Theoretical Foundations of Defeasible Description Logics

We extend description logics (DLs) with non-monotonic reasoning features. We start by investigating a notion of defeasible subsumption in the spirit of defeasible conditionals as studied by Kraus, Lehmann and Magidor in the propositional case. In particular, we consider a natural and intuitive semantics for defeasible subsumption, and investigate KLM-style syntactic properties for both preferential and rational subsumption. Our contribution includes two representation results linking our semantic constructions to the set of preferential and rational properties considered. Besides showing that our semantics is appropriate, these results pave the way for more effective decision procedures for defeasible reasoning in DLs. Indeed, we also analyse the problem of non-monotonic reasoning in DLs at the level of entailment and present an algorithm for the computation of rational closure of a defeasible ontology. Importantly, our algorithm relies completely on classical entailment and shows that the computational complexity of reasoning over defeasible ontologies is no worse than that of reasoning in the underlying classical DL ALC.

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1 Introduction

Description logics (DLs) [1] are central to many modern AI and database applications since they provide the logical foundation of formal ontologies. Yet, as classical formalisms, DLs do not allow for the proper representation of and reasoning with defeasible information, as shown up in the following example, adapted from Giordano et al.’s [46]: Students do not get tax invoices; employed students do; employed students who are also parents do not. From a naïve (classical) formalisation of this scenario, one concludes that the notion of employed student is an oxymoron, and, consequently, the concept of employed student is unsatisfiable. But while concept unsatisfiability has been investigated extensively in ontology debugging and repair [61, 71], our research problem here goes beyond that, as will become clear in the upcoming sections.

Endowing DLs with defeasible reasoning features is therefore a promising endeavour from the point of view of applications of knowledge representation and reasoning. Indeed, the past 25 years have witnessed many attempts to introduce defeasible-reasoning capabilities in a DL setting, usually drawing on a well-established body of research on non-monotonic reasoning (NMR). These comprise the so-called preferential approaches [21, 22, 24, 37, 40, 46, 47, 51, 52, 66, 67], circumscription-based ones [9, 10, 72], amongst others [2, 3, 8, 43, 53, 54, 55, 62, 65, 74].

Preferential extensions of DLs turn out to be particularly promising, mainly because they are based on an elegant, comprehensive and well-studied framework for non-monotonic reasoning in the propositional case proposed by Kraus, Lehmann and Magidor [56, 59] and often referred to as the KLM approach. Such a framework is valuable for a number of reasons. First, it provides for a thorough analysis of some formal properties that any consequence relation deemed as appropriate in a non-monotonic setting ought to satisfy. Such formal properties, which resemble those of a Gentzen-style proof system (see Section 3.1), play a central role in assessing how intuitive the obtained results are and enable a more comprehensive characterisation of the introduced non-monotonic conditional from a logical point of view. Second, the KLM approach allows for many decision problems to be reduced to classical entailment checking, sometimes without blowing up the computational complexity compared to the underlying classical case. Finally, it has a well-known connection with the AGM-approach to belief revision [45, 69] and with frameworks for reasoning under uncertainty [7, 44]. It is therefore reasonable to expect that most, if not all, of the aforementioned features of the KLM approach should transfer to KLM-based extensions of DLs, too.

Following the motivation laid out above, several extensions to the KLM approach to description logics have been proposed recently [21, 24, 27, 29, 32, 33, 37, 40, 46, 47, 51, 52, 63, 75], each of them investigating particular constructions and variants of the preferential approach. However, here our aim is to provide a comprehensive study of the formal foundations of preferential defeasible reasoning in DLs. By that we mean (i) defining a general and intuitive semantics; (ii) showing that the corresponding representation results (in the KLM sense of the term) hold, linking our semantic constructions with the KLM-style set of properties, and (iii) presenting an appropriate analysis of entailment in the context of ontologies with defeasible information with an associated decision procedure that is implementable.

