1 Abstract
We propose a logical analysis of the concept of typicality, central in human cognition (Rosch,1978). We start from a previously proposed extension of the basic Description Logic with a typicality operator that allows to consistently represent the attribution to classes of individuals of properties with exceptions (as in the classic example (i)typical birds fly, (ii) penguins are birds but (iii)typical penguins don’t fly). We then strengthen this extension in order to separately reason about the typicality with respect to different aspects (e.g., flying, having nice feather: in the previous example, penguins may not inherit the property of flying, for which they are exceptional, but can nonetheless inherit other properties, such as having nice feather).
2 Introduction
In [Giordano et al.2015] it is proposed a rational closure strengthening of . This strengthening allows to perform non monotonic reasoning in in a computationally efficient way. The extension, as already the related logic proposed in [Giordano et al.2013a] and the weaker (monotonic) logic presented in [Giordano et al.2009], allows to consistently represent typical properties with exceptions that could not be represented in standard .
For instance, in all the above logics one can say that:
SET 1:
Typical students don’t earn money
Typical working students do earn money
Typical apprentice working students don’t earn money
without having to conclude that there cannot exist working students nor apprentice working students. On the contrary, in standard typicality cannot be represented, and these three propositions can only be expressed by the stronger ones:
SET 2:
Students don’t earn money (Student EarnMoney)
Working students do earn money (Worker Student EarnMoney)
Apprentice working students don’t earn money (Worker Apprentice Student EarnMoney)
These propositions are consistent in only if there are no working students nor apprentice working students.
In all the extensions of mentioned above one can represent the set of propositions in by means of a typicality operator that, given a concept (e.g. Student) singles out the most typical instances of : so, for instance, refers to the typical instances of the concept Student. The semantics of is given by means of a preference relation that compares the typicality of two individuals: for any two and , means that is more typical than . Typical instances of a concept are those minimal with respect to (formally, as we will see later, , where s.t. ).
The operator has all the properties that, in the analysis of Kraus Lehmann and Magidor [Kraus, Lehmann, and Magidor1990] any non monotonic entailment should have. For instance, satisfies the principle of cautious monotonicity, according to which if , then ). The precise relations between the properties of and preferential entailment are established in [Giordano et al.2009].
Although the extensions of with the typicality operator allow to express of propositions, the resulting logic is monotonic, and it does not allow to perform some wanted, non monotonic inferences. For instance, it does not allow to deal with irrelevance which is the principle that from the fact that typical students are young, one would want to derive that typical blond students also are young, since being blond is irrelevant with respect to youth. As another example, when knowing that an individual, say John, is a student, and given of propositions, one would want to conclude that John is a typical student and therefore does not earn money. On the other hand, when knowing that John is a working student, one would want to conclude that he is a typical working student and therefore does earn money. In other words one would want to assume that an individual is a typical instance of the most specific class it belongs to, in the absence of information to the contrary.
These stronger inferences all hold in the strengthening of presented in [Giordano et al.2013a, Giordano et al.2015]. In particular, [Giordano et al.2015] proposes an adaptation to of the well known mechanism of rational closure, first proposed by Lehman and Magidor in [Lehmann and Magidor1992]. From a semantic point of view, this strengthening of corresponds to restricting one’s attention to minimal models, that minimize the height (rank) of all domain elements with respect to (i.e. that minimize the length of the chains starting from all individuals). Under the condition that the models considered are canonical, the semantic characterization corresponds to the syntactical rational closure. This semantics supports all the above wanted inferences, and the nice computational properties of rational closure guarantee that whether the above inferences are valid or not can be computed in reasonable time.
The main drawback of rational closure is that it is an allornothing mechanism: for any subclass of it holds that either the typical members of inherit all the properties of or they don’t inherit any property. Once the typical members of are recognized as exceptional with respect to for a given aspect, they become exceptional for all aspects. Consider the classic birds/penguins example, expressed by propositions:
SET 3:
Typical birds have nice feather
Typical birds fly
Penguins are birds
Typical penguins do not fly
In this case, since penguins are exceptional with respect to the aspect of flying, they are nontypical birds, and for this reason they do not inherit any of the typical properties of birds.
On the contrary, given of propositions, one wants to conclude that:

(**) Typical penguins have nice feather
This is to say that one wants to separately reason about the different aspects: the property of flying is not related to the property of having nice feather, hence we want to separately reason on the two aspects.
