9.  RDFS Axioms and Entailments

9.1  RDF and RDFS Axioms

In SWCLOS, all RDF and RDFS axioms are implemented. For exmaple, the following example shows RDF axiomatic triples.

gx-user(2): (typep rdf:type rdf:Property)
t
t
gx-user(3): (typep rdf:subject rdf:Property)
t
t
gx-user(4): (typep rdf:predicate rdf:Property)
t
t
gx-user(5): (typep rdf:object rdf:Property)
t
t
gx-user(6): (typep rdf:first rdf:Property)
t
t
gx-user(7): (typep rdf:rest rdf:Property)
t
t
gx-user(8): (typep rdf:value rdf:Property)
t
t
gx-user(9): (typep rdf:nil rdf:List)
t
t


The following example shows a part of RDFS axiomatic triples.

gx-user(14): (rdfs:domain rdf:type)
#<rdfs:Class rdfs:Resource>
gx-user(15): (rdfs:domain rdfs:domain)
#<rdfs:Class rdf:Property>
gx-user(16): (rdfs:domain rdfs:range)
#<rdfs:Class rdf:Property>
gx-user(17): (rdfs:domain rdfs:subPropertyOf)
#<rdfs:Class rdf:Property>
gx-user(18): (rdfs:domain rdfs:subClassOf)
#<RDFSclass rdfs:Class>
gx-user(19): (rdfs:domain rdf:subject)
#<rdfs:Class rdf:Statement>

gx-user(24): (rdfs:range rdf:type)
#<rdfsClass rdfs:Class>
gx-user(25): (rdfs:range rdfs:domain)
#<rdfsClass rdfs:Class>
gx-user(26): (rdfs:range rdfs:range)
#<rdfsClass rdfs:Class>
gx-user(27): (rdfs:range rdfs:subPropertyOf)
#<rdfs:Class rdf:Property>
gx-user(28): (rdfs:range rdfs:subClassOf)
#<rdfsClass rdfs:Class>

gx-user(31): (rdfs:subClassOf rdf:Alt)
#<rdfs:Class rdfs:Container>
gx-user(32): (rdfs:subClassOf rdf:Bag)
#<rdfs:Class rdfs:Container>
gx-user(33): (rdfs:subClassOf rdf:Seq)
#<rdfs:Class rdfs:Container>
gx-user(34): (rdfs:subClassOf rdfs:ContainerMembershipProperty)
#<rdfs:Class rdf:Property>

gx-user(35): (rdfs:subPropertyOf rdfs:isDefinedBy)
(#<rdf:Property rdfs:seeAlso>)

gx-user(36): (typep rdf:XMLLiteral rdfs:Datatype)
t
t
gx-user(37): (rdfs:subClassOf rdf:XMLLiteral)
#<rdfs:Class rdfs:Literal>
gx-user(38): (rdfs:subClassOf rdfs:Datatype)
#<rdfsClass rdfs:Class>


The followings also show important axiomatic triples on RDFS.

gx-user(39): (typep rdfs:Resource rdfs:Class)
t
t
gx-user(40): (typep rdfs:Class rdfs:Resource)
t
t
gx-user(41): (typep rdfs:Resource rdfs:Resource)
t
t
gx-user(42): (typep rdfs:Class rdfs:Class)
t
t
gx-user(43): (typep rdf:XMLLiteral rdfs:Class)
t
t
gx-user(44): (typep rdfs:Datatype rdfs:Class)
t
t


The class-instance relation in RDF is directly defined with rdf:type property. The notion of RDF class is different from the class notion in CLOS, but the subsumption relation among objects is the same as that in CLOS. The straightforward mapping of rdf:type relation to CLOS class instance relation conveys gains and benefits such as the potential of CLOS is available. However, this straightforward mapping causes only one obstacle to implement RDF on top of CLOS, namely rdfs:Class that is an instance of rdfs:Class itself. Allegro Common Lisp rejects such membership loop among classes, whereas cl:standard-object is an instance of itself in CLOS. Therefore, to work around this obstacle, the internal proxy class, rdfsClass, is introduced as a class of rdfs:Class and simultaneously as a subclass of rdfs:Class. Note that this twisted relation of class/instance and superclass/subclass between rdfs:Class and rdfsClass in CLOS produces a membership loop of rdfs:Class as well as the relation between rdfs:Resource and rdfs:Class in CLOS. See the followings.

gx-user(2): (cl:typep rdfs:Resource rdfs:Class)
t
gx-user(3): (cl:subtypep rdfs:Class rdfs:Resource)
t
t
gx-user(4): (cl:typep rdfs:Resource rdfs:Resource)
t
gx-user(5): (cl:typep rdfs:Class 'rdfsClass)
t
gx-user(6): (cl:subtypep 'rdfsClass rdfs:Class)
t
t
gx-user(7): (cl:typep rdfs:Class rdfs:Class)
t


Note that the membership loop on rdfs:Class is really implemented in CLOS semantics by means of a trick called twisted relation between rdfs:Class and its direct class rdfsClass.