In the remainder of the paper, we shall take the following route: After providing the required background on the DL we consider in this work as well as fixing the notation (Section 2), we introduce the notion of defeasible subsumption along with a set of KLM-inspired properties it ought to satisfy (Section 3). In particular, using an intuitive semantics for the idea that “usually, an element of the class  is also an element of the class ”, we provide a characterisation (via representation results) of two important classes of defeasible statements, namely preferential and rational subsumption. In Section 4, we start by investigating two obvious candidates for the notion of entailment in the context of defeasible DLs, namely preferential and modular entailment. These turn out not to have all properties seen as important in a non-monotonic DL setting, mimicking a similar result in the propositional case [59]. Therefore, we propose a notion of rational entailment and show that it is the definition of consequence we are looking for. We take this definition further by exploring the relationship that rational entailment has with both Lehmann and Magidor’s [59] definition of rational closure and the more recent algorithm by Casini and Straccia [37] for its computation (Section 5). After a discussion of, and comparison with, related work (Section 6), we conclude with a summary of our contributions and some directions for further exploration. Proofs of our results can be found in the appendix.

2 Logical preliminaries

Description Logics (DLs) [1] are decidable fragments of first-order logic with interesting properties and a variety of applications. There is a whole family of description logics, an example of which is  and on which we shall focus in the present paper.777For the reader not conversant with Description Logics but familiar with modal logics, there are results in the literature relating some families of description logics to systems of modal logic. For example, a well-known result is the one linking the DL  with the normal modal logic K [70].

The (concept) language of is built upon a finite set of atomic concept names , a finite set of role names  (a.k.a. attributes) and a finite set of individual names  such that , and are pairwise disjoint. In our scenario example, we can have for instance , , and , with the respective obvious intuitions. With we denote atomic concepts, with role names, and with individual names. Complex concepts are denoted with and are built using the constructors  (complement), (concept conjunction), (concept disjunction), (value restriction) and (existential restriction) according to the following grammar rules:

With we denote the language of all concepts, which is understood as the smallest set of symbol sequences generated according to the rules above. When writing down concepts of , we follow the usual convention and omit parentheses whenever they are not essential for disambiguation. Examples of concepts in our scenario are and .

The semantics of is the standard set-theoretic Tarskian semantics. An interpretation is a structure , where is a non-empty set called the domain, and is an interpretation function mapping concept names  to subsets of , role names  to binary relations over , and individual names  to elements of the domain , i.e., , , and .

Figure 1 depicts an interpretation for our scenario example with domain , and interpreting the elements of the vocabulary as follows: , , , , , , , , , , , .

    

  

Figure 1: A DL interpretation.

Let be an interpretation and define , for . We extend the interpretation function  to interpret complex concepts of as follows:

For the interpretation  in Figure 1, we have and .

Given , a statement of the form is called a subsumption statement, or general concept inclusion (GCI), read “ is subsumed by ”. A concrete example of GCI is . is an abbreviation for both and . An TBox  is a finite set of GCIs. Given , and , an assertional statement (assertion, for short) is an expression of the form or , read, respectively, “ is an instance of ” and “ is related to  via ”. Examples of assertions are and . An ABox  is a finite set of assertional statements. We denote statements with . Given  and , with we denote an  knowledge base, a.k.a. an ontology, an example of which is given below:

An interpretation satisfies a GCI (denoted ) if . (And then if .) satisfies an assertion (respectively, ), denoted (respectively, ), if (respectively, ).

In the interpretation  in Figure 1, we have , and .

We say that an interpretation is a model of a TBox  (respectively, of an ABox ), denoted (respectively, ) if for every in  (respectively, in ). We say that  is a model of a knowledge base  if and . It can be verified that the interpretation in Figure 1 is not a model of the example knowledge base above. (Actually, it is not hard to see that the knowledge base above admits no model.)

A statement  is (classically) entailed by a knowledge base , denoted , if every model of  satisfies . If for all interpretations , we say is a validity and denote this fact with .

The focus of the present paper being on defeasibility for description logic TBoxes only, we henceforth assume the ABox is empty. (We are currently in the process of extending our approach to description logic knowledge bases, with ABoxes included into the mix.) It is easy to see that, for  as above, we have .

For more details on Description Logics in general and on  in particular, the reader is invited to consult the Description Logic Handbook [1] and the introductory textbook on Description Logic [4].

3 Foundations for defeasibility in DLs

In this section, we lay the formal foundations of our approach to defeasible reasoning in DL ontologies. For the most part, we build on the so-called preferential approach to non-monotonic reasoning [56, 59, 73].

