Adding the Power-Set to Description Logics

02/26/2019 ∙ by Laura Giordano, et al. ∙ Università del Piemonte Orientale Universita Degli Studi Di Udine 0

We explore the relationships between Description Logics and Set Theory. The study is carried on using, on the set-theoretic side, a very rudimentary axiomatic set theory Omega, consisting of only four axioms characterizing binary union, set difference, inclusion, and the power-set. An extension of ALC, ALC^Omega, is then defined in which concepts are naturally interpreted as sets living in Omega-models. In ALC^Omega not only membership between concepts is allowed---even admitting circularity---but also the power-set construct is exploited to add metamodeling capabilities. We investigate translations of ALC^Omega into standard description logics as well as a set-theoretic translation. A polynomial encoding of ALC^Omega in ALCIO proves the validity of the finite model property as well as an ExpTime upper bound on the complexity of concept satisfiability. We develop a set-theoretic translation of ALC^Omega in the theory Omega, exploiting a technique originally proposed for translating normal modal and polymodal logics into Omega. Finally, we show that the fragment LC^Omega of ALC^Omega, which does not admit roles and individual names, is as expressive as ALC^Omega.

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1 Introduction

Concept and concept constructors in Description Logics (DLs) allow to manage information built-up and stored as collection of elements of a given domain. In this paper we would like to take the above statement seriously and put forward a DL doubly linked with a (very simple, axiomatic) set theory. Such a logic will be suitable to manipulate concepts (also called classes in OWL [26]) as first-class citizens, in the sense that it will allow the possibility to have concepts as instances (a.k.a. elements) of other concepts. From the set-theoretic point of view this is the way to proceed, as stated in the following quotation from the celebrated Naive Set Theory ([15]):

Sets, as they are usually conceived, have elements or members. An element of a set may be a wolf, a grape, or a pidgeon. It is important to know that a set itself may also be an element of some other set. […] What may be surprising is not so much that sets may occur as elements, but that for mathematical purposes no other elements need ever be considered.
P. Halmos

Also in the Description Logic arena the idea of enhancing the language of description logics with statements of the form , with and concepts is not: assertions of the form , with a concept name, are allowed in OWL-Full [26]. Here, we do not consider roles, i.e. relations among individuals (also called properties in OWL), as possible instances of concepts. However, we would like to push the usage of membership a little forward, allowing not only the possibility of stating that an arbitrary concept can be thought as an instance of another one (), or even as an instance of itself (), but also opening to the possibility of talking about all possible sub-concepts of , that is adding memberships to the power-set of .

In order to realize our plan we introduce a DL, to be dubbed , whose two parents are and a rudimentary (finitely axiomatized) set theory .

Considering an example taken from [28, 22], using membership axioms, we can represent the fact that eagles are in the red list of endangered species, by the axiom and that Harry is an eagle, by the assertion . We could further consider a concept , consisting of those lists that cannot be modified (if not by, say, a specifically enforced law) and, for example, it would be reasonable to ask but, more interestingly, we would also clearly want .

The power-set concept, , allows to capture in a natural way the interactions between concepts and metaconcepts. Considering again the example above, the statement “all instances of species in the Red List are not allowed to be hunted” can be represented by the concept inclusion axiom: , meaning that all the instances in the (as the class ) are collections of individuals of the class . Notice, however, that is not limited to include but can include a much larger universe of sets (e.g. anything belonging to ).

Motik has shown in [22] that the semantics of metamodeling adopted in OWL-Full leads to undecidability already for -Full, due to the free mixing of logical and metalogical symbols. In [22], limiting this free mixing but allowing atomic names to be interpreted as concepts and to occur as instances of other concepts, two alternative semantics (the Contextual -semantics and the Hilog -semantics) are proposed for metamodeling. Decidability of extended with metamodeling is proved under either of the two proposed semantics.