Here we propose a strengthening of the semantics used for rational closure in [Giordano et al.2015] that only used a single preference relation by allowing, beside , several preference relations that compare the typicality of individuals with respect to a given aspect. Obtaining a strengthening of rational closure is the purpose of this work. This puts strong constraints on the resulting semantics, and defines the horizon of this work. In this new semantics we can express the fact that, for instance, is more typical than with respect to the property of flying but is more typical that with respect to some other property, as the property of having nice feather. To this purpose we consider preference relations indexed by concepts that stand for the above mentioned aspects under which we compare individuals. So we will write that to mean that is preferred to for what concerns aspect : for instance means that is more typical than with respect to the property of flying.
We therefore proceed as follows: we first recall the semantics of the extension of with a typicality operator which was at the basis of the definition of rational closure and semantics in [Giordano et al.2013b, Giordano et al.2015]. We then expand this semantics by introducing several preference relations, that we then minimize obtaining our new minimal models’ mechanism. As we will see this new semantics leads to a strengthening of rational closure, allowing to separately reason about the inheritance of different properties.
3 The operator and the General Semantics
Let us briefly recall the logic which is at the basis of a rational closure construction proposed in [Giordano et al.2015] for . The intuitive idea of is to extend 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: , and , where is a concept name and a role name. A KB is a pair (TBox, ABox). TBox contains a finite set of concept inclusions . ABox contains a finite set of assertions of the form and , where are individual constants.
The semantics of is defined 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: means that is more typical than . Typical members of a concept , instances of , are the members of that are minimal with respect to (such that there is no other member of more typical than ). In rational models is further assumed to be modular: for all , if then either or . These rational models characterize .
Definition 1 (Semantics of [Giordano et al.2015])
A model of is any structure where: is the domain; is an irreflexive, transitive, and modular relation over that satisfies the finite chain condition(there is no infinite descending chain, hence if , also ); is the extension function that maps each concept name to , each role name to and each individual constant to . For concepts of , is defined in the usual way. For the operator, we have .
As shown in [Giordano et al.2015], 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 .
A model satisfies a knowledge base =(TBox,ABox) if it satisfies its TBox (and for all inclusions in TBox, it holds ), and its ABox (for all in ABox, , and for all in ABox, ). A query (either an assertion or an inclusion relation ) is logically (rationally) entailed by a knowledge base () if holds in all models satisfying .
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 [Giordano et al.2013b, Giordano et al.2015] the non monotonic mechanism of rational closure has been defined over , which extends to DLs the notion of rational closure proposed in the propositional context by Lehmann and Magidor [Lehmann and Magidor1992]. The definition is based on the notion of exceptionality. Roughly speaking holds (is included in the rational closure) of if (indeed, ) is less exceptional than . We briefly recall this construction and we refer to [Giordano et al.2013b, Giordano et al.2015] for full details. Here we only consider rational closure of TBox, defined as follows.
Definition 2 (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 TBox, it is possible to define a sequence of non increasing subsets of TBox ordered according to the exceptionality of the elements by letting and, for , s.t. does not occurr in . Observe that, being KB 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).
Rational closure builds on this notion of exceptionality:
Definition 3 (Rational closure of TBox)
Let KB = (TBox, ABox) be a DL knowledge base. The rational closure of TBox , where and are concepts.
As a very interesting property, in the context of DLs, the rational closure has a very interesting complexity: deciding if an inclusion belongs to the rational closure of TBox is a problem in ExpTime [Giordano et al.2015].
In [Giordano et al.2015] it is shown that the semantics corresponding to rational closure can be given in terms of minimal canonical models. With respect to standard models, in these models the rank of each domain element is as low as possible (each domain element is assumed to be as typical as possible). This is expressed by the following definition.
Definition 4 (Minimal models of (with respect to ))
Given and , we say that is preferred to () if: , for all 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 .
Furthermore, the models corresponding to rational closure are 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.
Definition 5 (Canonical model)
Given =(TBox,ABox),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))
The correspondence between minimal canonical models and rational closure is established by the following key theorem.
Theorem 1 ([Giordano et al.2015])
Let =(TBox,ABox) be a knowledge base and a query. We have that if and only if holds in all minimal canonical models of with respect to TBox (Definition 6).
4 Semantics with several preference relations
The main weakness of rational closure, despite its power and its nice computational properties, is that it is an allornothing mechanism that does not allow to separately reason on single aspects. To overcome this difficulty, we here consider models with several preference relations, one for each aspect we want to reason about. We assume this is any concept occurring in K: we call the set of these aspects (observe that may be nonatomic). For each aspect , expresses the preference for aspect : expresses the preference for flying, so if we know that , birds that do fly will be preferred to birds that do not fly, with respect to aspect fly, i.e. with respect to . All these preferences, as well as the global preference relation , satisfy the properties in Definition 7 below. We now enrich the definition of an model given above (Definition 1) by taking into account preferences with respect to all of the aspects. In the semantics we can express that for instance , whereas ( is preferred to for aspect but is preferred to for aspect ).
