A framework for a modular multi-concept lexicographic closure semantics
aa r X i v : . [ c s . A I] S e p A framework for a modular multi-conceptlexicographic closure semantics
Laura Giordano and Daniele Theseider Dupr´e
DISIT - Universit`a del Piemonte Orientale, Italy [email protected], [email protected]
Abstract.
We define a modular multi-concept extension of the lexicographic clo-sure semantics for defeasible description logics with typicality. The idea is that ofdistributing the defeasible properties of concepts into different modules, accord-ing to their subject, and of defining a notion of preference for each module basedon the lexicographic closure semantics. The preferential semantics of the knowl-edge base can then be defined as a combination of the preferences of the singlemodules. The range of possibilities, from fine grained to coarse grained modules,provides a spectrum of alternative semantics.
Kraus, Lehmann and Magidor’s preferential logics for non-monotonic reasoning [41,42],have been extended to description logics, to deal with inheritance with exceptions inontologies, allowing for non-strict forms of inclusions, called typicality or defeasibleinclusions , with different preferential and ranked semantics [29,17] as well as differ-ent closure constructions such as the rational closure [20,19,33,32], the lexicographicclosure [21], the relevant closure [18], and MP-closure [37].In this paper we define a modular multi-concept extension of the lexicographic clo-sure for reasoning about exceptions in ontologies. The idea is very simple: differentmodules can be defined starting from a defeasible knowledge base, containing a set D of typicality inclusions (or defeasible inclusions) describing the prototypical prop-erties of classes in the knowledge base. We will represent such defeasible inclusionsas T ( C ) ⊑ D [29], meaning that “typical C ’s are D ’s” or “normally C ’s are D ’s”,corresponding to conditionals C |∼ D in KLM framework.A set of modules m , . . . , m n is introduced, each one concerning a subject, anddefeasible inclusions belong to a module if they are related with its subject. By subject,here, we mean any concept of the knowledge base. Module m i with subject C i doesnot need to contain just typicality inclusions of the form T ( C i ) ⊑ D , but all defeasibleinclusions in D which are concerned with subject C i are admitted in m i . We call acollection of such modules a modular multi-concept knowledge base .This modularization of the defeasible part of the knowledge base does not definea partition of the set D of defeasible inclusions, as an inclusion may belong to morethan one module. For instance, the typical properties of employed students are relevantboth for the module with subject Student and for the module with subject
Employee . Laura Giordano and Daniele Theseider Dupr´e
The granularity of modularization has to be chosen by the knowledge engineer who canfix how large or narrow is the scope of a module, and how many modules are to beincluded in the knowledge base (for instance, whether the properties of employees andstudents are to be defined in the same module with subject
Person or in two differentmodules). At one extreme, all the defeasible inclusions in D can be put together in amodule associated with subject ⊤ (Thing). At the other extreme, which has been studiedin [36], a module m i is a defeasible TBox containing only the defeasible inclusions ofthe form T ( C j ) ⊑ D for some concept C i . In this paper we remove this restriction con-sidering general modules, containing arbitrary sets of defeasible inclusions, intuitivelypertaining some subject.In [36], following Gerard Brewka’s framework of Basic Preference Descriptionsfor ranked knowledge bases [14], we have assumed that a specification of the relativeimportance of typicality inclusions for a concept C i is given by assigning ranks to typ-icality inclusions. However, for a large module, a specification by hand of the rankingof the defeasible inclusions in the module would be awkward. In particular, a modulemay include all properties of a class as well as properties of its exceptional subclasses(for instance, the typical properties of penguins, ostriches, etc. might all be includedin a module with subject Bird ). A natural choice is then to consider, for each mod-ule, a lexicographic semantics which builds on the rational closure ranking to define apreference ordering on domain elements. This preference relation corresponds, in thepropositional case, to the lexicographic order on worlds in Lehmann’s model theoreticsemantics of the lexicographic closure [43]. This semantics already accounts for thespecificity relations among concepts inside the module, as the lexicographic closuredeals with specificity, based on ranking of concepts computed by the rational closure ofthe knowledge base.Based on the ranked semantics of the single modules, a compositional (preferen-tial) semantics of the knowledge base is defined by combining the multiple preferencerelations into a single global preference relation < . This gives rise to a modular multi-concept extension of Lehmann’s preference semantics for the lexicographic closure.When there is a single module, containing all the typicality inclusions in the knowledgebase, the semantics collapses to a natural extension to DLs of Lehmann’s semantics,which corresponds to Lehmann’s semantics for the fragment of ALC without universaland existential restrictions.We introduce a notion of entailment for modular multi-concept knowledge bases,based on the proposed semantics, which satisfies the KLM properties of a preferentialconsequence relation. This notion of entailment has good properties inherited from lexi-cographic closure: it deals properly with irrelevance and specificity, and it is not subjectto the “blockage of property inheritance” problem, i.e., the problem that property inher-itance from classes to subclasses is not guaranteed, which affects the rational closure[45]. In addition, separating defeasible inclusions in different modules provides a sim-ple solution to another problem of the rational closure and its refinements (includingthe lexicographic closure), that was recognized by Geffner and Pearl [27], namely, that“conflicts among defaults that should remain unresolved, are resolved anomalously”,giving rise to too strong conclusions. The preferential (not necessarily ranked) nature framework for a modular multi-concept lexicographic closure semantics 3 of the global preference relation < provides a simple way out to this problem, whendefeasible inclusions are suitably separated in different modules. ALC and its extensionwith typicality inclusions
Let N C be a set of concept names, N R a set of role names and N I a set of individualnames. The set of ALC concepts (or, simply, concepts) can be defined inductively asfollows: – A ∈ N C , ⊤ and ⊥ are concepts; – if C and D are concepts and R ∈ N R , then C ⊓ D, C ⊔ D, ¬ C, ∀ R.C, ∃ R.C areconcepts.A knowledge base (KB) K is a pair ( T , A ) , where T is a TBox and A is an ABox.The TBox T is a set of concept inclusions (or subsumptions) C ⊑ D , where C, D areconcepts. The ABox A is a set of assertions of the form C ( a ) and R ( a, b ) where C is aconcept, R ∈ N R , and a, b ∈ N I .An ALC interpretation [2] is a pair I = h ∆, · I i where: ∆ is a domain—a set whoseelements are denoted by x, y, z, . . . —and · I is an extension function that maps eachconcept name C ∈ N C to a set C I ⊆ ∆ , each role name R ∈ N R to a binary relation R I ⊆ ∆ × ∆ , and each individual name a ∈ N I to an element a I ∈ ∆ . It is extendedto complex concepts as follows: ⊤ I = ∆ ⊥ I = ∅ ( ¬ C ) I = ∆ \ C I ( C ⊓ D ) I = C I ∩ D I ( C ⊔ D ) I = C I ∪ D I ( ∀ R.C ) I = { x ∈ ∆ | ∀ y. ( x, y ) ∈ R I → y ∈ C I } ( ∃ R.C ) I = { x ∈ ∆ | ∃ y. ( x, y ) ∈ R I & y ∈ C I } . The notion of satisfiability of a KB in an interpretation and the notion of entailment aredefined as follows:
Definition 1 (Satisfiability and entailment).
