1 Introduction
Description logics (DLs) provide the logical foundation of formal ontologies of the OWL family.^{2}^{2}2https://www.w3.org/TR/owl2overview/ Among the various extensions proposed to enhance the representational capabilities of DLs, endowing them with nonmonotonic features is still a main issue, as documented by the past 20 years of technical development (see e.g. [8, 20, 27, 47] and references therein, and Section 6).
We recall that a typical problem that can be addressed using nonmonotonic formalisms is reasoning with ontologies in which some classes are exceptional w.r.t. some properties of their super classes.
Example 1.
We know that avian red blood cells, mammalian red blood cells, and hence also bovine red blood cells are vertebrate red blood cells, and that vertebrate red blood cells normally have a cell membrane. We also know that vertebrate red blood cells normally have a nucleus, but that mammalian red blood cells normally don’t. ∎
A classical formalisation of the ontology above would imply that mammalian red blood cells do not exist, since, being a subclass of vertebrate red blood cells, they would have a nucleus, but in the meantime, they are an atypical subclass that does not have a nucleus. Therefore, mammalian red blood cells would and would not have a nucleus at the same time. Unlike a classical approach, the use of a nonmonotonic formalism may allow us to deal with such exceptional classes.
Among the various proposals to inject nonmonotonicity into DLs, the preferential approach has recently gained attention [10, 12, 13, 14, 15, 16, 17, 22, 26, 28, 29, 30, 46] as it is based on one of the major frameworks for nonmonotonic reasoning in the propositional case, namely the KLM approach [37]. One of the main constructions in the preferential approach is Rational Closure (RC) [40]. RC has some interesting properties: the conclusions are intuitive, and the decision procedure can be reduced to a series of classical decision problems, sometimes preserving the computational complexity of the underlying classical decision problem.
In this paper, we address the concept subsumption decision problem under RC for nominal safe [35], a computationally tractable and practically important DL representative of the OWL 2 profile OWL 2 EL.^{3}^{3}3https://www.w3.org/TR/owl2profiles/ In fact, (i) nominal safe is the language [3], extended with the bottom concept (denoted ) and with the socalled ObjectHasValue construct (also denoted as in the DL literature), which is an existential quantification. Roughly, nominal safe is as , except for the fact that a nominal may only occur in concept expressions of the form , or in inclusion axioms of the form , stating that individual is an instance of concept (note that socalled role assertions can be also expressed via ); and (ii) in [35] it is shown that many OWL EL ontologies are nominal safe and that an reasoner is sufficient to decide the subsumption problem, decreasing the inference time significantly in practice.
In summary, our contributions are as follows.

We describe a subsumption decision procedure under RC for nominal safe that runs in polynomial time. Al feature of our approach is that our procedure relies entirely on a series of classical, monotonic subsumption tests, and, thus, any existing reasoner can be used as a black box to implement our method. Note that e.g., in [12] it is shown that the use of a DL reasoners as a black box for RC under is scalable in practice. We conjecture that this property holds with respect to RC under nominal safe as well.

