1 Introduction
Recently, a large amount of work has been done in order to extend the basic formalism of Description Logics (for short, DLs) with nonmonotonic reasoning features [26, 1, 10, 11, 13, 17, 20, 4, 2, 6, 25, 22]; the purpose of these extensions is that of allowing reasoning about prototypical properties of individuals or classes of individuals. In these extensions one can represent, for instance, knowledge expressing the fact that the hematocrit level is usually under 50%, with the exceptions of newborns and of males residing at high altitudes, that have usually much higher levels (even over 65%). Furthermore, one can infer that an individual enjoys all the typical properties of the classes it belongs to. As an example, in the absence of information that Carlos and the son of Fernando are either newborns or adult males living at a high altitude, one would assume that the hematocrit levels of Carlos and Fernando’s son are under 50%. This kind of inferences apply to individual explicitly named in the knowledge base as well as to individuals implicitly introduced by relations among individuals (the son of Fernando).
In spite of the number of works in this direction, finding a solution to the problem of extending DLs for reasoning about prototypical properties seems far from being solved. The most well known semantics for nonmonotonic reasoning have been used to the purpose, from default logic [1], to circumscription [2], to Lifschitz’s nonmonotonic logic MKNF [10, 25], to preferential reasoning [13, 4, 17], to rational closure [6, 9].
In this work, we focus on rational closure and, specifically, on the rational closure for . The interest of rational closure in DLs is that it provides a significant and reasonable nonmonotonic inference mechanism, still remaining computationally inexpensive. As shown for in [6], its complexity can be expected not to exceed the one of the underlying monotonic DL. This is a striking difference with most of the other approaches to nonmonotonic reasoning in DLs mentioned above, with some exception such as [25, 22]. More specifically, we define a rational closure for the logic , building on the notion of rational closure in [23] for propositional logic. This is a difference with respect to the rational closure construction introduced in [6] for , which is more similar to the one by Freund [12] for propositional logic (for propositional logic, the two definitions of rational closure are shown to be equivalent [12]). We provide a semantic characterization of rational closure in in terms of a preferential semantics, by generalizing to the results for rational closure in presented in [18]. This generalization is not trivial, since lacks a crucial property of , the finite model property [19]. Our construction exploits an extension of with a typicality operator , that selects the most typical instances of a concept , . We define a minimal model semantics and a notion of minimal entailment for the resulting logic, , and we show that the inclusions belonging to the rational closure of a TBox are those minimally entailed by the TBox, when restricting to canonical models. This result exploits a characterization of minimal models, showing that we can restrict to models with finite ranks. We also show that the rational closure construction of a TBox can be done exploiting entailment in , without requiring to reason in , and that the problem of deciding whether an inclusion belongs to the rational closure of a TBox is in ExpTime.
Concerning ABox reasoning, because of the interaction between individuals (due to roles) it is not possible to separately assign a unique minimal rank to each individual and alternative minimal ranks must be considered. We end up with a kind of skeptical inference with respect to the ABox, whose complexity in ExpTime as well.
2 A nonmonotonic extension of
Following the approach in [14, 17], we introduce an extension of [19] with a typicality operator in order to express typical inclusions, obtaining the logic . The intuitive idea is to allow concepts of the form , whose intuitive meaning is that selects the typical instances of a concept . We can therefore distinguish between the properties that hold for all instances of (), and those that only hold for the typical such instances (). Since we are dealing here with rational closure, we attribute to properties of rational consequence relation [23]. We consider an alphabet of concept names , role names , transitive roles , and individual constants . Given , , and we define:
As usual, we assume that transitive roles cannot be used in number restrictions [19]. A KB is a pair (TBox, ABox). TBox contains a finite set of concept inclusions and role inclusions . ABox contains assertions of the form and , where .
The semantics of is formulated in terms of rational models: ordinary models of are equipped with a preference relation on the domain, whose intuitive meaning is to compare the “typicality” of domain elements, that is to say, means that is more typical than . Typical instances of a concept (the instances of ) are the instances of that are minimal with respect to the preference relation (so that there is no other instance of preferred to )^{1}^{1}1As for the logic in [15], an alternative semantic characterization of can be given by means of a set of postulates that are essentially a reformulation of the properties of rational consequence relation [23]..
In the following definition we introduce the notion of
Definition 1 (Semantics of )
A model ^{2}^{2}2In this paper, we follow the terminology in [23] for preferential and ranked models, and we use the term “model” to denote an an interpretation. is any structure where:

