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__NUMBEREDHEADINGS__ {{TR|title=Direct Semantics}} <div id="editors"> ; Editors: : [http://web.comlab.ox.ac.uk/people/Boris.Motik/ Boris Motik], Oxford University Computing Laboratory : [http://ect.bell-labs.com/who/pfps/ Peter F. Patel-Schneider], Bell Labs Research, Alcatel-Lucent : [http://web.comlab.ox.ac.uk/people/Bernardo.CuencaGrau/ Bernardo Cuenca Grau], Oxford University Computing Laboratory ; Contributors (alphabetical order): : [http://web.comlab.ox.ac.uk/people/ian.horrocks/ Ian Horrocks], Oxford University Computing Laboratory : [http://www.cs.man.ac.uk/~bparsia/ Bijan Parsia], University of Manchester : [http://www.cs.man.ac.uk/~sattler/ Uli Sattler], University of Manchester </div> ; Abstract: : <div id="abstract"><p>{{:Shared_Abstract}}</p><p>This document provides the direct model-theoretic semantics for OWL 2, which is compatible with the description logic ''SROIQ''. Furthermore, this document defines the most common inference problems for OWL 2.</p></div> ; Status of this Document: {{:Shared_Copyright}} <div id="docbody"> __TOC__ == Introduction == This document defines the direct model-theoretic semantics of OWL 2. The semantics given here is strongly related to the semantics of description logics [<cite>[[#ref-description-logics|Description Logics]]</cite>] and it extends the semantics of the description logic ''SROIQ'' [<cite>[[#ref-sroiq|SROIQ]]</cite>]. As the definition of ''SROIQ'' does not provide for datatypes and punning, the semantics of OWL 2 is defined directly on the constructs of the structural specification of OWL 2 [<cite>[[#ref-owl-2-specification|OWL 2 Specification]]</cite>] instead of by reference to ''SROIQ''. For the constructs available in ''SROIQ'', the semantics of ''SROIQ'' trivially corresponds to the one defined in this document. Since each OWL 1 DL ontology is an OWL 2 ontology, this document also provides a direct semantics for OWL 1 Lite and OWL 1 DL ontologies; this semantics is equivalent to the direct model-theoretic semantics of OWL 1 Lite and OWL 1 DL [<cite>[[#ref-owl-1-semantics-and-abstract-syntax|OWL 1 Semantics and Abstract Syntax]]</cite>]. Furthermore, this document also provides the direct model-theoretic semantics for the OWL 2 profiles [<cite>[[#ref-owl-2-profiles|OWL 2 Profiles]]</cite>]. The semantics is defined for OWL 2 axioms and ontologies, which should be understood as instances of the structural specification [<cite>[[#ref-owl-2-specification|OWL 2 Specification]]</cite>]. Parts of the structural specification are written in this document using the functional-style syntax. OWL 2 allows ontologies, anonymous individuals, and axioms to be annotated; furthermore, annotations themselves can contain additional annotations. All these types of annotations, however, have no semantic meaning in OWL 2 and are ignored in this document. OWL 2 declarations are used only to disambiguate class expressions from data ranges and object property from data property expressions in the functional-style syntax; therefore, they are not mentioned explicitly in this document. == Direct Model-Theoretic Semantics for OWL 2 == This section specifies the direct model-theoretic semantics of OWL 2 ontologies. === Vocabulary === A <span id="def_datatype_map">''datatype map''</span>, formalizing [[Syntax#def_datatype_map|datatype maps]] from the OWL 2 Specification [<cite>[[#ref-owl-2-specification|OWL 2 Specification]]</cite>], is a 6-tuple ''D'' = ( ''N_DT'' , ''N_LS'' , ''N_FS'' , ''&sdot;&nbsp;^DT'' , ''&sdot;&nbsp;^LS'' , ''&sdot;&nbsp;^FS'' ) with the following components: * ''N_DT'' is a set of datatypes (more precisely, names of datatypes) that does not contain the datatype ''rdfs:Literal''. * ''N_LS'' is a function that assigns to each datatype ''DT'' &isin; ''N_DT'' a set ''N_LS(DT)'' of strings called ''lexical forms''. The set ''N_LS(DT)'' is called the ''lexical space'' of ''DT''. * ''N_FS'' is a function that assigns to each datatype ''DT'' &isin; ''N_DT'' a set ''N_FS(DT)'' of pairs ( ''F'' , ''v'' ), where ''F'' is a ''constraining facet'' and ''v'' is an arbitrary data value called the ''constraining value''. The set ''N_FS(DT)'' is called the ''facet space'' of ''DT''. * For each datatype ''DT'' &isin; ''N_DT'', the ''interpretation function'' ''&sdot;&nbsp;^DT'' assigns to ''DT'' a set ''(DT)^DT'' called the ''value space'' of ''DT''. * For each datatype ''DT'' &isin; ''N_DT'' and each lexical form ''LV'' &isin; ''N_LS(DT)'', the ''interpretation function'' ''&sdot;&nbsp;^LS'' assigns to the pair ( ''LV'' , ''DT'' ) a ''data value'' ( ''LV'' , ''DT'' )''^LS'' &isin; ''(DT)^DT''. * For each datatype ''DT'' &isin; ''N_DT'' and each pair ( ''F'' , ''v'' ) &isin; ''N_FS(DT)'', the ''interpretation function'' ''&sdot;&nbsp;^FS'' assigns to ( ''F'' , ''v'' ) the set ( ''F'' , ''v'' )''^FS'' &sube; ''(DT)^DT''. The set of datatypes ''N_DT'' of a datatype map ''D'' is not required to contain all datatypes from the [[Syntax#def_OWL_2_datatype_map|OWL 2 datatype map]]; this allows one to talk about subsets of the OWL 2 datatype map, which may be necessary for the various profiles of OWL 2. If, however, ''D'' contains a datatype ''DT'' from the [[Syntax#def_OWL_2_datatype_map|OWL 2 datatype map]], then ''N_LS(DT)'', ''N_FS(DT)'', ''(DT)^DT'', ( ''LV'' , ''DT'' )''^LS'' for each ''LV'' &isin; ''N_LS(DT)'', and ( ''F'' , ''v'' )''^FS'' for each ( ''F'' , ''v'' ) &isin; ''N_FS(DT)'' are required to coincide with the definitions for ''DT'' in the [[Syntax#def_OWL_2_datatype_map|OWL 2 datatype map]]. A <span id="def_vocabulary">''vocabulary''</span> ''V'' = ( ''V_C'' , ''V_OP'' , ''V_DP'' , ''V_I'' , ''V_DT'' , ''V_LT'' , ''V_FA'' ) over a datatype map ''D'' is a 7-tuple consisting of the following elements: * ''V_C'' is a set of [[Syntax#def_class|''classes'']] as defined in the OWL 2 Specification [<cite>[[#ref-owl-2-specification|OWL 2 Specification]]</cite>], containing at least the classes ''owl:Thing'' and ''owl:Nothing''. * ''V_OP'' is a set of [[Syntax#def_object_property|''object properties'']] as defined in the OWL 2 Specification [<cite>[[#ref-owl-2-specification|OWL 2 Specification]]</cite>], containing at least the object properties ''owl:topObjectProperty'' and ''owl:bottomObjectProperty''. * ''V_DP'' is a set of [[Syntax#def_data_property|''data properties'']] as defined in the OWL 2 Specification [<cite>[[#ref-owl-2-specification|OWL 2 Specification]]</cite>], containing at least the data properties ''owl:topDataProperty'' and ''owl:bottomDataProperty''. * ''V_I'' is a set of [[Syntax#def_individual|''individuals'']] (named and anonymous) as defined in the OWL 2 Specification [<cite>[[#ref-owl-2-specification|OWL 2 Specification]]</cite>]. * ''V_DT'' is a set containing all datatypes of ''D'', the datatype ''rdfs:Literal'', and possibly other datatypes; that is, ''N_DT'' &cup; { ''rdfs:Literal'' } &sube; ''V_DT''. * ''V_LT'' is a set of [[Syntax#def_literal|''literals'']] ''LV''^^''DT'' for each datatype ''DT'' &isin; ''N_DT'' and each lexical form ''LV'' &isin; ''N_LS(DT)''. * ''V_FA'' is the set of pairs ( ''F'' , ''lt'' ) for each constraining facet ''F'', datatype ''DT'' &isin; ''N_DT'', and literal ''lt'' &isin; ''V_LT'' such that ( ''F'' , ( ''LV'' , ''DT₁'' )''^LS'' ) &isin; ''N_FS(DT)'', where ''LV'' is the lexical form of ''lt'' and ''DT₁'' is the datatype of ''lt''. Given a vocabulary ''V'', the following conventions are used in this document to denote different syntactic parts of OWL 2 ontologies: * <span class="name">OP</span> denotes an object property; * <span class="name">OPE</span> denotes an object property expression; * <span class="name">DP</span> denotes a data property; * <span class="name">DPE</span> denotes a data property expression; * <span class="name">C</span> denotes a class; * <span class="name">CE</span> denotes a class expression; * <span class="name">DT</span> denotes a datatype; * <span class="name">DR</span> denotes a data range; * <span class="name">a</span> denotes an individual (named or anonymous); * <span class="name">lt</span> denotes a literal; and * <span class="name">F</span> denotes a constraining facet. === Interpretations === Given a datatype map ''D'' and a vocabulary ''V'' over ''D'', an <span id="def_interpretation">''interpretation''</span> ''I'' = ( ''&Delta;_I'' , ''&Delta;_D'' , ''&sdot;&nbsp;^C'' , ''&sdot;&nbsp;^OP'' , ''&sdot;&nbsp;^DP'' , ''&sdot;&nbsp;^I'' , ''&sdot;&nbsp;^DT'' , ''&sdot;&nbsp;^LT'' , ''&sdot;&nbsp;^FA'' ) for ''D'' and ''V'' is a 9-tuple with the following structure: * ''&Delta;_I'' is a nonempty set called the <span id="def_object_domain">''object domain''</span>. * ''&Delta;_D'' is a nonempty set disjoint with ''&Delta;_I'' called the <span id="def_data_domain">''data domain''</span> such that ''(DT)^DT'' &sube; ''&Delta;_D'' for each datatype ''DT'' &isin; ''V_DT''. * ''&sdot;&nbsp;^C'' is the <span * id="def_class_interpretation_function">''class interpretation function''</span> that assigns to each class ''C &isin; V_C'' a subset ''(C)^C'' &sube; ''&Delta;_I'' such that ** ''(owl:Thing)^C'' = ''&Delta;_I'' and ** ''(owl:Nothing)^C'' = &empty;. * ''&sdot;&nbsp;^OP'' is