1 ©Amit Mitra & Amar Gupta RELATIONSHIPS Reading Assignment Supplementary module 5.

1©Amit Mitra & Amar Gupta

RELATIONSHIPS

Reading Assignment

Supplementary module 5


A RELATIONSHIP IS AN INTERACTION BETWEEN OBJECTS

product customer retailer

is sold to.. buys thru...

product customer retailerXX

CARTESIAN PRODUCT

product 1 customer 1 retailer 1




product 2 customer 1 retailer 1…...

Relationship Class

Relationship Instances

Order of relationship = No. of classes involvedDegree of relationship = No. Of object instances involved

EXAMPLE OF A RELATIONSHIPS BETWEEN OBJECTS

(3rd order, 3rd degree relationship)


object

object 1 object 2Instance of relationship

OBJECT CLASSA

object 1 object 2Instances of relationship

inverse of relationship

OBJECT CLASSB

OBJECT CLASSA

OBJECT CLASSB

Instance of inverse of relationship

object 4object 1 Instances of relationship

object 2

OBJECT CLASSB

OBJECT CLASSA



BIJECTIVE RELATIONSHIP(1 to 1)

INJECTIVE RELATIONSHIP(1 to many)

SURJECTIVE RELATIONSHIP(many to 1)

MANY to MANY

object 3Instance of inverse of

relationship

CAREFULL!!! (2nd order, 2nd degree relationship)

•Cardinality or Cardinality Ratios may be constrained

– Usually cardinality ratios in both directions are specified


PERSON

ORGANIZATION

many to many

manytomanyresolution

PERSON

ORGANIZATION

Personemployed InOrganization

0 to many

Employ0 to many

may b

e emp

loyed b

y 0 or more

[emp

loy 0 or more]

MANY-TO-MANY RESOLUTION


object instance 1

OBJECT CLASS

Object instance 2

Instance of reflexive relationship

Instance of irreflexive

relationship

RECURSIVE RELATIONSHIPS

(homogenous facts)

Idempotent relationship must loop back to the same object instance(1st order, 1st degree relationship)

(1st order, 2nd degree relationship)

(1st order relationships, degree unknown)

• Symmetrical: Inverse must be same as relationship– Eg: Person is relative of Person

• Asymmetrical: Inverse cannot be same as relationship– Eg: Person is parent of Person– Cartesian Product

• Antisymmetrical (for reflexive relationships only): Inverse cannot be same as relationship unless it loops back to the same object instance, in which case the relationship must be the same in both directions.– Eg: Arithmetic subtraction– Poymorphism of patterns of separation

SYMMETRY

Subtype of


Instance

Class

OBJECT

Instance

Class

OBJECTClass

Instance

OBJECT

INFORMATION

RELATIONSHIPRECURSIVEIRREFLEXIVE

Polymorphisms of Relationships

NON-RECURSIVE


RecursiveRelationship

Subtype of

IdempotentRelationship

Subtypeof

Reflexive Relationship

Asymmetrical Relationship

Subtype of

AntisymmetricalRelationship

SymmetricalSubset of subset of

Exhaustivepartition

Subtypeof

Symmetrical Relationship

Eg: •involve•Locate•relative of

Eg: •Part of•parent of

Polymorphisms of Relationships (2)

Eg: “Help” in Person help

PersonEg: “Self

Help”

Idempotent relationship

will always be symmetrical

IrreflexiveRelationship

Non-recursiveRelationship

Polymorphismof

Polymorphismof

Subtype of (inherited from recursive relationship)

Eg: Civil War

Eg: Distinct Parts of the nation at war with itself

Where relationships begin


HIGHER ORDER RELATIONSHIP



possible ways in which minimum and maximumoccurrence (cardinality) constraints may be defined

for a three-way relationship

m..n

m..n

m..n



m..n

Instances of this combination must be unique, otherwise it will not be an

object class

The symmetry of each combination must be considered

The length of the tuple is determined by its degree

Combinations may be recursive

Cardinality of polymorphisms of a relationship cannot be less constrained

than its parent (it may be more constrained)

Cardinality = Nil is the same as the exclusion constraint


Mutually inclusive and exclusive relationships

INSURANCECOVERAGE

ASSET

PERSON

cover

cover

MUTUALLYEXCLUSIVERELATIONSHIPS

X

= cover

Person Asset

Degree = 1cover

Insurance Coverage

=

co

ver..