9.2  RDFS Entailment

The RDFS entailment directs that every resource in RDF is typed to rdfs:Resource. The following shows some examples.

gx-user(13): (typep rdfs:Resource rdfs:Resource)
t
t
gx-user(14): (typep rdfs:Class rdfs:Resource)
t
t
gx-user(15): (typep rdf:nil rdfs:Resource)
t
t
gx-user(16): (typep rdfs:comment rdfs:Resource)
t
t
gx-user(17): (typep "This is a plain literal." rdfs:Resource)
t
t
gx-user(18): (typep 1 rdfs:Resource)
t
t
gx-user(19): (typep <http://www.somewhere> rdfs:Resource)
t
t


9.3  RDF Entailment Rules

There are two RDF entailment rules, rdf1 and rdf2 as follows.

gx-user(9): (defIndividual uuu (aaa yyy))
Warning: Entail by rdfs1: aaa rdf:type rdf:Property.
#<|rdfs:Resource| uuu>
gx-user(10): (typep aaa rdf:Property)
t
t
gx-user(11): (defIndividual vvv (aaa (xsd:integer 1)))
#<|rdfs:Resource| vvv>
gx-user(12): (get-form vvv)
(|rdfs:Resource| vvv (aaa "1"^^xsd:integer))
gx-user(13): (typep (slot-value vvv 'aaa) rdf:XMLLiteral)
t
t


In any input triple, a predicate is an instance of rdf:Property (entailment rule rdf1). When an object of an input triple is a well-typed XML literal, SWCLOS creates an data object that is an instance of rdf:XMLLiteral typed to the designated type (entailment rule rdf2).

9.4  RDFS Entailment Rules

There are 13 rules in RDFS as shown in RDFS Entailment Rules in RDF Semantics.

However a lisp string and number is interpreted as a plain literal in RDFS by SWCLOS. Therefore, rdfs1 rule is implicitly realized for a plain literal.

Note that hereafter if line numbers are renewed some younger number, it means the system is newly booted and refleshed.

gx-user(2): (typep "This is a string in lisp." rdfs:Literal)
t
t
gx-user(3): (typep 1 rdfs:Literal)
t
t


For a plain literal with language tag, SWCLOS creates an instance object of rdf:inLang, then the result for a literal with language tag also satisfies rdfs1 rule.

Note that the system is rebooted, before each of the following examples, otherwise uuu or aaa in the example are collides.

Rdfs2 is a domain entailment as follows. SWCLOS proactively performs this entailment rule. In other words, SWCLOS newly creates an instance or refines the existing instance according to this rule.

Rdfs3 is a range entailment same as domain rule rdfs2.

Rdfs4a rule entails a subject in any triple is an instance of rdfs:Resource. In a description of form for addForm input parameter, individual's type is designated by a classspecifier in the form. If 'rdf:Description' is indicated for class description, or cl:nil is passed for the resource object creation as no type indication, rdfs4a rule may be applicable. However, there is a sensitive question in SWCLOS. In RDF(S) semantics, rdfs:Resource is a type of every resource, despite that a resource is an instance of rdfs:Class or an instance of rdfs:Resource. On the other hand, in CLOS level at SWCLOS implementation, we have two possibilities for the instance of rdfs:Resource. It may be either as an instance of rdfs:Resource CLOS class (namely instance), or as an instance of rdfs:Class metaclass (namely class). As a default in SWCLOS, an instance of rdfs:Resource as CLOS instance of rdfs:Resource is created. Thus, SWCLOS changes the category of created object from instance to class later on, if a metaclass is indicated for the object at the proper definition.

Rdfs4b has the same logic and the same problem as rdfs4a for the range entailment. See the following example.

Rdfs5 is a transitivity rule on rdfs:subPropertyOf. We implemented it on properties.

Rdfs6 is very natural rule on rdfs:subPropertyOf as well as rdfs10 on rdfs:subClassOf.

gx-user(9): (subproperty-p uuu uuu)
t


Rdfs7 may be a little bit confusing.

In the form, (slot-value uuu 'bbb), although getting a slot value of bbb that is attached to uuu object was attempted, the value yyy is attached by slot named aaa, then CLOS signaled an slot missing alarm. The function “->” is for RDF graph traversing and getting the value at the final place arrived. It takes account of not only designated properties as travel path but also their super-properties.

Rdfs8 means that the default superclass of instance of rdfs:CLass metaclass is rdfs:Resource.

gx-user(2): (defResource uuu (rdf:type rdfs:Class))
#<rdfs:Class uuu>
gx-user(3): (subtypep uuu rdfs:Resource)
t
t


Rdfs9 subsumption rule and rdfs10 are natively equipped with in CLOS.

Rdfs11 transitivity rule of rdfs:subClassOf is also natively equipped with in CLOS.

Rdfs12 is a special rule for rdfs:ContainerMembershipProperty. SWCLOS proactively adds the entailment that an instance of rdfs:ContainerMembershipProperty is a subproperty of rdfs:member.

The last rule rdfs13 is for data type definition.

SWCLOS satisfies rdfs13 by creating an instance of rdfs:Datatype. However, the lisp system cannot interpret this type until the same name lisp type is defined.