We will also illustrate how to adapt our procedure to a relevant modification of RC for DLs, namely Defeasible Inheritancebased DLs [14]. We recall that Defeasible Inheritancebased DLs have been introduced to overcome some inference limitations of RC [13]: in fact, in [14, Appendix A] it is shown that Defeasible Inheritancebased DLs behave better than RC w.r.t. most of the “benchmark” examples illustrated there. A feature of our proposed procedure is that it runs in polynomial time and maintains the advantage of the previous point.
In the following, we will proceed as follows: for the sake of completeness, Section 2 introduces nominal safe and recaps salient notions about Rational Closure for [12, 13, 46]; Section 3 we describe a polynomial time procedure to decide subsumption under RC for defeasible ; in Section 4, we adapt our procedure for the Defeasible Inheritancebased . In Section 5 we address nominal safe and show how we polynomially reduce reasoning within it to defeasible , and thus inherit the computational complexity of reasoning of the subsumption decision problem from defeasible and its RC extensions. Section 6 discusses related work and Section 7 concludes and addresses future work.
2 Preliminaries
To make the paper selfcontained, in the following we briefly present the DLs , , and nominal safe , and an exponential time procedure to decide subsumption in the DL under RC via series of classical subsumption tests. The latter is important here as we will adapt it to decide the subsumption problem for nominal safe in polynomial time.
2.1 The DLs , , and nominal safe
is the DL with the addition of the empty concept [3]. It is a proper sublanguage of . Note that considering alone would not make sense in our case as ontologies are always conceptsatisfiable, while the notion of defeasible reasoning is built over a notion of conflict (see Example 1) which needs to be expressible in the language.
is extended with socalled nominal concepts (denoted with the letter in the DL literature, while nominal safe is with some restrictions on the occurrence of nominals.
Syntax.
The vocabulary is given by a set of atomic concepts , a set of atomic roles and a set of individuals . All these sets are assumed to be finite. concept expressions are built according to the following syntax:
An ontology (or TBox, or knowledge base) is a finite set of Generalised Concept Inclusion (GCI) axioms ( is subsumed by ), meaning that all the objects in the concept are also in the concept . We use the expression as shorthand for having both and .
The DL .
A concept of the form is called a nominal. is without nominals.
The DL nominal safe .
Nominal safe is with some restrictions on the occurrence of nominals and is defined as follows [35]. An concept is safe if has only occurrences of nominals in subconcepts of the form ; is negatively safe (in short, nsafe) if is either safe or a nominal. A GCI is safe if is nsafe and is safe. An ontology is nominal safe if all its GCIs are safe. It is worth noting that nominal safeness is a quite commonly used pattern of nominals in OWL EL ontologies, as reported in [35].
Semantics.
An interpretation is a pair , where is a nonempty set, called interpretation domain and is an interpretation function that

maps atomic concepts into a set ;

maps (resp. ) into a set (resp. );

maps roles into a set ;

maps individuals into an object .
The interpretation function is extended to complex concept expressions as follows:
An interpretation satisfies (is a model of) if , denoted . satisfies (is a model of) an ontology if it satisfies each axiom in it. An axiom is entailed by a if every model of is a model of , denoted as .
Remark 1.
Note that a socalled concept assertion : ( is an instance of concept ) and a role assertion : ( and are related via role ) can easily be represented in nominal safe via the mapping and . ∎
In the following, we recap here some salient facts related to nominal safe [35, Appendix A], which we will use once we present our entailment decision algorithm for nominal safe . Specifically, we can replace nominals in a nominal safe ontology with newly introduced concept names, yielding an ontology , such that supports the same entailments as [35]. Hence, an entailment decision procedure for suffices to decide entailment for nominal safe (but not for unrestricted ).
Consider an ontology . For each individual occurring in consider a new atomic concept . For an concept, GCI, or ontology, we define to be the result of replacing each occurrence of each nominal in with . The following proposition provides a sufficient condition to check entailment.
Proposition 1 ([35], Lemma 5 and Corollary 2).
Let be an ontology and an axiom that do not contain atomic concepts of the form . Then

if then ;

if for some then is not satisfiable.∎
The converse of Proposition 1 does not hold in general but holds for nominal safe .
Proposition 2 ([35], Theorem 4).
Let be a nominal safe ontology and a safe axiom that do not contain atomic concepts of the form . Then

if for all then is satisfiable;

