Defeasible reasoning in Description Logics: an overview on DL^N
aa r X i v : . [ c s . A I] S e p Defeasible reasoning in Description Logics: anoverview on DL N Piero A. Bonatti , Iliana M. Petrova , and Luigi Sauro Dep. of Electrical Engineering and Information Technologies, Universit`a di NapoliFederico II, Italy
Abstract. DL N is a recent approach that extends description logicswith defeasible reasoning capabilities. In this paper we provide an overviewon DL N , illustrating the underlying knowledge engineering requirementsas well as the characteristic features that preserve DL N from some recur-rent semantic and computational drawbacks. We also compare DL N withsome alternative nonmonotonic semantics, enlightening the relationshipsbetween the KLM postulates and DL N . In complex areas such as law and science, knowledge has been in centuries for-mulated by primarily describing prototypical instances and properties, and thenby overriding the general theory to include possible exceptions. For example,many laws are formulated by adding new norms that, in case of conflicts, maypartially or completely override the previous ones. Similarly, biologists have beenincrementally introducing exceptions to general properties. For instance, the hu-man heart is usually located in the left-hand half of the thorax. Still there areexceptional individuals, with so-called situs inversus , whose heart is located onthe opposite side. Eukariotic cells are those with a proper nucleus, by definition.Still they comprise mammalian red blood cells, that in their mature stage haveno nucleus. Also many modern applications and methodologies in Computer Science relyon some sort of overriding mechanism. In Object Oriented Programming the def-initions in a subclass may override any conflicting bindings belonging to its su-perclasses. Analogously, formal languages designed to describe role-based accesscontrol or other privacy policies generally allow to formulate default conditions,such as open and closed policies, conflict resolution methods such as denialstake precedence , and authorization inheritance with exceptions [14].Summarizing, the mentioned fields manifest to a large extent different formsof defeasible knowledge where general axioms can be recanted in special casesby employing some suitable overriding mechanism. Nevertheless, this naturalapproach cannot be directly adopted in designing Semantic Web ontologies. In All of these examples are introduced and discussed in [39,41]. If no explicit authorization has been specified for a given access request, then anopen policy permits the access while a closed policy denies it. Piero A. Bonatti, Iliana M. Petrova, and Luigi Sauro fact, the underlying descriptions logics (DLs), which are based on the monotonicsemantics of FOL, do not allow to express and reason on defeasible knowledgeand exceptions. Consequently, several authors advocated nonmonotonic logicsas a useful means to address this limitation and proposed different formalismsbased on circumscription [10,9,11], autoepistemic logic [24,25], typicality opera-tors [26,28,31], or rational closure [29,20,6], just to mention a few.In this context, DL N [7,15,13] is a recent family of nonmonotonic DL specif-ically designed to meet the knowledge engineering requirements that come fromthe aforesaid application domains. DL N is prototype oriented: it uses N C todenote the normal/prototypical instances of a concept C , and extends termino-logical axioms with prioritized defeasible inclusions (DIs) C ⊑ n D .A difficulty arising at a design level is that the notion itself of prototype canbe inherently ambiguous. Prototypes may represent in a frequentistic fashion theproperties that are shared by the majority of the instances, or they can differentlybe interpreted idealistically as platonic models that might not exist in the realworld due to their degree of perfection. DL N does not aim at encompassingall different, and philosophically interesting, notions of prototype; being plainlyapplication-oriented, it is rather inspired by what McCarthy calls communicationand database storage conventions [37]. In this perspective, a prototype N C ismeant to factorize the common features of the concept C and confine exceptionalsubclasses to an explicit detailed axiomatization (so as to reduce the size and costof knowledge bases and improve their readability). Thus, defeasible inclusions C ⊑ n D mean (roughly speaking): “ by default, all prototypical instances thatsatisfy C satisfy also D , unless stated otherwise ”, that is, unless some higherpriority axioms contradict this implication. If such a contradiction arises, then C ⊑ n D is overridden . The standard/prototypical instances of C are required tosatisfy all the DIs that are not overridden in C .As mentioned above, also other nonmonotonic logics support defeasible in-heritance with overriding in general. Nevertheless, in each of these previous ap-proaches, either some desiderable features are missing or some natural inferencesdo not hold. Moreover, they are generally based on complex semantics whichmake defeasible reasoning difficult to track. In this respect, DLN’s behavior iseasier to grasp, and is expected to facilitate knowledge engineers in formulatingand validate ontologies at a large scale, while producing the expected conclu-sions.Finally, apart from rational closure and restricted forms of typicality [30], de-feasible reasoning significantly increases the computational complexity of stan-dard reasoning tasks even in low-complexity description logics [27,8,12]. Con-versely, in all DL fragments of pratical interest, DL N does not manifest a highercomplexity with respect to the classical counterpart. Moreover, efficiency can befurther enhanced through a range of optimization techniques, including modu-larization [13]. For example, circumscription identifies, in case of conflicting nonmonotonic axioms,all optimal repairs and then computes the inferences that hold for all repairs.efeasible reasoning in Description Logics: an overview on DL N ⊤ ∆ I bottom ⊥ ∅ negation ¬ C ∆ I \ C I conjunction C ⊓ D C I ∩ D I disjunction C ⊔ D C I ∪ D I ∃ 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 some common constructs.