3.1 Defeasible subsumption relations and their KLM-style properties

In a sense, class subsumption (alias concept inclusion) of the form is the main notion in DL ontologies. Given its implication-like intuition, subsumption lends itself naturally to defeasibility: “provisionally, if an object falls under , then it also falls under ”, as in “usually, students are tax exempted”. In that respect, a defeasible version of concept inclusion is the starting point for an investigation of defeasible reasoning in DL ontologies. (We shall also address defeasibility of the entailment relation in later sections.)

Definition 1 (Defeasible Concept Inclusion)

Let . A defeasible concept inclusion axiom (DCI, for short) is a statement of the form .

A defeasible concept inclusion of the form is to be read as “usually, an instance of the class  is also an instance of the class ”. For instance, the DCI formalises the example above. Paraphrasing Lehmann [57], the intuition of  is that “if [the fact it belongs to]  were all the information about an object available to an agent, then [that it also belongs to]  would be a sensible conclusion to draw about such an object”. It is worth noting that , just as , is a ‘connective’ sitting between the concept language (object level) and the meta-language (that of entailment) and it is meant to be the defeasible counterpart of the classical subsumption .

Being (defeasible) statements, DCIs will also be denoted by Whenever a distinction between GCIs and DCIs is in order, we shall make it explicitly.

Definition 2 (Defeasible TBox)

A defeasible TBox (DTBox, for short) is a finite set of DCIs.

Given a TBox  and a DTBox , we let and refer to it as a defeasible knowledge base (alias defeasible ontology).

Example 1

The following defeasible knowledge base gives a formal specification for our student scenario:

In our semantic construction later on, it will also be useful to be able to refer to infinite sets of concept inclusions. Let therefore denote a defeasible theory, defined as a defeasible knowledge base but without the restriction on and to finite sets.

In order to assess the behaviour of the new connective and check it against both the intuition and the set of properties usually considered in a non-monotonic setting, it is convenient to look at a set of -statements as a binary relation of the ‘antecedent-consequent’ kind.

Definition 3 (Defeasible Subsumption Relation)

A defeasible subsumption relation is a binary relation .

The idea is to mimic the analysis of defeasible entailment relations carried out by Kraus et al. [56] in the propositional case, where entailment is seen as a binary relation on the set of propositional sentences. Here we shall adopt the view of subsumption as a binary relation on concepts of our description language.

Sometimes (e.g. in the structural properties below) we shall write in the infix notation, i.e., as . The context will make clear when we will be talking about elements of a relation or statements (DCIs) in a defeasible knowledge base. Whenever disambiguation is in order, we shall flag it to the reader.

Definition 4 (Preferential Subsumption Relation)

A defeasible subsumption relation  is a preferential subsumption relation if it satisfies the following set of properties, which we refer to as the (DL versions of the) preferential KLM properties:

The (Cons) property is a consequence of the adoption of a DL-based semantics, which enforces the non-emptiness of the domain, as will become clear in the next section. The rest of the properties in Definition 4 result from a translation of the properties for preferential consequence relations proposed by Kraus et al. [56] in the propositional setting. They have been discussed at length in the literature for both the propositional and the DL cases [21, 24, 48, 49, 56, 59] and we shall not repeat so here.

If, in addition to the preferential properties above, the relation  also satisfies rational monotonicity (RM) below, then it is said to be a rational subsumption relation:

Rational monotonicity is often considered a desirable property to have, one of the reasons stemming from the fact it is a necessary condition for the satisfaction of the principle of presumption of typicality [58, Section 3.1]. Such a principle is a simple yet intuitive formalisation of a form of reasoning we carry out when facing lack of information: we reason assuming that we are in the most typical possible situation, compatible with the information at our disposal. (More details will be provided in Section 4).

