Web Ontology Language (OWL) Abstract Syntax and Semantics
Appendix A. Correspondence between Abstract OWL and OWL DL (Informative)

Editor:: Peter F. Patel-Schneider, Bell Labs Research, Lucent Technologies

A.1 Correspondence between Abstract Syntax and OWL DL
- A.1.1 Lemma 1
- A.1.2 Lemma 2
- A.1.3 Lemma 3
- A.1.4 Lemma 4
- A.1.5 Correspondence Theorem
A.2 Correspondence between OWL DL and OWL Full

Appendix A. Proofs (Informative)

This appendix contains proofs of theorems contained in Section 5 of the document.

A.1 Correspondence between Abstract OWL and OWL DL

This section shows that the two semantics for OWL DL correspond on certain OWL ontologies. One semantics used in this section is the direct model theory for abstract OWL ontologies given in the direct model-theoretic semantics section of this document. The other semantics is the extension of the RDFS semantics given in the RDFS-compatible model-theoretic semantics section of this document.

Throughout this section qualified names are used as shorthand for URI references. The namespace identifiers used in such names, namely rdf, rdfs, xsd, and owl, should be used as if they are given their usual definitions.
Throughout this section VRDFS is the RDF and RDFS built-in vocabulary, i.e., rdf:type, rdf:Property, rdfs:Class, rdfs:subClassOf, …, minus rdfs:Literal; and VOWL is the OWL built-in vocabulary, i.e., owl:Class, owl:onProperty, …, minus owl:Thing and owl:Nothing.
Throughout this section D will be a datatyping scheme, i.e., a set of URI references that have class extensions that are subsets of LV and mappings from strings to their class extension.
Throughout this section T will be the mapping from abstract OWL ontologies to RDF graphs.

Recall that a separated OWL vocabulary is a set of URI references V with a disjoint partition, written V = VI + VC + VD + VOP + VDP where owl:Thing and owl:Nothing are in VC, rdfs:Literal is in VD, and all the elements of D are in VD. Further V is disjoint from VRDFS∪VOWL.

An OWL abstract KB with separated names over a separated OWL vocabulary V = VI + VC + VD + VOP + VDP is a set of OWL axioms and facts without annotations as in Section 2 where <individualID>s are taken from VI, <classID>s are taken from VC, <datatypeID>s are taken from VD, <individualvaluedPropertyIDs> are taken from VOP, and <datavaluedPropertyID>s are taken from VDP.

Let V = VI + VC + VD + VOP + VDP be a separated OWL vocabulary. Then T(V) is the RDF graph that contains exactly <v,rdf:type,owl:Thing > for v ∈ VI, <v,rdf:type,owl:Class > for v ∈ VC, <v,rdf:type,rdfs:Datatype > for v ∈ VD, <v,rdf:type,owl:ObjectProperty > for v ∈ VOP, and <v,rdf:type,owl:DatatypeProperty > for v ∈ VDP.

The theorem to be proved is:
Let V' be a separated OWL vocabulary. Let K,Q be abstract OWL ontologies with separated names over V'. Then K OWL entails Q iff T(K),T(V') OWL DL entails T(Q).

Actually, a slightly stronger correspondence can be shown, but this is enough for now, as the presence of annotations and imports causes even more complications.

A.1.1 Lemma 1

Lemma 1: Let V' = VI + VC + VD + VOP + VDP be a separated OWL vocabulary.
Let V = VI ∪ VC ∪ VD ∪ VOP ∪ VDP ∪ VRDFS ∪ VOWL.
Let I'= <R,EC,ER,S> be an abstract OWL interpretation of V'.
Let I = <R_I,P_I,EXT_I,S_I,L_I> be an OWL DL interpretation of V that satisfies T(V'). Let CEXT_I have its usual meaning, and, as usual, overload I to map any syntactic construct into its denotation.
If R=CEXT_I(I(owl:Thing)), S(v)=S_I(v) for v ∈ VI, EC(v)=CEXT_I(S_I(v)) for v∈VC∪VD, and ER(v)=EXT_I(S_I(v)) for v∈VOP∪VDP
then for d any abstract OWL description or data range over V',

I OWL DL satisfies T(d), and
for any A mapping all the blank nodes of T(d) into R_I where I+A OWL DL satisfies T(d)
1. CEXT_I(I(M(T(d)))) = EC(d),
2. if d is a description then I(M(T(d)))∈CEXT_I(I(owl:Class)), and
3. if d is a data range then I(M(T(d)))∈CEXT_I(I(rdfs:Datatype)).

