Rational closure for all description logics

Artificial Intelligence 274 (2019) 197–223

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Artificial Intelligence www.elsevier.com/locate/artint

Rational closure for all description logics P.A. Bonatti Università di Napoli Federico II, Italy

a r t i c l e

i n f o

Article history: Received 11 June 2018 Received in revised form 23 January 2019 Accepted 8 April 2019 Available online 11 April 2019 Keywords: Rational closure Stable rankings Disjoint union model property

a b s t r a c t Rational closure is one of the most extensively studied nonmonotonic extensions of description logics. Nonetheless, so far it has been investigated only for description logics that satisfy the disjoint model union property, or limited fragments that support nominals. In this paper we show that for sufficiently expressive description logics, the traditional correspondence between rational closure and ranked interpretations does not hold. Therefore, in order to extend rational closure to a wider class of description logics it is necessary to change the definition of rational closure, or alternatively abandon its standard semantics. Here we pursue the former approach, and introduce stable rational closure, based on stable rankings. The resulting nonmonotonic logic is a natural extension of the standard rational closure: First, its refined exceptionality criterion yields a closure that satisfies the KLM postulates. Second, when a knowledge base enjoys the disjoint model union property, then stable rational closure equals the old notion. In the other cases, stable rankings may raise the exceptionality level of some concepts. Stable rational closure has a model-theoretic semantics based on upward-closed models, that relax the canonical models adopted in the past, in order to deal with logics that do not satisfy the disjoint union model property. Unfortunately, stable rankings do not always exist, and are not necessarily unique. However, they can be effectively enumerated for all defeasible knowledge bases in SROIQ, using any algorithm for reasoning with ranked models. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Description logics (DLs) correspond – up to superficial syntactic differences – to a class of decidable fragments of firstorder logic. As such, their inferences are monotonic. There is recurring evidence in the literature that non-monotonic inferences, especially inheritance with exception and overriding, would greatly help in modeling biomedical knowledge, policies, and other important application domains for DLs [30,31,4]. Consequently, many nonmonotonic extensions of DLs have been proposed to address these needs, for example [1,2,15,27,5,13,22,23,4]. Despite all this work, spanning across many decades, nonmonotonic inferences are not yet supported by standards like OWL2 and its reasoning engines. Some reasons for this lack of support, related to expressiveness limitations and computational complexity, have been analyzed in [4,6]. Here we add to that list the lack of a common shared view on semantics, that hinders the standardization of nonmonotonic inferences. While OWL2 and its profiles all rely on the firm ground of classical logic and its Tarskian semantics, nonmonotonic logics are based on a wide range of semantic ideas: default logic, circumscription, autoepistemic logic, rational closure and the other semantics are all based on different interpretation structures, and on different definitions of which rational-but-not-classically-valid inferences a logic should support.

E-mail address: [email protected]. https://doi.org/10.1016/j.artint.2019.04.001 0004-3702/© 2019 Elsevier B.V. All rights reserved.

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P.A. Bonatti / Artificial Intelligence 274 (2019) 197–223

Many years ago, Kraus, Lehmann and Magidor tried to single out, in a propositional framework, a common set of desiderata for the logics that model what normally holds [25,26]. The result was a set of axioms called KLM postulates, equipped with a matching preferential semantics. The KLM postulates and their preferential semantics (suitably adapted to the first-order setting of DLs) are now taken by some authors as a must that every nonmonotonic DL should satisfy. Others see the appeal of the unification effort and the elegance of the postulates, but take a broader view, that takes into account not only the KLM postulates and the axiomatization of normality, but also the knowledge representation needs arising from applications, as in [4,6]. The author believes that lack of adoption shall be tackled with such broader views, and a dispassionate analysis of all the desiderata and their mutual conflicts. As part of this meta-level analysis, in this paper we study how general the KLM approach can be. We focus on the rational closure of DLs [8,23], one of the main nonmonotonic semantics that satisfy the internalized, first-order version of the KLM postulates. A logic can hope to become the basis for a nonmonotonic extension of OWL2 only if it covers all of OWL2’s constructs. However, as of today, the model theoretic semantics of the rational closure of DLs is only defined for fragments of OWL2 that enjoy the disjoint model union property (DMUP for short),1 with the partial exception of [16,14] that in some restricted cases support nominals. There is an immediate explanation for this limitation. In order to capture faithfully the inferences of rational closure, models are “saturated” so that every combination of concepts consistent with the knowledge base K is simultaneously satisfied. The models that satisfy this saturation condition are called canonical [23].2 Clearly, the saturation condition can always be fulfilled if the underlying classical logic satisfies the DMUP. When it doesn’t, there is no guarantee that a canonical model exists, even if K is consistent. Example 1.1. Consider K = {  (∀U .¬ A )  (∀U .¬ B )}, where U is the universal role, i.e. a binary relation that contains all pairs of individuals. The above axiom says that in each model of K either A is empty or B is empty. Of course when A is empty, B can be nonempty and viceversa. So both A and B are consistent with K, but no single model can satisfy them both. On the contrary, a canonical model is required, by definition, to contain both a normal instance of A and a normal instance of B. Therefore, K has no canonical model, although K is satisfiable. 2 We will address this issue by relaxing the saturation condition to what we call upward closure. Informally speaking, upward closure requires that, if an instance x of C is not as normal as C can allow (given K), then C must contain also an instance y more normal than x. Note that this condition is vacuously satisfied when C is empty, so, with reference to the above example, upward closure allows A and B to be empty (provided that when they are nonempty they contain a maximally normal instance). However, when we tried to prove that upward-closed models constitute a faithful semantic characterization of rational closure, we realized that something is wrong in the way rational closure has been extended to DLs. Since its introduction, rational closure has been associated to so-called ranked interpretations, that characterize rational consequence relations. However, this correspondence does not hold for description logics that do not enjoy the DMUP. The problem we found (illustrated in Section 4) has two possible explanations: 1. the ranked interpretation structures adopted so far are not adequate; 2. the definition of exceptionality ranking, on which rational closure is based, is not adequate. In this paper we explore the second alternative and study modified definitions of the rational closure of DLs, while preserving the ranked interpretations adopted so far (which satisfy all KLM postulates). In particular we replace the iterative definition of ranking with one based on a notion of stability, and study the properties of the rankings that satisfy it. The main results are the following: 1. upward-closed models constitute a semantics for the new rational closure based on stable rankings, for all description logics, including those that do not satisfy the DMUP; 2. we provide an axiomatic characterization of upward-closed models, that paves the way to efficient implementations by reducing reasoning with upward-closed models to classical DL reasoning; 3. stable rankings collapse to the classical ranking of rational closure when the underlying logic satisfies the DMUP (i.e. the new rational closure is a natural generalization of the old notion); 4. when the DMUP does not hold, stable ranking may give higher rank to some concepts; 5. unfortunately, when the DMUP does not hold, stable rankings do not always exist, and there may be multiple, incomparable rankings (so, for some knowledge bases, there is no unique notion of rational closure).3

1 A knowledge base K has the disjoint model union property if the union of two disjoint models of K is still a model of K. In general, this property is broken by constructs such as nominals, assertions about individuals (ABoxes), and the universal role. 2 In [16], the same saturation condition is relaxed by applying it only to the concepts that occur in the scope of the typicality operator T. However this measure does not always remove the problem illustrated below. 3 A similar problem affects the rational closure of ABoxes introduced in previous works, as discussed later.

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Table 1 Glossary. Rational closure

Stable closure Defeasible Inclusions, i.e. expressions of the form C < D

DI

∼

dfs(K)

the set of DIs in K

str(K)

the set of strong axioms of K (i.e. K \ dfs(K))

 K rnk

the materialization of K’s DIs r rK (I , x)

concept/DI rankings the rank of an individual x in I (based on the DIs satisfied by x)

Ei

K|ri

the DIs of K with rank  i

RC(K)

RCr

the rational closure of K (w.r.t. ranking r)

DMUP

Disjoint Model Union Property

|=

entailment w.r.t. all ranked models r

|=

entailment w.r.t. upward-closed models only

D

the set of DIs of interest (those occurring in K, and the queries)

C

the set relevant concepts (based on D)

The paper is organized as follows: In the next section we recall the most closely relevant works on the rational closure of description logics. Then Section 3 summarizes the basics of defeasible description logics and their rational closure. In Section 4 we illustrate the problems that arise when the DMUP does not hold. The rest of the paper elaborates on our alternative notion of rational closure. Section 5 introduces stable rankings, stable rational closure, and its model-theoretic semantics, based on upward-closed models. In Section 6 we introduce the axiomatic characterization of upward-closed models and discuss its applications. Section 7 illustrates the relationships between standard rational closure and the stable version. Section 8 discusses the conditions under which stable rankings exist and are unique, and proves decidability results for both properties, under the assumption that the knowledge base is in the defeasible extension of SROIQ. This section provides also a preliminary complexity analysis, comprising some upper complexity bounds. Moreover, we briefly discuss the skeptical and credulous types of reasoning that naturally arise from the multiplicity of stable rankings. Then, in Section 9 we illustrate an elegant method for defeasible ABox reasoning and instance checking based on nominals, enabled by the support of logics that do not enjoy the DMUP. The paper is concluded by a final discussion of our results. In order to enhance readability, most proofs have been moved to the appendix. For the same purpose, we summarize in Table 1 the notation adopted in the paper (whose formal definition will be given in the following sections). 2. Related work There is a vast literature on nonmonotonic Description Logics. Here we focus on the main versions of the rational closure of DLs and refer the reader to [4,6] for a wider list of logics and their mutual relationships. Two related approaches to the rational closure of the DL ALC have been introduced in [12,13]. The latter extends the former with a pre-processing based on inheritance networks, in order to remove a well-known limitation called inheritance blocking in [4,6] and all-or-nothing in [16]. The aforementioned approaches have a glitch in the treatment of strong axioms, pointed out in [4], that has been solved in [8]. These approaches extend DLs with defeasible inclusions, that are expressions C< ∼ D, formally introduced in the next section, whose informal meaning is: “normally, the instances of C are instances D”. The set of defeasible inclusions inferable under the above logics constitutes a rational consequence relation and is closed under the KLM postulates. The above logics are not able to restrict role ranges to normal individuals. Some recent developments focus on overcoming these limits, and on making the notion of normality dependent on context [10,9]. In particular, the logic dSROIQ extends SROIQ (the DL that provides the foundations for the standard OWL2) with several constructs that depend on normality relations over pairs of individuals (i.e. the usual preferential semantics is extended to roles). However, this framework is currently monotonic, and SROIQ does not enjoy the DMUP. The stable rankings introduced in the following and their semantics based on upward-closed models may constitute a basis for the rational closure of dSROIQ . Another work on modeling nonmonotonic quantification is [29]. This paper introduces a nonmonotonic semantics and corresponding reasoning methods for variants of the rational and the relevant closures of EL⊥ that apply defeasible properties to role fillers. A syntactically different approach, based on concepts TC that denote the typical instances of concept C , has been introduced in [20]. In this language, defeasible inclusions are expressed with inclusions of the form TC  D, whose informal reading is: “typical C are D”. Various semantics have been provided for this language; rational closure is dealt with in [23]. That paper lays out a model theoretic semantics for the rational closure of ALC , based on minimal canonical ranked models (illustrated in the next section), that has influenced also the latest semantics of defeasible inclusions. This framework has

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P.A. Bonatti / Artificial Intelligence 274 (2019) 197–223 Name

Syntax

Semantics

inverse role

R−

{(d, e) | (e, d) ∈ R I }

universal role

U

I × I

top



I

bottom

⊥

∅

nominal

{a}

{aI }

negation

¬C

I \ C I

conjunction

C D

C I ∩ DI

disjunction

CD

C I ∪ DI

∃ restriction

∃ R .C

{d ∈ I | ∃e ∈ I .[(d, e) ∈ R I ∧ e ∈ C I ]}

∀ restriction

∀ R .C

{d ∈ I | ∀e ∈ I .[(d, e) ∈ R I → e ∈ C I ]}

Fig. 1. Syntax and semantics of the constructs used in the examples.

