Composition, Parts and Wholes COMP60421 Sean Bechhofer University - - PowerPoint PPT Presentation

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Composition, Parts and Wholes

COMP60421 Sean Bechhofer University of Manchester sean.bechhofer@manchester.ac.uk

Composition or Aggregation

• Forming an object whole using other objects as parts
• Treating complex things as a single object
• What are the primary composition relationships?
• What inferences can we make?
• What might we have in our representation

languages to support this?

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http://www.flickr.com/photos/hartini/2429653007/

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Parts & wholes: Some examples

• Bristles are part of a toothbrush
• Wheels are part of a shopping trolley
• A car is partly iron
• A cappuccino is partly milk
• A meter is part of a kilometer
• Manchester is part of England
• A tree is part of a forest
• A slice of pie is part of the pie
• A book chapter is part of a book
• Stan Laurel is part of Laurel and Hardy
• These are different kinds of composition, with different characteristics and

properties.

• Confusing them may result in incorrect (or undesirable) inferences.

http://www.flickr.com/photos/aramisfirefly/4585596077

Properties of Composition

• Winston et. al. describe properties of composition
• Configuration/Functionality

– Do the parts bear a functional or structural relationship to one another

• r the object they constitute?

– functional/non-functional

• Homeomerous

– Are the parts the same kind of thing as the whole? – homeomerous/non-homeomerous

• Invariance

– Can the parts be separated from the whole? – separable/inseparable

• We can then discuss combinations of these properties.

– We’ll consider Odell’s classification

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Component-Integral Object

• A configuration of parts within a whole
• Bristles - toothbrush
• Scene - film
• A particular arrangement (not just haphazard)
• If components cease to support the overall pattern then different

associations may arise – Handle ripped from a door of the car.

• No longer a part but a piece

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functional non-homeomeric separable

Material-Object

• Parts can’t be removed
• Capuccino is partly milk
• Bread is partly flour
• Define what objects are made of.
• Component-Integral can be separated

– Car without a door handle still a Car

• Material-Object can’t

– Bread without flour not bread

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functional non-homeomeric non-separable

Portion-Object

• Cf Material-Object, but parts are the same kinds of thing
• Slice of bread is a portion of bread
• meter is part of a kilometer
• A slice of bread is bread. So slices in a loaf are similar
• Portions divided by standard measures

– meter/kilometer – hour/day

• Selective inheritance of properties
• Ingredients of bread are ingredients of slice of bread

– But with different quantities

• Slice, helping, segment, lump, drop etc.

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functional homeomeric separable

Place-Area

• Unlike Portion-Object, pieces cannot be removed
• Manchester part of England
• Peak part of a mountain
• Often between places and locations.
• Pieces similar in nature.

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functional homeomeric non-seperable

Member-Bunch

• No requirement for a particular structural or functional relationship
• Tree part of a Forest
• Employee part of the Union
• Ship part of a Fleet
• Member-Bunch is not subclass!!!

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non-functional non-homeomeric separable

Member-Partnership

• An invariant form of Member-Bunch
• Stan Laurel is part of Laurel and Hardy
• Fred and Ginger are a dancing couple
• Removal of member destroys the partnership

– a different partnership may result

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non-functional non-homeomeric non-seperable

Summary of Odell’s Compositional Relationships

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Functional Homeomeric Seperable Component-Integral

! " !

Material-Object

! " "

Portion-Object

! ! !

Place-Area

! ! "

Member-Bunch

" " !

Member-Partnership

" " "

Non Compositional Relationships

• Topological inclusion

– I am in the lecture theatre

• Classification inclusion

– Catch 22 is a Book – It’s an instance of Book, not a part of it, so not Member-Bunch

• Attribution

– Properties of an object can be confused with composition – Height of a Lighthouse isn’t part of it

• Attachment

– Earrings aren’t part of Ears – Toes are part of Feet – Sometimes attachments are parts, but not always

• Ownership

– A bicycle has wheels – I have a bicycle

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So what?

Transitivity

• We might expect part-whole or composition relationships to behave

transitively. – But this is generally only true with the same kind of composition.

