1 Introduction
Reasoning about exceptions in ontologies is nowadays one of the challenges the description logics community is facing, a challenge which is at the very roots of the development of nonmonotonic reasoning in the 80 s. Many nonmonotonic extensions of Description Logics (DLs) have been developed incorporating nonmonotonic features from most of the nonmonotonic formalisms in the literature [3, 20, 22, 35, 11, 9, 28, 15, 42, 21, 8, 36, 14, 34, 29, 30], or defining new constructions and semantics such as in [7].
We focus on the rational closure for DLs [15, 18, 14, 30, 13] and, in particular, on the construction developed in [30], which is semantically characterized by minimal (canonical) preferential models. While the rational closure provides a simple and efficient approach for reasoning with exceptions, exploiting polynomial reductions to standard DLs [24, 41], the rational closure does not allow an independent handling of the inheritance of different defeasible properties of concepts^{1}^{1}1By 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 [43]). so that, if a subclass of is exceptional for a given aspect, it is exceptional tout court and does not inherit any of the typical properties of . This problem was called by Pearl [44] “the blocking of property inheritance problem”, and it is an instance of the “drowning problem” in [6].
To cope with this problem Lehmann [39] introduced the notion of the lexicographic closure, which was extended to Description Logics by Casini and Straccia [17], while in [18] the same authors develop an inheritancebased approach for defeasible DLs. Other proposals to deal with this “all or nothing” behavior in the context of DLs are the the logic of overriding, , by Bonatti, Faella, Petrova and Sauro [7], a nonmonotonic description logic in which conflicts among defaults are solved based on specificity, and the work by Gliozzi [33], 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 lexicographic closure allows for stronger inferences with respect to rational closure, but computing the defeasible consequences in the lexicographic closure may require to compute several alternative bases [39], namely, consistent sets of defeasible inclusions which are maximal with respect to a (so called 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 consistent 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. Our construction only requires a polynomial number of calls to the underlying preferential reasoner to be computed.
To develop a semantic characterization of the skeptical closure, we introduce a bipreference semantics (BPsemantics), which is still in the realm of the preferential semantics for defeasible description logics [27, 11, 28], developed along the lines of the preferential semantics introduced by Kraus, Lehmann and Magidor [37, 38]. The BPsemantics has two preference relations and is a refinement of the rational closure semantics. We show that the BP semantics provides a characterization of the MPclosure, a variant of the lexicographic closure introduced in [25]. and exploit it to build a semantics for the Skeptical closure.
Schedule of the paper is the following. In Section 2 we recall the definition of the rational closure for in [30] and of its semantics. In Section 3 we define the Skeptical closure. In Section 4, we introduce the bipreference semantics and, in Section 5 we show that it provides a semantic characterization of the MPclosure, a sound approximation of a multipreference semantics in [25]. In Section 6, the BPsemantics is used to define a semantic characterization for the skeptical closure. Finally, in Section 7, we compare with related work and conclude the paper.
This work is based on the extended abstract presented at CILC/ICTCS 2017 [23], where the notion of skeptical closure were first introduced.
2 The rational closure for
We briefly recall the logic which is at the basis of a rational closure construction proposed in [30] for . The idea underlying is that of extending the standard with concepts of the form , whose intuitive meaning is that selects the typical instances of a concept , to distinguish between the properties that hold for all instances of concept (), and those that only hold for the typical such instances (). The language is defined as follows:
,
where is a concept name and a role name. A knowledge base is a pair , where the TBox contains a finite set of concept inclusions , and the ABox contains a finite set of assertions of the form and , for individual names, and role name.
The semantics of is defined in terms of rational models, extending to the preferential semantics by Kraus, Lehmann and Magidor in [37, 38]: ordinary models of are equipped with a preference relation on the domain, whose intuitive meaning is to compare the “typicality” of domain elements: means that is more typical than . The instances of are the instances of concept that are minimal with respect to . In rational models, which characterize , is further assumed to be modular (i.e., for all , if then either or ) and wellfounded^{2}^{2}2Since has the finite model property, this is equivalent to having the Smoothness Condition, as shown in [30]. We choose this formulation because it is simpler. (i.e., there is no infinite descending chain, so that, if , also ). Ranked models characterize . Let us shortly recap their definition.