the <span id="def_object_property_interpretation_function">''object property interpretation function''</span> that assigns to each object property ''OP &isin; V_OP'' a subset ''(OP)^OP'' &sube; ''&Delta;_I'' &times; ''&Delta;_I'' such that ** ''(owl:topObjectProperty)^OP'' = ''&Delta;_I'' &times; ''&Delta;_I'' and ** ''(owl:bottomObjectProperty)^OP'' = &empty;. * ''&sdot;&nbsp;^DP'' is the <span id="def_data_property_interpretation_function">''data property interpretation function''</span> that assigns to each data property ''DP &isin; V_DP'' a subset ''(DP)^DP'' &sube; ''&Delta;_I'' &times; ''&Delta;_D'' such that ** ''(owl:topDataProperty)^DP'' = ''&Delta;_I'' &times; ''&Delta;_D'' and ** ''(owl:bottomDataProperty)^DP'' = &empty;. * ''&sdot;&nbsp;^I'' is the <span id="def_individual_interpretation_function">''individual interpretation function''</span> that assigns to each individual ''a &isin; V_I'' an element ''(a)^I'' &isin; ''&Delta;_I''. * ''&sdot;&nbsp;^DT'' is the <span id="def_datatype_interpretation_function">''datatype interpretation function''</span> that assigns to each datatype ''DT &isin; V_DT'' a subset ''(DT)^DT'' &sube; ''&Delta;_D'' such that ** ''&sdot;&nbsp;^DT'' is the same as in ''D'' for each datatype ''DT'' &isin; ''N_DT'', and ** ''(rdfs:Literal)^DT'' = ''&Delta;_D''. * ''&sdot;&nbsp;^LT'' is the <span id="def_literal_interpretation_function">''literal interpretation function''</span> that is defined as ''(lt)^LT'' = ( ''LV'' , ''DT'' )''^LS'' for each ''lt'' &isin; ''V_LT'', where ''LV'' is the lexical form of ''lt'' and ''DT'' is the datatype of ''lt''. * ''&sdot;&nbsp;^FA'' is the <span id="def_facet_interpretation_function">''facet interpretation function''</span> that is defined as ( ''F'' , ''lt'' )''^FA'' = ( ''F'' , ''(lt)^LT'' )''^FS'' for each ( ''F'' , ''lt'' ) &isin; ''V_FA''. The following sections define the extensions of ''&sdot;&nbsp;^OP'', ''&sdot;&nbsp;^DT'', and ''&sdot;&nbsp;^C'' to object property expressions, data ranges, and class expressions. ==== Object Property Expressions ==== The object property interpretation function ''&sdot;&nbsp;^OP'' is extended to object property expressions as shown in Table 1. <div class="center"> {| border="2" cellpadding="5" |+ <span class="caption">Table 1.</span> Interpreting Object Property Expressions |- ! Object Property Expression ! Interpretation ''&sdot;&nbsp;^OP'' |- | class="name" | ObjectInverseOf( OP ) | { ( ''x'' , ''y'' ) <nowiki>|</nowiki> ( ''y'' , ''x'' ) &isin; ''(OP)^OP'' } |- |} </div> ==== Data Ranges ==== The datatype interpretation function ''&sdot;&nbsp;^DT'' is extended to data ranges as shown in Table 3. All datatypes in OWL 2 are unary, so each datatype ''DT'' is interpreted as a unary relation over ''&Delta;_D'' &mdash; that is, as a set ''(DT)^DT'' &sube; ''&Delta;_D''. OWL 2 currently does not define data ranges of arity more than one; however, by allowing for ''n''-ary data ranges, the syntax of OWL 2 provides a "hook" allowing implementations to introduce extensions such as comparisons and arithmetic. An ''n''-ary data range ''DR'' is interpreted as an ''n''-ary relation ''(DR)^DT'' over ''&Delta;_D'' &mdash; that is, as a set ''(DT)^DT'' &sube; ''(&Delta;_D)ⁿ'' <div class="center"> {| border="2" cellpadding="5" |+ <span class="caption">Table 3.</span> Interpreting Data Ranges |- ! Data Range ! Interpretation ''&sdot;&nbsp;^DT |- | class="name" | DataIntersectionOf( DR₁ ... DR_n ) | ''(DR₁)^DT'' &cap; ... &cap; ''(DR_n)^DT'' |- | class="name" | DataUnionOf( DR₁ ... DR_n ) | ''(DR₁)^DT'' &cup; ... &cup; ''(DR_n)^DT'' |- | class="name" | DataComplementOf( DR ) | ''(&Delta;_D)ⁿ'' \ ''(DR)^DT'' where ''n'' is the arity of ''DR'' |- | class="name" | DataOneOf( lt₁ ... lt_n ) | { ''(lt₁)^LT'' , ... , ''(lt_n)^LT'' } |- | class="name" | DatatypeRestriction( DT F₁ lt₁ ... F_n lt_n ) | ''(DT)^DT'' &cap; ( ''F₁'' , ''lt₁'' )''^FA'' &cap; ... &cap; ( ''F_n'' , ''lt_n'' )''^FA'' |- |} </div> ==== Class Expressions ==== The class interpretation function ''&sdot;&nbsp;^C'' is extended to class expressions as shown in Table 4. For ''S'' a set, ''<nowiki>#S</nowiki>'' denotes the number of elements in ''S''. <div class="center"> {| border="2" cellpadding="5" |+ <span class="caption">Table 4.