Logic …. Disjoint Properties As for disjoint classes, two properties can be disjoint (owl :...
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Transcript of Logic …. Disjoint Properties As for disjoint classes, two properties can be disjoint (owl :...
![Page 1: Logic …. Disjoint Properties As for disjoint classes, two properties can be disjoint (owl : propertyDisjointWith) Property p and p’ are disjoint if no.](https://reader035.fdocuments.us/reader035/viewer/2022081513/56649f315503460f94c4bd8b/html5/thumbnails/1.jpg)
Logic …
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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
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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
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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)
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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”)
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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”
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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
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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
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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
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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
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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
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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
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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
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…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
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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
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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
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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
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( 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
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( 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
=
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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
1 11