Given an
ALC interpretation I = h ∆, · I i :- I satisfies an inclusion C ⊑ D if C I ⊆ D I ;- I satisfies an assertion C ( a ) if a I ∈ C I ;- I satisfies an assertion R ( a, b ) if ( a I , b I ) ∈ R I .Given a KB K = ( T , A ) , an interpretation I satisfies T (resp., A ) if I satisfies allinclusions in T (resp., all assertions in A ). I is an ALC model of K = ( T , A ) if I satisfies T and A .Letting a query F to be either an inclusion C ⊑ D (where C and D are concepts)or an assertion ( C ( a ) or R ( a, b ) ), F is entailed by K , written K | = ALC F , if for all ALC models I = h ∆, · I i of K , I satisfies F . Laura Giordano and Daniele Theseider Dupr´e
Given a knowledge base K , the subsumption problem is the problem of deciding whetheran inclusion C ⊑ D is entailed by K . The instance checking problem is the problemof deciding whether an assertion C ( a ) is entailed by K . The concept satisfiability prob-lem is the problem of deciding, for a concept C , whether C is consistent with K (i.e.,whether there exists a model I of K , such that C I = ∅ ).In the following we will refer to an extension of ALC with typicality inclusions, thatwe will call
ALC + T as in [29], and to the rational closure of ALC + T knowledgebases ( T , A ) [33,32]. In addition to standard ALC inclusions C ⊑ D (called strict inclusions in the following), in ALC + T the TBox T also contains typicality inclusionsof the form T ( C ) ⊑ D , where C and D are ALC concepts. Among all rational closureconstructions for
ALC mentioned in the introduction, we will refer to the one in [33],and to its minimal canonical model semantics. Let us recall the notions of preferential,ranked and canonical model of a defeasible knowledge base ( T , A ) , that will be usefulin the following. Definition 2 (Interpretations for
ALC + T ). A preferential interpretation N is anystructure h ∆, <, · I i where: ∆ is a domain; < is an irreflexive, transitive and well-founded relation over ∆ ; · I is a function that maps all concept names, role names andindividual names as defined above for ALC interpretations, and provides an interpre-tation to all
ALC concepts as above, and to typicality concepts as follows: ( T ( C )) I = min < ( C I ) , where min < ( S ) = { u : u ∈ S and ∄ z ∈ S s.t. z < u } .When relation < is required to be also modular (i.e., for all x, y, z ∈ ∆ , if x < y then x < z or z < y ), N is called a ranked interpretation. Preferential interpretations for description logics were first studied in [29], while rankedinterpretations (i.e., modular preferential interpretations) were first introduced for
ALC in [17].A preferential (ranked) model of an
ALC + T knowledge base K is a preferential(ranked) ALC + T interpretation N = h ∆, <, · I i that satisfies all inclusions in K ,where: a strict inclusion or an assertion is satisfied in N if it is satisfied in the ALC model h ∆, · I i , and a typicality inclusion T ( C ) ⊑ D is satisfied in N if ( T ( C )) I ⊆ D I .Preferential entailment in ALC + T is defined in the usual way: for a knowledge base K and a query F (a strict or defeasible inclusion or an assertion), F is preferentiallyentailed by K ( K | = ALC + T F ) if F is satisfied in all preferential models of K .A canonical model for K is a preferential (ranked) model containing, roughly speak-ing, as many domain elements as consistent with the knowledge base specification K .Given an ALC + T knowledge base K = ( T , A ) and a query F , let us define S K as theset of all ALC concepts (and subconcepts) occurring in K or in F , together with theircomplements. We consider all the sets of concepts { C , C , . . . , C n } ⊆ S K consistentwith K , i.e., s.t. K = ALC + T C ⊓ C ⊓ · · · ⊓ C n ⊑ ⊥ . Definition 3 (Canonical model). . A preferential model M = h ∆, <, I i of K is canon-ical with respect to S K if it contains at least a domain element x ∈ ∆ s.t. x ∈ ( C ⊓ C ⊓ · · · ⊓ C n ) I , for each set { C , C , . . . , C n } ⊆ S K consistent with K . For finite, consistent
ALC + T knowledge bases, existence of finite (ranked) canonicalmodels has been proved in [32] (Theorem 1). In the following, as we will only consider framework for a modular multi-concept lexicographic closure semantics 5 finite ALC + T knowledge bases, we can restrict our consideration to finite preferentialmodels. In this section we introduce a notion of a multi-concept knowledge base, starting froma set of strict inclusions T , a set of assertions A , and a set of typicality inclusions D ,each one of the form T ( C ) ⊑ D , where C and D are ALC concepts.