is the domain;

is an irreflexive, transitive, wellfounded, and modular relation over ;

is the extension function that maps each concept to , and each role to . For concepts of , is defined as usual. For the operator, we have , where and s.t. .
We say that an irreflexive and transitive relation is:

modular if, for all , if then or [23];

wellfounded if, for all , for all , either or such that .
It can be proved that an irreflexive and transitive relation on is wellfounded if and only if there are no infinite descending chains of elements of (see Appendix 0.B).
In [23] it is shown that, for a strict partial order over a set , the modularity requirement is equivalent to postulating the existence of a rank function , such that is a totally ordered set. In the presence of the wellfoundedness condition above, the totally ordered set happens to be a wellorder, and we can introduce a rank function assigning an ordinal to each domain element in , and let if and only if . We call the rank of element in . Observe that, when the rank is finite, it can be understood as the length of a chain from to a minimal (i.e. an s.t. for no , ).
Notice that the meaning of can be split into two parts: for any of the domain , just in case (i) , and (ii) there is no such that . In order to isolate the second part of the meaning of , we introduce a new modality . The basic idea is simply to interpret the preference relation as an accessibility relation. The wellfoundedness of ensures that typical elements of exist whenever , by avoiding infinitely descending chains of elements. The interpretation of in is as follows:
Definition 2
Given a model , we extend the definition of with the following clause:
for every , if then
It is easy to observe that is a typical instance of if and only if it is an instance of and , that is to say:
Proposition 1
Given a model , given a concept and an element , we have that
Since we only use to capture the meaning of , in the following we will always use the modality followed by a negated concept, as in .
In the next definition of a model satisfying a knowledge base, we extend the function to individual constants; we assign to each individual constant a domain element .
Definition 3 (Model satisfying a knowledge base)
Given a model , we say that:

a model satisfies an inclusion if ; similarly for role inclusions;

satisfies an assertion if ;

