DL-LITER in the Light of Propositional Logic for Decentralized Data Management.
ABSTRACT This paper provides a decentralized data model and associated algorithms for peer data manage- ment systems (PDMS) based on the DL-LITER de- scription logic. Our approach relies on reducing query reformulation and consistency checking for DL-LITER into reasoning in propositional logic. This enables a straightforward deployment of DL- LITER PDMSs on top of SomeWhere, a scalable propositional peer-to-peer inference system. We also show how to use the state-of-the-art Minicon algorithm for rewriting queries using views in DL- LITER in the centralized and decentralized cases.
- [Show abstract] [Hide abstract]
ABSTRACT: Les systèmes d'inférence pair-à-pair (P2PIS) sont constitués de serveurs autonomes appelés pairs. Chaque pair gère sa propre base de connaissances (BC) et peut communiquer avec les autres pairs via des mappings afin de réaliser une inférence particulière au niveau global du système. Dans la première partie de la thèse, nous proposons un algorithme totalement décentralisé de calcul de conséquences par déduction linéaire dans les P2PIS propositionnels et nous étudions sa complexité communicationnelle, en espace et en temps. Nous abordons ensuite la notion d'extension non conservative d'une BC dans les P2PIS et nous exhibons son lien théorique avec la déduction linéaire décentralisée. Cette notion est importante car elle est liée à la confidentialité des pairs et à la qualité de service offerte par le P2PIS. Nous étudions donc les problèmes de décider si un P2PIS est une extension conservative d'un pair donné et de calculer les témoins d'une possible corruption de la BC d'un pair de sorte à pouvoir l'empêcher. La seconde partie de la thèse est une application directe des P2PIS au domaine des systèmes de gestion de données pair-à-pair (PDMS) pour le Web Sémantique. Nous définissons des PDMS basés sur la logique de description DL-LITER pour lesquels nous fournissons les algorithmes nécessaires de test de consistance et de réponse aux requêtes. Notre approche repose sur les P2PIS propositionnels car nous réduisons les problèmes de la reformulation des requêtes et de test de l'inconsistance à des problèmes de calcul de conséquences en logique propositionnelle. - [Show abstract] [Hide abstract]
ABSTRACT: This paper points out that the notion of non-conservative extension of a knowledge base (KB) is important to the distributed logical setting of peer-to-peer inference systems (P2PIS), a.k.a. peer-to-peer semantic systems. It is useful to a peer in order to detect/prevent that a P2PIS corrupts (part of) its knowledge or to learn more about its own application domain from the P2PIS. That notion is all the more important since it has connections with the privacy of a peer within a P2PIS and with the quality of service provided by a P2PIS. We therefore study the following tightly related problems from both the theoretical and decentralized algorithmic perspectives: (i) deciding whether a P2PIS is a conservative extension of a given peer and (ii) computing the witnesses to the corruption of a given peer's KB within a P2PIS so that we can forbid it. We consider here scalable P2PISs that have already proved useful to Artificial Intelligence and DataBases.Ai Communications 01/2009; 22:211-233. · 0.45 Impact Factor
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DL-LITERin the Light of Propositional Logic for Decentralized Data Management
N. Abdallah and F. Goasdou´ e
LRI: Univ. Paris-Sud, CNRS, and INRIA
{nada,fg}@lri.fr
M.-C. Rousset
LIG: Univ. of Grenoble, CNRS, and INRIA
Marie-Christine.Rousset@imag.fr
Abstract
This paper provides a decentralized data model
and associated algorithms for peer data manage-
ment systems (PDMS) based on the DL-LITERde-
scription logic. Our approach relies on reducing
query reformulation and consistency checking for
DL-LITER into reasoning in propositional logic.
This enables a straightforward deployment of DL-
LITERPDMSs on top of SomeWhere, a scalable
propositional peer-to-peer inference system. We
also show how to use the state-of-the-art Minicon
algorithm for rewriting queries using views in DL-
LITERin the centralized and decentralized cases.
1Introduction
Ontologies are the backbone of the Semantic Web by pro-
viding a conceptual view of data and services made available
worldwide through the Web. Description logics are the for-
mal foundations of the OWL ontology web language recom-
mendedbyW3C. Theycovera broadspectrumof logical lan-
guagesforwhichreasoningis decidablewith a computational
complexity varying depending on the set of constructors al-
lowed in the language. Answering conjunctive queries over
ontologiesis a reasoningproblemof majorinterest fortheSe-
mantic Web the associated decision problem of which is not
reducible to (un)satisfiability checking. The DL-Lite family
[Calvanese et al., 2007]has been specially designed for guar-
anteeing query answering to be polynomial in data complex-
ity. Thisis achievedbyaqueryreformulationapproachwhich
(1) computes the most general conjunctive queries which, to-
getherwiththeaxiomsintheTbox,entailtheinitialqueryand
(2) evaluates each of those query reformulations against the
Abox seen as a relational database. Such an approach has the
practical interest that it makes possible to use an SQL engine
for the second step, thus taking advantageof well-established
queryoptimizationstrategiessupportedbystandardrelational
data management systems. The reformulation step is neces-
sary for guaranteeing the completeness of the answers but is
a reasoning step independent of the data. A major result in
[Calvanese et al., 2007]is that DL-LITERis one of the max-
imal fragments of the DL-Lite family supporting tractable
query answering over large amounts of data. DL-LITERis a
fragment of OWL-DL1which extends RDFS2with interest-
ingcontructorssuch as inverseroles anddisjointnessbetween
concepts and between roles. RDFS is the first standard of the
W3C concerning the Semantic Web. Its use for associating
semantic metadata to web resources is rapidly spreading at a
large scale, as shown by the Billion Triple Track of the Se-
mantic Web Challenge (http://challenge.semanticweb.org/).