Definition 4. A modular multi-concept knowledge base K is a tuple hT , D , m , . . . , m k , A , s i , where T is an ALC
TBox, D is a set of typicality inclusions, such that m ∪ . . . ∪ m k = D , A is an ABox, and s is a function associating each module m i with aconcept, s ( m i ) = C i , the subject of m i . The idea is that each m i is a module defining the typical properties of the instancesof some concept C i . The defeasible inclusions belonging to a module m i with sub-ject C i are the inclusions that intuitively pertain to C i . We expect that all the typi-cality inclusions T ( C ) ⊑ D , such that C is a subclass of C i , belong to m i , but notonly. For instance, for a module m i with subject C i = Bird , the typicality inclusion T ( Bird ⊓ Live at SouthPole ) ⊑ Penguin , meaning that the birds living at the southpole are normally penguins, is clearly to be included in m i . As penguins are birds, alsoinclusion T ( Penguin ) ⊑ Black is to be included in m i , and, if T ( Bird ) ⊑ Flying - Animal and T ( FlyingAnimal ) ⊑ BigWings are defeasible inclusions in the knowl-edge base, they both may be relevant properties of birds to be included in m i . For thisreason we will not put restrictions on the typicality inclusions that can belong to a mod-ule. We will see later that the semantic construction for a module m i will be able toignore the typicality inclusions which are not relevant for subject C i and that there arecases when not even the inclusions T ( C ) ⊑ D with C subsumed by C i are admitted in m i .The modularization m , . . . , m k of the defeasible part D of the knowledge basedoes not define a partition of D , as the same inclusion may belong to more than onemodule m i . For instance, the typical properties of employed students are relevant forboth concept Student and concept
Employee and should belong to their related mod-ules (if any). Also, a granularity of modularization has to be chosen and, as we will see,this choice may have an impact on the global semantics of the knowledge base. At oneextreme, all the defeasible inclusions in D are put together in the same module, e.g., themodule associated with concept ⊤ . At the other extreme, which has been studied in [36],a module m i contains only the defeasible inclusions of the form T ( C i ) ⊑ D , where C i is the subject of m i (and in this case, the inclusions T ( C ) ⊑ D with C subsumed by C i are not admitted in m i ). In this regard, the framework proposed in this paper couldbe seen as an extension of the proposal in [36] to allow coarser grained modules, whilehere we do not allow for user-defined preferences among defaults.Let us consider an example of multi-concept knowledge base. Example 1.
Let K be the knowledge base hT , D , m , m , m , A , s i , where A = ∅ , T contains the strict inclusions: Laura Giordano and Daniele Theseider Dupr´e
Employee ⊑ AdultAdult ⊑ ∃ has SSN . ⊤ PhdStudent ⊑ StudentPhDStudent ⊑ AdultHas no Scolarship ≡ ¬∃ hasScolarship . ⊤ PrimarySchoolStudent ⊑ ChildrenPrimarySchoolStudent ⊑ HasNoClasses Driver ⊑ AdultDriver ⊑ ∃ has DrivingLicence . ⊤ and the defeasible inclusions in D are distributed in the modules m , m , m as follows.Module m has subject Employee , and contains the defeasible inclusions: ( d ) T ( Employee ) ⊑ ¬ Young ( d ) T ( Employee ) ⊑ ∃ has boss . Employee ( d ) T ( ForeignerEmployee ) ⊑ ∃ has Visa . ⊤ ( d ) T ( Employee ⊓ Student ) ⊑ Busy ( d ) T ( Employee ⊓ Student ) ⊑ ¬ Young
Module m has subject Student , and contains the defeasible inclusions: ( d ) T ( Student ) ⊑ ∃ has classes . ⊤ ( d ) T ( Student ) ⊑ Young ( d ) T ( Student ) ⊑ Has no Scolarship ( d ) T ( HighSchoolStudent ) ⊑ Teenager ( d ) T ( PhDStudent ) ⊑ ∃ hasScolarship . Amount ( d ) T ( PhDStudent ) ⊑ Bright ( d ) T ( Employee ⊓ Student ) ⊑ Busy ( d ) T ( Employee ⊓ Student ) ⊑ ¬ Young
Module m has subject V ehicle , and contains the defeasible inclusions: ( d ) T ( Vehicle ) ⊑ ∃ has owner . Driver ( d ) T ( Car ) ⊑ ¬ SportsCar ( d ) T ( SportsCar ) ⊑ RunFast ( d ) T ( Truck ) ⊑ Heavy ( d ) T ( Bicycle ) ⊑ ¬ RunFast
Observe that, in previous example, ( d ) and ( d ) belong to both modules m and m .An additional module might be added containing the prototypical properties of Adults. In this section, we define a semantics of modular multi-concept knowledge bases, basedon Lehmann’s lexicographic closure semantics [43]. The idea is that, for each module m i , a semantics can be defined using lexicographic closure semantics, with some minormodification.Given a modular multi-concept knowledge base K = hT , D , m , . . . , m k , A , s i ,we let rank ( C ) be the rank of concept C in the rational closure ranking of the knowl-edge base ( T ∪ D , A ) , according to the rational closure construction in [33]. In the ra-tional closure ranking, concepts with higher ranks are more specific than concepts with framework for a modular multi-concept lexicographic closure semantics 7 lower ranks. While we will not recall the rational closure construction, let us consideragain Example 1. In Example 1, the rational closure ranking assigns to concepts Adult , Employee , ForeignEmployee , Driver , Student , HighSchoolStudent , Primary - SchoolStudent the rank , while to concepts PhDStudent and
