Geoinformatics

1. Geoinformatics Logic …

2. p A B a1 b1 P’ Disjoint Properties subject object • As for disjoint classes, two properties can be disjoint (owl : propertyDisjointWith) • Property p and p’ are disjoint if no two triple (SPO) statements exist that use these properties as predicate with the same subject (S, domain) and object (O, range) individuals. Let’s define:Person hasMother WomanPerson hasFather Man • If hasMother and hasFather properties are declared to be disjoint • If a RyanT is a Person and AshleyTis a Woman, we cannot assert: RyanThasMotherAshleyT and RyanThasFatherAshleyT hasMother Person Woman RyanT AshleyT hasFather subject object

3. crystallizesTo Magma Rock magma2 meltsTo rhyolite5 Disjoint Properties … subject object • Let’s declare the meltsTo and crystallizesTo, or linearAttitude and planarAttitude, as pairs of disjoint properties • And define their domain and range: • Notice that the subjects and objects of each of these two pairs of universal statements are not the same • Also notice that the two disjoint properties are neither inverse nor symmetric • We cannot simultaneously make the following pair of assertions:

4. Individual (Assertional) Axioms • Individual axioms include those that assert that an individual belongs to a set, e.g., C(a) denotes that a is a member (particular; individual, instance) of set (universal) C StrikSlipFault (“San Andreas Fault”) Batholith (“Idaho Batholith”) That is, the “San Andreas Fault” and “Idaho Batholith” are members of the StrikeSlipFault and Batholith set, respectively. Also: (StrikeSlipFault Fault) (“San Andreas Fault”  Fault)

5. Assertions … • p (a, b) or a p b asserts that an individual a is related to another individual b with the relation (property) p • locatedIn (“San Andreas Fault”, “California”) asserts that individual San Andreas Fault is located in California, and intrudes (“IdahoBatholith”, “BeltSupergroup”)

6. Equality between Individuals • Equality or inequality between two individuals, a and b, is asserted as a  bor a  b, respectively • For example, we may want to state that “Boulder Batholith” is equivalent to “Boulder Intrusion” by asserting: “Boulder Batholith”  “Boulder Intrusion”

7. Domain and Range Restrictions • Domain and range are used to infer membership of instances to certain classes • Domain and range define the subject (source) and object (target) of a property (p), respectively Mineral ageDateIsotopicAge • The domain for the ageDate property in the above statement is the Mineral class, and its range is the IsotopicAge class • All instances of the Mineral class (e.g., a calcite crystal) have an ageDatethat is of the IsotopicAge type ageDate Mineral a calcite crystal an isotopic age IsotopicAge Source class or Domain Target class or Range

8. Logical expression of domain and range • Logically, the domain and range restrictions, which are kinds of class axioms, are modeled as: • P.T  D, and T P.R, respectively, using the general class inclusion () axiom, where P is the predicate (property, relation), D is the domain class, and R is the range classes • Restricting the domain of the folds property to the Fold class by means of the class axiom is expressed as: folds.T Fold For all instances, they must haveat least one occurrence of the folds property with the specified domain folds Fold a fold Domain

9. folds a plane PlanarStructure Range Restricting the Range • Restricting the range of the folds property to planar structures: T P.R T folds.PlanarStructure P.R means for all instances, if they have the property P, it must have the specified range (i.e., values come from the object (range) class R) Given the assertion: folds (SheepMountain, Bedding) we infer, through reasoning, that SheepMountain is of the Fold type, and Bedding is a planar structure • This means that the ‘Sheep Mountain’ specifically folds bedding which is a planar structures! • To restrict the range of the folds property to either linear, planar structures, or both, we write: T folds.PlanarStructurefolds.LinearStructure

10. Local Property Restriction • Property restriction puts a local constraint on the use of the property • A property restriction is a special kind of class description • It describes an anonymous class, namely a class of all individuals that satisfy the restriction • The restriction puts a condition for using the property by individuals of a class

11. Property Restriction … • Later we will learn that in OWL, local property restrictions are applied to a class by making the class either an owl:subclassOf or an owl:equivalentClass of the unnamed (i.e., anonymous) restriction class which bears the condition for membership by its restricted property • The restriction provides a necessary and sufficient condition for membership

12. Types of Property Restriction • OWL has two kinds of property restrictions: value constraints and cardinality constraints • There are four types of value restriction: • owl:allValuesFrom , P.C • owl:someValuesFrom, P.C • owl:hasValue • owl:selfRestriction