For scalability and robustness but also for data protection,
it is important to investigate a fully decentralized model of
the Semantic Web, viewed as a huge peer data management
system (PDMS). Each peer may have its own local ontology
for describing its data, and interacts with some other peers by
establishing mappings with their ontologies. The result is a
network of peers with no centralized knowledge and thus no
global control on the data and knowledge distributed over the
web.
Thecontributionofthis paperis a decentralizeddatamodel
andassociatedalgorithmsfordatamanagementintheSeman-
tic Web based on distributed DL-LITER. We revisit the cen-
tralized current approach of [Calvanese et al., 2007]for data
consistency checking and query answering by reformulation
in order to design corresponding decentralized algorithms.
We also extend the current work on DL-Lite by providing
both a centralized and a decentralized algorithm for rewriting
queries using views when queries and views are conjunctive
queries over DL-LITERontologies.
Our approach relies on reducing the above data manage-
mentproblemsfor DL-LITERintodecentralizedreasoningin
distributed propositional logic, in order to deploy DL-LITER
PDMSs on top of the SomeWhere platform. SomeWhere is
a propositional P2P inference system for which experiments
have demonstrated the scalability[Adjiman et al., 2006].
The paper is organizedas follows. In Section 2, we present
the distributed DL-LITER data model which is based on
bridgingdistributed DL-LITERontologieswith mappings. In
Section 3, we provide decentralized algorithms for query an-
swering by reformulation and for data consistency checking.
In Section 4, we investigate the problem of query rewriting
using views in DL-LITERin the centralized and decentral-
ized cases. Finally, we conclude with a short discussion on
related work in Section 5.
1http://www.w3.org/2004/OWL/
2http://www.w3.org/TR/rdf-schema/
2010
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2Distributed DL-LITER
DL-LITERconcepts and roles are of the following form:
B → A | ∃R, C → B | ¬B, R → P | P−, E → R | ¬R
where A denotes an atomic concept, P an atomic role, and
P−the inverse of P. B denotes a basic concept (i.e., an
atomic concept A or an unqualified existential quantification
on a basic role ∃R) and R a basic role (i.e., an atomic role
P or its inverse P−). Finally, C denotes a general concept
(i.e., a basic concept or its negation) and E a general role
(i.e., a basic role or its negation).
An interpretation I = (ΔI,.I) consists of a nonempty in-
terpretation domain ΔIand an interpretation function .Ithat
assigns a subset of ΔIto each atomic concept, and a binary
relation over ΔIto each atomic role. The semantics of non
atomic concepts and roles is defined as follows:
(P−)I= {(o2,o1) | (o1,o2) ∈ PI}
(∃R)I= {o1| ∃o2(o1,o2) ∈ RI}
(¬B)I= ΔI\BIand (¬R)I= ΔI× ΔI\RI
An interpretationI is a model of a concept C (resp. a role E)
if CI?= ∅ (resp. EI?= ∅).
DL-LITER knowledge bases.
base (KB) is made of a Tbox representing a conceptual view
of the domain of interest (i.e., an ontology), and either an
Abox (a local set of facts)[Calvanese et al., 2007]or view ex-
tensions (predefinedqueriesoverthe Tboxtogetherwith their
answers)[Calvanese et al., 2008b]for representing the data.
A DL-LITERTbox is a finite set of inclusion statements of
the form B ? C and/or R ? E. General concepts or roles
are only allowed on the right hand side of inclusion state-
ments whereas only basic concepts or roles may occur on the
left hand side of inclusion statements. Inclusions of the form
B1? B2or of the form R1? R2are called positive inclu-
sions (PIs), whereas inclusions of the form B1? ¬B2or of
the form R1? ¬R2are called negative inclusions (NIs). An
interpretation I = (ΔI,.I) is a model of an inclusion B ? C
(resp. R ? E) if BI⊆ CI(resp. RI⊆ EI). It is a model of
a Tbox if it satisfies all of its inclusion statements. A Tbox T
logically entails an inclusion statement α, written T |= α, if
every model of T is a model of α.