Employee ⊓ Student the rank . In fact, PhDStudent are exceptional students, as they have a scholarship,while employed students are exceptional students, as they are not young. Their rankis higher than the rank of concept
Student as they are exceptional subclasses of class
Student .Based on the concept ranking, the rational closure assigns a rank to typicality in-clusions: the rank of T ( C ) ⊑ D is equal to the rank of concept C . For each module m i of a knowledge base K = hT , D , m , . . . , m k , A , s i , we aim to define a canonicalmodel, using the lexicographic order based on the rank of typicality inclusions in m i .In the following we will assume that the knowledge base hT ∪ D , Ai is consistent in thelogic ALC + T , that is, it has a preferential model. This also guarantees the existenceof (finite) canonical models [32]. In the following, as the knowledge base K is finite,we will restrict our consideration to finite preferential and ranked models.Let us define the projection of the knowledge base K on module m i as the knowl-edge base K i = hT ∪ m i , Ai . K i is an ALC + T knowledge base. Hence a preferentialmodel N i = h ∆, < i , · I i of K i is defined as in Section 2 (but now we use < i , instead of < , for the preference relation in N i , for i = 1 , . . . , k ).In his seminal work on the lexicographic closure, Lehmann [43] defines a modeltheoretic semantics of the lexicographic closure construction by introducing an orderrelation among propositional models, considering which defaults are violated in eachmodel, and introducing a seriousness ordering ≺ among sets of violated defaults. Fortwo propositional models w and w ′ , w ≺ w ′ ( w is preferred to w ′ ) is defined in [43] asfollows: w ≺ w ′ iff V ( w ) ≺ V ( w ′ ) (1) w is preferred to w ′ when the defaults V ( w ) violated by w are less serious than the de-faults V ( w ′ ) violated by w ′ . As we will recall below, the seriousness ordering dependson the number of defaults violated by w and by w ′ for each rank.In a similar way, in the following, we introduce a ranked relation < i on the domain ∆ of a model of K i . Let us first define, for a preferential model N i = h ∆, < i , · I i of K i ,what it means that an element x ∈ ∆ violates a typicality inclusion T ( C ) ⊑ D in m i . Definition 5.
Given a module m i of K , with s ( m i ) = C i , and a preferential model N i = h ∆, < i , · I i of K i , an element x ∈ ∆ violates a typicality inclusion T ( C ) ⊑ D in m i if x ∈ C Ii , x ∈ C I and x D I . Notice that, the set of typicality inclusions violated by a domain element x in a modelonly depends on the interpretation · I of ALC concepts, and on the defeasible inclusionsin m i . Furthermore, differently from the usual notion of violation in Lehmann’s seman-tics, for a module m i with subject C i , we do not consider the violations of domainelements x C Ii (i.e., the domain elements x which are not C i -instances are assumednot to violate any default in m i ). Let V i ( x ) be the set of the defeasible inclusions of m i violated by domain element x , and let V hi ( x ) be the set of all defeasible inclusions in m i with rank h which are violated by domain element x . Laura Giordano and Daniele Theseider Dupr´e
In order to compare alternative sets of defaults, in [43] the seriousness ordering ≺ among sets of defaults is defined by associating with each set of defaults D ⊆ K a tupleof numbers h n , n , . . . , n r i , where r is the order of K , i.e. the least finite i such thatthere is no default with the finite rank r or rank higher than r (but there is at least onedefault with rank r − ). The tuple is constructed considering the ranks of defaults in therational closure. n is the number of defaults in D with rank ∞ and, for ≤ i ≤ k , n i is the number of defaults in D with rank r − i (in particular, n r is the number of defaultsin D with rank ). Lehmann defines the strict modular order ≺ among sets of defaultsfrom the natural lexicographic order over the tuples h n , n , . . . , n k i . This order givespreference to those sets of defaults containing a larger number of more specific defaults.As we have seen from equation (1), ≺ is used by Lehmann to compare sets of violateddefaults and to prefer the propositional models whose violations are less serious.We use the same criterion for comparing domain elements, introducing a serious-ness ordering ≺ i for each module m i . Considering that the defaults with infinite rankmust be satisfied by all domain elements, we will not need to consider their violation inour definition (that is, we will not consider n in the following).The set V i ( x ) of defaults from module m i which are violated by x , can be associatedwith a tuple of numbers t i,x = h| V r − i ( x ) | , . . . , | V i ( x ) |i . Following Lehmann, we let V i ( x ) ≺ i V i ( y ) iff t i,x comes before t i,y in the natural lexicographic order on tuples(restricted to the violations of defaults in m i ), that is: V i ( x ) ≺ i V i ( y ) iff ∃ l such that | V li ( x ) | < | V li ( y ) | and, ∀ h > l , | V hi ( x ) | = | V hi ( y ) | Definition 6.
