Description Logic S HOI N

S HOI N belongs to DLs of high expressive power. In short, it allows to define (i)

transitive and non-transitive roles, (ii) their inverse, (iii) comprises various constructors to create complex concept expressions, and (iv) allows to describe inclusion hierarchies over roles and concepts. These are syntactically defined as follows.

Syntax ofS HOI N

Definition 3.1 (S HOI N roles). Let VOP be a countable set of role names.5 The set of

S HOI N roles (or roles for short) is VOP ∪ {R− | R ∈ VOP}, where R− is the inverse role of R. A role transitivity axiom is of the form Tra(R)where R∈ VOP.

A role inclusion axiom is of the form RvS, for two roles R and S, called sub and super role, respectively. A role is simple if it is not transitive and none of its subroles is transitive.

5_{We use the subscript OP to indicate that roles are called object properties in OWL and for distinguish-} ing them later on from concrete roles (a.k.a. data properties).

Since there is also the notion of concrete roles, which will be introduced later when we introducing data ranges, a role R∈ VOP is also called an abstract role.

Definition 3.2 (S HOI N concepts). Let VC, VI be countable sets of concept names and individual names respectively, that may, under certain restrictions, have non-empty intersec- tions.6 The set of S HOI N concepts (or concepts for short) over the vocabulary VC, VI, and roles is inductively defined as follows.

(1) Every concept name A ∈ VC is a concept. (2) >(top),⊥(bottom) are concepts.

(3) If C, D are concepts, R is a role, S is a simple role, a ∈ VI is an individual name, and n, m are non-negative integers, then the following are also concepts:

• ¬C (negation), CuD (conjunction), CtD (disjunction), • {a1, . . . , am}(nominal),

• ∃R.C (existential restriction),∀R.C (universal restriction),

• ≥n S (min cardinality restriction),≤n S (max cardinality restriction).7 A concept is atomic if it is a concept name. A concept is complex otherwise.

The operators available in a DL to formulate complex concepts (roles) are also called concept (role) constructors. If every constructor available in a DLL₁is also in another DL

L2thenL1is said to be a sublanguage ofL2. For instance, the basic DLALC8is a sublan- guage of S HOI N because it contains top, bottom, negation, conjunction, disjunction, existential restriction, and universal restriction.

As will be seen when formal semantics are given to the symbols and constructors, the symbol > is used to denote the universal concept (having the same semantics as Ct ¬C) and ⊥ denotes its complement the empty concept (Cu ¬C). In fact, in DLs that allow for negation there are dualities regarding constructors: Conjunction is dual to disjunction under negation since CuD ⇔ ¬(¬Ct ¬D) (De Morgan’s law) where

⇔ means logically equivalent. Analogously, existential and universal restriction are dual under negation (∃R.C ⇔ ¬∀R.¬C and∀R.C ⇔ ¬∃R.¬C). Consequently, for each complex concept created from these constructors there is always an equivalent dual concept; hence, such DLs provide syntactic sugar.

Notational Conventions

Throughout this thesis, we adopt the following notational conventions in definitions and examples:

6_{With punning, a meta modeling technique allowing to reuse names in cases in which it is possible to} disambiguate the exact use of a name, VC, VIand the set of role names VOPneed not, under certain restric- tions, be mutually disjoint. For more details seehttp://www.w3.org/2007/OWL/wiki/Punning.

7_{Number restrictions are limited to simple roles in order to retain decidability [HST99].} 8_{ALCstands for Attributive Language with Complement.}

• Definitions and abstract examples are typeset in math mode. The upper-case let- ters A, B are used for atomic concepts (A, B∈ VC); C, D for concepts; R, S for roles where S sometimes denotes a simple role; the lower-case letters a, b for individual names (a, b∈ VI); m, n for natural numbers.

• Concrete examples are typeset in slanted font shape. Concept names start with an uppercase letter followed by lowercase letters (e.g., Customer, Person, Book ), role names start with a lowercase letter (e.g., hasOrdered, likes), and individual names are composed of uppercase letters (e.g., ALICE, BOB ).

