Reasoning about multiple aspects in DLs: Semantics and Closure Construction

01/18/2018 ∙ by Laura Giordano, et al. ∙ Università del Piemonte Orientale Università di Torino 0

Starting from the observation that rational closure has the undesirable property of being an "all or nothing" mechanism, we here propose a multipreferential semantics, which enriches the preferential semantics underlying rational closure in order to separately deal with the inheritance of different properties in an ontology with exceptions. We provide a multipreference closure mechanism which is sound with respect to the multipreference semantics.

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1 Introduction

Reasoning with exceptions has been widely studied within non-monotonic reasoning in Artificial Intelligence and in Description Logics (DLs). In particular, a lot of work has been devoted to extending DLs with non-monotonig formalisms to allow reasoning about prototypical properties of individuals

[36, 1, 15, 16, 20, 23, 24, 29, 7, 21, 5, 11, 34, 31, 30, 10, 3].

In this paper we propose an extension of rational closure [32] for dealing with multiple preferences. One of the main difficulties of rational closure to deal with inheritance of defeasible properties of concepts is the fact that one cannot reason property by property: if a subclass of a class is exceptional with respect to for a given property, it does not inherit any of the defeasible properties of . This is the “all or nothing” behavior of rational closure.

Consider the following version of the classic birds/penguins example:

Typical birds fly

Penguins are birds

Typical penguins do not fly

Typical birds have nice feather

By rational closure, penguins (being exceptional birds concerning the property of flying) do not inherit any of the typical properties of birds.

On the contrary, one could expect penguins to inherit the property of having nice feather, for which they are not exceptional. More generally, we would like to reason independently on the inheritance of the properties of one concept by a more specific one. This is what Lehmann calls “presumption of independence” [33]: even if typicality is lost with respect to one property, we may still presume typicality with respect to another, unless there is reason to the contrary.

In this paper, we address this problem from the semantic point of view. Starting from the preferential semantics underlying rational closure introduced by Lehmann and Magidor [32] for propositional logics and extended to the description logic in [24], we consider a preferential semantics which allows to reason about typicality with respect to different aspects. In this semantics, an individual can be more preferred than another when considering one aspect (e.g. being a good student), while less preferred when considering another aspect (e.g. being a good citizen). In this enriched preferential semantics, different preference relations among domain elements are introduced, each one describing the preference of an individual over another one with respect to a given aspect/property. We show that this semantics is a strengthening of rational closure.

We provide a syntactic construction which is built over the rational closure and that we call multipreference closure. The multipreference closure is proved to be a sound construction for reasoning with multiple preferences, thus providing a sound approximation of the multipreference semantics. As we will see, this construction is strongly related with the lexicographic closure, proposed by Lehmann [33] and extended to the description logic by Casini and Straccia [12], but it exploits a different specificity ordering, as it will become clear in Section 5.1, where we compare the two constructions. Another approach related to ours is the logic of overriding by Bonatti, Faella, Petrova and Sauro [3], a nonmonotonic description logic which also allows reasoning independently about different defeasible properties. A difference with is that leads to the inconsistency of prototypical concepts (thus requiring a repair of the KB) in those cases when a conflict among defaults cannot be solved by overriding. In our approach, as in the lexicographic closure, such conflicts are silently removed considering (skeptically) what holds in all the alternative bases (i.e., maximal consistent sets of defeasible inclusions), while computes a single base. We discuss the relations between our approach and in Section 5.1, where we also suggest that the MP-closure construction could be further approximated by a construction which only requires a polynomial number of entailment checks in , following the approach in [26].

We will proceed as follows. In Section 2 we recall the rational closure for description logics and its semantics. In Sections 3 and 3.1, we define the multipreference semantics by introducing the notions of enriched and strongly enriched models of a knowledge base. In Section 5 we develop the multipreference closure construction, we show that it is sound with respect to the semantics, and we compare it with the lexicographic closure, with the logic of overriding [3], and with related constructions. Sections 6 and 7 conclude the paper by assessing the contribution with respect to related work.

2 The Rational Closure and its Semantics

Let us briefly recall the logic which is at the basis of a rational closure construction proposed in [24] for . The intuitive idea of is to extend the standard description logic 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 instances ().

Let be a set of concept names, a set of role names and a set of individual names. The language is defined as follows: , and , where and . A knowledge base (KB) is a pair , where the TBox contains a finite set of concept inclusions , and the ABox contains a finite set of assertions of the form and , where are individual names. In the following we will call non-extended concepts the concepts of the language, which do not contain the operator.