</span> Interpreting Class Expressions |- ! Class Expression ! Interpretation ''&sdot;&nbsp;^C'' |- | class="name" | ObjectIntersectionOf( CE₁ ... CE_n ) | ''(CE₁)^C'' &cap; ... &cap; ''(CE_n)^C'' |- | class="name" | ObjectUnionOf( CE₁ ... CE_n ) | ''(CE₁)^C'' &cup; ... &cup; ''(CE_n)^C'' |- | class="name" | ObjectComplementOf( CE ) | ''&Delta;_I'' \ ''(CE)^C'' |- | class="name" | ObjectOneOf( a₁ ... a_n ) | { ''(a₁)^I'' , ... , ''(a_n)^I'' } |- | class="name" | ObjectSomeValuesFrom( OPE CE ) | { ''x'' <nowiki>|</nowiki> &exist; ''y'' : ( ''x'', ''y'' ) &isin; ''(OPE)^OP'' and ''y'' &isin; ''(CE)^C'' } |- | class="name" | ObjectAllValuesFrom( OPE CE ) | { ''x'' <nowiki>|</nowiki> &forall; ''y'' : ( ''x'', ''y'' ) &isin; ''(OPE)^OP'' implies ''y'' &isin; ''(CE)^C'' } |- | class="name" | ObjectHasValue( OPE a ) | { ''x'' <nowiki>|</nowiki> ( ''x'' , ''(a)^I'' ) &isin; ''(OPE)^OP'' } |- | class="name" | ObjectHasSelf( OPE ) | { ''x'' <nowiki>|</nowiki> ( ''x'' , ''x'' ) &isin; ''(OPE)^OP'' } |- | class="name" | ObjectMinCardinality( n OPE ) | { ''x'' <nowiki>|</nowiki> #{ ''y'' <nowiki>|</nowiki> ( ''x'' , ''y'' ) &isin; ''(OPE)^OP'' } &ge; n } |- | class="name" | ObjectMaxCardinality( n OPE ) | { ''x'' <nowiki>|</nowiki> #{ ''y'' <nowiki>|</nowiki> ( ''x'' , ''y'' ) &isin; ''(OPE)^OP'' } &le; n } |- | class="name" | ObjectExactCardinality( n OPE ) | { ''x'' <nowiki>|</nowiki> #{ ''y'' <nowiki>|</nowiki> ( ''x'' , ''y'' ) &isin; ''(OPE)^OP'' } = n } |- | class="name" | ObjectMinCardinality( n OPE CE ) | { ''x'' <nowiki>|</nowiki> #{ ''y'' <nowiki>|</nowiki> ( ''x'' , ''y'' ) &isin; ''(OPE)^OP'' and ''y'' &isin; ''(CE)^C'' } &ge; n } |- | class="name" | ObjectMaxCardinality( n OPE CE ) | { ''x'' <nowiki>|</nowiki> #{ ''y'' <nowiki>|</nowiki> ( ''x'' , ''y'' ) &isin; ''(OPE)^OP'' and ''y'' &isin; ''(CE)^C'' } &le; n } |- | class="name" | ObjectExactCardinality( n OPE CE ) | { ''x'' <nowiki>|</nowiki> #{ ''y'' <nowiki>|</nowiki> ( ''x'' , ''y'' ) &isin; ''(OPE)^OP'' and ''y'' &isin; ''(CE)^C'' } = n } |- | class="name" | DataSomeValuesFrom( DPE₁ ... DPE_n DR ) | { ''x'' <nowiki>|</nowiki> &exist; ''y₁'', ... , ''y_n'' : ( ''x'' , ''y_k'' ) &isin; ''(DPE_k)^DP'' for each 1 &le; ''k'' &le; ''n'' and ( ''y₁'' , ... , ''y_n'' ) &isin; ''(DR)^DT'' } |- | class="name" | DataAllValuesFrom( DPE₁ ... DPE_n DR ) | { ''x'' <nowiki>|</nowiki> &forall; ''y₁'', ... , ''y_n'' : ( ''x'' , ''y_k'' ) &isin; ''(DPE_k)^DP'' for each 1 &le; ''k'' &le; ''n'' imply ( ''y₁'' , ... , ''y_n'' ) &isin; ''(DR)^DT'' } |- | class="name" | DataHasValue( DPE lt ) | { ''x'' <nowiki>|</nowiki> ( ''x'' , ''(lt)^LT'' ) &isin; ''(DPE)^DP'' } |- | class="name" | DataMinCardinality( n DPE ) | { ''x'' <nowiki>|</nowiki> #{ ''y'' <nowiki>|</nowiki> ( ''x'' , ''y'' ) &isin; ''(DPE)^DP''} &ge; n } |- | class="name" | DataMaxCardinality( n DPE ) | { ''x'' <nowiki>|</nowiki> #{ ''y'' <nowiki>|</nowiki> ( ''x'' , ''y'' ) &isin; ''(DPE)^DP'' } &le; n } |- | class="name" | DataExactCardinality( n DPE ) | { ''x'' <nowiki>|</nowiki> #{ ''y'' <nowiki>|</nowiki> ( ''x'' , ''y'' ) &isin; ''(DPE)^DP'' } = n } |- | class="name" | DataMinCardinality( n DPE DR ) | { ''x'' <nowiki>|</nowiki> #{ ''y'' <nowiki>|</nowiki> ( ''x'' , ''y'' ) &isin; ''(DPE)^DP'' and ''y'' &isin; ''(DR)^DT'' } &ge; n } |- | class="name" | DataMaxCardinality( n DPE DR ) | { ''x'' <nowiki>|</nowiki> #{ ''y'' <nowiki>|</nowiki> ( ''x'' , ''y'' ) &isin; ''(DPE)^DP'' and ''y'' &isin; ''(DR)^DT'' } &le; n } |- | class="name" | DataExactCardinality( n DPE DR ) | { ''x'' <nowiki>|</nowiki> #{ ''y'' <nowiki>|</nowiki> ( ''x'' , ''y'' ) &isin; ''(DPE)^DP'' and ''y'' &isin; ''(DR)^DT'' } = n } |} </div> === Satisfaction in an Interpretation === An interpretation ''I'' = ( ''&Delta;_I'' , ''&Delta;_D'' , ''&sdot;&nbsp;^C'' , ''&sdot;&nbsp;^OP'' , ''&sdot;&nbsp;^DP'' , ''&sdot;&nbsp;^I'' , ''&sdot;&nbsp;^DT'' , ''&sdot;&nbsp;^LT'' , ''&sdot;&nbsp;^FA'' ) <span id="def_satisfies_axiom">''satisfies''</span> an axiom w.r.t. an ontology ''O'' if the axiom satisfies the relevant condition from the following sections. Satisfaction of axioms in ''I'' is defined w.r.t. ''O'' because satisfaction of key axioms uses the following function: <div class="indent"> ''ISNAMED_O(x)'' = ''true'' for ''x'' &isin; ''&Delta;_I'' if and only if ''(a)^I'' = ''x'' for some named individual ''a'' occurring in the [[Syntax#def_axiom_closure|axiom closure]] of ''O'' </div> ==== Class Expression Axioms ==== Satisfaction of OWL 2 class expression axioms in ''I'' w.r.t. ''O'' is defined as shown in Table 5. <div class="center"> {| border="2" cellpadding="5" |+ <span class="caption">Table 5.