=Paper= {{Paper |id=Vol-2157/paper7 |storemode=property |title=Reasoning About Exceptions in Ontologies: from the Lexicographic Closure to the Skeptical Closure |pdfUrl=https://ceur-ws.org/Vol-2157/paper7.pdf |volume=Vol-2157 |authors=Laura Giordano,Valentina Gliozzi |dblpUrl=https://dblp.org/rec/conf/cade/0001G18 }} ==Reasoning About Exceptions in Ontologies: from the Lexicographic Closure to the Skeptical Closure== https://ceur-ws.org/Vol-2157/paper7.pdf

Reasoning about exceptions in ontologies:
from the lexicographic closure to the skeptical closure

Laura Giordano1 and Valentina Gliozzi2
1
DISIT - Università del Piemonte Orientale, Alessandria, Italy, laura.giordano@uniupo.it
2
Dipartimento di Informatica, Università di Torino, Italy, valentina.gliozzi@unito.it

Abstract. Reasoning about exceptions in ontologies is nowadays one of the chal-
lenges the description logics community is facing. The paper describes a prefer-
ential approach for dealing with exceptions in Description Logics, based on the
rational closure. The rational closure has the merit of providing a simple and ef-
ficient approach for reasoning with exceptions, but it does not allow independent
handling of the inheritance of different defeasible properties of concepts. In this
work we outline a possible solution to this problem by introducing a variant of the
lexicographical closure, that we call skeptical closure, which requires to construct
a single base. A preliminary version of this work appeared in [22].

1 Introduction

Reasoning about exceptions in ontologies is nowadays one of the challenges the de-
scription logics community is facing, a challenge which is at the very roots of the de-
velopment of non-monotonic reasoning in the 80s. Many non-monotonic extensions of
Description Logics (DLs) have been developed incorporating non-monotonic features
from most non-monotonic formalisms in the literature [2, 19, 25, 10, 21, 31, 8, 14, 37, 7,
20, 32, 13, 26, 30, 28], or defining new constructions and semantics such as in [6, 9].
The paper is based on a preferential approach for dealing with exceptions in descrip-
tion logics, where a typicality operator is used to select the typical (or most preferred)
instances of a concept [25]. This approach, as the preferential approach in [10], has been
developed along the lines of the preferential semantics introduced by Kraus, Lehmann
and Magidor [33, 34].
We focus on the rational closure for DLs [14, 17, 13, 28, 12] and, in particular, on the
construction developed in [28], which is semantically characterized by minimal (canon-
ical) preferential models. While the rational closure provides a simple and efficient
approach for reasoning with exceptions, exploiting polynomial reductions to standard
DLs [27], the rational closure does not allow an independent handling of the inheritance
of different defeasible properties of concepts1 so that, if a subclass of C is exceptional
for a given aspect, it is exceptional tout court and does not inherit any of the typical
properties of C. This problem was called by Pearl [39] “the blocking of property inher-
itance problem”, and it is an instance of the “drowning problem” in [5].
1
By properties of a concept, here we generically mean characteristic features of a class of
objects (represented by a set of inclusion axioms) rather than roles (properties in OWL [38]).
Laura Giordano and Valentina Gliozzi

