Recently, in the domain of Description Logics (DLs) a large amount of work has been done in order to extend the basic formalism with nonmonotonic reasoning features. The aim of these extensions is to reason about prototypical properties of individuals or classes of individuals. In these extensions one can represent for instance knowledge expressing the fact that the heart is usually positioned in the left-hand side of the chest, with the exception of people with situs inversus, that have the heart positioned in the right-hand side. Also, one can infer that an individual enjoys all the typical properties of the classes it belongs to. So, for instance, in the absence of information that someone has situs inversus, one would assume that it has the heart positioned in the left-hand side. A further objective of these extensions is to allow to reason about defeasibile properties and inheritance with exceptions. As another example, consider the standard penguin example, in which typical birds fly, however penguins are birds that do not fly. Nonmonotonic extensions of DLs allow to attribute to an individual the typical properties of the most specific class it belongs to. In this example, when knowing that Tweety is a bird, one would conclude that it flies, whereas when discovering that it is also a penguin, the previous inference is retracted, and the fact that Tweety does not fly is concluded.
In the literature of DLs, several proposals have appeared [20, 2, 1, 7, 14, 5, 12, 3, 16, 8, 19]. However, finding a solution to the problem of extending DLs for reasoning about prototypical properties seems far from being solved.
In this paper, we introduce a general framework for nonmonotonic reasoning in DLs based on (i) the use of a typicality operator T; (ii) a minimal model mechanism (in the spirit of circumscription). The typicality operator T, introduced in , allows to directly express typical properties such as , , and , whose intuitive meaning is that normally, the heart is positioned in the left-hand side of the chest, that typical birds fly, whereas penguins do not. The T operator is intended to enjoy the well-established properties of preferential semantics, described by Kraus Lehmann and Magidor (henceforth KLM) in their seminal work [15, 17]. KLM proposed an axiomatic approach to nonmonotonic reasoning, and individuated two systems, preferential logic P and rational logic R, and their corresponding semantics. It is commonly accepted that the systems P and R express the core properties of nonmonotonic reasoning.
In [12, 11] nonmonotonic extensions of DLs based on the T operator have been proposed. In these extensions, the semantics of T is based on preferential logic P. Nonmonotonic inference is obtained by restricting entailment to minimal models, where minimal models are those that minimise the truth of formulas of a special kind. In this work, we present an alternative and more general approach. First, in our framework the semantics underlying the T operator is not fixed once for all: although we consider here only KLM’s P or R as underlying semantics, in principle one might choose any other underlying semantics for T based on a modal preference relation. Moreover and more importantly, we adopt a minimal model semantics, where, as a difference with the previous approach, the notion of minimal model is completely independent from the language and is determined only by the relational structure of models.
The semantic approach to nonmonotonic reasoning in DLs presented in this work is an extension of the one described in  within a propositional context. We then propose a rational closure construction for DL extended with the T operator as an algorithmic counterpart of our minimal model semantics, whenever the underlying logic for T is KLM logic R. Rational closure is a well-established notion introduced in  as a nonmonotonic mechanism built on the top of R in order to perform some further truthful nonmonotonic inferences that are not supported by R alone. We extend it to DLs in a natural way, so that, in turn, we can see our minimal model semantics as a semantical reconstruction of rational closure.
More in details, we take as the underlying DL and we define a nonmonotonic inference relation on the top of it by restricting entailment to minimal models: they are those ones which minimize the rank of domain elements by keeping fixed the extensions of concepts and roles. We then proceed to extend in a natural way the propositional construction of rational closure to for inferring defeasible subsumptions from the TBox (TBox reasoning). Intuitively the rational closure construction amounts to assigning a rank (a level of exceptionality) to every concept; this rank is used to evaluate defeasible inclusions of the form : the inclusion is supported by the rational closure whenever the rank of is strictly smaller than the one of . Our goal is to link the rational closure of a TBox to its minimal model semantics, but in general it is not possible. The reason is that the minimal model semantics is not tight enough to support the inferences provided by the rational closure. However we can obtain an exact corresponce between the two if we further restrict the minimal model semantics to canonical models: these are models that satisfy by means of a distinct element each intersection of concepts drawn from the KB that is satisfiable with respect to the TBox.
We then tackle the problem of extending the rational closure to ABox reasoning: we would like to ascribe defeasible properties to individuals. The idea is to maximise the typicality of an individual: the more is “typical”, the more it inherits the defeasible properties of the classes it belongs too (being a typical member of them). We obtain this by minimizing its rank (that is, its level of exceptionality), however, because of the interaction between individuals (due to roles) it is not possible to 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 the ABox. We prove that it is sound and complete with respect to the minimal model semantics restricted to canonical models.