• Engine part of the Car
• Pistons part of the Engine
• Pistons part of the Car
• Sean’s arm part of Sean
• Sean part of School of Computer Science
• Sean’s arm part of School of Computer Science

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X is part of Y, Y is part of Z, thus X is part of Z isPartOf isConstituentOf isPortionOf isMemberOf ...

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Transitivity

• In partonomies, we may want to identify direct parts

– Piston directPartOf Engine; Engine directPartOf Car – Piston is not directPartOf Car, but is a partOf Car

• I want to query for all the direct parts of the Car, but not the direct parts of its

direct parts. – So directPartOf shouldn’t be transitive

• Solution: provide a transitive superproperty
• Queries can use the superproperty to query transitive closure
• Assertions use the direct part of relationship
• A standard ontology design pattern, sometimes referred to as transitive

reduction. Property: isPartOf Characteristics: Transitive Property: isDirectPartOf SubPropertyOf: isPartOf

Aside: Transitivity and Subproperties

• Transitive property R is one s.t.

for any x,y,z, if x R y and y R z, then z R z

• Transitivity is not “inherited” by subproperties.
• Nor is a superproperty of a transitive

property necessarily transitive

16 Property: knows Property: hasFriend SubPropertyOf: knows Characteristics: Transitive Property: hasBestFriend SubPropertyOf: hasFriend

Arthur Beth Charlie Daphne knows: hasFriend: hasBestFriend:

A note on Inverses

• OWL allows us to define inverse relationships
• hasPet / isPetOf
• hasParent / isChildOf
• (x R y) iff (y inv-R x)
• Be careful about what you can infer about inverse relationships
• X SubClassOf (hasPart some Y)

– All X’s have a part which is a Y – Are all Y’s a part of some X?

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Composition

• Composition provides a mechanism for forming an object whole using its

parts

• By considering basis properties if this part-whole relationship, we can

identify different kinds of relationship

• The different relationships then help us in identifying when, for example, we

can (or can’t) apply transitivity.

• Explicitly separating these in our representation can avoid incorrect/invalid

inferences.

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Modelling Family History

COMP60421 Sean Bechhofer sean.bechhofer@manchester.ac.uk With thanks to Robert Stevens

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Family History

• In Week 1, we had an exercise involving Family History data, kindly donated by

Robert Stevens.

• The spreadsheet listed people, along with the occupations they held at particular

times and where the information had been derived from.

• How might we define an ontology to allow us to describe this data?
• What are the classes we need to represent?
• What are the properties or relationships that we need to describe?
• How can we map from the spreadsheet data into some populated ontology?
• What queries can we then ask?

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• Name of Person

– Given Name – Surname – Possibly Married name

• Date of Birth

– If known

• Occupation

– Year – Source – (Additional Notes)

• Sex?

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Modelling in OWL

• Recall that OWL allows us to describe

– Individuals. – Classes (of Individuals). – Relationships between Individuals or Properties of Individuals.

• What are our Individuals here?
• What are the Classes
• What are the Properties/Relations?

Basic Data

• Each Person has

– Given Name – Surname – Date of Birth

• Some Persons (Women) may also have

– Married Surname

• OWL provides Datatype properties that allow us to associate data values with

Individuals. – Strings, numbers etc.

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Occupations

• We are assuming that we have a hierarchy of occupations or roles (not all of the

things that people are listed as doing are necessarily occupations)

• This is a simple taxonomy.
• We might, at some point, be concerned about modelling this more completely, e.g.

through descriptions of the roles, but for current purposes, an asserted hierarchy is fine.

• However, a key question is how we associate people with the occupations/roles

that they are playing.

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Modelling Occupations: Attempt #1

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Class: Person Class: Role Class: Butcher SubClassOf: Role Individual: W.G.Bright Types: Person, Butcher

Modelling Occupations: Attempt #2

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Class: Person Class: Role Class: Butcher SubClassOf: Role ObjectProperty: hasRole Individual: Butcher-1 Types: Butcher Individual: W.G.Bright Types: Person Facts: hasRole Butcher-1

Named and Anonymous Individuals

• OWL allows us to make statements about particular named individuals.
• Fred has a cat called Tibbs.
• We can also refer to anonymous individuals
• Fred has a cat, but we don’t know anything

about it

• This representation of incomplete

information can be useful when we don’t know (or don’t care) about the particular individual.