To cope with this problem Lehmann [35] introduced the notion of the lexicographic
closure, which was extended to Description Logics by Casini and Straccia [16], while
in [17] the same authors develop an inheritance-based approach for defeasible DLs.
Other proposals to deal with this “all or nothing” behavior in the context of DLs are the
logic of overriding, DLN , by Bonatti, Faella, Petrova and Sauro [6], a nonmonotonic
description logic in which conflicts among defaults are solved based on specificity, and
the work by Gliozzi [29], who develops a semantics for defeasible inclusions in which
models are equipped with several preference relations.
In this paper we will consider a variant of the lexicographic closure. The lexico-
graphic closure allows for stronger inferences with respect to rational closure, but com-
puting the defeasible consequences in the lexicographic closure may require to compute
several alternative bases [35], namely, consistent sets of defeasible inclusions which are
maximal with respect to a (seriousness) ordering. We propose an alternative notion of
closure, the skeptical closure, which can be regarded as a more skeptical variant of the
lexicographic closure. It is a refinement of rational closure which allows for stronger
inferences, but it is weaker than the lexicographic closure and its computation does not
require to generate all the alternative maximally consistent bases. Roughly speaking,
the construction is based on the idea of building a single base, i.e. a single maximal con-
sistent set of defeasible inclusions, starting with the defeasible inclusions with highest
rank and progressively adding less specific inclusions, when consistent, but excluding
the defeasible inclusions which produce a conflict at a certain stage without considering
alternative consistent bases.
Schedule of the paper is the following. In section 2 we recall the definition of ratio-
nal closure for ALC in [28]. In section 3, we define the new closure and in Section 4
we conclude the paper with some discussion of related work.

2 The rational closure
We briefly recall the logic ALC + TR which is at the basis of a rational closure construc-
tion proposed in [28] for ALC, which extends to ALC the notion of rational closure
introduced by Lehmann and Magidor [34]. The idea underlying ALC + TR is that of
extending the standard ALC with concepts of the form T(C), whose intuitive meaning
is that T(C) selects the typical instances of a concept C, to distinguish between the
properties that hold for all instances of concept C (C ⊑ D), and those that only hold
for the typical such instances (T(C) ⊑ D). The ALC + TR language is defined as
follows: CR := A | ⊤ | ⊥ | ¬CR | CR ⊓ CR | CR ⊔ CR | ∀R.CR | ∃R.CR , and
CL := CR | T(CR ), where A is a concept name and R a role name. A KB is a pair
K = (T , A), where the TBox T contains a finite set of concept inclusions CL ⊑ CR
and the ABox A contains a finite set of assertions of the form CR (a) and R(a, b), for
a, b individual names.
The semantics of ALC + TR is defined in terms of rational models: ordinary models of
ALC are equipped with a preference relation < on the domain, whose intuitive meaning
is to compare the “typicality” of domain elements: x < y means that x is more typical
than y. The instances of T(C) are the instances of concept C that are minimal with
respect to <. We refer to [28] for a detailed description of the semantics and we denote
by |=ALC+TR entailment in ALC + TR .
Title Suppressed Due to Excessive Length

The rational closure construction assigns a rank to each concept of the KB (the high-
est the rank, the more specific is the concept). It is based on the notion of exceptionality.
Roughly speaking T(C) ⊑ D holds in the rational closure of K if C is less exceptional
than C ⊓ ¬D. We shortly recall the construction of the rational closure w.r.t. TBox.

Definition 1 (Exceptionality of concepts and inclusions). Let E be a TBox and C a
concept. C is exceptional for E if and only if E |=ALC+TR T(⊤) ⊑ ¬C.2 An inclusion
T(C) ⊑ D is exceptional for E if C is exceptional for E. The set of inclusions in TBox
which are exceptional for E will be denoted by E(E).

Given a TBox T , it is possible to define a sequence of non increasing subsets of TBox
ordered according to the exceptionality of the elements E0 ⊇ E1 ⊇ E2 . . . by letting
E0 = T and, for i > 0, Ei = E(Ei−1 ) ∪ {C ⊑ D ∈ TBox s.t. T does not occurr in C}.
Observe that, being KB finite, there is an n ≥ 0 such that, for all m > n, Em = En
or Em = ∅. A concept C has rank i (denoted rank (C) = i) for T , iff i is the least
natural number for which C is not exceptional for Ei . If C is exceptional for all Ei
then rank (C) = ∞ (C has no rank). The rank of a typicality inclusion T(C) ⊑ D is
rank (C). Rational closure builds on this notion of exceptionality:

Definition 2 (Rational closure of TBox [28]). Let K = (T , K) be a DL knowledge
base. A typicality inclusion T(C) ⊑ D is in the rational closure of K w.r.t. TBox if
either rank (C) < rank (C ⊓ ¬D) or rank (C) = ∞.

