemisation of a program is at most 1, then every term in the chase is of the form x1(…xn(c)…) with c constant.
![Page 58: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/58.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2312
(f) Arity at Most 1
Remark. If the arity of every function symbol in the skolemisation of a program is at most 1, then every term in the chase is of the form x1(…xn(c)…) with c constant.
Corollary. Every term occurring in the chase corresponds to a path in the dependency graph and a constant.
![Page 59: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/59.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2312
(f) Arity at Most 1
Remark. If the arity of every function symbol in the skolemisation of a program is at most 1, then every term in the chase is of the form x1(…xn(c)…) with c constant.
Corollary. Every term occurring in the chase corresponds to a path in the dependency graph and a constant.
y
z
w
v
![Page 60: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/60.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2312
(f) Arity at Most 1
Remark. If the arity of every function symbol in the skolemisation of a program is at most 1, then every term in the chase is of the form x1(…xn(c)…) with c constant.
Corollary. Every term occurring in the chase corresponds to a path in the dependency graph and a constant.
y
z
w
v
v(w(z(y(c))) y(v(c))
![Page 61: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/61.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2313
Ensuring Tractability
![Page 62: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/62.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2313
(a) The dependency graph is acyclic.
Ensuring Tractability
![Page 63: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/63.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2313
(a) The dependency graph is acyclic.
(f) The arity of all function symbols in the skolemisation of the program is at most 1.
Ensuring Tractability
![Page 64: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/64.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2313
(a) The dependency graph is acyclic.
(f) The arity of all function symbols in the skolemisation of the program is at most 1.
All skolem terms correspond to some path in the dependency graph and some constant
Ensuring Tractability
![Page 65: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/65.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2313
(a) The dependency graph is acyclic.
(f) The arity of all function symbols in the skolemisation of the program is at most 1.
(w) The number of variables per rule is bounded.
All skolem terms correspond to some path in the dependency graph and some constant
Ensuring Tractability
![Page 66: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/66.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2313
(a) The dependency graph is acyclic.
(f) The arity of all function symbols in the skolemisation of the program is at most 1.
(w) The number of variables per rule is bounded.
Rules can be applied in polynomial time
The number of facts is polynomial in the number of terms
All skolem terms correspond to some path in the dependency graph and some constant
Ensuring Tractability
![Page 67: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/67.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2313
(a) The dependency graph is acyclic.
(f) The arity of all function symbols in the skolemisation of the program is at most 1.
(w) The number of variables per rule is bounded.
Rules can be applied in polynomial time
The number of facts is polynomial in the number of terms
All skolem terms correspond to some path in the dependency graph and some constant
The number of paths in the dependency graph is polynomial
Ensuring Tractability
![Page 68: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/68.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2313
(a) The dependency graph is acyclic.
(f) The arity of all function symbols in the skolemisation of the program is at most 1.
(w) The number of variables per rule is bounded.
Rules can be applied in polynomial time
The number of facts is polynomial in the number of terms
All skolem terms correspond to some path in the dependency graph and some constant
The number of paths in the dependency graph is polynomial
Polynomiality
Ensuring Tractability
![Page 69: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/69.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2313
(a) The dependency graph is acyclic.
(f) The arity of all function symbols in the skolemisation of the program is at most 1.
(w) The number of variables per rule is bounded.
Rules can be applied in polynomial time
The number of facts is polynomial in the number of terms
All skolem terms correspond to some path in the dependency graph and some constant
The number of paths in the dependency graph is polynomial
(?)
Polynomiality
Ensuring Tractability
![Page 70: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/70.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch /2314
Braids
![Page 71: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/71.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 14/23
Braids
![Page 72: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/72.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
Braids
14/23
![Page 73: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/73.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
Braids
14/23
![Page 74: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/74.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 15/23
(a) The dependency graph is acyclic.
(f) The arity of all function symbols in the skolemisation of the program is at most 1.
(w) The number of variables per rule is bounded.
Rules can be applied in polynomial time
The number of facts is polynomial in the number of terms
All skolem terms correspond to some path in the dependency graph and some constant
The number of paths in the dependency graph is polynomial
Polynomiality
Ensuring Tractability
(?)
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 15/23
(a) The dependency graph is acyclic.
(f) The arity of all function symbols in the skolemisation of the program is at most 1.
(w) The number of variables per rule is bounded.
Rules can be applied in polynomial time
The number of facts is polynomial in the number of terms
All skolem terms correspond to some path in the dependency graph and some constant
The number of paths in the dependency graph is polynomial
Polynomiality
Ensuring Tractability
(b) The length of the braids in the dependency graph is bounded.
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 15/23
(a) The dependency graph is acyclic.
(f) The arity of all function symbols in the skolemisation of the program is at most 1.
(w) The number of variables per rule is bounded.
Rules can be applied in polynomial time
The number of facts is polynomial in the number of terms
All skolem terms correspond to some path in the dependency graph and some constant
The number of paths in the dependency graph is polynomial
Polynomiality
Ensuring Tractability
(b) The length of the braids in the dependency graph is bounded.
