Extending OWL with Integrity Constraints

Extending OWL with Integrity Constraints

Jiao Tao1 , Evren Sirin2 , Jie Bao1 , Deborah L. McGuinness1
taoj2@cs.rpi.edu, evren@clarkparsia.com
baojie@cs.rpi.edu, dlm@cs.rpi.edu
1 Tetherless World Constellation,

Department of Computer Science,
Rensselaer Polytechnic Institute,
Troy, NY 12180, USA.
2 Clark & Parsia,

Washington, DC 20001, USA.

The 23rd International Workshop on Description Logics (DL 2010)

Tao,Sirin,Bao,McGuinness (RPI,Clark Parsia) Extending OWL with Integrity Constraints May 2010, DL 2010 1 / 20

Outline

Background
OWA and nUNA in OWL
IC Requirements for OWL
Related Work
Contributions
Preliminaries
SROIQ
DCQ
IC Semantics for OWL
Epistemic Approach
Formalization of IC Semantics
IC Validation
DCQnot
From ICs to DCQnot
IC Validation Algorithm
Implementation
Conclusions and Future Work

Background OWA and nUNA in OWL

OWA and nUNA in OWL

Open World Assumption (OWA)
CWA vs. OWA: any statement that is not known to be true is false vs.
unknown
Example 1: K satisﬁes α although p does not have a known producer.
K = {Product(p)}, α : Product ∃hasProducer.Producer



OWA and nUNA in OWL

unknown
nonUnique Name Assumption (nUNA)
UNA vs. nUNA: two different names always refer to different entities
vs. may refer to the same entity
Example 2: K satisfies α although p has two producers.
K = {Product(p), hasProducer(p, m1 ), hasProducer(p, m2 )}
α : Product ≤ 1hasProducer. .



OWA and nUNA in OWL

unknown
nonUnique Name Assumption (nUNA)
UNA vs. nUNA: two different names always refer to different entities
vs. may refer to the same entity
Example 2: K satisfies α although p has two producers.
K = {Product(p), hasProducer(p, m1 ), hasProducer(p, m2 )}
α : Product ≤ 1hasProducer. .
Difficult to use OWL for integrity constraint(IC) validation!


Background IC Requirements for OWL


Strong needs to use OWL as an IC language! Especially in hybrid
settings where OWL KBs are integrated with relational databases and
ICs are needed to enforce the named individuals in KBs to have some
known values.




known values.
Typing Constraints
Individuals participating in a relation should be instances of certain
types, i.e., domain&range axioms and universal restrictions.




known values.
Typing Constraints
Participation Constraints
Instances of the constrained class should have a role assertion i.e.,
existential restrictions.




known values.
Typing Constraints
Participation Constraints
Instances of the constrained class should have a role assertion i.e.,
existential restrictions.
Uniqueness Constraints
An individual cannot participate in multiple role assertions with the
same role, i.e., functional property restrictions, cardinality restrictions.


Background Related Work

Related Work

Rule-based approach [4, 6]
⊥ ← DL[Product](x), not P(x, y )
P(x, y ) ← DL[hasProducer](x, y ), DL[Producer](y )
IC validation: KDL , ICRule |= ⊥
Limitations: ontology developers need to be familiar with rules



Related Work

Epistemic query-based approach [1]
KProduct(x) → ∃y .(KhasProducer(x, y ) ∧ KProducer(y ))
IC validation: Ans(ICEQL−Lite , KFOL )
Limitations: complexity results in expressive DLs are unknown



Related Work

Epistemic query-based approach [1]
KProduct(x) → ∃y .(KhasProducer(x, y ) ∧ KProducer(y ))
IC validation: Ans(ICEQL−Lite , KFOL )
Limitations: complexity results in expressive DLs are unknown
Approach based on epistemic extension of DLs [2, 3]
KProduct ∃KhasProducer.KProducer
IC validation: KDL |= ICEDL
Limitations: less expressive DLs, strict UNA


Background Related Work cont.