Degree =

1

Person

Asset

InsuranceCoverage

PERSON

CAR

CARINSURANCE

own

own

MUTUALLYINCLUSIVERELATIONSHIPS

= own

Cardinality = 0 or more

own

Person

CarCar

Insurance

Degree = 2

=

Card

inality =

0 or more

ow

n..Person

car

CarInsurance

Degree =

2

PERSON

CAR

WHEEL

owns

SUBSET OF

owns

=

owns

Cardinality cannot exceed cardinality of Person owns Wheel

owns Person

Car Wheel

=

Ow

n..

Person

Wheel

Car

CardinalityCannotexceed

Mutual ExclusionIf one relationship exists, the other(s) cannot

Mutual InclusionIf any one relationship exists, the other(s) also must SubsettingIf one relationship exists, the other must, but not vice-versa

A subtype must stay within the lawful state space of its parent


Some additional information on cardinality constraints• Instance level constraints

–Eg: upper bound on cardinality ratio of Person lives in House

–Each house might have a different capacity

• Upper bound constraints on cardinality of relationships creates the concept of Capacity

• “Instance of” is a subtype of the subtyping relationship constrained to a single member

DO

MA

IN

(Exh

au

stive) Qu

alitative/Q

uan

titativ

e Par

tition

OB

JEC

T

Qu

alita

tiveA

ttribu

te

Each Objec t is anaggregation of one or more

Each Attribute is a Property of a single

Attrib

ute

Each Q

ualitative Attribute

is a Subtype of an Attribute

Each Q

uantitative Attribute

is a Subtype of an Attribute

FO

RM

AT

UN

ITO

FM

EA

SU

RE

is exp

ressed b

y 1

or m

any

[expr

ess]

con

vert to

0 or 1

[con

vert from]

con

vert to

0 or 1

[con

vert from]

Each Quantitative A ttributeis a Subtype of a

QU

AL

ITA

TIV

E D

OM

AIN

QU

AN

TIT

AT

IVE

DO

MA

INis ex

pressed

by

1 o

r man

y

is expressed

in 1

or m

any

Qu

antita

tiveA

ttribu

te

Instan

ceId

entifier

Each Qualitative A ttributeis a Subtype of a

(Exh

au

stive) Ord

inal/N

om

inal P

artitio

n

Ord

ina

lD

oma

inN

omin

al

Dom

ain

Each Ordinal Attributeis a Subtype of a

Each Nominal Attribute is a Subtype of a

Ord

ina

lA

ttribu

teN

omin

al

Attrib

ute

is aS

ub

typeof

is aS

ub

type

of

is aS

ub

typeof

Ea

ch ob

ject m

usth

ave ex

actly 1

[Iden

tifiesexa

ctly 1]

is aS

ub

type

of

is aS

ub

type

of

is aS

ub

typeof

(Ex

hau

stive) D

ifference/R

atio

Sca

led P

artitio

n

Ord

ina

lD

oma

inN

omin

al

Dom

ain

Differen

ceS

caled

Attrib

ute

Ra

tioS

caled

Attrib

ute

is aS

ub

typeof

is aS

ub

type

of

Each Difference ScaledA ttribute is a Subtype of a

Eac h Ratio Scaled A ttributeis a Subtype of a

is aS

ub

type

ofis a

Su

bty

peof

is a S

ub

typ

e of

is a S

ub

typ

e ofV

AL

UE

Th

e two

setsare eq

ua

l

Is asubtype of

Must take only 1

[may be value of

none, or many]

Relation

ship

Iden

tifieris a

Su

bty

peof

ST

AT

E O

F O

BJE

CT

TH

E S

TA

TE

OF

AN

OB

JE

CT

IS A

CO

LL

EC

TIO

N O

FA

TT

RIB

UT

E V

AL

UE

S

• Instance level cardinality (or capacity) constraints cannot violate corresponding class level constraints• The lower bound of the Cardinality (and enumeration) domain is nil

–Cardinality ratios map to the domain of cardinality quotients (obviously!)• Lower bound = nil = Optional relationship• Lower bound = 1 = Mandatory relationship• Often upper bound may be “many”

– “Many” is a finite value: Open bound , delimited by infinity

• (example of a finite, bounded/delimited pattern, in which the delimiter is infinite)