This semantic richness allows to obtain a strengthening of rational closure in which typicality with respect to every aspect is maximized. Since we want to compare our approach to rational closure, we keep the language the same than in . In particular, we only have one single typicality operator . However, the semantic richness could motivate the introduction of several typicality operators by which one might want to explicitly talk in the language about the typicality w.r.t. aspect , or , and so on. We leave this extension for future work.
Definition 7 (Enriched rational models)
Given a knowledge base K, we call an enriched rational model a structure , where , are defined as in Definition 1, and are preference relations over , with the properties of being irreflexive, transitive, satisfying the finite chain condition, modular (for all , if then either or ).
For all and for it holds that s.t. there is no s.t. and s.t. there is no s.t. and .
satisfies the further conditions that if:
(a) there is such that , and there is no such that or;
(b) there is s.t. , and for all s.t. , there is s.t. and .
In this semantics the global preference relation is related to the various preference relations relative to single aspects . Given (a) when is preferred to for a single aspect , and there is no aspect for which is preferred to . (b) captures the idea that in case two individuals are preferred with respect to different aspects, preference (for the global preference relation) is given to the individual that satisfies all typical properties of the most specific concept (if is more specific than , then ), as illustrated by Example 1 below.
We insist in highlighting that this semantics somewhat complicated is needed since we want to provide a strengthening of rational closure. For this, we have to respect the constraints imposed by rational closure. One might think in the future to study a semantics in which only (a) holds.We have not considered such a simpler semantics since it would no longer be a strengthening of the semantics corresponding to rational closure, and is therefore out of the focus of this work.
In order to be a model of an model must satisfy the following constraints.
Definition 8 (Enriched rational models of K)
Given a knowledge base K, and an enriched rational model for ,
is a model of if it satisfies both its TBox
and its ABox, where satisfies TBox if for all inclusions :
if does not occur in , then
if occurs in , and is , then both
(i) and (ii) .
satisfies ABox if
(i) for all in ABox, , (ii) for all in
ABox,
Example 1
Let , , . . We consider an model of , that we don’t fully describe but which we only use to observe the behavior of two Penguins , with respect to the properties of (not) flying and having nice feather. In particular, let us consider the three preference relations: .
Suppose (because , as all typical penguins, does not fly whereas exceptionally does) and there is no other aspect such that , and in particular it does not hold that (because for instance both have a nice feather). In this case, obviously it holds that (since (a) is satisfied).
Consider now a more tricky situation in which again holds (because for instance does not fly whereas flies), ( is a typical penguin for what concerns Flying) but this time holds (because for instance has a nice feather, whereas has not). So is preferred to for a given aspect whereas is preferred to for another aspect. However, enjoys the typical properties of penguins, and violates the typical properties of birds, whereas enjoys the typical properties of birds and violates those of penguins. Being concept Penguin more specific than concept Bird, we prefer to , since we prefer the individuals that inherit the properties of the most specific concepts of which they are instances. This is exactly what we get: by (b) holds.
Logical entailment for is defined as usual: a query (with form or ) is logically entailed by if it holds in all models of , as stated by the following definition. The following theorem shows the relations between and . Proofs are omitted due to space limitations.
Theorem 2
If then also . If does not occur in the other direction also holds: If then also .
The following example shows that alone is not strong enough, and this motivates the minimal models’ mechanism that we introduce in the next section. In the example we show that alone does not allow us to perform the stronger inferences with respect to rational closure mentioned in the Introduction (and in particular, it does not allow to infer (**), that typical penguins have a nice feather).
Example 2
Consider the above Example 1. As said in the Introduction, in rational closure we are not able to reason separately about the property of flying or not flying, and the property of having or not having a nice feather. Since penguins are exceptional birds with respect to the property of flying, in rational closure which is an allornothing mechanism, they do not inherit any of the properties of typical birds. In particular, they do not inherit the property of having a nice feather, even if this property and the fact of flying are independent from each other and there is no reason why being exceptional with respect to one property should block the inheritance of the other one. Does our enriched semantics enforce the separate inheritance of independent properties?
