OWL 1.1 Web Ontology Language
Model-Theoretic Semantics

W3C Member Submission 19 December 2006

This version:: http://www.w3.org/submissions/2006/SUBM-owl11-semantics-20061219/
Latest version:: http://www.w3.org/submissions/owl11-semantics/
Authors:: Bernardo Cuenca Grau, The University of Manchester; Boris Motik, The University of Manchester
Contributors:: Ian Horrocks, The University of Manchester; Bijan Parsia, The University of Manchester; Peter F. Patel-Schneider, Bell Labs Research, Lucent Technologies; Ulrike Sattler, The University of Manchester

Abstract

OWL 1.1 extends the W3C OWL Web Ontology Language with a small but useful set of features that have been requested by users, for which effective reasoning algorithms are now available, and that OWL tool developers are willing to support. The new features include extra syntactic sugar, additional property and qualified cardinality constructors, extended datatype support, simple metamodeling, and extended annotations. This document provides a model-theoretic semantics for OWL 1.1.

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1 Introduction
2 Model-Theoretic Semantics for OWL 1.1
References

1 Introduction

This document defines the formal semantics of OWL 1.1. The semantics given here follows the principles for defining the semantics of description logics [Description Logics] and is compatible with the description logic SROIQ presented in [SROIQ]. Unfortunately, the definition of SROIQ given in [SROIQ] does not provide for datatypes and metamodeling. Therefore, the semantics of OWL 1.1 is defined in a direct model-theoretic way, by interpreting the constructs of the functional-style syntax from [OWL 1.1 Specification]. For the constructs available in SROIQ, the semantics of SROIQ trivially corresponds to the one defined in this document.

OWL 1.1 does not have an RDF-compatible semantics. Ontologies expressed in OWL RDF are given semantics by converting then into the functional-style syntax and interpreting the result as specified in this document.

OWL 1.1 allows for annotations of ontologies and ontology axioms. Annotations, however, have no semantic meaning in OWL 1.1 and are ignored in this document.

Since OWL 1.1 is an extension of OWL DL, this document also provides a formal semantics for OWL DL and it is equivalent to the definition given in [OWL Abstract Syntax and Semantics].

2 Model-Theoretic Semantics for OWL 1.1

A vocabulary (or signature) V = ( N_C , N_Po , N_Pd , N_I , N_D , N_V ) is a 6-tuple where

N_C is a set of OWL classes,
N_Po is a set of object properties,
N_Pd is a set of data properties,
N_I is a set of individuals, and
N_D is a set of datatypes each associated with a positive integer datatype arity,
N_V is a set of well-formed constants.

Since OWL 1.1 allows punning [Metamodeling] in the signature, we do not require the sets N_C , N_Po , N_Pd , N_I , N_D , and N_V to be pair-wise disjoint. Thus, the same name can be used in an ontology to denote a class, a datatype, a property (object or data), an individual, and a constant. The set N_D is defined as it is because a datatype is defined by its name and the arity, and such a definition allows one to reuse the same name with different arities.

The semantics of OWL 1.1 is defined with respect to a concrete domain, which is a tuple D = ( Δ_D , .^D ) where

Δ_D is a fixed set called the data domain,
.^D assigns to each constant v ∈ N_V an element v^D of Δ_D, and
.^D assigns to each datatype d ∈ N_D with arity n an n-ary relation d^D over Δ_D.

The set of datatypes N_D in each OWL 1.1 vocabulary must include a unary datatype rdfs:Literal interpreted as 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].

The set 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 set Δ_D; however, the choice of the actual set is not actually relevant for the definition of the semantics, as long as it contains the interpretations of all datatypes that one can "reasonably think of." This allows the implementations to support datatypes other than the ones mentioned in the previous paragraphs without affecting the semantics.

Given a vocabulary V and a concrete domain D, an interpretation I = ( Δ_I , .^Ic , .^Ipo , .^Ipd , .^Ii ) is a 5-tuple where

Δ_I is a nonempty set disjoint with Δ_D, called the object domain;
.^Ic is the class 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 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 .^Ipo to object property expressions as shown in Table 1.

Table 1. Interpreting Property Expressions
Property	Interpretation
InverseObjectProperty(R)	{ ( x , y ) \| ( y , x ) ∈ R^Ipo }

We extend the interpretation function .^D to data ranges as shown in Table 2.

