Reasoning about exceptions in ontologies has led to the development of many non-monotonic extensions of Description Logics (DLs), incorporating non-monotonic features from most of NMR formalisms in the literature, starting from [41, 2, 20, 25, 14, 7, 16, 39] and notably including extensions of rule-based languages [22, 21, 33, 31, 28, 10], and new constructions and semantics [17, 5, 4].
Preferential approaches [34, 37], have been extended to description logics, to deal with inheritance with exceptions in ontologies, allowing for non-strict forms of inclusions, called typicality or defeasible inclusions, with different preferential semantics [25, 14] and closure constructions [16, 15, 27].
In this paper, we propose a “concept-aware multipreference semantics” for reasoning about exceptions in ontologies taking into account preferences with respect to different concepts and integrating them into a preferential semantics which allows a standard interpretation of defeasible inclusions. The intuitive underlying idea is that the relative typicality of two domain individuals usually depends on the aspects we are considering for comparison: Bob may be a more typical as sport lover than Jim but, vice-versa, Jim may be a more typical swimmer than Bob. This leads to consider a multipreference semantics in which there is a different preference relation among individuals for each aspect (or concept) . For instance, for concepts and , and would capture the knowledge above about their relative typicality.
Our approach is strongly related with Gerard Brewka’s proposal of preferred subtheories , later generalized within the framework of Basic Preference Descriptions for ranked knowledge bases . We extend to DLs the idea of having ranked or stratified knowledge bases (ranked TBoxes here) and to define preorders (preferences) on worlds (here, preferences among domain elements in a DL interpretation). Furthermore, we associate ranked TBoxes with concepts. The ranked TBox for concept describes the prototypical properties of -elements. For instance, the ranked TBox for concept Horse describes the typical properties of horses, of running fast, having a long mane, being tall, having a tail and a saddle. These properties are defeasible and not all horses should satisfy all of them.
The ranked TBox for determines a preference relation on the domain, defining the relative typicality of domain elements with respect to aspect . We then combine such preferences into a global preference relation to define a concept-wise multipreference semantics, in which all conditional queries can be evaluated as usual in preferential semantics. For instance, we may want to check whether typical Italian employees have a boss, given the preference relation , but no preference relation for concept ; or to check whether employed students are normally young or have a boss, given the preference relations and , resp., for employees and for students.
We introduce a notion of multipreference entailment and prove that it satisfies the KLM properties of preferential consequence relations. This notion of entailment deals properly with irrelevance and specificity, is not subject to the “blockage of property inheritance” problem, which affects rational closure , i.e., if a subclass is exceptional with respect to a superclass for a given property, it does not inherit from that superclass any other property.
To prove the feasibility of our approach, we develop a proof method for reasoning under the proposed multipreference semantics for the description logic , the fragment of OWL2 EL supported by ELK. We reformulate multipreference-entailment as a problem of computing preferred answer sets and, as a natural choice, we develop an encoding of the cw-multipreferential extension of in asprin , exploiting a fragment of Krötzsch’s Datalog materialization calculus .
As a consequence of the soundness and completeness of this reformulation of multipreference entailment, we prove that concept-wise multipreference entailment is in for ranked knowledge bases.
2 Preliminaries: The description logics
We consider the description logic  of the family . 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 as follows: , where , and . Observe that union, complement and universal restriction are not constructs. 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) of the form , where are concepts, and of role inclusions of the form , where . The ABox is a set of assertions of the form and where is a concept, , and .
An interpretation for 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 notions of satisfiability of a KB in an interpretation and of entailment are defined as usual:
Definition 1 (Satisfiability and entailment)
Given an interpretation :
- satisfies an inclusion if ;
- satisfies a role inclusions 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 .
A subsumption (resp., an assertion , ), is entailed by , written , if for all models of , satisfies .
3 Multiple preferences from ranked TBoxes
To define a multipreferential semantics for we extend the language with a typicality operator , as done for in . In the language extended with the typicality operator, an additional concept is allowed (where is an concept), whose instances are intended to be the prototypical instances of concept . Here, we assume that can only occur on the left hand side of concept inclusion, to allow typicality inclusions of the form , meaning that “typical C’s are D’s” or “normally C’s are D’s”. Such inclusions are defeasible, i.e., admit exceptions, while standard inclusions are called strict, and must be satisfied by all domain elements.