A DL-LITERAbox consists of a finite set of membership
assertions on atomic concepts and roles of the form A(a) and
P(a,b), stating respectivelythat a is an instanceofA andthat
the pair of constants (a,b) is an instance of P. The interpre-
tation function of an interpretation I = (ΔI,.I) is extended
to constants by assigning to each constant a a distinct object
aI∈ ΔI(i.e., the so called unique name assumption holds).
An interpretation I is a model of the membership assertion
A(a) (resp. P(a,b)) if aI∈ AI(resp., (aI,bI) ∈ PI). It is a
model of an Abox if it satisfies all of its assertions.
When the extensional knowledge is modeled using view
extensions, the KB is of the form ?T ,V,E? such that E is a
set of facts of the form v(¯t) where v is a view of V.
A DL-LITER knowledge
Queries and views over a DL-LITER KB.
sider (unions of) conjunctive queries of the form q(¯ x) :
∃¯ y conj(¯ x, ¯ y) where conj(¯ x, ¯ y) is a conjunction of atoms,
the variables of which are only the free variables ¯ x and the
existential variables ¯ y, and the predicates of which are either
We con-
atomic conceptsor roles of the KB. The arity of a queryis the
number of its free variables, e.g., 0 for a boolean query.
Given an interpretation I = (ΔI,.I), the semantics qIof
a boolean query q is defined as true if [∃¯ y conj(∅, ¯ y)]I=
true, and false otherwise, while the semantics qIof a query
q of arity n ≥ 1 is the relation of arity n defined on (ΔI) as
follows: qI= {¯ e ∈ (ΔI)n| [∃¯ y conj(¯ e, ¯ y)]I= true}.
A view v is defined by a query v(¯ x) : ∃¯ y conj(¯ x, ¯ y), and
has an extension E(v) which is a set of facts of the form v(¯t).
Following the open world assumption, we adopt the sound
semantics, i.e., for every interpretation I for each v(¯t) ∈
E(v),¯tI∈ vI.
A model of a KB K = ?T ,A? (resp. K = ?T ,V,E?) is an
interpretation I that is a model of both T and A (resp. of T ,
V and E). A KB K is consistent if it has at least one model.
K logically entails a membership assertion β, written K |=
β, if every model of K is a model of β.
(Certain) answers of a query over a DL-LITERKB.
defining the answers of a query over a KB, it is needed to
distinguish the case where the extensions of the query predi-
cates are given in an Abox, from the case where they just can
be (partially) inferred from extensions of views. In the latter
case, they are called the certain answers.
The answer set of a non boolean query q over K = ?T ,A?
is defined as: ans(q,K) = {¯t ∈ Cn| K |= q(¯t)} where C is
the set of the constants appearing in the KB, and q(¯t) is the
closed formula obtained by replacing in the query definition
the free variables in ¯ x by the constants in¯t.
The certain answer set of a non boolean query q over K =
?T ,V,E?, is defined as: cert(q,K) = {¯t ∈ Cn| K |= q(¯t)}.
By convention, the (certain) answer set of a boolean query
is {()}, () is the empty tuple, if K |= q(), and ∅ otherwise.
For
DL-LITERPDMSs
{Pi}i=1..n, where the index i models the identifier of the
peer Pi(e.g., its IP address). Each peer Pimanages its own
DL-LITERKB Kiwritten in terms of its own vocabulary,
i.e., atomic concepts and roles. We will note Ai(resp. Pi) the
atomic concept A (resp. the atomic role P) of Pi.
Mappings are here inclusion assertions (PIs and/or NIs) in-
volving concepts and/or roles of two different peers. For sim-
plifying the presentation, we consider that mappings are in
both KBs.
From a logical viewpoint, a PDMS S = {Pi}i=1..n is
a standard (yet distributed) DL-LITERKB K =
i.e., in contrast with other approaches ([Calvanese et al.,
2008a], [Franconi et al., 2004], [Serafini et al., 2005]) we
adopt a standard logical semantics for the mappings.
A DL-LITERPDMS S is a set of peers
?n
i=1Ki,
3
We first recall the Answer, Consistent, and PerfectRef
algorithms of [Calvanese et al., 2007] that are used for an-
swering queries over a DL-LITERKB K = ?T ,A? in the
centralized case (Section 3.1). Then we provide their decen-
tralized versions (in Sections 3.3 and 3.4). They are based
on the propositional encodingsummarized in Section 3.2 and
the use of the DeCa algorithm [Adjiman et al., 2006] which
is the decentralizedalgorithmfor propositionalreasoning im-
plemented in the SomeWhere platform.
Decentralized Query Answering
2011
Page 3
3.1
Given a union of conjunctive queries Q over a KB K =
?T ,A?, Answer (Algorithm 1) first checks whether K is in-
consistent (line 1). In that case, it returns all the tuples of
the arity of Q that can be generated from the constants occur-
ring in A (line 2). Otherwise, it gets ans(Q,K) by evaluat-
ing against A considered as a relational database the union of
conjunctive queries obtained by reformulation of Q (line 3).