been extended to SHIQ in [19]. Other recent papers focus on overcoming the limitations of the above semantics due to the all-or-nothing property, and the context-independent nature of normality, e.g. [18]. In general, none of the above semantics can be extended to more expressive logics, as discussed later in Section 3.2. The problem is that canonical models (and their analogues, such as R K ∪ , illustrated in Section 3.2) do not necessarily exist when the underlying DL does not satisfy the DMUP. To the best of our knowledge, only two approaches partially address this issue, and apply to knowledge bases that do not necessarily enjoy the DMUP (because they may contain nominals):

• the rational closure of ELO ⊥ , under the restriction that nominals may occur only in expressions of the form ∃ R .{a} and {a}  C [14]; • the rational closure of SROEL, whose semantics restricts the saturation condition of canonical models to the concepts that occur in T (however, this does not always guarantee that a relaxed canonical model exists) [16,17]. In the following, we are going to approach in a more general way the rational closure of logics that do not enjoy the DMUP. 3. Preliminaries Here we briefly recall the basics of defeasible description logics and their rational closure, focussing on the details that are directly needed in this work. The reader is referred to [3] for a detailed description of DLs. 3.1. Defeasible description logics The syntax of description logics is based on a vocabulary of symbols consisting of a set NC of concept names, a set NR of role names, and a set NI of individual names (all countably infinite). In expressive DLs, such atomic expressions can be combined by means of a large number of constructors to obtain compound expressions; Fig. 1 illustrates those that will be used throughout the examples of this paper.4 The upper part of the table describes inverse roles and the universal role U (that is supported by SROIQ, i.e. the DL that provides the foundation of the OWL2 standard). The lower part of the table describes compound concepts. In the following, metavariables C , D will range over (possibly compound) concept expressions, metavariables R , S will range over role expressions, and metavariables a, b will denote individual names. The semantics of defeasible DLs is defined in terms of (ranked) interpretations I = (I , ·I ,
4 Note, however, that our framework applies also to richer DLs supporting all of OWL2 constructs, fixpoint operators, and all of the other operators not occurring in Fig. 1. Our results hold for all DLs, with the exception of decidability results, that clearly require the monotonic fragment of the given defeasible DL to be decidable as well. 5 Some authors use an equivalent definition based on well-founded weak – or modular – orders.

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several role property assertions such as transitivity, functionality etc. The set of assertions of the form C (a) and R (a, b) contained in a knowledge base constitute its ABox (assertion box), while concept inclusions belong to the so-called TBox (terminological box). All of the above axioms (i.e. those supported by classical, monotonic description logics) will be called strong or classical axioms in the following. Additionally, defeasible DLs support defeasible inclusions (DI, for short) that are expressions C < ∼D whose intended meaning is: “the instances of C are normally instances of D”. A defeasible knowledge base (DKB, for short) is a finite set of axioms. Let the sets of defeasible and strong axioms of a DKB K be denoted respectively by < dfs(K) = {C < ∼ D | (C ∼ D ) ∈ K} ,

str(K) = K \ dfs(K) . We say that a DKB K is in a description logic DL (such as ALC or SROIQ, for instance) if its strong axioms and all the concepts occurring in dfs(K) belong to DL. Concerning axiom semantics, a ranked interpretation I satisfies (i) a concept inclusion C  D if C I ⊆ D I , (ii) an assertion I I C (a) if aI ∈ C I , and (iii) a DI C < ∼ D if min (C ) ⊆ D , where

minI (C ) = {x ∈ C I | ∀ y ∈ C I , y ≮I x} . Informally speaking, minI (C ) represents the set of maximally normal instances of C . If a ranked interpretation I satisfies an axiom α , then we write I |= α , and say that I is a model of α . We say that I satisfies a DKB K (in symbols, I |= K) if I satisfies all the axioms in K. In that case, we say that I is a model of K. We say that α is entailed by K if α is satisfied by all models of K; then we write K |= α . Some DLs enjoy a nice technical property called disjoint model union property (DMUP). A description logic DL has the DMUP iff all the knowledge bases in DL have the DMUP. In turn, a DKB K has the DMUP if, for all disjoint models6 I and J of K, the DKB K is satisfied also by their union I  J , defined by:

I J = A I J = R I J = hI J =
I ∪ J ; AI ∪ AJ ; RI ∪ RJ ; hI ∪ hJ ; {(x, y ) | hI J (x) < hI J ( y )}

( A ∈ NC ) ( R ∈ NR )

(where hI and hJ are the functions associated to
(C 1  D 1 ) ∨ (C 2  D 2 ) ⇔   ∀U .(¬C 1  D 1 )  ∀U .(¬C 2  D 2 ) ¬(C  D ) ⇔   ∃U .(C ¬ D ) . Thus, using U , it is possible to assert that ( A  ⊥) ∨ ( B  ⊥). Now, if I and J are disjoint models of this disjunction where, respectively, A and B are nonempty, then their union I  J is not a model of the disjunction, that – consequently – does not enjoy the DMUP. Note the analogy with Example 1.1. We will use similar examples a few more times along the paper. On the positive side, it is known that expressive monotonic DLs without nominals nor U , such as SHIQ, enjoy the DMUP. And if a classical DL has the DMUP, then its defeasible extension has the DMUP, too. This can be formalized as follows: Proposition 3.1. If str(K) has the DMUP, then K has the DMUP, too.7 Proof. Let I and J be disjoint models of K and let U = I  J . By hypothesis, U satisfies str(K). We are left to prove that U I J U satisfies an arbitrary (C < ∼ D ) ∈ K. The reader may easily verify that min (C ) ⊆ min (C ) ∪ min (C ). Moreover, since both I J U I J I J U I and J satisfy C < ∼ D by assumption, min (C ) ⊆ D and min (C ) ⊆ D hold. It follows that min (C ) ⊆ D ∪ D = D , that is, U satisfies C < ∼ D. 2

6 7

Two interpretations I and J are disjoint if I ∩ J = ∅. A similar result, formulated in a less general form, is proved in [7, Theorem 4].

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The semantics of the rational closure of DLs introduced so far implicitly rely on the DMUP. In the main part of this paper we will discuss how to deal with defeasible DLs that do not enjoy the DMUP. 3.2. The rational closure of description logics The logic introduced in the previous section is monotonic, that is, if K |= α and K ⊆ K , then also K |= α . So we have not yet addressed the needs for nonmonotonic inference reported in Section 1. Rational closure provides a nonmonotonic strengthening of |=. Here we briefly recall the basic definitions developed in [8,23]8 and refer the reader to these papers for a deeper introduction to rational closure. Rational closure is based on a notion of exceptionality of concepts and DIs. Roughly speaking a concept is exceptional if it cannot possibly contain any of the most normal individuals. The exceptionality of DIs corresponds to the exceptionality of their left-hand side. Formally: < Definition 3.2 (Exceptionality, E (·)). A concept C is exceptional in a defeasible knowledge base K iff K |=  < ∼ ¬C . A DI C ∼ D is exceptional in K iff C is exceptional in K. The set of DIs of K that are exceptional in K is denoted by E (K), that is: < < E (K) = {C < ∼ D | (C ∼ D ) ∈ K ∧ K |=  ∼ ¬C } .

In the rest of this section, let K be an arbitrary but fixed defeasible knowledge base. This helps in keeping notation simpler, since most of the notions used henceforth should have K as a parameter. The above notion of exceptionality can be used to assign concepts and DIs to an exceptionality ranking, in the following way. The sets of exceptional concepts and DIs can be further partitioned by re-evaluating exceptionality with respect to E (K) only, and so on. The resulting hierarchy is formalized through the sequence Ei i defined below. Definition 3.3 (Ei , rnk). Define the non-increasing sequence of defeasible knowledge bases E0 , E1 , . . . , Ei , . . . as follows:

E0 = K , Ei = E (Ei −1 ) ∪ str(K)

( i > 0) .

The rank of a concept C , denoted by rnk(C ), is the minimal integer i such that C is not exceptional in Ei . If no such i exists then we write rnk(C ) = ∞. Note that since K is finite, then for some integer i, Ei = Ei +n for all n > 0. We denote such Ei with E∞ . Clearly, rnk(C ) = ∞ iff C ∈ E (E∞ ). Rational closure is based on the ranking rnk. Informally speaking, C < ∼ D must be inferred whenever C is less exceptional than C ¬ D, that is, rnk(C ) < rnk(C ¬ D ). Moreover, since rnk(C ) = ∞ implies that K |= C  ⊥ (a well-known result, that here we recall in Lemma A.2), the complete definition is the following: Definition 3.4 (Rational closure of K).

RC(K) = {C < ∼ D | rnk(C ) < rnk(C ¬ D ) ∨ rnk(C ) = ∞} . The set RC(K) is the set of DIs that can be inferred from K. RC is not monotonic in K, and is closed under (an object-level, or internalized, version of) the KLM postulates (originally formulated for consequence relations [25,26]). The DL version of the postulates is illustrated in Fig. 2; for each postulate, if the premises are in RC(K), then also the conclusion is. The postulates are also satisfied by all ranked interpretations I , in the sense that if the premise is satisfied by I , then I satisfies also the conclusion. The connection between ranked interpretations and the consequence relations that satisfy the KLM postulates is tight: every such relation R is the set of defeasible inclusions that are satisfied by some ranked interpretation I (we say that R corresponds to I ). This result holds in the original propositional framework, as well as for the DLs that satisfy the DMUP (but we are going to prove that it does not always extend to the DLs that do not enjoy the DMUP). Given the above discussion, it is not surprising that the two main semantics for RC are based on selected subclasses of the ranked models of K [7,23]. Both semantics rely on the DMUP, and have been applied only to fragments of SHIQ , with the partial exception of ELO and SROEL mentioned in Section 2. The semantics introduced in [7] is based on the disjoint union R K ∪ of all the models of K with a fixed countable domain (suitably “renamed” to obtain disjointness). Here we report the semantics introduced in [23], which is easier to relax to address the problem illustrated in Example 1.1.

8

The latter paper applies to ALC the same semantics initially introduced in [21] for Lehmann’s rational closure.

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(REF)

(AND)

(OR)

(RM)

C< ∼C C< ∼D

C< ∼E

C< ∼D E C< ∼E

D< ∼E

CD< ∼E C< ∼E

(LLE)

(RW)

(CM)

C< ∼D E< ∼D C< ∼D C< ∼E C< ∼D

203

(|= C ≡ E )

(|= D  E ) C< ∼E

C D< ∼E

C< / ¬D ∼

C D< ∼E Fig. 2. The KLM postulates for DL.

