1. OWL and Inference: Practical examples Sean Bechhofer
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 (owl:differentFrom and owl:AllDifferent) 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(a:bus+driver complete intersectionOf(a:person
restriction(a:drives someValuesFrom (a:bus))))
Class(a:driver complete intersectionOf(a:person
restriction(a:drives someValuesFrom (a:vehicle))))
Class(a:bus partial a:vehicle)
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 Cat owners have cats as pets
Class(a:cat+owner complete intersectionOf(a:person
restriction(a:has_pet someValuesFrom (a:cat))))
SubPropertyOf(a:has_pet a:likes)
Class(a:cat+liker complete intersectionOf(a:person
restriction(a:likes someValuesFrom (a:cat))))
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(a:driver complete intersectionOf(a:person
restriction(a:drives someValuesFrom (a:vehicle))))
Class(a:driver partial a:adult)
Class(a:grownup complete intersectionOf(a:adult a:person))
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 Sheep only eat grass Grass is a plant
Class(a:sheep partial
restriction(a:eats allValuesFrom (a:grass))
a:animal)
Class(a:grass partial a:plant)
DisjointClasses(unionOf(restriction(a:part_of someValuesFrom (a:animal)) a:animal)
unionOf(a:plant restriction(a:part_of someValuesFrom (a:plant))))
Class(a:vegetarian complete intersectionOf(
restriction(a:eats allValuesFrom (complementOf(restriction(a:part_of someValuesFrom (a:animal)))))
restriction(a:eats allValuesFrom (complementOf(a:animal))) a:animal))
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(a:giraffe partial a:animal
restriction(a:eats allValuesFrom (a:leaf)))
Class(a:leaf partial restriction(a:part_of someValuesFrom (a:tree)))
Class(a:tree partial a:plant)
DisjointClasses(unionOf(restriction(a:part_of someValuesFrom (a:animal)) a:animal)
unionOf(a:plant restriction(a:part_of someValuesFrom (a:plant))))
Class(a:vegetarian complete intersectionOf(
restriction(a:eats allValuesFrom (complementOf(restriction(a:part_of someValuesFrom (a:animal)))))
restriction(a:eats allValuesFrom (complementOf(a:animal))) a:animal))
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 Old ladies must have a pet.
Class(a:old+lady complete
intersectionOf(a:person a:female a:elderly))
Class(a:old+lady partial intersectionOf(
restriction(a:has_pet allValuesFrom (a:cat))
restriction(a:has_pet someValuesFrom (a:animal))))
Class(a:cat+owner complete intersectionOf(a:person
restriction(a:has_pet someValuesFrom (a:cat))))
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(a:cow partial a:vegetarian)
DisjointClasses(unionOf(restriction(a:part_of someValuesFrom (a:animal)) a:animal)
unionOf(a:plant restriction(a:part_of someValuesFrom (a:plant))))
Class(a:vegetarian complete intersectionOf(
restriction(a:eats allValuesFrom (complementOf(restriction(a:part_of someValuesFrom (a:animal)))))
restriction(a:eats allValuesFrom (complementOf(a:animal))) a:animal))
Class(a:mad+cow complete intersectionOf(a:cow
restriction(a:eats someValuesFrom (intersectionOf(restriction(a:part_of someValuesFrom (a:sheep))
a:brain)))))
Class(a:sheep partial a:animal
restriction(a:eats allValuesFrom (a:grass)))
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(a:Daily+Mirror type(owl:Thing))
Individual(a:Mick type(a:male)
value(a:drives a:Q123+ABC)
value(a:reads a:Daily+Mirror))
Individual(a:Q123+ABC type(a:van) type(a:white+thing))
Class(a:white+van+man complete
intersectionOf(a:man restriction(a:drives someValuesFrom (intersectionOf(a:van a:white+thing)))))
Class(a:white+van+man partial restriction(a:reads allValuesFrom (a:tabloid)))
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 he’s 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(a:Spike type(owl:Thing)
value(a:is_pet_of a:Pete))
Individual(a:Pete
type(owl:Thing))
ObjectProperty(a:has_pet domain(a:person) range(a:animal))
ObjectProperty(a:is_pet_of inverseOf(a:has_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 Walt has pets Huey, Dewey and Louie
Individual(a:Walt type(a:person)
value(a:has_pet a:Huey)
value(a:has_pet a:Louie)
value(a:has_pet a:Dewey))
Individual(a:Huey type(a:duck))
Individual(a:Dewey type(a:duck))
Individual(a:Louie type(a:duck))
DifferentIndividuals(a:Huey a:Dewey a:Louie)
Class(a:animal+lover complete intersectionOf(a:person restriction(a:has_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 don’t 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 Minnie is elderly, female and has a pet, Tom
Individual(a:Minnie type(a:female) type(a:elderly) value(a:has_pet a:Tom))
Individual(a:Tom type(owl:Thing))
ObjectProperty(a:has_pet domain(a:person) range(a:animal))
Class(a:old+lady complete
intersectionOf(a:person a:female a:elderly))
Class(a:old+lady partial intersectionOf(
restriction(a:has_pet allValuesFrom (a:cat))
restriction(a:has_pet someValuesFrom (a:animal))))
Minnie is elderly, female and has a pet, Tom
Minnie must be a person
Minnie is be an old lady
All Minnie’s 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 Morgan’s 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(some (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(a:Person partial)
Class(a:Academic partial a:Person)
Class(a:Happy partial a:Person)
Class(a:Lecturer partial a:Academic)
Class(a:Professor partial a:Academic)
Class(a:Student partial a:Person)
ObjectProperty(a:hasFriend)
ObjectProperty(a:isFriendOf
inverseOf(a:hasFriend))
DisjointClasses(a:Student a:Academic)
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(a:arthur type(a:Student) type(a:Happy))
Individual(a:bob type(a:Student) type(complementOf(a:Happy)))
Individual(a:charlie type(a:Professor) type(a:Happy))
Individual(a:diane type(a:Professor) type(complementOf(a:Happy)))
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(a:Patricia value(a:hasFriend a:Arthur))
Individual(a:Quentin value(a:hasFriend a:Charlie) value(a:hasFriend a:Bob))
Individual(a:Richard value(a:hasFriend a:Charlie))
Individual(a:Roberta value(a:hasFriend a:Bob))
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(all (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 don’t 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 So Xanthe is now an instance of
Individual(a:Xanthe type(restriction(a:hasFriend cardinality(1))) value(a:hasFriend a:Arthur))
Individual(a:Yolanda type(restriction(a:hasFriend cardinality(2))) value(a:hasFriend a:Charlie) value(a:hasFriend a:Bob))
Individual(a:Zaphod type(restriction(a:hasFriend cardinality(1))) value(a:hasFriend a:Charlie))
Individual(a:Zeke type(restriction(a:hasFriend cardinality(1))) value(a:hasFriend a:Bob))
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 William has no friends. But William is an instance of:
Individual(a:William type(restriction(a:hasFriend 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 he’s an instance of
restriction(all friends X)
For any X (even Nothing).
Universal quantification over an empty collection is trivially true.

28. UNA revisited Fred has exactly one friend.
Individual(a:Fred
type(restriction(a:hasFriend cardinality(1)))
value(a:hasFriend a:John)
value(a:hasFriend a:Jack))
Individual(a:John)
Individual(a:Jack)
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.