Proof

The proof of the lemma is by a structural induction. Throughout the proof, let IOT = CEXT_I(I(owl:Thing)), IOC = CEXT_I(I(owl:Class)), IDC = CEXT_I(I(rdfs:Datatype)), IOOP = CEXT_I(I(owl:ObjectProperty)), IODP = CEXT_I(I(owl:DatatypeProperty)), and IL = CEXT_I(I(rdf:List)).

To make the induction work, it is necessary to show that for any d a description or data range with sub-constructs T(d) contains triples for each of the sub-constructs that do not share any blank nodes with triples from the other sub-constructs. This can easily be verified from the rules for T.

If p∈VOP then I satisfies p∈IOOP. Then, as I is an OWL DL interpretation, I satisfies <p,I(owl:Thing)>∈EXT_I(I(rdfs:domain)) and <p,I(owl:Thing)>∈EXT_I(I(rdfs:range)). Thus I satisfies T(p). Similarly for p∈VDP.

Base Case: v ∈ VC, including owl:Thing and owl:Nothing: As v∈VC and I satisfies T(V), thus I(v)∈CEXT_I(I(owl:Class)). Because I is an OWL DL interpretation CEXT_I(I(v))⊆IOT, so <I(v),I(owl:Thing)>∈EXT_I(I(rdfs:subClassOf)). Thus I OWL DL satisfies T(v). As M(T(v)) is v, thus CEXT_I(I(M(T(v))))=EC(v). Finally, from above, I(v)∈IOC.
Base Case: v ∈ VD, including rdfs:Literal: As v∈VD and I satisfies T(V), thus I(v)∈CEXT_I(I(rdfs:Datatype)). Because I is an OWL DL interpretation CEXT_I(I(v))⊆LV, so <I(v),I(rdfs:Literal)>∈EXT_I(I(rdfs:subClassOf)). Thus I OWL DL satisfies T(v). As M(T(v)) is v, thus CEXT_I(I(M(T(v))))=EC(v). Finally, from above I(v)∈IDC.
Base Case: d=oneOf(i₁…i_n), where the i_j are individual IDs: As i_j∈VI for 1≤j≤n and I satisfies T(V), thus I(i_j)∈IOT. The second comprehension principle for sequences then requires that there is some l∈IL that is a sequence of I(i₁),…,I(i_n) over IOT. For any l that is a sequence of I(i₁),…,I(i_n) over IOT the comprehension principle for oneOf requires that there is some y∈CEXT_I(I(rdfs:Class)) such that <y,l> ∈ EXT_I(IS(owl:oneOf)). From the third characterization of oneOf, y∈IOC. Therefore I satisfies T(d). For any I+A that satisfies T(d), CEXT_I(I+A(M(T(d)))) = {I(i₁),…,I(i_n)} = EC(d). Finally, I+A(M(T(d)))∈IOC.
Base Case: d=oneOf(v₁…v_n), where the vi are typed data values: As v_j is an abstract syntax typed data value for 1≤j≤n I(v_j)∈LV. The second comprehension principle for sequences then requires that there is some l∈IL that is a sequence of I(v₁),…,I(v_n) over LV. For any l that is a sequence of I(v₁),…,I(v_n) over LV the comprehension principle for oneOf requires that there is some y∈CEXT_I(I(rdfs:Class)) such that <y,l> ∈ EXT_I(IS(owl:oneOf)). From the second characterization of oneOf, y∈IOC. Therefore I satisfies T(d). For any I+A that satisfies T(d), CEXT_I(I+A(M(T(d)))) = {I(i₁),…,I(i_n)} = EC(d). Finally, I+A(M(T(d)))∈IDC.
Base Case: d=restriction(p value(i)), with p∈VOP∪VDP and i an individualID: As p∈VOP∪VDP, from above I satisfies T(p). As I satisfies T(V'), I(p)∈IOOP∪IODP. As i∈VI and I satisfies T(V'), I(i)∈IOT. From a comprehension principle for restriction, I satisfies T(d). For any A such that I+A satisfies T(d), CEXT_I(I+A(M(T(d)))) = { x∈IOT | <x,I(i)> ∈ EXT_I(p) } = { x∈R | <x,S(i)> ∈ ER(p) } = EC(d). Finally, I+A(M(T(d)))∈IOC.
Base Case: d=restriction(p value(i)), with p∈VOP∪VDP and i a typed data value.: Similar.
Base Case: d=restriction(p minCardinality(n)), with p∈VOP∪VDP and n a non-negative integer: Similar.
Base Case: d=restriction(p maxCardinality(n)), with p∈VOP∪VDP and n a non-negative integer: Similar.
Base Case: d=restriction(p Cardinality(n)), with p∈VOP∪VDP and n a non-negative integer: Similar.