This paper is meant to illustrate DL N and its main features, extending pre-vious discussions of DL N ’s properties with some recent contributions to rationalclosure. The paper is organized as follows. In the next section we briefly recallthe basics of monotonic description logics. In Section 3 we introduce DL N andprovide a few examples of knowledge bases and inference. Section 4 compares DL N with the other major nonmonotonic DLs in terms of practical engineeringrequirements, and in terms of logical properties, centred around the KLM pos-tulates. The paper is concluded by a summary and a list of interesting topics forfurther work. Description logics are a family of formal languages representing the logical foun-dations of the W3C Ontology Web Language (OWL2). They offer a variegatedset of logical constructors and axioms that balance between expressiveness andcomputational complexity according to the application needs. Due to space limi-tations, we refer to [5] for a comprehensive overview. Here, we just introduce theDL fragment
ALC , which allows to understand the examples that will follow.An alphabet or signature consists of a set N C of concept names , a set N R of role names , and (possibly) a set N I of individual names (all countably infinite).Thereafter, metavariables A , B will range over concept names, R and S overroles, and a , b and d over individual names. The term predicate will refer to ageneric element of N C ∪ N R .In DLs, a wide range of operators allow to inductively formulate compoundconcepts . The logic ALC , in particular, compound concepts are defined by thefollowing grammar:
C, D ::= A | ⊤ | ⊥ | ¬ C | C ⊓ D | C ⊔ D | ∃ R.C | ∀
R.C .
Note, however, that our framework applies also to more expressive DLs such as
SROIQ ( D ) that constitutes the foundation of the full standard OWL2.The semantics of DLs is defined in terms of interpretations I = h ∆ I , · I i . The domain ∆ I is a non-empty set of individuals and the interpretation function · I Piero A. Bonatti, Iliana M. Petrova, and Luigi Sauro maps each concept name A ∈ N C to a subset A I of ∆ I , each role name R ∈ N R to a binary relation R I on ∆ I , and each individual name a ∈ N I to an individual a I ∈ ∆ I . The extension of · I to ALC compound concepts is inductively definedas shown in the third column of Figure 1. An interpretation I is called a model of a concept C if C I = ∅ .A (general) TBox is a finite set of concept inclusions (CIs) C ⊑ D . As usual,we use C ≡ D as an abbreviation for C ⊑ D and D ⊑ C . An ABox is a finiteset of concept assertions C ( a ) and role assertions R ( a, b ). An interpretation I satisfies (i) a CI C ⊑ D if C I ⊆ D I , (ii) an assertion C ( a ) if a I ∈ C I , and (iii)an assertion R ( a, b ) if ( a I , b I ) ∈ R I . Then, I is a (classical) model of a TBox T (resp. an ABox A ) if I satisfies all the members of T (resp. A ).In this paper, we will sometimes mention some important DLs that havebeen extensively studied in the literature and constitute the foundation of se-mantic web standards. The logic EL supports only ⊤ , ⊓ , and ∃ . Its extension EL ⊥ supports also ⊥ . The logic EL ++ further adds concrete domains and someexpressive role inclusions (see [2] for further details).The logic DL-lite R [17] supports inclusions shaped like C ⊑ D and C ⊑ ¬ D ,where C and D range over concept names and unqualified existential restric-tions such as ∃ R and ∃ R − (where R − is the inverse of role R ). EL ++ andDL-lite R , respectively, constitute the foundation of the OWL2 profiles OWL2-EL and OWL2-QL. Both play an important role in applications; their inferenceproblems are tractable (the same holds for some extensions of DL-lite R , see [1]).Finally, we will use in Section 4.2 boolean combinations of assertions andinclusions. Although these axioms are not directly allowed in DLs, they can besimulated in SROIQ ( D ) through the