satisfies an assertion if .
Given a KB=(TBox,ABox), we say that: satisfies TBox if satisfies all inclusions in TBox; satisfies ABox if satisfies all assertions in ABox; satisfies KB (or, is a model of KB) if it satisfies both its TBox and its ABox.
As a difference with the approach in [17], we do no longer assume the unique name assumption (UNA), namely we do not assume that each is assigned to a distinct element . In [17], in which we compare models that might have a different interpretation of concepts and that are not canonical, UNA avoids that models in which two named individuals are mapped into the same domain element are preferred to those in which they are mapped into distinct ones. UNA is not needed here as we compare models with the same domain and the same interpretation of concepts, while assuming that models are canonical (see Definition 9) and contain all the possible domain elements “compatible” with the KB.
The logic , as well as the underlying , does not enjoy the finite model property [19].
Given a KB, we say that an inclusion is entailed by KB, written KB , if holds in all models satisfying KB; similarly for role inclusions. We also say that an assertion , with , is entailed by KB, written KB , if holds in all models satisfying KB.
Let us now introduce the notions of rank of a concept.
Definition 4 (Rank of a concept )
Given a model , we define the rank of a concept in the model as . If , then has no rank and we write .
Proposition 2
For any , we have that satisfies if and only if .
It is immediate to verify that the typicality operator itself is nonmonotonic: does not imply . This nonmonotonicity of allows to express the properties that hold for the typical instances of a class (not only the properties that hold for all the members of the class). However, the logic is monotonic: what is inferred from KB can still be inferred from any KB’ with KB KB’. This is a clear limitation in DLs. As a consequence of the monotonicity of , one cannot deal with irrelevance. For instance, if typical VIPs have more than two marriages, we would like to conclude that also typical tall VIPs have more than two marriages, since being tall is irrelevant with respect to being married. However, KB, , does not entail KB , even if the property of being tall is irrelevant with respect to the number of marriages. Observe that we do not want to draw this conclusion in a monotonic way from , since otherwise we would not be able to retract it when knowing, for instance, that typical tall VIPs have just one marriage (see also Example 1). Rather, we would like to obtain this conclusion in a nonmonotonic way. In order to obtain this nonmonotonic behavior, we strengthen the semantics of by defining a minimal models mechanism which is similar, in spirit, to circumscription. Given a KB, the idea is to: 1. define a preference relation among models, giving preference to the model in which domain elements have a lower rank; 2. restrict entailment to minimal models (w.r.t. the above preference relation) of KB.
Definition 5 (Minimal models)
Given and we say that is preferred to () if (i) , (ii) for all concepts , and (iii) for all , whereas there exists such that . Given a KB, we say that is a minimal model of KB with respect to if it is a model satisfying KB and there is no model satisfying KB such that .
The minimal model semantics introduced above is similar to the one introduced in [17] for . However, it is worth noticing that the notion of minimality here is based on the minimization of the ranks of the worlds, rather then on the minimization of formulas of a specific kind. Differently from [17], here we only compare models in which the interpretation of concepts is the same. In this respect, the minimal model semantics above is similar to the minimal model semantics FIMS, introduced in [16] to provide a semantic characterization to rational closure in propositional logic. In FIMS, the interpretation of propositions in the models to be compared is fixed. In contrast, in the alternative semantic characterization VIMS, models are compared in which the interpretation of propositions may vary. Although fixing the interpretation of propositions (or concepts) can appear to be rather restrictive, for the propositional case, it has been proved in [16] that the two semantic characterizations (VIMS and FIMS) are equivalent under suitable assumptions and, in particular, under the assumption that in FIMS canonical models are considered. Similarly to FIMS, here we compare models by fixing the interpretation of concepts, and we also restrict our consideration to canonical models, as we will do in section 5^{3}^{3}3 Note that our language does not provide a direct way for minimizing roles. On the other hand, fixing roles does not appear to be very promising. Indeed, for circumscribed KBs, it has been proved in [2] that allowing role names to be fixed makes reasoning highly undecidabe. For the time being we have not studied the issue of allowing fixed roles in our minimal model semantics for . .