Related Work cont.

Approach reuses OWL as an IC language



Related Work cont.

IC validation: T , A |=MM C



Related Work cont.

Limitations:



Related Work cont.

Limitations:
Unintuitive
Example 4: α is satisﬁed although the producer of p is unknown.
K = {Product(p), ∃hasProducer.Producer(p)}
α : Product ∃hasProducer.Producer



Related Work cont.

Limitations:
Unintuitive
Example 4: α is satisﬁed although the producer of p is unknown.
K = {Product(p), ∃hasProducer.Producer(p)}
α : Product ∃hasProducer.Producer
Maintenance burden
Example 5: α is satisﬁed although we are not sure if the producer of p
is m1 or m2 .
K = {Product(p), (∃hasProducer.{m1 , m2 })(p), Producer(m1 ),
Producer(m2 ), O(p), O(m1 ), O(m2 )}
α : Product ∃hasProducer.(Producer O)


Background Contributions

Contributions

The main contributions of this paper include:
An IC semantics for OWL
A sound and complete IC validation algorithm
A prototype implementation


Preliminaries SROIQ

Description Logics SROIQ [5]

SROIQ-interpretation I = (∆, ·I ): ·I maps a concept C to a subset
of ∆, a role R to a subset of ∆ × ∆, a named individual NI to an
element of ∆.


Preliminaries SROIQ


element of ∆.
I satisﬁes an axiom α, denoted I |= α, if
Type Axiom Condition on I
TBox C D C I ⊆ DI
R1 R2 I
R1 ⊆ R2 I

R1 . . . Rn R I ◦ . . . ◦ RI ⊆ RI
R1 n
RBox Ref(R) ∀x ∈ ∆ : x, x ∈ R I
Irr(R) ∀x ∈ ∆ : x, x ∈ R I
Dis(R1 , R2 ) I I
R1 ∩ R2 = ∅


Preliminaries SROIQ


element of ∆.
TBox C D C I ⊆ DI
R1 R2 I
R1 ⊆ R2 I

R1 . . . Rn R I ◦ . . . ◦ RI ⊆ RI
R1 n
Irr(R) ∀x ∈ ∆ : x, x ∈ R I
Dis(R1 , R2 ) I I
R1 ∩ R2 = ∅

I is a model of K if it satisﬁes all the axioms in K.


Preliminaries SROIQ


element of ∆.
TBox C D C I ⊆ DI
R1 R2 I
R1 ⊆ R2 I

R1 . . . Rn R I ◦ . . . ◦ RI ⊆ RI
R1 n
Irr(R) ∀x ∈ ∆ : x, x ∈ R I
Dis(R1 , R2 ) I I
R1 ∩ R2 = ∅

I is a model of K if it satisﬁes all the axioms in K.
K entails α, written as K |= α, if I |= α for all models I ∈ Mod(K).


Preliminaries DCQ

Distinguished Conjunctive Query (DCQ)

Syntax:
Query atom: q ← C (x) | R(x, y ) | ¬R(x, y ) | x = y | x = y
Conjunctive Query (CQ): Q ← q | Q1 ∧ Q2
Distinguished Conjunctive Query (DCQ): CQ containing only
distinguished variables1

1
A distinguished variable can be mapped to only known individuals, i.e., an element
from NI .

Preliminaries DCQ

Distinguished Conjunctive Query (DCQ)

Syntax:
Query atom: q ← C (x) | R(x, y ) | ¬R(x, y ) | x = y | x = y
Conjunctive Query (CQ): Q ← q | Q1 ∧ Q2
Distinguished Conjunctive Query (DCQ): CQ containing only
distinguished variables1
Semantics:
Given a KB K, a DCQ Q, and an assignment σ : NV → NI , we say
that K |=σ Q if:

K |=σ q iﬀ K |= σ(q)
σ
K |= Q1 ∧ Q2 iﬀ K |=σ Q1 and K |=σ Q2

where σ(Q) denotes the application of σ to Q.
Ans(Q, K) = {σ | K |=σ Q}
K |= Q if Ans(Q, K) = ∅
1
A distinguished variable can be mapped to only known individuals, i.e., an element
from NI .