universal role U . For example, ¬ ( C ⊑ D )and ( C ⊑ D ) ∨ ( C ⊑ D ) can be expressed as ⊤ ⊑ ∃ U. ( C ⊓ ¬ D ) and ⊤ ⊑ ( ∀ U. ( ¬ C ⊔ D )) ⊔ ( ∀ U. ( ¬ C ⊔ D )), respectively. DL N Given a classical description logic language DL , let DL N be the extension of DL with a new concept name N C for each DL concept C . N C is called a normalityconcept and denotes the normal instances of C .A DL N knowledge base is a disjoint union KB = S ∪ D such that – S is a finite set of DL concept inclusions and assertions; – D is a finite set of defeasible inclusions (DIs, for short) C ⊑ n D where C and D are DL N concepts.Thereafter, given a DI δ = C ⊑ n D , by pre ( δ ) and con ( δ ) we denote C and D ,respectively. A knowledge base is canonical if pre ( δ ) does not contain normalityconcepts, for all δ ∈ D .Roughly, C ⊑ n D means: “ the normal instances of C are instances of D ,unless stated otherwise by some higher priority axioms ”. As mentioned in theintroduction DIs have an utilitarian purpose. They are meant to factorize the efeasible reasoning in Description Logics: an overview on DL N common properties that hold for normal entities, so as to minimize the amountof knowledge that must be explicitly encoded.Defeasible inclusions are prioritized by a strict partial order ≺ over D . Theintended meaning of δ ≺ δ is that δ has higher priority than δ and, in caseof conflicts, it is preferable to sacrifice δ . DL N solves automatically only theconflicts that can be settled using ≺ . Any other conflict shall be resolved bythe knowledge engineer (typically by adding specific DIs). Here, we focus ona priority relation which is determined by so-called specificity . Roughly speak-ing, specificity states that, in case of conflicts, the specific properties of pre ( δ )override those of the more general concept pre ( δ ): δ ≺ δ iff S | = pre ( δ ) ⊑ pre ( δ ) and S 6| = pre ( δ ) ⊑ pre ( δ ) . (1)Note that DL N is largely parametric with respect to which priority relation isused. An alternative choice could be, for example, a priority relation based onthe ranking function of rational closure adopted in [20].Due to space limitations, we refer to [7] for the model-theoretic semantics of DL N . Here, we present only its reduction to classical reasoning.Let KB = S ∪D be a DL N knowledge base and α a query of interest, that canbe either a CI or an assertion. By KB |≈ α we mean that α is a DL N semanticconsequence of KB .The classical reduction of |≈ requires some preliminary notions: – For all DIs δ ∈ D and all normality concepts N C ∈ Σ , let δ N C = (cid:0) N C ⊓ pre ( δ ) ⊑ con ( δ ) (cid:1) ; – for all sets of DL axioms S ′ and all DIs δ , let S ′ ↓ ≺ δ denote the result ofremoving from S ′ all the axioms δ N C such that δ ’s priority is not higherthan δ ’s: S ′ ↓ ≺ δ = S ′ \ { δ N C | N C ∈ Σ ∧ δ δ } ; – finally, let δ , . . . , δ |D| be an arbitrary linearization of ( D , ≺ ), which meansthat { δ , . . . , δ |D| } = D and for all i, j = 1 , . . . , |D| , if δ i ≺ δ j then i < j .Then, KB |≈ α holds iff KB Σ | = α , where Σ is the set of normality conceptsoccurring in both KB and α , and KB Σ is the classical knowledge base resultingfrom the following inductive construction (where i = 1 , , . . . , |D| ): S Σ = S ∪ (cid:8) N C ⊑ C | N C ∈ Σ (cid:9) (2) S Σi = S Σi − ∪ (cid:8) δ N Ci | N C ∈ Σ and S Σi − ↓ ≺ δ i ∪{ δ N Ci } 6| = N C ⊑ ⊥ (cid:9) (3) KB Σ = S Σ |D| . (4)In informal terms, the first step extends S with the axioms N C ⊑ C statingthat the normal instances of C are a fortiori instances of C . The constructionproceeds by processing the DIs δ i ∈ D in decreasing priority order; if adding δ i to the (higher priority) δ j ≺ δ i that have been previously selected does not makeN C inconsistent, as stated by (3), then δ N Ci is included in KB Σ , otherwise δ N Ci is discarded (overridden). Piero A. Bonatti, Iliana M. Petrova, and Luigi Sauro
Example 1.