if then .∎
Note that Proposition 2 fails if the use of nominals is not safe.
Example 2 ([35], Remark 2).
Consider
It is easily verified that is satisfiable and that . However, for
we have that is satisfiable, , but . ∎
2.2 Rational Closure in
We briefly recap RC for the DL (see, e.g. [12]), which in turn is based on its original formulation for Propositional Logic [40].
The DL is the DL extended with concept negation, i.e. concept expressions of the form and semantics . Note that by using the negation and the conjunction we can introduce also, e.g. the disjunction , i.e. is a macro for ), that is, it is interpreted as .
A defeasible GCI axiom is of the form , that is read as ‘Typically, an instance of is also an instance of ’. We extend ontologies with a , i.e. a finite set of defeasible GCIs and denote an ontology as , where is a TBox and is a DBox.
Example 3.
We can formalise the information in Example 1 in and, thus, in with the following ontology , with^{5}^{5}5The acronyms stand for: BRBC  Bovine Red Blood Cells; MRBC  Mammalian Red Blood Cells; ARBC  Avian Red Blood Cells; VRBC  Vertebrate Red Blood Cells; hasN  has Nucleus; hasCM  has Cell Membrane.
∎
Given a KB , RC satisfies some basic desiderata: the axioms in and are included into the set of the derivable axioms, that moreover is closed under the following properties.
Reflexivity (Ref), Left Logical Equivalence (LLE), Right Conjunction (And), Left Disjunction (Or), and Right Weakening (RW) are all properties that correspond to wellknown properties of the classical subsumption relation . Cautious Monotonicity (CM) and Rational Monotonicity (RM) are constrained forms of Monotonicity that are useful and desirable in modelling defeasible reasoning. (CM) guarantees that our inferences are cumulative, that is, whatever we can conclude about typical s (e.g. that they are in ), we can add such information to () and still derive all the information associated to typical s (). The stronger principle (RM) is necessary to model the principle of presumption of typicality, that is, if the typical elements of a class satisfy certain properties (e.g. ) and we are not informed that the typical elements of do not satisfy the properties of (), then we can assume that the typical elements of satisfy all the properties characterising the typical elements of (). We refer to [37, 45] for a deeper explanation of the meaning of such properties and why they are desirable for modelling defeasible reasoning.
RC is a form of inferential closure that satisfies all the properties above; it is based on the semantic notion of ranked interpretation and on the directly connected notion of ranked entailment, which we illustrate next.
Definition 1 (Ranked interpretation).
A ranked interpretation is a triple , where and are as in the classical DL interpretations, while is a modular preference relation over , that is, a strict partial order satisfying the following property:
 Modularity:

is modular if and only if there is a ranking function s.t. for every , iff . ∎
In the definition above, means that the object is considered more typical than the object . The order allows us to partition the domain of a ranked interpretation into a sequence of layers, where for every object , iff and iff .^{6}^{6}6Given a set and the order defined over , From this partition, we can define the height of an individual as
The lower the height, the more typical the individual in the interpretation is taken to be. We can also extend this to a level of typicality for concepts: the height of a concept in an interpretation , , as the lowest (most typical) layer in which the concept’s extension is nonempty: i.e.
If in a model there is no individual satisfying a concept , we set .
Definition 2 (Ranked model).
An interpretation satisfies (is a model of) (denoted ) iff , and satisfies (is a model of) iff (denoted ). satisfies (is a model of) iff for all axioms . ∎
Hence, is satisfied by iff all the most typical individuals in are also in . We say that two ontologies are rank equivalent iff they are satisfied by exactly the same ranked models, and that an ontology is rank satisfiable iff there is at least a ranked model that satisfies it.
Remark 2.
Note that from the above definition of the satisfiability of an axiom we obtain the following correspondence: for every ranked model ,
This allows for the translation of every classical axiom into a defeasible axiom . Note also that such a translation is not feasible in , as is not supported in . ∎
Now, the definition of ranked entailment follows directly from the notion of a ranked model. So, let be the class of the ranked models of an ontology .
Definition 3 (Ranked Entailment).
Given an ontology and a defeasible axiom , rationally entails (denoted ) iff . ∎
The main drawbacks of ranked entailment are that it is too weak from the inferential point of view and does not satisfy the (RM) property [12, 40]. RC is a kind of entailment that extends Ranked Entailment, allowing us to overcome these limitations. It is based on a notion of exceptionality that is built on Ranked Entailment.
Definition 4 (Exceptionality).
A concept is exceptional w.r.t. an ontology iff . That is to say, is exceptional w.r.t. iff, for every ranked model ,
An axiom is exceptional w.r.t. iff is exceptional. ∎
Intuitively, a concept is exceptional w.r.t. an ontology iff it is not possible to have it satisfied by any typical individual (i.e., an individual in the layer , that corresponds to ) in any ranked model of the ontology. Iteratively applied, the notion of exceptionality allows to associate to every concept a rank value w.r.t. an ontology in the following way (called RC ranking procedure).

A concept has rank () iff it is not exceptional w.r.t. (that is, for some model of ). In this case we set for every defeasible axiom having as antecedent. The set of the axioms in with rank is denoted as .

For , has rank iff it does not have rank and it is not exceptional wrt . If , then we set . The set of the axioms in with rank is denoted as .