Definition 1 (Semantics of [30])
An interpretation of is any structure where: is the domain; is an irreflexive, transitive, modular and wellfounded relation over . is an interpretation function that maps each concept name to , each role name to and each individual name to . For concepts of , is defined in the usual way in interpetations [2]. In particular:
For the operator, we have .
The notion of satisfiability of a KB in an interpretation is defined as usual. Given an interpretation :
 satisfies an inclusion if ;
 satisfies an assertion if ;
 satisfies an assertion if .
A model satisfies a knowledge base if it satisfies all the inclusions in its TBox and all the assertions in its ABox . A query (either an assertion or an inclusion relation ) is logically (rationally) entailed by a knowledge base () if is satisfied in all the models of .
As shown in [30], the logic enjoys the finite model property and finite models can be equivalently defined by postulating the existence of a function , where assigns a finite rank to each world: the rank of a domain element is the length of the longest chain from to a minimal (s. t. there is no with ). The rank of a concept in is .
Although the typicality operator itself is nonmonotonic (i.e. does not imply ), the logic is monotonic: what is logically entailed by is still entailed by any with .
In [32, 30] a non monotonic construction of rational closure has been defined for , extending the construction of rational closure introduced by Lehmann and Magidor [38] to the description logic . Its definition is based on the notion of exceptionality. Roughly speaking holds in the rational closure of if is less exceptional than . We shortly recall this construction of the rational closure of a TBox and we refer to [30] for full details.
Definition 2 (Exceptionality of concepts and inclusions)
Let be a TBox and a concept. is exceptional for if and only if . An inclusion is exceptional for if is exceptional for . The set of inclusions which are exceptional for will be denoted by .
Given a TBox , it is possible to define a sequence of non increasing subsets of the TBox ordered according to the exceptionality of the elements by letting and, for , s.t. does not occurr in . Observe that, being knowledge base finite, there is an such that, for all or . A concept has rank (denoted ) for TBox, iff is the least natural number for which is not exceptional for . If is exceptional for all then ( has no rank). The rank of a typicality inclusion is . Observe that, for , contains less specific defeasible properties then .
Example 1
Let be the knowledge base with TBox:
stating that typical students do not pay taxes, but typical working students (which are students) do pay taxes and that typical students are smart. It is possible to see that
, .
In particular, the rank of concept is , as is nonexceptional for : there is a model of the KB containing a domain element with rank , which is an instance of ( satisfies all the inclusions in ). Instead, has rank , as it is exceptional for : it is not possible to find a domain element in some model of such that is an instance of and has rank . In fact, such a would be a typical and hence, it would be an instance of by the second inclusion. But, as a is a Student as well, it should satisfy the first defeasible inclusion as well and be an instance of , which is impossible. Hence, any instance of cannot have rank .
It is easy to see that the rank of the concepts , and is ; that the rank of concepts , and is ; and that the rank of concept is .
Rational closure builds on this notion of exceptionality:
Definition 3 (Rational closure of TBox)
Let be a DL knowledge base. The rational closure of TBox is defined as:
where and are concepts.
In Example 1, is in the rational closure of the TBox, as ; so is .
Exploiting the fact that entailment in can be polynomially encoded into entailment in , it is easy to see that deciding if an inclusion belongs to the rational closure of TBox is a problem in ExpTime and requires a polynomial number of entailment checks to an knowledge base. In [30] it is also shown that the semantics corresponding to rational closure can be given in terms of minimal canonical models. In such models the rank of domain elements is minimized to make each domain element to be as typical as possible. Furthermore, canonical models are considered in which all possible combinations of concepts are represented. This is expressed by the following definitions.
Definition 4 (Minimal models of )
Given and , we say that is preferred to () if: , for all (nonextended) concepts , for all , it holds that whereas there exists such that .
Given a knowledge base , we say that is a minimal model of (with respect to TBox) if it is a model satisfying and there is no model satisfying such that .
The models corresponding to rational closure are required to be canonical. This property, expressed by the following definition, is needed when reasoning about the (relative) rank of the concepts: it is important to have them all represented by some instance in the model.
Definition 5 (Canonical model)
Given , a model satisfying is canonical if for each set of concepts consistent with , there exists (at least) a domain element such that .
Definition 6 (Minimal canonical models (with respect to TBox))
is a minimal canonical model of , if it is a canonical model of and it is minimal with respect (see Definition 4) among the canonical models of .
