1 Introduction
The need for extending Description Logics (DLs) with nonmonotonic features has led, in the last decade, to the development of several extensions of DLs, obtained by combining them with the most wellknown formalisms for nonmonotonic reasoning [2, 49, 3, 18, 27, 22, 38, 12, 8, 14, 47, 7, 39, 13, 31, 6, 32] to deal with defeasible reasoning and inheritance, to allow for prototypical properties of concepts and to combine DLs with nonmonotonic rulebased languages under the answer set semantics [22], the wellfounded semantics [21], the MKNF semantics [47, 39], as well as in Datalog +/ [37]. Systems integrating Answer Set Programming (ASP) [24, 23] and DLs have been developed [50].
In this paper we study a rational extension of the logic , introduced by Krötzsch [41]. It is a lowcomplexity DL of the family [1] that includes local reflexivity, role conjunction and concept products and is at the basis of OWL 2 EL. The rational extension of is based on Kraus, Lehmann and Magidor (KLM) preferential semantics [40], and, specifically, on ranked models [43]. We call the logic and we define notions of rational and minimal entailment for it. Also, we develop a Datalog calculus for instance checking and subsumption under rational entailment and exploit it to construct the rational closure of a knowledge base using stratified negation.
The semantics of ranked interpretations for DLs was first studied in [12], where a rational extension of is developed allowing for defeasible concept inclusions of the form . In this work, following [28, 32], we extend the language of with typicality concepts of the form , whose instances are intended to be the typical elements. Typicality concepts can be used to express defeasible inclusions of the form (“the typical elements are ”). Here, however, as in [10, 26], we allow for typicality concepts to freely occur in concept inclusions. In this respect, the language with typicality that we consider is more general than the language in [32], where may only occur on the left hand side of inclusions as well as in assertions. For this language, we define a Datalog translation for which builds on the materialization calculus in [41], and, for reasoning about typicality, exploits the properties of ranked models, by suitably encoding the typicality operator and its properties. We show that instance checking for can be computed in polynomial time under the rational entailment. Following the approach developed in [41] for , the materialization calculus is used to define a Datalog calculus for subsumption in .
The rational closure of a knowledge base has been introduced by Lehmann and Magidor [43] to allow for stronger inferences with respect to preferential and rational entailment, and several constructions of rational closure have been proposed for [14, 16, 13, 32, 46]. Such constructions are defined for knowledge bases containing strict or defeasible inclusions and, for the construction in [32], which allows for defeasible inclusions of the form (where and are concepts), minimal canonical ranked models have been shown to provide a semantic characterization of rational closure for , thus generalizing to DLs the canonical model result in [43]. Based on the same construction introduced in [32] for , in this paper we use the Datalog calculus for plus stratified negation to compute in polynomial time the Rational Closure for simple KBs, where the typicality operator only occurs on the left hand side of inclusions.
However, not for all simple knowledge bases the rational closure is consistent. Indeed, the minimal canonical model semantics does not provide a general semantic characterization of the rational closure for with typicality, as a KB may have alternative minimal canonical models with incompatible rankings, or no canonical model at all. In particular, we show that instance checking in under the minimal canonical model semantics is hard. We observe that, even in cases when the KB has no minimal canonical model, the rational closure, when consistent, may still provide meaningful consequences.
A preliminary version of some results in the paper appeared in [34], and a version including the calculus for rational entailment in [35]. In this paper we also exploit the calculus for computing the rational closure of a (simple) KB in polynomial time. Furthermore, we provide a lower bound on the complexity of instance checking under the minimal canonical model semantics, thus strengthening the result in [34, 35].
2 A rational extension of
In this section we recall the logic introduced in [33, 35] extending the notion of concept in by adding typicality concepts. We refer to [41] for a detailed description of the syntax and semantics of . We let be a set of concept names, a set of role names and a set of individual names. A concept in is defined as follows:
where and . We introduce a notion of extended concept as follows:
where is a concept. Hence, any concept of is also an extended concept; a typicality concept is an extended concept and can occur in conjunctions and existential restrictions, but it cannot be nested.
A KB is a triple . contains a finite set of general concept inclusions (GCI) , where and are extended concepts; (as in [41]) contains a finite set of role inclusions of the form , generalized role inclusions of the form , role conjunction axioms and concept product axioms and , where and are extended concepts, and are role names in . contains individual assertions of the form and , where , and is an extended concept. Restrictions are imposed on the use of roles as in [41] (and, in particular, all the roles occurring in Self concepts and in role conjunctions must be simple roles, roughly speaking, roles which do not include the composition of other roles).