Recall that situs inversus refers to humans whose heart is on theright-hand side of the thorax, differently from typical humans whose heart is onthe opposite side. If we stipulate that no heart can be simultaneously located onboth sides, then a simple axiomatization is:
Human ⊑ n ∃ has heart . LH (5) SI ⊑ Human (6) SI ⊑ ∃ has heart . RH (7) ∃ has heart . LH ⊑ ¬∃ has heart . RH (8) where LH (resp. RH ) denotes left-positioned (resp. right-positioned) hearts and SI stands for situs inversus.Since S ⊆ KB Σ , by (7) and (8) we have that the instances of SI have theirheart on the right-hand side (and hence not on the left-hand side): KB |≈ SI ⊑ ∃ has heart . RH (9) KB |≈ SI ⊑ ¬∃ has heart . LH . (10)Then, let Σ = { N Human } . It is straightforward to see that KB Σ consists ofthe strong axioms (6) – (8), plus N Human ⊑ Human (11)
Human ⊓ N Human ⊑ ∃ has heart . LH . (12) Consequently, we have that
KB |≈ N Human ⊑ ∃ has heart . LH . (13)Moreover, as a classical consequence of the above inferences, one can furtherconclude that people with situs inversus are not standard humans: KB |≈ SI ⊑ ¬ N Human . (14) Conversely, if Σ = { N SI } , the iterative construction of KB Σ adds in first stepthe axiom N SI ⊑ SI , (15) then the DI (5) is overridden, since adding Human ⊓ N SI ⊑ ∃ has heart . LH (16) would make, together with axioms (6), (7), (8), and (15), N SI inconsistent.Consequently, N SI is simply a consistent subclass of SI that does not satisfyany further property.Now, extend KB with the additional DI: Human ⊑ n ∃ has organ . Nose . (17) Note that (17) and (5) have both maximal priority (indeed, they are incompa-rable by specificity). It is easy to see that (17) is overridden neither in N
Human nor in N SI , therefore both of the following inferences are valid: KB |≈ N Human ⊑ ∃ has organ . Nose (18)
KB |≈ N SI ⊑ ∃ has organ . Nose . (19) In other words, the property of having a nose is inherited even if (14) make SI exceptional w.r.t. Human . ⊓⊔ efeasible reasoning in Description Logics: an overview on DL N Example 2.
Consider the following variant of Nixon’s diamond [7]:
Quaker ⊑ n Pacifist , (20) Republican ⊑ n ¬ Pacifist , (21) RepQuaker ⊑ Republican ⊓ Quaker . (22) Note that the two DIs (20) and (21) are incomparable under specificity. More-over, since they both can be individually satisfied by N
RepQuaker , without mak-ing it inconsistent, none of them is overridden in N
RepQuaker . It follows thatN
RepQuaker must satisfy both DIs and consequently
RepQuaker is associated toan inconsistent prototype:
KB |≈ N RepQuaker ⊑ ⊥ . (23) Note that even if the prototype of
RepQuaker is inconsistent, the knowledgebase is consistent, as well as many normality concepts. In particular, we have
KB 6|≈ N Quaker ⊑ ⊥ and
KB 6|≈ N Republican ⊑ ⊥ .Moreover, consequence (23) cannot be resolved by logic since (20) and (21)are perfectly symmetric w.r.t. N
RepQuaker . Removing the inconsistency is upto the knowledge engineer which has to choose how to repair KB . For instance,adding RepQuaker ⊑ n Pacifist (resp.
RepQuaker ⊑ n ¬ Pacifist ) resolves theconflict in favor of the first (resp. second) DI. ⊓⊔ Remark 1.