The correspondence between minimal canonical models and rational closure is established by the following key theorem.
Theorem 2.1 ([30])
Let be a knowledge base and a query. Let be the rational closure of w.r.t. TBox. We have that if and only if holds in all minimal canonical models of with respect to TBox.
Furthermore: the rank of a concept in any minimal canonical model of is exactly the rank assigned by the rational closure construction, when is finite. Otherwise, the concept is not satisfiable in any model of the TBox.
Example 2
Considering again the KB in Example 1, we can see that defeasible inclusions and are satisfied in all the minimal canonical models of . In fact, for the first inclusion, in all the minimal canonical models of , has rank , while has rank . Thus, in all the minimal canonical models of each typical Italian student must be an instance of .
Instead, wethe deseafible inclusion is not minimally entailed from and, consistently, this inclusion does not belong to the rational closure of . Indeed, the concept is exceptional for , as it violates the defeasible property of students that, normally, they do not pay taxes (). For this reason, does not inherit “any” of the defeasible properties of . This problem is a well known problem of rational closure, called by Pearl [44] “the blocking of property inheritance problem”, and it is an instance of the “drowning problem” in [6].
To overcome this weakness of the rational closure, Lehmann introduced the notion of lexicographic closure [39], which strengthens the rational closure by allowing, roughly speaking, a class to inherit as many as possible of the defeasible properties of more general classes, giving preference to the more specific properties. The lexical closure has been extended to the description logic by Casini and Straccia in [17]. In the example above, the property of students of being smart would be inherited by working students, as it is consistent with all other (strict or defeasible) properties of working students. In the general case, however, there may be exponentially many alternative bases to be considered, which are all maximally preferred, and the lexicographic closure has to consider all of them to determine which defeasible inclusions can be accepted. In the next section we propose an approach weaker than the lexicographic closure, which leads to the construction of a single base.
3 The Skeptical Closure
Given a concept , one wants to identify the defeasible properties of the elements (if any). Assume that the rational closure of the knowledge base has already been constructed and that is the (finite) rank of concept in the rational closure^{3}^{3}3When , the defeasible inclusion belongs to the rational closure of TBox for any . Hence, we assume also belongs to the skeptical closure, and we defer considering this case until Definition 9. So far, we always assume to be finite.. The typical elements are clearly compatible (by construction) with all the defeasible inclusions in , but they might satisfy further defeasible inclusions with lower rank, i.e. those included in .
For instance, in the example above, concept has rank , and for working students all the defeasible inclusions in set above apply (in particular, that typical working students pay taxes). As for , the defeasible inclusion is not compatible with this property of typical students, while the defeasible property is, as there may be typical students which are Smart.
In general, there may be alternative maximal sets of defeasible inclusions compatible with , among which one would prefer those that maximize the sets of defeasible inclusions with higher rank. This is indeed what is done by the lexicographic closure [39], 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 (degree of seriousness), and where situations which violate a number of defaults with a certain rank are considered to be less plausible than situations which violate a lower number of defaults with the same rank. In general, there may be exponentially many alternative sets of defeasible inclusions (called bases in [39]) which are maximal and consistent for a given concept, and the lexicographic closure has to consider all of them to determine if a defeasible inclusion is to be accepted or not. As a difference, in the following, we define a construction which skeptically builds a single set of defeasible inclusions compatible with . The advantage of this construction is that it only requires a polynomial number of calls to the underlying preferential reasoner.
Let a concept with rank in the rational closure. In order to see which are the defeasible inclusions compatible with (beside those in ), we first single out the defeasible inclusions which are individually consistent with and . This is done while building the set of the defeasible inclusions which are not overridden by those in . As the set might not be globally consistent with , for the presence of conflicting defaults, we will consider the sets of defaults in with the same rank, going from to and we will add them to , if consistent. When we find an inconsistent subset, we stop. In this way, we extend with all the defeasible inclusions which are not conflicting and can be inherited by instances, even though the construction of rational closure has excluded them from .
Let be the set of typicality inclusions in the TBox which are individually compatible with (with respect to ), that is
For instance, in Example 1, for , which has rank , we have that
is the set of defeasible inclusions compatible with and . The defeasible inclusion is not included in as it is not (individually) compatible with .
