Metamodeling is a feature in ontologies, which allows for having classes as members of other classes. While providing very exible modeling power, it also comes with decidability issues. Because of this, metamodeling are often outright forbidden or unsupported in modeling frameworks. Some work exists that provides explicit support for metamodeling by augmenting the ontology. We introduce a new approach that provides metamodeling support in ontologies by means of a transformation. It rewrites the metamodeling features automatically into rules (M-rules) and strips them from the ontology. The resulting reduced ontology O0 together with the M-rules then form a hybrid knowledge base, which can be processed using NoHR. This framework allows for e ective reasoning and querying over the combined knowledge of ontology and rules.
Metamodeling is a process for specifying conceptual modeling requirements,
where classes have other classes as an instance, called metaconcepts. Relations
between metaconcepts are called metaproperties. The bene t for adding
metamodeling in ontologies provide explicit availability of knowledge. This can allow
for more understandable (less subject to misinterpretation) and reusable
models. Consider an example adopted from [
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However, in species domain contrary to speci c properties of some species, one can also articulate expressions about species themselves i.e. \GoldenEagle is listed in the IUCN Red List of endangered species" states that GoldenEagle as a whole class is an endangered species, and not just Harry. To formally model this expression, we can set GoldenEagle as an instance of new class EndangeredSpecies. Making GoldenEagle a type of EndangeredSpecies is concise and clear. Note that making GoldenEagle a subclass of EndangeredSpecies would not be correct, as it results in incorrect conclusions like \Harry is an EndangeredSpecies". In the aassertion above, EndangeredSpecies is a metaconcept for GoldenEagle.
OWL Full can express this type of metamodeling, but it is so expressive that
it leads to undecidability [
This section provides a brief overview of the OWL 2 QL ontology language under
Metamodeling Semantics [
The Metamodeling Semantics for OWL 2 QL is based on an interpretation structure, constituted by a tuple = ( ;: I ;: E ;: R;: A;: T ), where is the disjoint union of the two non-empty sets o (the object domain) and (the alue domain), and interpretation functions are partial, which means a single object can simultaneously be an individual :I , a class :E , an object property :R, a data property :A, and a data type :T . An interpretation I is a pair ( ; I) for ontology O, where is the interpretation structure and I is the interpretation function. The semantics of logically implied axioms are de ned in accordance with the notion of axiom satisfaction with the interpretation I. Moreover, I is said to be a model of O if there exists at least an interpretation that is a model of O. Finally, an axiom , is logically implied by O, denoted as O j= , if it holds for every model of O.
NoHR Hybrid Knowledgebase We adopt hybrid knowledge bases consisting
of well-founded MKNF [
KH
KB1+; : : : ; KBn+; notB1 ; : : : ; notBm
where H is the rule head and Bi with arity 1 i m are rst order atoms in
the rule body. A rule with an empty body is a Fact. A rule is a DL-safe rule, if all
variables in the rule occur in at least one non-DL atom. If all rules are DL-safe,
then K is DL-safe. The semantic transformation of K is based on the MKNF
formula (K), to which the semantics of MKNF can be applied. As de ned
in [
It is not easy to do metamodeling in ontologies, as there is little language support for ontological metamodeling. However, OWL 2 provides support for a basic level of metamodeling in ontologies via the use of Punning, but standard reasoners give no semantic support for those punned entities and treat them as di erent type of entities. Moreover, metamodeling features in ontology can easily pose semantic discrepancy in O, that results in undecidability in query answering.
To tackle these issues, our approach is to give semi-metamodeling support to ontologies by expressing the metamodeling information in non-monotonic Mrules (short for metamodeling-rules). For this we remove the metamodeling portion from the original ontology and transform it into the sine M Ontology (O0) and also generate M-rules for the removed portion.
De nition 1. A sine M Ontology (O0) is a subset of a metamodelled-ontology M O, without Clashing Axioms.
De nition 2. Consider entities C1 and c1 in an M-ontology. If there exists some axiom of the form (at-the-least) C1 v C2 and A(c1), then these are called clashing axioms, where C1 c1 and appears more than one type of position with the same IRI.
Our framework makes use of two languages: MS for OWL 2 QL as an ontology language, and NoHR as reasoner for hybrid knowledge base.
