WellFounded Semantics for Description Logic
Programs in the Semantic Web
THOMAS EITER
Technische Universit
¨
at Wien
GIOVAMBATTISTA IANNI
Universit
`
a della Calabria
THOMAS LUKASIEWICZ
University of Oxford and Technische Universit
¨
at Wien
ROMAN SCHINDLAUER
Technische Universit
¨
at Wien
The realization of the Semantic Web vision,in which computational logic has a prominent role,has
stimulated a lot of research on combining rules and ontologies,which are formulated in di®erent
formalisms.In particular,combining logic programming with the Web Ontology Language (OWL),
which is a standard based on description logics,emerged as an important issue for linking the
Rules and Ontology Layers of the Semantic Web.Nonmonotonic description logic programs (or
dlprograms) were introduced for such a combination,in which a pair
(L;P) of a description logic
knowledge base
L and a set of rules P with negation as failure is given a modelbased semantics that
generalizes the answer set semantics of logic programs.In this paper,we reconsider dlprograms
and present a wellfounded semantics for them as an analog for the other main semantics of
logic programs.It generalizes the canonical de¯nition of the wellfounded semantics based on
unfounded sets,and,as we show,lifts many of the wellknown properties from ordinary logic
programs to dlprograms.Among these properties:our semantics amounts to a partial model
approximating the answer set semantics,which yields for positive and strati¯ed dlprograms a
total model coinciding with the answer set semantics;it has polynomial data complexity provided
the access to the description logic knowledge base is polynomial;under suitable restrictions,it
has lower complexity and even ¯rstorder rewritability is achievable.The results add to previous
evidence that dlprograms are a versatile and robust combination approach,which moreover is
implementable using legacy engines.
Authors’ addresses:T.Eiter,T.Lukasiewicz,and R.Schindlauer:Institut f
¨
ur Informationssysteme,Technische
Universit
¨
at Wien,Favoritenstraße 911,1040 Wien,Austria;email:
feiter,lukasiewicz,romang@kr.tuwien.ac.at.
T.Lukasiewicz:Computing Laboratory,University of Oxford,Wolfson Building,Parks Road,Oxford OX1 3QD,
UK;email:thomas.lukasiewicz@comlab.ox.ac.uk.G.Ianni:Dip.di Matematica,Universit
`
a della Calabria,P.te
P.Bucci,Cubo 30B,87036 Rende,Italy;email:ianni@unical.it.This paper signiﬁcantly extends and revises
a paper that has appeared in:Proc.RuleML2004,pp.81–97,Hiroshima,Japan.LNCS 3323,Springer,2004.
This work has been partially supported by the Austrian Science Fund (FWF) under projects P17212,P20840,
and P20841,by the German Research Foundation (DFG) under the Heisenberg Programme,by the Italian Re
search Ministry (MIUR) under project INTERLINK II04CG8AGG,by the EPSRC grant EP/E010865/1,by the
European Commission under the IST REWERSE NoE IST2003506779,ONTORULE (ICT 231875),and the
Marie Curie Fellowship HPMFCT2001001286 (disclaimer:the authors are solely responsible for information
communicated and the EUCommission is not responsible for any views expressed),and by the Regione Calabria
and the EU under FESR 20072013 (project PIADLVSYSTEMs.r.l.).
Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use
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c
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ACMTransactions on Computational Logic,Vol.V,No.N,April 2010,Pages 1–36.
2 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
Categories and Subject Descriptors:I.2.4 [Knowledge Representation Formalisms and Methods]:Represen
tation languages;I.2.3 [Deduction and TheoremProving]:Inference engines,Logic programming,Nonmono
tonic reasoning and belief revision;F.4.1 [Mathematical Logic]:Computational logic
General Terms:Theory,Languages
Additional Key Words and Phrases:answer set semantics,description logic programs,description
logics,normal logic programs,semantic web,wellfounded semantic.
1.INTRODUCTION
During the last years,the Semantic Web [BernersLee et al.2001;Fensel et al.2002] has
been gaining momentumas a backbone for future information systems.Alayered architec
ture has been conceived to materialize this vision,with the World Wide Web Consortium
(W3C) being a steering force behind.This vision comprises lowlevel syntactic data levels
to highlevel semantic layers for which computational logic plays a prominent role.The
W3C develops standards,including the Resource Description Framework (RDF) for the
Data Layer of the architecture and the Web Ontology Language (OWL),which is based
on Description Logics,for the Ontology Layer;the Rule Interchange Format (RIF) Work
ing Group currently aims at a standard exchange format for the Rules Layer rather than a
common semantics,given the plethora of existing languages and types of rules.
It has been realized that rule bases and ontologies,formulated in different languages,
need to be combined in order to have,on the one hand,the expressive capabilities that
are needed to model certain scenarios,and on the other hand to make interoperability of
knowledge bases in different languages possible.However,due to an impedance mismatch
between rule and ontology formalisms,such a combination is nontrivial.Many proposals
have been made,cf.[Drabent et al.2009;Eiter et al.2008;Motik et al.2006;Rosati
2006;Lukasiewicz 2007] and references therein,which also give taxonomies to distinguish
different types of combinations and discuss fundamental technical issues.
Roughly,there are heterogeneous and homogeneous combinations,respectively depend
ing on whether the rule and the ontology predicates are distinguished in the integration or
not;among the heterogeneous ones are loose couplings,in which rule bodies may contain
queries to the ontology,and tight integrations,in which a modelbased semantics refers to
the semantics of the original rule language and to the FOL models of the ontology [Drabent
et al.2009].
An advanced approach of loose coupling are description logic programs (or dlprograms)
[Eiter et al.2004;2008],which are of the form
KB =(L;P),where L is a knowledge
base in a description logic,and
P is a ﬁnite set of description logic rules (or dlrules).Such
dlrules are similar to usual rules in logic programs with negation as failure,but may also
contain queries to
L in their bodies which are given by special atoms (on which possibly
default negation may apply).For example,a rule
cand(X;P) ÃpaperArea(P;A);DL[Referee](X);DL[expert](X;A)
may express that
X is a candidate reviewer for a paper P,if the paper is in area
A,and X
is known to be a referee and an expert for area
A.Here,the latter two are queries to
the description logic knowledge base
L,which has a concept Referee and role expert in
its signature.For the evaluation,the precise deﬁnition of Referee and expert within
L is
fully transparent,and only the logical contents at the level of inference counts.Thus,dl
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 3
programs fully support encapsulation and privacy of
L (which applications may request).
1
Another important feature of dlrules is that queries to L also allow for specifying an
input from
P,and thus for a ﬂow of information from
P to L,besides the ﬂow of informa
tion from L to
P,given by any query to L.Hence,dlprograms allow for building rules
on top of ontologies,but also (to some extent) building ontologies on top of rules.This
is achieved by dynamic update operators through which the extensional part of
L can be
hypothetically modiﬁed and then subsequent querying can be performed (thus constituting
a formof subjunctive queries [Grahne and Mendelzon 1995]).For example,the rule
paperArea(P;A) ÃDL[keyword ]kw;inArea](P;A)
intuitively says that paper P is in area A,if P is in A according to the description logic
knowledge base
L,where the extensional part of the keyword role in
L (which is known to
inﬂuence inArea) is augmented by the facts of a binary predicate kw fromthe program.In
this way,further knowledge can be supplied to
L before querying.Using this mechanism,
also more involved relationships between concepts and/or roles in
L can be exploited.
Eiter et al.[2004;2008] faithfully extended the answer set semantics [Gelfond and Lifs
chitz 1991] for ordinary normal programs,which is one of the most widely used semantics
for nonmonotonic logic programs,to dlprograms.They deﬁned weak and strong an
swer sets of dlprograms,which coincide with usual answer sets in the case of ordinary
normal programs.The description logic knowledge bases in dlprograms are speciﬁed in
the wellknown description logics
SHIF(D) and
SHOIN(D) which underly OWL Lite
and OWL DL [Horrocks and PatelSchneider 2004;Horrocks et al.2003],respectively,but
may be easily adapted to description logics in the upcoming OWL2 standard [Cuenca Grau
et al.2008].The resulting formalism is very expressive and facilitates advanced applica
tions like closedworld reasoning,default logic,nondeterministic model generation etc.
However,under a dataoriented perspective,similar as in deductive databases,also the
wellfounded semantics [van Gelder et al.1991] is of great importance for the Web.Besides
the answer set semantics,it is the most widely used semantics for nonmonotonic logic
programs.Differently fromthe answer set semantics,the wellfounded semantics remains
agnostic in the presence of conﬂicting information and leaves truth values undeﬁned,rather
than to reason by cases in different worlds;on the other hand,it assigns the truth value false
to a maximal set of atoms that cannot become true during the evaluation of a given program.
The wellfounded semantics has several attractive features;perhaps most important is that
it extends the perfect model semantics of stratiﬁed programs and that it has polynomial
time complexity (measured by the data size),while the answer set semantics is intractable;
indeed,efﬁcient implementations are available (e.g.,XSB
2
).The wellfounded semantics
assigns a coherent meaning to all logic programs,while some programs may have no
answer sets:moreover,it is a skeptical approximation of the answer set semantics,in
the sense that every wellfounded consequence of a given ordinary normal program
P is
contained in every answer set of
P.For the Web context,the signiﬁcance of the well
founded semantics is evidenced by the fact that several reasoners in this area use it for
nonmonotonic negation,including
Flora2
3
and OntoBroker
4
that are based on FLogic,
1
Here,“extensional” is meant as “knowledge about individuals”,as opposed to “terminological knowledge”,
which concerns knowledge about classes of individuals and their properties.
2
http://xsb.sourceforge.net/
3
http://flora.sourceforge.net/
4
http://www.ontoprise.de/en/home/products/ontobroker/
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
4 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
and IRIS and MINS,
5
which target the WSMLRule language [de Bruijn et al.2006].
Motivated by these observations,in this paper,we consider the issue of the wellfounded
semantics for dlprograms.Our main contributions can be summarized as follows:
—
We deﬁne the wellfounded semantics for normal dlprograms.Observe that we
explicitly opt for generalizing Van Gelder et al.’s [1991] ﬁxpoint characterization of the
wellfounded semantics for ordinary normal programs based on greatest unfounded sets.
Such a characterization adheres to the intuitive deﬁnition of wellfounded semantics,and,
in this respect,is preferable to alternative algebraic deﬁnitions;however,technical issues
require careful thought.Our proposal is the ﬁrst deﬁnition of wellfounded semantics for
such a language that is directly based on the intuitive notion of unfounded set;other related
proposals [Drabent et al.2007;Knorr et al.2007] allow either only limited interaction
between the rule and the ontology part,or use alternating ﬁxpoints (see Section 9).It is
important to point out that the dlprograms under the wellfounded semantics considered
here are modularly deﬁned and not restricted to a speciﬁc underlying description logic;
they are easily adapted to the description logics of the upcoming OWL 2 proposal.
6
—
We then prove some appealing semantic properties of the wellfounded semantics for
dlprograms.In particular,it generalizes the wellfounded semantics for ordinary normal
programs.Moreover,for general dlprograms,the wellfounded semantics is a partial
model,and for positive (resp.,stratiﬁed) dlprograms,it is a total model and the canonical
least (resp.,iterative least) model of these dlprograms.
—
Generalizing a result by Baral and Subrahmanian [1993],we then show that the
wellfounded semantics for dlprograms can be characterized in terms of the least and
the greatest ﬁxpoint of an operator
°
2
KB
,which is deﬁned using a generalized Gelfond
Lifschitz transformof dlprograms relative to an interpretation.
—
We also show that,similarly as for ordinary normal programs,the wellfounded
semantics for dlprograms approximates the strong answer set semantics for dlprograms.
Furthermore,we prove that when the wellfounded semantics of a dlprogramis total,then
it is the only strong answer set.
—
As for computation,we show how the wellfounded semantics of dlprograms
KB
can be computed by ﬁnite sequences of ﬁnite ﬁxpoint iterations,using the operator
°
KB
and the immediate consequence operator
T
KB
of positive dlprograms KB.
—
We then give a characterization of the combined complexity of the wellfounded
semantics for dlprograms,over both
SHIF(D) and SHOIN(D).Like for ordinary
normal programs,it is lower or equal to the complexity under the answer set semantics for
SHIF(D).Relative to program complexity [Dantsin et al.2001],for
SHIF(D),literal
inference is EXPcomplete under the wellfounded semantics and coNEXPcomplete un
der the strong answer set semantics [Eiter et al.2004].However,in case of
SHOIN(D)
the problemis P
NEXP
complete under both semantics [Eiter et al.2008].
—
We also characterize the data complexity of literal inference fromdlprograms under
the wellfounded semantics,which does not increase much compared to the data complex
ity of query answering in the underlying description logics:For dlprograms over both
SHIF(D) and
SHOIN(D),the problemis
P
NP
complete under data complexity.
5
http://irisreasoner.org/,http://tools.stiinnsbruck.at/mins/
6
http://www.w3.org/TR/2008/WDowl2profiles20081202/
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 5
—
We then delineate several data tractable cases.In detail,we show that when all dl
queries in a dlprogram can be evaluated in polynomial time (e.g.,for certain dlqueries
over Horn
SHIQ[Hustadt et al.2005]),then reasoning fromdlprograms under the well
founded semantics is complete for P under data complexity,and thus has the same data
complexity as for ordinary normal programs.Furthermore,when the evaluation of dl
queries in a dlprogram is ﬁrstorder rewritable (e.g.,for certain dlqueries over DLLite
[Calvanese et al.2007]),and the dlprogram is acyclic,then reasoning from dlprograms
under the wellfounded semantics is ﬁrstorder rewritable,and thus in LOGSPACE under
data complexity.Hence,in the latter case,efﬁcient evaluation by means of commercial,
SQLexpressive relational database systems is possible.
The rest of this paper is organized as follows.In Section 2,we revisit some basic
concepts of nonmonotonic logic programs and description logics.Section 3 recalls dl
programs and their answer set semantics from [Eiter et al.2008].In Section 4,we intro
duce the wellfounded semantics for dlprograms,and in Section 5,we analyze its semantic
properties.Sections 6 and 7 contain complexity characterizations and data tractable cases,
respectively,while Section 8 brieﬂy reports on a prototype implementation.After a discus
sion of related work in Section 9,we give in Section 10 a brief summary and an outlook
on future research issues.Detailed proofs of all results are given in Appendices A–D.
2.PRELIMINARIES
In this section,we recall normal programs under the wellfounded semantics,as well as
the expressive description logics
SHIF(D) and SHOIN(D).
2.1 Normal Programs
We now recall the syntax of normal programs and their wellfounded semantics.
2.1.1 Syntax.
As for the syntax of normal programs,we assume a functionfree ﬁrst
order vocabulary © = (P;C),consisting of two nonempty ﬁnite sets
C and P of constant
and predicate symbols,respectively,and a set X of variables.We adopt the convention that
variables start with an uppercase letter,while constant and predicate symbols start with a
lowercase letter.Atermis either a variable from
X or a constant symbol from©.An atom
is of the form
p(t
1
;:::;t
n
),where p 2 P,and
t
1
;:::;t
n
are terms.A classical literal (or
literal)
l is an atom
a or a negated atom:a.Anegationasfailure (NAF) literal is an atom
a or a defaultnegated atom
not a.A normal rule (or rule)
r is of the form
a Ãb
1
;:::;b
k
;not b
k+1
;:::;not b
m
;m>k >0;
(1)
where
a;b
1
;:::;b
m
are atoms.We refer to
a as the head of
r,denoted
H(r),while the
conjunction
b
1
;:::;b
k
;not b
k+1
;:::;not b
m
is the body of
r;its positive (resp.,nega
tive) part is
b
1
;:::;b
k
(resp.,not b
k+1
;:::;not b
m
).We deﬁne
B(r) = B
+
(r) [ B
¡
(r),
where
B
+
(r) = fb
1
;:::;b
k
g and
B
¡
(r) = fb
k+1
;:::;b
m
g.We say
r is a fact iff m= 0.