The other nonmonotonic semantics of DLs silently “hide” unresolvedconflicts, by deriving none of the conflicting properties. The result is a gap inthe knowledge base. For example, Nixon was notoriously not a pacifist, and untilthe conflict is resolved, this information is not accessible to reasoners. In otherexamples, such knowledge gaps may have important consequences [7]. Unlikethe other nonmonotonic logics, DL N helps knowledge engineers in identifyingthe gaps caused by unresolved conflicts . Searching for inconsistent prototypesis analogous to the classical KB debugging activity consisting in identifyinginconsistent concepts, and all engines support it. Example 3.
In several countries (e.g. Mexico, Norway and Brazil) military ser-vice is mandatory for male citizens (except for special cases such as mentaldisorders). After military training, citizens become reservists , and shall join thearmy again in case of war. This can be formalized with the following DIs:
MaleCitizen ⊑ n HasMilitaryTraining (24)
MaleCitizen ⊓ HasMilitaryTraining ⊑ n Reservist . (25)The exceptions to the above rules include minors: MinorMaleCitizen ⊑ MaleCitizen (26)
MinorMaleCitizen ⊑ ¬
HasMilitaryTraining . (27)Axiom (27) should prevent (25) from being applied to minors, that is, it should not be possible to conclude that N MinorMaleCitizen ⊑ Reservist (indeed,this is what happens with DL N ). ⊓⊔ Piero A. Bonatti, Iliana M. Petrova, and Luigi Sauro
For what concerns the computational complexity of DL N , notice that theiterative construction of KB Σ requires, at each step i ∈ { , . . . , |D|} , (i) torestrict S Σi to the DIs that have a higher priority than δ i , and (ii) to evaluate | Σ | consistency checks. If the priority relation is based on specificity, checkingwhether δ j ≺ δ i consists in solving two subsumption problems and hence it hasthe same complexity as entailment in DL . Also the second point simply comesdown to classical reasoning in the underlying DL . Consequently, DL N entailmenthas the same complexity as in DL . In general, considering that different priorityrelations can be used, the following characterization holds. Theorem 1.
Let DL be a DL fragment such that subsumption (resp. instance)checking in DL belongs to a complexity class C , and deciding the preferencerelation ≺ belongs to P C . If DL supports ⊓ in the left-hand side of inclusions,then subsumption (resp. instance) checking in DL N is in P C . Since P P equals P, the entailment problem KB |≈ α is tractable in low com-plexity description logics such as ( DL-lite ( HN ) horn ) N [1] and ( EL ++ ) N . Similarly,Theorem 1 tells us that SROIQ N reasoning is in P N2ExpTime for suitable priorityrelations.
The DL N family of logics results from a utilitarian way of approaching nonmono-tonic logic design. The main goal of this approach is addressing the practicalneeds of ontology and policy designers, that have been illustrated with severalexamples in the literature on biomedical ontologies and semantic web policies.Here is a summary of the main shortcomings addressed by DL N (see [7] for moredetails and explanations): – Inheritance blocking . Most of the logics grounded on preferential semanticsand rational closure block the inheritance of all default properties towardsexceptional subclasses (as opposed to overriding only the properties that aremodified in those subclasses). DL N ’s overriding mechanism does not sufferfrom this drawback (see Example 1). – Undesired CWA effects . Many nonmonotonic DLs extend default propertiesto as many individuals as possible, thereby introducing CWA (i.e. closed-world assumption) effects that clash with the intended behavior of ontologies.For instance, exceptional concepts (such as SI in Example 1) collapse to thelist of constants that are explicitly asserted to be in the concept (and if nosuch constants exist, exceptional concepts become inconsistent). DL N doesnot introduce any CWA effect because it does not force individuals to benormal, unless explicitly stated otherwise. P C is the class of all problems that can be solved by a deterministic Turing machinein polynomial time using an oracle for C .efeasible reasoning in Description Logics: an overview on DL N – Control on priorities . Since priorities are not fixed a priori in DL N , knowledgeengineers can adapt them to their needs. In principle, it is possible to overrideDIs based on temporal criteria (which may be useful in legal ontologies andontology versioning), define default conflict resolution criteria, and even userational closure’s specificity-based axiom ranking. The logics derived