OWL and Inference: Practical examples Sean Bechhofer

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1 Why did that happen? OWL and Inference: Practical examples Sean Bechhofer Syntax The examples in this session are presented using both DL syntax and Manchester Syntax. You should be familiar (and comfortable) with the various alternative syntactic presentations: Manchester Syntax DL Syntax FOL 1

2 Syntax Examples OWL: Class: TA Academic, Student DL: TA v Academic u Student ΦΟΛ: x.ta(x) Academic(x) Student(x) OWL: ObjectProperty: employs Range: Person DL: Top v employs.person ΦΟΛ: x.( y.employs(y,x)) Person(x) OWL: Class: CarOwner owns some Car DL: CarOwner owns.car ΦΟΛ: x.carowner(x) ( y.owns(x,y) Car(y)) 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 2

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. 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 in the past have assumed UNA. 3

4 Classes The following examples illustrate reasoning with classes and class definitions. We show examples of Inferred Subsumptions Inconsistency Bus Drivers are Drivers Class: bus_driver that drives some bus Class: driver that drives some vehicle adult Class: bus vehicle A bus driver is a 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. 4

5 Cat Owners like Cats Class: cat_owner that has_pet some cat ObjectProperty: has_pet SubPropertyOf: likes Class: cat_liker that likes some 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 Drivers are Grown Ups Class: driver that drives some vehicle adult Class: grownup adult and Drivers are defined as s that drive cars (complete definition) We also know that drivers are adults (partial definition) So all drivers must be adult s (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. 5

6 Sheep are Vegetarians Class: sheep animal, eats only grass Class: grass plant Disjoint: animal or (part_of some animal), plant or (part_of some plant) Class: vegetarian animal that eats only (not (animal)) and eats only (not (part_of some 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. Giraffes are Vegetarians Class: giraffe animal, eats only leaf Class: leaf part_of some tree Class: tree plant Disjoint: animal or (part_of some animal), plant or (part_of some plant) Class: vegetarian animal that eats only (not (animal)) and eats only (not (part_of some 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 6

7 Old Ladies own Cats Class: old_lady elderly and female and has_pet some animal and has_pet only cat Class: cat_owner that has_pet some 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. Class: cow vegetarian Mad Cows are inconsistent Disjoint: animal or (part_of some animal), plant or (part_of some plant) Class: vegetarian animal that eats only (not (animal)) and eats only (not (part_of some animal)) Class: mad_cow cow that eats some (brain that part_of some sheep) Class: sheep animal, eats only 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. 7

8 Individuals The following examples illustrate reasoning with individuals. We look at why particular individuals can be inferred to be members of particular classes. The Daily Mirror is a Tabloid Individual: Daily_Mirror owl:thing Individual: Mick male Facts: reads Daily_Mirror, drives Q123ABC Individual: Q123ABC van, white_thing Class: white_van_man man that drives some (van and white_thing) reads only tabloid Mick drives a white van Mick must be a 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. 8

9 Pete is a Person, Spike is an Animal Individual: Spike owl:thing Facts: is_pet_of Pete Individual: Pete owl:thing ObjectProperty: has_pet Domain: Range: animal InverseOf: is_pet_of SubPropertyOf: likes 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. Walt loves animals Individual: Walt Facts: has_pet Dewey, has_pet Huey, has_pet Louie Individual: Huey duck DifferentFrom: Dewey,, Louie, Individual: Dewey duck DifferentFrom: Huey,, Louie, Individual: Louie duck DifferentFrom: Dewey,, Huey, Class: animal_lover that has_pet min 3 owl:thing 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 in the definition of animal lover (as the domain restriction will allow us to draw this inference). 9

10 Tom is a Cat Individual: Minnie elderly, female Facts: has_pet Tom Individual: Tom owl:thing ObjectProperty: has_pet Domain: Range: animal InverseOf: is_pet_of SubPropertyOf: likes Class: old_lady elderly and female and has_pet some animal and has_pet only cat Minnie is elderly, female and has a pet, Tom Minnie must be a 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. General Patterns of Inference The next slides and examples illustrate some of the interactions between quantification and boolean operators. De Morgan s laws and Distribution We also discuss the ramifications of open world reasoning. 10