Algorithm 1: The original Answer algorithm
Answer(Q,K)
Input: a union of conjunctive queries Q and a KB K = ?T ,A?
Output: ans(Q,K)
(1) if not Consistent(K)
(2)
return Alltup(Q,K)
(3) else return (S
Consistent (Algorithm 2) builds a boolean query qunsat
thatchecksthatthe DL-LITERformulaethatmustbedisjoint,
according to the intentional knowledge modeled in T , indeed
have disjoint instances in A. qunsatis obtained as the union
of the first-orderlogic (FOL) translations of the NI-closure of
T , denoted cln(T ), i.e., the set of all the NIs entailed by T .
The FOL translations of NIs are defined by:
δ(B1? ¬B2) = ∃x γ1(x) ∧ γ2(x) such that
γi(x) = Ai(x) if Bi= Ai
γi(x) = ∃yiPi(x,yi) if Bi= ∃Pi
γi(x) = ∃yiPi(yi,x) if Bi= ∃P−
δ(R1? ¬R2) = ∃x,y ρ1(x,y) ∧ ρ2(x,y) such that
ρi(x,y) = Pi(x,y) if Ri= Pi
ρi(x,y) = Pi(y,x) if Ri= P−
Algorithm 2: The original Consistent algorithm
Consistent(K)
Input: a KB K = ?T ,A?
Output: true if K is satisfiable, false otherwise
(1) qunsat = ⊥ (i.e., qunsatis false)
(2) foreach α ∈ cln(T )
(3)
qunsat = qunsat∨ δ(α)
(4) if qdb(A)
(5)
return true
(6) else return false
Finally, PerfectRef (Algorithm 3) reformulates each
conjunctive query q in Q by using the PIs in T as rewrit-
ing rules. PIs are seen as logical rules that can be applied in
backward-chaining to query atoms in order to expand them.
More specifically, a PI I is applicable to an atom A(x) of a
query if I has A in its right-hand side, and a PI I is appli-
cable to an atom P(x1,x2) of a query if (i) x2 =
right-hand side of I is ∃P; or (ii) x1=
side of I is ∃P−; or (iii) I is a role inclusion assertion and its
right-handside is either P or P−. Note that denotes here an
unbounded existential variable of a query.
The following definition (Definition 32 from [Calvanese
et al., 2007]) defines the result gr(g,I) of the (backward)
application of the PI I to the atom g, which is the core of
PerfectRef (loop (a), lines 5 to 7).
Existing DL-LITERalgorithms: reminder
qi∈QPerfectRef(qi,T ))db(A)
i
i
unsat= ∅
and the
and the right-hand
Definition 1 (Backward application of a PI to an atom)
Let I be an inclusion assertion that is applicable to the atom
g. Then, gr(g,I) is the atom defined as follows:
- if g = A(x) and I = A1 ? A, then gr(g,I) = A1(x)
- if g = A(x) and I = ∃P ? A, then gr(g,I) = P(x, )
- if g = A(x) and I = ∃P−? A, then gr(g,I) = P( ,x)
- if g = P(x, ) and I = A ? ∃P, then gr(g,I) = A(x)
- if g = P(x, ) and I = ∃P1 ? ∃P, then gr(g,I) = P1(x, )
- if g = P(x, ) and I = ∃P−
- if g = P( ,x) and I = A ? ∃P−, then gr(g,I) = A(x)
- if g = P( ,x) and I = ∃P1 ? ∃P−, then gr(g,I) = P1(x, )
- if g = P( ,x) and I = ∃P−
- if g = P(x1,x2) and either I = P1 ? P or I = P−
then gr(g,I) = P1(x1,x2)
- if g = P(x1,x2) and either I = P1 ? P−or I = P−
then gr(g,I) = P1(x2,x1)
The subtle pointof PerfectRef is the needof simplifying
the produced reformulations (loop (b), lines 8 to 10), so that
some PIs that were not applicable to a reformulation become
applicable to its simplifications. A simplification amounts to
unify two atoms of a reformulation using their most general
unifier(usingreduce, line 10) andthen to switch the possibly
new unbounded existential variables to (using τ, line 10).
Algorithm 3: The original PerfectRef algorithm
PerfectRef(q,T )
Input: a conjunctive query q and a Tbox T
Output: a union of conjunctive queries
(1) PR := {q}
(2) repeat
(3)
PR?:= PR
(4)
foreach q ∈ PR?
