Paraconsistent and Approximate Semantics
for the OWL 2 Web Ontology Language
?
Linh Anh Nguyen
Institute of Informatics, University of Warsaw
Banacha 2, 02-097 Warsaw, Poland
nguyen@mimuw.edu.pl
Abstract. We introduce a number of paraconsistent semantics, including three-valued and
four-valued semantics, for the description logic SROIQ, which is the logical foundation of
OWL 2. We then study the relationship between the semantics and paraconsistent reasoning
in SROIQ w.r.t. some of them through a translation into the traditional semantics. We
also present a formalization of rough concepts in SROIQ.
1 Introduction
The Web Ontology Language (OWL) is a family of knowledge representation languages
for authoring ontologies. It is considered one of the fundamental technologies under-
pinning the Semantic Web, and has attracted both academic and commercial interest.
OWL has a formal semantics based on description logics (DLs), which are formalisms
concentrated around concepts (classes of individuals) and roles (binary relations between
individuals), and aim to specify concepts and concept hierarchies and to reason about
them.
1
DLs belong to the most frequently used knowledge representation formalisms
and provide a logical basis to a variety of well known paradigms, including frame-based
systems, semantic networks and semantic web ontologies and reasoners. The extension
OWL 2 of OWL, based on the DL SROIQ [6], became a W3C recommendation in
October 2009.
Some of the main problems of knowledge representation and reasoning involve vague-
ness, uncertainty, and/or inconsistency. There are a number of approaches for dealing
with vagueness and/or uncertainty, for example, by using fuzzy logic, rough set theory,
or probabilistic logic. See [7] for references to some works on extensions of DLs using
these approaches. A way to deal with inconsistency is to follow the area of paraconsistent
reasoning. There is a rich literature on paraconsistent logics (see, e.g., [5] and references
there).
Rough set theory was introduced by Pawlak in 1982 [17, 18] as a new mathematical
approach to vagueness. It has many interesting applications and has been studied and
extended by a lot of researchers (see, e.g., [21, 20, 19]). In rough set theory, given a
similarity relation on a universe, a subset of the universe is described by a pair of subsets of
the universe called the lower and upper approximations. In [22, 8] Schlobach et al. showed
how to extend DLs with rough concepts. By treating the similarity relation R as a role,
the lower and upper approximations of a concept C are expressed, respectively, as ∀R.C
?
This is a revised and extended version of the conference paper [14], which was partially supported by
grant N N206 399334 from the Polish Ministry of Science and Higher Education.
1
There is a rich literature on DLs. For good surveys consult [1], in particular papers [13, 2] as well as
the bibliography provided there.
2 L.A. Nguyen
and ∃R.C, provided that the properties of R (like reflexivity, symmetry, transitivity)
are encoded as axioms of the logic. Note that DL is closely related to modal logic and
characterizations of the lower and upper approximations using modal operators have
been studied earlier (e.g., in [26]). In [7] Jiang et al. gave some details about the rough
version of the DL ALC. In general, a traditional DL can be used to express and reason
about rough concepts if similarity relations are used as roles and the properties of the
similarity relations are expressible and used as axioms of the logic.
A number of researchers have extended DLs with paraconsistent semantics and para-
consistent reasoning methods [12, 23, 16, 10, 9, 27, 15]. The work [16] studies a constructive
version of the basic DL ALC. The remaining works except [15] are based on the well-
known Belnap’s four-valued logic [3, 4]. Truth values in this logic represent truth (
t
),
falsity (
f
), the lack of knowledge (
u
) and inconsistency (
i
). However, there are serious
problems with using Belnap’s logic for Semantic Web (see [11, 25, 15]). In [15] together
with Sza las we gave a three-valued paraconsistent semantics for the DL SHIQ, which is
related to the DL SHOIN used for OWL 1.1.
Both rough concepts and paraconsistent reasoning are related to approximation.