3.2 Preferential semantics and representation results

In this section, we present our semantics for preferential and rational subsumption by enriching standard DL interpretations  with an ordering on the elements of the domain . The intuition underlying this is simple and natural, and extends similar work done for the propositional case by Shoham [73], Kraus et al. [56], Lehmann and Magidor [59] and Booth et al. [12, 13, 14] to the case for description logics. This is not the first extension of this kind, as evidenced by the work of Boutilier [16], Baltag and Smets [5, 6], Giordano et al. [46, 48, 49, 50, 51, 52], Britz et al. [19, 20, 21, 22, 24] and Britz and Varzinczak [26, 27, 30, 29, 31, 32]. However, this is the first comprehensive semantic account of both preferential and rational subsumption relations, with accompanying representation results, based on the standard semantics for description logics.

Informally, our semantic constructions are based on the idea that objects of the domain can be ordered according to their degree of normality [16] or typicality [13, 14, 22, 46]. Paraphrasing Boutilier [16, pp. 110–116],

Surely there is no inherent property of objects that allows them to be judged to be more or less normal in absolute terms. These orderings are purely ‘subjective’ (in the sense that they can be thought of as part of an agent’s belief state) and the space of orderings deemed plausible by the agent may (among other things) be determined by e.g. empirical data. By using orderings in this way, we encode our (or the agent’s) expectations about the objects corresponding to their perceived regularity or typicality. Those objects not violating our expectations are considered to be more normal than the objects that violate some.

Hence we do not require that there exists something intrinsic about objects that makes one object more normal than another. Rather, the intention is to provide a framework in which to express all conceivable ways in which objects, with their associated properties and relationships with other objects, can be ordered in terms of typicality, in the same way that the class of all standard DL interpretations constitute a framework representing all conceivable ways of representing the properties of objects and their relationships with other objects. Just as the latter are constrained by stating subsumption statements in a knowledge base, the possible orderings that are considered plausible are encoded by writing down DCIs.

That said, we are ready for the definition of the first semantic construction the present work relies on.

Definition 5 (Preferential Interpretation)

A preferential interpretation is a tuple , where is a (standard) DL interpretation (which we denote by  and refer to as the classical interpretation associated with ), and  is a strict partial order on  (i.e., is irreflexive and transitive) satisfying the smoothness condition (for every , if , then ).888Given , with we denote the set for every .

Figure 2 depicts a preferential interpretation in our scenario example where  and  are as in the interpretation  shown in Figure 1, and , , , represented by the dashed arrows in the picture. (For the sake of presentation, in the picture we omit the transitive -arrows.)

    

  

Figure 2: A preferential interpretation.

Preferential interpretations provide us with a simple and intuitive way to give a semantics to DCIs.

Definition 6 (Satisfaction)

Let be a preferential interpretation, , and . The satisfaction relation  is defined as follows:

  • if ;

  • if .

If , then we say satisfies . satisfies a defeasible knowledge base , written , if for every , in which case we say  is a preferential model of . We say is satisfiable w.r.t.  if there is a model  of  s.t. .

It is easy to see that the addition of the -component preserves the truth of all classical subsumption statements holding in the remaining structure:

Lemma 1

Let be a preferential interpretation. For every , if and only if .

It is worth noting that, due to smoothness of , every (classical) subsumption statement is equivalent, with respect to preferential interpretations, to some DCI.

lemmarestatableClassicalStatements For every preferential interpretation , and every , if and only if .

The following result, of which the proof can be found in Appendix A, will come in handy later on.

lemmarestatableClosureDisjointUnionPref Preferential interpretations are closed under disjoint union.

An obvious question that can now be raised is: “How do we know our preferential semantics provides an appropriate meaning to the notion of defeasible concept inclusion?” The following definition will help us in answering this question:

Definition 7 (-Induced Defeasible Subsumption)

Let be a preferential interpretation. Then is the defeasible subsumption relation induced by .

The first important result we present here, which also answers the above raised question, shows that there is a full correspondence between the class of preferential subsumption relations and the class of defeasible subsumption relations induced by preferential interpretations. It is the DL analogue of a representation result proved by Kraus et al. for the propositional case [56, Theorem 3] and its proof can be found in Appendix B.

theoremrestatableRepResultPreferential[Representation Result for Preferential Subsumption] A defeasible subsumption relation is preferential if and only if there is a preferential interpretation  such that .