A normal program (or program)
P is a ﬁnite set of rules.We say
P is positive iff no rule
in
P contains defaultnegated atoms.
Example 2.1
All variables X 2 X and constants c 2 © are terms;
supplied(cpu;S) and
vendor(V ) are atoms.An example rule is
r =avoid(V ) Ã vendor(V );not rebate(V ),
which may encode that vendors without rebate are avoided.Then,
H(r) = avoid(V ),
B
+
(r) = fvendor(V )g,and B
¡
(r) = frebate(V )g.
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
6 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
2.1.2 WellFounded Semantics.
The wellfounded semantics of normal programs P
has many different equivalent deﬁnitions [van Gelder et al.1991;Baral and Subrahmanian
1993].We recall here the one based on unfounded sets,via the operators
U
P
,T
P
,and W
P
.
Let P be a program.Ground terms,atoms,literals,etc.,are deﬁned as usual.We denote
by
HB
P
the Herbrand base of P,that is,the set of all ground atoms with predicate and
constant symbols from
P (if P contains no constant symbol,then choose an arbitrary one
from
©),and by ground(P) the set of all ground instances of rules in
P (relative to HB
P
).
For literals l =a (resp.,l =:a),we use::l to denote
:a (resp.,a),and for sets of
literals S,we deﬁne
::S = f::l j l 2Sg and S
+
= fa2S j a is an atom
g.In particular,
::HB
P
is the set of all negated ground atoms with predicate and constant symbols from
P;we let
Lit
P
= HB
P
[::HB
P
.A set
I µLit
P
of ground literals is consistent iff
I\::I =;;any such
I is a (threevalued) interpretation relative to P.
A set U µHB
P
is an unfounded set of P relative to
I µLit
P
,if for every a2U and
every
r 2ground(P),if
H(r) =a,either (i):b 2I [::U for some atom
b 2B
+
(r) (i.e.,
either
:b 2 I or b 2 U),or (ii) b 2I for some atom b 2B
¡
(r).There exists the greatest
unfounded set of
P relative to
I,denoted U
P
(I).Intuitively,if I complies with the rules
of P (i.e.,no rule is falsiﬁed),then all atoms in
U
P
(I) can be safely switched to false and
the resulting interpretation still complies with the rules of
P.
The two operators T
P
and W
P
on consistent I µLit
P
are then deﬁned by:
—
T
P
(I) =fH(r) j r 2ground(P);B
+
(r) [::B
¡
(r) µIg;
—
W
P
(I) =T
P
(I) [::U
P
(I).
The operator W
P
is monotonic,and thus has a least ﬁxpoint,denoted
lfp(W
P
),
7
which
is the wellfounded semantics of
P,denoted WFS(P).A ground atom a2HB
P
is well
founded (resp.,unfounded) relative to
P,if
a (resp.,:a) is in lfp(W
P
).Intuitively,start
ing with
I =;,rules are applied to obtain new positive and negated facts (via
T
P
(I)
and::U
P
(I),respectively).This process is repeated until no longer possible.
The unfounded set of a partial interpretation I intuitively collects all those atoms that
cannot become true when extending
I with further information.An atom
b is unfounded
iff there is no rule with b in its head and with a body that can be made true.For example,
an atomnot appearing in any head is clearly unfounded.One crucial point in the deﬁnition
of unfounded set is that falsity of rule bodies can be testiﬁed by unfounded atoms belonging
to the same unfounded set,giving a notion of “selfsupportedness”.
Example 2.2
Consider the ground programP =fpÃnot q;q Ãp;pÃnot rg.
For I =;,we have T
P
(I) =;and
U
P
(I) =frg:p cannot be unfounded because of the
ﬁrst rule and condition (ii),and hence
q cannot be unfounded because of the second rule and
condition (i).Thus,
W
P
(I) =f:rg.Since
T
P
(f:rg) =fpg and U
P
(f:rg) =frg,it then
follows that
W
P
(f:rg) =fp;:rg.Since
T
P
(fp;:rg) =fp;qg and U
P
(fp;:rg) =frg,
it then follows
W
P
(fp;:rg) =fp;q;:rg.Thus,
lfp(W
P
) =fp;q;:rg.That is,r is un
founded relative to P,and the other atoms are wellfounded.
2.2 Description Logics
In this section,we recall the Description Logics
SHIF(D) and SHOIN(D),which
provide the logical underpinning of OWL Lite and OWL DL,respectively (see [Horrocks
7
As usual,for a generic operator T,let T
0
(A) = Aand T
i+1
(A) = T(T
i
(A)) for any integer i > 0.If T is
monotonic,then T has a least ﬁxpoint,denoted lfp(T),and lfp(T) = T
1
(;) =
S
i>0
T
i
(;) if T is compact.
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 7
and PatelSchneider 2004;Horrocks et al.2003] for further details and background).
Intuitively,Description Logics (DLs) model a domain of interest in terms of concepts
and roles,which represent classes of individuals and binary relations on individuals,re
spectively.In particular,a DL knowledge base encodes subset relationships between
classes of individuals,subset relationships between binary relations on individuals,the
membership of individuals to classes,and the membership of pairs of individuals to binary
relations on classes.Other important ingredients of
SHIF(D) (resp.,SHOIN(D)) are
datatypes (resp.,datatypes and individuals) in concept expressions.
2.2.1 Syntax.
We ﬁrst describe the syntax of SHOIN(D).We assume a set
E of
elementary datatypes and a set
Vof data values.Adatatype theory
D=(¢
D
;¢
D
) consists
of a datatype (or concrete) domain
¢
D
and a mapping ¢
D
that assigns to every elementary
datatype a subset of
¢
D
and to every data value an element of
¢
D
.The mapping
¢
D
is
extended to all datatypes by
fv
1
;:::g
D
=fv
D
1
;:::g.Let
ª = (A[ R
A
[ R
D
;I [ V) be
a vocabulary,where
A,R
A
,R
D
,and I are pairwise disjoint (denumerable) sets of atomic
concepts,abstract roles,datatype (or concrete) roles,and individuals,respectively.We
denote by
R
¡
A
the set of inverses R
¡
of all
R2R
A
.
A role is an element of R
A
[R
¡
A
[R
D
.Concepts are inductively deﬁned as follows.
Every atomic concept
C2A is a concept.If o
1
;o
2
;:::are individuals from
I,then
fo
1
;o
2
;:::g is a concept (called oneOf).If C and D are concepts,then also
(C u D),
(C t D),and:C are concepts (called conjunction,disjunction,and negation,respec
tively).If
C is a concept,R is an abstract role from R
A
[R
¡
A
,and
n is a nonnegative
integer,then
9R:C,8R:C,
>nR,and 6nR are concepts (called exists,value,atleast,and
atmost restriction,respectively).If
D is a datatype,U is a datatype role from R
D
,and n
is a nonnegative integer,then
9U:D,8U:D,
>nU,and 6nU are concepts (called datatype
exists,value,atleast,and atmost restriction,respectively).We use
> and?to abbreviate
the concepts Ct:C and
Cu:C,respectively,and we eliminate parentheses as usual.
We next deﬁne axioms and knowledge bases as follows.An axiom is an expression of
one of the following forms:
(1)
CvD,called concept inclusion axiom,where
C and D are concepts;
(2)
RvS,called role inclusion axiom,where either R;S 2R
A
or
R;S 2R
D
;
(3)
Trans(R),called transitivity axiom,where R 2 R
A
;
(4)
C(a),called concept membership axiom,where C is a concept and
a2I;
(5)
R(a;b) (resp.,U(a;v)),called role membership axiom,where R2R
A
(resp.,U 2R
D
) and a;b 2I (resp.,
a2I and v is a data value);and
(6)
a=b (resp.,a6=b),or =(a;b) (resp.,6=(a;b)),called equality (resp.,inequality) ax
iom,where
a;b 2I.
A (description logic) knowledge base
L is a ﬁnite set of axioms.For decidability,num
ber restrictions in
L must be simple abstract roles [Horrocks et al.1999].Observe that
in
SHOIN(D),concept and role membership axioms can also be expressed through
concept inclusion axioms.That the individual
a is an instance of the concept
C can be ex
pressed by the concept inclusion axiom
fag vC,and that the pair (a;b) (resp.,(a;v)) is an
instance of the role
R(resp.,U) can be expressed by fagv9R:fbg (resp.,
fag v9U:fvg).
The syntax of SHIF(D) is the one of SHOIN(D),but without the oneOf constructor
and with the atleast and atmost constructors limited to
0 and 1.
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
8 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
The following example introduces a DL knowledge base for a product database,which
is also used in some subsequent examples.
Example 2.3 (Product Database)
A small computer store obtains its hardware from sev
eral vendors.It uses the following DLknowledge base
L
1
:each potential vendor (the mem
bers of the Shop concept) has in stock some type of parts (encoded via the role providerOf).
The computer store has contracts for getting supplies of speciﬁc parts from speciﬁc ven
dors (encoded via the role contractorFor).
L
1
contains information about the product range
that is provided by each vendor and about possible rebate conditions (we assume here that
choosing two or more parts from the same seller causes a discount).Also,for some parts,
a shop may already be contracted as supplier.
> 1 providerOf v Shop;
> v 8providerOf:Part;
contractorFor v providerOf;
> 2 contractorFor v Discount;
Shop(s
1
);
Shop(s
2
);
Shop(s
3
);
Part(harddisk);
Part(cpu);
Part(case);
providerOf (s
1
;cpu);
providerOf (s
1
;case);
providerOf (s
2
;harddisk);
providerOf (s
2
;cpu);
providerOf (s
3
;harddisk);
providerOf (s
3
;case);
contractorFor(s
3
;case);
for S = fs
1
;s
2
;s
3
;harddisk;cpu;caseg,
c 6= c
0
for each pair c and c
0
of different constants appearing in S.
Here,the ﬁrst two axioms determine
Shop and Part as domain and range of the property
providerOf,respectively.The third axiom states the relationship between
contractorFor
and providerOf,while the fourth constitutes the concept
Discount by putting a cardinality
constraint on contractorFor.
2.2.2 Semantics.
We now deﬁne the semantics of SHIF(D) and
SHOIN(D) in
terms of general ﬁrstorder interpretations,as usual.
An interpretation I =(¢
I
;¢
I
) with respect to a datatype theory D=(¢
D
;¢
D
) consists
of a nonempty (abstract) domain
¢
I
disjoint from ¢
D
,and a mapping ¢
I
that assigns to
each
C2Aa subset of ¢
I
,to each
o 2I an element of ¢
I
,to each abstract role
R2R
A
a
subset of
¢
I
£¢
I
,and to each datatype role
U 2R
D
a subset of
¢
I
£¢
D
.The mapping
¢
I
is extended to all concepts and roles as usual [Horrocks and PatelSchneider 2004].
The satisfaction of a DL axiom F in the interpretation
I = (¢
I
;¢
I
) with respect to
D=(¢
D
;¢
D
),denoted I j=F,is deﬁned as follows:(1)
I j=CvDiff C
I
µD
I
;(2)
I j=
RvS iff R
I
µS
I
;(3)
I j=Trans(R) iff R
I
is transitive;(4)
I j=C(a) iff a
I
2C
I
;(5)
I j=R(a;b) iff (a
I
;b
I
) 2R
I
(resp.,
I j=U(a;v) iff (a
I
;v
D
) 2U
I
);and (6)
I j=a=b iff
a
I
=b
I
(resp.,
I j=a6=b iff a
I
6=b
I
).The interpretation
I satisﬁes the axiom
F,or I
is a model of F,iff
I j=F.The interpretation
I satisﬁes a DL knowledge base
L,or
I
is a model of L,denoted I j=L,iff
I j=F for all F 2L.We say that L is satisﬁable
(resp.,unsatisﬁable) iff
L has a (resp.,no) model.An axiom
F is a logical consequence
of L,denoted Lj=F,iff every model of
L satisﬁes F.A negated axiom
:F is a logical
consequence of L,denoted Lj=:F,iff every model of
L does not satisfy F.
Some important reasoning problems related to DL knowledge bases L are the follow
ing:decide (1) whether a given
L is satisﬁable;(2) given
L and a concept
C,whether
L 6j= Cv?;(3) given
L and two concepts
C and D,whether L j= CvD;(4) given
L,
an individual
a2I,and a concept C,whether L j= C(a);(5) given
L,two individuals
a;b 2I (resp.,an individual a2I and a data value v),and an abstract role
R2R
A
(resp.,
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 9
a datatype role U 2R
D
),whether
L j= R(a;b) (resp.,
L j= U(a;v)),and (6) given
L and
two individuals
a;b 2I,whether
L j= a = b or whether L j= a 6= b.
Here,(1) is a special case of (2),as
L is satisﬁable iff
L 6j= >v?.Furthermore,(2) and
(3) can be reduced to the complement of each other,as
Lj=Cu:Dv?iff
Lj=CvD.
Finally,in
SHOIN(D),as concept and role membership axioms can also be expressed
through concept inclusion axioms (see above),(4) and (5) are special cases of (3).
Example 2.4 (Product Database cont’d)
Consider again L
1
of Example 2.3.We observe
that,for example,
Discount v Shop is not a logical consequence of L
1
.Furthermore,
>2 providerOf (s
3
) is a logical consequence of L
1
,while Discount(s
3
) is not.
3.DESCRIPTION LOGIC PROGRAMS
In this section,we recall description logic programs (or simply dlprograms) under the
answer set semantics from[Eiter et al.2004;2008],which combine DLs (under the general
ﬁrstorder semantics) and normal programs under the answer set semantics.They consist of
a DL knowledge base
L and a ﬁnite set of generalized rules (called dlrules) P.Such rules
are similar to usual rules in logic programs with negation as failure,but may also contain
queries to
L in their bodies,possibly default negated.In such a query,it is asked whether
a certain DL axiom or its negation logically follows from
L.In [Eiter et al.2004;2008],
we considered dlprograms that may also contain classical negation and not necessarily
monotonic queries to
L.Here,we consider only the case where classical negation is absent
and all queries to
L are monotonic.The former is in line with the traditional wellfounded
semantics in the ordinary case,while the latter makes the development of a wellfounded
semantics for dlprograms simpler,putting the focus on the most relevant fragment of
dlprograms.Indeed,most atoms with queries to
L are in fact monotonic (naturally,a dl
programmay still contain NAFliterals).Furthermore,nonmonotonic queries to
Lmay be
emulated by atoms with monotonic queries under wellfounded semantics (cf.Section 5).
3.1 Syntax
We now deﬁne the syntax of dlprograms.As in Section 2.1,we assume a functionfree
ﬁrstorder vocabulary
©=(P;C),consisting of two nonempty ﬁnite sets
C and P of con
stant and predicate symbols,respectively,and a set
X of variables.A term is either a
constant symbol from
C or a variable from
X.As in Section 2.2,we assume a description
logic vocabulary
ª=(A[ R
A
[ R
D
;I [ V),where
A,R
A
,
R
D
,I,and Vare pairwise
disjoint (denumerable) sets of atomic concepts,abstract roles,datatype roles,individuals,
and data values,respectively.We assume that
A[ R
A
[ R
D
is disjoint from
P,while
I
P
µ C µ I [V,where
I
P
is the set of all constant symbols appearing in
P.