IC Semantics for OWL Epistemic Approach

Epistemic Approach

ICs are epistemic in nature [8]: about what the KB knows.



Epistemic Approach

We adopt this epistemic view of ICs and propose an IC semantics for
OWL, which is similar to how the semantics of epistemic DL ALCK
[2] and MKNF DL ALCKN F [3] are deﬁned, but is still diﬀerent from
them:



Epistemic Approach

We adopt this epistemic view of ICs and propose an IC semantics for
OWL, which is similar to how the semantics of epistemic DL ALCK
[2] and MKNF DL ALCKN F [3] are deﬁned, but is still diﬀerent from
them:
weak UNA vs. strict UNA
expressive DLs, i.e., SROIQ, vs. less expressive DLs, i.e., ALC.


IC Semantics for OWL Formalization of IC Semantics

IC-interpretation

An IC-interpretation is a pair I, U where I is a SROIQ
interpretation deﬁned over ∆I and U is a set of SROIQ
interpretations.



IC-interpretation

interpretations.
IC-interpretations of concepts, roles, individuals:

C I,U = {x I | x ∈ NI s.t. ∀J ∈ U, x J ∈ C J }
R I,U = { x I , y I | x, y ∈ NI s.t. ∀J ∈ U, x J , y J ∈ R J }
(R − )I,U = { x I , y I | y I , x I ∈ R I,U }
(C D)I,U = C I,U ∩ D I,U , (¬C )I,U = NI C I,U
(≥ nR.C )I,U = {x I | x ∈ NI s.t. #{y I | x I , y I ∈ R I,U , y I ∈ C I,U } ≥ n}
(∃R.Self)I,U = {x I | x ∈ NI s.t. x I , x I ∈ R I,U }, {a}I,U = {aI }.



IC-interpretation

interpretations.
IC-interpretations of concepts, roles, individuals:

C I,U = {x I | x ∈ NI s.t. ∀J ∈ U, x J ∈ C J }
R I,U = { x I , y I | x, y ∈ NI s.t. ∀J ∈ U, x J , y J ∈ R J }
(R − )I,U = { x I , y I | y I , x I ∈ R I,U }
(C D)I,U = C I,U ∩ D I,U , (¬C )I,U = NI C I,U
(≥ nR.C )I,U = {x I | x ∈ NI s.t. #{y I | x I , y I ∈ R I,U , y I ∈ C I,U } ≥ n}
(∃R.Self)I,U = {x I | x ∈ NI s.t. x I , x I ∈ R I,U }, {a}I,U = {aI }.

IC-interpretations use the CWA w.r.t. NI (standard DL
interpretations: OWA).



Weak UNA

Weak UNA: two named individuals with different identifiers are
assumed to be different by default unless their equality is required to
satisfy the axioms in the KB.



Weak UNA

How weak UNA is formalized?



Weak UNA

J = I if:
For every concept C, J |= C (a) implies I |= C (a);
For every role R, J |= R(a, b) implies I |= R(a, b);
EJ ⊂ EI
where EI = { a, b | a, b ∈ NI s.t. I |= a = b}.



Weak UNA

J = I if:
EJ ⊂ EI
ME models: the models of K with minimal equality (ME) between NI :
ModME (K) = {I ∈ Mod(K) | J , J ∈ Mod(K), J = I}



Weak UNA

J = I if:
EJ ⊂ EI
ME models: the models of K with minimal equality (ME) between NI :
ModME (K) = {I ∈ Mod(K) | J , J ∈ Mod(K), J = I}
IC-semantics uses the weak UNA (standard DL semantics: nUNA).