Starting from [22], many approaches to metamodeling have been proposed in the literature including membership among concepts. Most of them [9, 17, 20, 14] are based on a Hilog semantics, while [25, 23] define extensions of OWL DL and of (respectively), based on semantics interpreting concepts as well-founded sets. None of these proposals includes the power-set concept in the language.

Here, we propose an extension of with power-set concepts and membership axioms among concepts, whose semantics is naturally defined using sets (not necessarily well-founded) living in -models. We first prove that is decidable by defining, for any knowledge base , a polynomial translation into , exploiting a technique—originally proposed and studied in [8]—consisting in identifying the membership relation with the accessibility relation of a normal modality. Such an identification naturally leads to a correspondence between the power-set operator and the modal necessity operator , a correspondence used here to translate -type concepts. We show that the translation enjoys the finite model property and exploit it in the proof of completeness of the translation. From the translation in we also get an ExpTime upper bound on the complexity of concept satisfiability in . Interestingly enough, our translation has strong relations with the first-order reductions in [13, 17, 20].

We further exploit the correspondence between and the accessibility relation of a normal modality in another direction, to provide a polynomial set-theoretic translation of in the set theory . Our aim is to understand the real nature of the power-set concept in , as well as showing that a description logic with just the power-set concept, but no roles and no individual names, is as expressive as .

We proceed step by step by first defining a set-theoretic translation of based on Schild’s correspondence result [27] and on the set-theoretic translation for normal polymodal logics in [8]. Then, we consider the fragment of containing union, intersection, (set-)difference, complement, and power-set (but neither roles nor named individuals) and we show that this fragment, that we call , has an immediate set-theoretic translation into , where the power-set concept is translated as the power-set in . Finally, we provide an encoding of the whole into . This encoding shows that is as expressive as and also provides, as a by-product, a set-theoretic translation of where the membership relation is used to capture both the roles and the membership relation in . The full path leads to a set-theoretic translation of both the universal restriction and power-set concept of in the theory using the single relational symbol .

The outline of the paper is the following. In Section 2 we recall the definition of the description logics and , and of the set theory . In Section 3, we introduce the logic . In Section 4, we provide a translation of the logic into the description logic . In Section 5, we develop set-theoretic translations for and and an encoding of into . Sections 6 describes related work and Section 7 concludes the paper.

2 Preliminaries

2.1 The description logics and

Let be a set of concept names, a set of role names and a set of individual names. The set of concepts can be defined inductively as follows:

- , and are concepts in ;

- if and , then are concepts in .

A knowledge base (KB) is a pair , where is a TBox and is an ABox. The TBox is a set of concept inclusions (or subsumptions) , where are concepts in . The ABox is a set of assertions of the form and where is a concept, , and .

An interpretation for (see [2]) is a pair where: is a domain—a set whose elements are denoted by —and is an extension function that maps each concept name to a set , each role name to a binary relation , and each individual name to an element . It is extended to complex concepts as follows: , , , , , and

The notion of satisfiability of a KB in an interpretation is defined as follows:

Definition 1 (Satisfiability and entailment)

Given an interpretation :

- satisfies an inclusion if ;

- satisfies an assertion if and an assertion if .

Given a KB ,an interpretation satisfies (resp. ) if satisfies all inclusions in (resp. all assertions in ); is a model of if satisfies and .

Letting a query be either an inclusion (where and are concepts) or an assertion , is entailed by , written , if for all models of , satisfies .

Given a knowledge base , the subsumption problem is the problem of deciding whether an inclusion is entailed by . The instance checking problem is the problem of deciding whether an assertion is entailed by . The concept satisfiability problem is the problem of deciding, for a concept , whether is consistent with (i.e., whether there exists a model of , such that ).

In the following we will also consider the description logic allowing inverse roles and nominals. For a role , its inverse is a role, denoted by , which can be used in existential and universal restrictions with the following semantics: if and only if For a named individual , the nominal is the concept such that: .