The semantics of is defined in terms of ranked models similar to those introduced in [32]: 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 (i.e., for all , if then either or ) and well-founded 111Since has the finite model property, this is equivalent to having the Smoothness Condition, as shown in [24]. We choose this formulation because it is simpler. (i.e., there is no infinite -descending chain, so that, if , also ). Ranked models characterize .

Definition 1 (Semantics of [24])

A model of is any structure where: is the domain; is an irreflexive, transitive, modular and well-founded relation over . is an interpretation function that maps each concept name to , each role name to and each individual name to . For concepts of , is defined in the usual way. For the operator, we have .

As shown in [24], 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 if it satisfies its TBox (and for all inclusions , it holds), and its ABox (for all , and, for all , ). 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 [27, 24] a non monotonic construction of rational closure has been defined for , extending the notion of rational closure proposed in the propositional context by Lehmann and Magidor [32]. 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 [27, 24] 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 T-inclusion is exceptional for if is exceptional for . The set of T-inclusions of which are exceptional for will be denoted as .

Given a TBox, it is possible to define a sequence of non increasing subsets of a 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 be an knowledge base. The rational closure, , of the TBox , is defined as:

=

, where and are concepts.

A good property of rational closure is that, for , deciding if an inclusion belongs to the rational closure of TBox is a problem in ExpTime [24].

In [24] 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 such 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 (non-extended) 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 , a model satisfying is canonical if for each set of concepts consistent with , there exists (at least) a domain element such that .

Definition 6 (Minimal canonical models (with respect to TBox))

is a minimal canonical model of , if it is a canonical model of and it is minimal with respect (see Definition 4) among the canonical models of .

The correspondence between minimal canonical models and rational closure is established by the following key theorem.

Theorem 2.1 ([24])

Let be a knowledge base and a query. Let be the rational closure of w.r.t. TBox. We have that if and only if holds in all minimal canonical models of with respect to TBox.

3 Enriched Preferential Semantics

The main weakness of rational closure, despite its power and its nice computational properties, is that it is an all-or-nothing mechanism that does not allow to separately reason on single aspects. As mentioned in the introduction, to overcome this difficulty, here we consider models with several preference relations, one for each aspect we want to reason about. We assume an aspect can be any concept occurring in on the right end side of some typicality inclusion : we call the set of these aspects. Observe that may be non-atomic; it is an arbitrary non-extended concept. For each aspect , the relation expresses the preference for aspect being true: expresses the preference for flying, so if it holds that , birds that do fly will be preferred to birds that do not fly, with respect to aspect fly, i.e. with respect to .

Notice that the preferences with respect to aspects might be conflicting. It can be that, for instance, is preferred to for aspect (), whereas is preferred to for aspect (). In the example of birds, we can have that , whereas .

With this semantic richness we aim 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 as in . In particular, we only include a single typicality operator . However, the semantic richness could motivate the introduction of several typicality operators by which one could explicitly refer within the language to the typicality w.r.t. aspect , or , and so on. We leave this extension for future work.

Let us now enrich the definition of an model given above (Definition 1) by taking into account preferences with respect to the aspects, as well as a global preference relation .

Definition 7 (Enriched rational interpretation)

An enriched rational interpretation is a structure , where and are a domain and an interpretation function (as in Definition 1), are irreflexive, transitive, modular and well-founded preference relations over . Furthermore, satisfies the condition:

(a) If there is some such that , and there is no such that , then .

Last, we let: s.t. there is no s.t. and .

In the semantics above the global preference relation is related to the various preference relations , relative to single aspects , that we call indexed preference relations. Given condition (a), holds when is preferred to for a single aspect , and there is no aspect for which is preferred to . This allows to define preferences among elements having the same rank in the minimal canonical models of the rational closure. As it will become clear, this brings us towards the direction of a refinement of the semantics of rational closure.

Let s.t. there is no s.t. . In order to be a model of , an enriched rational model must satisfy the following conditions.

Definition 8 (Enriched rational models of K)

Given a knowledge base ,A, an enriched rational model (or enriched model) for is an enriched interpretation of which satisfies and , where:

satisfies the TBox if

  • for all strict inclusions (where does not occur in ), ;

  • for all typicality inclusions , ;

  • for all typicality inclusions , .

satisfies the ABox if: (i) for all , ; (ii) for all , .