(5)(a) foreach g ∈ q
(6)
if I is applicable to g
(7)
PR := PR ∪ {q[g/gr(g,I)]}
(8)(b) foreach g1,g2 ∈ q
(9)
if g1and g2unify
(10)
PR := PR ∪ {τ(reduce(q,g1,g2))}
(11) until PR?= PR
1? ∃P, then gr(g,I) = P1( ,x)
1? ∃P−, then gr(g,I) = P1( ,x)
1 ? P−
1 ? P
3.2
The propositionalencodingof a DL-LITERTbox T , denoted
Φ(T ), is the formula of propositional logic (PL) that corre-
sponds to the union of the PL encoding of every inclusion
assersion I of T : Φ(T ) =?
I∈TΦ(I).
The PL encoding of a concept inclusion B ? C, denoted
Φ(B ? C) is inductivelydefined by {Φ(B) ⇒ Φ(C)} where
Φ(B) = A when B = A, Φ(B) = P∃when B = ∃P,
Φ(B) = P∃−whenB = ∃P−, Φ(C) = Φ(B) whenC = B,
and Φ(C) = ¬Φ(B) when C = ¬B.
The PL encoding of a role inclusion R ? E, denoted
Φ(R ? E), is defined as follows:
Φ(P ? Q)={P ⇒ Q,P−⇒ Q−,P∃⇒ Q∃,P∃−⇒ Q∃−}
Φ(P−? Q)={P−⇒ Q,P ⇒ Q−,P∃−⇒ Q∃,P∃⇒ Q∃−}
Φ(P ? Q−)={P ⇒ Q−,P−⇒ Q,P∃⇒ Q∃−,P∃−⇒ Q∃}
Φ(P−? Q−)={P−⇒ Q−,P ⇒ Q,P∃⇒ Q∃,P∃−⇒ Q∃−}
Φ(P ? ¬Q)={P ⇒ ¬Q,P−⇒ ¬Q−}
Φ(P−? ¬Q)={P−⇒ ¬Q,P ⇒ ¬Q−}
Φ(P ? ¬Q−)={P ⇒ ¬Q−,P−⇒ ¬Q}
Φ(P−? ¬Q−)={P−⇒ ¬Q−,P ⇒ ¬Q}
Note also that in the following Φ(E) = Φ(R) when E = R,
Φ(E) = ¬Φ(R) when E = ¬R, Φ(R) = P when R = P,
and Φ(R) = P−when R = P−.
The PL encoding of the distributed Tbox?n
DL-LITER PDMS is the distributed propositional theory
?n
i=1Φ(Ti) obtained by the encoding of each local Tbox Ti.
Propositional encoding of a DL-LITERTbox
i=1Ti of a
2012
Page 4
DECA is a message-based algorithm implemented in the
SomeWhere platform ([Adjiman et al., 2006]) which com-
putes in a decentralized manner the logical consequences of
propositional clausal theories distributed in a P2P system.
More precisely, by a copy of DECA running locally on each
peer and transmitting forth and back messages conveying lit-
erals and clauses, DeCAi(li) (denoting DECA running on
the peer Piand triggered with an input literal liof the Pivo-
cabulary) produces the set of all the proper prime implicates
of liw.r.t. the distributed theory?n
prime implicates of {li} ∪?n
cates of?n
i=1Φ(Ti) alone.
i=1Φ(Ti), i.e., the set of
i=1Φ(Ti), which are not impli-
3.3Decentralized Consistency Checking
Our approach relies on decentralizing the computation of the
NI-closure of a distributed Tbox?n
PDMS without empty roles (the Tbox does not entail a NI
P ? ¬P) by exploiting a property transfer of the propo-
sitional encoding (Theorem 1) and then by using DECA.
The subtle point is that in a decentralized setting, we have
to launch the computation of the NI-closure from each peer
and thus possibly start from local PIs and NIs that could lead
to the derivation of new NIs by interacting with NIs and PIs
of other peers. For doing so, we define (Definition 2) and
compute with DECA the NI-closure of a peer w.r.t a PDMS
without empty roles.
i=1Ti of a DL-LITER
Theorem 1 (NI-entailment reduced to PL entailment)
Let T be the distributed Tbox of a DL-LITERPDMS without
empty roles, and Φ(T ) its PL encoding. Let X and Y be
both distinct basic concepts or distinct basic roles:
cln(T ) |= X ? ¬Y iff Φ(T ) |= ¬Φ(X) ∨ ¬Φ(Y ).
The proof is by induction on the number of rules defining the
NI-closure (Definition 9 in [Calvanese et al., 2007]) used for
producing X ? ¬Y , for the if direction, and on the smallest
length of the resolutionprooffor producingΦ(X) ⇒ Φ(¬Y )
for the converse direction.
Definition 2 (NI-closure of a peer w.r.t. a PDMS) Let T =
?n
i=1Tibe the distributed Tbox of a DL-LITERPDMS S =
{Pi}i=1..nwithout emptyroles. The NI-closureofPiw.r.t.S,
denoted cln(Pi), is obtained from Φ(T ) using DECA as fol-
lows:
• for every PI Z ? Y ∈ Tisuch that Z is in the vocabu-
lary of Piand Y in that of Pj(j may be i), Z ? ¬X ∈
cln(Pi) for any ¬Φ(X) ∈ DeCAj(Φ(Y )).