–These constraints are inherited by all counts

Person Home

Each person may own only 1 home

own0..1

Each person must own 1 home and no more

1..1

Each person must own 1, and may own more, homes

1..M 0..M

[Each home may be owned by 1 or more persons]


Relationships may be compositions of objects

House Town

Located in 1[location of 1 or more]Person Live in 1

[lived in by 0 or more]

OBJECTCLASS

OBJECTCLASS

OBJECTCLASS

RELATIONSHIPRELATIONSHIP

House TownLocated in 1

[location of 1 or more]

LIVE IN 1[LIVED IN BY 0 OR MORE]

Person Live in 1[lived in by 0 or more]

An object class may glue relationships end-to-end

Together, two relationships imply the third

Live inExplicitly asserting all three would denormalize and replicate information

• Structure also adds information to a collection: the entire composition considered together is a polymorphism of “Live in”

Could have been a

complex network of objects and

relationships

(remember the metamodel of Pattern)

•Which objects join which?•In a network of associations how do we distinguish objects from relationships?•The truth is that it does not matter; they are all objects and only the overall structure of association matters

•This is the essence of the pattern



13

©Amit Mitra & Amar Gupta©Amit Mitra & Amar Gupta


OBJECT

RELATIONSHIP

OBJECT

PROCESSRECURSIVE




Cardinality ratios of compositions

1. If any relationship in a chain of relationships in a composition is optional, the composite relationship will be optional

2. 1 to 1 relationships in tandem result in a composite 1 to 1 relationship

3. A 1 to 1 relationship in tandem with a 1 to many relationship results in a composite 1 to many relationship

4. A Many to 1 relationship in tandem with a 1 to 1 relationship results in a composite many to 1 relationship

5. A 1 to 1 relationship in tandem with a many to 1 relationship results in a composite many to 1 relationship

6. A Many to 1 relationship in tandem with a Many to 1 relationship results in a composite many to 1 relationship

7. A 1 to Many relationship in tandem with a 1 to Many relationship may result in a composite many to many relationship

8. A Many to 1 relationship in tandem with a 1 to Many relationship results in a composite many to many relationship


Mutability

•The Principle of Parsimony (of information)– Specify only what you must

• Basis for innovation!!

– Liskov’s principle• A subtype may always substitute its parent in a rule, but not necessarily vice-versa

(“It must be possible to substitute any object instance of a subclass for any object instance of a superclass without affecting the semantics of a program written in terms of the superclass”)MODIFICATION:“It is possible to substitute any Pattern with its subclass without affecting the semantics of the pattern”





ApartmentLivingSpace

Apartment House

Subtype of

object

•The concept of the essence of a pattern and the meaning of “Essential” emerge thus•Eg: Is a check a paper document or a payment?•What are the essential parts of a composition? What may we remove/change without changing the meaning of the composition?

•A car without its radio is still a car, but is a car without wheels still a car? What about a car without its chassis?•What is the essence of the meaning of “house”: A house with a roof only, a roof and outside walls, or also inside walls, all of these?

•Larger scope and complexity require more abstraction•The Universal Perspective must be abstract

•The goal of a process is its product•A network like the Universal Perspective provides many paths to a goal

•Changing the essential meaning of a goal is a paradigm shift•Apply the Principle of Parsimony with the Universal Perspective to help innovate or to automate innovation (subtypes of universal meanings may substitute each other as resources and products of processes

19


TRANSITIVITY

PERSON

HOUSE

WALL

Owns 0 or more[owned by 0 or more]

Consists of 1 or more[part of 0 or 1]

TRANSITIVE

Owns 0 or more[owned by 0 or more](automatically implied)

Transitive relationship

Transitivity, or the lack of it has several flavors


2nd Degree intransitivity

Transitivity





•Transitive: Together, a set of relationships imply another•One relationship must be eliminated to normalize knowledge•The degree of transitivity is the max. number of relationships in a transitive chain•If the set of relationships is a set of processes, the last process should be eliminated

Live inTransitive triad

•Intransitive: A relationship is barred if a set of relationships exists

•Atransitive:No inclusion or exclusion constraints between relationships•Is “friend of” transitive, atransitive or intransitive?