The semantics of [23] is based on so-called canonical models. They are defined in terms of a set of relevant concepts C that contains all the concepts occurring in K and in the queries of interest, plus their negations. A canonical model I shall contain an individual x ∈ (C 1 . . . C n )I for all K-consistent sets of concepts C 1 , . . . , C n contained in C. This “saturation” property can always be achieved in SHIQ, and it could be extended in a straightforward way to all DLs that enjoy the DMUP. It has been proved that a DI belongs to RC(K) iff it is satisfied by all the canonical models that are minimal with respect to a suitable model preference relation. Neither R K ∪ nor canonical models are guaranteed to exist when the DMUP does not hold. However, this is probably not simply a problem of the chosen semantics, as discussed in the next section. 4. Analysis of the RC for description logics In this section we illustrate a problem that arises when the DMUP does not hold. For this purpose, we exploit the expressiveness of the universal role U . Example 4.1 (The problem of axiom forgetting). Consider the following K:

δ1 : A 1 < ∼B

δ2 : A 2 < ∼ A 1 ∀U .(¬ A 1  ¬ B ) . I 9 < < There are models I of K where A I 1 = ∅ and  ∼ A 1 . On the contrary, K |=  ∼ ¬ A 2 (because if A 2 = ∅, then I  δ1 ). Consequently, A 1 has rank 0, while the rank of A 2 must be established by checking whether K \ {δ1 } |=  < ∼ ¬ A 2 . This I I I I entailment does not hold (e.g. let A 1 = A 2 = (¬ B ) =  ). So in this example A 2 has rank 1, and E∞ = ∅. Then A 2 < ∼ ⊥ does not belong to the rational closure of K. However, in all ranked models of K, A 2 is empty (otherwise δ1 could not be possibly satisfied). It follows that K |= A 2 < ∼ ⊥ and that the correct rank of A 2 should be ∞. This is not detected by rational closure because, after the first step, the rank is evaluated using only a subset of K (i.e. Ei −1 ). In particular, in this example, the rank of A 2 is evaluated after discarding δ1 . Consequently, the effects of the DIs with lower rank on higher ranks are systematically disregarded. 2

The above example shows that the ranking induced by the sequence Ei i does not always correspond to a ranked interpretation, so the standard semantic characterization of rational consequence relations does not completely extend to description logics. There are two possible interpretations of this example. 1. Ranked interpretation structures are not adequate. With more structured (e.g. multimodal) structures it may be possible to prevent the interference of axioms across different ranking levels by relegating them into different modal contexts. 2. The definition of ranking is not adequate. The problem is all in the fact that exceptionality is evaluated with respect to Ei −1 rather than all of K, so some axioms are erroneously disregarded. In this paper, we explore the second interpretation, and study an alternative, “correct” definition of rational closure. Before illustrating our revised version of rational closure, we briefly illustrate the sequence of dead-ends that led to its formulation. We first need the following notion:

9

I I E.g. let A I and A I 1 =B = 2 = ∅.

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Definition 4.2. The materialization of the DIs of a DKB K is

= K

{¬

C  D | (C < ∼ D ) ∈ K} .

 = . If dfs(K) = ∅ then we stipulate that K One may be tempted to address the problem of axiom forgetting by replacing E (·) with the following E  (·):

 < E  (Ei −1 ) = {C < ∼ D | K |= Ei −1 ∼ ¬C } . This version of the exceptionality check takes all of K into account (so it does not forget any axiom). However, the inclusion  Ei −1 < ∼ ¬C does not give us any information about the most normal instances of Ei −1 alone. It turns out that, since Ei −1 is < a subset of K, E  (Ei −1 ) is insensitive to its input (it is always equivalent to the set of DIs C < ∼ D such that K |=  ∼ ¬C , see Lemma A.1.(2) in the appendix). As a consequence, the ranking sequence would always reach its fixpoint in one step. A  better approach consists in replacing < ∼ with , as in the following function E (·):

 E  (Ei −1 ) = {C < ∼ D | K |= Ei −1  ¬C } .

(1)

The idea is that, after all, the individuals that satisfy all of Ei −1 are indeed the most normal instances at level i − 1, so the monotonic inclusion  suffices to verify whether any of those normal instances can possibly belong to C . The problem with this definition is that it does not match any of the current semantics, nor our relaxed notion of canonical model. In order to describe precisely this problem we first need the technical machinery introduced in the following section, so a detailed explanation is deferred to Appendix B. Here we only anticipate that: 1. RC , i.e. the notion of rational closure induced by E  , is complete but not sound with respect to the class of all ranked models of K. The problem is that some ranked interpretations do not have enough normal individuals to satisfy all the DIs in RC . This problem arises also when the DL enjoys the DMUP, and it has been tackled in the past by considering only a restricted class of ranked models, namely, the minimal canonical models mentioned in the previous section, that contain all possible normal types of individuals. 2. However, the class of all minimal canonical models is too narrow: the set of DIs that are valid with respect to this class models, in general, may be larger than RC , that is, RC is not always complete with respect to the minimal canonical models of K. 3. Our replacement for canonical models, i.e. upward-closed models (that will be introduced in the next section), extend the class of canonical models but not enough to solve the problem: RC is sound but still not complete with respect to this class of models. Our diagnosis is that the entailment |= in (1) is too weak. It should be based only on the class of upward-closed interprer tations. The resulting, stronger entailment (that we will denote with |= ) depends on the ranking, though, because upward closure does; so the ranking and the entailment based on saturated interpretations depend on each other. For this reason, in the next section, we shall take an approach which is commonly needed in nonmonotonic reasoning, where such cyclic dependencies easily arise: we shall introduce a suitable notion of stability, and study the rankings that satisfy it. 5. Stable rational closure Our revised notion of rational closure is based on a set of relevant concepts C analogous to the one for canonical models (cf. Section 3.2). In our framework, however, C is subject to milder restrictions: given any set of DIs of interest D ⊇ dfs(K), our results hold for all C that include (at least) the concepts C and C ¬ D for each (C < ∼ D ) ∈ D. Accordingly, we say that C is appropriate for K iff for all DIs (C < ∼ D ) ∈ K, both C and C ¬ D belong to C. In the following, when we discuss the ranking functions for a given DKB K, we will implicitly assume that the domain C of the rankings is appropriate for K. Definition 5.1 (Ranking functions for K). A concept ranking function for a DKB K is an ordinal-valued function r : C → ω + 1, where ω is the first infinite ordinal and C is appropriate for K. With a slight abuse of notation, r is extended to DIs and individuals as follows. For all C ∈ C and all D, let

r (C < ∼ D ) = r (C ) and denote the set of DIs of K with rank greater than or equal to i with:

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K|ri = {δ ∈ dfs(K) | r (δ)  i } . The K-rank of an individual x in an interpretation I is



rK (I , x) =

|r I } if such i exists min{ i | x ∈ K i

ω

otherwise.

In other words, individuals are ranked based on the DIs they satisfy: more precisely, rK (I , x) = i if i is the least rank such that x satisfies (the materialization of) all the DIs δ with r (δ)  i. The rational closure induced by a ranking function r can now be defined by analogy with RC(·). Definition 5.2 (Rational closure w.r.t. r).





RCr = C < ∼ D | {C , C ¬ D } ⊆ C ∧ r (C ) < r (C ¬ D ) ∨ r (C ) = ω



.

We obtain a semantics for RCr by replacing the notion of minimal canonical model with a relaxed definition, that allows all concepts with finite rank to be occasionally empty. The important property of minimal canonical models is that the most normal instances of a concept C ∈ C are – informally speaking – as normal as C can possibly be. This property constitutes the core of our semantics, and is formalized as follows: r

Definition 5.3 (Upward-closed models and |= ). An interpretation I is upward-closed (with respect to K and a ranking function r for K) iff for all concepts C ∈ C and all x ∈ C I such that rK (I , x) > r (C ), there exists y ∈ C I such that y
1. 2. 3. 4.

I = {x, y }, y
I I < To see that I is indeed a model of K, note that minI ( A 1 ) = {x} ⊆ B I 1 and min ( A 2 ) = { y } ⊆ B 2 . To see that I  A 1 A 2 ∼

B 2 , note that minI ( A 1 A 2 ) = {x}  B I 2.

 = , but not to However, I is not upward-closed. The K-rank of x is rK (I , x) = rnkK (I , x) = 1 (since x belongs to K|rnk 1  I K|rnk = (¬ A  B ) (¬ A  B ) ) while rnk ( A A ) = 0. The upward-closure conditions requires ( A A 1 1 2 2 1 2 1 2 ) to contain an 0 element z
 The reader may easily verify that for all ranking functions r for K, K |r0 = ¬ A  B. It follows that the ranked interpretation I specified below is an upward-closed model of K w.r.t. r, (because rK (I , x) = 0, therefore the upward-closure condition is vacuously satisfied): 1. I = {x}, 2. A I = B I = ∅.

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For similar reasons, all the ranked models J of K where A J = ∅ are upward-closed, too. Conversely it can be seen that for all r, A must be empty in all upward-closed models of K w.r.t. r, otherwise B should be nonempty, too (in order to satisfy A< ∼ B); but then the strong axiom would not be satisfied. On the contrary, B can be nonempty in some upward-closed models, like the interpretation J with J = {x}, A J = ∅, and B J = {x}.10 Note that K has an upward-closed model because upward-closure – unlike the notion of canonical model – allows A to be empty. 2 Of course, so far the ranking r is completely unrestricted. We want it to be the result of an iterative construction analogous r to the sequence Ei i . However, we replace the exceptionality test with a variant of (1) where |= is replaced with |= to 11 address the semantic difficulties sketched in Section 4 and illustrated in Appendix B. Definition 5.6 (Stable rankings). A ranking function r is stable (w.r.t. K) iff the domain C of r is appropriate for K, and for all concepts C ∈ C and ordinals i  ω ,

 |ri  ¬C ; 1. if i < r (C ) or r (C ) = ω , then K |= K r

 |rr (C )  ¬C . 2. if r (C ) = ω , then K  K r

In the following we will use the phrase “stable rational closure” (of a DKB K) to refer to any RCr such that the ranking function r is stable w.r.t. K. r

Remark 5.7. The stability of r depends on |= that in turn depends on r. So there is no a priori certainty that a stable rank exists, nor that it is unique. We will discuss this issue later on. We first illustrate a non-problematic example. Example 5.8. Consider again the K in Example 5.5. The features of the upward-closed models of K illustrated in that r r   example imply that for all ranking functions r, K  K |r0  ¬ B and for all i  ω , K |= K |ri  ¬ A. It follows immediately from Definition 5.6 that all the stable rankings r of K agree that r ( B ) = 0 and r ( A ) = ω . As a consequence, A < ∼ ⊥ belongs to all the stable rational closures RCr of K. 2 Stability suffices to enforce the postulates in Fig. 2. More precisely: Theorem 5.9. If r is stable, then RCr is closed under the KLM postulates restricted to D (the set of DIs of interest introduced at the beginning of this section). Moreover, if r is stable, then its upward-closed models faithfully capture the corresponding stable rational closure RCr . Theorem 5.10 (Soundness and completeness). Suppose that r is stable w.r.t. K, and that {C , C ¬ D } ⊆ C. Then r < (C < ∼ D ) ∈ RCr iff K |= C ∼ D .

The proof of this theorem is quite articulated and needs a number of small lemmas that characterize the properties of upward closed models. Here we summarize the basic ideas behind the proof and refer the interested reader to Appendix A for the formal details. Proof. (Sketch) Roughly speaking, we can assume without loss of generality that if I is an upward-closed model of K w.r.t. a stable ranking r, then for all C ∈ C:

 r (C ) < ω implies minI (C ) = C I ∩ K |rr (C ) I

(2)

 (the formal ground for this claim lies in Lemma A.8, Proposition A.6, and Lemma A.4). Moreover, if r (C ) < r ( D ), then K |rr (C ) I r  and D I must be disjoint, since stability requires that K |= K |rr (C )  ¬ D. It follows immediately from (2) that

r (C ) < r ( D ) implies minI (C ) ⊆ (¬ D )I . Now we are ready to prove soundness and completeness.

10 11

This property will be useful in the next example. r The role of |= will be clearer in the proof of completeness, see Theorem 5.10.

(3)

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r < Soundness. Suppose that (C < ∼ D ) ∈ RCr ; we have to show that K |= C ∼ D. By definition of RCr there are two possibilities: (i) r (C ) = ω , or (ii) r (C ) < r (C ¬ D ). In case (i) it can be proved that C is empty in all the upward-closed models of K r w.r.t. r (Lemma A.3), consequently K |= C < ∼ D. In case (ii), by (3) we have that for all upward-closed models I of K w.r.t. r r, minI (C ) ⊆ ¬(C ¬ D )I = (¬C )I ∪ D I , that is, minI (C ) ⊆ D I . It follows immediately that K |= C < ∼ D. r < < Completeness. Assume that (C ∼ D ) ∈ / RCr ; we have to show that K  C ∼ D. The assumption implies that r (C ) < ω and r  r (C )  r (C ¬ D ) (by definition of RCr ). Then, by stability, K  K |rr (C )  ¬(C ¬ D ). So there exist an upward-closed model12

 I of K and x ∈ I such that: (i) x ∈ K |rr (C ) , (ii) x ∈ C I , (iii) x ∈ / D I . By properties (i), (ii), and (2), x ∈ minI (C ). It follows, by r < (iii), that I  C < ∼ D. Then I witnesses that K  C ∼ D. This completes the proof.

2

Remark 5.11. Theorem 5.10 depends on the set of relevant concepts C, just like the semantics of RC based on canonical models. Actually, all of the above results, and those in Section 7, hold also if C is the set of all possible concepts. This C would yield the cleanest possible semantics, and support all queries at once. However, the axiomatic characterization that we give in Section 6 becomes infinite, and currently we do not know any effective way of reasoning with such an infinite theory. 6. Axiomatic characterization of upward closure r

The entailment relation |= based on upward-closed models characterizes stable rational closure (cf. Theorem 5.10) and r is essential for checking stability. In order to reason with upward-closed models and compute |= , here we provide an r axiomatic characterization of upward-closed interpretations that reduces the computation of |= to the computation of |=. The latter can then be decided in several ways. The reduction is based on DIs. Theorem 6.1. An interpretation I is upward-closed w.r.t. K and a ranking r for K iff for all concepts C ∈ C such that r (C ) <

r  I |= C < ∼ K|r (C ) .

ω,

 Proof. (Sketch) The “only if” part holds because upward-closure implies almost immediately that minI (C ) ⊆ K |rr (C ) I , for all

C ∈ C such that r (C ) < ω . Concerning the “if” part, first note that no instance x ∈ I such that rK (I , x) > r (C ) can be an I r   instance of K |rr (C ) I , by definition of rK . Therefore, if I |= C < ∼ K|r (C ) , then such x cannot belong to min (C ). It follows that whenever x ∈ C I and rK (I , x) > r (C ), there must be another instance y ∈ C I such that y
By the above theorem, |= can be reduced to |= by extending K with a suitable set of DIs. More formally: Corollary 6.2. For all rankings r for K, r r  K |= α iff K ∪ {C < ∼ K|r (C ) | C ∈ C ∧ r (C ) < ω} |= α .

r

Remark 6.3. Corollary 6.2 provides an effective way of checking whether K |= α holds, by reducing reasoning with upwardr closed models (|= ) to reasoning with ranked models (|=) that, in turn, can be implemented in at least two different ways. A first approach consists in exploiting the logic dSROIQ introduced in [9], that generalizes the logic illustrated in Section 3.1, provided that its strong axioms are in SROIQ and that the concepts C , D occurring in any DI C < ∼ D are SROIQ concepts. So, if K and α satisfy these syntactic restrictions, then the tableaux system introduced in [9] can be used to decide r r  whether K ∪ {C < ∼ K|r (C ) | C ∈ C} |= α holds, and hence whether K |= α holds. Recall that SROIQ is the DL that provides the logical foundation for the standard OWL2, therefore requiring that K and α belong to SROIQ is not really restrictive from a practical perspective. This is not the only way of reasoning with ranked models. A reduction to classical DL reasoning is introduced in [19] (see Proposition 2.9 and Corollary 2.10). This method enables the use of standard, well-engineered reasoners. Moreover, it proves that entailment w.r.t. ranked models has the same complexity as classical reasoning, for all the DLs rangining from ALC to SROIQ. This result (Corollary 2.10 of [19]) and Corollary 6.2 imply the following result.

r

12 Here is where the use of |= in the definition of stability plays an essential role. Without it I could be any model, and we could not apply property (2) later in the proof.

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Corollary 6.4. Let DL range from ALC to SROIQ, let K range over the defeasible knowledge bases in DL, and let C , D range over r DL concepts. Then, for all rankings r for K, deciding whether K |= C < ∼ D holds has the same complexity as (classical) subsumption in DL.

Proof. The composition of the reduction provided by Corollary 6.2 and the reduction used for Corollary 2.10 of [19] yields r a reduction of |= to classical subsumption. So, in order to complete the proof, we only need a reduction from classical r r subsumption to |= . Such reduction can be immediately obtained by noting that if dfs(K) = ∅, then K |= C < ∼ D holds iff C  D is classically entailed by K.

2

7. Relationships with RC Stable rankings yield a natural generalization of rational closure. The old and new notions of ranking are tightly related. In order to investigate the relationships between rational closure and stable rational closure, in this section we assume an arbitrary but fixed DKB K, and let Ei i denote the corresponding sequence of knowledge bases (cf. Definition 3.2). We start by stating and discussing the two results of this section. Then we outline a sketch of their proofs (the detailed proofs are in Appendix A). The first result tells us that stable rankings assign to each concept in C ∈ C a rank greater than, or equal to rnk(C ). In other words: Theorem 7.1 (Stable rankings are stronger). If r is stable w.r.t. K, then for all i < ω , dfs(Ei ) ⊆ K|ri . In some cases, stable rankings actually assign some concept C to a rank strictly higher than rnk(C ). This is how the axiom forgetting problem is solved. Example 7.2. It can be verified that the K in Example 4.1 has a stable rank r such that

r( A1) = 0 r( A2) = r( A1 ¬B ) = ω . The details are tedious but straightforward, and are left to the reader. Then rnk( A 2 ) = 1 < r ( A 2 ) = ω .

2

It is not by chance that the above example uses a K that does not enjoy the DMUP. When the DMUP holds, then rnk is the only stable rank (and the old and new notions of rational closure coincide). Theorem 7.3. Let r be stable w.r.t. K, and suppose that str(K) enjoys the DMUP. Then 1. for all i < ω , K|ri = dfs(Ei ); 2. for all concepts C , either r (C ) = rnk(C ), or r (C ) = ω and rnk(C ) = ∞; < 3. for all {C , C ¬ D } ⊆ C, (C < ∼ D ) ∈ RC(K) iff (C ∼ D ) ∈ RCr . In the rest of this section we sketch the proofs of the above two theorems. The proof of Theorem 7.1 is a simple induction  on i, relying on the following property, that relates the conditions that define Ei +1 and K |ri+1 (when r is stable):

r Ei |=  < ∼ ¬C implies K |= K|i  ¬C . r

r

This implication can be proved by observing that: Ei ⊆ K (so we can leverage the monotonicity of |=), |= is weaker than |= r (since upward-closed models are ranked models), and  < ∼ ¬C implies K|i  ¬C in all models of K (by Lemma A.12). The proof of Theorem 7.3 is more articulated. Point 2 follows easily from point 1, and point 3 follows easily from point 2. So the key result is point 1, that can be proved by induction on i using Theorem 7.1 and the converse of the above property, namely:

|r  ¬C implies Ei |=  < ¬C . K |= K ∼ i r

The proof of this implication relies on the observation that if K enjoys the DMUP, then every model I of str(K) ∪ K|rω that satisfies a concept C can be extended to an upward-closed model U of all K that satisfies C (Lemma A.13). With this r r lemma, every counterexample to Ei |=  < ∼ ¬C can be turned into a counterexample to K |= K|i  ¬C (which proves the

above implication). The DMUP allows to construct U by taking the union of a set of models I D , each of which provides a normal instance for some D ∈ C (with r ( D ) = ω ), as required by upward-closure.

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8. On the existence and uniqueness of stable rankings Of course, since under the DMUP all stable rankings equal rnk (by Theorem 7.3) it follows that: Corollary 8.1. If str(K) enjoys the DMUP then there exists a unique stable ranking w.r.t. K. If the DMUP does not hold, however, then we have no characterization of the existence of a stable rank as neat as Corollary 8.1. In this section we collect all the knowledge we have about this matter. Basically, the only positive result is the following: Theorem 8.2. If the set of relevant concepts C is finite and K is a SROIQ DKB, then checking whether a knowledge base K has a stable ranking is decidable, and the set of stable rankings of K can be effectively computed. If C is infinite, instead, then the above computability properties are still an open question. The proof of Theorem 8.2 relies on the observation that the domain of the ranking functions of interest is C itself, while their range is bounded by |dfs(K)| (because there can be at most one exceptionality level for each DI in K). Therefore, the number of all possible ranking functions is bounded by |dfs(K)||C| , so it is possible to enumerate all possible candidate rankings and check whether they satisfy the stability conditions with a polynomial number of entailment tests for each ranking. In turn these tests can be effectively decided as explained in Remark 6.3. This observation is also the starting point for the complexity bounds reported at the end of this section. Stable rankings do not necessarily exist (even if C is finite): Theorem 8.3. Some DKB K have no stable ranking. Proof. A K with no stable ranking over C = { A , A 1 , A 2 , A 3 , A ¬ B } is the following:

A< ∼B



 ( A i  ⊥) ∨ ¬( A i  A ¬ B ) → ( A (i mod 3)+1  A ¬ B ) (i = 1, 2, 3) .

(4) (5)

Informally speaking, A ¬ B denotes the set of all instances that violate (4). Therefore (5) creates a circular chain of dependencies between A 1 , A 2 , and A 3 , such that each A j ( j = 2, 3, 1) may contain a normal individual – i.e. one that satisfies (4) – if and only if its predecessor A i (i = 1, 2, 3) is nonempty and contains no such normal individuals. Consequently, each A j may have rank 0 iff its predecessor A i has rank > 0. Since the cycle has an odd number of such dependencies, A j may have rank 0 iff its rank is > 0. This irresolvable conflict prevents K from having any stable rank. Before starting to prove our claim, we note a few facts that hold for all rankings r for K:

 (i) K |r0 = ¬ A  B.

r  |r0  ¬ A, so r ( A ) = 0.13 (ii) K  K r (iii) By (ii), r ( A < ∼ B ) = 0, therefore K|i = , for all 1  i  ω .

 i < < r < (iv) Given (iii), for all C ∈ C with r (C ) > 0, C < ∼ K|r (C ) equals the tautology C ∼ . Moreover, A ∼ K|0 equals A ∼ ¬ A  B r  < that is equivalent to (4). It follows that K ∪ {C < ∼ K|r (C ) | C ∈ C ∧ r (C ) < ω} is logically equivalent to K ∪ { A i ∼ ¬ A  B | r ( A i ) = 0 (i = 1, 2, 3)}, therefore, by Corollary 6.2, r K |= α iff K ∪ { A i < ∼ ¬ A  B | r ( A i ) = 0 (i = 1, 2, 3)} |= α .

(6)

Now, to see that indeed K has no stable ranking r, we assume that such a ranking exists and derive a contradiction for all possible cases up to symmetries: a) r ( A 1 ) = r ( A 2 ) = r ( A 3 ) > 0 b) r ( A 1 ) = 0, r ( A 2 ) = r ( A 3 ) > 0 c) r ( A 1 ) = r ( A 3 ) = 0.

  |r0  ¬ A 1 (cf. (i)), consequently (by (6)) K  K |r0  ¬ A 1 so the stability condition for A 1 is In case a) we have K  K violated and we get a contradiction. A witness that the above entailment does not hold is the following model I , where I = {a, b, c } and a
13

The interpretation I such that I = {x}, A I = B I = {x}, and A I i = ∅ for i = 1, 2, 3 is always an upward-closed model of K, and it witnesses that

r  K K |r0  ¬ A.

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A I = {a, c } B I = {a} AI 1 = {b } AI 2 = {c } AI 3 = {c } .