We deﬁne dlqueries and dlatoms,which are used in rule bodies to express queries to
the DL knowledge base
L,as follows.A dlquery
Q(t) is either
(a)
a concept inclusion axiomF or its negation:F;or
(b)
of the forms C(t) or:C(t),where
C is a concept,and t is a term;or
(c)
of the forms R(t
1
;t
2
) or
:R(t
1
;t
2
),where
R is a role,and t
1
and
t
2
are terms;or
(d)
of the forms =(t
1
;t
2
) or 6=(t
1
;t
2
),where t
1
and t
2
are terms.
Note here that t is the empty argument list in (a),
t =t in (b),and t =(t
1
;t
2
) in (c) and (d),
and terms are deﬁned as above.A dlatom has the form
DL[S
1
op
1
p
1
;:::;S
m
op
m
p
m
;Q](t);m>0;
(2)
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
10 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
where each
S
i
is either a concept,a role,or a special symbol
µ 2f=;6=g;op
i
2f];
¡
[g;
p
i
is a unary predicate symbol,if S
i
is a concept,and a binary predicate symbol,otherwise;
and
Q(t) is a dlquery.We call
p
1
;:::;p
m
its input predicate symbols.Intuitively,
op
i
=]
(resp.,op
i
=
¡
[) increases
S
i
(resp.,:S
i
) by the extension of p
i
.A dlrule
r is of the
form (1),where any
b
1
;:::;b
m
2B(r) may be a dlatom.A dlprogram
KB =(L;P)
consists of a DL knowledge base
L and a ﬁnite set of dlrules P.We say
KB =(L;P) is
positive iff P is positive.
Example 3.1 (Product Database cont’d)
Consider the dlprogram KB
1
=(L
1
;P
1
),with
L
1
as in Example 2.3 and
P
1
given as follows,choosing vendors for needed parts relative
to possible rebates:
(1)
vendor(s
2
);vendor(s
1
);vendor(s
3
);
(2)
needed(cpu);needed(harddisk);needed(case);
(3)
avoid(V ) Ãvendor(V );not rebate(V );
(4)
rebate(V ) Ãvendor(V );DL[contractorFor ]buy
cand;Discount](V );
(5)
buy
cand(V;P) Ãvendor(V );not avoid(V );DL[providerOf ](V;P);needed(P);
not exclude(P);
(6)
exclude(P) Ãbuy
cand(V
1
;P);buy
cand(V
2
;P);V
1
6= V
2
;
(7)
exclude(P) ÃDL[contractorFor](V;P);needed(P);
(8)
supplied(V;P) ÃDL[contractorFor ] buy
cand;contractorFor](V;P);needed(P).
Rules (3)–(5) choose a possible vendor (
buy
cand) for each needed part,taking into ac
count that the selection might affect the rebate condition (by feeding the possible vendor
back to
L
1
,where the discount is determined).Rules (6) and (7) assure that each hardware
part is bought only once,considering that for some parts a contractor might already be
chosen.Rule (8) eventually summarizes all purchasing results.
3.2 Answer Set Semantics
We nowdeﬁne the answer set semantics of dlprograms and summarize some of its seman
tic properties.We ﬁrst deﬁne (Herbrand) interpretations and the satisfaction of dlprograms
in interpretations.The latter hinges on deﬁning the truth of ground dlatoms in interpreta
tions.In the sequel,let
KB =(L;P) be a dlprogramover the vocabulary
©=(P;C).
The Herbrand base of P,denoted HB
P
,is the set of all ground atoms with (a) predicate
symbols in
P that occur in P and (b) constant symbols in
C.An interpretation
I relative
to P is any subset of
HB
P
.Such an I is a model of a ground atom or dlatom
a (or I
satisﬁes a) under L,denoted
I j=
L
a,if the following holds:
—
if a2HB
P
,then I j=
L
a iff a2I;
—
if a is a ground dlatom DL[¸;Q](c),where
¸ = S
1
op
1
p
1
;:::;S
m
op
m
p
m
,then
I j=
L
a iff
L(I;¸) j= Q(c),where
L(I;¸) = L[
S
m
i=1
A
i
(I) and,for 1 6 i 6 m,
A
i
(I) =
½
fS
i
(e) j p
i
(e) 2Ig;if op
i
=];
f:S
i
(e) j p
i
(e) 2Ig;if op
i
=
¡
[.
We say I is a model of a ground dlrule
r iff I j=
L
H(r) whenever
I j=
L
B(r),that is,
I j=
L
a for all a2B
+
(r) and
I 6j=
L
a for all a2B
¡
(r).We say
I is a model of a dl
programKB =(L;P),denoted
I j=KB,iff
I j=
L
r for all r 2ground(P).We say
KB is
satisﬁable (resp.,unsatisﬁable) iff it has some (resp.,no) model.
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 11
Observe that the above satisfaction of dlatoms a in Herbrand interpretations I also in
volves negated concept inclusion axioms
:(CvD),negated concept membership axioms
:C(a),and negated role membership axioms
:R(a;b) and:U(a;v).For this reason,we
slightly extend the standard syntax and semantics of
SHIF(D) and SHOIN(D) by also
allowing such negated axioms.
8
The notions of satisfaction,satisﬁability,and entailment
are naturally extended to handle such negated axioms.In particular,a ﬁrstorder interpre
tation
I = (¢
I
;¢
I
) satisﬁes a negated axiom
:F,where F is equal to (CvD) (resp.,
C(a),R(a;b),U(a;v)),denoted I j=:F,iff
C
I
6µD
I
(resp.,a
I
62 C
I
,
(a
I
;b
I
) 62
R
I
,(a
I
;v
D
) 62 U
I
).Entailment (for dlatoms) in the slight extensions of
SHIF(D)
and SHOIN(D) can then be reduced to entailment in
SHIF(D) and SHOIN(D)
[Eiter et al.2008],respectively.
Aground dlatoma is monotonic relative to KB =(L;P) iff
I µI
0
µHB
P
implies that
if
I j=
L
a then I
0
j=
L
a.In this paper,we focus on monotonic ground dlatoms relative to a
dlprogram (which seem to be most natural),but one can also deﬁne nonmonotonic ones
(see [Eiter et al.2004;2008] and Section 9 for further discussion).
Like ordinary positive programs,every positive dlprogram KB is satisﬁable and has a
unique least model,denoted
M
KB
,which naturally characterizes its semantics.
The strong answer set semantics of general dlprograms is then deﬁned by a reduction to
the least model semantics of positive ones as follows,using a generalized transformation
that removes all defaultnegated atoms in dlrules.For dlprograms
KB =(L;P),the
strong dltransform of
P relative to L and an interpretation
I µHB
P
,denoted sP
I
L
,is
the set of all positive dlrules obtained from
ground(P) by (i) deleting every dlrule
r
such that
I j=
L
a for some a2B
¡
(r),and (ii) deleting from each remaining dlrule
r
the negative body.Notice that
sP
I
L
generalizes the GelfondLifschitz reduct P
I
[Gelfond
and Lifschitz 1991].Let
KB
I
denote the dlprogram (L;sP
I
L
).Since KB
I
is positive,
it has a unique least model.A strong answer set (or simply answer set) of
KB is an
interpretation I µHB
P
that coincides with the unique least model of
KB
I
.
Example 3.2 (Product Database cont’d)
The dlprogramKB
1
=(L
1
;P
1
) of Example 3.1
has the following three strong answer sets (only relevant atoms are shown):
fsupplied(s
3
;case);supplied(s
2
;cpu);supplied(s
2
;harddisk);rebate(s
2
);...g;
fsupplied(s
3
;case);supplied(s
3
;harddisk);rebate(s
3
);...g;
fsupplied(s
3
;case);...g.
Since the contractor
s
3
was already ﬁxed for the part case,two possibilities for a discount
remain (
rebate(s
2
) or rebate(s
3
);
s
1
is not offering the needed part harddisk,and the
shop will not give a discount only for the part
cpu).
We nowsummarize some properties.The strong answer sets of a dlprogram
KB =(L;P)
without dlatoms coincide with the ordinary answer sets of
P [Gelfond and Lifschitz 1991].
Moreover,strong answer sets of a general dlprogram KB are also minimal models of
KB.Finally,positive and stratiﬁed dlprograms have exactly one strong answer set,which
coincides with their canonical minimal model.(For stratiﬁed dlprograms,see Section 5.)
8
Actually,OWL 2 follows a similar pattern,allowing for negative property membership assertions,cf.
http://www.w3.org/TR/2008/WDowl2quickreference20081202/.Negative role mem
bership axioms can also be easily emulated using qualiﬁed role expressions,cf.[Eiter et al.2008];for DLs
with limited expressiveness,
¡
[ can be simply restricted to concepts.
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
12 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
4.WELLFOUNDED SEMANTICS
In this section,we deﬁne the wellfounded semantics for dlprograms.We do this by
generalizing the wellfounded semantics for ordinary normal programs.More speciﬁcally,
we generalize the deﬁnition based on unfounded sets as given in Section 2.
We ﬁrst deﬁne the notion of an unfounded set for dlprograms KB =(L;P).This is
not that easy technically:ﬁrst,truth and falsity of dlatoms depend on
L,besides
P.Sec
ond,establishing deﬁnite falsity of a positive classical atom b in a rule body is as easy as
checking that
:b appears in the current interpretation.Instead,for proving that a positive
dlatom is deﬁnitely false,it is necessary to consider a more general sufﬁcient condition,
which accounts for any possible further expansion of the current interpretation.These
considerations lead to the following notion of unfounded set for dlprograms.
Deﬁnition 4.1 (Unfounded Set)
Let I µLit
P
be consistent.A set
U µHB
P
is an un
founded set of KB relative to I iff the following holds:
9
(¤)
for every a2U and every
r2ground(P),if
H(r) =a,either (i)
:b 2I [::U for some
ordinary atom
b
2
B
+
(
r
)
,or (ii)
b
2
I
for some ordinary atom
b
2
B
¡
(
r
)
,or (iii) for
some dlatom
b 2B
+
(r),it holds that
S
+
6j=
L
b for every consistent
S µ Lit
P
with
I [::U µS,or (iv) for some dlatom
b 2B
¡
(r),I
+
j=
L
b.
What is new here are conditions (iii) and (iv).Intuitively,(iv) says that
not b is for sure
false,regardless of how I is further expanded,while (iii) says that
b will never become
true,if we expand
I in a way such that all unfounded atoms are kept false.The following
examples illustrate the concept of an unfounded set for dlprograms.
Example 4.2
Consider KB
2
=(L
2
;P
2
),where L
2
=fS vCg and
P
2
is as follows:
p(a) ÃDL[S ]q;C](a);q(a) Ãp(a);r(a) Ãnot q(a);not s(a):
Here,
S
1
=fp(a);q(a)g is an unfounded set of
KB
2
relative to
I =;,since p(a) is un
founded due to (iii),while
q(a) is unfounded due to (i).The set S
2
=fs(a)g is trivially an
unfounded set of
KB
2
relative to I,since no rule deﬁning
s(a) exists.
Relative to I =fq(a)g,S
1
is not an unfounded set of KB
2
(for
p(a),the condition fails)
but S
2
is.The set
S
3
=fr(a)g is another unfounded set of KB
2
relative to I.The greatest
unfounded set of
KB
2
relative to I is
S
2
[S
3
= fs(a);r(a)g.
Example 4.3
Consider a variant KB
3
=(L
3
;P
3
) of the dlprogram KB
2
=(L
2
;P
2
) of
Example 4.2 where
L
3
=L
2
=fS vCg,and P
3
is obtained from P
2
by negating the dl
literal in
P
2
,i.e.,it contains the rules
p(a) Ãnot DL[S ]q;C](a);q(a) Ãp(a);r(a) Ãnot q(a);not s(a):
Then,
S
1
= fp(a);q(a)g is not an unfounded set of
KB
3
relative to I =;(for the rule
deﬁning p(a),conditions (i)–(iii) are void,and condition (iv) fails),but
S
2
=fs(a)g is.
Relative to I =fq(a)g,however,both
S
1
and S
2
as well as S
3
=fr(a)g are unfounded
sets of
KB
3
.
9
Note that queries in dlatoms of [Eiter et al.2004;2008] are based on positive (founded) atoms p(a) in an
interpretation;unfounded atoms p(a) (i.e.,negative literals:p(a)) can be easily taken into account by using a
predicate p
0
deﬁned by p
0
(X) Ãnot p(X) and then updating the DL knowledge base with
¡
[p
0
.
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 13
Example 4.4
Among the unfounded sets of
KB
1
=(L
1
;P
1
) in Example 3.1 relative to
I
0
=;,there is fbuy
cand(s
1
;harddisk),
buy
cand(s
2
;case),
buy
cand(s
3
;cpu)g due
to (iii),since the dlatom in rule (5) of
P
1
will never evaluate to true for these pairs.This
reﬂects the intuition that the concept
providerOf narrows the choice for buying candidates.
The following lemma shows that the set of unfounded sets of
KB relative to I is closed
under union,which implies that KB has a greatest unfounded set relative to
I.
Lemma 4.5
Let KB =(L;P) be a dlprogram,and let
I µLit
P
be consistent.Then,the
set of unfounded sets of
KB relative to I is closed under union.
Based on the above result it turns out that KB has a greatest unfounded set relative to
I.
We now generalize the operators
T
P
,U
P
,and W
P
to dlprograms as follows.
Deﬁnition 4.6 (
T
KB
,U
KB
,W
KB
)
The operators T
KB
,U
KB
,and W
KB
on all consistent
IµLit
P
are as follows:
—
a 2 T
KB
(I) iff a 2 HB
P
and some r 2 ground(P) exists such that (a)
H(r) =a,
(b) I
+
j=
L
b for all b 2 B
+
(r),(c)
:b 2 I for all ordinary atoms b 2 B
¡
(r),and
(d)
S
+
6j=
L
b for each consistent
S µ Lit
P
with
I µ S,for all dlatoms b 2 B
¡
(r);
—
U
KB
(I) is the greatest unfounded set of KB relative to I;and
—
W
KB
(I) =T
KB
(I) [::U
KB
(I).
Note that T
KB
(I)\U
KB
(I) =;,and thus W
KB
(I) is indeed welldeﬁned.The follow
ing result shows that the three operators are all monotonic.
Lemma 4.7
Let KB be a dlprogram.Then,T
KB
,U
KB
,and W
KB
are monotonic.
Thus,in particular,W
KB
has a least ﬁxpoint,denoted
lfp(W
KB
).The wellfounded
semantics of dlprograms can thus be deﬁned as follows.
Deﬁnition 4.8 (Wellfounded Semantics)
Let KB =(L;P) be a dlprogram.The well
founded semantics of KB,which we denote as
WFS(KB),is deﬁned as lfp(W
KB
).An
atom
a2HB
P
is wellfounded (resp.,unfounded) relative to KB iff
a (resp.,:a) belongs
to WFS(KB).
The following examples illustrate the wellfounded semantics of dlprograms.
Example 4.9
Consider KB
2
of Example 4.2.For
I
0
=;,we have T
KB
2
(I
0
) =;and
U
KB
2
(I
0
) =fp(a);q(a);s(a)g.Hence,W
KB
2
(I
0
) =f:p(a);:q(a);:s(a)g (=I
1
).In the
next iteration,
T
KB
2
(I
1
) =fr(a)g and U
KB
2
=fp(a);q(a);s(a)g.Thus,
W
KB
2
(I
1
) =
f:p(a);:q(a);r(a);:s(a)g (=
I
2
).Since I
2
is total and W
KB
2
is monotonic,it follows
W
KB
2
(I
2
) =I
2
and hence WFS(KB
2
) =f:p(a);:q(a);r(a);:s(a)g.Accordingly,we
ﬁnd that
r(a) is wellfounded and all other atoms are unfounded relative to
KB
2
.Note
that KB
2
has the unique answer set
I =fr(a)g.