Terminological Knowledge

No matter which particular DL is considered, concepts and roles are used in terminologi- cal axioms to express how concepts or roles relate to each other in the domain of interest. In the most general case, these axioms have the form

CvD (RvS) or C ≡D (R ≡S) .

The former are called inclusions whereas the latter are called equalities. Informally, an inclusion states that the right-hand side concept (role) subsumes the left-hand side (i.e., a super concept/role that is more general than the sub concept/role). In logical terms, an inclusion C vD says that in order to be a member of D it is sufficient to be a member of C and that it is necessary to be a member of D to be a member of C. In fact, an inclusion can be understood as an implication C → D; we will come back to this later when detailing OWL 2 RL inSection 3.3.3. This is analogous the case for role inclusions R v S. Equalities, on the other hand, express that two concepts (roles), even though they might differ in intension, have the same extension. This means that an equality C ≡ D expresses necessary and sufficient conditions for concept membership in either direction; analogous for role equalities R ≡ S. Therefore, an equality X1 ≡ X2 is an abbreviation for two symmetric inclusion axioms X1 ≡ X2 ⇔ X1 v X2∧X2 v X1 where X1, X2are either concepts or roles.

An inclusion C v D where the left-hand side might possibly be a complex concept is called a general concept inclusion axiom (GCI). An equality A≡C whose left-hand side is an atomic concept A∈ VCis called a concept definition. Equalities of this kind are most often used to introduce symbolic names for complex concepts.

We will now formally define the most general form of collections of these axioms. Definition 3.3(RBox). An RBox Ris a finite set of role inclusion axioms of the form R v S and transitivity axioms Tra(R)where R, S are roles.

Definition 3.4(TBox). A TBoxT is a finite set of general concept inclusion axioms of the form C vD where C, D are concepts.

An atomic concept occurring on the left-hand side of a concept definition in T is defined whereas an atomic concept that only occurs on the right-hand side inT is prim- itive. Analogously, we can speak of primitive and defined roles. As a matter of fact, instances of primitive concepts can only be declared explicitly; how this can be done follows below.

Unfortunately, the presence of GCIs in the TBox, which is correspondingly called general then, causes worst-case computational complexity of terminological reasoning to become intractable. Therefore, a restricted form of the TBox is often considered where reasoning has tractable complexity for several DLs such as ALC [Lut99]. More precisely, an acyclic TBoxT is a TBox such that

• T contains only concept definitions,

• there is at most one definition for each concept name A ∈VC inT, and

• there does not exist a concept definition A ≡C inT where A occurs either directly or indirectly in C (i.e., if the definition of A does not transitively use itself).

If the last item does not hold thenT is called cyclic. The characteristic of acyclic TBoxes is that they are unequivocal regarding each defined concept. Furthermore, the extension of each defined concept is uniquely determined by the extension of primitive concepts. As a result, it is possible to compile away an acyclic TBox by a technique called unfold- ing: iteratively replace defined concepts occurring on the right-hand side of a concept definition by its definition unless only primitive concepts occur on the right-hand side of each concept definition.

Assertional Knowledge

The second part of knowledge representation means expresses the state of affairs in a domain: In the ABox one uses concepts and roles to make assertions about individuals. These assertions have the form

C(a), R(a, b), a =b, a6= b .

A concept assertion C(a)states that a is a member of C. A role assertion R(a, b) is used to state that b is a filler of the role R for a. More intuitively, a is related to b through R.

In contrast to the relational data model [Cod70, Cod90], DLs usually do not adopt the Unique Name Assumption (UNA). In the absence of the UNA the same individual might have different names (i.e., there can be aliases). This means that given only two distinct individual names a, b without having further knowledge about them one can neither conclude that they identify different individuals nor the same individual. Hence, the absence of the UNA necessitates the latter two types of assertions. An individual equality a=b asserts that the individual names a and b represent the same individual (i.e., a and b are different names identifying the same individual) and an individual inequality a 6=b asserts the opposite (i.e., that a and b represent distinct individuals in the domain). It should be clear that (in)equalities are dispensable under the UNA.