Rough concepts deal with concept approximation, while paraconsistent reasoning is a
kind of approximate reasoning. We can combine them to deal with both vagueness and
inconsistency. In this paper, we study rough concepts and paraconsistent reasoning in the
DL SROIQ. As rough concepts can be expressed in SROIQ using the usual way, we
just briefly formalize them. We concentrate on defining a number of different paraconsis-
tent semantics for SROIQ, studying the relationship between them, and paraconsistent
reasoning in SROIQ w.r.t. some of such semantics through a translation into the tra-
ditional semantics. Our paraconsistent semantics for SROIQ are characterized by four
parameters for:
– using two-, three-, or four-valued semantics for concept names
– using two-, three-, or four-valued semantics for role names
– interpreting concepts of the form ∀R.C or ∃R.C (two ways)
– using weak, moderate, or strong semantics for terminological axioms.
Note that, with respect to DLs, three-valued semantics has been studied earlier only
for SHIQ [15]. Also note that, studying four-valued semantics for DLs, Ma and Hitzler
[9] did not consider all features of SROIQ. For example, they did not consider concepts
of the form ∃R.Self and individual assertions of the form ¬S(a, b).
The rest of this paper is structured as follows. In Section 2 we recall notations and
semantics of SROIQ. In Section 3 we formalize rough concepts in SROIQ. We present
our paraconsistent semantics for SROIQ in Section 4 and study the relationship between
them in Section 5. In Section 6 we give a faithful translation of the problem of conjunctive
query answering w.r.t. some of the considered paraconsistent semantics into a version that
uses the traditional semantics. Section 7 concludes this work.
2 The Description Logic SROIQ
In this section we recall notations and semantics of the DL SROIQ [6]. Assume that our
language uses a finite set C of concept names, a subset N ⊆ C of nominals, a finite set R
of role names including the universal role U, and a finite set I of individual names. Let
Paraconsistent and Approximate Semantics for OWL 2 3
R
−
def
= {r
−
| r ∈ R \ {U }} be the set of inverse roles. A role is any member of R ∪ R
−
.
We use letters like R and S for roles.
An interpretation I = h∆
I
, ·
I
i consists of a non-empty set ∆
I
, called the domain of
I, and a function ·
I
, called the interpretation function of I, which maps every concept
name A to a subset A
I
of ∆
I
, where A
I
is a singleton set if A ∈ N, and maps every role
name r to a binary relation r
I
on ∆
I
, with U
I
= ∆
I
× ∆
I
, and maps every individual
name a to an element a
I
∈ ∆
I
. Inverse roles are interpreted as usual, i.e., for r ∈ R, we
define (r
−
)
I
def
= (r
I
)
−1
= {hx, yi | hy, xi ∈ r
I
}.
A role inclusion axiom is an expression of the form R
1
◦. . .◦R
k
v S. A role assertion is
an expression of the form Ref(R), Irr(R), Sym(R), Tra(R), or Dis(R, S), where R, S 6= U.
Given an interpretation I, define that:
I |= R
1
◦ . . . ◦ R
k
v S if R
I
1
◦ . . . ◦ R
I
k
⊆ S
I
I |= Ref(R) if R
I
is reflexive
I |= Irr(R) if R
I
is irreflexive
I |= Sym(R) if R
I
is symmetric
I |= Tra(R) if R
I
is transitive
I |= Dis(R, S) if R
I
and S
I
are disjoint,
where the operator ◦ stands for composition. By a role axiom we mean either a role
inclusion axiom or a role assertion. We say that a role axiom ϕ is valid in I and I
validates ϕ if I |= ϕ.
An RBox is a set R = R
h
∪ R
a
, where R
h
is a finite set of role inclusion axioms and
R
a
is a finite set of role assertions. It is required that R
h
is regular and R
a
is simple. In
particular, R
a
is simple if all roles R, S appearing in role assertions of the form Irr(R)
or Dis(R, S) are simple roles w.r.t. R
h
. These notions (of regularity and simplicity) will
not be exploited in this paper and we refer the reader to [6] for their definitions. An
interpretation I is a model of an RBox R, denoted by I |= R, if it validates all role
axioms of R.