What is perhaps surprising about this result is that no additional properties based on the syntactic structure of the underlying DL are necessary to characterise the defeasible subsumption relations induced by preferential interpretations. We provide below a few properties involving the use of quantifiers that are satisfied by all preferential subsumption relations. (See Section 5 for more on properties explicitly mentioning DL-specific constructs.)

The first two are ‘existential’ and ‘universal’ versions of cautious monotonicity (CM):

The third one is a rephrasing of the Rule of Necessitation in modal logic [42]. It guarantees the absence of so-called spurious objects [25] in the original preferential semantics for DLs by Britz et al. [23, 24]. That is, if is unsatisfiable, then so is (cf. Lemma 1).

In addition to preferential interpretations, we are also interested in the study of modular interpretations, which are preferential interpretations in which the -component is a modular ordering:

Definition 8 (Modular Order)

Given a set , is modular if it is a strict partial order, and its associated incomparability relation , defined by if neither nor , is transitive.

If is modular, then is an equivalence relation.

Definition 9 (Modular Interpretation)

A modular interpretation is a preferential interpretation such that  is modular.

Intuitively, modular interpretations allow us to compare any two objects w.r.t. their plausibility. Those that are incomparable are viewed as being equally plausible. As such, modular interpretations are special cases of preferential interpretations, where plausibility can be represented by any smooth strict partial order.

The main reason to consider modular interpretations is that they provide the semantic foundation of rational subsumption relations. This is made precise by our second important result below, which shows that the defeasible subsumption relations induced by modular interpretations are precisely the rational subsumption relations. Again, this is the DL analogue of a representation result proved by Lehmann and Magidor for the propositional case [59, Theorem 5] and its proof can be found in Appendix C.

theoremrestatableRepResultRational[Representation Result for Rational Subsumption] A defeasible subsumption relation is rational if and only if there is a modular interpretation  such that .

Analogous to the case for cautious monotonicity above, the following ‘existential’ and ‘universal’ versions of rational monotonicity are satisfied by all rational subsumption relations:

It is worth pausing for a moment to emphasise the significance of these two results (Theorems 

7 and 9). They provide exact semantic characterisations of two important classes of defeasible subsumption relations, namely preferential and rational subsumption, in terms of the classes of preferential and modular interpretations, respectively. As we shall see in Section 4, these results form the core of the investigation into an appropriate notion of entailment for defeasible DL ontologies.

4 Rationality in entailment

From the standpoint of knowledge representation and reasoning, a pivotal question is that of deciding which statements are entailed by a knowledge base. We shall devote the remainder of the paper to this matter, and in this section we lay out the formal foundations for that.

4.1 Preferential entailment

In the exploration of a notion of entailment for defeasible ontologies, an obvious starting point is to consider a Tarskian definition of consequence:

Definition 10 (Preferential Entailment)

A statement  is preferentially entailed by a defeasible knowledge base , written , if every preferential model of  satisfies .

As usual, this form of entailment is accompanied by a corresponding notion of closure.

Definition 11 (Preferential Closure)

Let  be a defeasible knowledge base. With we denote the preferential closure of .

Intuitively, the preferential closure of a defeasible knowledge base  corresponds to the ‘core’ set of statements, classical and defeasible, that should hold given those in . Hence, preferential entailment and preferential closure are two sides of the same coin, mimicking an analogous result for preferential reasoning in the propositional [56] case.

Recall (cf. the discussion following Definition 2) that a defeasible theory is a defeasible knowledge base without the restriction to finite sets. When assessing how appropriate a notion of entailment for defeasible ontologies is, the following definitions turn out to be useful, as will become clear in the sequel:

Definition 12 (-Induced Defeasible Subsumption)

Let be a defeasible theory. Then (1)  is the DTBox induced by  and (2)  is the defeasible subsumption relation induced by .

So, the DTBox induced by is the set of defeasible subsumption statements contained in , together with the defeasible versions of the classical subsumption statements in . The defeasible subsumption relation induced by  is simply the defeasible subsumption relation corresponding to .

Definition 13

A defeasible theory  is called preferential if the subsumption relation induced by it satisfies the preferential properties in Definition 4.