Let us define:
,
so that KB .
Proposition 3 (Existence of minimal models)
Let KB be a finite knowledge base, if KB is satisfiable then it has a minimal model.
Proof
Let be a model of KB, where we assume that determines and is the set of ordinals. Define the relation
if and and
where is also determined by a rank on ordinals. Define further . Let us define finally , where and is defined by the ranking, for any :
Observe that is welldefined for any concept and
is also welldefined (a set of ordinals has always a least element). We now show that KB. Since is the same as in , it follows immediately that .
We prove that . Let . Suppose by absurdity that , this means that . Let , such that . exists. Similarly, let , such that . We then have , as is minimal. Thus we get that against the fact that is a model of KB.
The following theorem says that reasoning in has the same complexity as reasoning in , i.e. it is in ExpTime. Its proof is given by providing an encoding of satisfiability in into satisfiability , which is known to be an ExpTimecomplete problem.
Theorem 2.1
Satisfiability in is an ExpTimecomplete problem.
The proof can be found in Appendix 0.A.
3 Rational Closure for
In this section, we extend to the notion of rational closure proposed by Lehmann and Magidor [23] for the propositional case. Given the typicality operator, the typicality inclusions (all the typical ’s are ’s) play the role of conditional assertions in [23]. Here we define the rational closure of the TBox. In Section 6 we will discuss an extension of rational closure that also takes into account the ABox.
Definition 6 (Exceptionality of concepts and inclusions)
Let be a TBox and a concept. is said to be exceptional for if and only if . A Tinclusion is exceptional for if is exceptional for . The set of Tinclusions of which are exceptional in will be denoted as .
Given a DL KB=(TBox,ABox), it is possible to define a sequence of non increasing subsets of TBox by letting and, for , s.t. does not occurr in . Observe that, being KB finite, there is an such that, for all or . Observe also that the definition of the ’s is the same as the definition of the ’s in Lehmann and Magidor’s rational closure [21], except for that here, at each step, we also add all the “strict” inclusions (where does not occur in ).
Definition 7 (Rank of a concept)
A concept has rank (denoted by ) for KB=(TBox,ABox), iff is the least natural number for which is not exceptional for . If is exceptional for all then , and we say that has no rank.
The notion of rank of a formula allows to define the rational closure of the TBox of a KB. Let be the entailment in . In the following definition, by KB we mean , where does not include the defeasible inclusions in KB.
Definition 8 (Rational closure of TBox)
Let KB=(TBox,ABox) be a DL knowledge base. We define, , the rational closure of TBox, as , where and are arbitrary concepts.
Observe that, apart form the addition of strict inclusions, the above definition of rational closure is the same as the one by Lehmann and Magidor in [23]. The rational closure of TBox is a nonmonotonic strengthening of . For instance, it allows to deal with irrelevance, as the following example shows.
Example 1
Let TBox = . It can be verified that . This is a nonmonotonic inference that does no longer follow if we discover that indeed comic actors are not charming (and in this respect are untypical actors): indeed given TBox’= TBox , we have that .
Furthermore, as for the propositional case, rational closure is closed under rational monotonicity [21]: from and it follows that .
Although the rational closure is an infinite set, its definition is based on the construction of a finite sequence of subsets of TBox, and the problem of verifying that an inclusion is in ExpTime. To prove this result we need to introduce some propositions.
First of all, let us remember that rational entailment is equivalent to preferential entailment for a knowledge base only containing positive nonmonotonic implications (see [23]). The same holds in preferential description logics with typicality. Let be the logic that we obtain when we remove the requirement of modularity in the definition of . In this logic the typicality operator has a preferential semantics [21], based on the preferential models of P rather then on the ranked models [23]. An extension of with typicality based on preferential logic P has been studied in [14]. As a TBox of a KB in is a set of strict inclusions and defeasible inclusions (i.e., positive nonmonotonic implications), it can be proved that:
Proposition 4
Given a with empty ABox, and an inclusion we have
Proof
(sketch) The (if) direction is trivial, thus we consider the (only if) one. Suppose that , let a preferential model of , where is transitive, irreflexive, and wellfounded, which falsifies . Then for some element and . Define first a model , where the relation is defined as follows:
It can be proved that:

