Recent research has shown that the semantics of the standard SW annotation language RDF (as well as its ontological extension RDFS) and that of the standard SW ontology language OWL DL are not compatible with each other. Pan and Horrocks [5] propose a sub-language of RDFS, called RDFS(FA), which provides a clear connection between RDF and OWL DL. This paper proposes OWL FA, an extension of OWL DL with the metamodeling architecture of RDFS(FA). It also investigates some reasoning tasks for OWL FA.
Introduction so that if
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OWL Full can be regarded as a not so successful attempt at integrating RDF with OWL DL. Firstly,
there are at least three known problems in extending the RDF(S) Model Theory (RDF MT) [
In this paper, we propose OWL FA, an extension of OWL DL with the metamodeling architecture
of RDFS(FA), which is a sub-language of RDFS that provides a clear connection with OWL DL.
Intuitively, RDFS(FA) stratifies the one-layer metamodeling architecture into a multiple-layer metamodeling
architecture, so as to overcome the problem of dual roles that RDFS has. The satisfiability-preserving
bi-directional mapping between RDFS(FA) axioms in 0-1 strata and OWL DL axioms suggests that we
could extend OWL DL with the metamodeling architecture of RDFS(FA) [
OWL FA Intuitively, OWL FA introduces a stratum number in class constructors and axioms to indicate the strata they belong to. Let i ≥ 0 be an integer. OWL FA consists of an alphabet of distinct class names VCi (for stratum i), datatype names VD, abstract property names VAPi (for stratum i), datatype property names VDP and individual (object) names (VI); together with a set of constructors (with subscriptions) to construct OWL FA-classes and OWL FA-properties.
OWL FA has a model theoretic semantics, which is defined in terms of interpretations. Given an OWL FA alphabet V, a set of built-in datatype names B ⊆ VD and an integer k ≥ 1, an OWL FA interpretation is a pair J = (ΔJ , ·J ), where ΔJ is the domain (a non-empty set) and ·J is the interpretation function, which satisfy the following conditions (where 0 ≤ i ≤ k): 1. ΔJ = S0≤i≤k−1 ΔAiJ ∪ ΔD, where ΔAiJ is the domain for stratum i and ΔD is the datatype domain; 2. ΔAiJ+1 = 2ΔAiJ ∪ 2ΔAiJ ×ΔAiJ and ΔD ∩ ΔAiJ = ∅; 3. ∀a ∈ VI : aJ ∈ ΔA0J and ∀C ∈ VCi+1 : CJ ⊆ ΔAiJ ; 4. ∀R ∈ VAPi+1 : RJ ⊆ ΔAiJ × ΔAiJ and ∀T ∈ VDP : T J ⊆ ΔA0J × ΔD; 5. S∀d∈B V (d) ⊆ ΔD, where V (d) is the value space of d; 6. ∀d ∈ VD, if d ∈ B, then3 (a) dJ = V (d), where V (d) is the value space of d, (b) if v ∈ L(d), then (“v”ˆˆd)J = L2V (d)(v), where L(d) is lexical space of d and L2V (d) is the lexical-to-value mapping of d, (c) if v 6∈ L(d), then (“v”ˆˆd)J is undefined; otherwise, dJ ⊆ ΔD and “v”ˆˆd ∈ ΔD.
In the rest of the paper, we assume that i is an integer such that 1 ≤ i ≤ k. The interpretation function can be extended to give semantics to OWL FA-properties and OWL FA-classes. Let RN ∈ VAPi be an abstract property name in stratum i and R an abstract property in stratum i. Valid OWL FA abstract properties are defined by the abstract syntax: R ::= RN | R−, where for some x, y ∈ ΔAiJ−1, hx, yi ∈ RJ iff hy, xi ∈ R−J . Valid OWL FA datatype properties are datatype property names.
Now we define the OWL FA-class descriptions. Let CN ∈ VCi be an atomic class name in stratum i, R an OWL FA-property in stratum i, o ∈ VI an individual, T ∈ VDP a datatype property name, and C, D OWL FA-classes in stratum i. Valid OWL FA-classes are defined by the abstract syntax: C ::= >i | ⊥ | CN | ¬iC | C ui D | C ti D | {o} | ∃iR.C | ∀iR.C | 6i nR | >i nR (if i = 1) ∃1T.d | ∀1T.d | 61 nT | >1 nT
Constructor
top bottom concept name general negation conjunction disjunction nominals exists restriction value restriction atleast restriction atmost restriction datatype exists restriction datatype value restriction datatype atleast restriction datatype atmost restriction
DL Syntax
Semantics >i ΔAiJ−1 ⊥ ∅ CN CNJ ⊆ ΔAiJ−1 ¬iC ΔAiJ−1 \ CJ C ui D CJ ∩ DJ C ti D CJ ∪ DJ
{o} {o}J = {oJ } ∃iR.C {x ∈ ΔAiJ−1 | ∃y.hx, yi ∈ RJ ∧ y ∈ CJ } ∀iR.C {x ∈ ΔAiJ−1 | ∀y.hx, yi ∈ RJ → y ∈ CJ } >i mR {x ∈ ΔAiJ−1 | ]{y | hx, yi ∈ RJ } ≥ m} 6i mR {x ∈ ΔAiJ−1 | ]{y | hx, yi ∈ RJ } ≤ m} ∃1T .d {x ∈ ΔA0J | ∃t.hx, ti ∈ T J ∧ t ∈ dJ } ∀1T .d {x ∈ ΔA0J | ∀t.hx, ti ∈ T J → t ∈ dJ } >1 mT {x ∈ ΔA0J | ]{t | hx, ti ∈ T J } ≥ m} 61 mT {x ∈ ΔA0J | ]{t | hx, ti ∈ T J } ≤ m}
Table 1. OWL FA classes The semantics of OWL FA-classes are presented in Table 1 (page 3). C is satisfiable iff there exist an interpretation J s.t. CJ 6= ∅; C subsumes D iff for every interpretation J we have CJ ⊆ DJ .