The rational closure construction that we propose has not just a theoretical interest and a simple minimal model semantics, we show that it is also feasible since its complexity is “only” ExpTime in the size of the knowledge base (and the query), thus not worse than the underlying monotonic logic. In this respect it is less complex than other approaches to nonmonotonic reasoning in DLs [12, 2] and comparable in complexity with the approaches in [5, 4, 19], and thus a good candidate to define effective nonmonotonic extensions of DLs.
2 The operator and the General Semantics
Let us briefly recall the DLs and introduced in [9, 10], respectively. The intuitive idea is to extend the standard allowing concepts of the form , where does not mention , 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 concept (), and those that only hold for the typical instances of () that we call T-inclusions, where is a concept not mentioning . Formally, the language is defined as follows.
We consider an alphabet of concept names , of role names , and of individual constants . Given and , we define , and . A KB is a pair (TBox, ABox). TBox contains a finite set of concept inclusions . ABox contains assertions of the form and , where .
The semantics of and is defined respectively in terms of preferential and rational111We use the expression “rational model” rather than “ranked model” which is also used in the literature in order to avoid any confusion with the notion of rank used in rational closure. models: ordinary models of are equipped by 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 members of a concept , that is members of , are the members of that are minimal with respect to this preference relation (s.t. there is no other member of more typical than ). Preferential models, in which the preference relation is irreflexive and transitive, characterize the logic , whereas the more restricted class of rational models, so that is further assumed to be modular, characterizes .
Definition 2.2 (Semantics of )
A model of is any structure where: is the domain; is an irreflexive and transitive relation over that satisfies the following Smoothness Condition: for all , for all , either or such that , where and s.t. ; is the extension function that maps each concept to , and each role to . For concepts of , is defined in the usual way. For the operator, we have .
Definition 2.3 (Semantics of )
A model of is an model as in Definition 2.2 in which is further assumed to be modular: for all , if then either or .
Definition 2.4 (Model satisfying a Knowledge Base)
Given a model , is extended to assign a distinct element222We assume the well-established unique name assumption. of the domain to each individual constant of . satisfies a knowledge base =(TBox,ABox), if it satisfies both its TBox and its ABox, where: - satisfies TBox if for all inclusions in TBox, it holds ; - satisfies ABox if: (i) for all in ABox, , (ii) for all in ABox, .
In  it has been shown that reasoning in is ExpTime complete, that is to say adding the operator does not affect the complexity of the underlying DL . We are able to extend the same result also for (we omit the proof due to space limitations):
Theorem 2.5 (Complexity of )
Reasoning in is ExpTime complete.
From now on, we restrict our attention to and to finite models. Given a knowledge base and an inclusion , we say that it is derivable from (we write ) if holds in all models satisfying .
The rank of a domain element in is the length of the longest chain from to a such that for no it holds that .
Finite models can be equivalently defined by postulating the existence of a function , and then letting iff .
Given a model , the rank of a concept in is . If , then has no rank and we write .
It is immediate to verify that:
For any , we have that satisfies iff .
As already mentioned, although the typicality operator itself is nonmonotonic (i.e. does not imply ), the logics and are monotonic: what is inferred from can still be inferred from any with . In order to define a nonmonotonic entailment we introduce the second ingredient of our minimal model semantics. As in , we strengthen the semantics by restricting entailment to a class of minimal (or preferred) models, more precisely to models that minimize the rank of worlds. Informally, given two models of , one in which a given has rank 2 (because for instance , and another in which it has rank 1 (because only ), we would prefer the latter, as in this model is “more normal” than in the former. We call the new logic .
Let us define the notion of query. Intuitively, a query is either an inclusion relation or an assertion of the ABox, and we want to check whether it is entailed from a given KB.
Definition 2.9 (Query)
A query is either an assertion or an inclusion relation . Given a model , a query holds in if , whereas a query holds in if .
In analogy with circumscription, there are mainly two ways of comparing models with the same domain: 1) by keeping the valuation function fixed (only comparing and if and in the two models respectively coincide); 2) by also comparing and in case . In this work we consider the semantics resulting from the first alternative, whereas we leave the study of the other one for future work (see Section 5 below). The semantics we introduce is a fixed interpretations minimal semantics, for short .
Definition 2.10 ()
Given and we say that is preferred to () if , , and for all , whereas there exists such that .