2.2 The theory

The first-order theory consists of the following four (simple) axioms, written in the language whose relational symbols are and , and whose functional symbols are , , :

In any -model everything is supposed to be a set. Hence, a set will have (only) sets as its elements and circular definitions of sets are not forbidden—i.e., for example, there are models of in which there are sets admitting themselves as elements. Moreover, not postulating in any link between membership and equality—in axiomatic terms, having no extensionality (axiom)—, there exist -models in which there are different sets with equal collection of elements.

The most natural -model—in which different sets are, in fact, always extensionally different—is the collection of well-founded sets , where:

             

In every system of set-theoretic equations of the form:

where is one among for and , finds a unique solution. If we drop the index-ordering restriction on variables appearing in the right-hand-side of set-theoretic equations (thereby allowing equations such as ), in order to guarantee the existence of solutions in the model we need to work with universes larger than . The most natural (and minimal) among them is a close relative of and goes under the name of (see [1, 24]).

Finally, a further enrichment of both and is obtained by adding atoms, that is copies of the empty-set, to be denoted by and collectively represented by . The resulting universes will be denoted by and .

A complete discussion relative to universes of sets to be used as models of goes beyond the scope of this paper. However, it is convenient to point out that, in all cases of interest for us here, an especially simple view of -models can be given using finite graphs. Actually, or can be seen as the collection of finite graphs (either acyclic or cyclic, respectively), where sets are represented by nodes and arcs depict the membership relation among sets (see [24]). Given one such membership graph it is convenient to single out a special node (the point of the graph), to isolate the specific set for which the description is introduced.

In the next section, we will regard the domain of a DL interpretation as a transitive set in a universe of an -model, i.e. will be a set of sets in (a universe of a model of) the theory rather than as a set of individuals, as customary in description logics.

3 The description logic

We start from the observation that in concepts are interpreted as sets (namely, sets of domain elements) and we generalize by allowing concepts to be interpreted as sets of the set theory . In addition, we extend the language of by introducing the power-set as a new concept constructor, and allowing membership relations among concepts in the knowledge base. We call the resulting extension of .

As before, let , , and be the set of individual names, concept names, and role names in the language, respectively. In building complex concepts, in addition to the constructs of , we also consider the difference and the power-set Pow constructs. The set of concepts are defined inductively as follows:

- , and are concepts;

- if are concepts and , then the following are concepts:

While the concept can be easily defined as in , this is not the case for the concept . Informally, the instances of concept are all the subsets of the instances of concept , which are “visible” in (i.e. which belong to) .

Besides usual assertions of the forms and with , allows in the ABox concept membership axioms and role membership axioms of the forms and , respectively, where and are concepts and is a role name.

Considering again the example from the Introduction, the additional expressivity of the language, in which general concepts (and not only concept names) can be instances of other concepts, allows for instance to represent the fact that polar bears are in the red list of endangered species, by the axiom . We can further represent the fact the polar bears are more endangered than eagles by adding a role and the role membership axiom - . The inclusion means that any element of (such as ) is a subset of , i.e., each polar bear cannot be hunted. As shown in [22], the meaning of the sentence “all the instances of species in the Red List are not allowed to be hunted” could be captured by combining the -semantics with SWRL [18], but not by the -semantics alone.

We define a semantics for by extending the semantics in Section 2.1 to capture the meaning of concepts (including concept ) as elements (sets) of the domain , chosen as a transitive set (i.e. a set satisfying ) in a model of . Roles are interpreted as binary relations over the domain . Individual names are interpreted as elements of a set of atoms from which the sets in are built.

Definition 2

An interpretation for is a pair over a set of atoms where:

  • the non-empty domain is a transitive set chosen in the universe of a model of over the atoms in ;111 In the following, for readability, we will denote by , , , (rather than , , ) the interpretation in a model of the predicate and function symbols , , , .

  • the extension function maps each concept name to an element ; each role name to a binary relation ; and each individual name to an element .

The function is extended to complex concepts of , as in Section 2.1 for , but for the two additional cases:  and  .