Table 2. Interpreting Data Ranges
Data Range	Interpretation
DataOneOf(v₁ ... v_n)	{ v₁^D , ... , v_n^D }
DataComplementOf(DR)	( Δ_D )ⁿ \ DR^D where n is the arity of DR
DatatypeRestriction(DR f v)	the n-ary relation over Δ_D obtained by applying the facet f with value v to the data range DR as specified in [XML Schema Datatypes]

We extend the class interpretation function .^Ic to classes as shown in Table 3. With #S we denote the number of elements in a set S.

Table 3. Interpreting Classes
Class	Interpretation
owl:Thing	Δ_I
owl:Nothing	empty set
ObjectComplementOf(C)	Δ_I \ C^Ic
ObjectIntersectionOf(C₁ ... C_n)	C₁^Ic ∩ ... ∩ C_n^Ic
ObjectUnionOf(C₁ ... C_n)	C₁^Ic ∪ ... ∪ C_n^Ic
ObjectOneOf(a₁ ... a_n)	{ a₁^Ii , ... , a_n^Ii }
ObjectSomeValuesFrom(R C)	{ x \| ∃ y : ( x, y ) ∈ R^Ipo and y ∈ C^Ic }
ObjectAllValuesFrom(R C)	{ x \| ∀ y : ( x, y ) ∈ R^Ipo implies y ∈ C^Ic }
ObjectHasValue(R a)	{ x \| ( x, a^Ii ) ∈ R^Ipo }
ObjectExistsSelf(R)	{ x \| ( x, x ) ∈ R^Ipo }
ObjectMinCardinality(n R C)	{ x \| #{ y \| ( x, y ) ∈ R^Ipo and y ∈ C^Ic } ≥ n }
ObjectMaxCardinality(n R C)	{ x \| #{ y \| ( x, y ) ∈ R^Ipo and y ∈ C^Ic } ≤ n }
ObjectExactCardinality(n R C)	{ x \| #{ y \| ( x, y ) ∈ R^Ipo and y ∈ C^Ic } = n }
DatatSomeValuesFrom(U₁ ... U_n DR)	{ x \| ∃ y₁, ..., y_n : ( x, y_k ) ∈ U_k^Ipd for each 1 ≤ k ≤ n and ( y₁, ..., y_n ) ∈ DR^D }
DatatAllValuesFrom(U₁ ... U_n DR)	{ x \| ∀ y₁, ..., y_n : ( x, y_k ) ∈ U_k^Ipd for each 1 ≤ k ≤ n implies ( y₁, ..., y_n ) ∈ DR^D }
DataHasValue(U v)	{ x \| ( x, v^D ) ∈ U^Ipd }
DataMinCardinality(n U DR)	{ x \| #{ y \| ( x, y ) ∈ U^Ipd and y ∈ DR^D } ≥ n }
DataMaxnCardinality(n U DR)	{ x \| #{ y \| ( x, y ) ∈ U^Ipd and y ∈ DR^D } ≤ n }
DataExactCardinality(n U DR)	{ x \| #{ y \| ( x, y ) ∈ U^Ipd and y ∈ DR^D } = n }

Satisfaction of OWL 1.1 axioms in an interpretation I is defined as shown in Table 4. With o we denote the composition of binary relations.