It turns out that the defeasible subsumption relation induced by the preferential closure of a defeasible knowledge base is exactly the intersection of the defeasible subsumption relations induced by the preferential defeasible theories containing .

lemmarestatableLemmaPrefClosure Let  be a defeasible knowledge base. Then

It follows immediately that the preferential closure of a defeasible knowledge base is preferential, and induces the smallest defeasible subsumption relation induced by a preferential defeasible theory containing .

Preferential entailment is not always desirable, one of the reasons being that it is monotonic, courtesy of the Tarskian notion of consequence it relies on (see Definition 10). In most cases, as witnessed by the great deal of work in the non-monotonic reasoning community, a move towards rationality is in order. Thanks to the definitions above and the result in Theorem 9, we already know where to start looking for it.

Definition 14 (Modular Entailment)

A statement  is modularly entailed by a defeasible knowledge base , written , if every modular model of  satisfies .

As is the case for preferential entailment, modular entailment is accompanied by a corresponding notion of closure.

Definition 15 (Modular Closure)

Let  be a defeasible knowledge base. With we denote the modular closure of .

Definition 16

A defeasible theory  is called rational if it is preferential and is also closed under the rational monotonicity rule (RM).

For modular closure we get a result similar to Lemma 13. lemmarestatableLemmaModClosure Let  be a defeasible knowledge base. Then

That is, the modular closure of a defeasible knowledge base induces the smallest defeasible subsumption relation induced by a rational defeasible theory containing . However, the modular closure of a defeasible knowledge base is not necessarily rational. That is, if one looks at the set of statements (in particular the -ones) modularly entailed by a knowledge base as a defeasible subsumption relation, then it need not satisfy the rational monotonicity property. This is so because modular entailment coincides with preferential entailment, as the following result, adapted from a well-known similar result in the propositional case [59, Theorem 4.2], shows.

Lemma 2

.

As a result, modular entailment unfortunately falls short of providing us with an appropriate notion of non-monotonic entailment. In what follows, we overcome precisely this issue.

4.2 Semantic rational entailment

In this section, we introduce a definition of semantic entailment which, as we shall see, is appropriate in the light of the discussion above. The constructions we are going to present are inspired by the semantic characterisation of rational closure by Booth and Paris [15] in the propositional case. We shall give a corresponding proof-theoretic characterisation of our version of semantic entailment in Section 5.1.

We focus our attention on a subclass of modular orders, referred to as ranked orders:

Definition 17 (Ranked Order)

Given a set , the binary relation is a ranked order if there is a mapping satisfying the following convexity property:

  • for every , if for some , then, for every such that , there is a for which ,

and s.t. for every , iff .

It is easy to see that a ranked order is also modular: is a strict partial order, and, since two objects are incomparable (i.e., ) if and only if , is a transitive relation. By constraining our preference relations to the ranked orders, we can identify a subset of the modular interpretations we refer to as the ranked interpretations.

Definition 18 (Ranked Interpretation)

A ranked interpretation is a modular interpretation s.t. is a ranked order.

We now provide two basic results about ranked interpretations. First, all finite modular interpretations are ranked interpretations.

lemmarestatablefinitemodular A modular interpretation s.t. is finite is a ranked interpretation.

Next, for every ranked interpretation , the function is unique.

propositionrestatableuniquerank Given a ranked interpretation , there is only one function satisfying the convexity property and s.t. for every , iff .

Proposition 18 allows us to use the function to define the notions of height and layers.

Definition 19 (Height & Layers)

Given a ranked interpretation , its characteristic ranking function , and an object , is called the height of  in .

For every ranked interpretation , we can partition the domain  into a sequence of layers , where, for every object , we have iff .

Intuitively, the lower the height of an object in an interpretation , the more typical (or normal) the object is in . We can also think of a level of typicality for concepts: the height of a concept in  is the index of the layer to which the restriction of the concept’s extension to its -minimal elements belong, i.e., if . As a convention, if , that is, if , then .

The following result (proved in Appendix D) will be useful for some of the proofs in later sections of the paper:

[Finite-Model Property]theoremrestatableFiniteModelProperty Defeasible  has the finite-model property. In particular, every defeasible  knowledge base that has a modular model, has also a finite ranked model.

Given a set of ranked interpretations, we can introduce a new form of model merging, ranked union.

