Logic …

Disjoint PropertiesAs 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 disjointIf 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

Womanobject

Person RyanT

subject

hasMotherAshleyThasFather

Disjoint Properties …Let’s declare the meltsTo and crystallizesTo, or linearAttitude and

planarAttitude, as pairs of disjoint propertiesAnd 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:

Rockobject

Magmasubject

crystallizesTorhyolite5meltsTomagma2

SPO: Magma crystallizesTo Rock

SP’O Magma meltsTo Rock

magma2 crystallizesTo rhyolite5magma2 meltsTo andesite5

Individual (Assertional) AxiomsIndividual 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 IndividualsEquality 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 RestrictionsDomain and range are used to infer membership of instances to

certain classesDomain and range define the subject (source) and object (target) of

a property (p), respectivelyCountry hasCapital CityCountry is the domain and City is range for hasCapital

Mineral ageDate IsotopicAgeThe domain for the ageDate property in the above statement is the

Mineral class, and its range is the IsotopicAge classAll 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

IsotopicAgeMineralageaMica

ageDate

Target class orRange

Source class orDomain

Local Property RestrictionProperty 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:hasValueowl:selfRestriction

owl : allValuesFrom, P.C Provides a value restriction for the range of a propertyT 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 rangei.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 Ci.e., the range for P is class C

A Ca b

P

Range

P.C ExampleThe set of individuals that are related by property P only to

individuals of class ClocatedIn.Nevada, is the set of individuals located only in

Nevada, and not anywhere elseNevadaCity City locatedIn.Nevada

The connective reads: ‘for all, if any’, meaning that the occurrence can be many or zeroTo 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

City Nevadaa blocatedIn

Range

IgneousRockMagma

crystallizeFrom

RangeRock

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 Ca b

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., LavaC

e.g., Minerala b

crystallize

P

LavaC

e.g., Mineralb

crystallize

P

CoolingLavaa

Range

Range

Owl : hasValue & owl : selfRestrictionThe 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)

Rock Minerala b

1 hasMineral

Qualified Number RestrictionThe 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).Minerali.e., Rock has one or more mineralsCar ( 1 hasWheel).Wheel

( n P) owl : maxCardinalityThe 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

SAtomicShellElectron

b

2 hasElectron

Shella

( n P).C and ( n P).CThe 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 : cardinalityThe owl : cardinality can be expressed as the intersection

of the owl : maxCardinality and owl : minCardinality

For this case, there are exactly n propertiesFor example, monomineralic rock is a rock with exactly

one kind of mineral:

MonomineralicRock Rock ( 1 hasMineral).Mineral ( 1 hasMineral).Mineral

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