Definition 3.5(ABox). An ABoxAis a finite set of assertions of the form C(a), R(a, b), a=

Knowledge Base

Depending on the context, slightly different definitions of a knowledge base (KB) can be found in the literature. One possibility is to state that it consists of a TBoxT, an RBoxR, and an ABoxA, denoted withK = (T,R,A). More often, however, a KB is defined to be made up by a TBox and an ABox only. This is achieved by defining the TBox to also contain role inclusion axioms (of an RBox) in addition to concept inclusions. Finally, one can define a KB as the union of a TBox and an ABox. Formally,

K:= (T,A) or K:= T ∪ A . (3.1)

To simplify expositions, we will adopt the definition as a pair by default but sometimes prefer the latter for convenience.

Given a DLL, a knowledge baseKis anL-knowledge base if and only if each complex concept and complex role occurring in the axioms and assertions inKis built using the constructors available in L. Analogously, we can speak ofL-concepts, L-inclusions,L- assertions,L-TBoxes, andL-ABoxes.

Finally, we use the general term syntactic construct to refer to any of the different types of axioms and assertions defined by some DL. The term syntactic instance is used to refer to a concrete instance of any of the available syntactic constructs. Consequently, we can say that a knowledge base is a set of syntactic instances.

Semantics ofS HOI N

Now that we have seen how the DL S HOI N is syntactically defined, “meaning” is given to concepts, roles, assertions, and axioms in a formal way. This is achieved in terms of model-theoretic semantics [Tar56], enabling to interpret each of them in a non- empty domain of interest.

Definition 3.6 (Semantics of S HOI N). An interpretation for S HOI N is a tuple I = (∆I,·I) where∆I is the non-empty interpretation domain. The interpretation function ·I as- signs each individual name a ∈ VI to an individual aI ∈ ∆I, each concept name A ∈ VC to a subset AI ⊆∆I, and each role name R∈ VOPNto a subset RI ⊆∆I ×∆I. The interpretation function is extended to transitive, inverse roles and complex concepts as shown inTable 3.1. If C and R is a concept and role, respectively, then CIand RIis called the extension of C and R, respectively, inI. An individual name a∈ VI represents an instance of a concept C if aI ∈CI inI.

An interpretation I satisfies a GCI C v D if CI ⊆ DI; analogous for role inclusion axioms, seeTable 3.1. An interpretationI satisfies a TBox T, writtenI |= T, if it satisfies all axioms inT. Such an interpretation is called a model ofT.

An interpretationI satisfies ABox assertions C(a)if aI ∈CI, R(a, b)if(aI, bI) ∈ RI, a =b if aI = bI, and a 6=b if aI 6= bI. I is a model of an ABoxA, writtenI |= A, if it satisfies all assertions inA.

An interpretation I is a model of the knowledge base K = (T,A), written I |= K, if it is a model of both T and A. If there exists a model I for K then K is said to be consistent (or satisfiable); synonymously, we say thatAis consistent w.r.t.T .

Table 3.1: Syntax and Semantics of S HOI N concept expressions and roles and corre- sponding OWL constructs

DL OWL

Ex. Syntax Semantics

A AI ⊆_∆I _Class > >I _=∆I _Thing ⊥ ⊥I =∅ Nothing R RI ⊆_∆I_×∆I _{ObjectProperty} S Tra(R) RI = (RI)+ TransitiveProperty CuD (CuD)I =CI∩DI intersectionOf CtD (CtD)I =CI∪DI unionOf ¬C (¬C)I =∆I\CI complementOf ∃R.C {x | ∃y.(x, y) ∈ RI and y∈CI} someValuesFrom ∀R.C {x | ∀y.(x, y) ∈ RI implies y∈ CI} allValuesFrom H R1v R2 RI₁ ⊆ RI₂ subPropertyOf O {a1, . . . , an} {a1, . . . , an}I = {aI1} ∪ · · · ∪ {aIn} oneOf? ∃R.{a} {x | (x, aI) ∈RI} hasValue I R− {(x, y) | (y, x) ∈ RI} inverseOf N ≥n S {x | ]{y | (x, y) ∈SI} ≥n} minCardinality ≤n S {x | ]{y | (x, y) ∈SI} ≤n} maxCardinality F Fun(S) (x, y) ∈SI and(x, z) ∈SIimplies y=z FunctionalProperty ]N denotes the cardinality of the set N.