frominheritance networks, preferential semantics, and rational closure can onlysupport their fixed, specificity-based overriding criterion. – Default role fillers . Should role values be restricted to normal individuals?Sometimes, this kind of inference is desirable, sometimes it is not, cf. [7].Some logics are completely unable to apply default properties to role values. Some others cannot switch this inference off when it is not desired. Only DL N and ALC + T min make it possible to control this kind of inference. Due to itsexplicit priorities, DL N is also able to encode a design pattern that makesrole ranges normal whenever this does not override any explicit DI. – Inconsistent prototype detection . We argued that when conflicts cannot besettled by priorities, silent conflict resolution is not a desirable feature: knowl-edge engineers should be involved because there is no universally correctautomated resolution criterion (cf. Example 2). Only DL N and probabilis-tic description logics (and ALC + T min , in some very specific cases) detectinconsistent prototypes and make them evident, as advocated in Remark 1. – Unique deductive closure . As a result of automated conflict resolution, severalnonmonotonic logics yield multiple deductive closures, corresponding to allthe alternative ways of solving each conflict. DL N is one of the logics thathas a unique closure. – Generality . Nonmonotonic extensions should be applicable to all descriptionlogics, or at least to the standard OWL2-DL (i.e. the logic
SROIQ ( D )).Typicality logics and rational closure, instead, are limited to logics thatsatisfy the disjoint union model property . Recently, it has been shown thatfor expressive DLs that do not enjoy this property, syntactic inference doesnot match semantics [6]. The same paper introduces stable rational closure that solves the generality problem for rational closure, but re-introduces theissue of multiple (or non existent) deductive closures. It is currently not clearhow to design a logic that satisfies the KLM postulates, is fully general, andyields a unique closure for all knowledge bases. – Low complexity . DL N preserves the tractability of these reasoning tasks forall low-complexity DLs, including the rich tractable logics EL ++ and DL-lite ( HN ) Horn . Currently, no other nonmonotonic DL enjoys this property to thesame extent. Rational closure has been proved to be tractable for EL ex-tended with ⊥ [23,38]. Some logics, such as [20,19,22,31,28], preserve theasymptotic complexity of ExpTime-complete DLs like ALC . More generally, DL N preserves the asymptotic complexity of all the DLs that belong to a This is the case for rational closure. Recently, in [38], a solution has been proposedfor EL with ⊥ . It is unclear how to extend it to more expressive DLs, and it is notpossible to “turn off” the application of default rules to role fillers.0 Piero A. Bonatti, Iliana M. Petrova, and Luigi Sauro deterministic complexity class that contains P. For nondeterministic com-plexity classes C , an upper bound is P C . DL N has been designed to address the above practical issues. In general, theutilitarian approach led us to make DL N neutral with respect to the inferencesthat are not always desired: when possible, DL N gives knowledge engineers theability of switching those inferences on and off. The final result of this investiga-tion is a logic that enjoys a unique set of properties, as shown by the summaryin Table 1. Table 1.
Summary of comparisons with nonmonotonic DLsCIRC DEF AEL TYP RAT PR
Features [12,11] [3,4] [25] [31,28] [20,19] [22] [36] DL N no inheritanceblocking X X X X X X no CWA effects
X X X X X fine-grainedcontrolon role ranges X (1) X detects in-consistentprototypes X (1) X (2) X unique deductiveclosure X X X preserves tractabil-ity X generality X X X X X implicit specificity
X X X X other priorities
X X X (1) Partially supported.(2) Inconsistency may propagate to the entire KB.
We deliberately refrained from adding a priori any requirements that are notdirectly motivated by applications, such as the KLM postulates. Interestingly, DL N satisfies many of those postulates, though, as illustrated in the next section. DL N and the KLM axioms In this section we analyze the logical properties of DL N through the KLM postu-lates. In [33,35,34], Kraus, Lehmann, and Magidor argued that in order to reasonabout what normally holds in the world, it is desirable to make nonmonotonicconsequence relations closed under certain properties, called KLM postulates . efeasible reasoning in Description Logics: an overview on DL N Table 2.