</span> Satisfaction of Class Expression Axioms in an Interpretation |- ! Axiom ! Condition |- | class="name" | SubClassOf( CE₁ CE₂ ) | ''(CE₁)^C'' &sube; ''(CE₂)^C'' |- | class="name" | EquivalentClasses( CE₁ ... CE_n ) | ''(CE_j)^C'' = ''(CE_k)^C'' for each 1 &le; ''j'' &le; ''n'' and each 1 &le; ''k'' &le; ''n'' |- | class="name" | DisjointClasses( CE₁ ... CE_n ) | ''(CE_j)^C'' &cap; ''(CE_k)^C'' = &empty; for each 1 &le; ''j'' &le; ''n'' and each 1 &le; ''k'' &le; ''n'' such that ''j'' &ne; ''k'' |- | class="name" | DisjointUnion( C CE₁ ... CE_n ) | ''(C)^C'' = ''(CE₁)^C'' &cup; ... &cup; ''(CE_n)^C'' and<br /> ''(CE_j)^C'' &cap; ''(CE_k)^C'' = &empty; for each 1 &le; ''j'' &le; ''n'' and each 1 &le; ''k'' &le; ''n'' such that ''j'' &ne; ''k'' |- |} </div> ==== Object Property Expression Axioms ==== Satisfaction of OWL 2 object property expression axioms in ''I'' w.r.t. ''O'' is defined as shown in Table 6. <div class="center"> {| border="2" cellpadding="5" |+ <span class="caption">Table 6.</span> Satisfaction of Object Property Expression Axioms in an Interpretation |- ! Axiom ! Condition |- | class="name" | SubObjectPropertyOf( OPE₁ OPE₂ ) | ''(OPE₁)^OP'' &sube; ''(OPE₂)^OP'' |- | class="name" | SubObjectPropertyOf( ObjectPropertyChain( OPE₁ ... OPE_n ) OPE ) | &forall; ''y₀'' , ... , ''y_n'' : ( ''y₀'' , ''y₁'' ) &isin; ''(OPE₁)^OP'' and ... and ( ''y_n-1'' , ''y_n'' ) &isin; ''(OPE_n)^OP'' imply ( ''y₀'' , ''y_n'' ) &isin; ''(OPE)^OP'' |- | class="name" | EquivalentObjectProperties( OPE₁ ... OPE_n ) | ''(OPE_j)^OP'' = ''(OPE_k)^OP'' for each 1 &le; ''j'' &le; ''n'' and each 1 &le; ''k'' &le; ''n'' |- | class="name" | DisjointObjectProperties( OPE₁ ... OPE_n ) | ''(OPE_j)^OP'' &cap; ''(OPE_k)^OP'' = &empty; for each 1 &le; ''j'' &le; ''n'' and each 1 &le; ''k'' &le; ''n'' such that ''j'' &ne; ''k'' |- | class="name" | ObjectPropertyDomain( OPE CE ) | &forall; ''x'' , ''y'' : ( ''x'' , ''y'' ) &isin; ''(OPE)^OP'' implies ''x'' &isin; ''(CE)^C'' |- | class="name" | ObjectPropertyRange( OPE CE ) | &forall; ''x'' , ''y'' : ( ''x'' , ''y'' ) &isin; ''(OPE)^OP'' implies ''y'' &isin; ''(CE)^C'' |- | class="name" | InverseObjectProperties( OPE₁ OPE₂ ) | ''(OPE₁)^OP'' = { ( ''x'' , ''y'' ) <nowiki>|</nowiki> ( ''y'' , ''x'' ) &isin; ''(OPE₂)^OP'' } |- | class="name" | FunctionalObjectProperty( OPE ) | &forall; ''x'' , ''y₁'' , ''y₂'' : ( ''x'' , ''y₁'' ) &isin; ''(OPE)^OP'' and ( ''x'' , ''y₂'' ) &isin; ''(OPE)^OP'' imply ''y₁'' = ''y₂'' |- | class="name" | InverseFunctionalObjectProperty( OPE ) | &forall; ''x₁'' , ''x₂'' , ''y'' : ( ''x₁'' , ''y'' ) &isin; ''(OPE)^OP'' and ( ''x₂'' , ''y'' ) &isin; ''(OPE)^OP'' imply ''x₁'' = ''x₂'' |- | class="name" | ReflexiveObjectProperty( OPE ) | &forall; ''x'' : ''x'' &isin; ''&Delta;_I'' implies ( ''x'' , ''x'' ) &isin; ''(OPE)^OP'' |- | class="name" | IrreflexiveObjectProperty( OPE ) | &forall; ''x'' : ''x'' &isin; ''&Delta;_I'' implies ( ''x'' , ''x'' ) &notin; ''(OPE)^OP'' |- | class="name" | SymmetricObjectProperty( OPE ) | &forall; ''x'' , ''y'' : ( ''x'' , ''y'' ) &isin; ''(OPE)^OP'' implies ( ''y'' , ''x'' ) &isin; ''(OPE)^OP'' |- | class="name" | AsymmetricObjectProperty( OPE ) | &forall; ''x'' , ''y'' : ( ''x'' , ''y'' ) &isin; ''(OPE)^OP'' implies ( ''y'' , ''x'' ) &notin; ''(OPE)^OP'' |- | class="name" | TransitiveObjectProperty( OPE ) | &forall; ''x'' , ''y'' , ''z'' : ( ''x'' , ''y'' ) &isin; ''(OPE)^OP'' and ( ''y'' , ''z'' ) &isin; ''(OPE)^OP'' imply ( ''x'' , ''z'' ) &isin; ''(OPE)^OP'' |- |} </div> ==== Data Property Expression Axioms ==== Satisfaction of OWL 2 data property expression axioms in ''I'' w.r.t. ''O'' is defined as shown in Table 7. <div class="center"> {| border="2" cellpadding="5" |+ <span class="caption">Table 7.</span> Satisfaction of Data Property Expression Axioms in an Interpretation |- ! Axiom ! Condition |- | class="name" | SubDataPropertyOf( DPE₁ DPE₂ ) | ''(DPE₁)^DP'' &sube; ''(DPE₂)^DP'' |- | class="name" | EquivalentDataProperties( DPE₁ ... DPE_n ) | ''(DPE_j)^DP'' = ''(DPE_k)^DP'' for each 1 &le; ''j'' &le; ''n'' and each 1 &le; ''k'' &le; ''n'' |- | class="name" | DisjointDataProperties( DPE₁ ... DPE_n ) | ''(DPE_j)^DP'' &cap; ''(DPE_k)^DP'' = &empty; for each 1 &le; ''j'' &le; ''n'' and each 1 &le; ''k'' &le; ''n'' such that ''j'' &ne; ''k'' |- | class="name" | DataPropertyDomain( DPE CE ) | &forall; ''x'' , ''y'' : ( ''x'' , ''y'' ) &isin; ''(DPE)^DP'' implies ''x'' &isin; ''(CE)^C'' |- | class="name" | DataPropertyRange( DPE DR ) | &forall; ''x'' , ''y'' : ( ''x , ''y ) &isin; ''(DPE)^DP'' implies ''y'' &isin; ''(DR)^DT'' |- | class="name" | FunctionalDataProperty( DPE ) | &forall; ''x'' , ''y₁'' , ''y₂'' : ( ''x'' , ''y₁'' ) &isin; ''(DPE)^DP'' and ( ''x'' , ''y₂'' ) &isin; ''(DPE)^DP'' imply ''y₁'' = ''y₂'' |- |} </div> ==== Datatype Definitions ==== Satisfaction of datatype definitions in ''I'' w.r.t. ''O'' is defined as shown in Table 8. <div class="center"> {| border="2" cellpadding="5" |+ <span class="caption">Table 8.