A Tableau Algorithm for Possibilistic Description Logic
ALC
Guilin Qi1 and Jeff Z. Pan2
1 Institute AIFB, University of Karlsruhe, Germany
gqi@aifb.uni-karlsruhe.de
2 Department of Computing Science, The University of Aberdeen
jpan@csd.abdn.ac.uk
Abstract. Uncertainty reasoning and inconsistency handling are two important
problems that often occur in the applications of the Semantic Web. Possibilistic
description logics provide a flexible framework for representing and reasoning
with ontologies where uncertain and/or inconsistent information is available. Al-
though possibilistic logic has become a popular logical framework for uncertainty
reasoning and inconsistency handling, its role in the Semantic Web is underesti-
mated. One of the challenging problems is to provide a practical algorithm for
reasoning in possibilistic description logics. In this paper, we propose a tableau
algorithm for possibilistic description logic ALC. We show how inference ser-
vices in possibilistic ALC can be reduced to the problem of computing the in-
consistency degree of the knowledge base. We then give tableau expansion rules
for computing the inconsistency degree of a possibilistic ALC knowledge. We
show that our algorithm is sound and complete. The computational complexity of
our algorithm is analyzed. Since our tableau algorithm is an extension of a tab-
leau algorithm for ALC, we can reuse many optimization techniques for tableau
algorithms ofALC to improve the performance of our algorithm so that it can be
applied in practice.
1 Introduction
Uncertainty reasoning and inconsistency handling are two important problems that of-
ten occur in the applications of the Semantic Web, such as the areas like medicine
and biology [21,16]. Recently, there is an increasing interest to extend Web Ontology
Language OWL to represent uncertain knowledge. Most of the work is based on De-
scription Logics (DL) that provide important formalisms for representing and reasoning
with ontologies. A DL knowledge base is then extended by attaching each axiom in it
with a degree of belief. The degree of belief can have several meanings depending on
the semantics of the logic. For example, in probabilistic description logics, the degree
of belief can be explained as degree of overlap between two concepts (see [21]) or
probability of a concept given another one (see [9,16]), and in possibilistic descrip-
tion logics, the degree of belief is explained as the necessity degree or certainty degree
(see [11,18]). Inconsistency handling in DL is another problem that has attracted a lot
of attention. Inconsistency can occur due to several reasons, such as modeling errors,
J. Domingue and C. Anutariya (Eds.): ASWC 2008, LNCS 5367, pp. 61–75, 2008.
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62 G. Qi and J.Z. Pan
migration or merging ontologies, and ontology evolution. When an ontology is incon-
sistent, an ontology language which has first-order features, such as a description logic,
cannot be applied to infer non-trivial conclusions.
Let us consider an uncertain medical ontology which is modified from the medical
example given in [16].
Example 1. Given an ontology consisting of the following terminological axioms at-
tached with weights:
ax
1
: (Heartpatient  HighBloodPressure, 1)
ax
2
: (PacemakerPatient  ¬HighBloodPressure, 1)
ax
3
: (HeartPatient  MalePacemakerPatient, 0.4)
ax
4
: (HeartPatient  ∃HasHealthInsurance.PrivateHealth, 0.9)
ax
5
: (PacemakerPatient(Tom), 0.8).
Suppose we use possibilistic logic, then ax
1
means that ”it is absolute certain that heart
patients suffers from high blood pressure”, ax
2
means that ”it is absolute certain that
pacemaker patients do not suffer from high blood pressure”, ax
3
says that ”it is a little
certain that heart patient are male pacemaker patient”, ax
4
says ”it is highly certain that
heart patients have a private insurance”, and finally ax
5
states that ”it is quite certain
that Tom is a pacemaker patient”. Suppose we learn that Tom is a heart patient with
degree 0.5 (ax
6
: (HeartPatient(Tom),0.5)), i.e., it is somewhat certain that Tom is
a heart patient, and we add this axiom to the ontology, then the ontology will become
inconsistent. From this updated ontology, we may want to query if Tom suffers from
high blood pressure and to ask to what degree we can infer this conclusion?
Possibilistic description logics, first proposed by Hollunder in [11], are extension of
description logics with possibilistic semantics. It is well-known that possibilistic logic is
a powerful logical framework for dealing with uncertainty and handling inconsistency.
Possibilistic description logics inherit these two nice properties and have very promising
applications in the Semantic Web. A possibilistic DL knowledge base consists of a set
of weighted axioms of the form (φ, α), where φ is a DL axiom such as an assertional
axiom of the form C(a) and α is an element of the semi-open real interval (0,1] or of a
finite total ordered scale. A weighted axiom (φ, α) encodes the constraint N(φ) ≥ α,
where N is a necessity measure [7], with the intended meaning that the necessity degree
of φ is at least α. One critical difference between possibilistic description logics and
probabilistic description logics is that the weight attached to an axiom is not absolute
and can be be replaced by another number as long as the ordering between two weights
is not changed. Therefore, possibilistic description logics provide a more flexible way
to represent uncertain information than probabilistic description logics. In Example 1,
if the weight of ax
4
is changed to 0.85, the changed possibilistic DL knowledge base
is query-equivalent to the original one. That is, the answer to an arbitrary query over
the original knowledge bases is the same that over the changed knowledge base. In
contrast, if we apply the probabilistic DLs in [16], axioms in Example 1 are considered
as conditional constraints which are interpreted as conditional probabilities of a concept
w.r.t. another concept. So the weights of axioms should be precisely given.