By iterating the previous step a finite number of times, we finally reach a (possibly empty) subset s.t. all the axioms in are exceptional w.r.t. .^{7}^{7}7Since is finite, we must reach such a point. If we define the rank value of the axioms in as , and the set is denoted as .
As a consequence, according to the procedure above, can be partitioned into a finite sequence , (), where may be possibly empty. This semantic procedure allows us to give a rank value to every concept and every defeasible subsumption. Using the rank values, we can define the notion of RC as follows:
Definition 5 (Rational Closure).
is in the RC of an ontology iff
∎
Informally, the above definition says that is in the rational closure of if the most typical instances of happen to be all instances of , and not of .
Given an ontology , distinct ways of defining models of characterising its RC can be found in [12] (summarised also here in A and presented also in [46, Section 4.1]) and in [29]. Both such kinds of models can be described as minimal models of the ontology . Paraphrasing Definition 23 in [29], we can define a preference relation among ranked interpretations in the following way.
Definition 6 ([29], Definition 23).
Let and be two ranked interpretations s.t. and for every concept . is preferred to () iff for every , , and there is a s.t. . An interpretation is minimal w.r.t. an ontology if it is a model of and there is no model of s.t. .
The reason behind the use of minimal models in characterising RC is in the direct connection between minimality and the presumption of typicality: in minimal models we maximise the amount of typicality for each individual in the domain, modulo the satisfaction of the ontology. We will go back to the role of minimality in Section 5.
The type of reasoning we are primarily interested in modelling is subsumption checking in under RC, that is, deciding whether a defeasible subsumption is or is not a consequence under RC of an ontology . In [12] a detailed decision procedure for subsumption checking in under RC is described, which we recap here.^{8}^{8}8The procedure is based on the one by Casini and Straccia [13] and paired with a proper semantics; the latter needed to be modified slightly since it does not always give back the expected result in case . The procedure presented in [12] have been presented (and peerreviewed) also in [46]. This will be useful, as our subsumption decision procedure for defeasible will be a variant of it. The key step in translating the semantic procedure into a correspondent one, based on classical decision steps, is given by the following proposition.
Proposition 3.
For every concept and every ontology , if
(1) 
then is exceptional w.r.t. . ∎
By Proposition 3, checking exceptionality can be done by using a classical DL reasoner for . Now, consider an defeasible ontology and a defeasible GCI . In order to decide whether is in the RC of an ontology , we perform two steps: the first one is a ranking procedure, that transforms the initial ontology into a rank equivalent ontology , where is partitioned into a sequence , with each containing the defeasible axioms with rank ; the second one uses to decide whether an axiom is in the RC of .
Specifically, define the function that, given any ontology , returns exceptional axioms as
(2) 
The function gives back axioms in that are exceptional w.r.t. (see also [12, Section 6]).
Now in order to decide whether is in the RC of , we execute the following two steps shown below, which we will call RC.Step 1 and RC.Step 2. Note that RC.Step 1 will correspond to procedure , while RC.Step 2 is encoded in procedure , both presented in Section 3 later on. Also, the execution of RC.Step 1, i.e., procedure , can be followed e.g. in Example 4), which also illustrates why Steps 1.1 and 1.2 may need to be repeated more than once to extract all the needed information.
 RC.Step 1

Let and . Repeat Steps 1.1 and 1.2 until .
 Step 1.1

Given , construct the sequence
Since is finite, the iteration will terminate with (a possibly empty) fixedpoint of .
 Step 1.2

For a defeasible GCI , define the rank of and of concept as
Define
It follows that is partitioned into sets , for some , with possibly empty. Finally, define
RC.Step 1 terminates after building a sequence of ontologies and ranking functions , for some , once we reach the point where . Let partition into for some . Furthermore, let , , and for every .
Once we have applied RC.Step 1, Proposition 3 holds also in the opposite direction.
Proposition 4.
Given an ontology , obtained from the application of to an ontology , then for every concept ,
if and only if is exceptional w.r.t. . ∎
 RC.Step 2