We define a semantics for based on ranked models [43]. As done in [32] for , we define the semantics of by adding to interpretations [41] a preference relation on the domain, which is intended to compare the “typicality” of domain elements. The typical instances of a concept , i.e., the instances of , are the instances of that are minimal with respect to . The properties of the relation are defined in agreement with the properties of the preference relation in Lehmann and Magidor’s ranked models [43]. A semantics for DLs with defeasible inclusions based on ranked models was first proposed in [12].
Definition 1
A interpretation is any structure where:

is a domain; is an interpretation function that maps each concept name to a set , each role name to a binary relation , and each individual name to an element ; the extension of to complex concepts is defined as usual:
; ; ;
= ;
= ;
= ; 
is an irreflexive, transitive, wellfounded and modular relation over ;

the interpretation of concept is defined as: , where and s.t. .
Furthermore, an irreflexive and transitive relation is wellfounded if, for all , for all , either or such that . It is modular if, for all , implies or . The wellfoundedness condition guarantees that if, for a nonextended concept , there is a element in , then there is a minimal element in (i.e., implies ).
In the following, we will refer to interpretations as ranked interpretations. Indeed, as in [43], modularity in preferential models can be equivalently defined by postulating the existence of a rank function , where is a totally ordered set. The preference relation can be defined from as follows: if and only if . Hence, in the following, we will assume that a rank function is always associated with any model . We also define the rank of a concept in the model as (if , then has no rank and we write ).
The semantics of the typicality operator defined above is the same as the one in [32] for . Similarly to other concept constructors, the typicality operator can be used in TBox and ABox with different restrictions, depending on the description logic. In [32], can only occur on the lefthand side of concept inclusions (namely, in typicality inclusions of the form ). Here, we call simple KBs the ones which respect this restriction, but, as in [10, 26], we also consider the general case with no restrictions on the occurrences of typicality concepts in concept inclusions and assertions. Instead, as in , we do not allow negation, union and universal restriction which are allowed in . Given an interpretation , the notions of satisfiability and entailment are defined as usual:
Definition 2 (Satisfiability and rational entailment)
An interpretation satisfies:
a concept inclusion if ;
a role inclusion if ;
a generalized role inclusion if
a role conjunction axiom if ;
a concept product axiom if ;
a concept product axiom if ;
an assertion if ;
an assertion if .
Given a KB , an interpretation satisfies (resp., , ) if satisfies all axioms in (resp., , ), and we write (resp., , ). An interpretation is a model of (and we write ) if satisfies all the axioms in , and .
Let a query be either a concept inclusion (where and are extended concepts) or an individual assertion. is rationally entailed by , written , if for all models of , satisfies . In particular, the instance checking problem (under rational entailment) is the problem of deciding whether an assertion (, or ) is rationally entailed by .
Given the correspondence of typicality inclusions with conditional assertions , it can be easily seen that each ranked interpretation satisfies the following semantic conditions, which are related to Lehmann and Magidor’s postulates of rational consequence relation [43] reformulated in terms of typicality:
It is easy to show that these semantic properties (where the interpretation of , which is not in the language, is the usual one) hold in all the ranked models. A similar formulation of the semantic properties in terms of defeasible inclusions has been previously provided by Britz et al. in [12], for the ranked semantics and a proof can be found in [46]. Another reformulation of the properties in terms of a selection function semantics was given in [28] for the preferential semantics and in [32] for the rational semantics. In particular, observe that property can be reformulated as:
if , then
and, in this form, it is a rephrasing of property , in the semantics with selection function of the operator studied in [32] (Appendix A) for . This property has a syntactic counterpart in in the axiom , where is the universal role (), which holds in all the ranked models. Observe that this axiom, as well as the property , is weaker than the Lehmann and Magidor’s postulate in [43], which would rather be reformulated, for a knowledge base , as:
and does not hold in (while it will hold under minimal entailment).
Consider the following example of knowledge base, stating that: typical Italians have black hair; typical students are young; they hate math, unless they are nerd (in which case they love math); all Mary’s friends are typical students. We also have the assertions stating that Mary is a student, that Mario is an Italian student, and is a friend of Mary, Luigi is a typical Italian student, Paul is a typical young student, and Tom is a typical nerd student.
Example 1
:
:
The fact that concepts can occur anywhere (apart from being nested in a operator) can be used, e.g., to state that typical working students inherit properties of typical students (), in a situation in which typical students and typical workers have conflicting properties (e.g., as regards paying taxes). Also, we could state that there are typical students who are Italian: .