Let be a (finite) set of distinguished concepts . Inspired to Brewka’s framework of basic preference descriptions , we introduce a ranked TBox for each concept , describing the typical properties of -elements. Ranks (non-negative integers) are assigned to such inclusions; the ones with higher rank are considered more important than the ones with lower rank.
A ranked knowledge base over is a tuple , where is a set of standard concept and role inclusions, is an ABox and, for each , is a ranked TBox of defeasible inclusions, , where each is a typicality inclusion of the form , having rank , a non-negative integer.
Consider the ranked KB (with empty ), where contains
and where the defeasible inclusions are as follows:
The ranked Tbox can be used to define an ordering among domain elements comparing their typicality as horses. For instance, given two horses Spirit and Buddy, if Spirit has long mane, no saddle, has a tail and runs fast, it is intended to be more typical than Buddy, a horse running fast, with saddle and long mane, but without tail, as having a tail (rank 2) is a more important property for horses wrt having a saddle (rank 0).
In order to define an ordering for each concept , where means that is at least as typical as w.r.t. (in the example, and, actually, ), among the preference strategies considered by Brewka, we adopt strategy , which considers the number of formulas satisfied by a domain element for each different rank.
Given a ranked knowledge base , where for all , let us consider an interpretation satisfying all the strict inclusions in and assertions in . For each , to define a preference ordering on , we first need to determine when a domain element satisfies/violates a typicality inclusion for . We say that satisfies in , if or , while violates in , if and . Note that, any element which is not an instance of trivially satisfies all conditionals . For a domain element , let be the set of typicality inclusions in with rank which are satisfied by : .
Definition 3 ( )
Given a ranked knowledge base as above and an interpretation , the preference relation associated with in is defined as follows:
|or such that and, ,|
A strict preference relation and an equivalence relation can be defined as usual letting: iff ( and not ), and iff ( and .
Informally, gives higher preference to domain individuals violating less typicality inclusions with higher rank. Definition 3 exploits Brewka’s strategy in DL context. In particular, all , , i.e., all elements not belonging to are assigned the same rank, the least rank as they trivially satisfy all the typical properties of ’s. As, for a ranked knowledge base, the strategy defines a total preorder on worlds  and, for each , we have applied this strategy to the materializations of the typicality inclusions in the ranked TBox , the relation is a total preorder on the domain . Then, the strict preference relation is a strict modular partial order, i.e., an irreflexive, transitive and modular relation (where, by modularity: for all , if then or ); is an equivalence relation.
As has the finite model property , we can restrict our consideration to interpretations with a finite domain. In principle, we would like to consider, for each concept , all possible domain elements compatible with the inclusions in , and compare them according to relation. This leads to restrict the consideration to models of that we call canonical, in analogy with the canonical models of rational closure . For each concept occurring in , let us consider a new concept name , (representing the negation of ) such that . Let be the set of all such and , and let the set of all subsumptions . A set of concepts in is consistent with if .
Given a ranked knowledge base an interpretation is canonical for if satisfies and, for any set of concepts consistent with , there exists a domain element such that, for all , , if , and , if .
Existence of canonical interpretations is guaranteed for knowledge bases which are consistent under the preferential (or ranked) semantics for typicality. with typicality is indeed a fragment of the description logic with typicality, for which existence of canonical models of consistent knowledge bases is proved in .
In agreement with the preferential interpretations of typicality logics, we further require that, if there is some -element in a model, then there is at least one -element satisfying all typicality inclusions for (i.e., a prototypical -element).
An interpretation is -compliant for if, I satisfies and, for all such that , there is some such that satisfies all defeasible inclusions in .
In a canonical and -compliant interpretation for , for each , the relation on the domain provides a preferential interpretation for the typicality concept as , in which all typical satisfy all typicality inclusions in .
Existence of a -compliant canonical interpretation is not guaranteed for an arbitrary knowledge bases. For instance, a knowledge whose typicality inclusions conflict with strict ones (e.g, and ) has no -compliant interpretation. However, existence of -compliant interpretations is guaranteed for knowledge bases which are consistent under the preferential (or ranked) semantics for typicality (see Appendix, Proposition 16), and can be tested in polynomial time in Datalog .
For a ranked knowledge base , and a given interpretation , the strict modular partial order relations over , defined according to Definition 3 above, determine the relative typicality of domain elements w.r.t. each concept . Clearly, the different preference relations do not need to agree, as seen in the introduction.