Consider a model in which we have , where is a bird (not a penguin) that flies and has a nice feather (), is a penguin that does not fly and has a nice feather (), is a penguin that does not fly and has no nice feather (). Suppose it holds that , , , , and , , . It can be verified that this is an model, satisfying (since the only typical Penguin is , instance of HasNiceFeather).
Unfortunately, this is not the only model of . For instance there can be equal to except from the fact that does not hold, nor holds. It can be easily verified that this is also an model of in which does not hold (since now also is a typical Penguin, and is not an instance of HasNiceFeather).
This example shows that although there are models satisfying well suited inclusions, the logic is not strong enough to limit our attention to these models. We would like to constrain our logic in order to exclude models like . Roughly speaking, we want to eliminate because it is not minimal: although the model as it is satisfies , so does not need to be preferred to to satisfy (neither with respect to nor with respect to ), intuitively we would like to prefer to (with respect to the property HasNiceFeather, whence in this case with respect to the global ), since does not falsify any of the inclusions with HasNiceFeather, whereas z does. This is obtained by imposing the constraint of considering only models minimal with respect to all relations , defined as in Definition 10 below. Notice that the wanted inference does not hold in minimal canonical models corresponding to rational closure: in these models does never hold (the two elements have the same rank) and this semantics does not allow us to prefer to . By adopting the restriction to minimal canonical models, we obtain a semantics which is stronger than rational closure (and therefore enforces all conclusions enforced by rational closure) and, furthermore, separately allows to reason on different aspects.
Before we end the section, similarly to what done above, let us introduce a rank of a domain element with respect to an aspect. We will use this notion in the following section.
Definition 9
The rank of a domain element with respect to in is the length of the longest chain from to a minimal (s.t. for no ). To refer to the rank of an element with respect to the preference relation we will simply write .
The notion just introduced will be useful in the following. Since and are clearly interdefinable (by the previous definition and by the properties of it easily follows that in all enriched models , iff , and iff ), we will shift from one to other whenever this simplifies the exposition.
5 Nonmonotonicity and relation with rational closure
We here define a minimal models mechanism starting from the enriched models of the previous section. With respect to the minimal canonical models used in [Giordano et al.2015] we define minimal models by separately minimizing all the preference relations with respect to all aspects (steps (i) and (ii) in the definition below), before minimizing (steps (iii) and (iv) in the definition below). By the constraints linking to the preference relations , this leads to preferring (with respect to the global ) the individuals that are minimal with respect to all for all aspects , or to aspects of most specific categories than of more general ones. It turns out that this leads to a stronger semantics than what is obtained by directly minimizing .
Definition 10 (Minimal Enriched Models)
Given two enriched models and we say that is preferred to with respect to the single aspects (and write ) if , , and:

(i) for all , for all : ;

(ii) for some , for some ,
We let the set there is no such that .
Given and , we
say that is overall preferred to (and write )
if, , and:

(iii) for all , ;

(iv) for some ,
We call a minimal enriched model of if it is a model of and there is no model of such that .
minimally entails a query if holds in all minimal models of . We write . We have developed the semantics above in order to overcome a weakness of rational closure, namely its allornothing character. In order to show that the semantics hits the point, we show here that the semantics is stronger than the one corresponding to rational closure. Furthermore, Example 3 below shows that indeed we have strengthened rational closure by making it possible to separately reason on the different properties. Since the semantic characterization of rational closure is given in terms of rational canonical models, here we restrict our attention to enriched rational models which are canonical.
Definition 11 (Minimal canonical enriched models of K)
An enriched model is a minimal canonical enriched model of if it satisfies , it is minimal (with respect to Definition 10) and it is canonical: for all the sets of concepts s.t. , there exists (at least) a domain element such that .
We call the semantics obtained by restricting attention to minimal canonical enriched models. In the following we will write:
to mean that holds in all minimal canonical enriched models of .
The following example shows that this semantics allows us to correctly deal with the wanted inferences of the Introduction, as (**). The fact that the semantics is a genuine strengthening of the semantics corresponding to rational closure is formally shown in Theorem 3 below.
Example 3
Consider any minimal canonical model of the same used in Example 1.It can be easily verified that in there is a domain element which is a penguin that does not fly and has a nice feather (). First, it can be verified that (by Definition 7, and since by minimality of and , and ). Furthermore, for all penguin that has not a nice feather, (again by Definition 7, and since by minimality of and , ). From this, in all minimal canonical models of it holds that , i.e., , which was the wanted inference (**) of the Introduction.
The following theorem is the important technical result of the paper:
Theorem 3
The minimal models semantics is stronger than the semantics for rational closure. Let (). If then .