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Individual: Fred Types: Person Facts: hasPet Tibbs Individual: Tibbs Types: Cat Individual: Fred Types: Person that hasPet some Cat

Modelling Occupations: Attempt #3

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Class: Person Class: Role Class: Butcher SubClassOf: Role ObjectProperty: hasRole Individual: W.G.Bright Types: Person that hasRole some Butcher

Modelling Occupations: Attempt #4

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Class: Person Class: Role Class: Butcher SubClassOf: Role Class: RolePlayed ObjectProperty: playsRole ObjectProperty: hasRole Individual: W.G.Bright Types: Person that playsRole some (RolePlayed that hasRole some Butcher)

Modelling Sex: Male and Female

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• People are Male or Female

– For the purposes of this model we will take a simplified view and not consider transgender/androgeny etc. – Thus every person is either Male or Female and not both

• A. Subclasses of Person Male, Female, with Individuals being asserted as instances
• f those classes
• Disjointness and Covering Axioms
• B. Two distinct Individuals Male, Female with a functional ObjectProperty hasSex

C.Classes Male, Female, a functional ObjectProperty hasSex with Individuals being asserted to be related to anonymous Individuals of those classes.

• Disjointness and Covering Axioms

Modelling Male and Female

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• Incomplete information?
• Extensibility?
• Shared “maleness”?
• Definitions of Man and Woman?
• How can we tell if people are Male
• r Female in the data?

Modelling Dates

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• The data states a date for the occupation

– Start date? – End date? – Some kind of duration?

• A simple approach here is to provide a DatatypeProperty that associates the year

(as an integer) with the RolePlayed instance – An advantage of considering the RolePlayed as an object. – Other data types are available. e.g. for dates

• Limitations?

Provenance

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• It is often important to maintain the provenance of information.

– Where does this fact come from?

• The data includes this for most of the facts

– Census Records – Birth Records – Death Records – Marriage Records

• We can provide additional attributes on the RolePlayed Individuals stating where

the information came from

• As with Sex there are possible choices

– Distinct Individuals: Census, BirthRecord, DeathRecord etc. – Classes for Source types – Classes for Source types with Anonymous Individuals. – Would we ever want to name these?

Named or Anonymous RolePlayed Object?

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Individual: William_George_Bright_1901 Types: playsRole some (RolePlayed and hasRole some Butcher and hasYear value "1929"^^integer and hasSource some MarriageRecord) Individual: James_Bright_1809 Facts: playsRole rolePlayed_001 Individual: rolePlayed_001 Types: RolePlayed and (hasRole some Seaman) Facts: hasYear “1839”^^integer

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Partonomic Relationships and SEP Triples

COMP60421 Sean Bechhofer University of Manchester sean.bechhofer@manchester.ac.uk Material kindly provided by Patrice Seyed, University of Buffalo

Partonomy

• Representing and reasoning on relationships such subclass-of (for

generalization) and part-of (for aggregation) is crucial for medical information systems.

• Is part-of transitive?

– usage must be consistent

• Generally transitive for anatomy

– The femur is part of the leg, leg is part of the body therefore femur is part of the body

• Part-Whole specialization

– inheritance of roles along a part-whole taxonomies

• Fracture of the Femur is a fracture of the Leg
• Representation and Reasoning must handle cases where this sort of

inference does not hold – Transplantation of the Aorta is not a Transplantation of the Heart – Amputation of a Finger is not Amputation of the Hand. – Myocarditis is not an Inflammation of the entire Heart (Pancarditis).

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Part-Whole

• Mechanisms to support this include
• Right Identities

– e.g. GRAIL – hasLocus inherited across isPartOf

• Explicit use of Disjunction

– Heart Disease is a Disease located in Heart or some part of Heart

• Structure-Entire-Part Triples

– Separate classes for the the whole of the parts (S), the whole (E) or parts (P).

• Each approach places different requirements on the representation and

potentially introduce different cognitive complexities.