Caveats.1. Fixed query size. 2. Horn rule set.
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 16/23
Evaluation
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 17/23
A1 ⊓ … ⊓ An ⊑ B ↦ A1(x) ∧ … ∧ An(x) → B(x)A ⊑ B1 ⊔ … ⊔ Bn ↦ A(x) → B1(x) ∨ … ∨ Bn(x)
A ⊑ ∀R . B ↦ A(y) ∧ R(x, y) → B(x)A ⊑ ∃R . B ↦ A(x) → ∃y . R(x, y) ∧ B(y)
R ⊑ S ↦ R(x, y) → S(x, y)R ∘ S ⊑ V ↦ R(x, y) ∧ S(y, z) → S(x, z)
R1 ⊓ … ⊓ Rn ⊑ S ↦ R1(x, y) ∧ … ∧ Rn(x, y) → S(x, y)A(a) ↦ A(a)
R(a, b) ↦ R(a, b)
SRI Axioms
![Page 79: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/79.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 17/23
A1 ⊓ … ⊓ An ⊑ B ↦ A1(x) ∧ … ∧ An(x) → B(x)A ⊑ B1 ⊔ … ⊔ Bn ↦ A(x) → B1(x) ∨ … ∨ Bn(x)
A ⊑ ∀R . B ↦ A(y) ∧ R(x, y) → B(x)A ⊑ ∃R . B ↦ A(x) → ∃y . R(x, y) ∧ B(y)
R ⊑ S ↦ R(x, y) → S(x, y)R ∘ S ⊑ V ↦ R(x, y) ∧ S(y, z) → S(x, z)
R1 ⊓ … ⊓ Rn ⊑ S ↦ R1(x, y) ∧ … ∧ Rn(x, y) → S(x, y)A(a) ↦ A(a)
R(a, b) ↦ R(a, b)
SRI Axioms
Remark 1. Deciding CQ entailment for SRI ontologies is 2ExpTime-Hard and in 3ExpTime.
![Page 80: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/80.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 17/23
A1 ⊓ … ⊓ An ⊑ B ↦ A1(x) ∧ … ∧ An(x) → B(x)A ⊑ B1 ⊔ … ⊔ Bn ↦ A(x) → B1(x) ∨ … ∨ Bn(x)
A ⊑ ∀R . B ↦ A(y) ∧ R(x, y) → B(x)A ⊑ ∃R . B ↦ A(x) → ∃y . R(x, y) ∧ B(y)
R ⊑ S ↦ R(x, y) → S(x, y)R ∘ S ⊑ V ↦ R(x, y) ∧ S(y, z) → S(x, z)
R1 ⊓ … ⊓ Rn ⊑ S ↦ R1(x, y) ∧ … ∧ Rn(x, y) → S(x, y)A(a) ↦ A(a)
R(a, b) ↦ R(a, b)
SRI Axioms
Remarks. 1. Every rule in a SRI ontology has at most 3 variables. 2. Every function symbol in the skolemisation of a SRI ontology is of arity one.
Remark 1. Deciding CQ entailment for SRI ontologies is 2ExpTime-Hard and in 3ExpTime.
![Page 81: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/81.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 17/23
A1 ⊓ … ⊓ An ⊑ B ↦ A1(x) ∧ … ∧ An(x) → B(x)A ⊑ B1 ⊔ … ⊔ Bn ↦ A(x) → B1(x) ∨ … ∨ Bn(x)
A ⊑ ∀R . B ↦ A(y) ∧ R(x, y) → B(x)A ⊑ ∃R . B ↦ A(x) → ∃y . R(x, y) ∧ B(y)
R ⊑ S ↦ R(x, y) → S(x, y)R ∘ S ⊑ V ↦ R(x, y) ∧ S(y, z) → S(x, z)
R1 ⊓ … ⊓ Rn ⊑ S ↦ R1(x, y) ∧ … ∧ Rn(x, y) → S(x, y)A(a) ↦ A(a)
R(a, b) ↦ R(a, b)
SRI Axioms
Remarks. 1. Every rule in a SRI ontology has at most 3 variables. 2. Every function symbol in the skolemisation of a SRI ontology is of arity one.
Remark 1. Deciding CQ entailment for SRI ontologies is 2ExpTime-Hard and in 3ExpTime.
![Page 82: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/82.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 17/23
A1 ⊓ … ⊓ An ⊑ B ↦ A1(x) ∧ … ∧ An(x) → B(x)A ⊑ B1 ⊔ … ⊔ Bn ↦ A(x) → B1(x) ∨ … ∨ Bn(x)
A ⊑ ∀R . B ↦ A(y) ∧ R(x, y) → B(x)A ⊑ ∃R . B ↦ A(x) → ∃y . R(x, y) ∧ B(y)
R ⊑ S ↦ R(x, y) → S(x, y)R ∘ S ⊑ V ↦ R(x, y) ∧ S(y, z) → S(x, z)
R1 ⊓ … ⊓ Rn ⊑ S ↦ R1(x, y) ∧ … ∧ Rn(x, y) → S(x, y)A(a) ↦ A(a)
R(a, b) ↦ R(a, b)
SRI Axioms
Remarks. 1. Every rule in a SRI ontology has at most 3 variables. 2. Every function symbol in the skolemisation of a SRI ontology is of arity one.