Example 4.10
Now consider KB
3
of Example 4.3.For I
0
=;,we have
T
KB
3
(I
0
) =;
and U
KB
3
(I
0
) =fs(a)g.Hence,
W
KB
3
(I
0
) =f:s(a)g (=I
1
).In the next iteration,we
have
T
KB
3
(I
1
) =;and U
KB
3
(I
1
) =fs(a)g.Then,
W
KB
3
(I
1
) =I
1
and WFS(KB
3
) =
f:s(a)g;atoms(a) is unfounded relative to
KB
3
.Note that KB
3
has no answer set.
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
14 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
Example 4.11
Consider again U
KB
1
(I
0
=;) of Example 4.4.Then,
W
KB
1
(I
0
) consists
of:U
KB
1
(I
0
) and all facts of
P
1
.This input to the ﬁrst iteration along with (iii) ap
plied to rule (8) adds those
supplied atoms to
U
KB
1
(I
1
) that correspond to the (negated)
buy
cand atoms of
U
KB
1
(I
0
).Then,T
KB
1
(I
1
) contains exclude(case) which forces ad
ditional
buy
cand atoms into U
KB
1
(I
2
),regarding (i) and rule (5).The same unfounded
set thereby includes
rebate(s
1
),stemming from rule (4).As a consequence,avoid(s
1
) is
in
T
KB
1
(I
3
).Eventually,the ﬁnal WFS(KB
1
) is not able to make any positive assumption
about choosing a new vendor (
buy
cand),but it is clear about s
1
being deﬁnitely not able
to contribute to a discount situation,since a contractor for
case is already chosen in
L
1
,
and s
1
offers only a single further part.
5.SEMANTIC PROPERTIES
In this section,we explore the semantic properties of the wellfounded semantics for dl
programs,and their relationship to the strong answer set semantics.An immediate result
is that it conservatively extends the wellfounded semantics for ordinary normal programs.
Theorem5.1
Let KB =(L;P) be a dlprogramwithout dlatoms.Then,the wellfounded
semantics of
KB coincides with the wellfounded semantics of
P.
The next result shows that the wellfounded semantics of a dlprogram
KB =(L;P) is
a partial model of
KB.Here,a consistent I µLit
P
is a partial model of
KB iff some
consistent
J µLit
P
exists such that (i)
I µJ,(ii)
J
+
is a model of KB,and (iii) J is
total,that is,for all
a 2 HB
P
,either a 2 J or
:a 2 J.Intuitively,a partial model
I,
which expresses a threevalued interpretation,is such that it can be completed obtaining a
(twovalued) model
I
0
µHB
P
of KB.
Theorem5.2
Let KB be a dlprogram.Then,
WFS(KB) is a partial model of KB.
Importantly,the wellfounded semantics for dlprograms can be characterized in terms
of the least and the greatest ﬁxpoint of a monotonic operator
°
2
KB
similarly to the well
founded semantics for ordinary normal programs [Baral and Subrahmanian 1993].We
then use this characterization to derive further properties of the wellfounded semantics for
dlprograms.
Deﬁnition 5.3
For a dlprogramKB =(L;P),let the operator °
KB
on
I µHB
P
be
°
KB
(I) =M
KB
I
;
which is the least model of the positive dlprogram
KB
I
=(L;sP
I
L
) (recall that sP
I
L
is the
strong dltransformof
P relative to L and I fromSection 3.2).
The next result shows that °
KB
is antimonotonic,like its counterpart for ordinary nor
mal programs [Baral and Subrahmanian 1993].Note that this result holds only if all dl
atoms in
P are monotonic;this,however,is clearly ensured if in dlatoms only the update
operators
] and
¡
[ can occur.
Proposition 5.4
Let KB =(L;P) be a dlprogram.Then,°
KB
is antimonotonic.
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WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 15
Hence,the operator °
2
KB
(I) =°
KB
(°
KB
(I)),for all I µHB
P
,is monotonic and thus
has a least and a greatest ﬁxpoint,denoted
lfp(°
2
KB
) and
gfp(°
2
KB
),respectively.We can
use these ﬁxpoints to characterize the wellfounded semantics of
KB.
Theorem5.5
Let KB =(L;P) be a dlprogram.Then,an atom
a2HB
P
is wellfounded
(resp.,unfounded) relative to KB iff
a2lfp(°
2
KB
) (resp.,a62gfp(°
2
KB
)).
Example 5.6
Consider again the dlprogramKB
1
of Example 3.1.Then:
°
KB
1
(;) = F[ favoid(c),
rebate(c) j c 2 fs
1
;s
2
;s
3
gg[
fexclude(c) j c 2 fharddisk;case;cpugg [
fbuy
cand(s
1
;cpu),buy
cand(s
1
;case),
buy
cand(s
2
;harddisk),
buy
cand(s
2
;cpu),
buy
cand(s
3
;harddisk),buy
cand(s
3
;case),
supplied(s
1
;cpu),supplied(s
1
;case),supplied(s
2
;cpu),
supplied(s
2
;harddisk),
supplied(s
3
;harddisk),supplied(s
3
;case)g,
°
2
KB
1
(;) = F [fexclude(case),
supplied(s
3
;case)g.
where F = fvendor(s
1
);vendor(s
2
);vendor(s
3
);
needed(cpu),needed(harddisk),
needed(case) g is the set of facts in the rule part.The set
lfp(°
2
KB
1
) contains the atoms
avoid(s
1
),
supplied(s
3
;case),and exclude(case),while gfp(°
2
KB
1
) does not contain
rebate(s
1
).Thus,
WFS(KB
1
) contains the literals avoid(s
1
),supplied(s
3
;case),and
:rebate(s
1
),corresponding to the result of Example 4.11 (and,moreover,to the intersec
tion of all answer sets of
KB
1
).
The next theoremshows that the wellfounded semantics for dlprograms approximates
their strong answer set semantics.That is,every wellfounded ground atomis true in every
answer set,and every unfounded ground atomis false in every answer set.
Theorem5.7
Let KB =(L;P) be a dlprogram.Then,every strong answer set of
KB
includes all atoms a2HB
P
that are wellfounded relative to
KB and no atom
a2HB
P
that is unfounded relative to KB.
A ground atom a is a cautious (resp.,brave) consequence under the strong answer set
semantics of a dlprogram
KB iff a is true in every (resp.,some) strong answer set of
KB.
Hence,under the strong answer set semantics,we have the following result.
Corollary 5.8
Let KB =(L;P) be a dlprogram.Then,under the strong answer set se
mantics,every wellfounded atom
a2HB
P
relative to KB is a cautious consequence
of
KB,and no unfounded atom
a2HB
P
relative to KB is a brave consequence of a
satisﬁable
KB.
If the wellfounded semantics of a dlprogram
KB=(L;P) is total,that is,contains
either
a or:a for every a2HB
P
,then it speciﬁes the only strong answer set of
KB.
Theorem5.9
Let KB =(L;P) be a dlprogram.If every atom
a2HB
P
is either well
founded or unfounded relative to KB,then the set of all wellfounded atoms
a2HB
P
relative to KB is the only strong answer set of
KB.
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
16 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
Regarding the meaning of the wellfounded semantics for dlprograms compared to the
(strong) answer set semantics,similar intuitions apply as in the case of ordinary logic
programs.For instance,the wellfounded semantics remains agnostic in case of cyclic
negation,while the answer set semantics either gets inconsistent (in case of odd cycles) or
branches into different cases;for more discussion,we refer to [van Gelder et al.1991].
Like in the case of ordinary normal programs,the wellfounded semantics for positive
and stratiﬁed dlprograms is total and coincides with their least model semantics and it
erative least model semantics,respectively.This result can be elegantly proved using the
characterization of the wellfounded semantics given in terms of the
°
2
KB
operator.
According to [Eiter et al.2004;2008],a stratiﬁcation of a dlprogram KB =(L;P)
is a mapping
¹:HB
P
[DL
P
!f0;1;:::;kg,where
DL
P
is the set of all dlatoms in
ground(P) and
k > 0,such that
(i)
for each r2ground(P),¹(H(r))>¹(l
0
) for each
l
0
2B
+
(r),and ¹(H(r))>¹(l
0
) for
each
l
0
2B
¡
(r),and
(ii)
¹(a) >¹(l) for each input literal
l of each a2DL
P
.
A dlprogram KB is stratiﬁed,if some stratiﬁcation of KB exists.For more background
and the deﬁnition of the iterated least model semantics,we refer to [Eiter et al.2004;2008].
Theorem5.10
Let KB =(L;P) be a dlprogram.If
KB is positive (resp.,stratiﬁed),
then (a)
WFS(KB) is a total model,that is,
WFS(KB)
+
[(::WFS(KB))
+
=HB
P
,
and (b)
WFS(KB)\HB
P
is the least model (resp.,the iterative least model) of
KB,
which coincides with the unique strong answer set of
KB.
Example 5.11
The dlprogramKB
2
in Example 4.2 is stratiﬁed (intuitively,the recursion
through negation is acyclic) while
KB
3
in Example 4.3 is not.The result computed in
Example 4.9 veriﬁes the conditions of Theorem5.10.
We ﬁnally show that we can limit ourselves to dlprograms in dlquery form,where dl
atoms equate designated predicates.Formally,a dlprogram
KB = (L;P) is in dlquery
form,if each
r 2P involving a dlatomis of the form
aÃb,where b is a dlatom.Any dl
program
KB =(L;P) can be transformed into a dlprogram
KB
dl
=(L;P
dl
) in dlquery
form.Here,
P
dl
is obtained from P by replacing every dlatom a(t) = DL[¸;Q](t),
t = t
1
;:::;t
n
,by
p
a
(t),and by adding the dlrule
p
a
(X) Ãa(X) to
P,where p
a
is a new
predicate symbol,and
X = X
1
;:::;X
n
is a list of distinct variables.Informally,
p
a
is an
abbreviation for a.
The following result now shows that KB
dl
and KB are equivalent under the well
founded semantics.Intuitively,this means that the wellfounded semantics tolerates abbre
viations in the sense that they do not change the semantics of a dlprogram.This normal
form is particularly useful for the computation of the wellfounded semantics,as it allows
to eliminate dlatoms fromarbitrary rules and to move themto special rules.Another good
property is that the transformation to normal formpreserves stratiﬁcation.
Theorem5.12
Let KB =(L;P) be a dlprogram.Then,
WFS(KB) =WFS(KB
dl
)\
Lit
P
.
We close this section with a brief comment on dlprograms with nonmonotonic dlatoms
[Eiter et al.2008].The latter also have the form (2),but
op
i
may be
¡
\,where
S
i
¡
\p
i
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 17
Table I.Complexity of literal entailment fromdlprograms
KB =(L;P) under the wellfounded semantics.
L in SHIF(D) L in SHOIN(D)
General Complexity
EXPcomplete
P
NEXP
complete
Data Complexity
P
NP
complete P
NP
complete
increases:S
i
by the complement of
p
i
.Given that
p
i
is the complement of p
i
,
S
i
¡
\p
i
is
equivalent to S
i
¡
[
p
i
.The predicate
p
i
can be deﬁned with a rule
p
i
(X) Ãnot DL[S
0
i
]
p
i
;S
0
i
](X),where
S
0
i
is a fresh concept resp.role name,provided that the DL knowledge
base is under unique names assumption (i.e.,different constant denote different objects)
and satisﬁable.In this way,any dlprogram
KB =(L;P) with satisﬁable
Lcan be rewritten
to the most relevant fragment that we consider here;for unsatisﬁable
L,the rewriting is also
usable (though
p
i
may not be the complement of p
i
).
6.COMPUTATION AND COMPLEXITY
In this section,we show how the wellfounded semantics of dlprograms
KB can be com
puted by ﬁnite sequences of ﬁnite ﬁxpoint iterations,using the operator
°
KB
and the imme
diate consequence operator
T
KB
of positive dlprograms KB.We also analyze the general
and the data complexity of reasoning from dlprograms under the wellfounded semantics
(as for data complexity,we assume that the size of data includes the factual part of both
the rules and the DL knowledge base).Our complexity results are compactly summarized
in Table 6.In detail,deciding literal entailment froma dlprogram
KB =(L;P) with L in
SHIF(D) (resp.,SHOIN(D)) under the wellfounded semantics is complete for EXP
(resp.,
P
NEXP
) in general,and complete for
P
NP
(for both DLs) under data complexity.
In fact,the P
NP
upper bound for data complexity extends to all description logics
L for
which literal inference
I j=
L
a is decidable in polynomial time with an NP oracle under
data complexity.
6.1 Fixpoint Iteration
The wellfounded semantics of dlprograms KB can be computed by two ﬁnite ﬁxpoint
iterations,via the operator
°
KB
,using in turn ﬁnite ﬁxpoint iterations for computing the
least models of positive dlprograms,via their immediate consequence operator.
More concretely,for any positive dlprogram KB =(L;P),the least model of KB,de
noted M
KB
,coincides with the least ﬁxpoint of the immediate consequence operator
T
KB
[Eiter et al.2004],which is deﬁned as follows for every
I µHB
P
:
T
KB
(I) =fH(r) j r 2ground(P);I j=
L
`for all`2B(r)g:
In order to compute the wellfounded semantics of a normal dlprogramKB = (L;P),
that is,WFS(KB) = lfp(°
2
KB
) [:(HB
P
¡gfp(°
2
KB
)),we compute the least and the
greatest ﬁxpoint of
°
2
KB
as the limits of the two ﬁxpoint iterations
lfp(°
2
KB
) = A
1
=
S
i>0
A
i
;where A
0
=;;and A
i+1
= °
2
KB
(A
i
);for i > 0;and
gfp(°
2
KB
) = O
1
=
T
i>0
O
i
;where O
0
= HB
P
;and O
i+1
= °
2
KB
(O
i
);for i > 0;
respectively,which are both reached within
jHB
P
j many steps.Recall that the operator
°
KB
is deﬁned by °
KB
(I) =M
KB
I
(with
KB
I
=(L;sP
I
L
)),for all I µHB
P
.As argued
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
18 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
above,
M
KB
I
coincides with lfp(T
KB
I
),for all
I µHB
P
.To compute °
KB
(I),for all
I µHB
P
,we thus compute the least ﬁxpoint of
T
KB
I
as the limit of the ﬁxpoint iteration
lfp(T
KB
I
) = S
1
=
S
i>0
S
i
;where S
0
=;;and S
i+1
= T
KB
I
(S
i
);for i > 0;
which is also reached within
jHB
P
j many steps.
6.2 General Complexity
We recall that for a given ordinary normal program,computing the wellfounded model
needs exponential time in general (measured in the program size [Dantsin et al.2001]),
and also reasoning from the wellfounded model has exponential time complexity.Fur
thermore,evaluating a ground dlatom
a of the form (2) for
KB =(L;P) given an inter
pretation I
p
of its input predicates
p = p
1
;:::;p
m
(that is,deciding whether I j=
L
a holds
for each
I that coincides on p with I
p
) is complete for EXP (resp.,coNEXP) for
L in
SHIF(D) (resp.,SHOIN(D)) [Eiter et al.2004],where EXP (resp.,NEXP) denotes
exponential (resp.,nondeterministic exponential) time;this is inherited fromthe complex
ity of deciding whether a knowledge base in
SHIF(D) (resp.,
SHOIN(D)) is satisﬁ
able [Tobies 2001;Horrocks and PatelSchneider 2004].