• for every NI Z ? ¬Y ∈ Ti
– if Z is inthevocabularyofPiandY inthatofPj(j
may be i), Z ? ¬X ∈ cln(Pi) for any ¬Φ(X) ∈
DeCAj(¬Φ(Y ))
– if Y is in the vocabulary of Pi and Z is in that
of Pj(j may be i), X ? ¬Y ∈ cln(Pi) for any
¬Φ(X) ∈ DeCAj(¬Φ(Z)).
The decentralized version of the original Consistent al-
gorithm, denoted Consistentiwhen running on peer Pi, is
simply obtained by replacing foreach α ∈ cln(T ) in Line
2 of Algorithm 2 by foreach α ∈ cln(Pi), and where each
conjunctive query of qunsatdoes not have to be evaluated by
Piagainst the (unknown) global Abox of the whole PDMS.
Indeed, by construction of qunsat, each of its conjunctive
queries has two conjuncts, one from Piand another from Pj
(j may be i), the latter providingin its atomic concept or role
the identifier j of the peer to contact for the evaluation.
Theorem 2 states the correctness of locally running
ConsistentioneachpeerPiforglobalconsistencychecking
of a PDMS without empty roles.
Theorem 2 (Correctness of PDMS consistency checking)
Let S = {Pi}i=1..nbe a DL-LITERPDMS without empty
roles. S is consistent iff Consistentireturns true for every
i = 1..n.
The proof relies first on Theorem 1 showing the equivalence
between logical entailement from a Tbox of a NI with en-
tailment in PL of the corresponding propositional encoding.
Then, both Lemma 12 in [Calvanese et al., 2007] and the
completenessof DECA(provedin[Adjimanet al., 2006]) en-
sure that cln(Pi) defined in Definition 2 contains all the NIs
entailed by the PDMS and involving a concept or role in the
vocabularyofthepeerPi. Finally,itis easytosee thatbyrun-
ning Consistentifor every peer Piof the PDMS, we obtain
all the NIs entailed by the PDMS. Therefore Theorem 15 and
Lemma 16 in[Calvanese et al., 2007]ensure that consistency
checking can be made by evaluating the union of conjunc-
tive queries in qunsatagainst the relevant part of the Abox. It
is exactly what running Consistention all the peers in the
PDMS does in a decentralized manner.
3.4 Decentralized Query Reformulation
Ourapproachrelies onthe propositionalencodingandtheuse
of DECA for decentralizing the backward-closure w.r.t. the
PIs of each atom in the query.
backward-closure of an atom w.r.t. the PIs as the iteration
of the one-step backward application of PIs (Definition 1).
Proposition 1 states the termination of this iterative process.
Definition 3 defines the
Definition 3 (Backward-closure of an atom w.r.t. PIs) Let
PI be a set of PIs, g an atom, and A a set of atoms.
Wedefinethe
backward-closure
as
cl gr(g,PI)=
?
cl gri({g},PI) is recursively defined as follows:
of
g
w.r.t.
PI
i≥1cl gri({g},PI)
where
• cl gr1(A,PI) = {gr(g,I) | g ∈ A, I ∈ PI and I is
applicable to g}
• cl gri+1(A,PI) = cl gr1(cl gri(A,PI),PI)
Proposition 1 (Termination of backward-closure w.r.t. PIs)
The backward-closure of an atom w.r.t. a set of PIs
is finite, i.e., there exists a constant n such that
cl gr(g,PI) =?n
i=1cl gri({g},PI).
The proof corresponds to the termination proof of
PerfectRef (Lemma 34 in[Calvanese et al., 2007]).
Theorem3 is the equivalentforthe PIs of the transferprop-
erty of the propositional encoding for the NIs stated in The-
orem 1. Its proof is also by induction (number of one-step
applications of a PI and smallest length of resolution proofs).
2013
Page 5
Theorem 3 (Backward-closure reduced to PL entailment)
Let T be a DL-LITERTbox the PIs of which form the set
PI. Let g,g?be atoms, and V1,V2propositional variables.
g?∈ cl gr(g,PI) iff Φ(T ) ∪ {¬V1} |= ¬V2where:
- g = A(x), g?= A?(x), V1= A, and V2= A?;
- g = A(x), g?= P(x, ), V1= A, and V2= P∃;
- g = A(x), g?= P( ,x), V1= A, and V2= P∃−;
- g = P(x,y), g?= Q(x,y), V1= P, and V2= Q;
- g = P(x,y), g?= Q(y,x), V1= P, and V2= Q−;
- g = P(x, ), g?= A(x), V1= P∃, and V2= A;
- g = P(x, ), g?= Q(x, ), V1= P∃, and V2= Q∃;
- g = P(x, ), g?= Q( ,x), V1= P∃, and V2= Q∃−;
- g = P( ,x), g?= A(x), V1= P∃−, and V2= A;
- g = P( ,x), g?= Q(x, ), V1= P∃−, and V2= Q∃;
- g = P( ,x), g?= Q( ,x), V1= P∃−, and V2= Q∃−.