Given a knowledge base , we say that is a minimal model of with respect to if it is a model satisfying and there is no model satisfying such that .
Next, we extend the notion of minimal model by also taking into account the individuals named in the ABox.
Definition 2.11 (Model minimally satisfying )
Given =(TBox,ABox), let and be two models of which are minimal w.r.t. Definition 2.10. We say that is preferred to with respect to ABox () if for all individual constants occurring in ABox, and there is at least one individual constant occurring in ABox such that . minimally satisfies in case there is no satisfying such that .
We say that minimally entails a query () if holds in all models that minimally satisfy .
3 A Semantical Reconstruction of Rational Closure in DLs
In this section we provide a definition of the well known rational closure, described in , in the context of Description Logics. We then provide a semantic characterization of it within the semantics described in the previous section.
Let be a DL knowledge base and a concept. is said to be exceptional for iff .
Let us now extend Lehmann and Magidor’s definition of rational closure to a DL knowledge base. First, we remember that the T operator satisfies a set of postulates that are essentially a reformulation of KLM axioms of rational logic R: in this respect, in  it is shown that the T-assertion is equivalent to the conditional assertion of KLM logic R. We say that a T-inclusion is exceptional for if is exceptional for . The set of T-inclusions which are exceptional for will be denoted as . Also in this case, it is possible to define a sequence of non-increasing subsets of by letting and, for , s.t. does not occurr in . Observe that, being finite, there is a such that for all or .
A concept has rank (denoted by ) for 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 knowledge base .
[Rational closure of TBox] Let =(TBox,ABox) be DL knowledge base. We define the rational closure of TBox of where
It is worth noticing that Definition 3.3 takes into account the monotonic logical consequences with respect to . This is due to the fact that the language here is richer than that considered by Lehmann and Magidor, who only considers the set of conditionals that, as said above, correspond to T-inclusions . The above Definition 3.3 also takes into account classical inclusions that belong to our language.
In the following we show that the minimal model semantics defined in the previous section can be used to provide a semantical characterization of rational closure.
First of all, we can observe that as it is cannot capture the rational closure of a TBox. For instance, consider the knowledge base (TBox,) of the penguin example, where TBox contains the following inclusions: , , . We observe that . Indeed in there can be a model in which , , , , , and . is a model of , and it is minimal with respect to (indeed it is not possible to lower the rank of nor of nor of unless we falsify ). Furthermore, is a typical black penguin in (since there is no other black penguin preferred to it) that flies. On the contrary, it can be verified that . Things change if we consider applied to models that contain a distinct domain element for each combination of concepts consistent with . We call these models canonical models. In the example, if we restrict our attention to models that also contain a which is a black penguin that does not fly, that is to say , , , and and can therefore be assumed to be a typical penguin, we are able to conclude that typically black penguins do not fly, as in rational closure. Indeed, in all minimal models of that also contain with , , , and , it holds that .
From now on, we restrict our attention to canonical minimal models.
Given a knowledge base and a query , we call the set of all concepts occurring (even as subconcepts) either in or in , as well as of their complements. In order to define canonical minimal models, we consider the set of all consistent sets of concepts that are consistent with . A set of concepts is consistent with if .
Definition 3.4 (Canonical minimal model w.r.t. and )
Given and a query , a minimal model satisfying is said to be canonical w.r.t. and if it contains at least a distinct domain element s.t. for each combination in consistent with .
We can prove the following results:
Let be a minimal canonical model of . For all concepts , it holds that .
The proof can be done by induction on the rank of concept .
Given , we have that if and only if holds in all canonical minimal models with respect to and .
This thoerem directly follows from Proposition 3.5. Due to space limitations we omit the proofs.
4 Rational Closure Over the ABox
In this section we extend the notion of rational closure defined in the previous section in order to take into account the individual constants in the ABox. We therefore address the question: what does the rational closure of a knowledge base allow us to infer about a specific individual constant occurring in the ABox of ? We propose the algorithm below to answer this question and we show that it corresponds to what is entailed by the minimal model semantics presented in the previous section. The idea of the algorithm is that of considering all the possible minimal consistent assignments of ranks to the individuals explicitly named in the ABox. Each assignment adds some properties to named individuals which can be used to infer new conclusions. We adopt a skeptical view of considering only those conclusions which hold for all assignments. The equivalence with the semantics shows that the minimal entailment captures a skeptical approach when reasoning about the ABox.
Definition 4.1 (Rational closure of ABox)
Let be the individuals explicitly named in the ABox. Let all the possible rank assignments (ranging from to ) to the individuals occurring in ABox.