Axiom Satisfaction, IC Satisfaction, Extended KB
An IC-interpretation I, U satisﬁes an axiom α, denoted as I, U |= α,if
Type Axiom Condition on I, U
TBox C D C I,U ⊆ D I,U
I,U I,U
R1 R2 R1 ⊆ R2
I,U I,U
R1 . . . Rn R R1 ◦ . . . ◦ Rn ⊆ R I,U
RBox Ref(R) ∀x ∈ NI : x I,U , x I,U ∈ R I,U
Irr(R) ∀x ∈ NI : x I,U , x I,U ∈ R I,U
I,U I,U
Dis(R1 , R2 ) R1 ∩ R2 = ∅



I,U I,U
R1 R2 R1 ⊆ R2
I,U I,U
R1 . . . Rn R R1 ◦ . . . ◦ Rn ⊆ R I,U
I,U I,U
Dis(R1 , R2 ) R1 ∩ R2 = ∅
The IC-satisfaction of α by K, i.e., K |=IC α, is deﬁned as:
K |=IC α iﬀ ∀I ∈ U, I, U |= α, where U = ModME (K)



I,U I,U
R1 R2 R1 ⊆ R2
I,U I,U
R1 . . . Rn R R1 ◦ . . . ◦ Rn ⊆ R I,U
I,U I,U
Dis(R1 , R2 ) R1 ∩ R2 = ∅
The IC-satisfaction of α by K, i.e., K |=IC α, is deﬁned as:
K |=IC α iﬀ ∀I ∈ U, I, U |= α, where U = ModME (K)
An extended KB is a pair K, C where K is a SROIQ KB
interpreted with the standard OWL semantics and C is a set of
SROIQ axioms interpreted with IC semantics. We say that K, C is
valid if ∀α ∈ C, K |=IC α.

IC Validation DCQnot

Distinguished Conjunctive Query with not (DCQnot )

DCQnot is the extension of DCQ with the NAF operator not .


IC Validation From ICs to DCQnot

From ICs to DCQnot

Translation from ICs to DCQnot : translate an IC axiom α into a
query Q such that when α is satisﬁed, Q is false.


IC Validation From ICs to DCQnot

From ICs to DCQnot

Translation from ICs to DCQnot : translate an IC axiom α into a
query Q such that when α is satisﬁed, Q is false.
Translation rules:
Concept Translation:

Tc (Ca , x) := Ca (x)
Tc (¬C , x) := not Tc (C , x)
Tc (C1 C2 , x) := Tc (C1 , x) ∧ Tc (C2 , x)
Tc (≥ nR.C , x) := (R(x, yi ) ∧ Tc (C , yi )) not (yi = yj )
1≤i≤n 1≤i<j≤n

Tc (∃R.Self, x) := R(x, x)
Tc ({a}, x) := (x = a)


IC Validation From ICs to DCQnot cont.

From ICs to DCQnot cont.
Translation rules cont.:
Axiom Translation:
T (C1 C2 ) := Tc (C1 , x) ∧ not Tc (C2 , x)
T (R1 R2 ) := R1 (x, y ) ∧ not R2 (x, y )
T (R1 . . . Rn R) := R1 (x, y1 ) ∧ . . . Rn (yn−1 , yn ) ∧ not R(x, yn )
T (Ref(R)) := not R(x, x)
T (Irr(R)) := R(x, x)
T (Dis(R1 , R2 )) := R1 (x, y ) ∧ R2 (x, y )


IC Validation From ICs to DCQnot cont.