By condition (3), the domain elements satisfying all the defeasible inclusions concerning aspect will be preferred with respect to to those falsifying some of them.

We call the description logic extending with typicality under the enriched semantics. Logical entailment in is defined as usual: a query (with form or ) is logically entailed by (written ) if holds in all the enriched models of .

The following example shows that, at least in some cases, condition allows to establish the expected preference between individuals.

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 birds, has a nice feather whereas does not) and there is no other aspect such that , and in particular it does neither hold that (because for instance, as all typical penguins, both and do not fly), nor that . In this case, obviously it holds that , since condition (a) in Definition 7 is satisfied.

However, the enriched semantics does not provide a refinement of the rational closure.

Example 2

Let us compare the domain element in a model with a domain element . We have that , and . Hence, condition (a) cannot help to conclude anything about the global relation concerning and (and, in particular, we cannot conclude ). However, in all the models of the rational closure, we would prefer to .

Observe that, in this last example, violates the defeasible properties of Birds of flying and having a nice feather, while violates the more specific defeasible property of Penguin of not flying.

3.1 S-Enriched models

In order to deal with cases as Example 2 and in order to obtain a strengthening of rational closure, we strengthen the definition of enriched model, by introducing an additional condition beside condition (a). In particular, we define a subset of enriched models, that we call strongly enriched (S-enriched) models, as they enforce the respect for “specificity” also in cases when enriched models do not. In addition to the constraints linking the global preference relation to the indexed preference relations , which leads to preferring (with respect to the global ) the individuals that are minimal with respect to the aspects , we add a further constraint which leads to prefer the individuals violating defeasible properties of less specific concepts with respect to individuals violating defeasible properties of more specific concepts. It turns out that this leads to a stronger semantics, which is able to capture wanted inferences, such as those in Example 2, and which provides a strengthening of rational closure semantics in Section 2.

In order to define S-enriched models of a knowledge base , we strengthen the definition of satisfiability of a TBox as follows.

Definition 9 (S-enriched rational models of K)

Given a knowledge base , an enriched interpretation is an S-enriched rational model for if satisfies the TBox and the ABox , where:

satisfies if

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

  • for all typicality inclusions , ;

  • for all typicality inclusions , .

  • If there is s.t. and and,
    for all s.t. and , there is s.t. ,  , and ,
    then .

satisfies A if: (i) for all , , (ii) for all , .

We call the logic based on the semantics of S-enriched models and we define logical entailment in as usual: a query is logically entailed by in (written ) if holds in all the S-enriched models of .

We call condition “specificity condition”, as it captures the idea that, in case two individuals are preferred one another with respect to different aspects, preference (with respect to the global preference relation ) should be given to the individual that falsifies typical properties of concepts with lower ranks. Violating a default property of a less specific concept is less serious than violating a default property of a more specific concept .

The idea is that, in S-enriched models, provides a strengthening of the preference relation in ranked models of rational closure. In particular, further preferences are determined among the elements having the same rank in the models of rational closure, both by exploiting the preference relations with respect to single aspects (condition ), and by exploiting the specificity criterium (condition ). Observe that is only a sufficient condition for . It is not required to be a necessary condition, and additional pairs might be needed for to satisfy modularity.

The above semantics allows us to model a form of inheritance in which the defeasible properties of concepts (classes) are inherited by more specific concepts, unless they are overridden by the properties of more specific ones. Also, the overriding of some defeasible property of a concept should not cause the overriding of all the defeasible properties of that concept, and the inheritance of a more specific property should win over the inheritance of a less specific one. These criteria are incorporated among the desirable principles considered by Lehmann in [33] namely, the “presumption of typicality”, the “presumption of independence”, “priority to typicality” and “respect of specificity”, which underly the lexicographic closure definition. Similar criteria are also at the basis of the nonmonotonic description logic [3], whose definition explicitly uses the notion of overriding. We will provide a detailed comparison with the lexicographic closure and with in Section 5.1.

With reference to Example 2, we can see that, with this notion of S-enriched semantics, we are able to give preference to a domain element which is a penguin that does not fly and has not nice feathers (thus violating the defeasible property of birds) with respect to an element corresponding to penguin which has not nice feathers but flies, violating the more specific property of penguins.

For we can prove the following theorem, showing the relations between and (the extension of with the typicality operator defined in Section 2). It is a an immediate consequence of the fact that a S-enriched model is a model.

