ontologies and large databases - inria...ontologies and large databases querying algorithms for the...

Ontologies and Large Databases Querying Algorithms for the Web of Data Jean-François Baget ([email protected]) Artificial Intelligence meets the Web of Data, Montpellier, 2013

EQUIPE PROJET

GraphIK INRIA Sophia-Antiplis

LIRMM

mailto:[email protected]

ONTOLOGY-BASED DATA

ACCESS FOR THE WEB OF DATA

Query

Ontology-based Query Answering

Knowledge Base

Data/Facts

Answers ?

Ontology

Data / Facts

Relational Database RDF (Semantic Web)

rdf:type

ex:parent

ex:a

F

rdf:type

ex:b

M

a b

a c

c ?

… …

parentOf

b

…

Male Fem.

a

…

ex:c

Abstraction in first-order logic

x( parentOf(a,b) parentOf(a,c)

parentOf(c,x) F(a) M(b) )

Etc.

ex:parent

Or in graphs / hypergraphs

1 2 a b p

1 2

c p

p

1

2

F M

A conjunction of facts ≈ a fact

A (boolean) CQ and a fact have the same form

Body Head

X, Y, Z :

tuples of variables

Any conjunction of

atoms (on variables

and constants)

x y (siblingOf(x,y) z (parentOf(z,x) parentOf(z,y)))

Same as Tuple Generating Dependencies

[+ Equality Generating Dependencies, Negative Constraints]

See also Datalog+/-

Same as the logical translation of Conceptual Graph rules

Generalize light Description Logics used for OBDA (DL-Lite, EL, …)

X Y ( B[X, Y] Z H[X, Z] )

Ontology: Existential Rules

Simplified form: siblingOf(x,y) parentOf(z,x) parentOf(z,y)

F = siblingOf(a,b)

Then h(head) can be « added » to F

Generating Fresh Variables

R = x y (siblingOf(x,y) z (parentOf(z,x) parentOf(z,y)))

F’= z0 (siblingOf(a,b) parentOf(z0,a) parentOf(z0,a))

h ={(x,a), (y,b)}

h: body F

A rule body head is applicable to a fact F if

there is a homomorphism h: body F

1

S

a

b

2

P

P 2 S

1

2

2

1 x

y

1

z

1

S

a

b

2

P

P

2

2

1

1 z0 with renaming existential variables of head

Basic Decision Problem

Given a KB K = (F, R) and a (Boolean) conjunctive query Q,

is Q entailed by K ?

Conjunctive Query Entailment

Existential Rules

(without equality)

Conjunctive

Query

Knowledge Base

Fact(s)

K = ( F, R )

Q

Forward vs Backward Chaining

F Q

R

F Q R

FC

BC

Fact saturation (« bottom-up »)

Query rewriting (« top-down »)

[Decomposition into 2 steps: DL-Lite]

F, R |= Q iff there is a homomorphism from Q’ to F, where Q’ is obtained by a

rewriting sequence from Q with R

F, R |= Q iff there is a homomorphism from Q to F’, where F’ is obtained by

a derivation sequence from F withR

Decidability Issues

Entailment is not decidable

Many decidable classes exhibited in databases and KR

Three generic kinds of properties ensuring decidability:

- Saturation by Forward Chaining halts (« finite expansion set », fes)

- Query rewriting by Backward Chaining halts (« finite unification

set », fus)

- Saturation by Forward Chaining may not halt but the generated

facts have a tree-like structure (« bounded treewidth set », bts)

These properties are not recognizable [Baget+ KR 10]

but they provide generic algorithms

Datalog

weakly-acyclic

wa-GRD

jointly

acyclic

acyclic

GRD

w-sticky

w-sticky-join

Inclusion

dependency

atomic body

domain-r

sticky join

sticky

guarded frontier-1

frontier-

guarded

weakly-

guarded

weakly fg

jointly fg

glut fg

FES

FUS

BTS

(Partial) Inclusion Map of

Decidable Classes

FINITE EXPANSION SETS

00 MOIS 2011 EMETTEUR - NOM DE LA PRESENTATION - 11

Breadth-First FC / Chase

F0 = F

Fi (i >1) obtained from Fi-1 by performing all possible rule applications on Fi-1