The KLM postulates in DL N Name Rule schema Sound in DL N REF α ∈ KBKB |≈ α X CT KB |≈ α KB ∪ { α } |≈ γ KB |≈ γ X CM KB |≈ α KB |≈ γ KB ∪ { α } |≈ γ X LLE
KB ∪ { α } |≈ γ | = α ≡ β KB ∪ { β } |≈ γ X RW KB |≈ α α | = γ KB |≈ γ X OR KB ∪ { α } |≈ γ KB ∪ { β } |≈ γ KB ∪ { α ∨ β } |≈ γ under extra axiomsRM KB |≈ γ KB 6|≈ ¬ α KB ∪ { α } |≈ γ under extra axioms KB is a canonical DL N knowledge base; α and β range over DL assertions and (strong) concept and role inclusions; γ ranges over DL N assertions and DL N concept/role inclusions;nonstandard DL axioms α ∨ β , ¬ β can be simulated, e.g. with the universal role; |≈ denotes the nonmonotonic consequence relation of DL N and | = classical inference. Although these postulates are not necessarily desiderata, due to the loose corre-spondence between their motivations and DL N ’s goals and semantics (cf. [7,15]),we regard them as a useful technical tool for comparison, since the validity ofthe postulates has been extensively investigated in most nonmonotonic logics.There exist several versions of the postulates; all of them contain postulatesthat are incompatible with DL N ’s novel way of highlighting unresolved con-flicts through inconsistent prototypes, for debugging purposes. So, hereafter, weassume that all unresolved conflicts have been fixed , as recommended by thisknowledge engineering methodology.The first version of the postulates – illustrated in Table 2 – is the verbatiminstantiation of the original, meta-level postulates. A consequence relation thatsatisfies the KLM postulates is called rational . It is called preferential if it sat-isfies all rules but RM, and cumulative if it satisfies all rules but RM and OR.With respect to this version of the postulates, DL N ’s consequence relation ( |≈ )is cumulative . Obviously, REF and RW always hold, because DL N is closed under classical infer-ence. Moreover rules CT, CM, and LLE are sound by [15, Thm 1].2 Piero A. Bonatti, Iliana M. Petrova, and Luigi Sauro It is interesting to note that the rational closure of DLs itself is not rationalw.r.t. this version of the postulates, e.g. it fails to satisfy the OR rule [15]. Onthe contrary, DL N can be made fully rational by making it a little more similarto typicality logic and rational closure, in the following respect. The semantic oftypicality and rational closure forces each consistent concept to have a normalinstance, through the smoothness property of preferential models and the notionof canonical model [29]. A similar condition can be enforced in DL N through theaxioms ¬ (N C ⊑ ⊥ ), for all consistent C occurring in KB or in the query. Theknowledge bases so extended satisfy all the postulates in Table 2. The aboveresults provide an immediate comparison with the consequence relations of Cir-cumscribed DLs (that are preferential) and those of Default and AutoepistemicDLs (that are not cumulative).
Name Rule schema SoundREF n C ⊑ n C X CT n C ⊑ n D C ⊓ D ⊑ n EC ⊑ n E CM n C ⊑ n D C ⊑ n EC ⊓ D ⊑ n E LLE n C ⊑ n E S | = C ≡ DD ⊑ n E RW n C ⊑ n D S | = D ⊑ EC ⊑ n E OR n C ⊑ n E D ⊑ n EC ⊔ D ⊑ n E RM n C ⊑ n E C n ¬ DC ⊓ D ⊑ n E S is the strong part of a knowledge base Table 3.