Table 4. Satisfaction of Axioms in an Interpretation
Axiom	Condition
SubClassOf(C D)	C^Ic ⊆ D^Ic
EquivalentClasses(C₁ ... C_n)	C_j^Ic = C_k^Ic for each 1 ≤ j , k ≤ n
DisjointClasses(C₁ ... C_n)	C_j^Ic ∩ C_k^Ic is empty for each 1 ≤ j , k ≤ n and j ≠ k
DisjointUnion(A C₁ ... C_n)	A^Ic = C₁^Ic ∪ ... ∪ C_n^Ic and C_j^Ic ∩ C_k^Ic is empty for each 1 ≤ j , k ≤ n and j ≠ k
SubObjectPropertyOf(R S)	R^Ipo ⊆ S^Ipo
SubObjectPropertyOf(SubObjectPropertyChain(R₁ ... R_n) S)	R₁^Ipo o ... o R_n^Ipo ⊆ S^Ipo
EquivalentObjectProperties(R₁ ... R_n)	R_j^Ipo = R_k^Ipo for each 1 ≤ j , k ≤ n
DisjointObjectProperties(R₁ ... R_n)	R_j^Ipo ∩ R_k^Ipo is empty for each 1 ≤ j , k ≤ n and j ≠ k
ObjectPropertyDomain(R C)	{ x \| ∃ y : (x , y ) ∈ R^Ipo } ⊆ C^Ic
ObjectPropertyRange(R C)	{ y \| ∃ x : (x , y ) ∈ R^Ipo } ⊆ C^Ic
InverseObjectProperties(R S)	R^Ipo = { ( x , y ) \| ( y , x ) ∈ S^Ipo }
FunctionalObjectProperty(R)	( x , y₁ ) ∈ R^Ipo and ( x , y₂ ) ∈ R^Ipo imply y₁ = y₂
InverseFunctionalObjectProperty(R)	( x₁ , y ) ∈ R^Ipo and ( x₂ , y ) ∈ R^Ipo imply x₁ = x₂
ReflexiveObjectProperty(R)	x ∈ Δ_I implies ( x , x ) ∈ R^Ipo
IrreflexiveObjectProperty(R)	x ∈ Δ_I implies ( x , x ) is not in R^Ipo
SymmetricObjectProperty(R)	( x , y ) ∈ R^Ipo implies ( y , x ) ∈ R^Ipo
AntisymmetricObjectProperty(R)	( x , y ) ∈ R^Ipo implies ( y , x ) is not in R^Ipo
TransitiveObjectProperty(R)	R^Ipo o R^Ipo ⊆ R^Ipo
SubDataPropertyOf(U V)	U^Ipd ⊆ V^Ipd
EquivalentDataProperties(U₁ ... U_n)	U_j^Ipd = U_k^Ipd for each 1 ≤ j , k ≤ n
DisjointDataProperties(U₁ ... U_n)	U_j^Ipd ∩ U_k^Ipd is empty for each 1 ≤ j , k ≤ n and j ≠ k
DataPropertyDomain(U C)	{ x \| ∃ y : (x , y ) ∈ U^Ipd } ⊆ C^Ic
DataPropertyRange(U DR)	{ y \| ∃ x : (x , y ) ∈ U^Ipd } ⊆ DR^D
FunctionalDataProperty(U)	( x , y₁ ) ∈ U^Ipd and ( x , y₂ ) ∈ U^Ipd imply y₁ = y₂
SameIndividual(a₁ ... a_n)	a_j^Ii = a_k^Ii for each 1 ≤ j , k ≤ n
DifferentIndividuals(a₁ ... a_n)	a_j^Ii ≠ a_k^Ii for each 1 ≤ j , k ≤ n and j ≠ k
ClassAssertion(a C)	a^Ii ∈ C^Ic
ObjectPropertyAssertion(R a b)	( a^Ii , b^Ii ) ∈ R^Ipo
NegativeObjectPropertyAssertion(R a b)	( a^Ii , b^Ii ) is not in R^Ipo
DataPropertyAssertion(U a v)	( a^Ii , v^D ) ∈ U^Ipd
NegativeDataPropertyAssertion(U a v)	( a^Ii , v^D ) is not in U^Ipd

Let O be an OWL 1.1 ontology with vocabulary V. O is consistent if an interpretation I exists that satisfies all the axioms of the axiom closure of O (the axiom closure of O is defined in [OWL 1.1 Specification]); such I is then called a model of O. A class C is satisfiable w.r.t. O if there is a model I of O such that C^Ic is not empty. O entails an OWL 1.1 ontology O' with vocabulary V if every model of O is also a model of O'; furthermore, O and O' are equivalent if O entails O' and O' entails O.

References

[Description Logics]: The Description Logic Handbook. Franz Baader, Diego Calvanese, Deborah McGuinness, Daniele Nardi, Peter Patel-Schneider, editors. Cambridge University Press, 2003; and Description Logics Home Page.
[Metamodeling]: On the Properties of Metamodeling in OWL. Boris Motik. In Proceedings of ISWC-2005
[OWL 1.1 Specification]: OWL 1.1 Web Ontology Language: Structural Specification and Functional-Style Syntax. Peter F. Patel-Schneider, Ian Horrocks, and Boris Motik, eds., 2006.
[OWL Abstract Syntax and Semantics]: OWL Web Ontology Language: Semantics and Abstract Syntax. Peter F. Patel-Schneider, Pat Hayes, and Ian Horrocks, Editors, W3C Recommendation, 10 February 2004.
[SROIQ]: The Even More Irresistible SROIQ. Ian Horrocks, Oliver Kutz, and Uli Sattler. In Proc. of the 10th Int. Conf. on Principles of Knowledge Representation and Reasoning (KR 2006). AAAI Press, 2006.
[XML Schema Datatypes]: XML Schema Part 2: Datatypes Second Edition. Paul V. Biron and Ashok Malhotra, eds. W3C Recommendation 28 October 2004.

OWL 1.1 Web Ontology Language Model-Theoretic Semantics