Inductive Case: d=complementOf(d'): From the induction hypothesis, I satisfies T(d'). As d' is a description, from the induction hypothesis there is a mapping, A, that maps all the blank nodes in T(d') into domain elements such that I+A satisfies T(d') and I+A(M(T(d'))) = EC(d') and I+A(M(T(d')))∈IOC. The comprehension principle for complementOf then requires that there is a y∈IOC such that I+A satisfies <y,e>∈EXT_I(I(owl:complementOf)) so I satisfies T(d). For any I+A that satisfies T(d), CEXT_I(I+A(M(T(d)))) = IOT-CEXT_I(I+A(M(T(d)))) = R-EC(d') = EC(d). Finally, I+A(M(T(d)))∈IOC.
Inductive Case: d = unionOf(d₁ … d_n): From the induction hypothesis, I satisfies d_i for 1≤i≤n so there is a mapping, A_i, that maps all the blank nodes in T(d_i) into domain elements such that I+A_i satisfies T(d_i). As the blank nodes in T(d_i) are disjoint from the blank nodes of T(d_j) for i≠j, I+A₁+…+A_n, and thus I, satisfies T(d_i)∪…∪T(d_n). Each d_i is a description, so from the induction hypothesis, I+A₁+…+A_n(M(T(d_i)))∈IOC. The first comprehension principle for sequences then requires that there is some l∈IL that is a sequence of I+A₁+…+A_n(M(T(d₁))),…, I+A₁+…+A_n(M(T(d_n))) over IOC. The comprehension principle for unionOf then requires that there is some y∈IOC such that <y,l>∈EXT_I(I(owl:unionOf)) so I satisfies T(d).
For any I+A that satisfies T(d), I+A satisfies T(d_i) so CEXT_I(I+A(d_i)) = EC(d_i)). Then CEXT_I(I+A(M(T(d)))) = CEXT_I(I+A(d₁))∪…∪CEXT_I(I+A(d_n)) = EC(d₁)∪…∪EC(d_n) = EC(d). Finally, I(M(T(d)))∈IOC.
Inductive Case: d = intersectionOf(d₁ … d_n): Similar.
Inductive Case: d = restriction(p x₁ x₂ … x_n): As p∈VOP∪VDP, from above I satisfies T(p). From the induction hypothesis, I satisfies restriction(p x_i) for 1≤i≤n so there is a mapping, A_i, that maps all the blank nodes in T(restriction(p x_i)) into domain elements such that I+A_i satisfies T(restriction(p x_i)). As the blank nodes in T(restriction(p x_i)) are disjoint from the blank nodes of T(restriction(p x_j)) for i≠j, I+A₁+…+A_n, and thus I, satisfies T(restriction(p x₁ … x_n)). Each restriction(p x_i) is a description, so from the induction hypothesis, M(T(restriction(p x_i)))∈IOC. The first comprehension principle for sequences then requires that there is some l∈IL that is a sequence of I+A₁+…+A_n(M(T(restriction(p x_i))),…, I+A₁+…+A_n(M(T(restriction(p x_i))) over IOC. The comprehension principle for intersectionOf then requires that there is some y∈IOC such that <y,l>∈EXT_I(I(owl:intersectionOf)) so I satisfies T(d).
For any I+A that satisfies T(d) I+A satisfies T(d_i) so CEXT_I(I+A(d_i)) = EC(d_i)). Then CEXT_I(I+A(M(T(d)))) = CEXT_I(I+A(restriction(p x_i)))∩…∩ CEXT_I(I+A(restriction(p x_n))) = EC(restriction(p x_i))cap;…∩EC(restriction(p x_i)) = EC(d). Finally, I(M(T(d)))∈IOC.
Inductive Case: d = restriction(p allValuesFrom(d')), with p∈VOP∪VDP and d' a description: As p∈VOP∪VDP, from above I satisfies T(p). From the induction hypothesis, I satisfies T(d'). As d' is a description, from the induction hypothesis, any mapping, A, that maps all the blank nodes in T(d') into domain elements such that I+A satisfies T(d') has I+A(M(T(d'))) = EC(d') and I+A(M(T(d')))∈IOC. As p∈VOP∪VDP and I satisfies T(V'), I(p)∈IOOP∪IODP. The comprehension principle for allValuesFrom restrictions then requires that I satisfies the triples in T(d) that are not in T(d') or T(p) in a way that shows that I satisfies T(d).
For any I+A that satisfies T(d), CEXT_I(I+A(M(T(d)))) = {x∈IOT | ∀ y∈IOT : <x,y>∈EXT_I(p) → y∈CEXT_I(M(T(d')))} = {x∈R | ∀ y∈R : <x,y>∈ER(p) → y∈EC(d')} = EC(d). Finally, I+A(M(T(d)))∈IOC.
Inductive Case: d = restriction(p someValuesFrom(d)) with p∈VOP∪VDP and d' a description: Similar.
Inductive Case: d = restriction(p allValuesFrom(d)) with p∈VOP∪VDP and d' a data range: Similar.
Inductive Case: d = restriction(p someValuesFrom(d)) with p∈VOP∪VDP and d' a data range: Similar.