We find the consistent with (, ABox), where:
- for all in ABox, we define s.t. , in , and s.t. in TBox ;
- let for all just calculated;
- is consistent with (, ABox) if ABox is consistent in .
We consider the minimal consistent i.e. those for which there is no consistent wih (, ABox) s.t. for all , and for a , .
We define the rational closure of ABox, denoted as , the set of all assertions derivable in from ABox for all minimal consistent rank assignments , i.e:
Theorem 4.2 (Soundness of )
Given =(TBox, ABox), for all individual constant in ABox, we have that if then holds in all minimal canonical models of .
Proof. [Fact 0] For any minimal canonical model of = (TBox, ABox) there is a minimal rank assignment consistent with respect to (, ABox), such that for all in ABox and all : if ABox then holds in . This can be proven as follows. Let be a minimal canonical model of . Let be the rank assignment corresponding to : s.t. for all in ABox . Obviously is minimal. Furthermore, ABox . Indeed, ABox by hypothesis. To show that we reason as follows: for all let . If clearly holds in . On the other hand, if : by hypothesis hence by the correspondence between rank of a formula in the rational closure and in minimal canonical models (see Proposition 3.5) also , but since , , therefore . By definition of , and since by Theorem 3.6, , holds in and therefore also . Hence, if ABox then holds in .
Let , and suppose for a contradiction that there is a minimal canonical model of s.t. does not hold in . By Fact 0 there must be a s.t. ABox , but this contradicts the fact that . Therefore must hold in all minimal canonical models of .
Theorem 4.3 (Completeness of )
Given =(TBox, ABox), for all individual constant in ABox, we have that if holds in all minimal canonical models of then .
Proof. We show the contrapositive. Suppose , i.e. there is a minimal consistent with (, ABox) s.t. ABox . We build a minimal canonical model of such that does not hold in as follows. Let where is maximal and consistent with and in ABox . We define the rank of each domain element as follows: , and . We then define in the obvious way: iff .
We then define as follows. First for all in ABox we let . For the interpretation of concepts we reason in two different ways for and . For , for all atomic concepts , we let iff . then extends to boolean combinations of concepts in the usual way. It can be easily shown that for any boolean combination of concepts , iff . For , we start by considering a model such that ABox and . This model exists by hypothesis. For all atomic concepts , we let in iff in . Of course for any boolean combination of concepts , iff .
In order to conclude the model’s construction, for each role , we define as follows. For , iff : . For , iff in . For , , iff there is an of such that in and, for all concepts , we have iff . is extended to quantified concepts in the usual way. It can be shown that for all for all (possibly) quantified , iff , and that for all in , for all quantified , iff .
satisfies ABox: for in ABox this holds by construction. For , this holds since in , hence in .
satisfies TBox: for elements , this can be proven as in Theorem 3.6. For this holds since it held in . For the inclusion this is obvious. For , for all we reason as follows. First of all, if rank() then and the inclusion trivially holds. On the other if rank(), , and therefore in , hence in , and we are done.
does not hold in , since it does not hold in . Last, is minimal: if it was not so there would be . However it can be shown that we could define a consistent with ( ABox) and preferred to , thus contradicting the minimality of , against the hypothesis. We have then built a minimal canonical model of in which does not hold. The theorem follows by contraposition.
Consider the standard penguin example. Let = (TBox, ABox), where TBox = , and ABox = .
Computing the ranking of concepts we get that , , , . It is easy to see that a rank assignment with is inconsistent with as would contain , , and . Thus we are left with only two ranks and with respectively and .
The set contains, among the others, , . It is tedious but esay to check that is consistent and it is the only minimal consistent one (being preferred to , thus both and belong to .
This example shows the need of considering multiple ranks of individual constants: normally computer science courses (CS) are taught only by academic members (A), whereas business courses (B) are taught only by consultants (C), consultants and academics are disjointed, this gives the following TBox: , , . Suppose the ABox contains: , , , and let = (TBox, ABox). Computing rational closure of TBox, we get that all atomic concepts have rank 0. Any rank assignment with , is inconsistent with since the respective will contain both and , from which both and follow, which gives an inconsistency.
There are two minimal consistent ranks: , such that , and , such that . We have that ABox and ABox . According to the skeptical definition of neither , nor belongs to , however belongs to .
Let us conclude this section by estimating the complexity of computing the rational closure of the ABox:
Theorem 4.6 (Complexity of rational closure over the ABox)
Given a knowledge base (TBox,ABox), an individual constant and a concept , the problem of deciding whether is ExpTime-complete.