Proof.(Sketch) By contraposition suppose that . Then there is a minimal canonical enriched model of and an such that . All consistent sets of concepts consistent with w.r.t. are also consistent with with respect to , and viceversa (by Theorem 2).By definition of canonical, there is also a canonical model of be this model. If does not contain the operator, we are done: in , as in , there is such that , hence does not hold in , and . If occurs in , and , we still need to show that also in , as in , . We prove this by showing that for all if in , then also in . The proof is by induction on .
(a): let and . Since does not violate any inclusion, also in (by minimality of ) for all preference relations , and also . This cannot hold for , for which (otherwise would violate , against the hypothesis). Hence in .
(b): let , i.e. . As in and the rank of in is , there must be a such that whereas in . Before we proceed let us notice that by definition of , as well as by what stated just above on the relation between rank of a concept and , . We will use this fact below. We show that, for any inclusion that is violated by , it holds that , so that, by (b), .
Let violated by , i.e. such that . Since satisfies , there must be in with . As , by inductive hypothesis, in . As , . Since it can be shown that , , and by condition (b), it holds that in .
With these facts, since holds in , also in , hence does not hold in , and .
The theorem follows by contraposition.
6 Conclusions and Related Works
A lot of work has been done in order to extend the basic formalism of Description Logics (DLs) with nonmonotonic reasoning features [Straccia1993, Baader and Hollunder1995, Donini, Nardi, and Rosati2002, Eiter et al.2004, Giordano et al.2007, Giordano et al.2013a, Ke and Sattler2008, Britz, Heidema, and Meyer2008, Bonatti, Lutz, and Wolter2009, Casini and Straccia2010, Motik and Rosati2010, Krisnadhi, Sengupta, and Hitzler2011, Knorr, Hitzler, and Maier2012, Casini et al.2013]. The purpose of these extensions is to allow reasoning about prototypical properties of individuals or classes of individuals.
The interest of rational closure for DLs is that it provides a significant and reasonable skeptical nonmonotonic inference mechanism, while keeping the same complexity as the underlying logic. The first notion of rational closure for DLs was defined by Casini and Straccia [Casini and Straccia2010]. Their rational closure construction for directly uses entailment in over a materialization of the KB. A variant of this notion of rational closure has been studied in [Casini et al.2013], and a semantic characterization for it has been proposed. In [Giordano et al.2013b, Giordano et al.2015] a notion of rational closure for the logic has been proposed, building on the notion of rational closure proposed by Lehmann and Magidor [Lehmann and Magidor1992], together with a minimal model semantics characterization.
It is well known that rational closure has some weaknesses that accompany its wellknown qualities, both in the context of propositional logic and in the context of Description Logics. Among the weaknesses is the fact that one cannot separately reason property by property, 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 . Among the strengths of rational closure there is its computational lightness, which is crucial in Description Logics. To overcome the limitations of rational closure, in [Casini and Straccia2011, Casini and Straccia2013] an approach is introduced based on the combination of rational closure and Defeasible Inheritance Networks, while in [Casini and Straccia2012] a lexicographic closure is proposed, and in [Casini et al.2014] relevant closure, a syntactic stronger version of rational closure.To address the mentioned weakness of rational closure, in this paper we have proposed a finer grained semantics of the semantics for rational closure proposed in [Giordano et al.2015], where models are equipped with several preference relations. In such a semantics it is possible to relativize the notion of typicality, whence to reason about typical properties independently from each other. We are currently working at the formulation of a syntactic characterization of the semantics which will be a strengthening of rational closure. As the semantics we have proposed provides a strengthening of rational closure, a natural question arises whether this semantics is equivalent to the lexicographic closure proposed in [Lehmann1995]. In particular, lexicographic closure construction for the description logic has been defined in [Casini and Straccia2012]. Concerning our Example 3 above, our minimal model semantics gives the same results as lexicographic closure, since can be derived from the lexicographic closure of the TBox and holds in all the minimal canonical enriched models of TBox. However, a general relation needs to be established.
An approach related to our approach is given in [Gil2014], where it is proposed an extension of with several typicality operators, each corresponding to a preference relation. This approach is related to ours although different: the language in [Gil2014] allows for several typicality operators whereas we only have a single typicality operator. The focus of [Gil2014] is indeed different from ours, as it does not deal with rational closure, whereas this is the main contribution of our paper.
Acknowledgement: This research is partially supported by INDAMGNCS Project 2016 ”Ragionamento Defeasible nelle Logiche Descrittive”.
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