11 De Morgan s Laws De Morgan s laws tell us how the various boolean and quantification operators relate to one another. The laws hold because of the way in which the interpretations are defined along with the corresponding laws in the underlying set theory. Thus we have: ( A) A (A u B) A t B (A t B) A u B R.A R. A R.A R. A De Morgan s Laws OWL has boolean operators and, or, not Due De Morgan s laws, we can see that these are not all necessary. Operators such as union or intersection can be defined in terms of the others. A u B (A u B) ( A t B) Or in OWL: (A and B) not((not A) or (not B)) Quantification can also be rewritten: R.A R.A ( R. A) Or in OWL: R only B not (R some (not B)) 11

12 Interpretation Example Revisited Δ = {v, w, x, y, z} A I = {v, w, x} B I = {x, y} R I = {(v, w), (v, x), (y, x), (x, z)} v w A I A u B = ( A t B) 8 R B = ( R. B) x y z B I Distribution Rules Distribution rules relate and and or: A u (B t C) (A u B) t (A u C) A t (B u C) (A t B) u (A t C) Or in OWL A and (B or C) (A and B) or (A and C) A or (B and C) (A or B) and (A or C) 12

13 Distribution Rules Distribution rules for existential quantification are similar to those for conjunction and disjunction: p some (A or B) (p some A) or (p some B) There are also inferences that can be drawn between expressions involving existential quantification and conjunction: p some (A and B) (p some A) and (p some B) (p some A) and (p some B) (p some A) or (p some B) (p some A) and (p some B) p some (A or B) The above inferences are not logical equivalences. Union is distributive in existentials, intersection is not. A Simple Ontology Class: :Professor :Academic Class: :Academic :Person DisjointWith: :Student Class: :Lecturer :Academic Class: :Student :Person DisjointWith: :Academic Class: :Happy :Person Class: :Person ObjectProperty: :isfriendof InverseOf: :hasfriend ObjectProperty: :hasfriend InverseOf: :isfriendof We have some basic classes, Person, Academic, Professor and Student. There is also a subclass of Happy Persons. Students and Academics are disjoint. 13

14 Example Individuals Individual: arthur Happy, Student Individual: bob Student, not Happy Individual: charlie Professor, Happy Individual: diane Professor, not Happy Professor Charlie Diane Student Bob Arthur Happy Note we can infer that Professors and Students are disjoint due to the disjointness axiom concerning Academics. Example Individuals Individual: patricia Facts: hasfriend arthur Individual: quentin Facts: hasfriend charlie, hasfriend bob Individual: richard Facts: hasfriend charlie Individual: roberta Facts: hasfriend bob These individuals now provide witnesses for the non-equivalence of definitions. (p some A) and (p some B) [A] p some (A and 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]. 14

15 Distribution Rules Distribution rules for universal quantification are similar to those for and/or: p only (A and B) (p only A) and (p only B) There are also inferences that can be drawn between expressions involving universal quantification and disjunction: (p only A) and (p only B)) (p only A) or (p only B) p only (A and B) (p only A) or (p only B) (p only A) or (p only B)) p only (A or B) Again, the above inferences are not logical equivalences. Intersection is distributive in existentials, union is not. Closed and Open Worlds In our previous examples, we find that Patricia is not an instance of hasfriend only (Student and 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. 15

16 Example Individuals Individual: xanthe hasfriend exactly 1 owl:thing Facts: hasfriend arthur Individual: yolanda hasfriend exactly 2 owl:thing Facts: hasfriend bob, hasfriend charlie Individual: zaphod hasfriend exactly 1 owl:thing Facts: hasfriend charlie Individual: zeke hasfriend exactly 1 owl:thing Facts: hasfriend 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 hasfriend only intersectionof(student Happy) Billy No Mates Individual: william hasfriend exactly 0 owl:thing William has no friends. But William is an instance of: hasfriend only (Student and Happy) hasfriend only (Student or Happy) In fact he s an instance of hasfriend only X For any X (even Nothing!). Universal quantification over an empty collection is trivially true. 16

17 (Lack of) UNA revisited Individual: fred(a:fred hasfriend exactly 1 owl:thing Facts: hasfriend john, hasfriend jack Individual: john Individual: 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. 17