Why did that happen?

1
Why did that happen?

• OWL and Inference Practical examples
• Sean Bechhofer

2
Inference and Classes

• The people and pets example ontology contains a
  number of classes and properties intended to
  illustrate particular aspects of reasoning in
  OWL.
• We can make inferences about relationships
  between those classes, in particular subsumption
  between classes
• Recall that A subsumes B when it is the case that
  any instance of B must necessarily be an instance
  of A.

A
B
3
Inference and Individuals

• In addition, the model contains a number of
  individuals
• We can make inferences about the individuals, in
  particular inferring that particular individuals
  must be instances of particular classes.
• This can be because of subsumption relationships
  between classes, or because of the relationships
  between individuals.
• The following slides examine some of the
  inferences we can make in the example model and
  discuss why those inferences come about.

4
Unique Name Assumption

• The Unique Name Assumption (UNA) says that any
  two individuals with different names are
  different individuals.
• OWL semantics does not make the UNA
• There are mechanisms in the language
  (owldifferentFrom and owlAllDifferent) that
  allow us to assert that individuals are
  different.
• However, many DL reasoners (including the one we
  use here) assume UNA.
• For the following examples, there is a tacit
  assumption that all individuals are asserted to
  be distinct.

5
Classes

• The following examples illustrate reasoning with
  classes and class definitions.
• We show examples of
• Inferred Subsumptions
• Inconsistency

6
Bus Drivers are Drivers
Class(abusdriver complete intersectionOf(aperso
n restriction(adrives someValuesFrom
(abus)))) Class(adriver complete
intersectionOf(aperson restriction(adrives
someValuesFrom (avehicle)))) Class(abus
partial avehicle)

• A bus driver is a person that drives a bus
• A bus is a vehicle
• A bus driver drives a vehicle, so must be a
  driver
• The subclass is inferred due to subclasses being
  used in existential quantification.

7
Cat Owners like Cats
Class(acatowner complete intersectionOf(aperson
restriction(ahas_pet someValuesFrom
(acat)))) SubPropertyOf(ahas_pet
alikes) Class(acatliker complete
intersectionOf(aperson restriction(alikes
someValuesFrom (acat))))

• Cat owners have cats as pets
• has pet is a subproperty of likes, so anything
  that has a pet must like that pet.
• Cat owners must like a cat.
• The subclass is inferred due to a subproperty
  assertion

8
Drivers are Grown Ups
Class(adriver complete intersectionOf(aperson
restriction(adrives someValuesFrom
(avehicle)))) Class(adriver partial
aadult) Class(agrownup complete
intersectionOf(aadult aperson))

• Drivers are defined as persons that drive cars
  (complete definition)
• We also know that drivers are adults (partial
  definition)
• So all drivers must be adult persons (e.g.
  grownups)
• An example of axioms being used to assert
  additional necessary information about a class.
  We do not need to know that a driver is an adult
  in order to recognize one, but once we have
  recognized a driver, we know that they must be
  adult.

9
Sheep are Vegetarians
Class(asheep partial restriction(aeats
allValuesFrom (agrass)) aanimal) Class(agras
s partial aplant) DisjointClasses(unionOf(restri
ction(apart_of someValuesFrom (aanimal))
aanimal)
unionOf(aplant restriction(apart_of
someValuesFrom (aplant)))) Class(avegetarian
complete intersectionOf( restriction(aeats
allValuesFrom (complementOf(restriction(apart_of
someValuesFrom (aanimal)))))
restriction(aeats allValuesFrom
(complementOf(aanimal))) aanimal))

• Sheep only eat grass
• Grass is a plant
• Plants and parts of plants are disjoint from
  animals and parts of animals
• Vegetarians only eat things which are not animals
  or parts of animals
• Note the complete definition, which means that we
  can recognise when things are vegetarians.

10
Giraffes are Vegetarians
Class(agiraffe partial aanimal
restriction(aeats allValuesFrom (aleaf)))
Class(aleaf partial restriction(apart_of
someValuesFrom (atree))) Class(atree partial
aplant) DisjointClasses(unionOf(restriction(apa
rt_of someValuesFrom (aanimal)) aanimal)
unionOf(aplant
restriction(apart_of someValuesFrom
(aplant)))) Class(avegetarian complete
intersectionOf( restriction(aeats
allValuesFrom (complementOf(restriction(apart_of
someValuesFrom (aanimal)))))
restriction(aeats allValuesFrom
(complementOf(aanimal))) aanimal))

• Giraffes only eat leaves
• Leaves are parts of trees, which are plants
• Plants and parts of plants are disjoint from
  animals and parts of animals
• Vegetarians only eat things which are not animals
  or parts of animals
• Similar to the previous example with the
  additional inference provided by the existential
  restriction in the definition of leaf

11
Old Ladies own Cats
Class(aoldlady complete intersectionOf(apers
on afemale aelderly)) Class(aoldlady partial
intersectionOf( restriction(ahas_pet
allValuesFrom (acat)) restriction(ahas_pet
someValuesFrom (aanimal)))) Class(acatowner
complete intersectionOf(aperson
restriction(ahas_pet someValuesFrom (acat))))

• Old ladies must have a pet.
• All pets that old ladies have must be cats.
• An old lady must have a pet that is a cat.
• An example of the interaction between an
  existential quantification (asserting the
  existence of a pet) and a universal
  quantification (constraining the types of pet
  allowed).
• This also illustrates that an ontology is one
  view on the world you may disagree with my
  modelling but I am being explicit about it.