Based on Theorem 3, the decentralized computation of
cl gr(g,PI) is straighforward using DECA: if g is built
from the vocabulary of the peer Pi, g?∈ cl gr(g,PI) iff
¬V2∈ DeCAi(¬V1) for the same values of g, g?, V1, and V2
of the corresponding cases of Theorem 3.
The decentralized version of PerfectRef, denoted
PerfectRefiwhen running on peer Pi, is given in Algo-
rithm 4. For each atom in the query, it computes first (in the
decentralizedmannerexplainedpreviously)thesetofallofits
reformulations, and then it produces a first set of reformula-
tions ofthe originalquerybybuildingall theconjunctionsbe-
tween the atomic reformulations(denoted?n
at Line 5). Those reformulations are then possibly simpli-
fied by unifying some of their atoms (Lines 8 to 11), and the
reformulationprocess is iterated on those newly producedre-
formulations until no simplification is possible (general loop
starting on Line 4).
Algorithm4: ThedecentralizedPerfectRef algorithmrun-
ning on the peer Piof the PDMS S
PerfectRefi(q)
Input: a conjunctive query q over the Tbox Tiof the peer Pi
Output: a union of conjunctive queries over the Tbox T of the
PDMS S
(1) PR := {q}
(2) PR?:= PR
i=1cl gr(gi,PI)
(3) while PR?= ∅
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11) PR = PR ∪ PR?∪ PR??
(12) return PR
(a) foreach q?= g1∧ g2∧ ... ∧ gn ∈ PR?
PR??= ?n
i=1cl gr(gi,PI)
PR?= ∅
(b) foreach q??∈ PR??
foreach g?
if g?
2unify
PR?:= PR?∪ {τ(reduce(q??,g?
1,g?
2∈ q??
1and g?
1,g?
2))}
The following theorem states the correctness of the decen-
tralized reformulation algorithm PerfectRefi.
Theorem 4 (Correctness of PerfectRefi) Let
T
=
?n
i=1Ti be a Tbox of a PDMS. Let q be a con-
junctive query over Ti. PerfectRefi(q) returns the same
set of conjunctive queries as PerfectRef(q,T ).
Its proof results (1) from the observation that the cen-
tralized version of PerfectRefi(in which cl gr(gi,PI) is
computed by iterating the one-step application of PIs on each
atom giof the query) produces the same results than the orig-
inal PerfectRef,and (2)fromTheorem3 andthe complete-
ness of DECA, ensuringthe decentralizedcomputationof the
whole set cl gr(gi,PI).
In contrast with the original Answer algorithm, the global
consistency of the PDMS cannot be checked at query time
since the queried peer Pidoes not know all the peers in the
PDMS. However, it can get the identifiers id1,...,idk of
the peers involved in a reformulation of the query (to con-
tact them) from the identifiers used in the atomic concept and
role names involved in that reformulation. Algorithm 5 de-
scribes the decentralized Answerialgorithm that checks in a
decentralized manner whether?k
tent and computes the set of corresponding answers by eval-
uating each reformulation against the relevant Aboxes.
j=1(Tidj∪ Aidj) is consis-
Algorithm 5: The decentralized Answer algorithm running
on the peer Piof the PDMS S
Answeri(Q)
Input: a union of conjunctive queries Q over the KB Ki = ?Ti,Ai?
of Pi
Output: ans(Q,K) where K = ?T ,A? is the KB of the PDMS S
(1) q =S
(2) if Consistentidjreturns true for every peer Pidjinvolved in q
return qdb(Sk
(4) else return the singleton {⊥}
In that algorithm ⊥ replaces AllTup(Q,K) of the original
Answer algorithm.
The interest of Algorithm 5 is to provide well-founded an-
swers, i.e.,answersthat canbe entailedfroma consistentsub-
set of the (possibly inconsistent) KB of the PDMS.
q?∈QPerfectRefi(q?)
(3)
j=1Aidj)
4Query Answering using Views by Rewriting
We provide algorithms for computing the certain answers of
a query over a (centralized or decentralized) DL-LITERKB
K = ?T ,V,E? where E is the extensionof views in V that are
conjunctivequeries overT . For doing so, we make use of the
scalable MiniCon [Pottinger and Halevy, 2001] algorithm
which produces the maximally-contained conjunctive rewrit-
ings of a conjunctive query q using a set V of conjunctive
views. A conjunctive rewriting of q is a conjunctive query
qvwhose body predicates are the head predicates of views in
V such that T ∪ V |= ∀¯ x(qv(¯ x) ⇒ q(¯ x)). In that setting
([Halevy, 2001]), the set of certain answers of a query can be
obtainedbyevaluatingagainstthe view extensionsthe (finite)
union of its maximally-containedconjunctive rewritings.