4 Combining multiple preferences into a global preference
We are interested in defining a notion of typical -element, and defining an interpretation of , which works for all concepts and not only for the distinguished concepts in . This can be used to evaluate subsumptions of the form when does not belong to the set of distinguished concepts . We address this problem by introducing a notion of multipreference concept-wise interpretation, which generalizes the notion of preferential interpretation  by allowing multiple preference relations and, then, combining them in a single (global) preference.
Let us consider the following example:
Let be the ranked KB (with empty ), containting the strict inclusions:
The ranked TBox contains the defeasible inclusions:
the ranked TBox contains the defeasible inclusions:
and the ranked TBox contains the inclusions:
We might be interested to check whether typical Italian students are young or whether typical employed students are young. This would require the typicality inclusions and to be evaluated. Nothing should prevent Italian students from being young (irrelevance). Also, we expect that neither we can conclude that typical employed students are young nor that they are not, as the properties of typical students and of typical employees are conflicting concerning age. However, we would like to conclude that typical employed students have a boss and have classes, as typical employed students should inherit the properties of typical students and of typical employee which are not overridden (i.e., there is no blocking of inheritance). As PhD students are students, class is more specific than class , and PhD students should inherit all the typical properties of Students, except having no scholarship , which is overridden by ().
To evaluate conditionals for any concept we introduce a concept-wise multipreference interpretation, that combines the preference relations into a single (global) preference relation and interpreting as . The relation should be defined starting from the preference relations also considering specificity.
Let us consider the simplest notion of specificity among concepts, based on the subsumption hierarchy (one of the notions considered for ).
Definition 7 (Specificity)
Given a ranked knowledge base over the set of concepts , and given two concepts , is more specific than (written ) if and .
Relation is irreflexive and transitive (see ). Alternative notions of specificity can be used, based, for instance, on the rational closure ranking.
We are ready to define a notion of multipreference interpretation.
Definition 8 (concept-wise multipreference interpretation)
A (finite) concept-wise multipreference interpretation (or cw-interpretation) is a tuple such that: (a) is a non-empty domain;
for each , is an irreflexive, transitive, well-founded and modular relation over ;
the (global) preference relation is defined from as follows:
is an interpretation function, as defined in interpretations (see Section 2), with the addition that, for typicality concepts: , where and s.t. .
Notice that relation is defined from based on a modified Pareto condition: holds if there is at least a such that and, for all , either holds or, in case it does not, there is some more specific than such that (preference in this case overrides ). We can prove the following (the proof can be found in Appendix A).
Given a cw-interpretation , relation is an irreflexive, transitive and well-founded relation.
In a cw-interpretation we have assumed the ’s to be any irreflexive, transitive, modular and well-founded relations. In a cw-model of , the preference relations ’s will be defined from the ranked TBoxes ’s according to Definition 3.
Definition 10 (cw-model of )
Let be a ranked knowledge base over and an interpretation for . A concept-wise multipreference model (or cw-model) of is a cw-interpretation such that: for all , is defined from and , according to Definition 3; satisfies all strict inclusions inclusions in and all assertions in .
As the preferences ’s, defined according to Definition 3, are irreflexive, transitive, well-founded and modular relations over , a cw-model is indeed a cw-interpretation. By definition satisfies all strict inclusions and assertions in , but is not required to satisfy all typicality inclusions in , unlike in preferential typicality logics .
Consider, in fact, a situation in which typical birds are fliers and typical fliers are birds ( and ). In a cw-model two domain elements and , which are both birds and fliers, might be incomparable wrt. , as is more typical than as a bird, while is more typical than as a flier, even if one of them is minimal wrt. and the other is not. In this case, they will be both minimal wrt. . In preferential logics, we would conclude that , which is not the case under the cw-semantics. This implies that the notion of cw-entailment (defined below) is not stronger than preferential entailment. It is also not weaker, as it allows to reason about specificity, irrelevance and does not suffers from inheritance blocking.
The notion of cw-entailment exploits canonical and -compliant cw-models of . A cw-model is canonical (-compliant) for if the interpretation is canonical (-compliant) for .
Definition 11 (cw-entailment)
An inclusion is cw-entailed from if is satisfied in all canonical and -compliant cw-models of .
It can be proved that this notion of entailment satisfies the KLM postulates of a preferential consequence relation (see Appendix, Proposition 19).