 It can be easily verified that I |= K and I  K |r0  ¬ A 1 . r In case b), stability requires that K ∪ { A 1 < ∼ ¬ A  B } |= K|0  ¬ A 3 . However this entailment is contradicted by the following counterexample I , where I = {a, b, c } and a
¬ A  B } |= ¬ A  B  ¬ A 1 holds, too. It follows by (6) that no matter how A 2 is ranked, K |= ¬ A  B  ¬ A 1 holds. But then the stability condition for A 1 is violated. 2 Other knowledge bases, instead, have multiple stable rankings, hence multiple, mutually incomparable closures. Theorem 8.4. Some DKB K have multiple stable rankings. Proof. Consider the following K:

A1 < ∼B

( A 2 A 3 )(a) ( A2  A1 ¬B ) ∨ ( A3  A1 ¬B ) . We are going to prove that this K admits two incomparable, stable rankings r1 and r2 over C = { A 1 , A 2 , A 3 , A 1 ¬ B }.14 Informally speaking, by the third axiom, either all instances of A 2 violate the unique DI of K, or the same holds for all the instances of A 3 . In the latter case, we obtain a stable ranking r1 such that r1 ( A 2 ) < r1 ( A 3 ); in the former case we obtain a stable ranking r2 such that r2 ( A 2 ) > r2 ( A 3 ). We start by proving that the rank of A 1 is 0 in all stable rankings r for K. Let I be any upward-closed model of K I (w.r.t. r). By the second and third axioms of K, aI ∈ A I 1 = ∅ so, by the first axiom, there exists x ∈ min ( A 1 ) such that

 r x ∈ B I . Then x belongs to K |r0 I = (¬ A 1  B )I as well as A I 1 . It follows easily that for all stable rankings r, K  K|0  ¬ A 1 , therefore r ( A 1 ) = 0. Next we prove that if r is a stable ranking, then r ( A 2 ) > 0 and r ( A 3 ) > 0 cannot hold simultaneously. This can be seen with two models I2 and I3 with domain {x, a} (x = a) such that aI2 = aI3 = a. In both models, let a belong to A 1 ¬ B r

I

I

and x belong to A 1 B. Moreover, let x ∈ A 2 2 and x ∈ A 3 3 . Note that if r ( A 2 ) > 0 and r ( A 3 ) > 0 hold, then I2 and I3

 are upward-closed and witness that K  K |r0  ¬ A i (i = 2, 3). As a consequence, the rank of A 2 and A 3 should be 0 (a contradiction). Now we consider the ranking such that r ( A 1 ) = r ( A 2 ) = 0, r ( A 3 ) = 1, and r ( A 1 ¬ B ) = 1. We are going to prove that r is  stable w.r.t. K. The model I2 of K introduced above is upward-closed w.r.t. r and provides a counterexample to K |r0  ¬ A i , for i = 1, 2. Thus the stability conditions for A 1 and A 2 are satisfied. Concerning A 3 , note that in all upward-closed models  I of K (w.r.t. r), minI ( A 2 ) must be included in K |r0 I = (¬ A 1  B )I . Therefore the third axiom of K can be satisfied r

 only if A 3  A 1 ¬ B, hence K |r0  ¬ A 3 . This stability condition confirms that r ( A 3 ) > 0. Moreover, I2 witnesses that

r    KK |r1  ¬ A 3 (since K |r1 = ). We conclude that the stability conditions for A 3 are satisfied. Finally, K |r0 = (¬ A 1  B ) is

14 We do not include B in C in order to remove irrelevant details from the proof. It is not hard to see that if B were included in C, then all the resulting stable rankings would give it rank 0.

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 clearly subsumed by the equivalent concept ¬( A 1 ¬ B ), and I2 witnesses that K  K |r1  ¬( A 1 ¬ B ); then the stability conditions for A 1 ¬ B are satisfied. By the above discussion, this ranking r is indeed stable w.r.t. K. Let r1 = r. By a symmetric argument, the ranking r2 such that r2 ( A 1 ) = r2 ( A 3 ) = 0 and r2 ( A 2 ) = r2 ( A 1 ¬ B ) = 1 is stable w.r.t. K, too (use I3 instead of I2 in the proof). This completes the proof of the theorem. As a side comment, it is not hard to verify that there can be no further rankings. 2 r

Therefore, the generalization of rational closure does not guarantee that the set of consequences of a knowledge base exists, nor that it is unique (unless the knowledge base enjoys the DMUP). However, stable rational closure provides a non-ambiguous closure to many knowledge bases that do not enjoy the DMUP, as shown in Example 5.8 and, later, in Example 9.1. While it seems difficult to find natural consistent DKB with no stable rankings, in the next section we will argue that multiple closures arise more easily, due to ABoxes. The multiplicity of stable rational closures naturally introduces the opportunity for different notions of entailment, by analogy with default logic, autoepistemic logic, and answer set programming. Namely, an axiom C < ∼ D is a skeptical (resp. credulous) consequence of a DKB K iff C < ∼ D belongs to all (resp. some) of the stable rational closures of K induced by the rankings with a given, appropriate domain C. Similar notions, in the context of rational closure, have already been discussed in [11], whose authors note that – however – some KLM postulates do not hold for skeptical entailment. The same remark applies to our framework: Example 8.5. Consider again the knowledge base in the proof of Theorem 8.4. Note that A 1 < ∼ B belongs to both of the < stable closures of the knowledge base, that is, RCr1 and RCr2 . Moreover, A 1 ∼ ¬ A 2 does not belong to RCr1 (since r1 ( A 1 ) = < < r1 ( A 2 ) = 0), so A 1 < ∼ ¬ A 2 is not skeptically entailed. Given the two premises A 1 ∼ B and A 1  ∼ ¬ A 2 , the Rational Monotony postulate (RM) states that A 1 A 2 < ∼ B should be entailed. However, this DI is not skeptically entailed because it does not belong to RCr2 , since all the upward-closed models of the knowledge base w.r.t. r2 satisfy A 2  A 1 ¬ B. We conclude that skeptical entailment does not satisfy Rational Monotony. 2 Actually, under the above definition, credulous entailment does not satisfy all the KLM postulates, either, as shown in the following example.15 Example 8.6. We adopt a knowledge base K inspired by the one used in the proof of Proposition 4 of [11]. K contains a single assertion R (a, b) and the defeasible inclusion:

< ∼ A ∀ R .¬ A .

(7)

Note that a and b cannot simultaneously satisfy (7), because if a satisfies its right-hand side, then b satisfies ¬ A and violates (7). We are going to show that, for this reason, the above DKB has two stable rankings such that credulous entailment does not satisfy the (AND) postulate. The question is whether {a} and {b} normally satisfy the right-hand side of (7) in a credulous sense, so we are going to consider the rankings over a set C that contains {a}, {b}, plus the subconcepts of (7) and their negation. In all stable rankings, (7) has rank 0. This knowledge base has two stable rankings r and s with the following properties:

• r ({a}) = 0, r ({b}) = 1; therefore, all upward-closed models (w.r.t. r) satisfy {a}  ∀ R .¬ A (by (7)) and, consequently, {b}  ¬ A (by R (a, b)); • s({b}) = 0, s({a}) = 1; therefore, all upward-closed models (w.r.t. s) satisfy {b}  A (by (7)). < We immediately note that {b} < ∼ ¬ A and {b} ∼ A belong to RCr and RCs , respectively, however no stable rational closure

contains {b} < ∼ ¬ A A, so the (AND) postulate is not satisfied by the above notion of credulous entailment.

2

Given that satisfying the KLM postulates is a primary purpose of rational closure, and given that the postulates are not fully satisfied by skeptical and credulous entailment, as shown by the above two examples, we do not analyze these two kinds of entailment any further in this paper, and focus on the computation of single stable rational closures. More precisely we consider the following decision problem:

15 The credulous approach of [11] satisfies the postulates because it amounts to choosing a single closure; similarly, in our framework, each stable rational closure satisfies the postulates.

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Definition 8.7 (Entailment under stable rational closure). The entailment problem for rational closure consists in deciding whether a given set of DIs D is included in some of the stable rational closures RCr of a given DKB K, where the domain C of r is the set of concepts C and C ¬ D such that (C < ∼ D ) ∈ D ∪ dfs(K). The rest of this section is devoted to a preliminary discussion of the complexity of the above problem and of the related problem of deciding whether a stable ranking r exists, given K and a finite C. As we pointed out after Theorem 8.2, a trivial algorithm for finding a stable ranking consists in generating each candidate ranking function, and checking whether any of them is stable. A better approach, from the viewpoint of complexity analysis, consists in guessing a stable ranking – if some exists – by means of a nondeterministic Turing machine N ? (where the question mark indicates that N is an oracle machine whose oracle has not yet been specified, cf. [28]). The guessing phase consists in writing on an auxiliary tape the graph of a ranking function r, that is, a list of pairs (C , r (C )) (one for each C ∈ C) such that r (C ) is nondeterministically chosen from [0, |dfs(K)| − 1] ∪ {ω}. The encoding of the graph of r takes polynomial space – bounded by O (||C|| + (|C| · log |dfs(K)|)), where ||C|| is the length of the encoding of C – so r can be guessed in polynomial time. After guessing r, N ? calls the oracle several times, to check whether r satisfies the stability conditions. By definition of stability, this requires to compute a subsumption check for each i in the range of r, and for each concept in C, that is, O (|dfs(K)| · |C|) different subsumptions. Each of them is decided by a single call to the oracle, consequently this verification phase is polynomial, too. It follows that the nondeterministic Turing machine N ? runs in polynomial time (using an oracle for subsumption), therefore: Theorem 8.8. Deciding whether a stable ranking for K and a finite C exists is in NPC , where C is the complexity of the subsumption problems involved in stability checking. An upper bound to the complexity of entailment under stable rational closure can be estimated through a simple modification of the Turing machine N ? used for the above proof. The new machine takes as input a set of DIs C i < ∼ D i (1  i  n), a

DKB K, and shall decide whether these DIs belong to a same rational closure RCr of K as specified in Definition 8.7. After guessing r and checking with the help of the oracle that r is stable w.r.t. the given K (as explained before), the modified Turing machine searches the encoding of r to determine the values of all r (C i ) and r (C i ¬ D i ). Then it only has to compare these value pairs and accept iff r (C i ) < r (C i ¬ D i ) for all i = 1, . . . , n. Searching r and comparing values takes polynomial time (since the encoding of r has polynomial size), so we obtain the same complexity bound as before:

Theorem 8.9. The entailment problem for stable rational closure is in NPC , where C is the complexity of the subsumption problems involved in stability checking. Note that when a single DI is given (n = 1), the above method computes credulous entailment, therefore this problem is in NPC , too. r The above theorems are the building blocks for more precise complexity estimates. By Corollary 6.4 (that proves that |= has the same complexity as classical subsumption), the oracle C mentioned in the above two theorems can be set to the complexity class of classical subsumption in the given DL. First consider DLs whose classical subsumption problem is in EXP16 (such as SHOQ). In this case, by the above argument, the complexity of finding a stable ranking is in NPEXP that equals EXP. Then, by Theorem 8.8: Corollary 8.10. Checking whether a DKB in SHOQ has a stable ranking over a given finite C is in EXP. The same holds for all DL whose subsumption problem is in EXP. In this case, deciding whether a stable ranking exists is not harder than subsumption checking in the monotonic fragment of the logic. Exact complexity characterizations (i.e. completeness results) are harder to find. To see why, consider that there exist many logics that – unlike SHOQ – enjoy the DMUP, still have an EXP-complete subsumption problem, like ALC with empty ABoxes. By Theorem 7.3, such logics always have a stable rank, so their stable rank existence problem is strictly less complex than subsumption checking. Exact complexity characterizations are an interesting topic for further research. Next consider EL and DL-lite (two logics whose subsumption problem is tractable). All we know about rational closure in these logics is that if the ABox is empty, then the rational closure of EL extended with ⊥ can be computed in polynomial time [14].17 Both EL (with ⊥) and DL-lite enjoy the DMUP, when the ABox is empty, so all the TBoxes in these logics have a stable ranking, by Theorem 7.3. Also in this case, deciding the existence of a stable rank is easier than monotonic subsumption. Things are slightly more complicated if classical subsumption checking belongs to a nondeterministic complexity class. For instance, if subsumption checking is NEXP-complete18 (as in SHOIQ), then deciding whether a stable ranking exists

16 17 18

k

EXP is the class of decision problems solvable in deterministic exponential time, that is, DTIME(2n ). This paper comprises also a method for reasoning with nominals under a nonstandard semantics. k

NEXP is the class of decision problems solvable in nondeterministic exponential time, that is, NTIME(2n ).