A preferential model N i = h ∆, < i , · I i of K i = hT ∪ m i , Ai , is a lexico-graphic model of K i if h ∆, · I i is an ALC model of hT , Ai and < i satisfies the followingcondition: x < i y iff V i ( x ) ≺ i V i ( y ) . (2)Informally, < C j gives higher preference to domain elements violating less typicalityinclusions of m i with higher rank. In particular, all x, y C Ii , x ∼ C i y , i.e., all ¬ C i -elements are assigned the same preference wrt < i , the least one, as they trivially satisfyall the typicality properties in m i . As in Lehmann’s semantics, in a lexicographic model N i = h ∆, < i , · I i of K i , the preference relation < i is a strict modular partial order, i.e.an irreflexive, transitive and modular relation. As well-foundedness trivially holds forfinite interpretations, a lexicographic model N i of K i is a ranked model of K i . Proposition 1.
A lexicographic model N i = h ∆, < i , · I i of K i = hT ∪ m i , Ai is aranked model of K i . A multi-concept model for K can be defined as a multi-preference interpretationwith a preference relation < i for each module m i . Definition 7 (Multi-concept interpretation).
Let K = hT , D , m , . . . , m k , A , s i bea multi-concept knowledge base. A multi-concept interpretation M for K is a tuple h ∆, < , . . . , < k , · I i such that, for all i = 1 , . . . , k , h ∆, < i , · I i is a ranked ALC + T interpretation, as defined in Section 2. framework for a modular multi-concept lexicographic closure semantics 9 Definition 8 (Multi-concept lexicographic model).
Let K = hT , D , m , . . . , m k , A , s i be a multi-concept knowledge base. A multi-concept lexicographic model M = h ∆, < , . . . , < k , · I i of K is a multi-concept interpretation for K , such that, for all i = 1 , . . . , k , N i = h ∆, < i , · I i is a lexicographic model of K i = hT ∪ m i , Ai . A canonical multi-concept lexicographic model of K is multi-concept lexicographicmodel of K such that ∆ and · I are the domain and interpretation function of somecanonical preferential model of hT ∪ D , Ai , according to Definition 3. Definition 9 (Canonical multi-concept lexicographic model).
Given a multi-conceptknowledge base K = hT , D , m , . . . , m k , A , s i , a canonical multi-concept lexico-graphic model of K , M = h ∆, < , . . . , < k , · I i , is a multi-concept lexicographic modelof K such that there is a canonical ALC + T model h ∆, < ∗ , · I i of hT ∪D , Ai , for some < ∗ . Observe that, restricting to the propositional fragment of the language (which doesnot allow universal and existential restrictions nor assertions), for a knowledge base K without strict inclusions and with a single module m , with subject ⊤ , containing allthe typicality inclusions in K , the preference relation < corresponds to Lehmann’slexicographic closure semantics, as its definition is based on the set of all defeasibleinclusions in the knowledge base. For multiple modules, each < i determines a ranked preference relation which can beused to answer queries over module m i (i.e. queries whose subject is C i ). If we wantto evaluate the query T ( C ) ⊑ D (are all typical C elements also D elements?) inmodule m i (assuming that C concerns subject C i ), we can answer the query using the < i relation, by checking whether min < i ( C I ) ⊆ D I . For instance, in Example 1, thequery “are all typical Phd students young?” can be evaluated in module m . The answerwould be positive, as the property of students of being normally young is inheritedby PhD Student. The evaluation of a query in a specific module is something that isconsidered in context-based formalisms, such as in the CKR framework [9], wherethere is a language construct eval ( X , c ) for evaluating a concept (or role) X in context c . The lexicographic orders < i and < j (for i = j ) do not need to agree. For instance,in Example 1, for two domain elements x and y , we might have that x < y and y < x , as x is more typical than y as an employee, but less typical than x as a student.To answer a query T ( C ) ⊑ D , where C is a concept which is concerned with morethan one subject in the knowledge base (e.g., are typical employed students young?),we need to combine the relations < i .A simple way of combining the modular partial order relations < i is to use Paretocombination. Let ≤ i be defined as follows: x ≤ i y iff y < i x . As < i is a modular partialorder, ≤ i is a total preorder. Given a canonical multi-concept lexicographic model M = h ∆, < , . . . , < k , · I i of K , we define a global preference relation < on ∆ as follows: x < y iff ( i ) for some i = 1 , . . . , k, x < i y and ( ∗ )( ii ) for all j = 1 , . . . , k, x ≤ j y, The resulting relation < is a partial order but, in general, modularity does not hold for < . Definition 10.
Given a canonical multi-concept lexicographic model M = h ∆, < , . . . , < k , · I i of K , the combined lexicographic interpretation of M , is a triple M P = h ∆, <, · I i , where < is the global preference relation defined by (*). We call M P a combined lexicographic model of K (shortly, an m cl -model of K ). Proposition 2.