The set of concepts is the smallest set such that:
– all concept names (including nominals) and >, ⊥ are concepts
– if C, D are concepts, R is a role, S is a simple role, and n is a non-negative integer,
then ¬C, C uD, C tD, ∀R.C, ∃R.C, ∃S.Self, ≥nS.C, and ≤nS.C are also concepts.
We use letters like A, B to denote concept names, and letters like C, D to denote
concepts.
Given an interpretation I, the interpretation function ·
I
is extended to complex
concepts as follows, where #Γ stands for the number of elements in the set Γ :
>
I
def
= ∆
I
⊥
I
def
= ∅ (¬C)
I
def
= ∆
I
\ C
I
(C u D)
I
def
= C
I
∩ D
I
(C t D)
I
def
= C
I
∪ D
I
(∀R.C)
I
def
= {x ∈ ∆
I
| ∀y[hx, yi ∈ R
I
implies y ∈ C
I
]}
(∃R.C)
I
def
= {x ∈ ∆
I
| ∃y[hx, yi ∈ R
I
and y ∈ C
I
]}
(∃S.Self)
I
def
= {x ∈ ∆
I
| hx, xi ∈ S
I
}
(≥ n S.C)
I
def
= {x ∈ ∆
I
| #{y | hx, yi ∈ S
I
and y ∈ C
I
} ≥ n}
(≤ n S.C)
I
def
= {x ∈ ∆
I
| #{y | hx, yi ∈ S
I
and y ∈ C
I
} ≤ n}.
4 L.A. Nguyen
A terminological axiom, also called a general concept inclusion (GCI), is an expression
of the form C v D. A TBox is a finite set of terminological axioms. An interpretation
I validates an axiom C v D, denoted by I |= C v D, if C
I
⊆ D
I
. We say that I is a
model of a TBox T , denoted by I |= T , if it validates all axioms of T .
We use letters like a and b to denote individual names. An individual assertion is
an expression of the form a
.
=6= b, C(a), R(a, b), or ¬S(a, b), where S is a simple role and
R, S 6= U . Given an interpretation I, define that:
I |= a
.
=6= b if a
I
6= b
I
I |= C(a) if a
I
∈ C
I
I |= R(a, b) if ha
I
, b
I
i ∈ R
I
I |= ¬S(a, b) if ha
I
, b
I
i /∈ S
I
.
We say that I satisfies an individual assertion ϕ if I |= ϕ. An ABox is a finite set of
individual assertions. An interpretation I is a model of an ABox A, denoted by I |= A,
if it satisfies all assertions of A.
A knowledge base is a tuple hR, T , Ai, where R is an RBox, T is a TBox, and A is
an ABox. An interpretation I is a model of a knowledge base hR, T , Ai if it is a model
of all R, T , and A. A knowledge base is satisfiable if it has a model.
A (conjunctive) query is an expression of the form ϕ
1
∧ . . . ∧ ϕ
k
, where each ϕ
i
is
an individual assertion. An interpretation I satisfies a query ϕ = ϕ
1
∧ . . . ∧ ϕ
k
, denoted
by I |= ϕ, if I |= ϕ
i
for all 1 ≤ i ≤ k. We say that a query ϕ is a logical consequence
of a knowledge base hR, T , Ai, denoted by hR, T , Ai |= ϕ, if every model of hR, T , Ai
satisfies ϕ.
Note that, queries are defined to be “ground”. In a more general context, queries
may contain variables for individuals. However, one of the approaches to deal with such
queries is to instantiate variables by individuals occurring in the knowledge base or the
query.