An OWL FA knowledge base Σ consists of Σ1, . . . , Σk. Each Σi consists of a TBox Ti, an RBox Ri and an ABox Ai. Due to space limitation, we only provide details of OWL FA ABox here; it is obvious to extend traditional class and property axioms to meta-class axioms and meta-property axioms by introducing stratum numbers. Let a, b ∈ VI be individuals, C1 a class in stratum 1, R1 an abstract property in stratum 1, l a literal, T ∈ VD a datatype property, X, Y classes or abstract properties in stratum i, E a class in stratum i + 1 and S an abstract property in stratum i+1. An OWL FA ABox A1 is a finite set of individual axioms of the following forms: a :1 C1, called class assertions, ha, bi :1 R1, called abstract property assertions, ha, li :1 T , called datatype property assertions, a = b, called individual equality axioms and , a 6= b, called individual inequality axioms. An interpretation J satisfies a :1 C1 if aJ ∈ C1J ; it satisfies ha, bi :1 R1 if haJ , bJ i ∈ R1J ; it satisfies ha, li :1 T if haJ , lJ i ∈ T J ; it satisfies a = b if aJ = bJ ; it satisfies a 6= b if aJ 6= bJ . An OWL FA ABox Ai is a finite set of axioms of the following forms: X : E, called meta-class assertions, hX, Y i : R, called meta-property assertions, or X =i−1 Y , called meta individual equality axioms. An interpretation J satisfies X : E if X J ∈ EJ ; it satisfies hX, Y i : R if hX J , Y J i ∈ RJ ; it satisfies X =i−1 Y if X J = Y J . Note that there are no meta-individual inequality axioms in OWL FA; it is more intuitive for users to apply disjoint class axioms.
According to the above definition, it is obvious that Σ1 is a SHOI N (D) knowledge base, and Σ2, . . . , Σk are S HIQ knowledge bases. Note that classes and property names in Σi are treated as individual names in Σi+1; therefore, class and property equality axioms in Σi can act as individual equality axioms in Σi+1. On the other hand, individual equalities explicitly asserted and implicitly entailed by number restrictions in Σi+1 can act as class and property equality axioms in Σi.
Definition 1. Let Σ = hΣ1, . . . , Σki be an OWL FA knowledge base, where each of Σ1, . . . , Σk is consistent. Σ∗ =hΣ1∗, . . . , Σk∗i, called the explicit knowledge base, is constructed by making all the implicit atomic class axioms, atomic property axioms and individual equality axioms explicit. 3 To simplify the presentation, we do not distinguish datatype names and datatype URIrefs here. As we have a finite set of vocabulary, we have the following Lemma.
Lemma 1. Given an OWL FA knowledge base Σ =hΣ1, . . . , Σki. Σ∗ can be constructed from Σ in finite steps.
Proof (sketch): When k = 1, it is easy to show that we can calculate the explicit knowledge base Σ1∗ in finite steps because the sets of names of classes (in stratum 1), roles (in stratum 1) and individuals are finite. When k > 1, let us assume that we can calculate the explicit knowledge bases Σ10, ..., Σi0 (where 1 ≤ i < k) from Σ1, . . . , Σi in finite steps. We add all the class and property equality axioms in Σ i0 to Σi+1. If the updated Σi+1 is consistent, we can make the implicit individual equality axioms (if any) explicit and add new class and property equality axioms into Σ i0. According to our assumption, we can calculate Σ 010, ..., Σ i00 in finite steps. As the individual names in Σi+1 are finite, we can calculate the explicit knowledge bases Σ1∗, . . . , Σi∗+1 in finite steps.
Theorem 1. Given an OWL FA knowledge base Σ =hΣ1, . . . , Σki and a class C in stratum i, C is satisfiable w.r.t. Σ iff C is satisfiable w.r.t. Σi∗.
Theorem 2. Given an OWL FA knowledge base Σ =hΣ1, . . . , Σki. Σ is satisfiable iff each Σi∗ (1 ≤ i ≤ k) is satisfiable.
Theorem 2 indicates we can reduce the OWL FA-knowledge base satisfiability problem to the OWL DL-knowledge base satisfiability problem. 3
Related Work
[
Conclusion and Outlook
In this paper, we propose the OWL FA ontology language as a metamodeling extension of OWL DL,
using the metamodeling architecture of RDFS(FA), which is very similar to that of UML. The syntax of
OWL FA is very similar to that of OWL DL; the former introduces a stratum number to attach to OWL
FA class constructors and axioms. The semantics of OWL FA is a natural extension of that of OWL DL,
dividing the abstract domain into k sub-domains for k strata. We have shown that OWL FA is decidable,
and its basic reasoning tasks can be reduced to that of OWL DL. In the future, we plan to implement the
construction of extended knowledge bases so that we can use OWL DL reasoners to reason with OWL
FA ontologies, and to evaluate it with, for example, the WordNet ontology.
4 In [