OWL 2 Web Ontology Language:
Model-Theoretic Semantics

1 Introduction

Since OWL2 is an extension of OWL DL, this document also provides a formalModel-Theoretic Semantics for OWL Lite and2

This section specifies the model-theoretic semantics of OWL DL and it is equivalent to2 ontologies in the definition givenfunctional-style syntax.

2.1 Vocabulary

2.2 Interpretations

Given a datatype is defined by its name and the arity,map D and such a definition allows one to reuse the same name with different arities. The semantics of OWL 2 is defined with respect to a concrete domain , which isa tuplevocabulary V over D, an interpretation Int = ( Δ _DInt , .Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) wherefor D and V is a 9-tuple with the following structure.

• Δ_Int is a nonempty set called the object domain.
• Δ_D is a fixednonempty set disjoint with Δ_Int called the data domain , .such that (DT)^DT ⊆ Δ_D for each datatype DT ∈ V_DT.
• ⋅ ^C is the class interpretation function that assigns to each constant vclass C ∈ N V an elementV _{D ofC} a subset (C)^C ⊆ Δ _{D ,Int} such that
  • (owl:Thing)^C = Δ_Int and
  • . D(owl:Nothing)^C = ∅.
• ⋅ ^OP is the object property interpretation function that assigns to each datatype dobject property OP ∈ N D with arity n an n -ary relation d D overV_OP a subset (OP)^OP ⊆ Δ _{D .Int} × Δ_Int such that
  • (owl:TopObjectProperty)^OP = Δ_Int × Δ_Int and
  • (owl:BottomObjectProperty)^OP = ∅.
• ⋅ ^DP is the set of datatypes N D indata property interpretation function that assigns to each OWL 2 vocabulary must includedata property DP ∈ V_DP a unary datatype rdfs:Literal interpreted assubset (DP)^DP ⊆ Δ _{D ; furthermore, it must also include the following unary datatypes: xsd:string , xsd:boolean , xsd:decimal , xsd:float , xsd:double , xsd:dateTime , xsd:time , xsd:date , xsd:gYearMonth , xsd:gYear , xsd:gMonthDay , xsd:gDay , xsd:gMonth , xsd:hexBinary , xsd:base64Binary , xsd:anyURI , xsd:normalizedString , xsd:token , xsd:language , xsd:NMTOKEN , xsd:Name , xsd:NCName , xsd:integer , xsd:nonPositiveInteger , xsd:negativeInteger , xsd:long , xsd:int , xsd:short , xsd:byte , xsd:nonNegativeInteger , xsd:unsignedLong , xsd:unsignedInt , xsd:unsignedShort , xsd:unsignedByte , and xsd:positiveInteger . These datatypes, as well as the well-formed constants from N V , are interpreted as specified in [ XML Schema Datatypes ]. (Note that the value spaces of primitive XML Schema Datatypes are disjoint, so "2"^^xsd:decimal D is different from "2"^^xsd:float" D .) The setInt} × Δ_D is a fixed set that must be large enough ; that is, it must contain the extension of each datatype from N D and, apart from that, an infinite number of other objects.such a definition is ambiguous, as it does not uniquely single out a particular setthat
  • (owl:TopDataProperty)^DP = Δ_Int × Δ_D ; however, the choice of the actual setand
  • (owl:BottomDataProperty)^DP = ∅.
• ⋅ ^I is not actually relevant for the definition of the semantics, as long as it containsthe interpretations of all datatypesindividual interpretation function that one can "reasonably think of." This allows the implementationsassigns to support datatypes other than the ones mentioned in the previous paragraphs without affecting the semantics. Giveneach individual a vocabulary∈ V _{and a concrete domain D ,I} an interpretationelement (a)^I = (∈ Δ _{I ,Int}.
• Ic⋅ ^DT, . Ipo⋅ ^LT, . Ipdand ⋅ ^FA are the same as in D, . Ii ) is a 5-tuple where Δ I is a nonempty set disjoint withand (rdfs:Literal)^DT = Δ_D , called the object domain ;.

Ic isThe following sections define the extensions of ⋅ ^OP, ⋅ ^DT, and ⋅ ^C to object property expressions, data ranges, and class expressions.

2.2.1 Object Property Expressions

The object property interpretation function that assigns to each OWL class A ∈ N Ic a subset A Ic of Δ I  ; . Ipo is the object property interpretation function that assigns to each object property R ∈ N Po a subset R Ipo of Δ I x Δ I  ; . Ipd is the data property interpretation function that assigns to each data property U ∈ N Pd a subset U Ipd of Δ I x Δ D  ; . Ii⋅ ^OP is the individual interpretation function that assigns to each individual a ∈ N I an element a Ii from Δ I . We extend the object interpretation function . Ipoextended to object property expressions as shown in Table 1.

We extend2.2.2 Data Ranges

The datatype interpretation function . D⋅ ^DT is extended to data ranges as shown in Table 2.3. Note that datatypes in OWL 2 are all unary; thus, each datatype DT is interpreted as a unary relation (DT)^DT over Δ_D. Data ranges, however, can be n-ary—this allows implementations to provide built-in predicates such as comparisons or arithmetic as an extension. Hence, an n-ary data range DR is interpreted as an n-ary relation (DR)^DT over Δ_D.