Clearly, although each defeasible inclusion in is compatible with , it might be the case that overall set is not compatible with , i.e.,
Let us consider the following variant of Example 1.
Example 3
Let be the knowledge base with the TBox:
Let . While concepts and have rank , concept has rank . In this example:
where is the set of strict inclusions in . The property that typical employed students are not young, overrides the property that students are typically young. Indeed the default is not individually compatible with . Instead, the defeasible properties and  are both individually compatible with , and
Nevertheless, the overall set is not compatible with . In fact, the two defeasible inclusions in are conflicting.
When compatible with , is the unique maximal basis with respect to the seriousness ordering in [39] (as defined for constructing the lexicographic closure).
When is not compatible with , we cannot use all the defeasible inclusions in to derive conclusions about typical elements. In this case, we can either just use the defeasible inclusions in , as in the rational closure, or we can additionally use a subset of the defeasible inclusions . This is essentially what is done in the lexicoghaphic closure, where (in essence) the most preferred subsets of are selected according to a lexicographic order, which prefers defaults with higher ranks to defaults with lower ranks. In our construction instead, we consider the subsets of the set defined above, by adding to all the defeasible inclusions in with rank (let us call this set ), provided they are (altogether) compatible with and . Then, we can add all the defeasible inclusions with rank which are individually compatible with w.r.t. (let us call this set ), provided they are altogether compatible with , and , and so on and so forth, for lower ranks. This leads to the construction below.
Definition 7
Given two sets of defeasible inclusions and , is globally compatible with w.r.t. if
Definition 8
Let be a concept such that ( finite). The skeptical closure of with respect to is the set of inclusions where:

is the set of defeasible inclusions with rank which are individually compatible with w.r.t. (for each finite rank );

is the least (for ) such that is globally compatible with w.r.t. , if such a exists; , otherwise.
Intuitively, contains, for each rank , all the defeasible inclusions having rank which are compatible with and with the more specific defeasible inclusions (having rank ). As is not included in the skeptical closure, it must be that i.e., the set contains conflicting defeasible inclusions which are not overridden by more specific ones. In this case, the inclusions in (and, similarly, all the defeasible inclusions with rank lower than ) are not included in the skeptical closure w.r.t. .
Example 4
For the knowledge base in Example 1, where has rank , we have , which is compatible with and . Hence, .
When a defeasible inclusion belongs to the skeptical closure of a TBox is defined as follows.
Definition 9
Let be a knowledge base and a query. is in the skeptical closure of if either in the rational closure of or .
Once the rational closure of TBox has been computed, the identification of the defeasible inclusions in requires a number of entailment checks which is linear in the number of defeasible inclusions in TBox. First, the compatibility of each defeasible inclusion in TBox with has to be checked to compute all the ’s. Then, a compatibility check for each rank of the rational closure is needed, to verify the compatibility of , for each from to in the worst case. The maximum number or ranks in the rational closure is bounded by the number of defeasible inclusions in TBox (but it might be significantly lower in practical cases). Hence, computing the skeptical closure for requires a number of entailment checks which is, in the worst case, .
Example 5
For the knowledge base in Example 1, we have seen that, for (with rank ), is (globally) compatible with w.r.t. , and . It is easy to see that , and that is in the skeptical closure of TBox. In this case, the typical property of students of being Smart is inherited by working students.
Example 6
For the knowledge base in Example 3, as we have seen, has rank , , and . In this case, as contains conflicting defaults about tax payment, is not (globally) compatible with and , so that .
Let us consider the following knowledge base from [25] to see that, in the skeptical closure, inheritance of defeasible properties, when not overridden for more specific concepts, applies to concepts of all ranks.
Example 7
Consider a knowledge base , where and contains the following inclusions:
.
Here, we expect that the defeasible property of birds having a nice feather is inherited by typical penguins, even though penguins are exceptional birds regarding flying. We also expect that typical baby penguins inherit the defeasible property of penguins that they do not fly, although the defeasible property is instead overridden for typical baby penguins, and that they inherit the typical property of birds of having nice feather. We have that , , as, in the rational closure construction:
,
,
In particular, for , we get
Also, is (globally) consistent with , and is (globally) consistent with . Hence, , . Furthermore,
is in the skeptical closure of TBox as .