So, let , , and be partitioned into .
 Step 2.1

For define the concept
Note that if then .
 Step 2.2

Finally, given , let be the first concept of the sequence such that . If there is no such let be . Then, we say that is derivable from iff .
is derivable from iff (see Remark 2).
With (resp. ) we will denote that (resp. ) is derivable from via RC.Step 2. In [12, 46] it is shown that RC.Step 1 is correct w.r.t. the semantic definition of ranking, and that RC.Step 2 is correct w.r.t. the semantic definition of RC (Definition 5). That is, respectively,
Proposition 5 ([12], Proposition 7).
Given an ontology and a concept , then holds. ∎
Proposition 6 ([12], Theorem 5).
Given an ontology , and concepts , then is in the RC of iff . ∎
Remark 3.
Note that an indispensable requirement of the above described defeasible subsumption procedure for under RC is to have a classical DL subsumption decision procedure supporting the empty concept, concept conjunction, negation and disjunction. ∎
From [12], the following propositions are immediate.
Proposition 7.
A classical GCI is in the RC of iff , where has been computed using RC.Step 1. ∎
Corollary 8 ([12], Corollary 2).
An ontology does not have a ranked model iff , where has been computed using RC.Step 1. ∎
3 Rational Closure in
We now present a subsumption decision procedure under RC for by adapting the procedure for under RC to . By Remark 3, as does not support concept negation and disjunction, the main problem we have to address is to find a way to overcome this limitation. Concretely, we will define alternative ways both to

express whether an concept is exceptional using a classical subsumption problem only;

express the subsumption problems in Steps 2.1 and 2.2 above in terms of subsumption problems only.
3.1 A Subsumption Decision Procedure for under RC
Consider a defeasible ontology . As for , we will define two procedures. The first one is a ranking procedure that transforms the initial ontology into a rank equivalent ontology , where is partitioned into a sequence , with each containing the defeasible axioms with rank . The second one uses to decide whether an axiom is in the RC of .
The Ranking Procedure.
Given an ontology , the ranking procedure is defined by means of two procedures: one for finding exceptional axioms and one for determining the rank value of axioms, as defined in Section 2.2.
In the following, given an ontology , and a new atomic concept (with indicating a set of defeasible subsumptions), we define as
(3) 
Informally, we introduce the atom as a way of representing the information that characterises the lowest rank. Hence, its introduction is aimed at the formalisation of the typicality of the lowest layer: is introduced to represent the individuals in that are in the lowest layer.
Remark 4.
The aim of the definition for is to replace the subsumption test in Proposition 3 with the subsumption test
for an ontology . ∎
We obtain an analogue of Proposition 3.
Proposition 9.
For every concept and every ontology , if
(4) 
where is a new atomic concept, then is exceptional w.r.t. . ∎
Procedure illustrates how to compute the exceptional axioms. algocf[h]
The procedure instead, shows how we implement RC.Step 1 in , which we comment shortly on next. algocf[h] We start by considering an ontology . Lines 810 loop until we reach a (possibly empty) fixedpoint of exceptional axioms. Then, each axiom in the fixed point of the exceptionality function is eliminated from (line 12) and we add to (line 13). We repeat the loop in lines 5  13 until no exceptional axioms can be found anymore (i.e., , for some ).
Remark 5.
Note that the loop in between lines 5  13 allows us to move all the strict knowledge possibly ‘hidden’ inside the DBox to the TBox. That is, there may be defeasible axioms in the DBox that are actually equivalent to classical axioms, and, thus, can be moved from the DBox to the TBox as classical inclusion axioms. Example 4 illustrates such a case. ∎
Lines 1517 determine the rank value of the remaining defeasible axioms not in . That is, set is the set of axioms of rank (), which are the axioms in .
The following can easily be shown.
Proposition 10.
Consider an ontology . Then returns the ontology , where is partitioned into a sequence , where , and all are equal to the sets , and obtained via RC.Step 1. ∎
Also, once we have applied the procedure , the proposition corresponding in the framework to Proposition 4 holds.
Proposition 11.
Given an ontology , obtained from the application of the procedure to an ontology , for every concept ,
if and only if is exceptional w.r.t. . ∎
Next, we describe some examples that illustrate the behaviour of the ranking procedure. The following example shows a case in which there is nondefeasible knowledge ‘hidden’ in a DBox and that more than one cycle of the lines 414 in is needed to extract this information.
Example 4.
Let be an ontology with
It can be verified that the execution of
Comments
There are no comments yet.