Centralized Case
First V iewConsistent(K) checks the
consistency of the KB It is a variant of the original
Consistent algorithm obtained by replacing:
- qunsat = qunsat ∨ δ(α) in Line 3 of Algorithm 2
by qunsat
=
qunsat ∨ MiniCon(δ(α),V),
MiniCon(δ(α),V)
provides
rewritings using V of the FOL translation δ(α) of the NI α,
- qdb(A)
unsatin Line(4)ofAlgorithm2bythe evaluationagainst
the view extensions: qdb(E)
unsat.
where
themaximally-contained
2014
Page 6
Algorithm 6 describes the CertainAnswer algorithm. If the
KB is inconsistent, the algorithm returns Alltup(Q,E), the
set of all the tuples of the arity of Q generated from the con-
stants in E. Otherwise, it computes (using MiniCon(q?,V))
the rewritings in terms of the views of the conjunctivequeries
q?returned by PerfectRef as the different ways of unfold-
ing the initial query using the PIs of T .
Algorithm 6: The CertainAnswer algorithm
CertainAnswer(Q,K)
Input: a union of conjunctive queries Q and a KB K = ?T ,V,E?
Output: cert(Q,K)
(1) if not V iewConsistent(K)
(2)
return Alltup(Q,E)
(3) else
(4)
Q?=S
return (S
q∈QPerfectRef(q,T )
q?∈Q?MiniCon(q?,V))db(E)
(5)
Theorem 5 (Correctness of CertainAnswer) Let K
?T ,V,E? be a DL-LITERconsistent KB and Q a union of
conjunctive queries. cert(Q,K) = CertainAnswer(Q,K).
=
For one direction, from q?∈ PerfectRef(q,T ) and qv ∈
Minicon(q?,V) we infer: T ∪ V |= ∀¯ x(qv(¯ x) ⇒ q(¯ x)),
and thus qvis a conjunctive rewriting of q, the evaluation of
which provides certain answers. For the converse direction:
if¯t is a certain answer, by adapting the notion of witness of a
tuple introduced in Lemma 39 of [Calvanese et al., 2007] to
E instead of A, we build from the witness of¯t a specific con-
junctive query qvover view atoms such that¯t is in its answer
set and we show by induction that this query is a rewriting
of a reformulation q?of q. Since Minicon computes all the
maximally-containedrewritings(Theorem1in[Pottingerand
Halevy, 2001]), qvis contained in one of them, and¯t will be
returned by Line 5 of Algorithm 6.
Decentralized
sketch the approach:
V iewConsistent and of CertainAnswer are denoted
V iewConsistentiand CertainAnsweriwhen running on
a peer Pi. They are extensions of the decentralized algo-
rithms Consistentiand Answeripresented in the previous
section. They use the function fetchV iews(q) where q is
a conjunctive query possibly involving the vocabulary of
different peers: fetchV iews(q) retrieves from those peers
the views a body atom of which can be unified with a body
atom of the query.
TheV iewConsistentialgorithmextendsConsistentiby
replacingthe evaluationofqunsatagainstthe relevantAboxes
by the evaluation of Qunsatagainst the relevant view exten-
sions, where Qunsatis obtained from qunsat(which is com-
puted as in Consistenti) as follows:
Qunsat=?
q∈qunsatMiniCon(q,fetchViews(q)).
The CertainAnswerialgorithm extends Answeriby re-
placing the evaluation against the relevant Aboxes of the
union q of conjunctive queries obtained by reformulation of
the initial query(Line 2 in Algorithm5) by evaluatingagainst
the relevant view extensions the following query Q:
Q =?
Case
Forspacelimitation,we just
the decentralized versions of
q?∈qMiniCon(q?,fetchV iews(q?)).
5 Conclusion
This papers builds on and extends existing work in data inte-
gration ([Calvanese et al., 2007; 2008b] and [Pottinger and
Halevy, 2001]). For view-based query answering in DL-
LITER we provide a centralized and a decentralized algo-
rithmtocomputethecertainanswers basedonrewritings. For
the decentralized case, our work extends the data model of
the SOMEOWL and SOMERDFS PDMSs ([Adjiman et al.,
2006;2007]). We followthe sameapproachbasedonlimiting
the data model allowing to reduce consistency checking and
queryreformulationto reasoningin propositionallogic. It is a
way of getting the decidability for query answering in PDMS
which is not guaranteed in general ([Halevy et al., 2003]),
while adopting a standard logical semantics, in contrast with
other works (e.g., ([Calvanese et al., 2008a],[Franconi et al.,
2004],[Serafini et al., 2005]). It is also a distinguishingpoint
from the approach in [Bertossi and Bravo, 2007] (based on
answer set programs) for defining consistent answers in pos-
sibly inconsistent PDMSs.
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