1. Part I: Set Constructs

2. RDF Schema (RDFS) RDF does not provide mechanisms to define domain classes and properties RDFS is a vocabulary that provides many constructs in addition to those of RDF RDFS has the same syntax as RDF The prefix for RDF Schema is rdfs, and its URIref is: http://www.w3.org/2000/01/rdf-schema#

3. RDFS: RDF Schema Set constructs are defined in the RDF Schema Compared to RDF which defines graphs As a schema for RDF, RDFS (which is written in RDF triples) provides information, i.e., meaning, about the data through inference (i.e., deriving inferred triples from those asserted)

4. RDFS Constructs  RDFS offers constructs for:  Type ( rdfs : Class )  Subsumption ( rdfs : subClassOf )  Properties ( rdfs : propertyOf )  Relationships between properties ( rdfs : subPropertyOf )  How a property is used with respect to classes and datatypes (i.e., rdfs : domain and rdfs : range )  For supplemental information and annotation of resources, it provides: rdfs : seeAlso rdfs : isDefinedBy rdfs : comment

5. rdfs: Class  Class is the basic building block of RDFS  rdfs : Class allows us to structure the knowledge based on type  An RDFS class is a resource whose rdf : type property value is the rdfs : Class of the RDFS vocabulary  This triple is constructed through the rdf : type predicate and the rdfs : class object  Here are some examples in the default namespace: : Fault rdf : type rdfs : Class. : Lake rdf : type rdfs : Class.  The rdf : type and rdfs : class refer to the membership to a set and the set itself, respectively, i.e., class is a set!

6. rdfs : subClassOf We can define a class B to be a rdfs : subClass of class A by using the rdfs:subClassOf property and rdfs : Class This construct allows building vocabularies through hierarchies Or in N3 format: : GneissicBandingrdfs : subClassOf: Foliation. rdfs : subClassOf allows inference through the type propagation rule As in sets, if x is a member of A, and A is a subclass of B, then x is a member of B

7. rdfs : subClassOf – another example We can define a class B to be a rdfs : subClass of class A by using the rdfs:subClassOf property and rdfs : Class This construct allows building vocabularies through hierarchies Or in N3 format: : KarstAquiferrdfs : subClassOf: Aquifer.

8. NormalFault subClassOf Fault <rdf:RDF xml:base="http://www.StructuralGeology.org/structure/" xmlns:owl="http://www.w3.org/2002/07/owl#" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:rdfs="http://www.w3.org/2000/01/rdf-schema#" xmlns:struc="http://www.StructuralGeology.org/structure/">

9. Alternative code

10. rdfs : domain and rdfs : range rdfs:domain assigns a property to a subject class rdfs : range assigns the value of property either to an object class or a datatype If we want the range to have several objects, we can use the rdf : Alt container as the range

11. rdfs: domain and rdfs:range of the heave property <rdf:RDF xml:base="http://www.StructuralGeology.org/structure/" xmlns:owl="http://www.w3.org/2002/07/owl#" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:rdfs="http://www.w3.org/2000/01/rdf-schema#" xmlns:xsd="http://www.w3.org/2001/XMLSchema" xmlns:struc="http://www.StructuralGeology.org/structure/">

12. rdfs: domain and rdfs:range of the pointBar property <rdf:RDF xml:base="http://www.Streams.org/River/" xmlns:owl="http://www.w3.org/2002/07/owl#" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:rdfs="http://www.w3.org/2000/01/rdf-schema#" xmlns:xsd="http://www.w3.org/2001/XMLSchema" xmlns:river="http://www.Streams.org.org/River/">

13. Instance (individual) An instance is an individual It may be an instance of one or more classes. Here is how an individual is declared: or alternatively Or, alternatively:

14. Instance - another example or alternatively Or, alternatively:

15. rdf:Property and rdfs:subProperty Properties are instances of the rdf:Property Properties are defined at the global scale, and do not belong to specific classes rdf:Property is a class, and its instances are properties Properties relate sets of individual pairs, and as such can be conceived as classes that can be subclassed, i.e., specialized The rdf:Property may have zero or more rdfs:subProperty subproperties e.g., highlyAltered is a subproperty of the altered property e.g., mildlyFunny is a subproperty of the funny property

16. rdfs:subPropertyOf Relationship propagate through This provides meaning to the properties (i.e., predicates) that link classes (resources) The rdfs:subPropertyOf provides a mechanism to extend properties, from a more general type to more specific i.e., allow hierarchy (taxonomy) of properties struc: displacerdfs:subPropertyOf struc:move

17. Inference of rdfs:subPropertyOf P is a subproperty of P’ if P’(x,y) whenever P(x,y)  This means that if a specific property holds for entities lower in the hierarchy, so does the general property above it. In other words: P rdfs:subPropertyOf P’ If x P y Then x P’ y struc:displace rdfs:subPropertyOf struc:move If sanAndreasFault displace RockUnit1 Then, SanAndreasFault move RockUnit1 P’ P A B x y move displace FaultRockUnit

18. rdfs:subPropertyOf – example rdf:about="http://www.w3.org/2002/07/owl#ObjectProperty"/>

19. Examples  ‘shear’ and ‘extend’ are more specific types of ‘displace’  ‘brittlyDeform’ is more specific than ‘deform’  ‘rotate’ is more specific than ‘deform’ shear rdfs:subPropertyOf displace If: Fault shear Grain Then: Fault displace Grain rotate rdfs:subPropertyOf deform If: Fault rotate Fold Then: Fault deform Fold displace shear FaultGrain xy deform rotate FaultFold xy

20. Set Intersection & Union in RDFS Although RDFS does not have explicit mechanisms to define intersection and union, we can express these set logical statements through design patterns that use RDFS primitives Recall that the intersection of set A and B (A  B) contains all elements which are in both A and B

21. Multiple rdfs:subClassOf and Intersection We can construct intersection with two subclasses! If we define set C to be the intersection of A and B (i.e., C  A  B), that is, if x is in C, then it is also in both A and B This means that C is a subclass of A and a subclass of B. C rdfs:subClassOf A. C rdfs:subClassOf B. A C B x C x

22. Inference  Using the rdfs:subClassOf inference rule: If: x rdf:type C Then x rdf:type A x rdf:type B  Therefore, given x is in C  A  B, we can infer membership in A and B from membership in C, but not the other way around! C x

23. Example: Pyroclastic Rocks  Pyroclastic rocks (e.g., tuff) have the properties of both volcanic and sedimentary rocks. : PyroclasticRockrdfs : subClassOf: VolcanicRock. : PyroclasticRockrdfs : subClassOf: DepositionalRock. : HuckleberryTuffrdf : type: PyroclasticRock.  We can infer that Huckleberry tuff in Wyoming is both depositional and volcanic. We infer the following two: : HuckleberryTuff rdf :type: VolcanicRock. : HuckleberryTuffrdf :type: DepositionalRock.  Notice that the inference is uni-directional, i.e., pyroclastic rock is both volcanic and depositional, but every depositional sedimentary or volcanic rock is not pyroclastic!

25. Pyroclastic rock subclass PyroclasticRock DepositionalRock VolcanicRock x

26. Vein: Both mineral and filled fracture : Vein rdfs : subClassOf Mineral. : Vein rdfs : subClassOf FilledFracture. : aCalciteVein rdf : type :Vein.  We can infer the following: : aCalciteVein rdf :type Mineral. : aCalciteVein rdf :type FilledFracture.  Notice that in this case, the instance of vein is both a mineral and a filled fracture, but any mineral or filled fracture is not a calcite vein! Mineral Vein FilledFracture aCalciteVein