Theorem 3.1

If then also . If does not occur in the other direction also holds: If then also .

The theorem is a an immediate consequence of the fact that a S-enriched model is a model. By contraposition, from the hypothesis that , there is an S-enriched model satisfying and falsifying . Since is also an model of , it follows that . For the second part, observe that, by contraposition, if , then there is an model of falsifying . We can define an S-enriched model of , , by letting, for all , . It is easy to see that satisfies condition (a) in Definition 7 as well as conditions (1)-(4) in Definition 9 and, hence, it is an S-enriched model of which falsifies .

4 Minimal S-enriched models and their relation with rational closure

As in the semantic characterization of the rational closure in Section 2, we restrict our consideration to minimal canonical models of the KB. We define minimal S-enriched models by first minimizing the rank of each domain element with respect to the indexed preference relations ’s, and then by minimizing the rank of the elements with respect to the global preference relation . Let be the rank of a domain element of the model with respect to the indexed relation .

Definition 10 (Minimal S-enriched models of (with respect to ))

Given two S-enriched models and ,

  • is preferred to w.r.t. the aspects (and write ) if , , and:

    • for all , ;

    • for some ,

  • is preferred to w.r.t. the global preference relation (and write ) if , , and

    • for all , ;

    • for some ,

We combine the two preference relations in the lexicographic order: We say that is preferred to (and write ) if , , and

  • either or

  • and .

Given a knowledge base , we say that is a minimal S-enriched model of (with respect to TBox) if it is an S-enriched model of and there is no model satisfying such that .

As the definition of the global preference depends on the indexed preferences , we first minimize with respects to the aspects and, then, with respect to the global preference .

In minimal models, each preference relation ranks the domain elements into two levels: the domain elements satisfying all the defeasible inclusion concerning aspect (having rank ), and the domain elements falsifying some defeasible inclusion concerning aspect (having rank ). This is similar to the interpretation given by Lehmann to single defaults in [33]).

Let us restrict our attention to minimal S-enriched models which are canonical.

Definition 11 (Minimal canonical S-enriched models of K)

A minimal canonical S-enriched model of is an S-enriched model of , which is minimal (with respect to Definition 10) and it is canonical, i.e., for each set of (non-extended) concepts s.t. , there exists (at least) a domain element such that .

In the following we will write: to mean that holds in all minimal canonical S-enriched models of .

The following example shows that this semantics allows us to correctly deal with the wanted inferences. and, in particular, that inheritance of defeasible properties, when not overridden for more specific concepts, applies to concepts of all ranks.

Example 3

Consider a knowledge base =(T ,A), where and contains the following inclusions:

.

As we have seen from Example 1, the defeasible property of birds having a nice feather is inherited by typical penguins, even though penguins are exceptional birds regarding flying. Here, we also expect that typical baby penguins inherit the defeasible property of penguins that they do not fly (by presumption of independence [33]), although the defeasible property is instead overridden for typical baby penguins.

Consider two domain elements and which are both baby penguins and have a non black feather. Suppose that flies and doesn’t. Then violates the defeasible property that penguins typically do not fly, while violates the defeasible property that birds typically fly. As and , condition neither allows to conclude , nor . However, violates a more specific defeasible property than and, hence, by the specificity condition of S-enriched models in Definition 9, we can conclude that holds. Indeed, the S-enriched minimal model semantics allows us to conclude that , as wanted.

We have developed the semantics above in order to overcome a weakness of rational closure, namely its all-or-nothing character. In order to show that the semantics hits the point, we prove that the semantics of minimal canonical S-enriched models is a refinement of the semantics of rational closure, i.e. that minimal entailment in strengthens reasoning under the rational closure.

Theorem 4.1

Let be a knowledge base. If then .

Proof

By contraposition suppose that . Then there is a minimal canonical S-enriched model of and an such that . All sets of concepts consistent with w.r.t. are also consistent with with respect to , and viceversa (by Theorem 3.1). By definition of canonical, is a canonical model of according to Definition 1 in Section 2. Also, there must be a minimal canonical model of obtained from by possibly lowering the ranks of domain elements. Let be such a 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 , , i.e. . We prove this by showing that for all if in , then also in . The proof is by induction on .

For the base case, let and . Since does not violate any inclusion, also in (by minimality of ) . This cannot hold for , for which (otherwise would violate , against the hypothesis). Hence holds in .