The following result shows that computing the wellfounded semantics of a dlprogram
KB =(L;P) over SHIF(D) is feasible in exponential time,and that reasoning fromsuch
programs under the wellfounded semantics is EXPcomplete;hardness holds even when
Lis empty or P contains only one rule.That is,the complexity of the wellfounded seman
tics for such programs does not increase over the one of ordinary normal programs.The
membership part follows from the above ﬁxpoint characterization of the wellfounded se
mantics of dlprograms and the EXPmembership of deciding
I j=
L
a for Lin SHIF(D),
while the hardness part follows fromthe EXPhardness of reasoning fromthe wellfounded
semantics of ordinary normal programs as well as the EXPhardness of deciding knowl
edge base satisﬁability in
SHIF(D).
Theorem6.1
Given a vocabulary ©and a dlprogramKB =(L;P) with Lin SHIF(D),
computing
WFS(KB) is feasible in exponential time.Furthermore,given additionally a
literal
l 2Lit
P
,deciding whether
l 2WFS(KB) holds is EXPcomplete.Hardness holds
even in the cases where (a)
L is empty or (b) P contains only one rule.
For dlprograms over SHOIN(D),the computation of the wellfounded semantics and
reasoning from it is expected to be more complex than for dlprograms over
SHIF(D),
since already evaluating a single dlatom is coNEXPhard.Computing the wellfounded
semantics is feasible,in a similar manner as in the case of
SHIF(D),in exponential
time using an oracle for evaluating dlatoms;to this end,an NP oracle is sufﬁcient.As
for the reasoning problem,this means that deciding whether
l 2WFS(KB) holds is in
EXP
NP
.A more precise account reveals the following strict characterization of the com
plexity,showing that reasoning from dlprograms
KB =(L;P) over SHOIN(D) under
the wellfounded semantics is complete for P
NEXP
,which is intuitively strictly contained
in
EXP
NP
,
10
and hardness holds even when
P is stratiﬁed.The membership part follows
10
In EXP
NP
,a NEXP oracle can be emulated,and computation trees with branching on the (emulated) oracle
answers can have double exponentially many paths and exponential depth;intuitively,ﬁnding the correct compu
tation path in such a tree needs exponentially many NEXP oracle calls.Still P
NEXP
=EXP
NP
is possible,e.g.,
if NEXP=EXP and NP=P.
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 19
fromthe above ﬁxpoint characterization of the wellfounded semantics of dlprograms and
the coNEXPmembership of deciding
I j=
L
a for Lin SHOIN(D),using a census tech
nique,which essentially allows to evaluate all dlatoms in advance in polynomial time with
an oracle for NEXP,while the hardness part follows from the P
NEXP
hardness of strong
answer set existence for stratiﬁed dlprograms [Eiter et al.2004].
Theorem6.2
Given a vocabulary ©,a dlprogramKB =(L;P) with L in SHOIN(D),
and a literal
l 2Lit
P
,deciding whether
l 2WFS(KB) holds is P
NEXP
complete.Hard
ness holds even in the case where
P is stratiﬁed.
The results in Theorems 6.1 and 6.2 also show that,like for ordinary normal programs,
inference under the wellfounded semantics is computationally less complex than under
the answer set semantics for dlprograms
(L;P) with L from SHIF(D),as cautious
reasoning from the strong answer sets such a dlprograms is complete for coNEXP;with
L fromSHOIN(D),the complexity is the same.[Eiter et al.2004;2008].
Analog complexity results for literal inference under the wellfounded semantics can
be derived for
L from other DLs;for the upcoming OWL2 proposal,an adjusted proof of
Theorem6.2 shows that the problemis in
P
2NEXP
(and presumably also complete for this
class),and for the OWL2 proﬁles EL,QL,and RL,an adjusted proof of Theorem 6.1 that
it is EXPcomplete.This is because deciding
I j=
L
a for Lin the DL SROIQunderlying
OWL2 is co2NEXPcomplete,as follows from[Kazakov 2008],and for
L in EL,QL,and
RL is polynomial.
11
6.3 Data Complexity
We now explore the data complexity of reasoning from dlprograms
KB =(L;P) under
the wellfounded semantics.Here,only the constant symbols in the vocabulary
©,the
concept and role membership axioms in
L,and the facts in P may vary,while the rest of
©,L,and P is ﬁxed.The following result,which follows from the above ﬁxpoint char
acterization of the wellfounded semantics of dlprograms,shows that the data complexity
of dlprograms does not increase much compared to the one of query answering in the
description logic where
L is from.
12
Proposition 6.3
Given a vocabulary ©,a dlprogram KB =(L;P) with L from a des
cription logic
L for which deciding I j=
L
a has data complexity in class
C,and a literal
l 2Lit
P
,deciding whether l 2WFS(KB) holds is in P
C
under data complexity.
Exploiting this,we derive that for both L = SHIF(D) and
L = SHOIN(D) the
problem is
P
NP
complete under data complexity;hardness holds even when
L is in ALE
and
P is stratiﬁed.Indeed,unsatisﬁability and instance checking in
SHOIN(D) (and
SROIQ(D)) are in coNP under data complexity (which follows from results in [Pratt
Hartmann 2008]);the hardness part is shown by a generic reduction fromTuring machines,
exploiting the coNPhardness proof for instance checking in
ALE by Donini et al.[1994].
11
As follows fromhttp://www.w3.org/TR/2008/WDowl2profiles20081202/.
12
Note that a slightly modiﬁed construction can be used to derive the data complexity of deciding consistency
and of cautious/brave reasoning under strong/weak answer sets.
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
20 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
Theorem6.4
Given a vocabulary ©,a dlprogram KB =(L;P) with L in SHIF(D)
and a literal
l 2Lit
P
,deciding whether l 2WFS(KB) holds is P
NP
complete under data
complexity.Hardness holds even in the case where (i)
L is in ALE and (ii)
P is stratiﬁed.
7.DATA TRACTABILITY
We nowdelineate special cases where reasoning fromdlprograms under the wellfounded
semantics can be done in polynomial time and in LOGSPACE in the data complexity.
7.1 Polynomial Case
We ﬁrst focus on the case where the evaluation of all dlatoms in a dlprogramcan be done
in polynomial time.In this case,reasoning from dlprograms under the wellfounded se
mantics is complete for P under data complexity,and thus has the same data complexity as
reasoning from ordinary normal programs under the wellfounded semantics.This result
is formally expressed by the following theorem,whose membership part follows imme
diately from Proposition 6.3 while the hardness part follows from the Pcompleteness of
reasoning fromthe wellfounded semantics of ordinary normal programs.
Theorem7.1
Given a vocabulary ©,a dlprogram KB =(L;P),and a literal l 2Lit
P
,
where every dlatom in
P can be evaluated in polynomial time,deciding whether
l 2
WFS(KB) is complete for P under data complexity.Hardness holds even if
L =;.
Since there is a current trend towards highly scalable query answering and reasoning
over ontologies,there are many recent DLs that allow for evaluating dlatoms in polyno
mial time.Among the most expressive ones is Horn
SHIQ [Hustadt et al.2005],which
is a fragment of the description logic behind OWL Lite,and which allows for reasoning
and conjunctive query answering in polynomial time under data complexity [Eiter et al.
2008].The following theorem shows that reasoning from dlprograms
KB =(L;P) un
der the wellfounded semantics,where
L is deﬁned in HornSHIQ,has the same data
complexity as in the ordinary case,when all concepts in dlqueries in
P are atomic.
Theorem7.2
Given a vocabulary ©,a dlprogram KB =(L;P),and a literal l 2Lit
P
,
where (i)
L is deﬁned in HornSHIQ,and (ii) all concepts C and D in dlqueries of one
of the forms among
CvD,:(CvD),C(t),and:C(t) in P are atomic (including
?
and >),deciding whether
l 2WFS(KB) is complete for P under data complexity.
Similarly,under data complexity,literal inference under the wellfounded semantics is
Pcomplete for dlprograms over knowledge bases in the OWL2 proﬁles EL,QL,and RL.
7.2 FirstOrder Rewritable Case
We next consider the case where the evaluation of every dlquery in a dlprogram
KB =
(L;P) is ﬁrstorder rewritable.In this case,if we make additional acyclicity assumptions
about
P,then reasoning from dlprograms under the wellfounded semantics is also ﬁrst
order rewritable,which implies that reasoning from dlprograms under the wellfounded
semantics can be done in LOGSPACE under data complexity.
Here,a dlquery Q(t) over L is ﬁrstorder rewritable iff it can be expressed in terms
of a ﬁrstorder formula
Á(t) over the set L
CR
of all concept and role membership axioms
in
L,that is,for every c,it holds that
L j= Q(c) iff I
L
CR
j= Á(c),where for any set
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 21
of atoms
F,we denote by I
F
the total Herbrand interpretation that satisﬁes exactly the
atoms in
F (i.e.,under the closed world assumption on
F).
13
The dlprogram
KB is ﬁrst
order rewritable iff the extension of every predicate
p(x) in WFS(KB) can be expressed
in terms of a ﬁrstorder formula
Á(x) over the set F of all concept and role membership
axioms in
Land all database facts in P,that is,for every
c,it holds that p(c) 2WFS(KB)
iff
I
F
j= Á(c).Informally,such dlatoms and predicates can be expressed in terms of SQL
queries over a relational database.The notion of acyclicity for dlprograms assures that
they are ﬁrstorder rewritable when all dlatoms are so.It is deﬁned as follows.Let
P
P
denote the set of all predicate symbols in P.We say
KB =(L;P) is acyclic iff a map
ping
·:P
P
!f0;1;:::;ng exists such that for every
r 2P,the predicate symbol p
of H(r),and every predicate symbol
q of some ordinary b 2B(r) or of an input argument
of some dlatom
b 2B(r),it holds
·(p) >·(q).
The next result shows that reasoning from acyclic dlprograms
KB = (L;P) under the
wellfounded semantics is ﬁrstorder rewritable (and thus literal inference can be decided in
LOGSPACE under data complexity),when (i) all dlqueries in
P are ﬁrstorder rewritable,
and (ii) if the operator
¡
[ occurs in P,then
L is deﬁned over a description logic that (ii.a)
is CWAsatisﬁable (that is,for every description logic knowledge base L
0
,the union of
L
0
and all negations of concept and role membership axioms that are not entailed by L
0
is
satisﬁable) and (ii.b) allows for ﬁrstorder rewritable concept and role memberships.
Theorem7.3
Let © be a vocabulary,KB = (L;P) an acyclic dlprogram,and
l 2 Lit
P
a literal,such that (i) every dlquery in
P is ﬁrstorder rewritable and (ii) if the operator
¡
[ occurs in P,then
L is deﬁned over a description logic that (ii.a) is CWAsatisﬁable,
and (ii.b) allows for ﬁrstorder rewritable concept and role memberships.Then,deciding
whether
l 2 WFS(KB) is ﬁrstorder rewritable.
In particular,reasoning from acyclic dlprograms
KB =(L;P) under the wellfounded
semantics is ﬁrstorder rewritable (and thus can be done in LOGSPACE under data com
plexity),when (i)
L is deﬁned in a description logic of the DLLite family [Calvanese et al.
2007] (in which knowledge base satisﬁability and conjunctive queries are both ﬁrstorder
rewritable) and (ii) we assume suitable restrictions on dlqueries in
P.
Theorem7.4
Given a vocabulary ©,an acyclic dlprogram
KB = (L;P),and a lit
eral
l 2Lit
P
,where (i)
L is deﬁned in a description logic of the DLLite family,and
(ii) all dlqueries in
P are of one of the forms
CvD,:(CvD),
C(t),and R(t;s),
where C is an atomic concept,and
D is an atomic or a negated atomic concept,deciding
whether
l 2WFS(KB) is ﬁrstorder rewritable.
Finally,we remark that the LOGSPACE feasibility generalizes fromﬁrstorder rewritable
dlatoms to one that can be evaluated in LOGSPACE,but omit further details.
8.IMPLEMENTATION
Based on the ideas of Section 6,we developed an experimental system for computing the
wellfounded semantics of a given dlprogram
KB =(L;P).It consists of three separate
13
Note that the notion of ﬁrstorder rewritability here does not mean that every knowledge base Lin a description
logic L can be expressed as an equivalent ﬁrstorder theory (which holds for most description logics).Note also
that the ﬁrstorder rewritability here corresponds to the ﬁrstorder reducibility in [Calvanese et al.2007].
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
22 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
modules:the answer set solver DLV [Leone et al.2006],the description logic reasoner
RACER [Haarslev and M
¨
oller 2001],and a module
W that computes WFS(KB) by ac
cessing DLV and RACER.
In a ﬁrst step,a programP
d
is computed fromP by replacing every dlatom
DL[¸;Q](t)
by an atom
p
DL[¸;Q]
(t),where p
DL[¸;Q]
is a fresh predicate.The program
P
d
is then
grounded using the grounding module of the DLV system.For that,optimizations per
formed by that module are properly disabled (otherwise,the result may not be sound for
our purposes).After appropriately reintroducing the dlatoms in the obtained program
grd(P
d
),the resulting program
P
00
= grd(P
d
)
0
is returned to the module W,which then
computes
lfp(°
2
(L;P
00
)
) and
gfp(°
2
(L;P
00
)
) as deﬁned in Section 6.1.Whenever the truth
value of a given dlatom has to be determined,
W invokes the RACER system;the latter
performs reasoning on
L and variants thereof.
It is worth mentioning that the RACER module has been embedded within a caching
module that shortcuts multiple (time consuming) similar queries;e.g.,the truth value of
DL[¸;C](a) can be quickly established if
DL[C](a) is true and this information is cached;
dually,if
DL[¸;C](a) is cached as false,subsequent queries
DL[C](a) can be answered
by a quick cache lookup.
The module W is also exploited for computing the answer set semantics of
KB.In
virtue of Theorem 5.7,one can indeed,provided
KB is consistent,compute WFS(KB)
and exploit this information for constraining atoms in
lfp(°
2
(L;P
d
)
) as true in any answer
set,while atoms
gfp(°
2
(L;P
d
)
) can be constrained to not appear in any answer set.One can
exploit constraints (i.e.,rules with empty head) in DLV programs for this,which allow
to prune models.An intermediate ordinary program
P
0
obtained from P can be enriched
with the constraint
Ãnot a for any atoma such that a2WFS(KB),and with a constraint
Ãa for any atom a such that
:a2WFS(KB).Notice that such constraints may also be
added only for a subset of
WFS(KB) (e.g.,the subset obtained after some steps in the
least/greatest ﬁxpoint iteration of
°
2
KB
).This technique proves to be useful for helping the
answerset programming solver to converge to solutions faster.
The prototype system
14
in fact supports both the answer set semantics and the well
founded semantics of dlprograms.More details about the architecture and the algorithms,
as well as optimization techniques,can be found in [Eiter et al.2005;Schindlauer 2006;
Eiter et al.2008].
9.RELATED WORK
9.1 Combinations of Rules and Ontologies
Many proposals to integrate rules and ontologies have been made in the last years (see
[Eiter et al.2008;Drabent et al.2009;Rosati 2006;Motik and Rosati 2007a] for recent
surveys).We focus here on important approaches regarding wellfounded semantics.
Donini et al.[1998] combined Datalog with the DL ALC into
ALlog.Arule may have
atoms C(X) where C is a concept in the body,which act as “constraints”;the variable
X
must however also occur in an ordinary body atom (DLsafety).More generally,Rosati’s
DL+log [2006] distinguishes DL and Datalog atoms,which may occur everywhere in a
rule,but not is restricted to Datalog atoms;for decidability,a further weaksafety condition
is imposed.Rosati deﬁned an answer set semantics for a KB
(T;P) by a reduction to
14
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ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 23
ordinary logic programming,which faithfully generalizes the semantics of
T and P.