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Partonomic Relationships

• How can we capture this sort of reasoning while staying within a language

with restricted expressiveness? – ALC does not offer transitive relationships

• S-E-P Triples
• Part-Whole reasoning is implicitly done via subsumption reasoning (when

and where transitive roles/property chains deemed expensive)

• SEP triples implicitly express part-whole relationships between anatomical

entities

• An(other) example of an Ontology Design Pattern

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SEP Triples

• For each entity X, we introduce XS, XE and XP.
• ’S’ stands for a Structure Class

– Individuals that are a part of a “whole” entity – HeartS : a part of the heart or a whole heart

• Two subclasses: ‘E’ and ‘P’ Class for Entire and Part Class

– HeartE : an entire heart – HeartP : a part of a heart

• Use appropriate subclass for definitions

5 HeartS HeartE HeartP

SEP Triples

• SEP triples given for all anatomical entities
• Body

– (BodyS, BodyE, BodyP)

• Heart

– (HeartS, HeartE, HeartP)

• Myocardium

– (MyocardiumS, MyocardiumE, MyocardiumP )

• Connecting axioms

– HeartS SubClassOf: BodyP

• Appropriate class then used for a definition.

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HeartS HeartE HeartP MyocardiumS MyocardiumE MyocardiumP BodyS BodyE BodyP

Entailment

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Entailment based

• n Partonomic

Relationships

Class: Carditis EquivalentTo: Inflammation and (hasLocus some HeartS) Class: Myocarditis EquivalentTo: Inflammation and (hasLocus some MyocardiumS)

HeartS HeartE HeartP MyocardiumS MyocardiumE MyocardiumP Carditis hasLocus Myocarditis hasLocus

Carditis is an inflammation of the Heart or one of its parts Myocarditis is an inflammation of the Myocardium or one of its parts

Non Entailment

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No Entailment!

Class: Pancarditis EquivalentTo: Inflammation and (hasLocus some HeartE) Class: Myocarditis EquivalentTo: Inflammation and (hasLocus some MyocardiumS)

HeartS HeartE HeartP MyocardiumS MyocardiumE MyocardiumP Pancarditis hasLocus Myocarditis hasLocus

Pancarditis is an inflammation of the entire Heart

SEP Triples

• All very nice, but the SEP approach doesn’t actually model the relational

structure of parthood. – Rather partonomic “level” -- whether things are a whole or a part.

• Alternatives introduce explicit relationships, but require additional

expressivity in the representation. – Effect on computational resources needed. – Potential effect on the cognitive complexity

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Alternative 1

• Introduce explicit definitions for the “structure” and “part” entities, using a

new transitive, reflexive relationship partOf

• But, needs transitive properties and disjunction.
• Also, reflexive nature of partOf results in equivalences.

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Class XE Class XS EquivalentTo: XE or (partOf some XE) Class XP EquivalentTo: (partOf some XE)

Alternative 2

• Replace the use of XS with X or (partOf some X)
• Replace the use of XE with X
• Replace axioms of the form XS SubClassOf: YP with

X SubClassOf (partOf some Y)

• Removes the SEP triples altogether, but requires disjunction

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Alternative 3

• Introduce a property properPartof, subproperty of partOf.

– Encapsulates “proper” subparts (e.g. not the whole)

• Replace the use of XS with X or (partOf some X)
• Replace the use of XE with X
• Replace axioms of the form XS SubClassOf: YP with

X SubClassOf (properPartOf some Y)

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Querying Structure

• Finding things that are located in the entire, part or structure

13 Entire Part Structure O hasLocus some HeartE hasLocus some HeartP hasLocus some HeartS A1 hasLocus some HeartE hasLocus some HeartP hasLocus some HeartS A2 hasLocus some Heart hasLocus some (partOf some Heart) hasLocus some (Heart

• r (partOf some Heart))

A3 hasLocus some Heart hasLocus some (properPartOf Heart) hasLocus some (partOf some Heart)

Partonomic Relationships

• SEP triples capture some aspects of part-whole.
• Original SEP approach doesn’t actually model the relational structure of

parthood. – Rather partonomic “level”

• Alternatives do, but require additional expressivity in the representation.

– Effect on computational resources needed.

• Choice may not always be good!

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