Exploiting the fact that entailment in ALC + TR can be polynomially encoded into
entailment in ALC, it is easy to see that deciding if an inclusion T(C) ⊑ D belongs to
the rational closure of TBox is a problem in E XP T IME [28].

Example 1. Let K be the knowledge base with the following TBox T :
T(Student ) ⊑ ¬Pay Taxes
T(WStudent ) ⊑ Pay Taxes
T(Student ) ⊑ Young
WStudent ⊑ Student
stating that typical students do not pay taxes and are young, while typical working stu-
dents (which are students) do pay taxes. We can see that Student has rank 0, while
WStudent has rank 1 (as working students falsify the first default) and:
E0 = T ; E1 = {T(WStudent )⊑ Pay Taxes, WStudent ⊑ Student};
and the defeasible inclusions T(Student ⊓ Italian) ⊑ ¬Pay Taxes and T(WStudent
⊓Italian) ⊑ Pay Taxes both belong, as expected, to the rational closure of K, as be-
ing Italian is irrelevant with respect to being or not a typical student. However, we
cannot conclude that T(WStudent) ⊑ Y oung , as concept WStudent is exceptional
w.r.t. Student concerning the property of paying taxes and, hence, it does not inherit
any defeasible property of Student.

In this example the rational closure is too weak to infer that typical working students, as
all typical students, are young. The lexicographic closure [35] strengthens the rational
2
Observe that, as the instances of concept ⊤ are all the domain elements, T(⊤) is the set of all
the preferred domain elements w.r.t. <
Laura Giordano and Valentina Gliozzi

closure by allowing to retain, roughly speaking, as many as possible of the defeasible
properties, lgiving preference to the more specific properties. In the example, the prop-
erty of students of being Young would be inherited by working students, as it is consis-
tent with all the other (strict or defeasible) properties of W Student (those in E1 ). In the
general case, there may be exponentially many alternative sets of defeasible inclusions
(bases) which are maximal and consistent for a given concept and the lexicographic clo-
sure considers all of them to conclude that a defeasible inclusion is accepted. Besides
specificity, the lexicographic closure also considers the number of defaults accepted, for
each rank, in the alternative bases and gives preference to those bases maximizing the
number of defaults with the highest rank. In the next section we propose an approach
weaker than the lexicographic closure, which leads to the construction of a single base.

3 From the lexicographic to the skeptical closure
Given a concept B, one wants to identify the defeasible properties of the B-elements.
Assume that the rational closure of the knowledge base K has already been constructed
and that k is the rank of concept B in the rational closure. The typical B elements
are clearly compatible with all the defeasible inclusions in Ek , but they might satisfy
other defeasible inclusions with lower rank, i.e. those included in E0 , E1 , . . . , Ek−1 .
In general, there may be alternative maximal sets of defeasible inclusions compatible
with B, among which one would prefer those that maximize the number of defeasible
inclusions with higher rank. This is indeed what is done by the lexicographic closure
[35], which considers alternative maximally preferred sets of defaults called “bases”,
which, roughly speaking, maximize the number of defaults of higher ranks with respect
to those with lower ranks (the so called degree of seriousness), and where situations
which violate more defaults with a certain rank are considered to be less plausible than
situations which violates less defaults with the same rank. As a difference, in the follow-
ing, we aim at defining a construction which skeptically builds a single set of defeasible
inclusions compatible with B.
Let S B be the set of typicality inclusions T(C) ⊑ D in K which are individually
compatible with B w.r.t. Ek , that is
S B = {T(C) ⊑ D ∈ TBox | Ek ∪ {T(C) ⊑ D} 6|=ALC+TR T(⊤) ⊑ ¬B}.
Clearly, although each defeasible inclusion in S B is compatible with B, it might be the
case that overall set S B is not compatible with B, i.e., Ek ∪ S B |=ALC+TR T(⊤) ⊑
¬B. When compatible with B, S B is the unique maximal basis with respect to the
seriousness ordering in [35] (as defined for constructing the lexicographic closure).
When S B is not compatible with B, we cannot use all the defeasible inclusions in
B
S to derive conclusions about typical B elements. In this case, we can either just use
the defeasible inclusions in Ek , as in the rational closure, or we can additionally use
a subset of the defeasible inclusions S B . For instance, we can additionally use all the
defeasible inclusions in S B with rank k − 1 (let us call this set Sk−1
B
), provided they are
(altogether) compatible with B and Ek . Then, we can, possibly, add all the defeasible
B
inclusions with rank k − 2 which are individually compatible with B w.r.t. Ek ∪ Sk−1
B B
(let us call them Sk−2 ), provided they are altogether compatible with B, Ek and Sk−1 ,
and so on and so forth, for lower ranks. This leads to the construction below.
Title Suppressed Due to Excessive Length

