Logic …

Disjoint Properties 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 Woman Person hasFather Man If hasMother and hasFather properties are declared to be disjoint If RyanT is a Person and AshleyT is a Woman, we cannot assert: RyanT hasMother AshleyT and RyanT hasFather AshleyT S P O S P’ O B object A a1 subject p P’ b1 Woman object Person RyanT subject hasMother AshleyT hasFather

Disjoint Properties … 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 the same Also notice that the two disjoint properties are neither inverse nor symmetric We cannot simultaneously make the following pair of assertions: Rock object Magma subject crystallizesTo rhyolite5meltsTomagma2 SPO: Magma crystallizesTo Rock SP’O Magma meltsTo Rock magma2 crystallizesTo rhyolite5 magma2 meltsTo andesite5

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”) City (“Los Angeles”) Note: The “San Andreas Fault” and “Idaho Batholith” are members of the StrikeSlipFault and Batholith set, respectively. Given: StrikSlipFault (“San Andreas Fault”) We infer that: (StrikeSlipFault  Fault)  (“San Andreas Fault”  Fault)

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”)

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

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 Country hasCapital City Country is the domain and City is range for hasCapital Mineral ageDate IsotopicAge 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., aMica) have an ageDate property that is of the IsotopicAge type; which in this case has a value of age IsotopicAge Mineral ageaMica ageDate Target class or Range Source class or Domain

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

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

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

owl : allValuesFrom,  P.C Provides a value restriction for the range of a property T   P.R all values of P come from the object (range) class R 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 the class C  P.C denotes the set of individuals a, such that for any individual b, if P relates a to b, then b is in C i.e., the range for P is class C A C ab P 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 CityNevada ab locatedIn Range IgneousRock Magma crystallizeFrom Range Rock

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 (at least one) values of the property P come from class C Like the  connective, the  connective provides a value restriction for the range of a property A C ab P Range

…  P.C denotes 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.Mineral means the set of individuals (not necessarily magma; could be water or cat) that crystallize some mineral, e.g., CoolingLava  (Lava   crystallize.Mineral) i.e., cooling lava is a lava that crystallizes at least one mineral e.g., Lava C e.g., Mineral a b crystallize P Lava C e.g., Mineral b crystallize P CoolingLava a Range

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

Cardinality Number Restrictions (  n P) owl : minCardinality 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 (class is not unspecified) that are related to at least n individuals by the property P (i.e., there must be at least n count of the property, where n is a non-negative integer) RockMineral ab  1 hasMineral

Qualified Number Restriction The three cardinalities are called unqualified because the 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 hasMineral property (  n R).C Rock  (  1 hasMineral).Mineral i.e., Rock has one or more minerals Car  (  1 hasWheel).Wheel

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

(  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 MeanderingRiver  River  (  1 meander).Meander Silicon-oxygen tetrahedra in tectosilicates share all (i.e., 4) of their oxygens (i.e., share exactly all four) Tectosilicate  Silicate  (  4 tetrahedraShares).Oxygen  (  4 tetrahedraShares).Oxygen  =

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