Rosati and Motik’s [2007b] hybrid MKNF KBs
K=(T;P) treat DL and Datalog atoms
uniformly,thus allowing the operator not to be applied to dlatoms.They resort to a trans
formation of
K into a formula
¼(K) in the logic of Minimal Knowledge and Negation
as Failure (MKNF) [Lifschitz 1991],lifting,in a sense,
DL+log KBs to a more general
and elegant framework.MKNF has the modal operators
KÁ and not Á,which intuitively
mean that
Á is necessarily known to be true respectively that
Á is not true,i.e.,
:Á can be
consistently assumed.Rosati and Motik’s semantics is based on MKNF models,which are
(pairs of) sets of possible worlds,and it naturally captures the answer set semantics of
P.
These approaches assign hybrid KBs a semantics in terms of twovalued models (resp.
sets of such models,in case of MKNF).We now brieﬂy discuss two proposals of well
founded semantics that build on them.
9.1.1 Hybrid Programs.
Drabent and Maluszynski [2007] introduced hybrid programs
(T;P) where
T,the ontology,is a set of DL axioms (in ﬁrstorder logic) and
P is a normal
logic programin which constraint expressions
C
1
;:::;C
m
may occur in rule bodies,where
each C
i
is a DNF over literals
p(X) and:p(X) with ontology predicates
p.In some sense,
hybrid programs are a variant of
DL+log under wellfounded semantics,but closer in spirit
to
ALlog.as ontology predicates cannot occur in rule heads.
The wellfounded semantics for hybrid programs is deﬁned,similar as the
DL+log se
mantics,by a reduction to ordinary logic programming,but under wellfounded semantics;
an operational semantics for query answering,based on an extension of SLDresolution
handling negation and constraints,has been implemented [Drabent et al.2007].
Different fromdlprograms,hybrid programs (T;P) allow only a unidirectional ﬂow of
information from
T to P,as ontology predicates cannot occur in rule heads,and they seem
to be more queryoriented than modeloriented.Query answering from positive hybrid
programs is,like for positive ordinary programs,reducible to (un)provability in classical
logic;this holds only for a fragment of the corresponding class of dlprograms.On the
other hand,hybrid programs allow for reasoning by cases from
T via simple rules.For dl
programs,this is not possible,but such reasoning may be shifted to dlatoms or supported
by more expressive dlatoms (e.g.,cqatoms [Eiter et al.2008]).
Noticeably,inconsistency of T spreads to P,and all ground queries are true.For ex
ample,if T is unsatisﬁable and
P = fq Ã p(a);q Ã:p(a);r Ã not qg,where
p
is an atomic concept,then both
r and q are concluded under hybrid programs semantics;
however,intuitively one may expect that
r is false,as it can never be true regardless of the
contents of
T.The corresponding dlprogram,with reasoning by cases of
p(a) expressed
by q ÃDL[p t:p](a),would conclude that
r is false under the wellfounded semantics.
9.1.2 Hybrid MKNF Knowledge Bases under WellFounded Semantics.
Knorr et al.
[2008] gave a wellfounded semantics for hybrid MKNF KBs
K = (T;P) where T is as
above and
P amounts to a normal logic program;a particular threevalued (partial) MKNF
model
wf (K) of the transformation
¼(K) is selected for the semantics of K.This (Kripke
style) model is determined via an alternating ﬁxpoint construction akin to the one for the
wellfounded semantics of ordinary logic programs.Most of the properties of the tradi
tional wellfounded model are preserved,including that computing
wf (K) is polynomial
in data complexity if entailment in the DL underlying
T has such complexity.
The approach of Knorr et al.bears some similarity to ours as it builds on a monotonic
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
24 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
consequence operator.However,the alternating ﬁxpoint construction has a strong technical
ﬂavor and may be less persuasive than a construction fromﬁrst principles with unfounded
sets.Similar as with hybrid programs above,
wf (K) may not exist if T itself or its interac
tion with
P is not consistent.The latter can be detected in the ﬁxpoint construction,while
inconsistency of
T is not expressible at the object level;in dlprograms,this is trivial (use
e.g.a rule
incons Ã DL[> v?]() ) and exploitable to express paraconsistent behavior.
Finally,the interfacing approach makes dlprograms more amenable for incorporating vari
ants of entailment from the ontology and (possibly heterogenous) other knowledge bases,
which seems more difﬁcult for the tight integration in the hybrid MKNF approach.
9.2 Logic Programming with Aggregates
Our dlprograms are related to extensions of logic programs with aggregates,for which
also a wellfounded semantics has been developed independently,e.g.,[Calimeri et al.
2005;Pelov et al.2007].Such programs allow aggregate atoms in rule bodies,which
in [Calimeri et al.2005] are roughly of the form
f(S) µ k,where
f(S) is an aggregate
function f such as min,max,
sum,or count,applied to a set of elements S that is speciﬁed
using a conjunction of ordinary atoms,
µ is a comparison operator,and
k a value.An
example is#countfX:h(X);p(X;a)g < 2,which evaluates to true if less than two
ground values for
X satisfy the given conjunction.Pelov et al.[2007] considered a notion
of aggregate where
f and µ are abstracted to aggregate functions and aggregate relations.
Intuitively,aggregate atoms work similarly as dlatoms over some given input from the
program,even though the underlying evaluation domain is completely different.Notice
ably,Calimeri et al.[2005] deﬁned a wellfounded semantics of nonmonotonic logic pro
grams
P with aggregates (assuming each is either monotone or antimonotone) based on
a notion of unfounded set,in the usual way [van Gelder et al.1991].According to their
deﬁnition,a set of (ordinary) ground atoms
X is unfounded w.r.t.a given (partial) inter
pretation
I,if for each rule r from the grounding of
P that has some atom from X in the
head,either (a) some antimonotone literal in the body of
r
is false w.r.t.
I
,or (b) some
monotone body literal of
r is false w.r.t.(I ¡ X) [::X;here,falsity of an aggregate
atom in a partial interpretation amounts to falsity in all its totalizations.The condition (a)
corresponds to our conditions (ii) and (iv) in Deﬁnition 4.1,while (b) corresponds to (i)
and (iii).Note that the two notions of unfoundedness coincide if
I\X =;.This is the
relevant case for
WFS(KB),as in the least ﬁxpointconstruction of
W
KB
,U
KB
(I) and I
(which is contained in
T
KB
(I)) will be always disjoint.Thus,Calimeri et al.’s notion of
unfounded set results in the same wellfounded semantics as our notion.
The notion of unfounded set was extended later by Faber [2005] to arbitrary aggregates,
by changing (a) and (b) to falsity of some literal in the body of
r w.r.t.I and w.r.t.(I ¡
X)[::X,respectively.To accommodate nonmonotonic dlatoms like those in [Eiter et al.
2004;2008],we can to the same effect change (iv) in Deﬁnition 4.1 to (iv
0
) for some dl
atomb 2B
¡
(r),
S
+
j=
L
b for every consistent
S µ Lit
P
with I [::U µS,and generalize
(b) of
T
KB
(I) to (b
0
) S
+
j=
L
b,for all consistent
S µ Lit
P
with I µ S and all b 2 B
+
(r).
The properties in Section 5 then naturally carry over to the extended setting (where strong
answer sets do not allow nonmonotonic dlatoms in positive rule bodies).
On the other hand,Pelov et al.deﬁned wellfounded semantics for logic programs with
aggregates on a purely algebraic basis without unfounded sets,using operators on bilattices
in the theory of approximating operators [Denecker et al.2004].Studying dlprograms and
their properties in an analog framework would be an interesting issue for further research.
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 25
10.CONCLUSION
In this paper,we presented a wellfounded semantics for nonmonotonic dlprograms [Eiter
et al.2004;2008],which combine logic programs and description logic knowledge bases
in a loose coupling by an interfacing approach.The semantics faithfully generalizes the
canonical wellfounded semantics for ordinary normal logic programs [van Gelder et al.
1991],and is,like the latter,deﬁned via greatest unfounded sets for dlprograms.The
proposal is distinct from other proposals of wellfounded semantics for combinations of
rules and description logics,such as [Drabent and Maluszynski 2007] and [Knorr et al.
2008],which provide a heterogenous but tight integration and a homogenous integration,
respectively,and which are not based on unfounded sets.By its nature,it is amenable
to realize nonmonotonic rules over ontologies by combining existing reasoning engines
which may be modularly replaced.
As we have shown,the proposed semantics retains a number of properties of the well
founded semantics for ordinary logic programs in the generalized context,including an
equivalent characterization in terms of a generalized GelfondLifschitz transform,and that
the wellfounded semantics is a partial model that approximates the (strong) answer set
semantics,while in the positive and stratiﬁed case,it is a total model that coincides with
the answer set semantics for dlprograms.Furthermore,we provided a complexity analysis,
which shows that our proposal also retains the good computational properties of the well
founded semantics.In particular,it is polynomial under data complexity provided that the
access to the description logic part is polynomial (as e.g.with the proﬁles EL,QL,and
RL in the upcoming OWL2 standard
15
);depending on the structure of the programand the
description logic class,one has even lower complexity and,in case of acyclic programs
and DLLite ontologies,one even achieves ﬁrstorder rewritability.
There are several directions for further work.One direction is optimization and efﬁcient
implementation of the wellfounded semantics,but also of restricted fragments like those
we considered,in particular the ones where ontology reasoning is ﬁrstorder expressible.
To this end,tightly integrated nonmonotonic logic programming and relational databases
engines,like the DLV
DB
system [Terracina et al.2008],may be fruitfully exploited for
evaluating programs with recursion.On the other hand,topdown evaluation methods for
efﬁcient query answering,as well as developing magic sets are intriguing issues.
Another direction are language extensions.The language we considered can be readily
extended to use cqatoms [Eiter et al.2009],which allow to query the ontology also with
conjunctive queries and unions thereof.In contrast,an extension to rules with disjunctive
heads seems less straightforward;many proposals for wellfounded semantics of disjunc
tive logic programs exist (see,e.g.,[Wang and Zhou 2005] and [Knorr and Hitzler 2007]
for discussion),but none is ultimately acknowledged and they have limited signiﬁcance in
practice.An extension to rules with explicit negation [Pereira and Alferes 1992] may be
targeted,which then also may use threevalued dlatoms,in line with the underlying logic.
Finally,an interesting direction would be to establish a similar formalism over multiple
ontologies,possible even in heterogeneous formats (e.g.,RDF and OWL).
15
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ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
26 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
APPENDIX
A.PROOFS FOR SECTION 4
PROOF OF OF LEMMA 4.5.
Suppose U
1
;U
2
µHB
P
are both unfounded sets of
KB
w.r.t.I.We now show that
(¤) holds for U =U
1
[U
2
.Let
a2U
1
and r 2ground(P)
with H(r) =a.Then,one of (i)(iv) holds for
U =U
1
,and thus one of (i)(iv) holds for
U =U
1
[U
2
.Similarly,for any
a2U
2
and any r 2ground(P) with H(r) =a,one of (i)
(iv) holds for
U =U
1
[U
2
.In summary,for any
a2U
1
[U
2
and any r 2ground(P) with
H(r) =a,one of (i)(iv) holds for
U =U
1
[U
2
.That is,
(¤) holds for U =U
1
[U
2
.
PROOF OF LEMMA 4.7.
It is sufﬁcient to show that T
KB
and U
KB
are monotonic.Let
J
1
µ J
2
µ Lit
P
be consistent.We ﬁrst show that
T
KB
is monotonic.If some
r 2
ground(P) exists such that conditions (a)–(d) in the deﬁnition of
T
KB
hold for
I =J
1
,
then for the same
r (a)–(d) hold for I =J
2
.That is,T
KB
(J
1
) µ T
KB
(J
2
).We next
prove that
U
KB
is monotonic.If
(¤) holds for an unfounded set U relative to I =J
1
,then
(¤) holds for U relative to I =J
2
.Hence,every unfounded set of
KB w.r.t.J
1
is also an
unfounded set of
KB w.r.t.J
2
.Thus,
U
KB
(J
1
) µ U
KB
(J
2
).
B.PROOFS FOR SECTION 5
PROOF OF THEOREM 5.2.
Let KB =(L;P).We have to show that there exists some
total interpretation
M¶WFS(KB) such that
M
+
is a model of KB,that is,satisﬁes all
instantiated rules of
P.Let
M=WFS(KB) [ (HB
P
¡(WFS(KB) [::WFS(KB))).
That is,
M is obtained fromWFS(KB) by assigning true to all ground atoms whose value
is unknown in
WFS(KB).We now show that
M
+
is a model of KB.
Each rule in ground(P) such that H(r) 2M is clearly satisﬁed in
M
+
.Consider
thus any rule r 2ground(P) such that
H(r) =2M.Then,::H(r) 2WFS(KB) and thus
H(r) 2U
KB
(WFS(KB)),and one of (i)–(iv) in
(¤) holds for
I =WFS(KB) and U =
U
KB
(WFS(KB)) there.Note that I [::U =I.Thus,if (i) or (ii) holds,clearly some
literal in
B(r) is false in M
+
,and hence r is satisﬁed by
M
+
.If (iii) holds,then S
+
6j=
L
b
for every consistent
S µ Lit
P
such that M µ S.Hence,in particular
M
+
6j=
L
b,and thus
b is false in
M
+
.Since b 2 B
+
(r),this means that r is satisﬁed by
M
+
.Finally,if (iv)
holds,then WFS(KB)
+
j=
L
b for some
b 2 B
¡
(r).By monotonicity,M
+
j=
L
b,and
thus b is true in
M
+
.Again,r is satisﬁed by M
+
.Since r was arbitrary,it follows that
M
+
is a model of KB.
PROOF OF PROPOSITION 5.4.
Let I µJ µHB
P
.Since every dlatom in
P is mono
tonic,it holds sP
J
L
µsP
I
L
.Hence,every model of
(L;sP
I
L
) is also a model of (L;sP
J
L
).
Thus,the least model of
(L;sP
J
L
) is a subset of every model of
(L;sP
I
L
),and thus in
particular also of the least model of
(L;sP
I
L
).That is,
°
KB
is antimonotonic.
PROOF OF THEOREM 5.5 (SKETCH).
The proof can be carried out by generalizing the
proof in [Van Gelder 1989] that the alternating ﬁxpoint partial model coincides with the
wellfounded partial model.One new aspect is to show that
°
KB
(I) is the set of all atoms
a2HB
P
that logically follow from KB and the negated atoms in
::(HB
P
¡ I).The
operator
S
P
(J) on all J µ::HB
P
in [Van Gelder 1989] then coincides with
°
KB
(I),
where I =HB
P
¡::J.Another new aspect is to show that our notion of unfounded set is
complete in the sense that no other atomoutside the greatest unfounded set can be assumed
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 27
false.This corresponds to showing that
W
?
µ°
KB
(W
+
);
(3)
where
W=lfp(W
KB
) and
W
?
=HB
P
¡(W
+
[(::W)
+
).Roughly,(3) can be proved
as follows.It can be shown that
W
+
µ°
KB
(W
+
) µ W
+
[W
?
.Towards a contradiction,
suppose that
U =W
?
¡°
KB
(W
+
) 6=;.Hence,for every
a2U and every
r 2ground(P)
with H(r) =a,it holds that either (i)
:b 2W[::U for some ordinary atom b 2B
+
(r),
or (ii)
b 2W for some ordinary atom b 2B
¡
(r),or (iii) for some dlatom
b 2B
+
(r),
we have that
°
KB
(W
+
) 6j=
L
b,and thus
S
+
6j=
L
b for every consistent
S µLit
P
with
W[::U µS,or (iv)
W
+
j=
L
b for some dlatom b 2B
¡
(r).Hence,
U is an un
founded set of KB relative to
W.But this contradicts
W=lfp(W
KB
).This shows that (3)
holds.
PROOF OF THEOREM 5.7.