Any atom with rank i can be obtained by a derivation sequence of

length ≤ max (|rule body|)i

F0

F1

Fk

…

Atoms with rank 0

Atoms with rank 1

Atoms with rank k

bound independent from |F|

F,R |= Q iff there is k s.t. Q maps to Fk

Note: k is not independent from F

fes ≠ « bounded » set of rules (cf. this notion in Datalog)

R is bounded if there is k s.t. for all F, Fk+1 Fk

(ex: acyclic GRD)

[Baget+ JAIR 2002]

= core chase

Finite Saturation (fes)

Def: R is a finite expansion set (fes) if for all fact F, there is F’ (finitely)

derived from F s.t. any rule application on F’ leads to a fact equivalent to

F’

R is fes iff for all fact F, there is k s.t. Fk+1 Fk

No existential

variables

Main Recognizable Classes with Finite

Saturation (fes)

Datalog

Acyclic position

dependency graph Weak-

acyclicity

aGRD Acyclic

GRD

Joint-

acyclicity

Acyclic existential

dependency graph

GRD with any

kind of fes s.c.c.

fes-

GRD

[Baget KR 04]

[Baget KR 04]

[Deutsch+ ICDT 03]

[Fagin+ ICDT 03]

[Krötzsch+ IJCAI 11]

Position dependency graph: nodes are positions in predicates

edges show how existential variables are propagated

Graph of Rule Dependencies: nodes are rules

(GRD – also chase graph) edges express that a rule may lead to trigger a rule

Super-

weak

acyclicity

[Marnette PODS 09]

Stratified

chase

graph

Acyclic chase graph

[Deutsch+ PODS 08]

[Deutsch+ PODS 08]

GRD with wa

strongly

connected

components

Semantic condition [Cuenca Grau+ KR 12]

MFA

FINITE UNIFICATION SETS

00 MOIS 2011 EMETTEUR - NOM DE LA PRESENTATION - 15

Remark: If Q1 is more general than Q2 (i.e. Q1 maps to Q2) then Q2 is useless

Def: R is a finite unification set (fus) if, for any query Q,

the set of most general rewritings of Q with R is finite

[ dropping « most general » would weaken the notion]

Finite Query Rewriting (fus)

[Baget+ DL 08, IJCAI 09]

Prop: [König+ RR 2011] When R is fus, there is a unique sound and

complete set of most general rewritings of Q

[unique if each query in the set is made non redundant]

E.g. necessary properties of

concepts / relations

Main Recognizable Classes with Finite Query

Rewriting (fus)

Atomic-

body Sticky

Domain-

restricted

Sticky-

join Each head atom contains

all or none of

the body variables

E.g. concept product

elephant(x) mouse(y) bigger-than(x,y)

[Cali+

PVLDB 2010]

[Baget+ IJCAI 09] [Baget+ IJCAI 09]

[Cali+ RR 10]

= linear Datalog+/- [Cali+ PODS 09]

Body restricted

to a single atom

Restricts multiple

occurrences

of body variables

that do not occur

in all head atoms

Each head atom contains all the

body variables

(GREEDY) BOUNDED

TREEWIDTH SETS

Width of a tree decomposition = max number of nodes in a bag (minus 1)

Treewidth of a graph = min width over all decomposition trees of this graph

r

a

p(a,b) q(b,z0) r(a,b,t0) p(b,t0) q(t0,z1) r(b,t0,t1) p(t0,t1)

Decomposition tree:

1) each node (term) appears in a bag

2) each hyperedge (atom) has all its nodes in a bag

3) for each node x, the subgraph induced by the bags containing x is connected

b

r

t0

z0 a b

node

hyper

edge

Decomposition Tree / Treewidth

p p p

q q

z1

t1

b z0 a b t0

t0 z1 b t0 t1

p(a,b)

q(b,z0)

r(a,b,t0)

p(b,t0)

r(b,t0,t1)

p(t0,t1)

q(t0,z1)

Bounded treewidth and decidability

Definition: A set R of existential rules is a bounded treewidth set (bts) if,

for any fact F, there exists a bound b such that, for any F’ derived from

F, treewidth(F’) < b

Theorem (basically [Cali+, KR 2008]): : If R is bts, then entailment is

decidable.