To see that the notion of skeptical closure is rather weak, let us slightly modify the KB in Example 3 (removing the last inclusion).
Example 8
Consider the TBox
As in Example 3, the rational closure assigns rank to concepts and and rank to . In this case,
,
;
;
.
.
As is not (globally) compatible with and , again . Therefore, the defeasible property that typical students are young is not inherited by typical employed students.
The skeptical closure is a weak construction: in Example 8 due to the conflicting defaults concerning tax payment for and (both with rank 0) also the property that typical students are young is not inherited by the typical employed students. Notice that, the property that typical working students are young would be accepted in the lexicographic closure of , as there are two bases, the one including and the other including , both containing . The skeptical closure is indeed weaker than the lexicographic closure (and, in particular, would be in the lexicographic closure as defined in [17]).
In the next section, we introduce a semantics based on two preference relations. We will show that this semantics characterizes a variant of the lexicographic closure introduced in [25] and exploit it to define a semantic construction for the weaker skeptical closure.
4 Refined, bipreference Interpretations
To capture the semantics of the skeptical closure, we build on the preferential semantics for rational closure of , introducing a notion of refined, bipreference interpretation (for short, BPinterpretation), which contains an additional notion of preference with respect to an interpretation. We let an interpretation to be a tuple , where the triple is a ranked interpretation as defined in Section 2. and is an additional preference relation over , with the properties of being irreflexive, transitive and wellfounded (but we do not require modularity of ). In BPinterpretations, represents a refinement of .
Definition 10 (BPinterpretation)
Given a knowledge base K, a bipreference interpretation (or BPinterpretation) is a structure , where is a domain, is an interpretation function as defined in Definition 1, where, in particular, , and and are preference relations over , with the properties of being irreflexive, transitive, wellfounded. Furthermore is modular.
The bipreference semantics, builds on a ranked semantics for the preference relation , providing a characterization of the rational closure of , and exploits it to define the preference relation which is not required to be modular. As we will see, this semantics provides a sound and complete characterization of a variant of the lexicographic closure, and we will use it as well to provide a semantic characterization of the skeptical closure. The BPsemantics has some relation with the multipreference semantics in [25]. However, it does non exploits multiple preferences w.r.t. aspects and it directly build on the preference relation . Also, in BPinterpretations, is not required to be modular.
Let be the ranking function associated in to the modular relation , which is defined as the ranking function in the models of the rational closure in Section 2. Similarly, the ranking function is extended to concepts by defining the rank of a concept in a BPinterpretation (w.r.t. the preference relation ) as .
Given a BPinterpretation and an element , we say that violates the typicality inclusion if . Let us define when a BPinterpretation is a model of a knowledge base :
Definition 11 (BPmodel of K)
Given a knowledge base K, a BPinterpretation is a BPmodel of if it satisfies both its TBox and its ABox, in the following sense:

for all strict inclusions in the TBox (i.e., does not occur in ), ;

for all typicality inclusions in the TBox, ;

satisfies the following specificity condition:
If
there is some which is violated by and,

for all , which is violated by and not by , there is a , which is violated by and not by , and such that ,
then ;


for all in ABox, ; and, for all in ABox, ;
While the satisfiability conditions (1), (2) and (4) are the same as in Section 2 for the ranked model , the specificity condition (3) requires the relation to satisfy the condition that, if violates defeasible inclusions more specific than those violated by , then (in particular, the condition means that concept is more specific than concept , as it has an higher rank in the rational closure).
In the definition above we do not impose the further requirement that, for all inclusions , holds. However, can easily see that this condition follows from condition (2) and from the property that holds.
Proposition 1
Given a knowledge base and a BPmodel of , .
Proof
We show that implies If . If , then for some , . As is a minimal canonical BPmodel of , by the correspondence with the rational closure, satisfies all the defeasible inclusions in . Instead, falsifies some defeasible inclusion with . As can only falsify defeasible inclusions with rank less then , by condition (3) in Definition 11, . Therefore, .
Corollary 1
Given a knowledge base and a BPmodel of , for all inclusions , holds.
Proof
We define logical entailment under the BPsemantics as follows: a query (of the form or ) is logically entailed by in (written ) if holds in all BPmodels of .
The following result can be easily proved for BPentailment:
Theorem 4.1
If then also . If does not occur in the other direction also holds: If then also