For any I µHB
P
,it holds that
I is a strong answer set of
KB iff
I is a ﬁxpoint of °
KB
.Since
lfp(°
2
KB
) µI µgfp(°
2
KB
) for every ﬁxpoint of
°
KB
,
it thus follows that
lfp(°
2
KB
) µI µgfp(°
2
KB
) for every strong answer set
I of
KB.Thus,
every such I includes every wellfounded and no unfounded atom
a2HB
P
relative to
KB.
PROOF OF THEOREM 5.9.
If every a2HB
P
is either wellfounded or unfounded rel
ative to KB,then
lfp(°
2
KB
) =gfp(°
2
KB
).Hence,
lfp(°
2
KB
) =I =gfp(°
2
KB
),for every ﬁx
point
I µHB
P
of °
KB
.That is,
lfp(°
2
KB
) =I =gfp(°
2
KB
) for every answer set
I of KB.
That is,the set of all wellfounded
a2HB
P
relative to KB is the only answer set of
KB.
PROOF OF THEOREM 5.10 (SKETCH).
We take advantage of the characterization of
WFS(KB) given in Theorem5.5.Assume ﬁrst
KB is positive.Then,for every
I µHB
P
,
it holds that s P
I
L
=P and thus
°
KB
(I) is the least model of KB.Thus,the only ﬁx
point of
°
KB
(and thus also the least and the greatest ﬁxpoint of
°
KB
) is the least model
of
KB,which in turn is the unique answer set of
KB.Suppose next
KB is stratiﬁed.Since
lfp(°
2
KB
) µI µgfp(°
2
KB
) holds for the unique answer set
I of KB,it is sufﬁcient to show
that neither (a)
lfp(°
2
KB
) ½I nor (b)
I ½gfp(°
2
KB
) holds for the unique answer set
I of
KB.This can be proved by contradiction along a stratiﬁcation
¸ of KB.
PROOF OF THEOREM 5.12.
We are given KB = (L;P),the corresponding KB
dl
=
(L;P
dl
),and an interpretation I over Lit
P
.Let us denote
I
dl
as
I [ fp
a
(c) j I j=
L
a(c)
for each ground dlatom appearing in ground(P)g.Also,deﬁne
G(I) = °
KB
(I) and
G
dl
(I) = °
KB
dl
(I).The proof relies on the following intermediate results.
Lemma B.1
Let I be any interpretation,and let J = G
dl
(I).Then,J = J
dl
.
The above follows from the fact that p
a
(c) Ã a(c) appears in sP
dl
I
L
,for each ground
dlatomappearing in
ground(P);so if J j=
L
a(c),then we will have
p
a
(c) 2 J.
Lemma B.2
For every interpretation I over Lit
P
,
G(I)
dl
= G
dl
(I
dl
).
The above holds since one can observe that sP
I
L
and sP
dl
I
dl
L
have the same rules,with
the only difference that each (positive) dlatom
a(c) in sP
I
L
is replaced with p
a
(c) in
sP
dl
I
dl
L
,and a rule of the form
p
a
(c) Ã a(c) is added;one can then easily observe that
G(I)
dl
and G
dl
(I
dl
) coincide.
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
28 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
Proposition B.3
lfp(G
2
)
dl
= lfp((G
dl
)
2
) and gfp(G
2
)
dl
= gfp((G
dl
)
2
).
Let I
0
=;.One shows ﬁrst by induction on k > 0 that for the kth powers of
G(I
0
)
and G
dl
(I
dl
0
),denoted by G
k
(I
0
) and
(G
dl
)
k
(I
dl
0
),we have
G
k
(I
0
)
dl
= (G
dl
)
k
(I
dl
0
):
(4)
The equality obviously holds for
k = 0.Given (4) holds for
k,then for k +1,we have
G
k+1
(I
0
)
dl
= (G(G
k
(I
0
)))
dl
:
Now,let
I = G
k
(I
0
).Then,by Lemma B.2,we have
G(I)
dl
= G
dl
(I
dl
);
since by the induction hypothesis,
G
k
(I
0
)
dl
= (G
dl
)
k
(I
dl
0
),we get
G(G
k
(I
0
))
dl
= G
dl
((G
dl
)
k
(I
dl
0
)) = (G
dl
)
k+1
(I
dl
0
);
which proves (4) for each
k > 0.Furthermore,we have that
I
dl
0
µ (G
dl
)
2k
(I
0
),for each k > 0:
(5)
Observe indeed that
G
dl
(I
0
) contains I
dl
0
,as well as (G
dl
)
2
(I
0
),and that
(G
dl
)
2
is mono
tonic.From (5) we conclude that
((G
dl
)
2k
)(I
dl
0
) and ((G
dl
)
2k
)(I
0
) converge to the same
limit,which is
lfp((G
dl
)
2
).On the other hand,
G
2k
(I
0
)
dl
converges to lfp(G
2
)
dl
.Thus,
we get lfp(G
2
)
dl
= lfp((G
dl
)
2
).
In a similar way,one can show that the greatest ﬁxpoints of G
2
and
(G
dl
)
2
are related:
indeed,by letting
I
0
= HB
P
,we have G
2k
(HB
P
)
dl
= (G
dl
)
2k
(HB
dl
P
),where
HB
dl
P
¶
(G
dl
)
2k
(HB
P
),thus (G
dl
)
2k
(HB
dl
P
) converges to
gfp((G
dl
)
2
).
C.PROOFS FOR SECTION 6
PROOF OF THEOREM 6.1.
We ﬁrst show that,given KB = (L;P) and I µ HB
P
,
computing °
KB
(I) is feasible in exponential time,which then implies that computing
lfp(°
2
KB
) and
gfp(°
2
KB
) (and thus also WFS(KB)) is feasible in exponential time.
The reduct
KB
I
=(L;sP
I
L
) is constructible in exponential time,since (i) ground(P) is
computable in exponential time and (ii)
I j=
L
a for each dlatom
a in ground(P) can be
decided in exponential time,by the complexity of deciding knowledge base satisﬁability
in
SHIF(D).Furthermore,computing the least model of KB
I
is feasible in exponential
time by computing
lfp(T
KB
I
) =
S
n
i=0
T
i
KB
I
(;) with
n = jHB
P
j,which requires at most
exponentially many applications of
T
KB
I
,each of which is computable in exponential time
(deciding
I j=
L
a for any dlatom a in ground(P) is feasible in exponential time,by the
complexity of deciding knowledge base satisﬁability in
SHIF(D)).
Therefore,we can compute lfp(°
2
KB
) = A
1
,by computing
A
0
,A
1
,...until A
i
=
°
2i
KB
(;) = °
2i+2
KB
(;) = A
i+1
holds for some
i.Since i is bounded by jHB
P
j and the latter
is exponential in the size of
© and KB,the positive part of
WFS(KB),that is,lfp(°
2
KB
),
is computable in exponential time.The negative part of
WFS(KB) is easily obtained
from gfp(°
2
KB
) = O
1
,which can be similarly computed in exponential time.Therefore,
computing
WFS(KB) is feasible in exponential time.
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 29
Hence,deciding whether
l 2WFS(KB) holds is in EXP.The EXPhardness of the
problemis immediate fromthe EXPhardness of deciding whether a given positive Datalog
program logically implies a given ground atom [Dantsin et al.2001] as well as from the
EXPhardness of deciding whether a knowledge base in
SHIF(D) is satisﬁable.
PROOF OF THEOREM 6.2.
For membership in P
NEXP
,an algorithm is not allowed to
use exponential work space (only polynomial space).Thus,differently fromthe situation in
the proof of Theorem6.1,we cannot simply compute the powers
°
j
KB
(;) and °
j
KB
(HB
P
),
because ground(P) is exponential.The idea is to move this probleminside an oracle call.
It is easy to see that we can compute WFS(KB) and decide l 2WFS(KB) in expo
nential time,if the answers for all dlatom evaluations
I
p
j=
L
a that we encounter during
the computation of the powers
°
j
KB
(;) and °
j
KB
(HB
P
) would be known.However,de
ciding
I
p
j=
L
a is coNEXPcomplete for a
SHOIN(D) knowledge base
L;as it is not
known whether coNEXP=NEXP,it is unclear how these answers can be computed in
side a NEXP oracle call itself.To surmount this problem,we apply a census technique that
provides enough information to the oracle for verifying a correct guess for all the answers.
The fact that I
p
j=
L
a is in coNEXP implies that if
I
p
6j=
L
a,then there is an expo
nential size “proof” witnessing this fact which a Turing machine can nondeterministically
generate and check in exponential time.Therefore,given a ground dlatom
a and an inte
ger k > 0,deciding whether there are at least
k different interpretations
I
1
p
;:::;I
k
p
of the
input predicates of
a such that I
j
p
6j=
L
a is in NEXP (candidate
I
1
p
;:::;I
k
p
and respective
proofs can be nondeterministically generated and checked in exponential time).Any
k for
which this can hold is bounded by a number n
a
which is exponential in the size of
KB and
©:each interpretation I
p
is a set of unary and/or binary ground facts,and only polynomi
ally many different such ground facts are possible with respect to
KB and ©;hence,the
number of different
I
p
is at most single exponential in the size of
KB and ©.
In order to decide whether l 2WFS(KB) holds,we can thus proceed as follows:
(1)
For each ground dlatoma in ground(P),compute by binary search on
[0;:::;n
a
],
using the NEXP oracle,the exact number of inputs
I
p
such that I
p
6j= a,denoted
f
a
.
(2)
Ask the oracle whether for each dlatom
a,there are f
a
different inputs
I
1
p
;:::;I
f
a
p
such that (a)
I
j
p
6j=
L
a,1 6 j 6 f
a
,and (b) for the computation of the powers
°
i
KB
(;)
resp.°
i
(HB
P
) where for each
I
p
j=
L
a the value compliant with
I
1
p
;:::;I
f
a
p
is taken (i.e.,
if the input of
I
p
to a is the same as for some I
j
p
,1 6 j 6 f
a
,then
I
p
j=
L
a is false,
otherwise true),it holds that l is contained in the limit
A
1
of the sequence °
2k
(;) if
l is a
positive literal resp.that b is not contained in the limit
O
1
of the sequence °
2k
(HB
P
) if
l = not b.
(3)
If the oracle answers yes,return yes,otherwise no.
Note that for the answer “yes”,(b) is only relevant if all tests in (a) succeed.Hence,
Step 3 correctly decides whether
l 2WFS(KB) holds.
Step 1 is feasible in polynomial time modulo the NEXP oracle calls,since the number
of ground dlatoms
a in ground(P) is polynomial and the binary search takes
O(log n
a
)
many steps,which is polynomial in the size of
KB and ©.The oracle query in Step 2 is
in NEXP,since the proper (unique) inputs
I
1
p
;:::;I
f
a
p
together with their witnesses can
be guessed and veriﬁed in exponential time (step (a) is feasible for each
a in exponential
time in total),and (b) is feasible in exponential time;In summary,this algorithmcorrectly
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
30 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
decides whether
l 2WFS(KB) holds in polynomial time with a
NEXP oracle.This proves
the membership part.
The P
NEXP
hardness is easily derived from Theorem 5.10 and the result that deciding
whether a stratiﬁed
KB in which classical negation
:may occur has some strong answer
set is P
NEXP
complete [Eiter et al.2004].Replace in a stratiﬁed
KB classical negative
literals:p(t) by positive literals
p(t),where
p is a fresh predicate,and add rules
f Ã
p(t);
p(t),where
f is a fresh propositional atom.Then,for the resulting dlprogram
KB
0
,
we have:f 2WFS(KB) iff KB has some strong answer set.
PROOF OF PROPOSITION 6.3.
We showthat,for KB =(L;P) where
Lis in a DL such
that evaluating
I j=
L
a for given I µHB
P
and ground dlatom a has a data complex
ity in class
C,computing °
KB
(I) is feasible in polynomial time with a
Coracle in the
data complexity.This then implies that computing
lfp(°
2
KB
) and
gfp(°
2
KB
) (and thus also
WFS(KB)) is feasible in polynomial time with a
Coracle in the data complexity.
The reduct KB
I
=(L;sP
I
L
) is constructible in polynomial time with a
Coracle,since
(i) ground(P) is computable in polynomial time and (ii)
I j=
L
a for each dlatom
a in
ground(P) is decidable using the Coracle.Furthermore,computing the least model of
KB
I
is feasible in polynomial time with a
Coracle by computing lfp(T
KB
I
) =
S
n
i=0
T
i
KB
I
(;) with n=jHB
P
j,which requires at most polynomially many applications of
T
KB
I
,each of which is computable in polynomial time with a Coracle.
Thus,we can compute lfp(°
2
KB
) =A
1
,by computing
A
0
,A
1
,...until A
i
=°
2i
KB
(;) =
°
2i+2
KB
(;) =A
i+1
holds for some
i.Since i is polynomially bounded by jHB
P
j,the positive
part of
WFS(KB),that is,lfp(°
2
KB
),is computable in polynomial time with a
Coracle.
The negative part of WFS(KB) is easily obtained from gfp(°
2
KB
) =O
1
,which can be
similarly computed in polynomial time with a
Coracle.Therefore,computing WFS(KB)
is feasible in polynomial time with a
Coracle in the data complexity,and thus deciding
whether
l 2WFS(KB) is in
P
C
in the data complexity.
PROOF OF THEOREM 6.4.
As for membership in P
NP
,we observe ﬁrst that instance
checking in
SHIF(D) is in coNP under data complexity.This follows from the results
in [Glimm et al.2008],which showed that the data complexity of answering conjunc
tive queries in
SHIQ is coNPcomplete,where the knowledge bases are also allowed
to contain negated role assertions.Thus,the same data complexity holds for
SHIQ(D).
Hence,deciding whether
I j=
L
a for interpretations
I,knowledge bases L in SHIF(D),
and dlatoms
a is clearly in coNP in the data complexity for a with queries of the form
C(b),:C(b),R(b;c),
:R(b;c),
U(b;v),and
:U(b;v).Furthermore,it is also in coNP in
the data complexity for all other types of dlatoms,since (i)
L
0
j=CvD iff
L
0
[f(C u
:D)(e);
A(d)g j=:A(d);(ii) L
0
j=:(CvD) iff
L
0
[ fCvD;
A(d)g j=:A(d);(iii)
L
0
j= =(b;c) iff
L
0
[f 6=(b;c);A(d)g j=:A(d);and (iv)
L
0
j= 6=(b;c) iff L
0
[f=(b;c);
A(d)g j=:A(d),where
d and e are fresh individuals,and Ais a fresh atomic concept.The
P
NP
membership follows then by Proposition 6.3.
Hardness for P
NP
of literal entailment froma stratiﬁed dlprogram
KB =(L;P) with L
in ALE is proved by a generic reduction from Turing machines
M,exploiting the coNP
hardness proof for instance checking in
ALE by Donini et al.[1994].Informally,the main
idea behind the proof is to use a dlatom to decide the result of the
jth oracle call made
by a polynomialtime bounded M with access to a NP oracle,where the results of the
previous oracle calls are known and input to the dlatom.By a proper sequence of dlatom
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 31
evaluations,the result of
M’s computation on input
v can then be obtained.
More concretely,let M be a polynomialtime bounded deterministic Turing machine
with access to a
NP oracle,and let v be an input for M.Since every oracle call can
simulate
M’s computation on v before that call,once the results of all the previous oracle
calls are known,we can assume that the input of every oracle call is given by
v and the
results of all the previous oracle calls.Since
M’s computation after all oracle calls can be
simulated within an additional oracle call,we can assume that the result of the last oracle
call is the result of
M’s computation on v.Finally,since without loss of generality all
computations of
M on inputs of size
s make l =p(s) oracle calls and since any input to an
oracle call can be enlarged by “dummy” bits,we can assume that the inputs to all oracle
calls have the same length
n=2 ¢ (k+l),where k is the size of v;we assume that the input
to the
m+1th oracle call (with m2f0;:::;l¡1g) has the form
v
k
1v
k¡1
1:::v
1
1c
0
1c
1
1:::c
m¡1
1c
m
0:::c
l¡1
0;
where
v
k
;v
k¡1
;:::;v
1
are the symbols of v in reverse order,which are all marked as
valid by a subsequent “
1”,c
0
;c
1
;:::;c
m¡1
are the results of the previous moracle calls,
which are all marked as valid by a subsequent “
1”,and c
m
;:::;c
l¡1
are “dummy” bits,
which are all marked as invalid by a subsequent “
0”.