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is in NPNEXP , that equals PNEXP [24] (i.e. the class of problems that can be solved in polynomial time by a deterministic Turing machine with oracle NEXP). By a similar argument, if the underlying DL is SROIQ, whose subsumption problem is N2EXP-complete,19 then deciding whether a stable ranking exists is in NPN2EXP = PN2EXP .20 As a consequence: Corollary 8.11. Let K be a DKB in a DL whose classical subsumption problem is in NEXP (resp. N2EXP). Then deciding whether K has a stable ranking over a given finite C is in PNEXP (resp. PN2EXP ). Clearly, analogous considerations – and upper bounds – apply to entailment. The details are straightforward and left to the reader. The results are summarized by the next corollary of Theorem 8.9: Corollary 8.12. The entailment problem for stable rational closure is in:

• • • •

P for all the TBoxes in EL⊥ ; EXP for ALC , SHIQ, SHOQ, and all the DLs whose classical subsumption problem is in EXP; PNEXP for SHOIQ, and all the DLs whose classical subsumption problem is in NEXP; PN2EXP for SROIQ, and all the DLs whose classical subsumption problem is in N2EXP.

A complete complexity analysis lies out of the scope of this paper, so we leave an exact characterization of the complexity of computing stable rankings and their consequences as an interesting topic for future work. Other interesting open questions concern the complexity – for given K and C – of deciding whether K has a unique stable ranking, and the complexity of computing all the stable rankings for K. 9. ABox reasoning using nominals In this section we briefly discuss the stable rational closure of defeasible knowledge bases with nominals and nonempty ABoxes. Nominals can be used to reason about individuals by leveraging the same construction designed for DI entailment, in a uniform way. Under stable rational closure, where nominals can be freely used, this approach is always applicable, although – as we will point out in the following – this is not necessarily the most efficient approach. Some previous works followed different approaches. The DLs of typicality infer default properties only for the individuals that are explicitly declared to be typical (with assertions like (TC )(a)), while [12,7] adopt a separate, specific construction for reasoning about individuals. With nominals, defeasible instance checking – that is, checking whether an individual a is normally a C – can always be 21 reduced to checking whether {a} < ∼ C belongs to RCr (where r is the given stable ranking of K). Example 9.1. Normally, red blood cells (hereafter RBC) have a nucleus, but mammalian RBC (MRBC for short) don’t have a nucleus in their mature stage. This knowledge can be formalized as follows:

MRBC  RBC RBC < ∼ ∃hasNucleus

MRBC < ∼ ¬∃hasNucleus

If we extend the above DKB with the ABox {RBC(c1), MRBC(c2)} then all the stable rankings r of the resulting DKB K are such that

r (RBC) = r ({c1}) = 0 , r (MRBC) = r ({c2}) = 1 . < As expected, RCr contains {c1} < ∼ ∃hasNucleus and {c2} ∼ ¬∃hasNucleus.

2

Technically, this approach to instance checking is particularly elegant, due to its uniformity with DI entailment, and infers default properties for all individuals, without explicitly marking them as typical. However, the above formalization does not remove the intrinsic difficulties related to individuals. In particular, as it happens in [12,7], the default properties of two individuals may be mutually incompatible and generate multiple stable closures, as in the following example.

nk

19

N2EXP is the class of decision problems solvable in nondeterministic double exponential time, that is NTIME(22 ). The proof of NPNEXP = PNEXP , illustrated in [24], works out of the box with the more powerful oracle N2EXP, thereby proving that NPN2EXP = PN2EXP . 20

21

Similar approaches occur in [4,16,19].

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Example 9.2. Let K consist of the following axioms:

A1 < ∼ ∀ R .¬ B

A2 < ∼B

A 1 (a) A 2 (b) R (a, b) . Note that either a violates the (materialization of) the first DI, or b violates the second. This K has two stable rankings such that (respectively):

• either r ({a}) = 0 and r ({b}) = 1, • or r ({a}) = 1 and r ({b}) = 0. The two rankings yield two stable rational closures where (respectively): < • either {a} < ∼ ∀ R .¬ B and {b} ∼ ¬ B, < • or {a} < ∼ ¬∀ R .¬ B and {b} ∼ B. 2

Thus, ABoxes are a natural cause for the existence of multiple alternative closures. In fact, the order-dependent ABox closures of [12,7] anticipated the existence of multiple stable rankings illustrated above – they are just different formalizations of the same phenomenon. It is worth noting that the ABox closures introduced by [12,7] are still of interest because they do not involve nominals, that may increase computational complexity. In some cases it is possible to get the benefits of the uniform approach based on nominals without paying the complexity price, by simulating nominals with ordinary concepts (see, for example, Sec. 5.2 of [4]). A complete analysis of the methods for reasoning about individuals and their complexity lies beyond the scope of this paper. It can be regarded as an optimization issue, given that the approach based on nominals can always be applied under stable rational closure (possibly at the cost of adding nominals to the given logic). Remark 9.3. With the syntax adopted in this paper, it is difficult to tell whether a role assertion R (a, b) normally holds. An interesting direction for further research consists in extending our framework to the richer language introduced in [9], that associates normality orderings to roles, as well. In such a framework, it may be possible to treat all assertions uniformly. 10. Discussion Summarizing, we observed that the previous definition of rational closure for description logics (denoted by RC(K)) does not match any semantics based on ranked models when the knowledge base K does not enjoy the DMUP (cf. the axiom forgetting problem illustrated in Example 4.1). More precisely, for some K, rational closure assigns a finite rank to concepts that – however – cannot be possibly satisfied in any ranked model of K (so the concept’s rank should be ∞, no matter what subclass of ranked interpretations is chosen as a semantics of rational closure). Consequently, the traditional correspondence between rational closure and ranked models fails for expressive description logics. This problem arises because, when the underlying DL is expressive enough, a decision on the rank of a concept C taken at step i may clash with the satisfiability of the DIs previously assigned to a rank j < i (that are not taken into account while evaluating the exceptionality level of C ). The bottom-up nature of the ranking sequence Ei i does not allow to rectify that decision. In order to avoid such backward effects, in this paper we explored a modified definition of rational closure called stable rational closure, based on a notion of stable ranking. We proved that the stable rational closures RCr induced by the stable rankings r of K constitute a natural generalization of RC(K) in the following sense: 1. each stable rational closure RCr is closed under the KLM postulates; 2. if K enjoys the DMUP, then K has exactly one stable rational closure RCr = RC(K) (and one stable ranking r). 3. when the DMUP does not hold, the only difference between rnk and stable rankings is that the latter may give some concepts a higher rank, in order to avoid the problems caused by axiom forgetting (cf. Example 7.2). Stable rational closures enjoy a model-theoretic semantic characterization based on a subclass of the standard ranked interpretations, namely, the upward-closed models of K. The notion of upward-closed model relaxes that of minimal canonical model, as it allows concepts to be occasionally empty. So upward-closed models solve the problems that arise when the DMUP does not hold (already observed in [19]), and the new semantics characterizes stable rational closures for all DLs, including those that do not satisfy the DMUP. This enables the use of nominals, that support a uniform and elegant treatment of ABoxes and instance checking, as explained in Sec. 9 (although this is not necessarily the most efficient approach).

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Unfortunately, when the DMUP does not hold, the stability condition cannot be always satisfied (cf. Theorem 8.3), and sometimes it can be satisfied in multiple, non comparable ways. In the proof of Theorem 8.4, we have shown a K that has two symmetric stable rankings. Due to their symmetry, there exists no logical (i.e. interpretation-neutral) criterion for choosing a “canonical” ranking among them. So the new rational closure behaves like Default Logic and Autoepistemic Logic: sometimes a knowledge base may have multiple deductive closures, sometimes it may have no deductive closure at all. We observed in Section 9 that ABoxes are a natural cause for the existence of multiple rankings. The multiplicity of stable rational closures immediately suggests a skeptical and a credulous type of entailment (cf. Section 8). However, none of them satisfies all of the KLM postulates. Since satisfying the postulates was a primary goal of rational closure, it seems fair to argue that neither skeptical nor credulous entailment is appropriate in this context. An interesting topic for further investigation, related to the general lack of a single closure, is the identification of classes of defeasible knowledge bases that do not enjoy the DMUP, but nonetheless have exactly – or at least – one stable closure. We carried out a preliminary complexity analysis of the basic decision problems related to stable rational closures. Roughly speaking – for the DLs considered so far – when classical subsumption in the given DL belongs to a deterministic time complexity class C , then deciding whether a stable ranking exists and deciding whether a set of DIs is included in some stable rational closure, belong to C , too. If C is a nondeterministic time complexity class, instead, then complexity may raise to PC . The technical difficulties dealt with in this work arise because the meta-level notion of nonmonotonic consequence relation analyzed by Kraus, Lehmann, and Magidor has been internalized, that is, turned into an object-level operator < ∼ (and, accordingly, the original model preference relation on interpretations has been transformed into a preference relation on individuals). Some authors think that such internalization is a natural variant of the original KLM framework, but our results suggest that the old meta-level notion of rational closure has its nice properties because it is impossible, at the object level, to axiomatize constraints involving two or more interpretations. The DMUP is an effective replacement of this condition for the internalized framework, but when the DMUP does not hold, it is very possible to assert constraints over two or more individuals that affect both the existence of a stable ranking and its uniqueness. Among the interesting directions for further research mentioned along the paper, we recall that the mismatch between rational closure and the semantics based on ranked models might be tackled with an alternative approach, based on finding an appropriate new class of interpretations that matches the standard construction of rational closure. Another interesting question is how to deal effectively with infinite sets C of relevant concepts. When C is finite, instead, a practical question is how to choose C. A similar problem has been already discussed in [4]. There is a wide range of options, for example:

• Adopting a minimal C for each query, containing only the query itself and the additional concepts required for appropriateness w.r.t. the given knowledge base K. With this approach a stable ranking must be computed for each new query, so this is likely to be the most expensive approach, from a computational perspective.

• Adopting a large C that covers the most frequent queries or the most recent ones. Such a C may be extended on demand, each time a new query (not covered by C) is submitted. This approach is likely to be way more efficient, if the sequence of given queries is repetitive enough. The ranking needs to be re-computed only when a novel query is submitted. A practical instance of this method consists in limiting the size of C, e.g. by having new entries replace one of C’s members when the limit size is reached. The discarded member could be – say – the oldest, or the less used, among the concepts that are not required by appropriateness. Clearly, a reliable assessment of the alternative methods for choosing C – such as the above ones – requires systematic experimental validation on real-world testbeds. It is also important to find methods for extending rankings incrementally, when C grows, as opposed to re-computing them from scratch. Additional important issues lie beyond the scope of this paper and require further work. For instance, here we do not address the expressiveness limitations identified in previous work and collected in [4,6]. Such limitations, that originally affect rational closure, are obviously inherited by stable rational closure, since the old and the new frameworks are equivalent when the DMUP holds. Some of the most important issues to be tackled include the following phenomena:

• inheritance blocking, which means that whenever a defeasible property of a superclass is overridden in a concept C , none of the defeasible properties of the superclass can be inherited by C ;

• default properties do not apply to role fillers; • the conflicts that cannot be resolved with specificity are silently removed; this behavior makes it difficult to identify some gaps and missing axioms in the knowledge base. We refer the reader to [4,6] for a detailed description of these and further relevant problems. All of these limitations need to be addressed before rational closure can be applied in practice. It should be seen whether the ongoing research on these issues, mentioned in Section 2, can be combined with the idea of stable rational closure, and how.