A.1.2 Lemma 2

Lemma 2: Let V', V, I', and I be as in Lemma 1. Let D be an OWL directive over V'. Then I satisfies T(D) iff I' satisfies D.

Proof

Case: D = Class(foo complete d₁ … d_n).

Let d=intersectionOf(d₁ … d_n). As d is a description over V', thus I satisfies T(d) and for any A mapping the blank nodes of T(d) such that I+A satisfies T(d), CEXT_I(I+A(M(T(d)))) = EC(d). Thus for some A mapping the blank nodes of T(d) such that I+A satisfies T(d), CEXT_I(I+A(M(T(d)))) = EC(d) and I+A(M(T(d))∈IOC.

If I' satisfies D then EC(foo) = EC(d). From above, there is some A such that CEXT_I(I+A(M(T(d)))) = EC(d) = EC(foo) = CEXT_I(I(foo)) and I+A(M(T(d))∈IOC. Because I satisfies T(V), I(foo))∈IOC, thus <I(foo),I+A(M(T(d)))> ∈ EXT_I(I(owl:sameClassAs)). Therefore I satisfies T(D).

If I satisfies T(D) then I satisfies T(intersectionOf(d₁ … d_n)). Thus there is some A as above such that <I(foo),I+A(M(T(d)))> ∈ EXT_I(I(owl:sameClassAs)). Thus EC(d) = CEXT_I(I+A(M(T(d)))) = CEXT_I(I(foo)) = EC(foo). Therefore I' satisfies D.

Case: D = Class(foo partial d₁ … d_n)

If I' satisfies D then EC(foo) ⊆ EC(d). From above, there is some A such that CEXT_I(I+A(M(T(d)))) = EC(d) ⊇ EC(foo) = CEXT_I(I(foo)) and I+A(M(T(d))∈IOC. Because I satisfies T(V), I(foo))∈IOC, thus <I(foo),I+A(M(T(d)))> ∈ EXT_I(I(rdfs:subClassOf)). Therefore I satisfies T(D).