Standard DL inferences hold for concepts and inclusions. For instance, we can conclude that Mario is a typical student (by (g)) and young (by (b)). However, by the properties of defeasible inclusions, Luigi, who is a typical Italian student, and Paul, who is a typical young student, both inherit the property of typical students of being math haters, respectively, by properties (SRM) and (SCM). Instead, as Tom is a typical nerd student, and typical nerd students are math lovers, this specific property of typical nerd students prevails over the less specific property of typical students of hating math. So we can consistently conclude that Tom is a .
A normal form for knowledge bases can be defined. A KB in is in normal form if it admits all the axioms of a KB in normal form:
(where , and ) and, in addition, it admits axioms of the form: and with . Extending the results in [1] and in [41], it is easy to see that, given a KB, a semantically equivalent KB in normal form (over an extended signature) can be computed in linear time. In essence, for each concept occurring in the KB, we introduce two new concept names, and . A new KB is obtained by replacing all the occurrences of with in all the inclusions and assertions, and adding the following additional inclusion axioms:
, , ,
Then the new KB undergoes the normal form transformation for [41]. The resulting KB is linear in the size of the original one.
Example 2
Considering the TBox in Example 1, inclusion
is transformed in the following set of inclusions:
Inclusion is mapped to the set of inclusions:
Then is transformed further (the normal form transformation for ) into: , and , while is split in two inclusions.
All the other axioms in the TBox, apart from (b) and (c), have to be transformed in normal form. Assertions are also subject to the normal form transformation. For instance, becomes , where is one of the concept names introduced above.
3 Minimal entailment
In Example 1, we cannot conclude that all typical young Italians have black hair (and in case Bob is a typical young Italian, he has black hair) using property (SRM) above, as we do not know whether there is some typical Italian who is also young. To support such a stronger nonmonotonic inference, a minimal model semantics is needed to select those interpretations where individuals are as typical as possible. Among models of a KB, we select the minimal ones according to the following preference relation over the set of ranked interpretations. An interpretation is preferred to () if: ; for all nonextended concepts ; for all , , and there exists such that . We say that an interpretation is a minimal model of if there is no model of such that .
We can see that in all the minimal models of the KB in Example 1, is an instance of the concept and the inclusion is satisfied, as nothing prevents a individual from having rank .
In particular, we consider the notion of minimal canonical model defined in [32] to capture rational closure of an KB extended with typicality. The requirement of a model to be canonical is used to guarantee that models contain enough individuals. Given a KB and a query , let be the set of all the (nonextended) concepts (and subconcepts) occurring in or together with their complements ( is finite). In the following, we assume that all concepts occurring in the query are included in .
Definition 3 (Canonical models)
A model of is canonical if, for each set of concepts consistent with (i.e., s.t. ), there exists (at least) a domain element such that .
Definition 4
is a minimal canonical model of if it is a canonical model of and it is minimal with respect to the preference relation .
Definition 5 (Minimal entailment)
Given a query , is minimally entailed by , written if, for all minimal canonical models of , satisfies .
We can show that instance checking in under minimal entailment is hard. The proof is based on a reduction of the minimal entailment problem of
positive disjunctive logic programs
, which has been proved to be a hard problem by Eiter and Gottlob in [19]. A similar reduction has been used to prove hardness of entailment for Circumscribed Left Local knowledge bases in [7], and in [33] to show that instance checking under the minimal model semantics (a different semantics) is a hard problem for knowledge bases. The reduction in [33] (Appendix B in the “Supplementary materials”) does not work for the minimal canonical model semantics, since the resulting knowledge base may have no canonical model, but a simplification of it does work.Let us recall the minimal entailment problem of positive disjunctive logic programs [19]. Let be a set of propositional variables. A clause is a formula , where each literal is either a propositional variable or its negation . A positive disjunctive logic program (PDLP) is a set of clauses , where each contains at least one positive literal. A truth valuation for is a set , containing the propositional variables which are true. A truth valuation is a model of if it satisfies all clauses in . For a literal , we write if and only if every minimal model (with respect to subset inclusion) of satisfies . The minimalentailment problem can be then defined as follows: given a PDLP and a literal , determine whether . In the following we sketch the reduction of the minimalentailment problem for a PDLP to the instance checking problem under minimal entailment, from a knowledge base constructed from .