5 Reasoning under the cw-multipreference semantics
In this section we consider the problem of checking cw-entailment of a typicality subsumption as a problem of determining preferred answer sets. Based on this formulation, that we prove to be sound and complete, we show that the problem is in . We exploit asprin  to compute preferred answer sets.
The idea is that, in principle, for checking we would need to consider all possible typical -elements in all possible canonical and -compliant cw-model of , and verify whether they are all instances of . However, we will prove that it is sufficient to consider, among all the (finite) cw-models of , the polynomial models that we can construct using the fragment of the materialization calculus for , by considering all alternative interpretations for a distinguished element , representing a prototypical -element. In preferred models, which minimize the violation of typicality inclusions by , it indeed represents a typical -element. An interesting result is that neither we need to consider all the possible interpretations for constants in the model nor to minimize violation of typicalities for them. Essentially, when evaluating the properties of typical employed students we are not concerned with the typicality (or atypicality) of other constants in the model (e.g., with typical cars, with typical birds, and with typical named individuals). Differently from the semantics in , which generalizes rational closure by allowing typicality concepts on the rhs of inclusions, we are not required to consider all possible alternative interpretation and ranks of individuals in the model. We will see, however, that we do not loose solutions (models) in this way.
In the following we first describe how answer sets of a base program, corresponding to cw-models of , are generated. Then we describe how preferred models can be selected, where represent a typical -element.
We will assume that assertions ( and ) are represented using nominals as inclusions (resp., and ), where a nominal is a concept containing a single element and . We also assume that the knowledge base is in normal form , where a typicality inclusion is in normal form when . Extending the results in , it can be proved that, given a KB, a semantically equivalent KB in normal form (over an extended signature) can be computed in linear time.
The base program for the (normalized) knowledge base and typicality subsumption is composed of three parts, .
is the representation of in Datalog according to , where, to keep a DL-like notation, we do not follow the convention where variable names start with uppercase; in particular, , , and , are intended as ASP constants corresponding to the same class/role names in . In this representation, , , are used for , , , and, for example, , are used for , . Additionally, is used for having rank , and the following definitions for distinguished concepts, typical properties, and valid ranks, will be used in defining preferences: For each distinguished concept , is included, where is an auxiliary individual name. Other auxiliary constants (one for each inclusion ) are needed, as in , to deal with existential rules.
contains the subset of the inference rules (1-29) for instance checking in  that is relevant for (reported in Appendix B), for example ; for , an additional rule is used: . Additionally, contains the version of the same rules for subclass checking (where represents , see again ), and then the following rule encodes specificity :
also contains the following rules:
Rule (a) generates alternative answer sets (corresponding to different interpretations) where may have the typical properties of the concepts it belongs. The constant , such that holds, represents a typical (a minimal element wrt. ) only in case it is an instance of (i.e., holds). Rule (b) establishes that, if there is an instance of concept in the interpretation, then must be an instance of (it models -compliance) and, by rule (c), is a typical instance of , i.e., it is minimal wrt. among -elements in the interpretation at hand. By rule (d), a typical instance of has all typical properties of . The rules (b)-(d) only allow to derive conclusions involving constants
contains (if necessary) normalized axioms defining in in terms of other concepts (e.g., definining as ) and the facts , ,
Given a normalized ranked knowledge base over the set of concepts , and a (normalized) subsumption , we can prove the following:
if there is an answer set of the ASP program , such that , then there is a -compliant cw-model for that falsifies the subsumption .
if there is a -compliant cw-model of that falsifies the subsumption , then there is an answer set of , such that .
We exploit the idea of identifying the minimal -elements in a canonical cw-model of , as the elements of the preferred answer sets of .
Let and be answer sets of . is preferred to if in (denoted as ) is globally preferred to in (denoted as ), that is, . defined according to Definition 8, point (c), provided that relations are defined according to Definition 3, by letting:
in , satisfies
where, in , satisfies if or ; and similarly for . The strict relation is defined accordingly.
Essentially, we compare and identifying the concepts of which is an instance in and in and evaluating which defaults are satisfied for in and in , using the same criteria used for comparing domain elements in Section 3.
The selection of preferred answer sets, the ones where is in , and then in , can be done in asprin with the following preference specification: where is defined by the preference program below. In such asprin programs, defining whether an answer set is preferred to according to preference is done by defining a predicate ; and are used to check whether the atoms in the preference specification hold in and , respectively. In the following, , correspond to , , , respectively, for and , comparing what for to what for it: