1 relations chapter 9 formal specification using z

1

Relations

Chapter 9

Formal Specification using Z

2

Formal definition of Relation

• The Z examples so far have been limited to one basic type. We need of relating one set to another.

• A relation is a set which is based in the idea of the Cartesian product.

3


• A Cartesian product is the pairing of values of two or more sets. The Cartesian product is written:

• X Y Z• For example• {red, green} {hardtop,softtop} =• {(red, hardtop), (red, softtop), (green, hardtop), (green,softtop)}

• This set consists of ordered paired called a tuple

4


• A binary relation R from the set X to the set Y is a subset of the Cartisian Product X Y. If (x,y) R, we write xRy and say x is related to y. In the case X=Y, we call R a binary relation on X.

• The domain of R is the set• { x X | (x,y) R from some y Y}• The range of R is the set• {y Y | (x,y) R for some x X}

5

Relation as a table

• A relation can be thought of as a table that lists the relationship of elements to other elements

Student Course

Bill Computer Science

Mary Maths

Bill Art

Beth History

Beth Computer Science

Dave Math

The table is a set of ordered pairs

6

Relation as a set of ordered Pairs

• If we let

• X = { Bill, Mary, Beth, Dave}

• Y = {ComputerScience, Math, Art, History}

• then the relation from the prevuous table is:

• R = {(Bill,ComputerScience), (Mary, Math), (Bill,Art), (Beth,ComputerScience), (Beth,History), (Dave, Math) }

• We say Mary R Math. The domain is the domain (first column) of R is the set X the range (second column) of R is the set Y. So far the relation has been defined by specifying the ordered pairs involved.

7

Relation defined by rule

• It is also possible to define a relation by giving a rule for membership of the relation.

• Let X = { 2,3,4} and Y = {3,4,5,6,7}• If we define R from X to Y by:• (x,y) R if x divides y (with zero remainder)• We get:• R = { (2,4), (2,6), (3,3), (3,6), (4,4) }

8

Relation written as table

• This can be written as a table .

X Y

2 4

2 6

3 3

3 6

4 4

The domain of R is the set {2,3,4} and the range is {3,4,5}

9

1 2

43

Let R be the relation on X = {1,2,3,4}

defined by :

(x,y) R if x y,

x,y X.

is shown

The digraph for R

R = {(1,1), (1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)}

The domain and the range are both equal to x. R is reflexive, transitive and antisymmetric.

10

1 2

43

Let R be the relation on X = {1,2,3,4}

defined by :

(x,y) R if x < y,

x,y X.

is shown

The digraph for R

R = { (1,2),(1,3),(1,4),(2,3),(2,4),(3,4)}

The domain and the range are both equal to x. R is reflexive, transitive and non-symmetric.

11

Types of relation

• A relation R on a set X is reflexive if (x,x) R for every x X.

• A relation R on a set X is symmetric if for all x,y X, if (x,y) R, then (y,x) R.

• A relation R on a set X is antisymmetric if for all x,y X, if (x,y) R, (y,x) and x y , then (y,x) R.

12

Types of relation

• A relation R on a set X is transitive if for all x,y,z X, if (x,y) R, and (y,z) R then (x,z) R.

13

Types of relation

• Let be a relation on the set A, then is reflexive if x x for all x A is symmetric if x y implies y x is transitive if x y and y z imply x z is antisymmetric if x y and y x imply

that x=y

14

Relations

• A special case of Cartesian product is a pair. A binary relation is set of pairs, of related values.

• [COUNTRY] the set of all countries

• [LANGUAGE] set of all languages

15

A relation on COUNTRY and LANGUAGE called SPEAKS could have values: {(France,French),(Canada,French),(Canada,English),(GB,English),(USA,English)}

FranceCanadaGBUSA

COUNTRY

French

English

SPEAKS relation

LANGUAGE

16

RELATIONS

• As can be seen from the SPEAKS relation, there are no restrictions requiring values of the two sets to be paired only one-to-one.

• We are free to choose the direction or order of the relation:

• SPOKEN: (LANGUAGE COUNTRY)

17

Declaring Relations in Z

• R: X Y ( equivalent to (X Y) )

• SPEAKS: COUNTRY LANGUAGE

• ( equivalent to (COUNTRY LANGUAGE) )

• In English we say:

• R relates X to Y.

• SPEAKS relates COUNTRY to LANGUAGE

18

MAPLETS

• A maplet relates x to y and is written:

• x y == (x,y)

• ‘x is related to y’ or ‘x maps to y’

• GB English SPEAKS

• is equivalent to

• (GB, English) SPEAKS

19

Membership

• To discover if a certain pair of values are related it is sufficient to see if the pair or maplet is an element of the relation:

• GB English SPEAKS

• OR

• (GB, English) SPEAKS

20

Infix Notation

• If we declare a relation using low-line characters as place holders to show the fact that is infix

_SPEAKS_ : COUNTRY LANGUAGE

• GB speaks English

• In general

xRy == x y R==(x,y) R

21

The DOMAIN and RANGE of the relation SPEAKSdom SPEAKS is set of countries where at least one language is spoken.ran SPEAKS is set of languages which are spoke in at least one country.