Definition 3. Given two sets of defeasible inclusions S and S ′ , S is globally compatible
with B w.r.t. S ′ if S ∪ S ′ 6|=ALC+TR T(⊤) ⊑ ¬B.

Definition 4. Let B be a concept such that rank (B) = k. The skeptical closure of K
with respect to B is the set of inclusions S sk,B = Ek ∪ Sk−1 B B
∪ Sk−2 ∪ . . . ∪ ShB where:
- SiB ⊆ Ei − Ei+1 is the set of defeasible inclusions with rank i which are individually
B B B
compatible with B w.r.t. Ek ∪ Sk−1 ∪ Sk−2 ∪ . . . ∪ Si+1 (for each finite rank i ≤ k);
B
- h is the least j (for 0 ≤ j ≤ k − 1) such that Sj is globally compatible with B w.r.t.
B B B
Ek ∪ Sk−1 ∪ Sk−2 ∪ . . . ∪ Sj+1 , if such a j exists; S sk,B = Ek , otherwise.
Intuitively, S sk,B contains, for each rank j, all the defeasible inclusions having rank
j which are compatible with B and with the more specific defeasible inclusions (with
B B
rank > j). As Sh−1 is not included in the skeptical closure, it must be that Ek ∪ Sk−1 ∪
B B B
Sk−2 ∪ . . . ∪ Sh ∪ Sh−1 |=ALC+TR T(⊤) ⊑ ¬B i.e., the set Sh−1 contains conflicting
defeasible inclusions which are not overridden by more specific ones. In this case, the
B
inclusions in Sh−1 (and all the defeasible inclusions with rank lower than h − 1) are not
included in the skeptical closure w.r.t. B. Let us now define entailment of a defeasible
inclusion from the skeptical closure of TBox.
Definition 5. Let T(B) ⊑ D be a defeasible inclusion and let k = rank (B) be the
rank of concept B in the rational closure. T(B) ⊑ D is in the skeptical closure of
TBox if S sk,B |=ALC+TR T(⊤) ⊑ (¬B ⊔ D).
After the rational closure of the TBox has been computed, the identification of the de-
feasible inclusions in S sk,B requires a number of entailment checks which is linear in
the number of defeasible inclusions in TBox: the individual compatibility of a defeasi-
ble inclusion of rank i in TBox has to be checked only once to compute SiB ; also, for
each rank i of the rational closure (in the worst case), a (global) compatibility check is
needed for SiB .
In Example 1 the inclusion T(WStudent ) ⊑ Young is in the skeptical closure of
TBox, as WStudent has rank 1 and inclusion T(Student ) ⊑ Young in E0 is compati-
ble with WStudent. No other inclusions with rank 0 are compatible with E1 .
Example 2. Let us consider, instead, the knowledge base K ′ with TBox:
T(Student ) ⊑ ¬Pay Taxes
T(Worker ) ⊑ Pay Taxes
T(Student ) ⊑ Young
WStudent ⊑ Student ⊓ Worker
the inclusion T(WStudent ) ⊑ Young is not in the skeptical closure of TBox ′ , as
S0WStudent is not compatible with WStudent (w.r.t. E1 ), due to the conflicting defaults
concerning tax payment for Worker and Student (both with rank 0). Hence, the de-
feasible property that typical students are young is not inherited by typical working
students.
Notice that, the property that typical working students are young is accepted in the
lexicographic closure of K ′ , as there are two bases (the one including T(Student ) ⊑
¬Pay Taxes and the other T(Worker ) ⊑ Pay Taxes), both containing T(Student )
⊑ Young. The skeptical closure is indeed weaker than the lexicographic closure.
Laura Giordano and Valentina Gliozzi