From ICs to DCQnot cont.
Translation rules cont.:
Axiom Translation:
T (C1 C2 ) := Tc (C1 , x) ∧ not Tc (C2 , x)
T (R1 R2 ) := R1 (x, y ) ∧ not R2 (x, y )
T (R1 . . . Rn R) := R1 (x, y1 ) ∧ . . . Rn (yn−1 , yn ) ∧ not R(x, yn )
T (Ref(R)) := not R(x, x)
T (Irr(R)) := R(x, x)
T (Dis(R1 , R2 )) := R1 (x, y ) ∧ R2 (x, y )
Example 6:
T (Product ∃hasProducer.Producer)
:= Tc (Product, x) ∧ not Tc (∃hasProducer.Producer, x)
:= Product(x) ∧ not (hasProducer(x, y ) ∧ Tc (Producer, y ))
:= Product(x) ∧ not (hasProducer(x, y ) ∧ Producer(y ))


IC Validation IC Validation Algorithm


Theorem: Given an extended KB K, C with expressivity SRI, SROIQ
( SROIQ, SROI resp.), we have that K |=IC α iﬀ K |= T (α) where
α ∈ C [9].




α ∈ C [9].

To limit the interactions between the disjunctive (in)equality axioms in K
and the cardinality restrictions in C, we can require:




α ∈ C [9].

To limit the interactions between the disjunctive (in)equality axioms in K
and the cardinality restrictions in C, we can require:
K: within SRI, no nominals & cardinality restrictions thus no
disjunctive (in)equality;
C: within SROI, no cardinality restrictions.


Implementation

Implementation

Prototype: IC validator 2 in the OWL 2 DL reasoner Pellet.
SPARQL-based [7] implementation.

2
http://clarkparsia.com/pellet/oicv-0.1.2.zip

Implementation

Implementation

The validation process is as follows:
Reading OWL IC axioms
Translating to DCQnot
Translating to SPARQL query
Executed by the SPARQL query engine

2

Implementation

Implementation

The validation process is as follows:
Reading OWL IC axioms
Translating to DCQnot
Translating to SPARQL query
Executed by the SPARQL query engine
Preliminary evaluation:
Dataset: LUBM(5), 100K individuals, 800K ABox axioms
IC validation time: 2s
Logical consistency checking time: 10s

2



In this paper, we have
described an IC semantics for OWL
presented a sound and complete IC validation algorithm
described the prototype IC validator




In this paper, we have
described an IC semantics for OWL
presented a sound and complete IC validation algorithm
described the prototype IC validator
In the future, we will:
look at the performance of IC validation in real-world datasets
explore the IC validation algorithms for the fullly expressive DLs


Questions

Questions?

Thanks for your attention!


Questions

D. Calvanese, G. D. Giacomo, D. Lembo, M. Lenzerini, and R. Rosati.
EQL-Lite: Eﬀective First-Order Query Processing in Description
Logics.
In Proc. IJCAI2007, pages 274–279, 2007.
F. M. Donini, M. Lenzerini, D. Nardi, W. Nutt, and A. Schaerf.
An Epistemic Operator for Description Logics.
AI, 100(1–2):225–274, 1998.
F. M. Donini, D. Nardi, and R. Rosati.
Description Logics of Minimal Knowledge and Negation as Failure.
ACM Trans. Comput. Logic, 3(2):177–225, 2002.
T. Eiter, G. Ianni, T. Lukasiewicz, R. Schindlauer, and H. Tompits.
Combining Answer Set Programming with Description Logics for the
Semantic Web.
AI, 172(12-13):1495–1539, 2008.
I. Horrocks, O. Kutz, and U. Sattler.
The Even More Irresistible SROIQ.

Questions

In Proc. KR2006, pages 57–67, 2006.
B. Motik.
A Faithful Integration of Description Logics with Logic Programming.
In Proc. IJCAI2007, pages 477–482, 2007.
E. Prud’hommeaux and A. Seaborne.
SPARQL Query Language for RDF, 2008.
R. Reiter.
On Integrity Constraints.
In Proc. TARK1988, pages 97–111, 1988.
J. Tao, E. Sirin, J. Bao, and D. McGuinness.
Integrity Constraints in OWL.
Technical report, Dept. of Computer Science, Rensselaer Polytechnic
Institute, Troy, NY., 2010.


Extending OWL with Integrity Constraints

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