FranceCanadaGBUSA

COUNTRY

(SOURCE)

French

English

SPEAKS relation

dom SPEAKS ran SPEAKS

LANGUAGE(TARGET)

22

Domain and Range

A relation relates values of a set called source or from-set to a set called target or to-set.

Below the source is X and the target Y.R: X Y

R = {(x1,y2), (x2,y2), (x4,y3), (x6, y3) ), (x4,y2)}

23

Domain

In most cases only a subset of of the source is involved in the relation. This subset is called the domain

dom R = {x1,x2,x4,x6}

24

Range

In most cases only a subset of of the target is involved in the relation. This subset is called the range

ran R = {y2,y3}

25

The DOMAIN and RANGE of the relation R dom R = {x1,x2,x4,x6} ran R = {y2,y3}

x1

x2

x4

x6

X

(SOURCE)

Y2

y3

R relation

dom R ran R

Y(TARGET)

x5

x3

y1

y4

26

Relational Image• To discover the set of values from the range

of a relation related to a set of values from the domain of the relation one can use the relational image:

• RS • pronounced the ‘relational image of S in R’.

For the languages spoken in France and Switzerland would be:

• SPEAKS {France, Switzerland} == {French,German, Italian, Romansch}

27

Constant value for a Relation

• Some relations have constant values. If the value of a relation is known, it can be given by an axiomatic definition. For example the infix relation greater-than-or-equal:

i,j:i jn: i=j+ n

28

Domain Restriction

• The domain restriction operator restricts a relation to that part where the domain is contained in a particular set. The relation R domain restricted to S, is written as:

• S R• The point of the operator can be thought

of as pointing left to the domain

29

Range Restriction

• The range restriction operator restricts a relation to that part where the range is contained in a particular set. The relation R range restricted to S, is written as:

• R S• The point of the operator can be

thought of as pointing right to the range

30

Domain Subtraction

• The Domain Subtraction operator restricts a relation to that part where the domain is not contained in a particular set. The relation R domain subtracted to S, is written as:

• S R• The point of the operator can be thought

of as pointing left to the domain

31

Range Subtraction

• The Range Subtraction operator restricts a relation to that part where the Range is not contained in a particular set. The relation R range subtracted to S, is written as:

• R S

32

Example of Domain Restriction

Holsholidays: COUNTRY DATE

• EU: COUNTRY

• The relation mapping only EU countries to their public holidays is:

• EU holidays

• The relation mapping only non-EU countries to their public holidays is:

• EU holidays

33

Composition

• Relations can be joined together in an operation called composition. Given:

• R: X Y and

• Q: Y Z

• We can have forward composition or backward composition

34

Forward Composition

• Given:

• R: X Y and

• Q: Y Z

• forward composition is defined as:

• R;Q: X Z

35

Forward Composition

XZY

R Q

R;Q

36

XZY

R Q

R;Q

For any pair (x,y) related by R forward composed with Q

x R;Q z

there is a y where R relates x to y and Q relates y to z;

y:Y xRy yQz

37

Backward Composition

• Given:

• R: X Y and

• Q: Y Z

• backward composition is defined as:

• QR: R;Q

38

Repeated Composition

• A homogeneous relation is one which relates values from a type to values of the same type. Such a relation can be composed with iteslf

• R: X X

• R ; R: X X

39

Composition Example

• Countries are related by the borders relation if the share a border:

• borders: COUNTRY COUNTRY

• e.g. France borders Switzerland

• Switzerland borders Austria

• Countries are related by borders composed with borders if they each share a border with a third country.

40

Composition Example

• The two facts:– 1. France borders Germany– 2. Germany borders Austria

• can be represented by:

• France borders;borders Austria

• We can write:

• Spain borders3 Denmark

• for

• Spain borders; borders; borders Denmark

41

Transitive Closure

• In general the transitive closure R+ as used in

• x R+ y• means that there is a repeated composition of

R which relates x to y. For example

• France borders+ India.• Means France is on the same land-mass as

India

42

Identity Relation

• The identity relation

• id X

• is the relation which maps all x’s on to themselves:

• id X == {x:X x x }

43

Reflexive Transitive Closure

• The repeated composition

R* = x R+ id X

includes the identity relation. The Reflexive Transitive Closure is similar to the Transitive Closure except that it includes the identity relation. For example

• x R* x is true, even if x R x is false, (id x)

• France borders* France is true

44

Relation Example

• Public holidays around the world can be described as follows:

• [COUNTRY] the set of all countries

• [DATE] the dates of a given year

Holsholidays: COUNTRY DATE

45

Relation Example

• An operation to discover whether a date d? is a public holiday in country c? is:

• REPLY ::= yes | no

EnquireHols

c?: COUNTRY

d?: DATE

rep! : REPLY

c? d?holidays rep! = yes)

c? d?holidays rep! = no)

46

Relation Example

• An operation to decree whether a public holiday in country c? on date d? is:

DecreeHols

c?: COUNTRY

d?: DATE

holidays’ = holidays c? d?