</span> Satisfaction of Datatype Definitions in an Interpretation |- ! Axiom ! Condition |- | class="name" | DatatypeDefinition( DT DR ) | style="text-align: left" | ''(DT)^DT'' = ''(DR)^DT'' |- |} </div> ==== Keys ==== Satisfaction of keys in ''I'' w.r.t. ''O'' is defined as shown in Table 9. <div class="center"> {| border="2" cellpadding="5" |+ <span class="caption">Table 9.</span> Satisfaction of Keys in an Interpretation |- ! Axiom ! Condition |- | class="name" | HasKey( CE ( OPE₁ ... OPE_m ) ( DPE₁ ... DPE_n ) ) | style="text-align: left" | &forall; ''x'' , ''y'' , ''z₁'' , ... , ''z_m'' , ''w₁'' , ... , ''w_n'' :<br /> &nbsp;&nbsp;&nbsp; if ''x'' &isin; ''(CE)^C'' and ''ISNAMED_O(x)'' and<br /> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; ''y'' &isin; ''(CE)^C'' and ''ISNAMED_O(y)'' and<br /> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; ( ''x'' , ''z_i'' ) &isin; ''(OPE_i)^OP'' and ( ''y'' , ''z_i'' ) &isin; ''(OPE_i)^OP'' and ''ISNAMED_O(z_i)'' for each 1 &le; ''i'' &le; ''m'' and<br /> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; ( ''x'' , ''w_j'' ) &isin; ''(DPE_j)^DP'' and ( ''y'' , ''w_j'' ) &isin; ''(DPE_j)^DP'' for each 1 &le; ''j'' &le; ''n''<br /> &nbsp;&nbsp;&nbsp; then x = y |- |} </div> ==== Assertions ==== Satisfaction of OWL 2 assertions in ''I'' w.r.t. ''O'' is defined as shown in Table 10. <div class="center"> {| border="2" cellpadding="5" |+ <span class="caption">Table 10.</span> Satisfaction of Assertions in an Interpretation |- ! Axiom ! Condition |- | class="name" | SameIndividual( a₁ ... a_n ) | ''(a_j)^I'' = ''(a_k)^I'' for each 1 &le; ''j'' &le; ''n'' and each 1 &le; ''k'' &le; ''n'' |- | class="name" | DifferentIndividuals( a₁ ... a_n ) | ''(a_j)^I'' &ne; ''(a_k)^I'' for each 1 &le; ''j'' &le; ''n'' and each 1 &le; ''k'' &le; ''n'' such that ''j'' &ne; ''k'' |- | class="name" | ClassAssertion( CE a ) | ''(a)^I'' &isin; ''(CE)^C'' |- | class="name" | ObjectPropertyAssertion( OPE a₁ a₂ ) | ( ''(a₁)^I'' , ''(a₂)^I'' ) &isin; ''(OPE)^OP'' |- | class="name" | NegativeObjectPropertyAssertion( OPE a₁ a₂ ) | ( ''(a₁)^I'' , ''(a₂)^I'' ) &notin; ''(OPE)^OP'' |- | class="name" | DataPropertyAssertion( DPE a lt ) | ( ''(a)^I'' , ''(lt)^LT'' ) &isin; ''(DPE)^DP'' |- | class="name" | NegativeDataPropertyAssertion( DPE a lt ) | ( ''(a)^I'' , ''(lt)^LT'' ) &notin; ''(DPE)^DP'' |- |} </div> ==== Ontologies ==== An interpretation ''I'' <span id="def_satisfies_ontology">''satisfies''</span> an OWL 2 ontology ''O'' if all axioms in the [[Syntax#def_axiom_closure|axiom closure]] of ''O'' (with anonymous individuals standardized apart as described in Section 5.6.2 of the OWL 2 Specification [<cite>[[#ref-owl-2-specification|OWL 2 Specification]]</cite>]) are satisfied in ''I'' w.r.t. ''O''. === Models === Given a datatype map ''D'', an interpretation ''I'' = ( ''&Delta;_I'' , ''&Delta;_D'' , ''&sdot;&nbsp;^C'' , ''&sdot;&nbsp;^OP'' , ''&sdot;&nbsp;^DP'' , ''&sdot;&nbsp;^I'' , ''&sdot;&nbsp;^DT'' , ''&sdot;&nbsp;^LT'' , ''&sdot;&nbsp;^FA'' ) for ''D'' is a <span id="def_model">''model''</span> of an OWL 2 ontology ''O'' w.r.t. ''D'' if an interpretation ''J'' = ( ''&Delta;_I'' , ''&Delta;_D'' , ''&sdot;&nbsp;^C'' , ''&sdot;&nbsp;^OP'' , ''&sdot;&nbsp;^DP'' , ''&sdot;&nbsp;^J'' , ''&sdot;&nbsp;^DT'' , ''&sdot;&nbsp;^LT'' , ''&sdot;&nbsp;^FA'' ) for ''D'' exists such that ''&sdot;&nbsp;^J'' coincides with ''&sdot;&nbsp;^I'' on all named individuals and ''J'' satisfies ''O''. Thus, an interpretation ''I'' satisfying ''O'' is also a model of ''O''. In contrast, a model ''I'' of ''O'' may not satisfy ''O'' directly; however, by modifying the interpretation of anonymous individuals, ''I'' can always be coerced into an interpretation ''J'' that satisfies ''O''. === Inference Problems === Let ''D'' be a datatype map and ''V'' a vocabulary over ''D''. Furthermore, let ''O'' and ''O₁'' 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''. Furthermore, <span id="def_variable">''variables''</span> are symbols that are not contained in ''V''. Finally, a <span id="def_boolean_conjunctive_query">''Boolean conjunctive query''</span> ''Q'' is a closed formula of the form <div class="indent"> <span class="name">&exist; x₁ , ... , x_n , y₁ , ... , y_m : [ A₁ &and; ... &and; A_k ]</span> </div> where each <span class="name">A_i</span> is an ''atom'' of the form <span class="name">C(s)</span>, <span class="name">OP(s,t)</span>, or <span class="name">DP(s,u)</span> with <span class="name">C</span> a class, <span