If I satisfies T(D) then I satisfies T(intersectionOf(d₁ … d_n)). Thus there is some A as above such that <I(foo),I+A(M(T(d)))> ∈ EXT_I(I(rdfs:subClassOf)). Thus EC(d) = CEXT_I(I+A(M(T(d)))) ⊇ CEXT_I(I(foo)) = EC(foo). Therefore I' satisfies D.

Case: D = EnumeratedClass(foo i₁ … i_n)

Let d=oneOf(i₁ … i_n). As d is a description over V' so I satisfies T(d) and for some A mapping the blank nodes of T(d) such that I+A satisfies T(d), CEXT_I(I+A(M(T(d)))) = EC(d).

Case: D= DisjointClasses(d₁ … d_n)

As d_i is a description over V' therefore I satisfies T(d_i) and for any A mapping the blank nodes of T(d_i) such that I+A satisfies T(d_i), CEXT_I(I+A(M(T(d_i)))) = EC(d_i).

If I satisfies T(D) then for 1≤i≤n there is some A_i such that I satisfies <I+A_i(M(T(d_i))),I+A_j(M(T(d_j)))> ∈ EXT_I(I(owl:disjointWith)) for each 1≤i<j≤n. Thus EC(d_i)∩EC(d_j) = {}, for i≠j. Therefore I' satisfies D.

If I' satisfies D then EC(d_i)∩EC(d_j) = {} for i≠j. For any A_i and A_j as above <I+A_i+A_j(M(T(d_i))),I+A_i+A_j(M(T(d_j)))> ∈ EXT_I(I(owl:disjointWith)), for i≠j. As at least one A_i exists for each i, and the blank nodes of the T(d_j) are all disjoint, I+A₁+…+A_n satisfies T(DisjointClasses(d₁ … d_n)). Therefore I satisfies T(D).

Case: D = EquivalentClasses(d₁ … d_n)

Similar.

Case: D = SubClassOf(d₁ d2)

Somewhat similar.

Case: D = IndividualProperty(p super(s₁) … super(sn) domain(d₁) … domain(d_m) range(r₁) … range(r_k) [inverse(i)] [Symmetric] [Functional] [InverseFunctional] [OneToOne] [Transitive])

As d_i for 1≤i≤m is a description over V' therefore I satisfies T(d_i) and for any A mapping the blank nodes of T(d_i) such that I+A satisfies T(d_i), CEXT_I(I+A(M(T(d_i)))) = EC(d_i). Similarly for r_i for 1≤i≤k.

If I' satisfies D, then, as p∈VOP, I satisfies I(p)∈IOOP. Then, as I is an OWL DL interpretation, I satisfies <I(p),I(owl:Thing)>∈EXT_I(I(rdfs:domain)) and <I(p),I(owl:Thing)>∈EXT_I(I(rdfs:range)). Also, ER(p)⊆ER(s_i) for 1≤i≤n, so EXT_I(I(p))=ER(p) ⊆ ER(s_i)=EXT_I(I(s_i)) and I satisfies <I(p),I(s_i)>∈EXT_I(I(rdfs:subPropertyOf)). Next, ER(p)⊆EC(d_i)×R for 1≤i≤m, so <z,w>∈ER(p) → z∈EC(d_i) and for any A such that I+A satisfies T(d_i), <z,w>∈EXT_I(p) → z∈CEXT_I(I+A(M(T(d_i)))) and thus <I(p),I+A(M(T(d_i)))>∈EXT_I(I(rdfs:domain)). Similarly for r_i for 1≤i≤k.

If I' satisfies D and inverse(i) is in D, then ER(p) and ER(i) are converses. Thus <u,v>∈ER(p) iff <v,u>∈ER(i) so <u,v>∈EXT_I(p) iff <v,u>∈EXT_I(i) and I satisfies <I(p),I(i)>∈EXT_I(I(owl:inverseOf)). If I' satisfies D and Symmetric is in D, then ER(p) is symmetric. Thus if <x,y>∈ ER(p) then <y,x>∈ER(p) so if <x,y> ∈ EXT_I(p) then <y, x>∈EXT_I(p). and thus I satisfies p∈CEXT_I(I(owl:Symmetric)). Similarly for Functional, InverseFunctional, and Transitive. Thus if I' satisfies D then I satisfies T(D).