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Acknowledgements The author is grateful to: Franz Baader and Laura Giordano for the stimulating conversations; Giovanni Casini for his friendly help in understanding the behavior of rational closure; the anonymous reviewers for their insightful and constructive comments that helped in improving the quality of the paper. Appendix A. Proofs A.1. Proofs for Section 5 Theorem 5.9. If r is stable, then RCr is closed under the KLM postulates restricted to D (the set of DIs of interest introduced at the beginning of Section 5). Proof. Assume that r is stable. We start by proving the following Claim: If F and G are in C and |= F  G, then r (G )  r ( F ). r  |ri  ¬G implies To prove the claim it suffices to note that its assumption implies |= ¬G  ¬ F . So, for all i  ω , K |= K

r  K |= K |ri  ¬ F . It follows, by stability, that r (G )  r ( F ). This proves the Claim. Now we demonstrate the closure of RCr with

respect to each postulate, assuming that their premises and conclusions are in D (so that for each such C < ∼ D, it holds that C and C ¬ D belong to C). r  |ri  ¬(C ¬C ). It follows, by stability, that (REF) Note that ¬(C ¬C ) is equivalent to , so for all i  ω , K |= K r (C ¬C ) = ω . Consequently, either r (C ) < r (C ¬C ) or r (C ) = ω . In both cases, (C < ∼ C ) ∈ RCr . (LLE) If |= C ≡ E, then C can be equivalently replaced with E in the stability conditions. It follows that r (C ) = r ( E ) and < r (C ¬ D ) = r ( E ¬ D ) (if the involved concepts are in C). It follows immediately that (C < ∼ D ) ∈ RCr iff ( E ∼ D ) ∈ RCr .

(RW) If |= C  E, then also |= C ¬ E  C ¬ D. By the Claim, r (C ¬ D )  r (C ¬ E ). It follows easily, by definition of < RCr , that if (C < ∼ D ) ∈ RCr then (C ∼ E ) ∈ RCr . (AND) If r (C ) = ω then the theorem trivially holds. Next, assume that r (C ) < ω and that the premises are in RCr . Then r (C ) < r (C ¬ D ) and r (C ) < r (C ¬ E ). Moreover, by the Claim,

r (C ¬ D )  r (C (¬ D  ¬ E )) r (C ¬ E )  r (C (¬ D  ¬ E )) therefore r (C ) < r (C (¬ D  ¬ E )) = r (¬(C E ), hence (C < ∼ D E ) ∈ RCr . (OR) If the first premise (C < ∼ E) is in RCr , then either r (C ) < r (C ¬ E ) or r (C ) = ω . In the former case, by stability,

 K |= K |rr (C )  ¬(C ¬ E ) . r

(8)

In the latter case, by the Claim, r (C ¬ E ) = ω , therefore (8) holds. Using a symmetric argument for the second premise, we derive that if both premises are in RCr , then (8) and the following entailment hold: r  K |= K |rr ( D )  ¬( D ¬ E ) .

(9)

  Now assume without loss of generality that r (C )  r ( D ) (the opposite case is symmetric). Then |= K |rr (C )  K |rr ( D ) , con-

r r   sequently, by (9), K |= K |rr (C )  ¬( D ¬ E ). Together with (8), this implies K |= K |rr (C )  ¬(C ¬ E ) ¬( D ¬ E ). By De Morgan’s laws and the distributivity of over , this entailment is equivalent to

r  K |= K |rr (C )  ¬((C  D ) ¬ E )

(10)

So, by stability, either r (C ) = ω or r (C ) < r ((C  D ) ¬ E ). In the first case, by the Claim and (10),

r (C  D )  r (C ) = r ((C  D ) ¬ E ) ; it follows easily that the consequent of (OR) is in RCr . Similarly, in the latter case, we can apply the Claim and obtain r (C  D )  r (C ) < r ((C  D ) ¬ E ), which implies that the consequent is in RCr . (CM) If r (C ) = ω then (by the Claim) r (C D ) = ω and (C D < ∼ E ) ∈ RCr . Otherwise (if r (C ) < ω ) assuming that the premises are in RCr , we have r (C ) < r (C ¬ D ) and r (C ) < r (C ¬ E ). This implies (by stability) that r  K |= K |rr (C )  ¬(C ¬ D )

(11)

 K |= K |rr (C )  ¬(C ¬ E ) .

(12)

r

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r r   By (11) and K  K |r r (C )  ¬C (stability), it follows that K  K |r r (C )  ¬(C D ), that is, r (C D )  r (C ). Then

r (C D )  r (C ) < r (C ¬ E )

 r (C D ¬ E )

by (12)

by the Claim

that is, (C D < ∼ E ) ∈ RCr . (RM) Assume the first premise is in RCr while C < ∼ ¬ D is not (i.e. r (C ) < ω and r (C ¬¬ D )  r (C )). Then

r (C D ) = r (C ¬¬ D )  r (C )

< r (C ¬ E )

(first premise)

 r (C D ¬ E ) It follows that (C D < ∼ E ) ∈ RCr .

(by the Claim).

2

Now we need several technical lemmas and propositions, in order to prove that RCr is sound and complete with respect r to |= . Lemma A.1. Let I be a ranked interpretation, let C be any concept, and K any DKB. 1. If minI () ⊆ C I then minI () = minI (C ). ). 2. If I |= K, then minI () = minI (K Proof. (Part 1) Suppose not; we shall derive a contradiction. By assumption, minI () ⊆ C I , and some of the following two cases hold: (a) There exists x ∈ minI (C ) such that x ∈ / minI (). Then there must be y
I . Suppose that this is not true; then I |= K but (Part 2) By Part 1, it suffices to prove that I |= K implies minI () ⊆ K there exists x such that x ∈ minI () ,

(13)

I . /K x∈

(14)

I I I I The latter implies that there must be (C < ∼ D ) ∈ K such that x ∈ (C ¬ D ) = C \ D . It follows, by (13), that x ∈ min (C ). Then x is a witness that minI (C )  D I , i.e. I  C < ∼ D. This contradicts the assumption that I |= K. 2

Next, we recall a well known property of the old notion of ranking. Lemma A.2. If rnk(C ) = ∞ and I is a model of E∞ , then C I = ∅.

∞ ) (Lemma A.1.(2)). Now, by definition of E∞ and Proof. Assume that rnk(C ) = ∞ and I |= E∞ . Then minI () = minI (E rank, E∞ |=  < ¬ C . It follows, by the above equation, that ∼

∞ ) ⊆ (¬C )I . minI (E

(15)

= ∅ (we shall derive a contradiction). There must be x0 ∈ min (C ). By (15), x ∈ / min (E∞ ), so there I I < must be some (C 1 ∼ D 1 ) ∈ E∞ such that x0 ∈ (C 1 ¬ D 1 ) . Note that x0 ∈ / min (C 1 ), otherwise I would not be a model Now suppose that C I

I

I

of E∞ . Then there must be x1 ∈ minI (C 1 ) such that x1
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Lemma A.3. Let r be stable w.r.t. K and C ∈ C. If r (C ) = ω and I is an upward-closed model of K, then C I = ∅.

  |rω  ¬C , so minI (K |rω ) ⊆ (¬C )I . This is the analogue of (15), and Proof. Suppose not, and let x ∈ C I . By stability, K |= K  from this point on the proof is similar to the proof of Lemma A.2 (replace E∞ with K |rω ). 2 r

 Lemma A.4. If I is upward-closed w.r.t. K and a ranking r for K, then for all concepts C ∈ C such that r (C ) < ω , minI (C ) ⊆ K |rr (C ) I .   Proof. Suppose not, and let x ∈ minI (C ) \ K |rr (C ) I . Then, by definition, rK (I , x) > r (C ) (because x ∈ /K |rr (C ) I ). By upward

closure, it follows that there exists y ∈ C I such that y
2

Proposition A.5. If r is stable and C , D ∈ C, then |= C  D implies r (C )  r ( D ). r  |rr (C )  ¬ D. Moreover, (i) is equivaProof. Suppose not, i.e. (i) |= C  D and (ii) r (C ) < r ( D ). By stability, (ii) implies K |= K

r  lent to |= ¬ D  ¬C ; it follows that K |= K |rr (C )  ¬C , which contradicts part 2 of stability.

2

Proposition A.6. Let r be a stable ranking, I be an upward-closed model of K w.r.t. r, and C ∈ C. If x ∈ C I , then rK (I , x)  r (C ). r   Proof. We prove the contrapositive. If rK (I , x) < r (C ), then for some i < r (C ), x ∈ (K |ri )I . By stability, K |= K |ri  ¬C . It

follows that x ∈ / CI.

2

Moreover, for technical reasons, we exploit a class of ranked interpretations that share a same normality relation, determined by individual ranks. Definition A.7 (rK-interpretations, r-models). Let r be a ranking function for a DKB K. An r K-interpretation is an interpretation I such that x r (C ) (since x does not satisfy a

 member of K |rr (C ) ). It follows that rK (J , y ) > r (C ). Note that rK depends only on (·)J , that equals (·)I by construction. Then rK (I , y ) > r (C ), too.

  |rr (C ) I , therefore y ∈ K |rr (C ) I . Then, by definition, rK (I , y )  r (C ) (a contraHowever, by (i) and Lemma A.4, minI (C ) ⊆ K diction). Finally, we prove that J is upward-closed. Pick any concept C ∈ C and any x ∈ C J such that rK (J , x) > r (C ). By con-

 struction, we also have x ∈ C I . Since C I = ∅ there exists y ∈ minI (C ), and since I is upward-closed, y ∈ K |rr (C ) I , by

Lemma A.4. Then, by definition, rK (J , y )  r (C ). It follows that rK (J , y ) < rK (J , x), hence y
Proof. If r (C )  r (C ¬ D ) then r (C ) = r (C ¬ D ) (by Proposition A.5). Then, by stability and the assumption that r (C ) < ω ,

r   KK |rr (C )  ¬(C ¬ D ). So there exist an upward-closed model I of K and x ∈ I such that: (i) x ∈ K |rr (C ) , (ii) x ∈ C I , (iii) I x∈ / D . By Lemma A.8 we can assume without loss of generality that I is an r K-interpretation. By (i), rK (I , x)  r (C ), so by Proposition A.6, rK (I , x)  rK (I , y ) for all y ∈ C I . Since I is an r K-interpretation, it follows r < that x ∈ minI (C ). As a consequence, by (iii), I  C < ∼ D, hence K  C ∼ D. 2

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r Lemma A.10 (Soundness). If r is stable w.r.t. K, and {C , C ¬ D } ⊆ C, then r (C ) < r (C ¬ D ) implies K |= C < ∼ D.

Proof. Suppose not, i.e. r (C ) < r (C ¬ D ) but there exists an upward-closed model I of K such that minI (C )  D I . Then r   there exists x ∈ minI (C ) \ D I . By Lemma A.4, x ∈ K |rr (C ) I . Moreover, by stability, K |= K |rr (C )  ¬(C ¬ D ). It follows that

x ∈ D I (a contradiction).

2

Theorem 5.10. Suppose that r is stable w.r.t. K, and that {C , C ¬ D } ⊆ C. Then r < (C < ∼ D ) ∈ RCr iff K |= C ∼ D .

Proof. A straightforward consequence of Lemmas A.3, A.10 and A.9.

2

A.2. Proofs for Section 6

 Lemma A.11. Let r be a ranking for K and suppose that for all C ∈ C with r (C ) < ω , minI (C ) ⊆ K |rr (C ) I . Then I is upward-closed w.r.t. K and r. Proof. Assume that I satisfies the lemma’s hypothesis. Let C be any concept in C, and let x be any individual such that (i) x ∈ C I and (ii) rK (I , x) > r (C ). Note that (ii) implies that r (C ) < ω . We have to prove that there exists y ∈ C I such that  y
2

Theorem 6.1 An interpretation I is upward-closed w.r.t. K and a ranking r for K iff for all concepts C ∈ C such that r (C ) < r  I |= C < ∼ K|r (C ) . Proof. An immediate consequence of Lemmas A.4 and A.11.