The proposed framework consist of three functions: 1. The rst function checks for clashing axioms in M O and generates the sine-M Ontology. To retain the same semantic knowledge between M O and sine M Ontology, we omit the clashing axioms appearing in the ABox, because omission of general concepts and roles stores in T Box will cause semantic inconsistencies and change the meaning of the represented domain. 2. The second function takes eliminated axioms with attached information as input and translates them into M-rules. The basic building blocks for Mrules are the OWL 2 QL assertions under MS, where an assertion is simply an axiom consisting of sets of individuals, classes or roles. The form of assertions for M-O that are suitable for metamodeling can have one of following forms: p1(e1; e2) and p2(e1; e2; e3), where p1 can be a class assertion C(x) or disjoint assertion C1 u C2 v ? on concepts and p2 is a role or property C(x; y) and e1; e2; e3 are expressions over VO.Currently we have introduced and focused on the following assertions notions for M-rules. We say that: { In mCAssert(e1; e2), e1 represents an object argument and e2 appears as class argument. { In isDisjCAssert(e1; e2), e1; e2 appear as class arguments. { In mRAssert(e1; e2; e3), represents information about Object and Data property.
In mOPAssert(e1; e2; e3), e2 and e3 represent individual arguments and e1 appears as property argument.
In mDPassert(e1; e2; e3), e2 and e3 represent individual arguments and e1 appears as literal. 3. The third function passes the O0 and M-rules to the NoHR reasoner for validation and for querying the hybrid knowledge base.
The basic structure and semantics of O0 after omission of clashing axioms remain the same as M O.
Lemma 1. O0 is said to be a subset of M O if there exists at least some syntactic correspondence between the signature of both O0 and M-O, for the existence of named entities, so signature(M O) \ signature(O0) 6= ;.
Lemma 2. Let M O be a metamodelling ontology and O0 be a sine M Ontology. A as a concept/unary predicate and R as a role/binary predicate, O0 is a semantic preserving representation model of M O.
M M
O j= A(a) O j= R(a; b) () ()
0 = A(a) O j
0 = R(a; b) O j Example 1. Consider the example mentioned in the introduction, which shows that EndangeredSpecies is a metaconcept for GoldenEagle. First, we strip the metamodeling part from the metamodelled ontology M O and generate O0:
In Table 1 we can see that the translation of M O to O0did not delete the EndangeredSpecies class, even though it is a metaclass in order to retain the same schematic structure and semantic knowledge in O0as M O. So, the translation function just eliminates the fact that makes EndangeredSpecies a metaclass and that is in this case an axiom of the form:
Then, the translation function transforms the eliminated part from M O into M-rules, which captures the metamodeling semantics of OWL 2 QL. To avoid ambiguity and misinterpretation of the arguments, we have used "is" and "Of" in our predicate i.e. isInstanceOf(?X, ?Y) which gives more clarity of roles of the arguments. After translation, metamodeling axioms are appended to the M-rules.
mCAssert(GoldenEagle,EndangeredSpecies) isInstanceOf(?C,?D) :- mCAssert(?C,?D),not isTmCAssert(?C,?D) isTmCAsert(?C,?D) :- mCAsert(?C,?B), mCAsert(?B,?D)
isInstanceOf(?X,GoldenEagle) :- GoldenEagle(?X) isInstanceOf(?X,EndangeredSpecies) :- EndangeredSpecies(?X)
EndangeredSpecies(?X) :- isInstanceOf(?X,EndangeredSpecies) We have used one of the notions introduced above, mCAssert(e1; e2), where m stands for meta and the argument e1 represents the class as an individual. isTmCAssert is the transitive closure on meta-classes and the key idea for its de nition is to stop cycles, which may occur when both arguments of mCAssert are the same or when the arguments form a loop. Next we de ned the isInstanceOf predicate in the rules. The purpose of this predicate is to show the instanceOf hierarchy chain between entities and captures the metamodeling semantics of entities. Rules 4 and 5 help rule 2 to capture all the instances that are linked with meta-Assertions and ease the meta-querying. On the other hand, rule 6 updates the instances listed for EndangeredSpecies in O0along with the instances listed in M-rules. To accommodate metaquerying, our translation function also checks the facts that come along with meta-asserted axioms like disjoint axioms. Suppose BaldEagle and GoldenEagle are disjoint in O0and are instances of EndangeredSpecies. Then, to capture the