Observe that . As is not guaranteed to be closed under union, intersection, etc., the interpretation of a concept is a set in but not necessarily an element of . However, given the interpretation of the power-set concept as the portion of the (set-theoretic) power-set visible in , it easy to see by induction that, for each , the extension of is a subset of .

Given an interpretation , the satisfiability of inclusions and assertions is defined as in interpretations (Definition 1). Satisfiability of (concept and role) membership axioms in an interpretation is defined as follows:

- satisfies if ;

- satisfies if .
With this addition, the notions of satisfiability of a KB and of entailment in (denoted ) can be defined as in Section 2.1.

The problem of instance checking in includes both the problem of verifying whether an assertion is a logical consequence of the KB and the problem of verifying whether a membership is a logical consequence of the KB (i.e., whether is an instance of ).

In the next section, we define a polynomial encoding of the language into the description logic .

4 Translation of into

To provide a proof method for , we define a translation of into the description logic , including inverse roles and nominals. In [8] the membership relation is used to represent a normal modality of a modal logic. In this section, vice-versa, we exploit the correspondence between and the accessibility relation of a modality, by introducing a new (reserved) role in to represent the inverse of the membership relation: in any interpretation , will stand for . The idea underlying the translation is that each element of the domain in an interpretation can be regarded as the set of all the elements such that .

The translation of a knowledge base of into can be defined as follows. First, we associate each concept of to a concept of by replacing all occurrences of the power-set constructor Pow with a concept involving the universal restriction (see below). More formally, we (inductively) define the translation of by simply recursively replacing every subconcept appearing in by , while the translation commutes with concept constructors in all other cases.

Semantically this will result in interpreting any (sub)concept by

which, recalling that stands for , will characterise the collection of subsets of visible in (i.e. subsets of which are also elements of ): , that is, , as expected.

4.1 Translating TBox, ABox, and queries

We define a new TBox, , by introducing, for each inclusion in , the inclusion in . Additionally, for each (complex) concept occurring in the knowledge base (or in the query) on the l.h.s. of a membership axiom or , we extend with a new individual name222The symbol should remind the e-xtension of . and we add the concept equivalence:

(1)

in . From now on, new individual names such as will be called concept individual names. This equivalence is intended to capture the property that, in all the models of , is in relation with all and only the instances of concept , i.e., for all , if and only if .

As in the case of the power-set constructor, this fact can be verified by analyzing the semantics of :

which, recalling that stands for and interpreting the nominal, will stand for

which, by the concept equivalence , is as to say that and have the same extension.

Remark 1

It is important to notice that every concept individual name of the sort introduced above—that is, every individual name whose purpose is that of providing a name to the extension of —, in general turns out to be in relation with other elements of the domain of (unless is an inconsistent concept and its extension is empty). This is in contrast with the assumption relative to other “standard” individual names , for which we will require (see below).

We define as the set of assertions containing:

- for each concept membership axiom in , the assertion ,

- for each role membership axiom in , the assertion ,

- for each assertion in , the assertion ,

- for each assertion in , the assertion and, finally,

- for each (standard) individual name , the assertion .

As noticed above, the last requirement forces all named individuals (in the language of the initial knowledge base ) to be interpreted as domain elements which are not in relation with any other element.

Let be the knowledge base obtained by translating into .

Example 1

Let be the knowledge base considered above:

and

.

By the translation above, we obtain:

          

          

entails in . In fact, from and - , it follows that, in all models of , - . Furthermore, from  and the assertion  , it follows that holds. Hence, . As this holds in all models of , is a logical consequence of . It is easy to see that follows from as well.

Let be a query of the form , or We assume that all the individual names, concept names and role names occurring in also occur in and we define a translation of the query as follows:

- if is a subsumption , then is the subsumption ;
- if is an assertion , then is the assertion ;
- if is a membership axiom (respectively, ), then is the assertion (respectively, ).

In the following we state the soundness and completeness of the translation of an knowledge base into .