If I satisfies T(D) then, for 1≤i≤n, <I(p),I(s_i)>∈EXT_I(I(rdfs:subPropertyOf)) so ER(p)=EXT_I(I(p)) ⊆ EXT_I(I(s_i))=ER(s_i). Also, for 1≤i≤m, for some A such that I+A satisfies T(d_i), <I(p),I+A(M(T(d_i)))>∈EXT_I(I(rdfs:domain)) so <z,w>∈EXT_I(p) → z∈CEXT_I(I+A(M(T(d_i)))). Thus <z,w>∈ER(p) → z∈EC(d_i) and ER(p)⊆EC(d_i)×R. Similarly for r_i for 1≤i≤k.

If I satisfies T(D) and inverse(i) is in D, then I satisfies <I(p),I(i)>∈EXT_I(I(owl:inverseOf)). Thus <u,v>∈EXT_I(p) iff <v,u>∈EXT_I(i) so <u,v>∈ER(p) iff <v,u>∈ER(i) and ER(p) and ER(i) are converses. If I satisfies D and Symmetric is in D, then I satisfies p∈CEXT_I(I(owl:Symmetric)) so if <x,y> ∈ EXT_I(p) then <y, x>∈EXT_I(p). Thus if <x,y>∈ ER(p) then <y,x>∈ER(p) and ER(p) is symmetric. Similarly for Functional, InverseFunctional, and Transitive. Thus if I satisfies T(D) then I' satisfies D.

Case: D = DataProperty(p super(s₁) … super(sn) domain(d₁) … domain(d_m) range(r₁) … range(r_l) [Functional])

Similar.

Case: D = EquivalentProperties(p₁ … p_n), for p_i∈VOP

As p_i∈VOP and I satisfies T(V'), I(p_i)∈IOP. If I satisfies T(D) then <I(p_i),I(p_j)> ∈ EXT_I(I(owl:samePropertyAs)), for each 1≤i<j≤n. Therefore EXT_I(p_i) = EXT_I(p_j), for each 1≤i<j≤n; ER(p_i) = ER(p_j), for each 1≤i<j≤n; and I' satisfies D.

If I' satisfies D then ER(p_i) = ER(p_j), for each 1≤i<j≤n. Therefore EXT_I(p_i) = EXT_I(p_j), for each 1≤i<j≤n. From the OWL DL definition of owl:samePropertyAs, <I(p_i),I(p_j)> ∈ EXT_I(I(owl:samePropertyAs)), for each 1≤i<j≤n. Thus I satisfies T(D).

Case: D = SubPropertyOf(p₁ p₂)

Somewhat similar, but simpler.

Case: D = SameIndividual(i₁ … i_n)

Similar to SamePropertyAs.

Case: D = DifferentIndividuals(i₁ … i_n)

Similar to SamePropertyAs.

Case: D = Individual(i type(t₁) … type(t_n) value(p₁ v₁) … value(p_n v_n))

If I satisfies T(D) then there is some A that maps each blank node in T(D) such that I+A satisfies T(D). A simple examination of T(D) shows that the mappings of A plus the mappings for the individual IDs in D, which are all in IOT, show that I' satisfies D.

If I' satisfies D then for each Individual construct in D there must be some element of R that makes the type relationships and relationships true in D. The triples in T(D) then fall into three categories. 1/ Type relationships to owl:Thing, which are true in I because the elements above belong to R. 2/ Type relationships to OWL descriptions, which are true in I because they are true in I', from Lemma 1. 3/ OWL property relationships, which are true in I' because they are true in I. Thus I satisfies T(D).

A.1.3 Lemma 3

Lemma 3: Let V' = VI + VC + VD + VOP + VDP be a separated OWL vocabulary. Let V = VI ∪ VC ∪ VD ∪ VOP ∪ VDP ∪ VRDFS ∪ VOWL. Then for every OWL DL interpretation I = < R_I, P_I, EXT_I, S_I, L_I > of V that satisfies T(V') there is an abstract OWL interpretation I' of V' such that for any OWL abstract KB K of V, I' abstract OWL satisfies K iff I OWL DL satisfies T(K).