3 Rough Concepts in Description Logic
Let I be an interpretation and R be a role standing for a similarity predicate. For x ∈ ∆
I
,
by the neighborhood of x w.r.t. R we understand the set of elements similar to x specified
by n
R
(x)
def
= {y ∈ ∆
I
| hx, yi ∈ R
I
}. The lower and upper approximations of a concept
C w.r.t. R, denoted respectively by C
R
and C
R
, are interpreted in I as follows:
(C
R
)
I
def
= {x ∈ ∆
I
| n
R
(x) ⊆ C
I
}
(C
R
)
I
def
= {x ∈ ∆
I
| n
R
(x) ∩ C
I
6= ∅}
In words, (C
R
)
I
consists of objects whose neighborhoods w.r.t. R completely belong
to C
I
, and (C
R
)
I
consists of objects whose neighborhoods contain at least one object
of C
I
. Intuitively, if the similarity predicate R reflects the perception ability of an agent
then
– x ∈ (C
R
)
I
means that all objects indiscernible from x are in C
I
– x ∈ (C
R
)
I
means that there are objects indiscernible from x in C
I
.
Paraconsistent and Approximate Semantics for OWL 2 5
The pair hC
R
, C
R
i is usually called the rough concept of C w.r.t. the similarity pred-
icate R. The following proposition is well known from the literature of rough DLs [22, 7].
Its proof is straightforward.
Proposition 3.1. Let I be an interpretation, C be a concept, and R be a role. Then
(C
R
)
I
= (∀R.C)
I
and (C
R
)
I
= (∃R.C)
I
. That is, ∀R.C and ∃R.C are the lower and
upper approximations of C w.r.t. R, respectively. C
One can adopt different restrictions on a similarity predicate R. It is expected that
the lower approximation is a subset of the upper approximation. That is, for every in-
terpretation I and every concept C, we should have that (C
R
)
I
⊆ (C
R
)
I
, or equiv-
alently, (∀R.C)
I
⊆ (∃R.C)
I
. The latter condition corresponds to seriality of R
I
(i.e.
∀x ∈ ∆
I
∃y ∈ ∆
I
R
I
(x, y)), which can be formalized by the global assumption ∃R.>.
Thus, we have the following proposition, which is clear from the view of the correspond-
ing theory of modal logics [24].
Proposition 3.2. Let I be an interpretation. Then (C
R
)
I
⊆ (C
R
)
I
holds for every
concept C iff I validates the terminological axiom > v ∃R.>. C
In most applications, one can assume that similarity relations are reflexive and sym-
metric. Reflexivity of a similarity predicate R is expressed in SROIQ by the role assertion
Ref(R).
2
Symmetry of a similarity predicate R can be expressed in SROIQ by the role
assertion Sym(R) or the role inclusion axiom R
−
v R. Transitivity is not always assumed
for similarity relations. If one decides to adopt it for a similarity predicate R, then it
can be expressed in SROIQ by the role assertion Tra(R) or the role inclusion axiom
R ◦ R v R. In particular, in SROIQ, to express that a similarity predicate R stands for
an equivalence relation we can use the three role assertions Ref(R), Sym(R), and Tra(R).
Example 3.3. Consider the domain of universities and the language with
– N = {University-of-Warsaw, Name-Linh-Anh-Nguyen}
– C = N ∪ {University, Institute, Academic-Teacher, Teacher, Course, Name}
– R = {has-name, is-part-of, works-at, teaches, similar-name, U }
– I = {UW, IIUW, IMUW, SemanticWeb, DataMining, LANguyen, ASzalas,
HSNguyen, “University of Warsaw”, “Institute of Informatics, University of War-
saw”, “Institute of Mathematics, University of Warsaw”, “Andrzej Sza las”, “Nguyen”,
“Anh Linh Nguyen”, “Linh Anh Nguyen”, “Hung Son Nguyen”}.
Let
– R = {works-at ◦ is-part-of v works-at,
Tra(is-part-of), Ref(similar-name), Sym(similar-name)}
– T = {∃works-at.University u ∃teaches.> v Academic-Teacher,
Academic-Teacher v Teacher}
– A = {University(UW), has-name(UW, “University of Warsaw”),
Institute(IIUW), is-part-of(IIUW, UW),
has-name(IIUW, “Institute of Informatics, University of Warsaw”),
2
Reflexivity of R can also be expressed by id v R, where id stands for the “identity” role.