A combined lexicographic model M P of K is a preferential interpreta-tion satisfying all the strict inclusions and assertions in K . A combined lexicographic model M P of K is a preferential interpretation as thosedefined for ALC + T in Definition 2 (and, in general, it is not a ranked interpretation).However, preference relation < in M P is not an arbitrary irreflexive, transitive andwell-founded relation. It is obtained by first computing the lexicographic preferencerelations < i for modules, and then by combining them into < . As M P satisfies allstrict inclusions and assertions in K but is not required to satisfy all typicality inclusions T ( C ) ⊑ D in K , M P is not a preferential ALC + T model of K as defined in Section2. Consider a situation in which there are two concepts, Student and
YoungPerson ,that are very related in that students are normally young persons and young persons arenormally students (i.e., T ( Student ) ⊑ YoungPerson and T ( YoungPerson ) ⊑ Stu - dent ) and suppose there are two modules m and m such that s ( m ) = Student and s ( m ) = YoungPerson . The two classes may have different (and even contradictory)prototypical properties, for instance, normally students are quiet (e.g., when they arein their classrooms), T ( Student ) ⊑ Quiet , but normally young persons are not quiet, T ( YoungPerson ) ⊑ ¬ Quiet . Considering the preference relations < and < , associ-ated with the two modules in a canonical multi-concept lexicographic model, we mayhave that, for two young persons Bob and John, which are also students, bob < john and john < bob , as Bob is quiet and John is not. Then, John and Bob are incompara-ble in the global relation < . Both of them, depending on the other prototypical proper-ties of students and young persons, might be minimal, among students, wrt the globalpreference relation < . Hence, the set min < ( Student I ) is not necessarily a subset of min < ( Student I ) . That is, typical students in the global relation may include instances(e.g., john ) which do not satisfy all the typicality inclusions for Student , as they are are(globally) incomparable with the elements in min < ( Student I ) . This implies that thenotion of m cl -entailment (defined below) cannot be stronger than preferential entailmentin Section 2. However, given the correspondence of m cl -models with the lexicographicclosure in the case of a single module with subject ⊤ , containing all the typicality inclu-sions in D , m cl -entailment can neither be weaker than preferential entailment.In general, for a knowledge base K and a module m i , with s ( m i ) = C i , the inclu-sion min < ( C Ii ) ⊆ min < i ( C Ii ) may not hold and, for this reason, a combined lexico- framework for a modular multi-concept lexicographic closure semantics 11 graphic interpretation may fail to satisfy all typicality inclusions. In this respect, canon-ical multi-concept lexicographic models are more liberal than KLM-style preferentialmodels for typicality logics [30], where all the typicality inclusions are required to besatisfied and, in the previous example, min < ( Student I ) ⊆ Quiet I must hold for thetypicality inclusion to be satisfied. In fact, the knowledge base above is inconsistent inthe preferential semantics and has no preferential model: from T ( Student ) ⊑ Young - Person and T ( YoungPerson ) ⊑ Student , it follows that T ( Student ) = T ( Young - Person ) should hold in all preferential models of the knowledge base, which is impos-sible given the conflicting typicality inclusions T ( Student ) ⊑ Quiet and T ( Young - Person ) ⊑ ¬ Quiet .To require that all typicality inclusions in K are satisfied in M P , the notion of m cl -model of K can be strengthened as follows. Definition 11. A T -compliant m cl -model (or m cl T -model) M P = h ∆, <, · I i of K is a m cl -model of K such that all the typicality inclusions in K are satisfied in M P , i.e., forall T ( C ) ⊑ D ∈ D , min < ( C I ) ⊆ D I . Observe that, m cl T -model M P = h ∆, <, · I i of K = hT , D , m , . . . , m k , A , s i isa KLM-style preferential model for the ALC + T knowledge base hT ∪ D , Ai , asdefined in Section 2. As a difference, the preference relation < in a m cl T -model isnot an arbitrary irreflexive, transitive and well-founded relation, but is defined from thelexicographic preference relations < i ’s according to equation (*).We define a notion of multi-concept lexicographic entailment ( m cl -entailment) inthe obvious way: a query F is m cl -entailed by K ( K | = m cl F ) if, for all m cl -models M P = h ∆, <, · I i of K , F is satisfied in M P . Notice that a query T ( C ) ⊑ D issatisfied in M P when min < ( C I ) ⊆ D I .Similarly, a notion of m cl T -entailment can be defined: K | = m cl T F if, for all m cl T -models M P = h ∆, <, · I i of K , F is satisfied in M P .As, for any multi-concept knowledge base K , the set of m cl T -models of K is a sub-set of the set of m cl -models of K , and there is some K for which the inclusion is proper(see, for instance, the student and young person example above), m cl T -entailment isstronger than m cl -entailment. It can be proved that both notions of entailment satisfythe KLM postulates of preferential consequence relations, which can be reformulatedfor a typicality logic, considering that typicality inclusions T ( C ) ⊑ D [29] stand forconditionals C |∼ D in KLM preferential logics [41,42]. See also [8] for the formulationof KLM postulates in the Propositional Typicality Logic (PTL).In the following proposition, we let “ T ( C ) ⊑ D ” mean that T ( C ) ⊑ D is m cl -entailed from a given knowledge base K . Proposition 3. m cl -entailment satisfies the KLM postulates of preferential consequencerelations, namely:(REFL) T ( C ) ⊑ C (LLE) If A ≡ B and T ( A ) ⊑ C , then T ( B ) ⊑ C (RW) If C ⊑ D and T ( A ) ⊑ C , then T ( A ) ⊑ D (AND) If T ( A ) ⊑ C and T ( A ) ⊑ D , then T ( A ) ⊑ C ⊓ D (OR) If T ( A ) ⊑ C and T ( B ) ⊑ C , then T ( A ⊔ B ) ⊑ C (CM) If T ( A ) ⊑ D and T ( A ) ⊑ C , then T ( A ⊓ D ) ⊑ C Stated differently, the set of the typicality inclusions T ( C ) ⊑ D that are m cl -entailedfrom a given knowledge base K is closed under conditions (REFL)-(CM) above. Forinstance, (LLE) means that if A and B are equivalent concepts in ALC and T ( A ) ⊑ C is m cl -entailed from a given knowledge base K , than T ( B ) ⊑ C is also m cl -entailedfrom K ; similarly for the other conditions (where inclusion C ⊑ D is entailed by K in ALC ). It can be proved that also m cl T -entailment satisfies the KLM postulates ofpreferential consequence relations.It can be shown that both m cl -entailment and m cl T -entailment are not stronger thanLehmann’s lexicographic closure in the propositional case. Let us consider again Exam-ple 1. Example 2.