Analogues of the KLM pos-tulates for DIs [7] Name Rule schema SoundREF N N C ⊑ C X CT N N C ⊑ D N( C ⊓ D ) ⊑ E N C ⊑ E partlyCM N N C ⊑ D N C ⊑ E N( C ⊓ D ) ⊑ E partlyLLE N N C ⊑ E C ≡ D N D ⊑ E RW N N C ⊑ D D ⊑ E N C ⊑ E X OR N N C ⊑ E N D ⊑ E N( C ⊔ D ) ⊑ E partlyRM N N C ⊑ E N C
6⊑ ¬ D N( C ⊓ D ) ⊑ E partly C , D , and E range over DL concepts Table 4. DL N Candidate inferencerules inspired by KLM postulates
Several logics internalize the nonmonotonic consequence relation and pushthe KLM postulates to the object level (e.g. [20,21,22,18,16,28,29]). The result-ing postulates for DL N are reported in Table 3. Their validity is clearly affectedby overriding. For instance, if the second premise of CM n were overridden, thenthere would be no logical ground for supporting the conclusion. However, if thepremises are not overridden, then all the postulates of Table 3 are valid in DL N .It is interesting to note that a similar phenomenon can be observed in Lehmannsaccount of default reasoning [34]. In Sec. 6 Lehmann exhibits a knowledge base efeasible reasoning in Description Logics: an overview on DL N with no (consistent) rational closure; however it has a lexicographic closure be-cause the latter ignores all overridden defaults. The second interesting remarkis that two of these postulates unconditionally hold in most practically interest-ing cases: (i) the OR n rule holds if the priority relation is specificity; (ii) LLE n holds whenever the priority relation is not sensitive to syntactic details i.e. treatslogically equivalent DIs in the same way (like specificity does).Another internalized version of the postulates, analogous to those satisfiedby typicality logics [28,29] is reported in Table 4. It can be shown that typicalityDLs satisfy these postulates only because the normality criterion is assumed tobe concept-independent (i.e. if John is more typical than Mary as a driver, thenhe must also be more typical than Mary as a worker, as a tax payer, and soon) [15]. DL N does not adopt this strong assumption: in DL N each concept mayhave its own notion of what is more normal or standard and – for this reason –it does not universally satisfy CT N and CM N . So an interesting open question iswhether the postulates of Table 4 can possibly be satisfied by a logic that doesnot rely on a unique, concept-independent normality relation.Again, DL N can be made fully rational (with respect to this version of thepostulates) by making it more similar to rational closure. Consider the restrictionof DL N where N does not explicitly occur in KB (N can be used only in queries).In practice, this means that role fillers cannot be forced to be normal, similarlyto what inevitably happens in rational closure and default DLs, due to thelimitations of these logics. Under this restriction, all postulates in Table 4 hold[15, Thm 4, 5, 6]. DL N addresses a number of drawbacks that affect the nonmonotonic seman-tics of DLs. It has been designed with a particular attention to practical issuesthat hinder the adoption of nonmonotonic semantics in OWL2 and its profiles,including expressiveness limitations, and complexity problems. Moreover, DL N reasoning can be easily reduced to classical reasoning, thereby leveraging thehigh-quality, well-engineered implementations of DL reasoning. In the light ofthese properties, DL N compares favorably to the other nonmonotonic DLs, assummarized in Table 1.Interestingly, even if KLM postulates played no role in DL N ’s design, DL N satisfies to a large extent the major meta-level and internalized versions of thepostulates. The postulates in Table 3 hold up to overriding, as in Lehmann’slexicographic closure. Moreover, DL N is flexible enough to satisfy all the otherpostulates by means of additional axioms or syntactic restrictions that make DL N more similar to typicality logics and rational closure.Many different directions deserve further investigation. From a semanticviewpoint, we mentioned in Section 4.2 that the KLM postulates OR and RMare sound in DL N under the assumption that consistent concepts have at leastone typical individual. Taking inspiration from typicality logics, variants of DL N can be investigated where this assumption is hard-coded in the semantics. This should be done with some care, though: if too many individuals were forced tobe normal, then the undesirable closed-world assumption effects described in [7],that affect typicality logics, might be introduced in DL N .The study of the logical properties of DL N can be refined by investigatingthe mutual relationships between DIs and their effects on normality concepts. Inparticular, it would be interesting to investigate hybrid versions of the postulates,whose premises are taken from Table 3 while consequents are taken from Table 4.Finally, one of the primary strengths of DL N is that it preserves the tractabil-ity of the low-complexity DLs underlying the OWL2-EL and OWL2-QL profiles.However, asymptotic tractability alone does not suffice for practical purposes. In[13], two optimization techniques have been successfully applied to obtain real-time query answering over large knowledge bases. One optimization is basedprecisely on a suitably modified module extraction algorithm, that so far con-stitutes the most effective optimization technique for DL N (excluding combinedapproaches). The other optimization, called optimistic method , reduces the num-ber of retractions (an expensive class of operations in incremental reasoning). DL N ’s module extractor, however, proved to be less effective for KBs that con-tain many explicit occurrences of the normality concepts, and for those withnonempty ABoxes (due to the lesser effectiveness of the underlying classicalmodule extractors in such contexts). To overcome these problematic cases, weplan to improve the module extractor for DL N by discarding the normality con-cepts (and related axioms) and assertions that are irrelevant to a given query. References
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