Proof

Let CEXT_I be defined as usual from I. The required abstract OWL interpretation will be I' = < CEXT_I(I(owl:Thing)), EC, ER, S > where S(n) = I(n) for n∈VI, EC(n) = CEXT_I(I(n)) for n∈VC∪VD, and ER(n) = EXT_I(I(n)) for n∈VOP∪VDP.
V', V, I', and I meet the requirements of Lemma 2, so for any directive D over V' I satisfies T(D) iff I' satisfies D.
Satisfying an abstract KB is just satisfying its directives and satisfying the translation of an abstract KB is just satisfying all the triples so I OWL DL satisfies T(K) iff I' abstract OWL satisfies K.

A.1.4 Lemma 4

Lemma 4: Let V' = VI + VC + VD + VOP + VDP be a separated OWL vocabulary. Let V = VI ∪ VC ∪ VD ∪ VOP ∪ VDP ∪ VRDFS ∪ VOWL. Then for every Abstract OWL interpretation I' = < U, EC, ER, S > of V' there is an OWL DL interpretation I of V that satisfies T(V') such that for any abstract OWL KB K of V', I OWL DL satisfies T(K) iff I' abstract OWL satisfies K.

Proof

Construct I = < R_I, P_I, EXT_I, S_I, L_I > as follows:
- Let IOC be the OWL descriptions over V'.
  Let OD be the OWL data ranges over V'.
  Let OP = VOP ∪ VDP.
  Let OL be the finite sequences over U, over IOC, and over LV plus rdf:nil.
  Let OX = VRDFS∪VOWL - {rdf:nil}.
- L_I be any mapping compatible with D.
- R_I = U ∪ IOC ∪ OD ∪ OP ∪ OL ∪ OX
- P_I = OP
- Let I(n) = S(n) for n∈VI.
  Let I(n) = n for n∈V-VI.
- For i∈U, CEXT_I(i)={}.
  For c∈IOC, CEXT_I(c)=EC(c).
  For d∈OD, CEXT_I(d)=EC(d).
  For r∈OP, EXT_I(r)=ER(r) and CEXT_I(r)={}.
  For l∈OL, CEXT_I(r)={}.
- CEXT_I(rdf:first)={}, CEXT_I(rdf:rest)={}
  CEXT_I(rdf:nil)={}, CEXT_I(rdf:List)=OL
  <x,y> ∈ EXT_I(rdf:first) iff x∈OL and y is its first element
  <x,y> ∈ EXT_I(rdf:rest) iff x∈OL and y is its tail
- owl:intersectionOf, owl:oneOf, owl:unionOf, owl:complementOf, owl:onProperty, owl:allValuesFrom, owl:someValuesFrom, owl:hasValue, owl:minCardinality, owl:maxCardinality, and owl:cardinality have empty class extensions and their extensions are as necessary to link the elements of IOC up with their parts.
CEXT_I(rdfs:range) = {};
EXT_I(rdfs:range) = { <x,y> : x∈OP y∈IOC ∧ ∀ z, <w,z> ∈ EXT_I(x) → z ∈ CEXT_I(y); } ∪ [RDFS ranges]
CEXT_I(rdfs:subClassOf) = {};
CEXT_I(rdfs:Datatype) = D;
EXT_I(rdfs:subClassOf) = { <x,y> : x,y∈IOC ∧ CEXT_I(x) ⊆ CEXT_I(y) } ∪ [RDFS subclasses]
CEXT_I(rdfs:subPropertyOf) = {};
EXT_I(rdfs:subPropertyOf) = { <x,y> : x,y∈OP ∧ EXT_I(x) ⊆ EXT_I(y) } ∪ [RDFS subproperties]
CEXT_I(rdf:type) = {}, EXT_I(rdf:type) is determined by CEXT_I. Then I is an OWL DL interpretation because the conditions for the class extensions in OWL DL match up with the conditions for class-like OWL abstract syntax constructs.
V', V, I', and I meet the requirements of Lemma 2, so for any directive D over V' I satisfies T(D) iff I' satisfies D.
Satisfying an abstract KB is just satisfying its directives and satisfying the translation of an abstract KB is just satisfying all the triples so I OWL DL satisfies T(K) iff I' abstract OWL satisfies K.

