Post on 22-Nov-2014
description
Georg Gottlob
Computing Laboratory
Department of Computer Science
University of Oxford
Joint work with
G. Orsi and A. Pieris
D
reports(john,ann)
reports(ann,tom)
reports(X,Y) employee(X)
reports(X,Y) employee(Y)
reports(X,Y) manager(Y,X)
manager(X,Y),manager(Y,Z) manager(X,Z)
Intensional DB
reports(john,ann) reports(ann,tom)
employee(john) employee(ann)
employee(tom) manager(ann,john)
manager(tom,ann) manager(tom,john)
Q = XYZ employee(X),manager(X,Y),manager(Y,Z)
D
reports(john,ann)
reports(ann,tom)
Intensional DB
reports(john,ann) reports(ann,tom)
employee(john) employee(ann)
employee(tom) manager(ann,john)
manager(tom,ann) manager(tom,john)
Unique
Least Herbrand Model
for
D υ
Q = XYZ employee(X),manager(X,Y),manager(Y,Z)
reports(X,Y) employee(X)
reports(X,Y) employee(Y)
reports(X,Y) manager(Y,X)
manager(X,Y),manager(Y,Z) manager(X,Z)
employee v supervisedBy
supervisedBy v employee
employee supervisedBy
employee v supervises-
supervises v employee
employee v supervises-
supervises v employee
?:- supervises(X,John), employee(X)
employee v supervises-
supervises v employee
?:- supervises(X,John), employee(X)
employee v supervises-
supervises v employee
?:- supervises(John,X)
employee v supervises-
supervises v employee
?:- supervises(John,X)
Usually done with tableaux algorithms
employee v supervises-
supervises v employee
DL axiom Datalog rule
Concept Inclusion
employee v person employee(X) person(X)
(Inverse) Role Inclusion
reports¡ v manager reports(X,Y) manager(Y,X)
Role Transitivity
trans(manager) manager(X,Y),manager(Y,Z) manager(X,Z)
Participation
employee v report employee(X) Y report(X,Y)
Disjointness
employee v :customer employee(X),customer(X)
Functionality
funct(reports) reports(X,Y),reports(X,Z) Y = Z
DL axiom Datalog(?) rule
D
Q = X (X)
D
B ( (B D [ ) B ² Q) D [ ² Q ,
B ¶ D Æ B ²
Q = X (X)
D
B ( (B ² D[) B ² Q) D [ ² Q ,
B ¶ D Æ B ²
DLs (DL-Lite, EL, Flogic Lite)
Datalog
DB
technology
+
constraints
υ
υ
dom(D) = {a,b,c,d}
chase(D, ) = ?
D
D dom(D) = {a,b,c,d}
chase(D, ) = D υ { r(b,z0)
D dom(D) = {a,b,c,d}
chase(D, ) = D υ { r(b,z0), r(d,z1)
D dom(D) = {a,b,c,d}
chase(D, ) = D υ { r(b,z0), r(d,z1)
z0 = z1
D dom(D) = {a,b,c,d}
chase(D, ) = D υ { r(b,z1), r(d,z1) }
see, e.g.,
Deutsch, Nash & Remmel
PODS 08
D
. . .
chase(D,)
M1
M2
h1 h2
D
. . .
chase(D,)
M1
M2
h1 h2
Q h
D υ ² Q , chase(D,) ² Q
Σ Σ Σ ΣΣ
Guarded
Linear
DL-Lite Sticky-join
EL
FO-rewritable
PTIME
Guarded
Linear
DL-Lite Sticky-join
EL
Linear
FO-rewritable
PTIME
Guarded
Linear
DL-Lite Sticky-join
EL
We focus on the linear fragment:
• first-order rewriting
• semantic query optimizations
Q
Q
Q
Q Q*
D
Q
Q Q*
DL-Lite: Popular family of DLs with highly tractable data complexity
DL-Lite TBox Datalog Representation
DL-Litecore
promoter v dealsWith
promoter v customer
DL-LiteR
promotes¡ v isAdvertised
DL-LiteF
funct( supervises- )
professor(X) Y teachesTo(X,Y)
professor(X),student(X)
promotes(X,Y) isAdvertised(Y,X)
supervises(Y,X),supervises Y,X Y = Z
D Q
D Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
SELECT promoter
FROM promotesTo PT, customer C
WHERE PT.promoted = C.name
UNION
SELECT promoter
FROM promotesTo PT1, promotesTo PT2
WHERE PT1.promoted = PT2.promoted
UNION
SELECT name
FROM promoter
Schema
Execute Q over D:
D
SELECT promoter
FROM promotesTo PT, customer C
WHERE PT.promoted = C.name
UNION
SELECT promoter
FROM promotesTo PT1, promotesTo PT2
WHERE PT1.promoted = PT2.promoted
UNION
SELECT name
FROM promoter
Schema
:
:
:
:
:
:
:
Q:
:
:
8-
EL: Popular DL for complex systems modelling (e.g., biology)
EL TBox Datalog Representation
v
u v
9
elephant(E),mouse(M) biggerThan(E,M)
person v 9father¡
9father v person
?:- 9father.{John} u person
person v 9father¡
9father v person
?:- 9father.{John} u person
?:- 9father¡ u 9father.{John} u person
person v 9father¡
9father v person
?:- 9father.{John} u person
?:- 9father¡ u 9father.{John} u person
9father¡, 9father.{John}, person
person v 9father¡
9father v person
?:- 9father.{John} u person
?:- 9father¡ u 9father.{John} u person
9father¡, 9father.{John}, person
father
person v 9father¡
9father v person
?:- 9father.{John} u person
?:- 9father¡ u 9father.{John} u person
9father¡, 9father.{John}, person
father
father
person v 9father¡
9father v person
?:- 9father.{John} u person
true:- 9father¡ u 9father.{John} u person
9father¡, 9father.{John}, person
father
father
Model
T
E
T , E
T
E
T , E
T
E
T , E