Person

Parent of

Person PersonParent of

Parent of

DISSALLOWED

(Degree can be infinite)(Eg: The transitivity of Ancestor of)


COMPOSITION x

Subtype of

TransitiveComposition

IntransitiveComposition

Subtype of

Tansitivity (2)

• Transitivity is about compositions• A Transitive relationship may be symmetrical or not

– Eg: Relative of (symmetrical) vs. Ancestor of (asymmetrical)• An intransitive relationship has more information than a transitive relationship

– It will always be asymmetrical• Eg: Parent of

May be symmetrical (eg: relative of)OrAsymmetrical (eg: ancestor of) May only be asymmetrical (eg: parent of)

Increasinf information

Increasinf information

22


Quantitative (arithmetic)Rule Expression

NominalRule Expression

OrdinalRule Expression

Subtype of

Subtype of

Inherit Classification information;Add ranking information

Inherit classification and ranking information;Add quantitative information

Quantitative (arithmetic)Rule Expression

NominalRule Expression

OrdinalRule Expression

Subtype of

Subtype of

• Ranking operations

•Quantitative operations(includes arithmetic operations, ranges, inequalities, equalities formulae etc)

•Boolean operations(including existence constraints like cardinality constraints)

OCCURRENCE AND EXISTENCE TO ARITHMETIC AND EQUATIONS


PRODUCTSALE

PRODUCT“constituent”of relationship

CUSTOMER

“constituent”of relationship

= Relationship

PLACE“constituent”of relationship

3rd ORDERRELATIONSHIP

sell

Sell to

Sell at

STATE SPACE OF A RELATIONSHIP


AN INSTANCE OF A 3RD ORDER MANIFOLD SHOWN IN THREE SPACE

products

places

customers

a particular product sold toa particular customer ata particular place

• An instance of an “ordinary” binary relationship would be a point in two-space

• Each item, product, place and customer may normalize different properties (like product feature, geographical footprint and customer age respectively)

– Each axis may unfold into a state space, each axis of which could in turn summarize a state space and so on


products

places

customers


A category is a region with various points inside it

Ovelappingcategories

MutuallyExclusive, orNon- overlappingcategories

Two dimensional region

CATEGORIES IN THREE SPACE


BOREL SET

Sale price

pro

du

cts

places

= INSTANCES OFBOREL SETS

ANOTHER INSTANCE OF A

BOREL SET

•A BOREL SET is the set of all possible regions in that space– In nominal state space it will be the set of all possible subsets– If we consider the “All” value, the Borel object will include all intervals

in state spaces of supertypes and subtypes


EXAMPLES OF 3RD ORDER “MARKET SEGMENT” BOREL SET IN THREE

SPACE

products

places

customers


A category is a region with various points inside it

Ovelappingcategories

MutuallyExclusive, orNon- overlappingcategories

categoryid

product place customer

seg 1 ALL NYC GE

seg 1 ALL NYC GM

seg 2 ALL NYC GM

seg 2 ALL NYC Toyota

More categories

The origin may represent shared values like “All”For spaces composed of axes that have a Nil value, such as ratio scaled spaces, the origin might represent the shared Nil value


BOREL OBJECT EXAMPLE

Note: At least one relationship, (or combination of relationships), with constituent objects -- product,customer ,sales channel -- must exist at each moment in time

PRODUCTSALE

PRODUCT“constituent”of relationship

CUSTOMER


SALESCHANNEL


MARKETSEGMENT

many to many

PRODUCTSALE

MARKETSEGMENT

PRODUCT SALE IN MARKET SEGMENT

1 to many

0 to many0 to many

0 to many

0 to many

manytomanyresolution

= Relationship

= Borel Object

= Resolution Object


(At least 1 relationship to a constituent entity must exist at any given time)

RELATIONSHIP(ASSOCIATIVE OBJECT)

OBJECT“constituent”of Borel Object

OBJECT

“constituent”of manifold

OBJECT

“constituent”of manifold

BOREL OBJECTOF

RELATIONSHIP

many to

many

RELATIONSHIP

BOREL OBJECT

RESOLUTIONOBJECT

1 to many

0 to many

0 to many

0 to many

0 to many

= Relationship

Degree of combination = 1 or more

Eg.: ProductTransfer/usageagreement

Eg.: Market segment

resolution object name <relationship name>

IN<borel object name>

BOREL OBJECT RULES

Every relationship implies a corresponding Borel Object

1 ©Amit Mitra & Amar Gupta RELATIONSHIPS Reading Assignment Supplementary module 5.

Documents

degree of relationship

mandatory relationship

injective relationship

subtyping relationship

surjective relationship

threeway relationship

relationship carefull

object class object