ω,

2

A.3. Proofs for Section 7

I . Lemma A.12. For each model I of a DKB K, there exists a model J of K such that J = I , (·)J = (·)I , and minJ () = K Proof. Obtain J from I by replacing
I ∧ y ∈ I ) ∨ (x, y ∈ I ∧ x
(Lemma A.1.(2)) so, by definition of
(16)

I . Now we have to show that K

= ∅ = J |= K. Since domain and extension function are preserved by J (and I is a model of K), J |= str(K). We are only left to prove that < J |= dfs(K). We assume the contrary, i.e. that for some (C < ∼ D ) ∈ K, J  C ∼ D, and derive a contradiction. By assumption, minJ (C )  D J = D I . Then there must be x0 ∈ minJ (C ) \ D I . Clearly x0 is also a member of C I , so it must be x0 ∈ / minI (C ), otherwise I would not be a model of K. Now note that

since x0 ∈ minJ (C ) \ minI (C ) it holds that x0 ∈  KI (because by def. of
(17)

  We prove its contrapositive. Assume that K  K |ri  ¬C and let I be an upward-closed model of K such that K |ri I  (¬C )I . Note that K ⊇ K|ri so I is a model of K|ri and, by Lemma A.12, K|ri has a model J such that (i) J |= str(K), and (ii)  minJ () = K |ri I  (¬C )I . By (i) and the induction hypothesis, J is also a model of Ei , so, by (ii), Ei   < ∼ ¬C . 2 r

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In order to prove that rational closure and its stable version coincide under the DMUP, we first need two lemmas: Lemma A.13 (“Canonical” models under the DMUP). Suppose that r is stable and that str(K) enjoys the DMUP. Every model I of str(K) ∪ K|rω that satisfies a concept C can be extended to an upward-closed model U of all K that satisfies C . r  Proof. Let I and C be as in the above statement. For all D ∈ C, if r ( D ) < ω then, by stability, K  K |rr ( D )  ¬ D. Then, there

exists an upward-closed model I D of K such that D I D = ∅. Select a set S of such models by choosing an I D for each D ∈ C (r (C ) = ω ), in such a way that the members of S are pairwise disjoint, and also disjoint from I . Now define U as follows:

U =



J J ∈S ∪{I }  P = J ∈S ∪{I } P J (where P

(


∪ {(x, y ) | ∃ D ∈ C : x ∈ min

is a concept name or a role name)

( D ) ∧ y ∈ minI ( D )} .

Note that, by construction, C U = ∅ (due to the elements imported from I ). Then we are left to prove that U |= K and U is upward-closed. To prove that U |= K, observe that U satisfies str (K) by the DMUP (because I and the members of S are all models of str (K)), so it suffices to prove that U |= dfs(K). U U U U Suppose not, and pick any (C 0 < ∼ D 0 ) ∈ K such that min (C 0 )  D 0 . Let x ∈ min (C 0 ) \ D 0 . Note that, by construction and by Lemma A.3, for all D ∈ C

if r ( D ) < ω, then D U = ∅ and minU ( D ) = U



J ∈S

minJ ( D ) ;

(18)

I

if r ( D ) = ω, then min ( D ) = min ( D ) .

(19)

So there are two possibilities:

(a) r (C 0 ) < ω . In this case, by (18), there must be some J ∈ S such that minJ (C 0 )  D J 0 . But then J would not be a model of K (a contradiction). r (b) r (C 0 ) = ω . In this case, by (19), minI (C 0 )  D I 0 . But then I would not be a model of K|ω (a contradiction).

We conclude from (a) and (b) that U must be a model of K. Finally, we show that U is upward-closed. Let D be any member of C, and suppose that y ∈ D U and rK (U , y ) > r ( D ). Note that r ( D ) < ω . Again, there are two possibilities:

(c ) y ∈ I . Then, by (18) and the definition of
(20)

To prove the claim, assume that K is satisfiable and Ei ⊇ K|ri . We prove the contrapositive of (20). Accordingly, suppose I r that Ei   < ∼ ¬C , and let I be a model of Ei (as well as K|i ⊆ Ei ) such that min ()  (¬C ). By Lemma A.1.(2), also

   minI (K |ri )  (¬C ), so I  K |ri  ¬C . Equivalently, (K |ri C )I = ∅.

  |ri C . It follows that K  K |ri  ¬C , By Lemma A.13, I can be extended to an upward-closed model of K that satisfies K and the Claim is proved. We are now ready to prove the Lemma. If K is unsatisfiable, then it is easy to see that for all i < ω , Ei = K and K|ri = dfs(K), so the theorem holds. If K is satisfiable, then the theorem holds for i = 0 by definition. Next assume, by induction hypothesis, that dfs(Ei ) = K|ri , and for all DIs δ let lhs(δ) denote the left-hand side of δ . r |r  ¬lhs(δ)} K|ri+1 = {δ ∈ (K|ri ) | K |= K i

⊆ {δ ∈ dfs(Ei ) | Ei |=  < ∼ ¬lhs(δ)}

r

(by stability) (by ind. hyp. and the Claim)

= E i +1 . From the above induction it follows that for all i < ω , dfs(Ei ) ⊇ K|ri . The Lemma follows immediately from Theorem 7.1.

2

P.A. Bonatti / Artificial Intelligence 274 (2019) 197–223

221

Theorem 7.3. Let r be stable w.r.t. K, and suppose that str(K) enjoys the DMUP. Then 1. for all i < ω , K|ri = dfs(Ei ); 2. for all concepts C , either r (C ) = rnk(C ), or r (C ) = ω and rnk(C ) = ∞; < 3. for all {C , C ¬ D } ⊆ C, (C < ∼ D ) ∈ RC(K) iff (C ∼ D ) ∈ RCr . Proof. Point 1 holds by Lemma A.14. To prove Point 2, note that for all C ∈ C with r (C ) < ω ,

|r  ¬C } (from stability) r (C ) = min{i | K  K i r

= min{i | Ei   < ∼ ¬C }

(using Point 1 and the proofs of (17) and (20))

= rnk(C ) . The same equations hold if rnk(C ) = ∞. Clearly, in all the other cases we have r (C ) = ω and rnk(C ) = ∞. Finally, Point 3 follows immediately from Point 2 and the definitions of RC(K) and RCr . 2 Lemma A.15. Let r be stable w.r.t. K, and let m = max{r (δ) | δ ∈ dfs(K) ∧ r (δ) < ω}. Then (a) m  |dfs(K)|; (b) for all C ∈ C such that r (C ) < ω , r (C )  m + 1. Proof. We are going to prove (a) by showing that “there are no holes in the ranking of DIs below claim holds:

ω”, that is, the following

Claim: If there exists δ ∈ dfs(K) such that r (δ) = i < ω , then for all j < i, there exists δ  ∈ dfs(K) with r (δ  ) = j. To prove the Claim, suppose that it does not hold, so there must be an index i < ω and δ ∈ dfs(K) such that r (δ) = i and

  no δ  ∈ dfs(K) has rank i − 1. Consequently, K|ri−1 = K|ri . By stability, K |= K |ri−1  ¬C and K  K |ri  ¬C . However, this is r r a contradiction (because K|i −1 = K|i ) and the Claim is proved. It follows immediately from the Claim that r (δ) cannot exceed |dfs(K)|, which proves (a). Note that (a) implies that for r r r all i > m, K|ri = Km +1 . Therefore, if m + 1 < i < ω , K|i −1 = K|i , and by analogy with the Claim’s proof it follows that no concept C ∈ C can have rank i. Point (b) immediately follows. 2 r

r

Theorem 8.2. If the set of relevant concepts C is finite and K is a SROIQ DKB, then checking whether a knowledge base K has a stable ranking is decidable, and the set of stable rankings of K can be effectively computed. Proof. By Lemma A.15, the set of stable rankings of K is contained in the set of functions C → [0, |dfs(K)| + 1] ∪ {ω}, which is finite (since C is finite) and can be effectively enumerated. For each ranking r in this set, it must be checked whether r is r  stable. This requires verifying whether K |= K |ri  ¬C holds, for all C ∈ C and all i  r (C ). By Corollary 6.2, these entailments r r  are equivalent to K ∪ {C < ∼ K|r (C ) | C ∈ C ∧ r (C ) < ω}  K|i  ¬C , that can be effectively checked with the tableaux system introduced in [9] or with the reduction to classical DL reasoning introduced by [19] (cf. Remark 6.3). 2 Appendix B. Drawbacks of E  Let Ei i be the ranking sequence induced by the exceptionality criterion E  (·) defined in (1), that is,

E0 = K    < Ei = {C < ∼ D | (C ∼ D ) ∈ Ei −1 ∧ K |= Ei −1  ¬C }

( i > 0) .

  ¬C }, if such i exists, and r (C ) = ∞ otherwise. The Let r be the resulting concept ranking, that is, r (C ) = min{i | K  E i corresponding notion of rational closure is RC (K) = {(C < ∼ D ) | r (C ) < r (C ¬ D ) ∨ r (C ) = ∞} . Like standard rational closure, RC (K) may contain inclusions that are not satisfied by some models of K. < Example B.1. Let K = { A 1 < ∼ B 1 , A 2 ∼ B 2 }. Note that all predicates have rank 0, and that r ( A 1 A 2 ) = 0 < r ( A 1 A 2 ¬ B 2 ) =  < 1. Consequently, ( A 1 A 2 < ∼ B 2 ) ∈ RC (K). However, A 1 A 2 ∼ B 2 is falsified by the model I of K such that:

222

1. 2. 3. 4.

P.A. Bonatti / Artificial Intelligence 274 (2019) 197–223

I = {x, y }, y
I I < To see that I is indeed a model of K, note that minI ( A 1 ) = {x} ⊆ B I 1 and min ( A 2 ) = { y } ⊆ B 2 . To see that I  A 1 A 2 ∼

B 2 , note that minI ( A 1 A 2 ) = {x}  B I 2.

2

The problem, in the above example, is the same that led to the introduction of canonical models in [23], namely: A I 2 is missing the most normal instance type admitted by K (in this case, an instance of A 2 that satisfies all the DIs with rank 0, including the first DI). In other words, I is not upward-closed. Recall that upward-closed models are our replacement for canonical models. Upward-closed models restrict the saturation condition of canonical models to nonempty concepts only, in order address Example 1.1. It can be proved that RC is sound w.r.t. upward-closed models. However, RC is not complete with respect to upward-closed models. To see this, consider the knowledge base K consisting of the following inclusions:  δ1 : A < ∼ A  δ2 : B < ∼B

 δ3 : A 1 < ∼ A B ¬∃U .( B (¬ A  A )) .

The reader may easily verify that A and B have rank 0, while A 1 must have higher rank, so E1 (K) = {δ3 }. Now we show  (K) = ∅. that A 1 has rank 1, therefore E∞   ¬ A 1 . The counterexample I has two elements, x and y, such that For this purpose, it suffices to show that K  E 1 x
A I = {x, y } ( A  )I = {x} B I = ( B  )I = A I 1 = { y} .

 I ∩ A I . This proves that K  E   Then minI ( A ) = {x} and minI ( B ) = minI ( A 1 ) = { y }. Note that I |= K and that y ∈ E 1 1 1

¬ A1 .

Note, however, that I is not upward-closed (rK (I , y ) > 0 = r ( B ) but y is the only instance of B). Indeed, we are now going to prove that A 1 must be empty in every upward-closed model of K. Suppose not, and let J be an upward-closed J J model of K such that A J 1 = ∅. Then there exists x ∈ min ( A 1 ). Clearly x shall satisfy the right-hand side of δ3 (otherwise J would not be a model of K). Accordingly, x must be an instance of A and B but not of A  , so x does not satisfy δ1 . Consequently, since δ1 ∈ E0 , rK (J , x) > 0 = r ( B ). Then, by upward-closure, there must be z0 ∈ B I such that z0
 J then there must be z1
 J , however, z shall be an instance of ¬ A  A  , so z must be an instance of  J . In order to satisfy E z ∈ minJ ( B ) ∩ E 0 0 B (¬ A  A  ). But then x cannot satisfy the right-hand side of δ3 , therefore J  K (a contradiction). This proves that RC is not complete w.r.t. the class of upward-closed models. Indeed, the above proof shows that A 1 < ∼⊥

holds in all upward-closed models of K, but it is not in RC (K) because r ( A 1 ) = 1 = ∞. r

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