12
Mad Cows are inconsistent
Class(acow partial avegetarian) DisjointClasses
(unionOf(restriction(apart_of someValuesFrom
(aanimal)) aanimal)
unionOf(aplant restriction(apart_of
someValuesFrom (aplant)))) Class(avegetarian
complete intersectionOf( restriction(aeats
allValuesFrom (complementOf(restriction(apart_of
someValuesFrom (aanimal)))))
restriction(aeats allValuesFrom
(complementOf(aanimal))) aanimal)) Class(amad
cow complete intersectionOf(acow
restriction(aeats someValuesFrom
(intersectionOf(restriction(apart_of
someValuesFrom (asheep))

abrain))))) Class(asheep partial
aanimal restriction(aeats allValuesFrom
(agrass)))

• Cows are naturally vegetarians
• A mad cow is one that has been eating sheeps
  brains
• Sheep are animals
• Thus a mad cow has been eating part of an animal,
  which is inconsistent with the definition of a
  vegetarian.

13
Individuals

• The following examples illustrate reasoning with
  individuals.
• We look at why particular individuals can be
  inferred to be members of particular classes.

14
The Daily Mirror is a Tabloid
Individual(aDailyMirror type(owlThing))
Individual(aMick type(amale) value(adrives
aQ123ABC) value(areads aDailyMirror))
Individual(aQ123ABC type(avan)
type(awhitething)) Class(awhitevanman
complete intersectionOf(aman
restriction(adrives someValuesFrom
(intersectionOf(avan awhitething)))))
Class(awhitevanman partial restriction(areads
allValuesFrom (atabloid)))

• Mick drives a white van
• Mick must be a person and an adult, so he is a
  man
• Mick is a man who drives a white van, so hes a
  white van man
• A white van man only reads tabloids, and Mick
  reads the Daily Mirror, thus the Daily Mirror
  must be a tabloid
• Here we see interaction between complete and
  partial definitions plus a universal
  quantification allowing an inference about a role
  filler.

15
Pete is a Person, Spike is an Animal
Individual(aSpike type(owlThing)
value(ais_pet_of aPete)) Individual(aPete
type(owlThing)) ObjectProperty(ahas_pet
domain(aperson) range(aanimal)) ObjectProperty(
ais_pet_of inverseOf(ahas_pet))

• Spike is the pet of Pete
• So Pete has pet Spike
• Pete must be a Person
• Spike must be an Animal
• Here we see an interaction between an inverse
  relationship and domain and range constraints on
  a property.

16
Walt loves animals
Individual(aWalt type(aperson)
value(ahas_pet aHuey) value(ahas_pet
aLouie) value(ahas_pet aDewey)) Individual(a
Huey type(aduck)) Individual(aDewey
type(aduck)) Individual(aLouie
type(aduck)) DifferentIndividuals(aHuey
aDewey aLouie) Class(aanimallover complete
intersectionOf(aperson restriction(ahas_pet
minCardinality(3))))

• Walt has pets Huey, Dewey and Louie
• Huey Dewey and Louie are all distinct individuals
• Walt has at least three pets and is thus an
  animal lover.
• Note that in this case, we dont actually need to
  include person in the definition of animal lover
  (as the domain restriction will allow us to draw
  this inference).

17
Tom is a Cat
Individual(aMinnie type(afemale)
type(aelderly) value(ahas_pet
aTom)) Individual(aTom type(owlThing)) Object
Property(ahas_pet domain(aperson)
range(aanimal)) Class(aoldlady complete
intersectionOf(aperson afemale
aelderly)) Class(aoldlady partial
intersectionOf( restriction(ahas_pet
allValuesFrom (acat)) restriction(ahas_pet
someValuesFrom (aanimal))))

• Minnie is elderly, female and has a pet, Tom
• Minnie must be a person
• Minnie is be an old lady
• All Minnies pets must be cats.
• Here the domain restriction gives us additional
  information which then allows us to infer a more
  specific type. The universal quantification then
  allows us to infer information about the role
  filler.

18
Break
19
General Patterns of Inference

• The next slides and examples illustrate some of
  the interactions between quantification and
  boolean operators.
• Distribution and De Morgans laws
• We also discuss the ramifications of open world
  reasoning.