2.2.3 Class Expressions

The class interpretation function . Ic⋅ ^C is extended to descriptionsclass expressions as shown in Table 3. With4. For S a set, #S we denotedenotes the number of elements in a setS.

2.3 Satisfaction of OWL 2 axiomsin an Interpretation

2.3.1 Class Expression Axioms

Satisfaction of OWL 2 class expression axioms in Int w.r.t. O is defined as shown in Table 5.

SubObjectPropertyOf(R S) R Ipo2.3.2 Object Property Expression Axioms

Satisfaction of OWL 2 object property expression axioms in Int w.r.t. O is defined as shown in Table 6.

2.3.3 Data Property Expression Axioms

Satisfaction of OWL 2 data property expression axioms in Int w.r.t. O is notdefined as shown in R Ipo SymmetricObjectProperty(R) (Table 7.

2.3.4 Keys

Satisfaction of keys in Int w.r.t. O is defined as shown in Table 8.

2.3.5 Assertions

Satisfaction of OWL 2 assertions in Int w.r.t. O is defined as shown in Table 9.

2.3.6 Ontologies

2.4 Models

An interpretation Int = ( Δ_Int , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) is a model of an OWL 2 ontology O if an interpretation Int' = ( Δ_Int , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I' , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) exists such that ⋅ ^I' coincides with ⋅ ^I on all named individuals and Int' satisfies O.

Thus, an interpretation Int satisfying O is also a model of O. In contrast, a model Int of O may not satisfy O directly; however, by modifying the interpretation of anonymous individuals, Int can always be coerced into an interpretation Int' that satisfies O.

2.5 Inference Problems

Let D be a datatype map and V a vocabulary over D. Furthermore, let O and O' be OWL 2 ontologies, CE, CE₁, and CE₂ class expressions, and a a named individual, such that all of them refer only to the vocabulary elements in V.

• O is satisfiable (or consistent) w.r.t. D if a model of O w.r.t. D and V exists.
• O entails O' w.r.t. D if every model of O w.r.t. D and V is also a model of O' w.r.t. D and V.
• O and O' are equivalent w.r.t. D if O entails O' w.r.t. D and O' entails O w.r.t. D.
• O and O' are equisatisfiable w.r.t. D if O is satisfiable w.r.t. D if and only if O' is satisfiable w.r.t D.
• CE is satisfiable w.r.t. O and D if a model Int = ( Δ_Int , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) of O w.r.t. D and V exists such that (CE)^C ≠ ∅.
• CE₁ is subsumed by CE₂ w.r.t. O and D if (CE₁)^C ⊆ (CE₂)^C for each model Int = ( Δ_Int , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) of O w.r.t. D and V.
• a is an instance of CE w.r.t. O and D if (a)^I ∈ (CE)^C for each model Int = ( Δ_Int , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) of O w.r.t. D and V.

3 Independence of the Semantics from the Datatype Map

The semantics of OWL 2 has been defined in such a way that the semantics of an OWL 2 ontology O does not depend on the choice of a datatype map, as long as the datatype map chosen contains all the datatypes occurring in O. This statement is made precise by the following theorem, which has several useful consequences:

• One can interpret an OWL 2 ontology O by considering only the datatypes explicitly occurring in O.
• When referring to various reasoning problems, the datatype map D need not be given explicitly, as it is sufficient to consider the datatype map implicitly defined by the ontology in question.
• OWL 2 reasoners can provide datatypes not explicitly mentioned in this specification without fear that this will change the semantics of OWL 2 ontologies not using these datatypes.

Theorem 1. Let O₁ and O₂ be OWL 2 ontologies over a vocabulary V and D = ( N_DT , N_LT , N_FA , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) a datatype map such that each datatype mentioned in O₁ and O₂ is either rdfs:Literal or it occurs in N_DT. Furthermore, let D' = ( yN_DT' , x ) ∈ R Ipo AsymmetricObjectProperty(R) ( xN_LT' , y ) ∈ R Ipo implies ( yN_FA' , x⋅ ^DT ' , ⋅ ^LT ' , ⋅ ^FA ' ) is not in R Ipo TransitiveObjectProperty(R) R Ipo o R Ipo ⊆ R Ipo SubDataPropertyOf(U V) U Ipdbe a datatype map such that N_DT ⊆ V Ipd EquivalentDataProperties(U 1 ... UN _{) U j Ipd = U k Ipd for each 1 ≤ jDT}', k ≤N _{DisjointDataProperties(U 1 ... ULT}(DT) = N _{) U j Ipd ∩ U k Ipd is emptyLT}'(DT) and N_FA(DT) = N_FA'(DT) for each 1 ≤ j , k ≤DT ∈ N_DT, and j ≠ k DataPropertyDomain(U C) { x | ∃ y : (x⋅ ^DT ', y ) ∈ U Ipd } ⊆ C Ic DataPropertyRange(U DR) { y | ∃ x : (x⋅ ^LT ', y ) ∈ U Ipd } ⊆ DR D FunctionalDataProperty(U) ( xand ⋅ ^FA ' are extensions of ⋅ ^DT, ⋅ ^LT, y 1 ) ∈ U Ipdand ( x⋅ ^FA, yrespectively. Then, O₁ entails O₂ ) ∈ U Ipd imply yw.r.t. D if and only if O₁ = yentails O₂ SameIndividual(aw.r.t. D'.

4 References