By the coNPhardness proof for instance checking in
ALE in [Donini et al.1994],for
the
NP oracle M
0
and any input b 2§
¤
,there exists a knowledge base
L
0
[L
b
in ALE,a
concept D in ALE,and an individual
f such that M
0
accepts b iff L
0
[L
b
6j= D(f),and
L
0
,L
b
,D,and f can be constructed in polynomial time from b.More concretely,
L
0
,L
b
,
and D are given as follows:
L
0
= fA(true);:A(false)g;
L
b
= fCl (f;c
1
);Cl (f;c
2
);:::;Cl (f;c
n
);
P
1
(c
1
;l
1
1+
);P
2
(c
1
;l
1
2+
);N
1
(c
1
;l
1
1¡
);N
2
(c
1
;l
1
2¡
);:::;
P
1
(c
n
;l
n
1+
);P
2
(c
n
;l
n
2+
);N
1
(c
n
;l
n
1¡
);N
2
(c
n
;l
n
2¡
)g;
D = 9Cl:((9P
1
::A) u(9P
2
::A) u(9N
1
:A) u(9N
2
:A)):
Note that the entailment problem
L
0
[L
b
6j= D(f) in ALE encodes the satisﬁability prob
lemfor a 2+2CNF formula
F =C
1
^C
2
^¢ ¢ ¢^C
n
,where
C
i
=A
i
1+
_A
i
2+
_:A
i
1¡
_:A
i
2¡
and the
A
i
j
’s are propositional symbols including true and false,which has been shown to
be NPhard by a reduction from3SAT in [Donini et al.1994].
Let the stratiﬁed dlprogramKB =(L;P) now be deﬁned as follows:
L = L
0
;
P =
S
l
j=0
P
j
;
where
P
j
=P
j
v
[P
j
q
[P
j
b
for every j 2f0;:::;lg.Informally,every set of dlrules
P
j
generates the input of the
j+1th oracle call,which includes the results of the ﬁrst j oracle
calls.Here,
P
l
prepares,for simplicity,the input of a “dummy” (nonhappening)
l+1th or
acle call which contains the result of the
lth (that is,the last) oracle call.More concretely,
the bitstring
a
¡2k
¢ ¢ ¢ a
2l¡1
is the input of the j+1th oracle call iff
b
j
¡2k
(a
¡2k
);:::;
b
j
2l¡1
(a
2l¡1
) are in the canonical model of KB.The components
P
j
v
,P
j
q
,and P
j
b
of P
j
,
with j 2f0;:::;lg,are deﬁned as follows:
(1)
P
0
v
writes v into the input of the ﬁrst oracle call,and every P
j
v
copies
v into the input
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
32 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
of the j+1th oracle call,for j 2f1;:::;lg:
P
0
v
= fb
0
¡2i
(v
i
) Ã j i 2f1;:::;kgg [fb
0
¡2i+1
(1) Ã j i 2f1;:::;kgg;
P
j
v
= fb
j
¡i
(x) Ãb
j¡1
¡i
(x) j i 2f1;:::;2kgg:
(2)
P
0
q
initializes the rest of the input of the ﬁrst oracle call with “dummy” bits,and
every
P
j
q
with j 2f1;:::;lg writes the result of the jth oracle call into the input of the
j+1th oracle call and carries over all the other result and dummy bits from the input of
the
jth oracle call (where
D=9Cl:((9P
1
::A) u(9P
2
::A) u(9N
1
:A) u(9N
2
:A))):
P
0
q
= fb
0
i
(0) Ã j i 2f0;:::;2l¡1g;
P
j
q
= fb
j
i
(x) Ãb
j¡1
i
(x) j i 2f0;:::;2l¡1g;i 62f2j¡2;2j¡1gg [
fb
j
2j¡2
(0) ÃDL[Cl]cl
j¡1
;P
1
]p
j¡1
1
;P
2
]p
j¡1
2
;N
1
]n
j¡1
1
;N
2
]n
j¡1
2
;D](f);
b
j
2j¡2
(1) Ãnot b
j
2j¡2
(0);
b
j
2j¡1
(1) Ãg:
(3)
Every P
j
b
with j 2f0;:::;lg realizes the polynomialtime reduction,which trans
forms any input
b
j
of the Turing machine
M
0
into the knowledge base L
b
j
in ALE,repre
sented as facts over the predicate symbols
cl
j
,p
j
1
,p
j
2
,n
j
1
,and n
j
2
.
Observe then that M accepts v iff the last oracle call returns “yes”.The latter is equivalent
to
b
l
2l¡2
(1) 2WFS(KB).In summary,M accepts v iff
b
l
2l¡2
(1) 2WFS(KB).
D.PROOFS FOR SECTION 7
PROOF OF THEOREM 7.1.
Membership in P follows from Proposition 6.3 and the as
sumption that all dlatoms can be evaluated in polynomial time,as
P
P
= P.Hardness
for P follows fromthe
Pcompleteness of literal inference fromordinary normal programs
under the wellfounded semantics (cf.[Dantsin et al.2001]).
PROOF OF THEOREM 7.2.
The statement of the theoremfollows fromTheorem7.1 and
the result that conjunctive query answering froma knowledge base in Horn
SHIQcan be
done in polynomial time in the data complexity [Eiter et al.2008],since all evaluations
of dlatoms can be reduced to this problem.Observe ﬁrst that,for
L in HornSHIQ,
any negated concept (resp.,role) membership axiom
:C(b) (resp.,:R(b;c)) in the in
put argument of a dlatom can be ignored in the actual evaluation of the dlquery,and
handled by evaluating an additional dlquery
C(b) (resp.,R(b;c)):if any of these (poly
nomially many) additional dlqueries evaluates to true,then the original dlquery evaluates
to true (since the description logic knowledge base along with the input of the dlatom is
unsatisﬁable),otherwise the original dlquery is simply evaluated ignoring
:C(b) (resp.,
:R(b;c)).This is due to the fact that knowledge bases in Horn
SHIQ have canoni
cal universal models [Eiter et al.2008].Observe then that dlqueries
C(b) and R(b;c)
are clearly conjunctive queries.Moreover,axioms
=(b;c) and 6=(b;c) are disallowed
in Horn
SHIQ and thus also cannot occur as dlqueries.Furthermore,all other dl
queries can be reduced to knowledge base unsatisﬁability:(i)
L
0
j=:C(b) iff
L
0
[fC(b)g
is unsatisﬁable;(ii)
L
0
j=:R(b;c) iff
L
0
[fR(b;c)g is unsatisﬁable;(iii)
L
0
j=C vD iff
L
0
[fC(e);D
0
(e);DuD
0
v?g is unsatisﬁable;and (iv)
L
0
j=:(CvD) iff
L
0
[fCvDg
is unsatisﬁable,where
e is a fresh individual,and D
0
is a fresh atomic concept.This can in
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 33
turn be reduced to conjunctive queries:
L
0
is unsatisﬁable iff
L
0
[fA
0
(d);AuA
0
v?g j=
A(d),where
d is a fresh individual,and A and A
0
are fresh atomic concepts.
PROOF OF THEOREM 7.3.
KB is acyclic,thus there exists
·:P
P
!f0;1;:::;ng
such that for every dlrule
r 2P,the predicate symbol p of H(r),and every predicate
symbol
q of some ordinary b 2B(r) or of an input argument of some dlatom
b 2B(r),
it holds that
·(p) >·(q).We call ·(p) the rank of p.By assumption,every dlquery in
P
can be expressed in terms of a ﬁrstorder formula over the set
A of all concept and role
membership axioms in
L.We now show by induction on
·(p) 2f0;1;:::;ng that each
predicate symbol p2P
P
can be expressed in terms of a ﬁrstorder formula over the set
F
of all concept and role membership axioms in
L and the database facts in P,constructed
frompredicate symbols of rank
0.
Basis:Each predicate p2P
P
of rank 0 can trivially be expressed in terms of a ﬁrstorder
formula over
F.
Induction:We have to consider the evaluation of a dlatomDL[¸;Q](c) and the deﬁnition
of a predicate
p2P
P
via the set of all rules in P with p in their head:
(i) Consider the dlatom DL[¸;Q](c) with ¸=¸
+
;¸
¡
,where
¸
+
=S
1
] p
1
;:::;S
l
] p
l
,
¸
¡
=S
l+1
¡
[p
l+1
;:::;S
m
¡
[p
m
,and
m>l >0.The dlquery Q(c) can be expressed
in terms of a ﬁrstorder formula
®(x) over
A,that is,
Lj=Q(c) iff I
A
j=®(c).Since
the underlying DL allows for ﬁrstorder rewritable concept and role memberships,every
S
i
in ¸
¡
,l <i 6m,can be expressed in terms of a ﬁrstorder formula
Ã
S
i
(y) over
A,
that is,Lj=S
i
(c) iff
I
A
j=Ã
S
i
(c) for every c.By the induction hypothesis,every in
put predicate
p
j
in ¸ can be expressed in terms of a ﬁrstorder formula
Ã
j
(x) over
F,
that is,p
j
(c) 2WFS(KB) iff
I
F
j=Ã
j
(c).We deﬁne the ﬁrstorder formula
±(x) for
DL[¸;Q](x) over
F as follows:
±(x) = ®
¸
+
(x) _
m
_
j=l+1
9y(Ã
¸
+
S
j
(y) ^Ã
j
(y));
(6)
where
¯
¸
+
is obtained from ¯ by replacing every S
i
(s) such that
S
i
occurs in ¸
+
by
S
i
(s) _Ã
i
1
(s) _ ¢ ¢ ¢ _Ã
i
k
i
(s),where
S
i
1
;:::;S
i
k
i
are all occurrences of S
j
in
¸
+
.
For example,suppose L=fC(a)g and
P =f p(c);q(b);r Ãp(x);r ÃDL[C ] p;C](x);s Ãnot DL[C ] p;C
¡
[q;C](x) g:
Then,both dlatoms in
P have the same query Q(x) (= C(x)) over
L which can be
expressed by the formula ®(x) =C(x) over A=fC(a)g,and the predicates
p and q can be
expressed by the formulas
Ã
p
(x) = p(x) and
Ã
q
(x) = q(x),respectively,over F =fC(a);
p(c);q(b)g.The dlatom
DL[C ] p;C](x) is thus translated into
±
1
(x) = ®
¸
+
(x) =
C(x) _ p(x) over F (note that
m = l),while the dlatom DL[C ] p;C
¡
[q;C](x) is
translated into
±
2
(x) = C(x) _ p(x) _ 9y ((C(y) _ p(y)) ^q(y)) over F.
Note that I
F
j=S
i
(c) iff S
i
(c) 2 L,for all
1 6i 6l.Hence,
I
F
j=S
i
(c) _Ã
i
1
(c) _¢ ¢ ¢ _Ã
i
k
i
(c)
iff
S
i
(c) 2L or
p
i
j
(c) 2 WFS(KB),for some 1 6j 6k
i
iff
S
i
(c) 2L[
S
l
i=1
A
i
(WFS(KB)) (recall
A
i
(I) fromSection 3.2)
iff
I
A
0
j=S
i
(c),where A
0
= A[
S
l
i=1
A
i
(WFS(KB)).
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
34 ¢ T.Eiter,G.Ianni,T.Lukasiewicz,and R.Schindlauer
It follows fromthis that
I
F
j=®
¸
+
(c) iff I
A
0
j=®(c) and
I
F
j=Ã
¸
+
S
j
(c) iff
I
A
0
j=Ã
S
j
(c),for all l <j 6m.
This in turn implies that
I
F
j=±(c) iff (i)
L[ A
0
j=Q(c),or
(ii)
L[ A
0
j=S
j
(d) and
p
j
(d) 2WFS(KB) for some l <j 6mand
d.
Let A
00
= A
0
[
S
m
j=l+1
A
j
(WFS(KB)).If
L [ A
00
6j=Q(c),then clearly both (i) and (ii)
are false;conversely,if
L[A
0
6j=Q(c) and
L[A
0
6j=S
j
(d) for every
p
j
(d) 2 WFS(KB)
where l <j 6m,then
L[A
00
6j=Q(c) holds since the underlying DL is CWAsatisﬁable.
In summary,this shows that I
F
j=±(c) iff
L [ A
00
j=Q(c) iff WFS(KB) satisﬁes
DL[¸;Q](c).That is,
±(x) is a ﬁrstorder formula for
DL[¸;Q](x) over F.
(ii) Consider next the set of all rules in P with p in their head.W.l.o.g.,the heads
p(x) of all
these rules coincide.Let ®(x) denote the disjunction of the existentially quantiﬁed bodies
of these rules,where the default negations in the rule bodies are interpreted as classical
negations.By the induction hypothesis,every body predicate in
®(x) can be expressed
in terms of a ﬁrstorder formula over
F,and the same holds for every dlatom in
®(x).
Let ®
0
(x) be obtained from ®(x) by replacing all but the predicates of rank
0 by these
ﬁrstorder formulas.Then,®
0
(x) is a ﬁrstorder formula over
F for p.
Continuing our example,the rules for r in P are translated into the ﬁrstorder formula
9xp(x) _9x±
1
(x) = 9xp(x) _9x(C(x) _p(x)) ´ 9x(C(x) _ p(x))
and the rule for
s into
9x:±
2
(x) = 9x:(C(x) _p(x) _9y ((C(y) _p(y)) ^q(y)))
over
fC(a);p(c);q(b)g.
PROOF OF THEOREM 7.4.
We apply Theorem 7.3.Observe ﬁrst that
L is deﬁned in
a description logic of the DLLite family in which knowledge base satisﬁability and con
junctive queries are both ﬁrstorder rewritable.Observe also that
L is deﬁned in a CWA
satisﬁable description logic [Calvanese et al.2007] (and thus Theorem 7.3 also allows
the operator
¡
[ to occur in P).Hence,all dlatoms with dlqueries of the form
C(t)
and R(t;s) are immediately ﬁrstorder rewritable.Furthermore,all other dlatoms are
also ﬁrstorder rewritable,since their dlqueries can be reduced to conjunctive queries as
follows:(i)
L
0
j=CvD iff
L
0
[fC(e);D
0
(e);D
0
v:D;A
0
(d);A
0
v:Agj=A(d),and
(ii)
L
0
j=:(C vD) iff
L
0
[ fCvD;A
0
(d);A
0
v:Agj=A(d),where
d and e are fresh
individuals,and A,A
0
,and
D
0
are fresh atomic concepts.By Theorem7.3,it thus follows
that deciding whether
l 2WFS(KB) is ﬁrstorder rewritable.
ACKNOWLEDGMENTS
We are grateful to Diego Calvanese,Magdalena Ortiz and Ulrike Sattler for providing valu
able information on complexityrelated issues about OWLDL related description logics,
and to Włodzimierz Drabent for interesting discussions.We further thank the reviewers
of this paper and its RuleML2004 preliminary version,whose useful and constructive
comments have helped to improve this work.
ACMTransactions on Computational Logic,Vol.V,No.N,April 2010.
WellFounded Semantics for Description Logic Programs in the Semantic Web ¢ 35
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Received March 2009;revised January 2010;accepted April 2010
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