is transitive and irreflexive

is wellfounded

if then

if then .
We can show that is a model of . This is obvious for inclusions that do not involve , as the interpretation is the same. Given an inclusion , if it holds in then it holds also in as . Moreover falsifies by , in particular (the only interesting case) when . To this regard, we know that , suppose by absurd that , since , we have that , thus there must be a with . But then by 4 and we get a contradiction. Thus and , that is falsifies in .
Observe that in model satisfies:
As a next step we define a modular model , where the relation is defined as follows. Considering where is wellfounded, we can define by recursion the following function from to ordinals:

if is minimal in

if the set is finite

if the set is infinite.
Observe that if then . We now define:
Notice that is clearly transitive, modular, and wellfounded; moreover implies . We can prove as before that is a model of and that it falsifies by . For the latter, we consider again the only interesting case when . Suppose by absurd that , since , we have that , thus there must be a with . But means that . We can conclude that it must be also , otherwise by (*) we would have which entails , a contradiction. We have shown that , thus a contradiction. Therefore and , that is falsifies in . We have shown that .
The proof above also extends to a KB with a nonempty ABox, but it must not contain positive typicality assertions on individuals.
Proposition 5
Let KB=(TBox,) be a knowledge base with empty ABox. iff , where , and are polynomial encodings in of KB, and , respectively.
Proof
By Proposition 4, we have that
iff
where is any (strict or defeasible) inclusion in .
To prove the thesis it suffices to show that for all inclusions in :
iff
for some polynomial encoding , and in .
The idea, on which the encoding is based, exploits the definition of the typicality operator introduced in [14], in terms of a GödelLöb modality as follows: is defined as where the accessibility relation of the modality is the preference relation in preferential models.
We define the encoding KB’=(TBox’, ABox’) of KB in as follows. First, ABox’=.
For each TBox, not containing , we introduce in TBox’.
For each occurring in the TBox, we introduce a new atomic concept and, for each inclusion TBox, we add to TBox’ the inclusion
Furthermore, to capture the properties of the modality, a new role is introduced to represent the relation in preferential models, and the following inclusions are introduced in TBox’:
The first inclusion accounts for the transitivity of . The second inclusion accounts for the smoothness (see [23, 14]): the fact that if an element is not a typical element then there must be a typical element preferred to it.
For the encoding of the inclusion : if is a strict inclusion in , then and ; if is a defeasible inclusion in , i.e. , then, we define and .
It is clear that the size of KB’ is polynomial in the size of the KB (and the same holds for and , assuming the size of and polynomial in the size of the KB). Given the above encoding, we can prove that:
iff
By contraposition, let us assume that . We want to prove that . From the hypothesis, there is a preferential model satisfying KB such that for some element , and . We build a model satisfying KB’ as follows:
;
, for all concepts in the language of ;
, for all roles ;
if and only if in the model .
By construction it follows that . Also, it can be easily verified that satisfies all the inclusions in KB’ and that and . Hence .
By contraposition, let us assume that . We want to prove that . From the hypothesis, we know there is a model satisfying KB’, such that and . We build a model satisfying KB such that some element of does not satisfy the inclusion . We let:
;
, for all concepts in the language of ;
, for all roles ;
if and only if (the transitive closure of ).
By construction, it is easy to show that and we can easily verify that satisfies all the inclusions in KB and that and .
The relation is transitive, as it is defined as the transitive closure of , but is not guaranteed to be wellfounded. However, we can modify the relation in to make it wellfounded, by shortening the descending chains.
For any , we let . Observe that for the elements in a descending chain , the set is monotonically increasing (i.e., ).
We define a new model by changing the preference relation in to as follows:
iff ( and ) or
( and and such that , )
In essence, for a pair of elements such that but and are instances of exactly the same boxed concepts () and is not the first element in the descending chain which is instance of all the boxed concepts in , we do not include the pair in (so that and will not be comparable in the preorder ). The relation is transitive and wellfounded. can be shown to be a model of KB, and to be an instance of but not of . Hence, .
Theorem 3.1 (Complexity of rational closure over TBox)
Given a TBox, the problem of deciding whether is in ExpTime.
Proof
Checking if can be done by computing the finite sequence of non increasing subsets of TBox inclusions in the construction of the rational closure. Note that the number of the is , where is the size of the knowledge base KB. Computing each , requires to check, for all concepts occurring on the left hand side of an inclusion in the TBox, whether . Regarding as a knowledge base with empty ABox, by Proposition 5 it is enough to check that , which requires an exponential time in the size of (and hence in the size of KB). If not already checked, the exceptionality of and of have to be checked for each , to determine the ranks of and of (which also can be computed in and requires an exponential time in the size of KB). Hence, verifying if is in ExpTime.
The above proof provides an ExpTime complexity upper bound for computing the rational closure over a TBox in and shows that the rational closure of a TBox can be computed simply using the entailment in .
4 Infinite Minimal Models with finite ranks
In the following we provide a characterization of minimal models of a KB in terms of their rank: intuitively minimal models are exactly those ones where each domain element has rank if it satisfies all defeasible inclusions, and otherwise has the smallest rank greater than the rank of any concept occurring in a defeasible inclusion of the KB falsified by the element. Exploiting this intuitive characterization of minimal models, we are able to show that, for a finite KB, minimal models have always a finite ranking function, no matter whether they have a finite domain or not. This result allows us to provide a semantic characterization of rational closure of the previous section to logics, like , that do not have the finite model property.
Given a model , let us define the set of defeasible inclusions falsified by a domain element , as .
Proposition 6
Let be a model of KB and , then: (a) if then ; (b) if then for every such that, for some , .
Proof
Observe that (a) follows from (b). Let us prove (b). Suppose for a contradiction that (b) is false, so that and for some such that, for some , , we have . We have also that . But KB, in particular , thus it must be , but , so that we get that a contradiction.
Proposition 7
Let KB and be a model of ; suppose that for any it holds:

(a) if then

(b) if then for every such that, for some , .
then KB.
Proof
Let , suppose that for some , it holds , then . By hypothesis, we have , against the fact that
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