We define a KB in as follows. We introduce a concept name for each variable (). Also, we introduce in an auxiliary concept , a concept name associated with the set of clauses , and a concept name associated with each clause in (). We let be an individual name, and we define as follows: , , and contains the following inclusions (where and are concepts associated with each literal occurring in , as defined below):
(1) (2) for all in
(3) for all in
(4) (5)
for each , , and where and (for ) are defined as follows:
where is the universal role.
Let us consider an arbitrary model of . Observe that, by (1), all the instances are instances. Hence, (being a typical ) must have rank greater than 0, and it will have rank 1 in all minimal canonical models. Inclusions (2) and (3) force the instances of to be the union of the instances of . Inclusions (4) and (5) force the instances of to be the intersection of the instances of .
The minimal canonical models of satisfying are intended to correspond to the (propositional) minimal interpretation satisfying . Roughly speaking, the concepts such that in correspond to the variables true in the interpretation satisfying . In any minimal canonical model of , either has rank 0 (and is not a typical ), or has rank 1 (and is a typical ). Also, a minimal model of in which the ranking of a set of ’s is 0, is preferred to the models in which the ranking of some of those ’s is higher (i.e., 1). This captures the subset inclusion minimality in the interpretations of the positive disjunctive logic program . The assertion in is required to select only those interpretations satisfying the set of disjunctions. Observe also that is an istance of for all (due to the Abox) but it may be a nontypical instance of .
In any minimal canonical model of : either or . Hence, for the two concepts in the definition of are disjoint and complementary, and is actually the concept representing the complement of .
Proposition 1
Given a set of clauses and a literal ,
if and only if
where is the KB associated with as above and is the concept associated with , i.e., if , and
if .
Proof
() We prove that if then . Let be a minimal canonical model of falsifying . We want to construct a (propositional) interpretation satisfying S and falsifying . Let be the set of all the variables such that is a typical element, i.e., . We show that is a minimal model of that falsifies . The fact that is a model of can be easily shown from the fact that iff , while iff i.e., iff . In fact when , must have rank lower than , i.e. rank . Hence, a literal is true in iff is satisfied in and it is false in if is satisfied in . From the facts that the concepts and are disjoint and complementary for , that inclusions (2)(5) are satisfied and that is satisfied as well, it follows that the interpretation satisfies all the clauses in .
Consider a clause in . As is satisfied in , must be satisfied as well, by (5). By (3) it is not the case that is satisfied in . There must be a for , such that is satisfied in . Hence, the literal must be satisfied in . Thus, is satisfied in . This holds for all the clauses in . Hence, satisfies .
In a similar way it can be seen that, as is falsified in , then the literal is falsified in . The minimality of can be proved by contradiction. If were not minimal, there would be a model of , with . First observe that, for any valuation we can define a concept obtained as the conjunction of the name concepts in the set and is consistent with . In fact, we can always add to a model of a new domain element with rank 1 satisfying as well as the inclusions (1)(5) (by properly defining the evaluation of the concepts ’s and of ), to obtain a new model of . In particular, in a canonical model, for each propositional valuation , there must be a domain element which is an instance of . For , let . There must be an element of which is an instance of and, furthermore, (if not, would not be satisfiable in for all , contradicting the fact that, by construction of , for each , is satisfiable in ). We can define a model such that , by only changing the rank of in to . We can easily see that is still a model of , a model in which, for all , is not satisfied, but in which inclusions (1)(5) and the ABox assertions are still satisfied (given that is a model of and that iff ). This contradicts the hypothesis that is minimal.
() We prove that if then .
Let be a (propositional) minimal interpretation satisfying and falsifying . We build a minimal canonical model of falsifying , as follows. Let be defined as follows (where and ):
; ;
; ; if and otherwise;
, for all ; ;
iff ; ;
iff ;
if then ;
if then ;
if then ;
Finally, for all such that , for each , we let iff some literal of is true in ; and, also, iff is an instance of all .
It turns out that, for all the variables , is an instance of while, for all the variables , is an instance of and all domain elements are instances of .
It is easy to see that is a model of , as it satisfies all the assertions and inclusions in . Also, is a canonical model of , as all the nonextended concepts occurring in have an instance in the model. To see that falsifies , we proceed by cases. Consider the case where and . In this case, and, by construction, , so that falsifies . Consider the case when and . It must be that and, by construction, . As there is no domain element with rank which is an instance of , . Hence, falsifies .
The minimality of among the canonical models of can be shown by contradiction. Suppose that is not minimal, then there is a canonical model of (with and ) such that . Hence there must be at least one concept which has rank 1 in and rank <