Proof: Direct consequence of [Courcelles90] + [Thomas88] for

compactness.

Datalog

weakly-acyclic

wa-GRD

jointly

acyclic

acyclic

GRD

w-sticky

w-sticky-join

Inclusion

dependency

atomic body

domain-r

sticky join

sticky

guarded frontier-1

frontier-

guarded

weakly-

guarded

weakly fg

jointly fg

glut fg

FES

FUS

BTS

(Partial) Inclusion Map of Decidable

Classes

An atom in the body

guards all the body

variables

Guard only the frontier

Guard only affected

variables

(i.e.possibly mapped

to new existentials)

Guard only affected variables

from the frontier Frontier: variables shared

by the body and the head

Some Recognizable bts (and not fes) Classes

of Rules

guarde

d [Cali+ KR’08]

weakly

guarded

[Cali+ KR’08] The frontier

has size 1

[Baget+ KR’10]

frontier

guarded

[Baget+ KR’10]

weakly

frontier

guarded

[Baget+ IJCAI’09]

frontier

1

r(x,y) r(y,z) s(x,y,z) r(y,u) r(z,u) r(x,y) r(y,z) r(x,z) r(z,u)

r(x,y) r(y,z)

r(y,u) r(z,u)

datalog

From BTS to GBTS

Recognizability: As fes and fus, bts is an unrecognizable

class.

Algorithms: Can we, as done for fes and fus, present a

generic algorithm for all bts subclasses?

• Problem 1: we have to be able to compute the

bound b = fC(F, R) for each of the bts subclasses.

• Problem 2: even in that case, the enumeration of

every interpretation of treewidth lesser than b (and of

sufficient depth) is not reasonable.

Solution: the GBTS (Greedy BTS) class

The GBTS class: main ideas

Goal: Ensure that we can greedily build a decomposition tree of

bounded width along the chase.

F = B0

T0 = Terms(F) {possible constants}

j(Bi) B1 = j(Bi)

T1 = Terms(B1) T0

B2 = j’(Bi’)

T2 = Terms(B2) T0

j’(Bi’

)

F1

k(Bq)

Is there a bag Bp such that k

maps terms of frontier(Rq) to Tp?

YES

B3 = k(Bq)

T3 = Terms(B3) T0

NO

The procedure halts and FAILS

The GBTS class: main ideas

Definition: A set of existential rules R is gbts if, for any fact F,

for any R-derivation sequence from F, this greedy algorithm

does not return FAILS (in that case, it effectively builds a

decomposition tree of width |F| + max(|Ri|)).

Question 1: Is gbts an « interesting subset » of bts?

Question 2: Is there a generic algorithm for gbts?

Question 3: What is its complexity?

Question 4: Is gbts recognizable?

Which known bts classes are gbts?

Datalog

weakly-

acyclic

wa-

GRD

jointly

acyclic

acyclic

GRD

w-sticky

w-sticky-join

Inclusion

dependency

atomic

body domain-r

sticky join

sticky

guarded

frontier-1

frontier-

guarded

weakly-

guarded

weakly fg

jointly fg

glut fg

FES

FUS

BTS

GBTS

Overview of the gbts algorithm

Step 1: a finite encoding of the chase

Let us suppose that R is gbts

F = B0

T0

B1

T1

B2

T2

B4

T4

B3

T3

B5

T5

B12

T12

B10

T10

B9

T9

B8

T8 B6

T6

B7

T7

B14

T14

Take Bi and Bj created

by a same rule R (resp. by i and j), s.t.

there is a bijection ij between Ti and Tj

with ij (t) = t’ iff t and t’ have been

generated from the same term of R.

B3

T3

B4

T4 3,4

If, moreover, ij determines an

isomorphism between the atoms

of the subtrees respectively

rooted in Bi and Bj

3,4

3,4

Then Bi and Bj are equivalent.

(and so are their children)

4,10

3,10 B10

T10

Consider an equivalence

class of bags.

Choose a (high) representant

B4

T4

Delete subtrees of all others! Blocked tree

Computing the blocked tree

To each bag we associate a pattern

A pattern of a bag (at step p) encodes all ways of mapping

a subset of any rule body to the fact obtained at step p,

while using some terms from this bag.

Equivalence of patterns at some step p implies that copies

(as defined before) under equivalent bags belong to the

chase.

Since there is finitely many non equivalent patterns, that

algorithm halts, and we obtain the blocked tree.

There exists an Atom-Tree

decomposition Qt of Q


Step 2: querying a blocked tree Let us suppose that we have built a finite blocked tree

F = B0

T0

B1

T1

B2

T2

B4

T4

B3

T3

B5

T5

B12

T12

B10

T10

B9

T9

B8

T8 B6

T6

B7

T7

B14

T14

3,4 3,4 4,10

This blocked tree encodes the chase.

Suppose Q maps to the chase.

Choose a set of bags (in the chase) that supports all atoms of Q.

F = B0

T0

B3

T3

B5

T5

B10

T10

B12

T12

Q0

Q3 Q1

0

Q5 Q1

2

B14

T14

Q1

4

3

0

10

5

14

12

There exists an ATD

mapping of Qt

is Valid

A complicated algorithm to check if Q maps to R-chase(F)

For every ATD Qt of Q

• For every ATD-mapping of Qt in chase(F)

If is Valid in chase(F)

Return YES

Return NO

Problem: we don’t have the chase, only the blocked tree…

For every ATD Qt of Q

• For every ATD-mapping of Qt in the blocked tree

If is Valid in the blocked tree

Return YES

Return NO


Step 2: querying a blocked tree


Step 2: querying a blocked tree We have to find an ATD mapping in the blocked tree…

F = B0

T0

B1

T1

B2

T2

B4

T4

B3

T3

B12

T12

B10

T10

B8

T8 B6

T6

B14

T14

3,4 3,4 4,10

F = B0

T0

B3

T3

B5

T5

B10

T10

B12

T12

Q0

Q3 Q1

0

Q5 Q1

2

B14

T14

Q1

4

3

0

10

5

14

12

14, 8 o 14

12, 6 o 12

5, 6 o 5


Step 2: querying a blocked tree … and prove it corresponds to a Valid one in one completion.

F = B0

T0

B1

T1

B2

T2

B4

T4

B3

T3

B10

T10

B8

T8 B6

T6

3,4 3,4 4,10

F = B0

T0

B3

T3

B5

T5

B10

T10

B12

T12

Q0

Q3 Q1

0

Q5 Q1

2

B14

T14

Q1

4

3

0

10

5

14

12

14, 8 o 14

12, 6 o 12

5, 6 o 5

Q0

Q3 Q1

0

Q5 Q1

2

Q1

4


Step 2: querying a blocked tree

Validation is backtrack-free

Generating the required bag is done at most in b ff steps

Overall complexity of the querying step of the algorithm

• 2q bq q b ff, where b is the size of the blocked tree

• The size of b is a double exponential in F and R

Complexity results

On the recognizability of gbts

F Body Head B0

F’

R

R’

Rnew: in(x, B0), in(y, z) -> in(x, z)

Suppose that there is a derivation from (F, R) that fails.

• There is a derivation R-derivation from F to G that does not fail,

and a failing mapping from some body B to G • There is a R’-derivation from F’ to G’,

and a failing mapping from some body B to G’ • There is a failing validation of some body B to the blocked tree of (F’, R’)

If a derivation from (F, R) fails, then a derivation from (U, R) fails

ON FORWARD AND

BACKWARD CHAININGS

- 36

FORWARD & BACWARD CHAININGS

Jean-François Baget – Derivation and Rewriting Lengths - 37

F1 F2

1 F3

2 Fn-1 Fn

n-1

Qn

n

…

F1

Q2 Q1

1

1 Q3

2 Qn-1 Qn n-1 …

Theorem [Salvat 96]: the following assertions are

equivalent.

• Q is semantic consequence of F and R

• Q maps to some F’ finitely R-derived from F

• Some finite R-rewriting Q’ of Q maps to F

FINITE EXPANSION & UNIFICATION SETS


INITIAL MISUNDERSTANDING

Definition [Cali & al. 2010]: A set of existential rules R has the bounded

derivation-depth property (BDDP) iff, for every facts F and Q, whenever Q can be deduced from F and R, Q is entailed by the R-saturation of F at rank ,

where = f(R, Q)

A formulation, expressed in Forward Chaining,

that is very similar to FES / BTS

BUT Known BDDP classes are FUS, not FES nor BTS !

Jean-François Baget – Derivation and Rewriting Lengths

OVERLOOKED LEMMAS [Salvat, 96]


F1 F2

1

Q2

2

Q1

1

1

F1 F2

1

Q2

2

Q1

1

1

F3

2

Q3

3

2

Fn-1

Qn-1

n-1

Fn

n-1

Qn

n

n-1

F3 2

Q3

3

2

Fn-1

Qn-1

n-1

Fn

n-1

Qn

n

n-1

…

… F1 F2

1

Q2

2

Q1

1

1

F1 F2

1

Q2

2

Q1

1

1

BOUNDED DERIVATION DEPTH & FUS (1)


Definition [Cali & al. 2010]: A set of existential rules R has the (Q) bounded

derivation-depth property (QBDDP) iff, for every facts F and Q, whenever Q can be deduced from F and R, Q is entailed by the R-saturation of F at rank ,

where = f(R, Q)

Property: If R has the QBDDP, then R is f.u.s. Moreover, for any Q, the depth

of the R-rewriting tree of Q is N (where N is the max size of Q and rule bodies).

Q’ Q

Q’

Q’

Q Q’’ For any Q’ rewritable from Q,

there is a Q’’ in the rewriting tree

of depth N

that is more general than Q’.

BOUNDED DERIVATION DEPTH & FUS (2)


Definition [Cali & al. 2010]: A set of existential rules R has the (Q)

bounded derivation-depth property (QBDDP) iff, for every facts F and Q, whenever Q can be deduced from F and R, Q is entailed by the R-saturation

of F at rank , where = f(R, Q)

Property: If R is f.u.s., then R has the QBDDP.

Q

= f(R, Q)

Q’ Q

F

Suppose Q can be deduced from F and R

Q’

BOUNDED DERIVATION DEPTH & FES


Definition [Oxford 2012]: A set of existential rules R has the (F) bounded

derivation-depth property (FBDDP) iff, for every facts F and Q, whenever Q can be deduced from F and R, Q is entailed by the R-saturation of F at

rank , where = f(R, F)

Property: R is f.e.s.iff R has the FBDDP.

() ()

F F1 F (UM) …

= f(R, F)

Q

F F

…

= f(R, F)

Q F+1

CONCLUSION (1)


Alternate Definition: A set of existential rules R has the bounded

deduction length property (BDLP) iff, for every facts F and Q, whenever Q can be deduced from F and R, there exists s.t. one of the following

equivalent assertions holds: • Q is entailed by an R-derivation of F of length .

• F entails an R-rewriting of Q of length .

Moreover: • If = f(F, R), we say that R has the Fact BDLP (FBDLP)

• If = f(Q, R), we say that R has the Query BDLP (QBDLP)

CONCLUSION (2)


Thank you

[email protected]

mailto:[email protected]

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