Proof. Let be the size of the knowledge base and let the size of the query be . As the number of inclusions in the knowledge base is , then the number of non-increasing subsets in the construction of the rational closure is . Moreover, the number of named individuals in the knowledge base is . Hence, the number of different rank assignments to individuals is such that both and are . Observe that . Hence, is , with and linear in , i.e., the number of different rank assignments is exponential in .
To evaluate the complexity of the algorithm for computing the rational closure of the ABox, observe that:
(i) For each , the number of sets is (which is linear in ). The number of inclusions in each is , as the size of is and the number of T-inclusions , with is . Hence, the size of set is .
(ii) For each , the consistency of (, ABox) can be verified by checking the consistency of ABox in , which requires exponential time in the size of the set of formulas ABox (which, as we have seen, is polynomial in the size of ). Hence, the consistency of each can be verified in exponential time in the size of .
(iii) The identification of the minimal assignments among the consistent ones requires the comparison of each consistent assignment with each other (i.e. comparisons), where each comparison between and requires steps. Hence, the identification of the minimal assignments requires steps.
(iv) To define the rational closure of ABox, for each concept occurring in or in the query (there are many concepts), and for each named individual , we have to check if is derivable in from ABox for all minimal consistent rank assignments . As the number of different minimal consistent assignments is exponential in , this requires an exponential number of checks, each one requiring exponential time in the size of the knowledge base . The cost of the overall algorithm is therefore exponential in the size of the knowledge base.
5 Conclusions and Related works
We have defined a rational closure construction for the Description Logic extended with a tipicality operator and provided a minimal model semantics for it based on the idea of minimizing the rank of objects in the domain, that is their level of “untypicality”. This semantics correspond to a natural extension to DLs of Lehmann and Magidor’s notion of rational closure. We have also extended the notion of rational closure to the ABox, by providing an algorithm for computing it that is sound and complete with respect to the minimal model semantics. Last, we have shown an ExpTime upper bound for the algorithm.
In future work, we will consider a further ingredient in the recipe for nonmonotonic DLs. In analogy with circumscription, we can consider a stronger form of minimization where we minimize the rank of domain elements, but we allow to vary the extensions of concepts. Nonmonotonic extensions of low complexity DLs based on the T operator have been recently provided . In future works, we aim to study the application of the proposed semantics to DLs of the and DL-Lite families, in order to define a rational closure for low complexity DLs.
 discusses the application of rational closure to DLs. The authors first describe a construction to compute rational closure in the context of propositional logic, then they adapt such a construction to the DL . As  extends to DLs a construction which, in the propositional case, is proved to be equivalent to Lehmann and Magidor’s rational closure, it may be conjectured that their construction is equivalent to our definition of rational closure in Section 3, which is the natural extension of Lehmann and Magidor’s definition.  keeps the ABox into account, and defines closure operations over individuals. They introduce a consequence relation among a KB and assertions, under the requirement that the TBox is unfoldable and the ABox is closed under completion rules, such as, for instance, that if ABox, then both and (for some individual constant ) must belong to the ABox too. Under such restrictions they are able to define a procedure to compute the rational closure of the ABox assuming that the individuals explicitly named are linearly ordered, and different orders determine different sets of consequences. They show that, for each order , the consequence relation is rational and can be computed in PSpace. In a subsequent work , the authors introduce an approach based on the combination of rational closure and Defeasible Inheritance Networks (INs).
The logic we consider as our base language is equivalent to the logic for defeasible subsumptions in DLs proposed by , when considered with as the underlying DL. The idea underlying this approach is very similar to that of : some objects in the domain are more typical than others. In the approach by , is more typical than if . The properties of correspond to those of in . At a syntactic level the two logics differ, so that in  one finds the defeasible inclusions instead of of , however it has be shown in  that the logic of preferential subsumption can be translated into by replacing with .
In  the semantics of the logic of defeasible inclusions is strenghtened by a preferential semantics. Intuitively, given a TBox, the authors first introduce a preference ordering on the class of all subsumption relations including TBox, then they define the rational closure of TBox as the most preferred relation w.r.t. , i.e. such that there is no other relation such that TBox and . Furthermore, the authors describe an ExpTime algorithm in order to compute the rational closure of a given TBox. However, they do not address the problem of dealing with the ABox. In  a plug-in for the Protégé ontology editor implementing the mentioned algorithm for computing the rational closure for a TBox for OWL ontologies is described.
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