20
Distribution Rules

• Distribution rules for existential quantification
  are similar to those for conjunction and
  disjunction
• restriction(some p unionOf(A B) ?
  unionOf(restriction(some (p A)) restriction(some
  (p B)))
• There are also inferences that can be drawn
  between expressions involving existential
  quantification and conjunction
• restriction(some p intersectionOf(A B) ?
  intersectionOf(restriction(some (p A))
  restriction(some (p B)))
• intersectionOf(restriction(some (p A))
  restriction(some (p B))) ? unionOf(restriction(s
  ome (p A)) restriction(some (p B)))
• intersectionOf(restriction(some (p A))
  restriction(some (p B))) ? restriction(some p
  unionOf(A B)
• The above inferences are not logical
  equivalences.
• Union is distributive in existentials,
  intersection is not.

21
A Simple Ontology
Class(aPerson partial) Class(aAcademic partial
aPerson) Class(aHappy partial
aPerson) Class(aLecturer partial
aAcademic) Class(aProfessor partial
aAcademic) Class(aStudent partial
aPerson) ObjectProperty(ahasFriend) ObjectPrope
rty(aisFriendOf inverseOf(ahasFriend)) Disjoin
tClasses(aStudent aAcademic)

• We have some basic classes, Person, Academic,
  Professor and Student.
• There is also a subclass of Happy Persons.
• Students and Academics are disjoint.

22
Example Individuals
Individual(aarthur type(aStudent)
type(aHappy)) Individual(abob type(aStudent)
type(complementOf(aHappy))) Individual(acharlie
type(aProfessor) type(aHappy))
Individual(adiane type(aProfessor)
type(complementOf(aHappy)))
Professor
Charlie
Diane
Happy
Student
Arthur
Bob

• Note we can infer that Professors and Students
  are disjoint due to the disjointness axiom
  concerning Academics.

23
Example Individuals
Individual(aPatricia value(ahasFriend
aArthur)) Individual(aQuentin value(ahasFriend
aCharlie) value(ahasFriend aBob)) Individual(a
Richard value(ahasFriend aCharlie)) Individual(a
Roberta value(ahasFriend aBob))

• These individuals now provide witnesses for the
  non-equivalence of definitions.
• restriction(some p intersectionOf(A B ) A
• intersectionOf(restriction(some (p A))
  restriction(some (p B))) B
• Quentin has a friend who is Happy (Charlie) and a
  friend who is a Student (Bob). Quentin is not
  known to have a friend who is both Happy and a
  Student.
• We are able to infer that Quentin is an instance
  of A, but not of B.

24
Distribution Rules

• Distribution rules for universal quantification
  are similar to those for and/or
• restriction(all p intersectionOf(A B) ?
  intersectionOf(restriction(all (p A))
  restriction(all (p B)))
• There are also inferences that can be drawn
  between expressions involving universal
  quantification and disjunction
• intersectionOf(restriction(all (p A))
  restriction(all (p B))) ? unionOf(restriction(al
  l (p A)) restriction(all (p B)))
• restriction(all p intersectionOf(A B) ?
  unionOf(restriction(all (p A)) restriction(all (p
  B)))
• unionOf(restriction(all (p A)) restriction(all (p
  B))) ? restriction(all p unionOf(A B)
• Again, the above inferences are not logical
  equivalences.
• Intersection is distributive in existentials,
  union is not.

25
Closed and Open Worlds

• In our previous examples, we find that Patricia
  is not an instance of
• restriction(all friends intersectionOf(Student
  Happy))
• This is due to the open world assumption (OWA).
• We cannot assume that if we dont know something
  then it is false.
• In this example, there may be other friends that
  Patricia has that are not Students.
• Reasoning in DLs is monotonic
• If we know that x is an instance of A, then
  adding more information to the model cannot cause
  this to become false.

26
Example Individuals
Individual(aXanthe type(restriction(ahasFriend
cardinality(1))) value(ahasFriend aArthur))
Individual(aYolanda type(restriction(ahasFriend
cardinality(2))) value(ahasFriend aCharlie)
value(ahasFriend aBob)) Individual(aZaphod
type(restriction(ahasFriend cardinality(1)))
value(ahasFriend aCharlie)) Individual(aZeke
type(restriction(ahasFriend cardinality(1)))
value(ahasFriend aBob))

• The above individuals have extra cardinality
  constraints that close the roles and allow us to
  make inferences about all the friends that they
  have.
• So Xanthe is now an instance of
• restriction(all friends intersectionOf(Student
  Happy))

27
Billy No Mates
Individual(aWilliam type(restriction(ahasFriend
cardinality(0))))

• William has no friends.
• But William is an instance of
• restriction(all friends intersectionOf(Student
  Happy))
• restriction(all friends unionOf(Student Happy))
• In fact hes an instance of
• restriction(all friends X)
• For any X (even Nothing).
• Universal quantification over an empty collection
  is trivially true.

28
UNA revisited
Individual(aFred type(restriction(ahasFriend
cardinality(1))) value(ahasFriend aJohn)
value(ahasFriend aJack)) Individual(aJohn)
Individual(aJack)

• Fred has exactly one friend.
• Fred has a friend John and friend Jack
• This allows us to infer that John and Jack must
  be the same person.
• Recall that this inference would not drawn by
  many DL reasoners (including the current
  implementation of RACER) instead in this case
  an inconsistency will be deduced.