Proposition 1 (Soundness of the translation)

The translation of an knowledge base into is sound, that is, for any query :

Proof

By contraposition, assume and let be a model of in that falsifies . Let be the transitive set living in a model of with universe 333Observe that, on the grounds of Proposition 2, we can safely assume to be essentially a transitive set in the standard model of , (i.e. the collection of hereditarily finite rational hypersets over atoms in ). As a matter of fact, the domain is (a graph) obtained duplicating sets (nodes) representing extensionally equal but pairwise distinct sets/elements.. We build an interpretation , which is going to be a model of falsifying in , by letting:

- ;

- for all , ;

- for all roles , ;

- for all , if and only if ;

- for all (standard) individual name , ;

- for all ,
The interpretation is well defined. First, the interpretation of a named concept is a subset of as expected. In fact, as and is a transitive set, each is an element (a set) , and hence an element in . Also, each element belongs to and hence to . It is easy to see that the interpretation of constant is an element . In fact, as the named individual has been added by the translation to the language of , there must be some membership axiom (or ) in , for some (respectively, for some and ). Considering the case that axiom is in , as is a model of , satisfies , so that must hold. However, as , it must be . Hence, by construction, . In case , it must hold that . As , then . In particular, .

We can prove by induction on the structural complexity of the concepts that, for all ,

(2)

For the base case, the property above holds for and , as and , and it also holds by construction for all concept names .

For the inductive step, let and let , for some . As is an interpretation, and and since by induction (2) holds for concepts and , we have and . Therefore, and, by definition of an interpretation, . It is easy to see that the vice-versa also holds, i.e., if then .

For the case , let , for some . As is an interpretation, for all , if then . By construction of , if and only if and, by inductive hypothesis, if and only if . Hence, for all such that , . Therefore, . As , and is transitive, then . Therefore, , and . As , . The vice-versa can be proved similarly.

Let us consider the case . Let for some . As and is an interpretation, there is a such that and . By inductive hypothesis, . Furthermore, by construction of , it must be that and . Hence, . The vice-versa can be proved similarly as well as all the other cases for the concept .

Using (2) we can now check that all axioms and assertions in are satisfied in .

For an inclusion axiom , the corresponding inclusion axiom is in . If for some , by (2) and, by the inclusion , . Hence, again by (2), .

Each assertion , is obtained from the translation of the assertion . From the fact that is satisfied by , i.e. , given property (2), it follows that .

For each assertion obtained from the translation of a membership axiom , from the fact that is a model of , we know that holds. By construction, and we have seen that . From , it follows that and, by property (2), .

For each assertion obtained from the translation of a role membership axiom , from the fact that is a model of , we know that and hold. We want to show that . As, by construction, and from , it follows that . By the definition of role interpretation in , .

For each assertion , for , it is easy to see that . As and an element of in is interpreted as an empty set, there is no such that . Hence, by definition of in the model , there is no such that .

We still need to show that axiom is satisfied in for all the concepts occurring in on the l.h.s. of membership axioms. Let . By property (2), and, by construction, . We want to show that , i.e. that . By the definition of in the interpretation , if and only if . But immediately follows from the previous conclusions that , as by construction.

To conclude the proof, it can be easily shown that, for any query , is satisfied in if and only if is satisfied in .

Before proving the completeness of the translation of into , we show that, if the translation of a knowledge base in has a model in , then it has a finite model as well.

Proposition 2

Let be a knowledge base in and let be its translation in . If has a model in , then it has a finite model.

Proof

We prove this result by providing an alternative (but equivalent) translation of in the description logic , using a single negated role .

extends with role complement operator, where, for any role , the role is the negation of role , where if and only if . In the translation, we exploit to capture non-membership, where if and only if (i.e., in set terms, ). Decidability of concept satisfiability in has been proved by Lutz and Sattler in [21]. The finite model property of a language with a single negated role can be proved as done in [10] (Section 2) for a logic with the “window modality”, by standard filtration, extended to deal with additional K-modalities (for the other roles) as in the proof in [3]. Indeed, as observed in [21], the “window operator”