Remark 1. Deciding CQ entailment for SRI ontologies is 2ExpTime-Hard and in 3ExpTime.
![Page 83: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/83.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 17/23
A1 ⊓ … ⊓ An ⊑ B ↦ A1(x) ∧ … ∧ An(x) → B(x)A ⊑ B1 ⊔ … ⊔ Bn ↦ A(x) → B1(x) ∨ … ∨ Bn(x)
A ⊑ ∀R . B ↦ A(y) ∧ R(x, y) → B(x)A ⊑ ∃R . B ↦ A(x) → ∃y . R(x, y) ∧ B(y)
R ⊑ S ↦ R(x, y) → S(x, y)R ∘ S ⊑ V ↦ R(x, y) ∧ S(y, z) → S(x, z)
R1 ⊓ … ⊓ Rn ⊑ S ↦ R1(x, y) ∧ … ∧ Rn(x, y) → S(x, y)A(a) ↦ A(a)
R(a, b) ↦ R(a, b)
SRI Axioms
Remarks. 1. Every rule in a SRI ontology has at most 3 variables. 2. Every function symbol in the skolemisation of a SRI ontology is of arity one.
Remark 1. Deciding CQ entailment for SRI ontologies is 2ExpTime-Hard and in 3ExpTime.
![Page 84: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/84.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 17/23
A1 ⊓ … ⊓ An ⊑ B ↦ A1(x) ∧ … ∧ An(x) → B(x)A ⊑ B1 ⊔ … ⊔ Bn ↦ A(x) → B1(x) ∨ … ∨ Bn(x)
A ⊑ ∀R . B ↦ A(y) ∧ R(x, y) → B(x)A ⊑ ∃R . B ↦ A(x) → ∃y . R(x, y) ∧ B(y)
R ⊑ S ↦ R(x, y) → S(x, y)R ∘ S ⊑ V ↦ R(x, y) ∧ S(y, z) → S(x, z)
R1 ⊓ … ⊓ Rn ⊑ S ↦ R1(x, y) ∧ … ∧ Rn(x, y) → S(x, y)A(a) ↦ A(a)
R(a, b) ↦ R(a, b)
SRI Axioms
Remarks. 1. Every rule in a SRI ontology has at most 3 variables. 2. Every function symbol in the skolemisation of a SRI ontology is of arity one.
Remark 1. Deciding CQ entailment for SRI ontologies is 2ExpTime-Hard and in 3ExpTime.
![Page 85: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/85.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 18/23
SRI Axioms
Remarks. 1. Every rule in an OWL ontology has at most 3 variables. 2. Every function symbol in the skolemisation of a SRI ontology has arity one
![Page 86: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/86.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 18/23
SRI Axioms
Remarks. 1. Every rule in an OWL ontology has at most 3 variables. 2. Every function symbol in the skolemisation of a SRI ontology has arity one
Corollary. To guarantee that tractable reasoning over a SRI ontology is possible we only need to verify the following: 1. Acyclicity. 2. Braid length in the dependency graph is bounded.
![Page 87: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/87.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 19/23
Evaluation Results
![Page 88: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/88.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 19/23
Evaluation Results
Ontologies 1576 225Acyclic 974 (61.8%) 170 (75.6%)
AcyclicityMOWL Corpus Oxford Ontology Repo
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 19/23
Evaluation Results
Ontologies 1576 225Acyclic 974 (61.8%) 170 (75.6%)
Acyclicity
Braid Length
1 2 3 4 5 6 11 22 23 25 Total851 153 56 61 11 1 1 2 7 1 114474 88 93 98 99 99 99 99.1 99.3 99.9 100
MOWL Corpus + Oxford(max. length
of a braid)(count)
(%)
MOWL Corpus Oxford Ontology Repo
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Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 20/23
Conclusions
![Page 91: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/91.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 21/23
More Results!
![Page 92: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/92.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 22/23
Conclusions
Efficient CQ Answering over a large subset of OWL 2 real-world ontologies is possible!
![Page 93: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/93.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch 22/23
Conclusions
Efficient CQ Answering over a large subset of OWL 2 real-world ontologies is possible!Future work: - Implementation of a chase based reasoner for these ontologies - Refine conditions
![Page 94: Tractable Query Answering for Expressive Ontologies and Rules* fileTractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch Tractable](https://reader035.fdocuments.in/reader035/viewer/2022070618/5e1470f02f18d869ed4b8813/html5/thumbnails/94.jpg)
Tractable Query Answering for Expressive Ontologies and Rules David Carral, Irina Dragoste, Markus Krötzsch
Tractable Query Answering for Expressive Ontologies and Rules*
David Carral, Irina Dragoste, Markus Krötzsch Knowledge-Based Systems Group at
23/23
*Published at ISWC 2017