13. owl : allValuesFrom, P.C • Provides a value restriction for the range of a property • The  connective corresponds to owl : allValuesFrom construct, which means: for all instances, if they have the property or relation P, it must have the specified range • i.e., the object values for the property come from a class C • P.C denotes the set of individuals a, such that for any individual b, ifP relates a to b, then b is in C • i.e., the range for P is class C Range A C a b P

14. City locatedIn Nevada a b Range P.C Example • The set of individuals that are related by property P only to individuals of class C • locatedIn.Nevada, is the set of individuals located only in Nevada, and not anywhere else NevadaCity City  locatedIn.Nevada • The  connective reads: ‘for all, if any’, meaning that the occurrence can be many or zero • To say that igneous rocks are those rocks that form only from crystallization out of a magma, we assert:IgneousRock Rock crystallizeFrom.Magma • Only cylindrical folds have axis: CylindricalFold Fold hasAxis.Axis • Non-cylindrical fold is one without any axis: NonCylindricalFold Fold CylindricalFold IgneousRock crystallizeFrom Magma Rock Range

15. owl:someValuesFrom, P.C • The  connective corresponds to the owl:someValuesFrom construct, which means: For all instances, they must have at least one occurrence of the property with the specified range • Some values of the property come from the class C • Like the  connective, the  connective provides a value restriction for the range of a property Range A C a b P

16. crystallize e.g., Lava Ce.g., Mineral a b … P • P.Cdenotes the set of individuals a, such that there exists an individual b, such that P relates a to b, and b is in C • It denotes the set of individuals that are related to some individuals of class C by property P • The  connective reads: ‘there exists at least one’ • crystallize.Mineralmeans the set of individuals (not necessarily magma; could be water) that crystallize some minerals, e.g., CoolingLava (Lava  crystallize.Mineral) • i.e., cooling lava is a lava that crystallizes at least one mineral crystallize Lava Ce.g., Mineral b P CoolingLava a

17. Owl : hasValue & owl : selfRestriction • The owl:hasValue is a special kind of the owl:someValuesFrom. It means that all instances must have the property with the exact value • For example, it is used when we want to restrict the range of the hasMoon property only to Saturn, i.e., only deal with the moons of Saturn, • We can restrict the hasMylonite property to the San Andreas Fault • The owl:selfRestriction makes a restriction on a property that relates an individual to itself, e.g., selfRising, selfAbsorption

18. Cardinality Number Restrictions ( n P)owl : minCardinality  1 hasMineral Rock Mineral a b • Cardinality restrictions specify the number of times a property can be used to describe an instance of a class • The unqualified number restriction ( n P)(owl : minCardinality) denotes the class of individuals that are related to at leastn individuals by the property P • (i.e., there must be at least n properties, where n is a non-negative integer)

19. Qualified Number Restriction • The three cardinalities are called unqualified becausethe class of individuals is unspecified, e.g., ( n P) • If qualified , i.e., ( n P).C, for example, the Rock class is related to at least 1 mineral from the Mineral class by the hasMineralproperty ( n R).C Rock  ( 1 hasMineral).Mineral • i.e., Rock has one or more minerals

20. ( n P) owl : maxCardinality • The unqualified number restriction ( n P) (owl : maxCardinality) denotes the class of individuals that are related to at mostn individuals by the property P (i.e., there can be at most n properties) • For example, instances of the s atomic subshell can have at most 2 electrons ( 2 hasElectron) (SAtomicShell Shell) ( 2 hasElectron).Electron  2 hasElectron SAtomicShell b Electron a Shell

21. ( n P).C and ( n P).C • The  n R.C and  n R.C are qualified number restrictions because the class C is specified • For example, we can state that cylindrical fold has at most one hinge line by: CylindricalFold Fold  ( 1 hingeline).Axis • Silicon-oxygen tetrahedra in tectosilicates share all (i.e., 4) of their oxygens Tectosilicate Silicate ( 4 tetrahedraShares).Oxygen  ( 4 tetrahedraShares).Oxygen   =

22. owl : cardinality • The owl : cardinality can be expressed as the intersection of the owl : maxCardinality and owl : minCardinality • For this case, there are exactly n properties • For example, monomineralic rock is a rock with exactly one kind of mineral: MonomineralicRock Rock ( 1 hasMineral).Mineral  ( 1 hasMineral).Mineral  1  1 1