For the inductive case, let , i.e. . As in and the rank of in is , there must be a such that whereas in , so that holds in the minimal S-enriched model .

Before we proceed let us notice that by definition of in Section 2, 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 such that and , it holds that , so that, by (4), .

Let such that and . As is a minimal model, is violated by , i.e. . Since satisfies , cannot be a typical -element and there must be in with . As , by inductive hypothesis, in . As , . Since it can be shown that , , and by condition (4), it holds that in .

With these facts, since holds in , also in , hence does not hold in , and .

The theorem follows by contraposition. ∎

Observe that, in the proof of Theorem 4.1, we have not used condition (a) of Definition 8. Indeed, we can show that the specificity condition in minimal S-enriched models (Definition 9) subsumes condition (a), dealing with multiple aspects. Let us consider a simplified notion of S-enriched model in which condition is omitted.

Proposition 1

Let be a minimal simplified S-enriched model of . We can show that if condition (4) holds, then condition (a) holds as well.

Proof

To see that condition (4) implies condition (a), suppose that the precondition of holds, i.e., that there is some such that in , and there is no such that . We show that follows using condition (4).

As and the model is minimal, in particular, it is minimal with respect to the aspects and there must be a defeasible inclusion s.t. satisfies it (), and violates it (). Additionally, for all (), , that is, all the defeasible inclusions satisfied by are also satisfied by . Therefore, the antecedent of condition (4), the “If part”, holds as there is no inclusion which is falsified by and satisfied by . Hence, by condition (4), follows. ∎

5 The multipreference-closure

As the minimal S-enriched semantics is a strengthening of the rational closure semantics, in this section we build on the rational closure to define a new notion of closure, that we call the multipreference-closure (MP-closure, for short). We show that the MP-closure provides a sound approximation of the minimal S-enriched semantics: reasoning in the MP-closure will allow to derive sound conclusions with respect to the minimal canonical S-enriched models semantics. The MP-closure can be regarded as a variant of the lexicographic closure [33] and we compare the MP-closure with the lexicographic closure and with the nonmonotonic description logic [3].

According to condition in Definition 9, the rank of concepts in a S-enriched model is used to determine the specificity of typicality inclusions and, thus, to determine the preference relation among domain elements. As the ranking of concepts in the rational closure approximates the ranking of concepts in minimal canonical S-enriched models of the KB, it can be used for determining the specificity of typicality inclusions in a closure thus providing a sound approximation of minimal entailment in the S-enriched semantics.

In particular, if is the rank of a concept in the rational closure, the most preferred -elements in minimal canonical S-enriched models must be among the -elements with rank . According to condition we prefer a -element with rank to another one with the same rank, if for all the defeasible inclusions falsified by and not by there is a more specific defeasible inclusion falsified by and not by . In essence, we need to identify those -elements with rank which satisfy a maximal subset of defeasible inclusions, containing inclusions being as specific as possible (something which is very similar to what the lexicographic closure construction [33] does).

In the following, we provide a construction (the MP-closure) to check the entailment of a subsumption query from a TBox and we show that the logical consequences under the MP-closure are sound with respect to the S-enriched semantics. Given a TBox , we compute the sequence of TBoxes according to the rational closure construction in Section 2. We let be the set of typicality inclusions contained in (i.e. those defeasible inclusions with rank ) and let be the set of typicality inclusions with rank . Observe that . Given a set of typicality inclusions, we let: , for all ranks in the rational closure, thus defining a partition of the set according to the rank. We introduce a preference relation among sets of typicality inclusions as follows: ( is preferred to ) if and only if there is an such that, and, for all , . The meaning of is that, considering the highest rank in which and do not contain the same defeasible inclusions, contains more defeasible inclusions in than .

Definition 12

Let be a concept such that and let . is a maximal set of defeasible inclusions compatible with in if:

  • and there is no such that and ( is preferred to ).

where is the materialization of , i.e., .

Informally, is a maximal set of defeasible inclusions compatible with and if there is no set which is consistent with and and is preferred to since it contains more specific defeasible inclusions. The construction is similar to that of the lexicographic closure [33, 12], although, in this case, the comparison of the sets of defeasible inclusions with the same rank (i.e. of and ) is based on subset inclusion rather than on the size of the sets, as in the lexicographic closure.

To check if a subsumption is derivable from the MP-closure of TBox we have to consider all the maximal sets of defeasible inclusions that are compatible with .

Definition 13

Let