Definition 20 (Ranked Union)

Given a countable set of ranked interpretations , a ranked interpretation is the ranked union of if the following holds:

  • , i.e., the disjoint union of the domains from , where each has the elements of its domain renamed as , , … so that they are all distinct in ;

  • iff ;

  • iff and ;

  • for every , .

The latter condition corresponds to imposing that iff .

Informally, the ranked union of a set of ranked interpretations is the result of merging all their layers of height into a single layer of height , for all .

lemmarestatableClosureDisjointUnionMod Given a set of ranked models of a defeasible knowledge base , their ranked union is itself a ranked model of .

Let be a defeasible knowledge base and let be a fixed countably infinite set. Define

The following result shows that the set suffices to characterise modular entailment (the proof is in Appendix D):

lemmarestatableCountablyInfiniteDomain For every and every , iff , for every .

Therefore, we can use just the set of interpretations in to decide the consequences of  w.r.t. modular entailment.

We can now use the set as a springboard to introduce what will turn out to be a canonical modular interpretation for . Using and ranked union we can define the following relevant model.

Definition 21 (Big Ranked Model)

Let be a defeasible knowledge base. The big ranked model of  is the ranked model that is the ranked union of the models in .

Given Lemma 20, we can state the following:

Corollary 1

is a ranked model of .

Armed with the definitions and results above, we are now ready to provide an alternative definition of entailment in the context of defeasible ontologies:

Definition 22 (Rational Entailment)

A statement  is rationally entailed by a knowledge base , written , if .

That such a notion of entailment indeed deserves its name is witnessed by the following result, a consequence of Corollary 1 and Theorem 9:

Corollary 2

Let be a defeasible knowledge base. is rational.

In conclusion, rational entailment is a good candidate for the appropriate notion of defeasible consequence we have been looking for. Of course, a question that arises is whether a notion of closure, in the spirit of preferential and modular closures, that is equivalent to it can be defined. In the next section, we address precisely this matter.

5 Rational closure for defeasible knowledge bases

We now turn our attention to the exploration, in a DL setting, of the well-known notion of rational closure of a defeasible knowledge base as studied by Lehmann and Magidor [59] for propositional logic. For the most part, we base our constructions on the work by Casini and Straccia [37, 40], amending it wherever necessary. (An alternative semantic characterisation of rational closure in DLs has also been proposed by Giordano et al. [51, 52].) As we shall see, rational closure provides a proof-theoretic characterisation of rational entailment and the complexity of its computation is no higher than that of computing entailment in the underlying classical DL.

5.1 Rational closure and a correspondence result

Rational closure is a form of inferential closure based on modular entailment , but it extends its inferential power. Such an extension of modular entailment is obtained by formalising the already mentioned principle of presumption of typicality [58, Section 3.1]. That is, under possibly incomplete information, we always assume that we are dealing with the most typical possible situation that is compatible with the information at our disposal. We first define what it means for a concept to be exceptional, a notion that is central to the definition of rational closure:

Definition 23 (Exceptionality)

Let be a defeasible knowledge base and . We say  is exceptional in if . A DCI is exceptional in  if is exceptional in .

A concept is considered exceptional in a knowledge base  if it is not possible to have a modular model of  in which there is a typical object (i.e., an object at least as typical as all the others) that is in the interpretation of . Intuitively, a DCI is exceptional if it does not concern the most typical objects, i.e., it is about less normal (or exceptional) ones. This is an intuitive translation of the notion of exceptionality used by Lehmann and Magidor [59] in the propositional framework, and has already been used by Casini and Straccia [37] and Giordano et al. [52] in their investigations into defeasible reasoning for description logics.

Applying the notion of exceptionality iteratively, we associate with every concept  a rank in , which we denote by . We extend this to DCIs and associate with every statement a rank, denoted :

  1. Let , if is not exceptional in , and let for every DCI having  in the LHS, with . The set of DCIs in  with rank  is denoted as .

  2. Let , if does not have a rank of  and it is not exceptional in the knowledge base composed of  and the exceptional part of , that is, . If , then let for every DCI . The set of DCIs in  with rank is denoted .

  3. In general, for , a concept  is assigned a rank of  if it does not have a rank of and it is not exceptional in . If , then , for every DCI