Fun(S)is a syntactic variant of≤1 S (listed for completeness sake).

Based onDefinition 3.6we can formally express the semantics of the UNA.

Definition 3.7(Unique Name Assumption). An interpretationIrespects the unique name assumption iff·I is an injection regarding interpretation of individual names; that is, ∀a, b ∈

VI: aI =bI implies a =b.

The cases where·I is a surjection or a bijection regarding individual names are re- spectively referred to as the parameter names assumption (i.e., no unnamed individuals) and standard names assumption (e.g., the identity function aI =a).

In contrast to database schema frameworks such as the ER model [Che76, SS77] that make the assumption of defining a single model, a KB does not define a single model. In fact, it can be seen as a set of constraints that may be satisfied by a possibly infinite set of models. This is easiest to understand when considering that no constraints at all – the empty KB – means that any model is possible. Adding more constraints usu- ally means fewer models, up to the point where no model remains possible due to the existence of contradictory constraints. The latter is called an inconsistent KB (seeDefini- tion 3.6). Speaking of constraints here, it is important not to confuse this with the notion of integrity constraints. “Rather than being statements about the world, [integrity] con- straints are statements about what the KB can be said to know” [Rei88]. In other words, integrity constraints are meant for enforcing the acceptable states (content) of the KB.

Yet one might want to interpret axioms in the KB both as integrity constraints and in the “standard” way [MHS09, TSBM10].

Moreover, different assumptions on the cardinality of the domain ∆I can be made; hence, on the size of models. In open environments such as the Web one typically as- sumes an open (infinite) domain∆I since one cannot assume to have complete knowl- edge. On the other hand, one can strictly close the domain by stating> v {a1, . . . , an} where a1, . . . , an are the named individuals that shall exist in the domain. An open versus closed domain has consequences, for instance, on the existential constructor: whereas ∃hasFriend.> can refer to new, otherwise unknown, individuals in an open domain, it does refer to an ai in a closed domain (and without further knowledge it is not known which particular ai this is). In contrast, a database model is considered to be a finite structure since a database is understood as a complete representation of an application domain: as noted above, the database is a particular model.

The aspect whether the domain is open versus closed is not to be confused with the open world assumption (OWA) versus the closed world assumption (CWA), which refer to different reasoning paradigms. The former means that given an assertion or axiom ψ and a KB K, if ψ is satisfied only in some models of K then neitherK |= ψnor K 6|= ψ

can be concluded. In such a case we can merely draw the conclusion that the truth- value of ψ is not known; K is underspecified w.r.t. ψ. In contrast, a failure to proof ψ to be true due to lack of sufficient information implies that it is false under the CWA. Formally,K 6|= ψimpliesK |= ¬ψ. This reflects common-sense reasoning in which one

conjectures ψ to be false if it is not true, which builds on the assumption that all relevant knowledge is available (as opposed to the assumption under the OWA that knowledge is incomplete). The CWA is common to databases. For instance, if the product database of a book seller does not contain a certain book then it is concluded that the book is not offered by the book store; under the OWA we can only conclude that it might be offered, but we cannot be certain based on the information stored in the database.

Finally, the syntactic constructs allowed inS HOI N can interact in such a way that a KB does not admit finite models (i.e., it admits only models with an infinite domain ∆I_{). For example, DLs allowing for inverse roles, functional roles (≤1 R), and cycles in}

the TBox [CGLN01]. In general, a DL where each concept or TBox admits a finite model is said to have the finite model property [EF95]. This becomes important for reasoning procedures and especially if one wants to compute and materialize the complete set of implicit axioms and assertions entailed by a KB. Obviously, this is impossible if the KB admits only infinite models.