14.  OWL APIs

14.1  Subsumption in OWL

We discriminate the substantial inclusiveness on concepts that is decided through rdfs:subClassOf, owl:intersectionOf, and owl:unionOf, against the non-substantial inclusiveness, that can be reasoned out by entailment rules and a subsumption inference engine. The former is calculated through the CLOS class-subclass relationship, and the latter is deduced by the subsumption algorithm described below in SWCLOS. The top concept in the OWL universe, owl:Thing, substantially subsumes every concept of OWL vocabulary in the CLOS class-subclass relation or OWL reasoning, but the bottom concept, owl:Nothing, is virtually subsumed by other concepts only through this algorithm. We extended and modified the basic structural subsumption algorithm, that is described in “The Description Logic Handbook” [DLH], to one that contains disjunction (owl:unionOf), negation (owl:complementOf), equivalency (owl:sameAs and owl:equivalentOf), functional and inverse-functional relation (owl:FunctionalProperty and owl:InverseFunctionalProperty), symetric relation (owl:SymmetricProperty), the value restriction (owl:allValuesFrom), the full existential restriction (owl:someValuesFrom), the filler restriction (owl:hasValue), and the number restriction (owl:maxCardinality, owl:minCardinality, and owl:cardinality).

The algorithm is summarized as follows, for two concepts C and D;

1. If C is the bottom, then it returns true.
2. If D is the top, then it returns true.
3. If D is the bottom, then it returns false.
4. If C is a subclass of D in CLOS, then it returns true.
5. If D is a subclass of C in CLOS, then it returns false.
6. If C and D is the same as individual, then it returns true.
7. If C and D have oneOf slots and C�fs members are subset of D�fs members, then it returns true, otherwise false.
8. For some (cc) of equivalent classes of C and some (dd) of equivalent classes of D;
   • If cc is a subclass of dd in CLOS, then it returns true.
   • If dd is a subclass of cc in CLOS, then it returns false.
   • If cc has unionOf slot values and every member is subsumed by dd, then it returns true.
   • If dd has unionOf slot values and cc is subsumed by some of union members of dd, then it returns true. This process is called union branching.
   • If dd has intersectionOf slot values, then the algorithm for intersection is performed. See the below.
   • If cc and dd is equal through owl:equivalentClass, owl:FunctionalProperty, or owl:InversefunctionalProperty, then it returns true.
   • If cc and dd are complement, then it returns false.
   • If cc and dd are disjoint, then it returns false.
   • Otherwise go to the next step below for transtivity checking as individuals.
9. For some (cc) of same individuals of C and some (dd) of same individuals of D;
   • If cc and dd shares a transitive property and cc is inferior to dd on the property, then return true.

The algorithm for intersections is summarized as follows.

1. Firstly, D is unfolded on the intersection of concepts. Namely, if an element of intersections is also the concept of intersections, all members of the intersection are retrieved and the flattened list of all intersection elements is made.
2. Secondly, C’s class precedence list (cpl), that is, a list of all super concepts of C including itself is computed.
3. If every concept member (except property restrictions) of intersections of unfolded D subsumes some concept member of C’s cpl, then go to the next step, otherwise false.
4. If no restrictions of unfolded D, then returns true. Otherwise, go to the next step, namely model generation and testing process.
5. Generate possible models of C and every models are checked if some of them satisfy the D’s restrictions.
   • Firstly, all slot definitions of C for RDF and OWL properties are retrieved.
   • For each of restrictions of unfolded D, the following steps are performed, and every step must be true, otherwise imediately returns with false.
     • In case of no slot definition of C and no cardinality restriction of D, if D’s restriction is allValuesFrom, then true, else if someValuesFrom or hasValue, then returns false in auto-epistemic-closed-world or true in open world.
     • In case that there is no slot definition of C and cardinality restriction of D exists,
     • In case that both C’s slot and D’s restriction exists, possible models of C are created from C’s slot definition of the current property under the cardinality restriction from D, and possible models of D are also created. Then, some of C’s models satisfy every D’s models, then true, otherwise returns false.

The algorithm of generation models and typed unification theory, which is used in model matching, is more complicated and difficult for beginers of SWCLOS. This advanced topic is shown in Phd theses.

The following demonstrates the function of this algorithm.

14.2  Membership in OWL

We extended the membership checking algorithm in OWL and it is implemented to gx:typep. The followings demonstrate some membership relations among classes and things in OWL after loading Wine Ontology.

gx-user(15): (typep vin:ElyseZinfandel vin:Zinfandel)
t
t
gx-user(16): (typep vin:ElyseZinfandel owl:Thing)
t
t
gx-user(17): (owl-thing-p vin:ElyseZinfandel)
t
gx-user(18): (typep vin:Zinfandel owl:Class)
t
t
gx-user(19): (typep vin:Zinfandel owl:Thing)
t
t
gx-user(20): (owl-class-p vin:Zinfandel)
t
gx-user(21): (owl-class-p owl:Class)
nil
gx-user(22): (owl-class-p owl:Restriction)
nil
gx-user(23): (owl-thing-p owl:Class)
nil
gx-user(24): (owl-thing-p owl:Restriction)
nil
gx-user(25): (owl-class-p owl:Thing)
t
gx-user(26): (owl-thing-p owl:Thing)
t
gx-user(27): (owl-class-p owl:Nothing)
t
gx-user(28): (owl-thing-p owl:Nothing)
nil


An individual (instance) that is typed to owl:Thing and a concept (class) that is typed to owl:Class belong to the OWL universe. However, note that owl:Class and owl:Restriction are not members in OWL universe. Also note that owl:Thing itself belongs to OWL universe but owl:Nothing does not, even though the class of owl:Nothing is owl:Class as well as owl:Thing, because owl:Nothing is complement of owl:Thing.

14.3  The Most Specific Concepts

Computing the most specific concepts in a set of concepts is basic operation in DL and SWCLOS. We use this routine very often in SWCLOS.

References