class="name">OP</span> an object property, <span class="name">DP</span> a data property, <span class="name">s</span> and <span class="name">t</span> individuals or some variable <span class="name">x_j</span>, and <span class="name">u</span> a literal or some variable <span class="name">y_j</span>. The following inference problems are often considered in practice. '''Ontology Consistency''': ''O'' is ''consistent'' (or ''satisfiable'') w.r.t. ''D'' if a model of ''O'' w.r.t. ''D'' and ''V'' exists. '''Ontology Entailment''': ''O'' ''entails'' ''O₁'' w.r.t. ''D'' if every model of ''O'' w.r.t. ''D'' and ''V'' is also a model of ''O₁'' w.r.t. ''D'' and ''V''. '''Ontology Equivalence''': ''O'' and ''O₁'' are ''equivalent'' w.r.t. ''D'' if ''O'' entails ''O₁'' w.r.t. ''D'' and ''O₁'' entails ''O'' w.r.t. ''D''. '''Ontology Equisatisfiability''': ''O'' and ''O₁'' are ''equisatisfiable'' w.r.t. ''D'' if ''O'' is satisfiable w.r.t. ''D'' if and only if ''O₁'' is satisfiable w.r.t ''D''. '''Class Expression Satisfiability''': ''CE'' is satisfiable w.r.t. ''O'' and ''D'' if a model ''I'' = ( ''&Delta;_I'' , ''&Delta;_D'' , ''&sdot;&nbsp;^C'' , ''&sdot;&nbsp;^OP'' , ''&sdot;&nbsp;^DP'' , ''&sdot;&nbsp;^I'' , ''&sdot;&nbsp;^DT'' , ''&sdot;&nbsp;^LT'' , ''&sdot;&nbsp;^FA'' ) of ''O'' w.r.t. ''D'' and ''V'' exists such that ''(CE)^C'' &ne; &empty;. '''Class Expression Subsumption''': ''CE₁'' is ''subsumed'' by a class expression ''CE₂'' w.r.t. ''O'' and ''D'' if ''(CE₁)^C'' &sube; ''(CE₂)^C'' for each model ''I'' = ( ''&Delta;_I'' , ''&Delta;_D'' , ''&sdot;&nbsp;^C'' , ''&sdot;&nbsp;^OP'' , ''&sdot;&nbsp;^DP'' , ''&sdot;&nbsp;^I'' , ''&sdot;&nbsp;^DT'' , ''&sdot;&nbsp;^LT'' , ''&sdot;&nbsp;^FA'' ) of ''O'' w.r.t. ''D'' and ''V''. '''Instance Checking''': ''a'' is an ''instance'' of ''CE'' w.r.t. ''O'' and ''D'' if ''(a)^I'' &isin; ''(CE)^C'' for each model ''I'' = ( ''&Delta;_I'' , ''&Delta;_D'' , ''&sdot;&nbsp;^C'' , ''&sdot;&nbsp;^OP'' , ''&sdot;&nbsp;^DP'' , ''&sdot;&nbsp;^I'' , ''&sdot;&nbsp;^DT'' , ''&sdot;&nbsp;^LT'' , ''&sdot;&nbsp;^FA'' ) of ''O'' w.r.t. ''D'' and ''V''. '''Boolean Conjunctive Query Answering''': ''Q'' is an ''answer'' w.r.t. ''O'' and ''D'' if ''Q'' is true in each model of ''O'' w.r.t. ''D'' and ''V'' according to the standard definitions of first-order logic. In order to ensure that ontology entailment, class expression satisfiability, class expression subsumption, and instance checking are decidable, the following restriction w.r.t. ''O'' needs to be satisfied: <div class="indent"> Each class expression of type <span class="nonterminal">MinObjectCardinality</span>, <span class="nonterminal">MaxObjectCardinality</span>, <span class="nonterminal">ExactObjectCardinality</span>, and <span class="nonterminal">ObjectHasSelf </span> that occurs in ''O₁'', ''CE'', ''CE₁'', and ''CE₂'' can contain only object property expressions that are [[Syntax#def_simple|simple]] in the [[Syntax#def_axiom_closure|axiom closure]] ''Ax'' of ''O''. </div> For ontology equivalence to be decidable, ''O₁'' needs to satisfy this restriction w.r.t. ''O'' and vice versa. These restrictions are analogous to the first condition from Section 11.2 of the OWL 2 Specification [<cite>[[#ref-owl-2-specification|OWL 2 Specification]]</cite>]. == Independence of the Direct Semantics from the Datatype Map in OWL 2 DL (Informative) == OWL 2 DL has been defined so that the consequences of an OWL 2 DL ontology ''O'' do 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, and it has several useful consequences: * One can apply the direct semantics to an OWL 2 DL 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 an implicit datatype map containing only the datatypes from the given ontology. * OWL 2 DL reasoners can provide datatypes not explicitly mentioned in this specification without fear that this will change the meaning of OWL 2 DL ontologies not using these datatypes. '''Theorem DS1.''' Let ''O₁'' and ''O₂'' be OWL 2 DL ontologies over a vocabulary ''V'' and ''D'' = ( ''N_DT'' , ''N_LS'' , ''N_FS'' , ''&sdot;&nbsp;^DT'' , ''&sdot;&nbsp;^LS'' , ''&sdot;&nbsp;^FS'' ) a datatype map such that each datatype mentioned in ''O₁'' and ''O₂'' is ''rdfs:Literal'', a datatype defined in the respective ontology, or it occurs in ''N_DT''. Furthermore, let ''D<nowiki>'</nowiki>'' = ( ''N_DT<nowiki>'</nowiki>'' , ''N_LS<nowiki>'</nowiki>'' , ''N_FS<nowiki>'</nowiki>'' , ''&sdot;&nbsp;^{DT&nbsp;<nowiki>'</nowiki>}'' , ''&sdot;&nbsp;^{LS&nbsp;<nowiki>'</nowiki>}'' , ''&sdot;&nbsp;^{FS&nbsp;<nowiki>'</nowiki>}'' ) be a datatype map such that ''N_DT'' &sube; ''N_DT<nowiki>'</nowiki>'', ''N_LS(DT)'' = ''N_LS<nowiki>'</nowiki>(DT)'', and ''N_FS(DT)'' = ''N_FS<nowiki>'</nowiki>(DT)'' for each ''DT'' &isin; ''N_DT'', and ''&sdot;&nbsp;^{DT&nbsp;<nowiki>'</nowiki>}'', ''&sdot;&nbsp;^{LS&nbsp;<nowiki>'</nowiki>}'', and ''&sdot;&nbsp;^{FS&nbsp;<nowiki>'</nowiki>}'' are extensions of ''&sdot;&nbsp;^DT'', ''&sdot;&nbsp;^LS'', and ''&sdot;&nbsp;^FS'', respectively. Then, ''O₁'' entails ''O₂'' w.r.t. ''D'' if and only if ''O₁'' entails ''O₂'' w.r.t. ''D<nowiki>'</nowiki>''. ''Proof.'' Without loss of generality, one can assume ''O₁'' and ''O₂'' to be in negation-normal form [<cite>[[#ref-description-logics|Description Logics]]</cite>]. Furthermore, since datatype definitions in ''O₁'' and ''O₂'' are acyclic, one can assume that each defined datatype has been recursively replaced with its definition; thus, all datatypes in ''O₁'' and ''O₂'' are from ''N_DT'' &cup; { ''rdfs:Literal'' }. The claim of the theorem is equivalent to the following statement: an interpretation ''I'' w.r.t. ''D'' and ''V'' exists such that ''O₁'' is and ''O₂'' is not satisfied in ''I'' if and only if an interpretation ''I<nowiki>'</nowiki>'' w.r.t. ''D<nowiki>'</nowiki>'' and ''V'' exists such that ''O₁'' is and ''O₂'' is not satisfied in ''I<nowiki>'</nowiki>''. The (&lArr;) direction is trivial since each interpretation ''I'' w.r.t. ''D<nowiki>'</nowiki>'' and ''V'' is also an interpretation w.r.t. ''D'' and ''V''. For the (&rArr;) direction, assume that an interpretation ''I'' = ( ''&Delta;_I'' , ''&Delta;_D'' , ''&sdot;&nbsp;^C'' , ''&sdot;&nbsp;^OP'' , ''&sdot;&nbsp;^DP'' , ''&sdot;&nbsp;^I'' , ''&sdot;&nbsp;^DT'' , ''&sdot;&nbsp;^LT'' , ''&sdot;&nbsp;^FA'' ) w.r.t. ''D'' and ''V'' exists such that ''O₁'' is and ''O₂'' is not satisfied in ''I''. Let ''I<nowiki>'</nowiki>'' = ( ''&Delta;_I'' , ''&Delta;_D<nowiki>'</nowiki>'' , ''&sdot;&nbsp;^{C&nbsp;<nowiki>'</nowiki>}'' , ''&sdot;&nbsp;^OP'' , ''&sdot;&nbsp;^{DP&nbsp;<nowiki>'</nowiki>}'' , ''&sdot;&nbsp;^I'' , ''&sdot;&nbsp;^{DT&nbsp;<nowiki>'</nowiki>}'' , ''&sdot;&nbsp;^{LT&nbsp;<nowiki>'</nowiki>}'' , ''&sdot;&nbsp;^{FA&nbsp;<nowiki>'</nowiki>}'' ) be an interpretation such that * ''&Delta;_D<nowiki>'</nowiki>'' is obtained by extending ''&Delta;_D'' with the value space of all datatypes in ''N_DT<nowiki>'</nowiki>'' \ ''N_DT'', * ''&sdot;&nbsp;^{C&nbsp;<nowiki>'</nowiki>}'' coincides with ''&sdot;&nbsp;^C'' on all classes, and * ''&sdot;&nbsp;^{DP&nbsp;<nowiki>'</nowiki>}'' coincides with ''&sdot;&nbsp;^DP'' on all data properties apart from ''owl:topDataProperty''. Clearly, ''<span class="name">DataComplementOf( DR )</span>^DT'' &sube; ''<span class="name">DataComplementOf( DR )</span>^{DT&nbsp;<nowiki>'</nowiki>}'' for each data range ''DR'' that is either a datatype, a datatype restriction, or an enumerated data range. The ''owl:topDataProperty'' property can occur in ''O₁'' and ''O₂'' only in tautologies. The interpretation of all other data properties is the same in ''I'' and ''I<nowiki>'</nowiki>'', so ''(CE)^C'' = ''(CE)^{C&nbsp;<nowiki>'</nowiki>}'' for each class expression ''CE'' occurring in ''O₁'' and ''O₂''. Therefore, ''O₁'' is and ''O₂'' is not satisfied in ''I<nowiki>'</nowiki>''. QED <div id="changelog"> == Appendix: Change Log (Informative) == === Changes Since Proposed Recommendation === No changes have been made to this document since the [http://www.w3.org/TR/2009/PR-owl2-direct-semantics-20090922/ Proposed Recommendation of 22 September, 2009]. === Changes Since Candidate Recommendation === This section summarizes the changes to this document since the [http://www.w3.org/TR/2009/CR-owl2-direct-semantics-20090611/ Candidate Recommendation of 11 June, 2009]. * An editorial comment was added to clarify the role played by the OWL 2 datatype map. === Changes Since Last Call === This section summarizes the changes to this document since the [http://www.w3.org/TR/2009/WD-owl2-semantics-20090421/ Last Call Working Draft of 21 April, 2009]. * Some minor editorial changes were made. </div> == Acknowledgments == {{:Shared_Acknowledgments}} The editors extend special thanks to Markus Krötzsch (FZI), Michael Schneider (FZI) and Thomas Schneider (University of Manchester) for their thorough reviews. {{:Regular_attendees}} {{:Past_Members}} == References == === Normative References === {{:References/ref-owl-2-specification}} === Nonnormative References === {{:References/ref-description-logics}} {{:References/ref-owl-1-semantics-abstract-syntax}} {{:References/ref-owl-2-profiles}} {{:References/ref-sroiq}} </div>

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