Let us add another module m with subject Citizen to the knowledge base K , plus the following additional axioms in T : Italian ⊑ Citizen French ⊑ CitizenCanadian ⊑ Citizen
Module m has subject Citizen , and contains the defeasible inclusions: ( d ) T ( Italian ) ⊑ DriveFast ( d ) T ( Italian ) ⊑ HomeOwner
Suppose the following typicality inclusion is also added to module m : ( d ) T ( PhDStudent ) ⊑ ¬ HomeOwner
What can we conclude about typical Italian PhD students? We can see that neither the in-clusion T ( PhDStudent ⊓ Italian ) ⊑ HomeOwner nor the inclusion T ( PhDStudent ⊓ Italian ) ⊑ ¬ HomeOwner are m cl -entailed by K .In fact, in all canonical multi-concept lexicographic models M = h ∆, < , . . . , < , · I i of K , all elements in min < (( P hDStudent ⊓ Italian ) I ) ( the minimal ItalianPhDStudent wrt < ), have scholarship, are bright, are not home owners (which aretypical properties of PhD students), have classes and are young (which are propertiesof students not overridden for PhD students).On the other end, all elements in min < (( PhDStudent ⊓ Italian ) I ) (i.e., the min-imal Italian PhDStudent wrt < ) have the properties that they drive fast and are homeowners. As < -minimal elements and < -minimal PhDStudent ⊓ Italian -elementsare incomparable wrt < , the < -minimal Italian PhD students will include them all.Hence, min < (( PhDStudent ⊓ Italian ) I ) HomeOwner I and min < (( PhDStudent ⊓ Italian ) I ) ( ¬ HomeOwner ) I .The home owner example is a reformulation of the example used by Geffner and Pearlto show that the rational closure of conditional knowledge bases sometimes gives toostrong conclusions, as “conflicts among defaults that should remain unresolved, are re-solved anomalously” [27]. Informally, if defaults ( d ) and ( d ) are conflicting forItalian Phd students before adding any default which makes PhD students exceptionalwrt Students (in our formalization, default ( d ) ), they should remain conflicting af-ter this addition. Instead, in the propositional case, both the rational closure [42] andLehmann’s lexicographic closure [43] would entail that normally Italian Phd studentsare not home owners. This conclusion is unwanted, and is based on the fact that ( d ) has rank , while ( d ) has rank in the rational closure ranking. On the other hand, framework for a modular multi-concept lexicographic closure semantics 13 T ( PhDStudent ⊓ Italian ) ⊑ ¬ HomeOwner is neither m cl -entailed from K , nor m cl T -entailed from K . Both notions of entailment, when restricted to the propositional case,cannot be stronger than Lehmann’s lexicographic closure.Geffner and Pearl’s Conditional Entailment [27] does not suffer from the above men-tioned problem as it is based on (non-ranked) preferential models. The same problem,which is related to the representation of preferences as levels of reliability, has also beenrecognized by Brewka [13] in his logical framework for default reasoning, leading toa generalization of the approach to allow a partial ordering between premises. The ex-ample above shows that our approach using ranked preferences for the single modules,but a non-ranked global preference relation < for their combination, does not sufferfrom this problem, provided a suitable modularization is chosen (in example above, ob-tained by separating the typical properties of Italians and those of students in differentmodules). The approach considered in Section 4 does not allow to reason with a hierarchy of mod-ules, but it considers a flat collection of modules m , . . . , m k , each module concerningsome subject C i . As we have seen, a module m i may contain defeasible inclusions re-ferring to subclasses of C i , such as PhDStudent in the case of module m with subject Student . When defining the preference relation < i the lexicographic closure semanticsalready takes into account the specificity relation among concepts within the module(e.g., the fact that PhDStudent is more specific than
Student ).However, nothing prevents us from defining two modules m i (with subject C i ) and m j (with subject C j ), such that concept C j is more specific than concept C i . For in-stance, as a variant of Example 1, we might have introduced two different modules m with subject Student and m with subject PhDStudent . As concept
PhDStudent ismore specific than concept
Student (in particular,
PhDStudent ⊑ Student is entailedfrom the strict part of knowledge base T in ALC ), the specificity information should betaken into account when combining the preference relations. More precisely, preference < should override preference < when comparing PhDStudent -instances.This is the principle followed by Giordano and Theseider Dupr´e [36] to define aglobal preference relation, in the case when each module with subject C i only containstypicality inclusions of the form T ( C i ) ⊑ D . A more sophisticated way to combine thepreference relations < i into a global relation < is used to deal with this case with respectto Pareto combination, by exploiting the specificity relation among concepts. While werefer therein for a detailed description of this more sophisticated notion of preferencecombination, let us observe that this solution could be as well applied to the modularmulti-concept knowledge bases considered in this paper, provided an irreflexive andtransitive notion of specificity among modules is defined.Another aspect that has been considered in the previously mentioned paper is thepossibility of assigning ranks to the defeasible inclusions associated with a given con-cept. While assigning a rank to all typicality inclusions in the knowledge base may beawkward, often people have a clear idea about the relative importance of the properties for some specific concept. For instance, we may know that the defeasible property thatstudents are normally young is more important than the property that student normallydo not have a scholarship. For small modules, which only contain typicality inclusions T ( C i ) ⊑ D for a concept C i , the specification of user-defined ranks of the C i ’s typicalproperties is a feasible option and a ranked modular preference relation can be definedfrom it, by using Brewka’s strategy from his framework of Basic Preference Descrip-tions for ranked knowledge bases [14]. This alternative may coexist with the use of thelexicographic closure semantics built from the rational closure ranking for larger mod-ules. A mixed approach, integrating user-specified preferences with the rational closureranking for the same module, might be an interesting alternative. This integration, how-ever, does not necessarily provide a total preorder among typicality inclusions, whichis our starting point for defining the modular preferences < i and their combination.Alternative semantic constructions should be considered for dealing with this case.According to the choice of fine