4 Conclusions and related work

We have introduced a weaker variant of the lexicographic closure [35, 16], which deals
with the problem of “all or nothing” affecting the rational closure without generating
alternative “bases”. Other refinements of the rational closure, also deal with this lim-
itation of the rational closure, are the relevant closure [11] and the inheritance-based
rational closure [15, 17], In particular, in [15, 17], a new closure operation is defined by
combining the rational closure with defeasible inheritance networks. The inheritance-
based rational closure, in Example 2, is able to conclude that typical working students
are young, relying on the fact that only the information related to the connection of
WStudent and Young (and, in particular, only the defeasible inclusions occurring on
the routes connecting WStudent and Young in the corresponding net) are used in the
rational closure construction for answering the query.
Another approach which deals with the above problem of “inheritance with excep-
tions” has been proposed by Bonatti et al. in [6], where the logic DLN captures a form
of “inheritance with overriding”: a defeasible inclusion is inherited by a more specific
class if it is not overridden by more specific (conflicting) properties. In Example 2, our
construction behaves differently from DLN , as in DLN the concept WStudent has an
inconsistent prototype, as working students inherit two conflicting properties by super-
classes: the property of students of paying taxes and the property of workers of paying
taxes. In the skeptical closure one cannot conclude that T(WStudent) ⊑ ⊥ and, using
the terminology in [6], the conflict is “silently removed”. In this respect, the skepti-
cal closure appears to be weaker than DLN , although it shares with DLN (and with
lexicographic closure) a notion of overriding.
Bozzato et al. in [9] present an extension of the CKR framework in which defeasible
axioms can be included in the global context and can be overridden by knowledge in
a local context. Exceptions have to be justified in terms of semantic consequence. A
translation of extended CHRs (with knowledge bases in SROIQ-RL) into Datalog
programs under the answer set semantics is also defined.
Concerning the multipreference semantics introduced in [29] (and further refined in
[23]) to provide a semantic strengthening of the rational closure, we have shown in [23]
that a variant of Lehmann’s lexicographic closure (which does not take into account
the number of defaults within the same level, but only their subset inclusion) provides
a sound approximation of the multipreference semantics. We expect that the skeptical
closure introduced in this work is still a sound, though weaker, approximation for the
multipreference semantics in [23].
Detailed comparisons and the study of the semantics underlying the skeptical clo-
sure will be subject of future work. The relationships among the above variants of ra-
tional closure for DLs and the notions of rational closure for DLs developed in the
contexts of fuzzy logic [18] and probabilistic logics [36] have to be investigated as well.
As it has been show in [3] for the propositional logic case, KLM preferential logics and
the rational closure [33, 34], the probabilistic approach [1], the system Z [39] and the
possibilistic approach [4, 3] are all related with each other, and similar relations might
be expected to hold for the non-monotonic extensions of description logics as well. Al-
though the skeptical closure has been defined based on the preferential extension of
Title Suppressed Due to Excessive Length

ALC, the same construction could be adopted for more expressive description logics,
provided the rational closure can be defined [24], as well as for the propositional case.
Acknowledgement: We thank the anonymous reviewers for their helpful comments.

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