correct semantics of O0in M-rules, we have to de ne a new predicate called isDisjCAssert. (1) (2) (3) (4) (5) (6) (7) (8) (9) isDisjCAssert(GoldenEagle,BaldEagle) isDisjcOf(?X,?Y) :- isDisjCAssert(?Y, ?X) isDisjCAssert(?Y, ?X) :- isDisjCAssert(?X, ?Y) The isDisjCAssert predicate ensures the semantic correspondence between the entities in the ontology and te atoms in rules. Rule 7 is the meta-assertion involving GoldenEagle and BaldEagle. Rule 9 ensures symmetry, and Rule 8 (wit 9) checks for disjoint assertions on instance level and makes GoldenEagle and BaldEagle di erent individuals. If the asserted meta-axioms come with role assertions, then the translation function will use mRAssert(e1; e2; ; e3). Suppose in Example 1 we have a property axiom, which says that animals are listed in EndangeredSpecies by IUCN, and assume IUCN a metaclass, which makes ListedBy a metaproperty. The translation of this axiom will be: mOPAssert(ListedBy,EndangeredSpecies,IUCN) (10) isOPInstance(?R,?X,?Y) :- mOPAssert(?R,?X,?Y), not isTmOPAssert(?R,?X,?Y) (11) isTmOPAssert(?R,?X,?Y) :- mOPAssert(?R,?X,?Z),mOPAssert(?R,?Z,?Y) (12)
ListedBy(?X,?Y) :- isOPInstance(ListedBy,?X,?Y) (13) Rule 11 shows that the isOPInstance predicate helps capturing those entities that are linked together via a relation and isTmOPAssert helps avoiding cyclic relations. We omit dealing with other OWL 2 QL axioms, since we assume that the three assertions treated in the M-rules above are su cient for most basic metamodeling tasks. 4
As query language we consider conjunctive queries. A conjunctive query q over O is an expression ask where B (query body), consisting of a conjunction of atoms over VO. We consider a metavariable in a query to be a variable that may refer to metaclasses and metaproperties.
De nition 3. A metaquery is an expression consisting of meta-predicates p and meta-variables v, where p can have other p as their arguments and v can appear in predicate positions.
The semantics of queries resort to the interpretation of queried ontology and rules according to metamodeling semantics. We expressed the queries in NoHR, that allow conjunction of predicates with variables in a query and to get the correct substitution for variables, queries need to be ground before being processed by the DL reasoner.
Example 2. Consider a conjunctive query q to Example 1, where q should retrieve all those Birds that are instances of EndangeredSpecies (ES) and live in Central Park Zoo (CPZ): isInstanceOf(?C, EndangeredSpecies), isInstanceOf(?X, ?C),Birds(?X), Lives in(?X, CPZ) With the use of O0and M-rules, we get the set qi0 = fq00; q10; q20g is a rewriting for q, where q0 checks for all the instances of EndangeredSpecies de ned by rules and will substitute ?C with GoldenEagle. q1 takes the substituion of ?C and checks for its instances from both O0and M-rules and will substitute ?X with Harry. q2 takes the substition of ?X = Harry and checks O0for the property Lives in, if proven true, it can derive the queried goal and the answer to qi0 is equivalent to q for any set of ground facts A.
The query q, according to De nition 3 is clearly a metaquery. The full power
of metaquyering, where T box and Abox atoms coexist, can be achieved with the
use of the generalized predicate isInstanceOf from the translation function. We
called isInstanceOf predicate generalized, as it tries to capture the semantics of
rst order (FO) constructs like C(x), C1 v C2 such that it does not require FO
constructs in the query. For instance, in the above example we ask to retrieve
all those birds of subclass Eagle who are instances of EndangeredSpecies (ES)
and live in Central Park Zoo (CPZ). This query can be answered by the query
above, with the fact that GoldenEagle is an instance of EndangeredSpecies and
GoldenEagle is a subclass of Eagle as well.
Higher Order Semantics [
Hybrid Integration of rule systems with ontology languages have shown
signi cant advances and plays a central role in the development of the Semantic
Web. Our focus here is on one of the integration techniques known as the hybrid
MKNF knowledge bases [
We showed that our approach not only allows metamodeling facility in ontology but it allows for writing a slightly restricted form of metaqueries as well. We plan to perform more experiments to evaluate our approach, that is, a comparison with other approaches that support metamodeling feature in OWL 2 QL ontologies. Also we plan to extend the notion of assertions to achieve the full power of meta-querying by using non-monotonic rules.