grained or coarse grained modules, to the choice ofthe preferential semantics for each module (e.g., based on user-specified ranking or onLehmann’s lexicographic closure, or on the rational closure, etc.), and to the presence ofa specificity relation among modules, alternative preferential semantics for modularizedmulti-concept knowledge bases can emerge. In this paper, we have proposed a modular multi-concept extension of the lexicographicclosure semantics, based on the idea that defeasible properties in the knowledge basecan be distributed in different modules, for which alternative preference relations canbe computed. Combining multiple preferences into a single global preference allowsa new preferential semantics and a notion of multi-concept lexicographic entailment( m cl -entailment) which, in the propositional case, is not stronger than the lexicographicclosure. m cl -entailment satisfies the KLM postulates of a preferential consequence relation.It retains some good properties of the lexicographic closure, being able to deal withirrelevance, with specificity within the single modules, and not being subject to the“blockage of property inheritance” problem. The combination of different preferencerelations provides a simple solution to a problem, recognized by Geffner and Pearl, thatthe rational closure of conditional knowledge bases sometimes gives too strong conclu-sions, as “conflicts among defaults that should remain unresolved, are resolved anoma-lously” [27]. This problem also affects the lexicographic closure, which is stronger thanthe rational closure. Our approach using ranked preferences for the single modules, buta non-ranked preference < for their combination, does not suffer from this problem,provided a suitable modularization is chosen. As Geffner and Pearl’s Conditional En-tailment [27], also some non-monotonic DLs, such as ALC + T min , a typicality DLwith a minimal model preferential semantics [31], and the non-monotonic descriptionlogic DL N [5], which supports normality concepts based on a notion of overriding, donot not suffer from the problem above.Reasoning about exceptions in ontologies has led to the development of many non-monotonic extensions of Description Logics (DLs), incorporating non-monotonic fea- framework for a modular multi-concept lexicographic closure semantics 15 tures from most of NMR formalisms in the literature. In addition to those already men-tioned in the introduction, let us recall the work by Straccia on inheritance reasoning inhybrid KL-One style logics [46] the work on defaults in DLs [3], on description logicsof minimal knowledge and negation as failure [24], on circumscriptive DLs [7,6], thegeneralization of rational closure to all description logics [4]. as well as the combinationof description logics and rule-based languages [26,25,44,40,39,34,10].Our multi-preference semantics is related with the multipreference semantics for ALC developed by Gliozzi [38], which is based on the idea of refining the rationalclosure construction considering the preference relations < A i associated with differ-ent aspects, but we follow a different route concerning the definition of the preferencerelations associated with modules, and the way of combining them in a single prefer-ence relation. In particular, defining a refinement of rational closure semantics is notour aim in this paper, as we prefer to avoid some unwanted conclusions of rational andlexicographic closure while exploiting their good inference properties.The idea of having different preference relations, associated with different typicalityoperators, has been studied by Gil [28] to define a multipreference formulation of thetypicality DL ALC + T min , mentioned above. As a difference, in this proposal weassociate preferences with modules and their subject, and we combine the differentpreferences into a single global one. An extension of DLs with multiple preferences hasalso been developed by Britz and Varzinczak [16,15] to define defeasible role quantifiersand defeasible role inclusions, by associating multiple preference relations with roles.The relation of our semantics with the lexicographic closure for ALC by Casiniand Straccia [20,22] should be investigated. A major difference is in the choice of therational closure ranking for
ALC , but it would be interesting to check whether their con-struction corresponds to our semantics in the case of a single module m with subject ⊤ , when the same rational closure ranking is used.Bozzato et al. present extensions of the CKR (Contextualized Knowledge Reposi-tories) framework by Bozzato et al. [9,10] in which defeasible axioms are allowed inthe global context and exceptions can be handled by overriding and have to be justi-fied in terms of semantic consequence, considering sets of clashing assumptions foreach defeasible axiom. An extension of this approach to deal with general contextualhierarchies has been studied by the same authors [11], by introducing a coverage rela-tion among contexts, and defining a notion of preference among clashing assumptions,which is used to define a preference relation among justified CAS models, based onwhich CKR models are selected. An ASP based reasoning procedure, that is completefor instance checking, is developed for
SROIQ -RL.For the lightweight description logic EL + ⊥ , an Answer Set Programming (ASP) ap-proach has been proposed [36] for defeasible inference in a miltipreference extensionof EL + ⊥ , in the specific case in which each module only contains the defeasible inclu-sions T ( C i ) ⊑ D for a single concept C i , where the ranking of defeasible inclusionsis specified in the knowledge base, following the approach by Gerhard Brewka in hisframework of Basic Preference Descriptions for ranked knowledge bases [14]. A speci-ficity relation among concepts is also considered. The ASP encoding exploits asprin [12], by formulating multipreference entailment as a problem of computing preferredanswer sets, which is proved to be Π p -complete. For EL + ⊥ knowledge bases, we aim at extending this ASP encoding to deal with the modular multi-concept lexicographicclosure semantics proposed in this paper, as well as with a more general framework,allowing for different choices of preferential semantics for the single modules and fordifferent specificity relations for combining them. For lightweight description logicsof the EL family [1], the ranking of concepts determined by the rational closure con-struction can be computed in polynomial time in the size of the knowledge base [35,23].This suggests that we may expect a Π p upper-bound on the complexity of multi-conceptlexicographic entailment. Acknowledgement:
We thank the anonymous referees for their helpful commentsand suggestions. This research is partially supported by INDAM-GNCS Project 2019.
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