A.1.5 Correspondence Theorem

Theorem 1: Let V' be a separated OWL vocabulary. Let K,Q be abstract OWL ontologies with separated names over V'. Then K OWL entails Q iff T(K),T(V') OWL DL entails T(Q).

Proof

Suppose K OWL entails Q. Let I be an OWL DL interpretation that satisfies T(K),T(V'). Then from Lemma 3, there is some abstract OWL interpretation I' such that for any abstract OWL ontology X over V', I satisfies T(X) iff I' satisfies X. Thus I' satisfies K. Because K OWL entails Q, I' satisfies Q, so I satisfies T(Q). Thus T(K),T(V') OWL DL entails T(Q).

Suppose T(K),T(V') OWL DL entails T(Q). Let I' be an abstract OWL interpretation that satisfies K. Then from Lemma 4, there is some OWL DL interpretation I that satisfies T(V') such that for any abstract OWL ontology X over V', I satisfies T(X) iff I' satisfies X. Thus I satisfies T(K). Because T(K),T(V') OWL DL entails T(Q), I satisfies T(Q), so I' satisfies Q. Thus K abstract OWL entails Q.

A.2 Correspondence between OWL DL and OWL Full

This section contains a proof sketch concerning the relationship between OWL DL and OWL Full. This proof has not been fully worked out. Significant effort may be required to finish the proof and some details of the relationship may have to change.

Let K be an RDF graph. An OWL interpretation of K is an OWL interpretation (from Section 5.2) that is an RDFS interpretation of K.

Lemma 5: Let V be a separated vocabulary. Then for every OWL intepretation I there is an OWL DL interpretation I' (as in Section 5.3) such that for K any OWL KB in the abstract syntax with separated vocabulary V, I is an OWL interpretation of T(K) iff I' is an OWL DL interpretation of T(K).

Proof sketch: As all OWL DL interpretations are OWL interpretations, the reverse direction is obvious.

Let I = < R_I, EXT_I, S_I, L_I > be an OWL interpretation that satisfies T(K). Let I' = < R_I', EXT_I', S_I', L_I' > be an OWL interpretation that satisfies T(K). Let R_I' = CEXT_I(I(owl:Thing)) + CEXT_I(I(owl:ObjectProperty)) + CEXT_I(I(owl:IndividualProperty)) + CEXT_I(I(owl:Class)) + CEXT_I(I(rdf:List)) + R_I, where + is disjoint union. Define EXT_I' so as to separate the various roles of the copies. Define S_I' so as to map vocabulary into the appropriate copy. This works because K has a separated vocabulary, so I can be split according the the roles, and there are no inappropriate relationships in EXT_I. In essence the first component of R_I' is OWL individuals, the second component of R_I' is OWL datatype properties, the third component of R_I' is OWL object properties, the fourth component of R_I' is OWL classes, the fifth component of R_I' is RDF lists, and the sixth component of R_I' is everything else.

Theorem 2: Let K,C be RDF graphs such that each of K, C, and K∪C is the translation of some OWL KB in the abstract syntax with separated vocabulary. Then K OWL Full entails C if K OWL DL entails C.

Proof: From the above lemma and because all OWL Full interpretations are OWL interpretations.

Comment: The only if direction cannot be proved without showing that OWL Full has no semantic oddities, which has not yet been done.

Web Ontology Language (OWL) Abstract Syntax and Semantics Appendix A. Correspondence between Abstract OWL and OWL DL (Informative)

Contents

Appendix A. Proofs (Informative)

A.1 Correspondence between Abstract OWL and OWL DL

A.1.1 Lemma 1

A.1.2 Lemma 2

A.1.3 Lemma 3

A.1.4 Lemma 4

A.1.5 Correspondence Theorem

A.2 Correspondence between OWL DL and OWL Full

Web Ontology Language (OWL) Abstract Syntax and Semantics
Appendix A. Correspondence between Abstract OWL and OWL DL (Informative)