47

Relation Example

• An operation to abolish a public holiday in country c? on date d? is:

AbolishHols

c?: COUNTRY

d?: DATE

holidays’ = holidays \c? d?

48

Relation Example

• An operation to find the dates ds! of all public holidays in country c? is:

DatesHols

c?: COUNTRY

ds?: DATE

ds! = holidays c?

49

Inverse of a relation

• The inverse of a relation R from X to Y

• R: X Y is written

• R~

• and it relate Y to X (mirror image). So if

• x R y

• then

• y R~x

50

Family Example

• [PERSON] the set of all persons

• father,mother: PERSON PERSON

• ‘x father y’ meaning x has father y

• ‘v mother w’ meaning v has mother w

51

Family Example: parent relation

• parent: PERSON PERSON

• can be defined as

• parent = father mother

52

Family Example: sibling

• sibling: PERSON PERSON• can be defined as• sibling = (parent ; parent ~) \ id PERSON• This is defined as the relation parent composed

with its own inverse, in other words the set of persons with the same parent. Note the inverse of parent that relates parent to child. A person is not usually counted as their own sibling, hence the use of the identity relation.

53children ZY

R Q

parent;parent~

sibling: PERSON PERSONcan be defined assibling = (parent ; parent ~) \ id PERSONchildren parents : PERSONchildren = {Pat, Bernard,Marion}parents = {Liam, Mary}

Pat

Bernard

Marion

Liam

Mary

Pat

Bernard

Marion

54children ZY

parentparent~

parent;parent~

(parent ; parent ~) = {(Pat, Pat), (Pat,Bernard), (Pat,Marion), (Bernard, Bernard), (Bernard,Pat), (Bernard,Marion), (Marion,Marion), (Marion,Pat), (Marion,Bernard)}

(parent ; parent ~) \ id PERSON = { (Pat,Bernard), (Pat,Marion),(Bernard,Pat), (Bernard,Marion), (Marion,Pat), (Marion,Bernard)}

Pat

Bernard

Marion

Liam

Mary

Pat

Bernard

Marion

55

Family Example: ANCESTOR

• ancestor: PERSON PERSON

• can be defined as

• ancestor = parent +

• The relation ancestor can be defined as the repeated composition of parent.

56

Chapter 9 Question 1

• Express the fact that the language Latin is not the official language of any country:

57


• Express the fact that the language Latin is not the official language of any country:

• Latin:LANGUAGE

• Latin ran speaks

58


• Express the fact that Switzerland has four languages

59


• Express the fact that Switzerland has four languages

• #speaks {Switzerland} = 4

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• Give a value to the relation speaksInEU which relates countries which are in the set EU to their languages.

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• Give a value to the relation speaksInEU which relates countries which are in the set EU to their languages.

• EU: COUNTRY

• speaksInEU : COUNTRY LANGUAGE

• speaksInEU = EU speaks

62


• Give a value to the relation grandParent

63


• Give a value to the relation grandParentgrandParent : PERSON PERSON

grandParent = parent ; parent

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• A person’s first cousin is defined as a child of the person’s aunt or uncle. Give a value for the relation firstCousin.

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• A person’s first cousin is defined as a child of the person’s aunt or uncle. Give a value for the relation firstCousin.

firstCousin : PERSON PERSON

firstCousin = (grandParent ; grandParent ~) \ sibling

66


Given

[PERSON] set of uniquely identifiable persons

[MODULE] set module numbers at a university

students,teachers,EU,inter: PERSON

offered: MODULE

studies: PERSON MODULE

teaches: PERSON MODULE

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[MODULE] set module numbers at a university

students,teachers,EU,inter: PERSON

offered: MODULE

studies: PERSON MODULE

teaches: PERSON MODULE

Explain the predicates:

EU inter =

EU inter = students

dom studies students

dom teaches teachers

ran studies offered

ran teaches = ran studies

68


Explain the effect of each predicates:

EU inter =

EU inter = students

dom studies students

dom teaches teachers

ran studies offered

ran teaches = ran studies

Students are either from the EU or overseas, but not from both.

Students study and teachers teach.

Only offered modules can be studied.

Those modules that are taught are studied.

69


Give an expression for the set of modules studied by a person p.

70


Give an expression for the set of modules studied by a person p.

studies {p}

71


Give an expression for the number of modules taught by a person q.

72


Give an expression for the number of modules taught by a person q.

#(teaches {q} )

73


• Explain what is meant by the inverse of the relation studies

studies~

74


• Explain what is meant by the inverse of the relation studies

studies~

• The inverse of the relation studies relates modules to persons studying them.

75


• Explain what is meant by the composition of the relations

studies ; teaches~

76


• Explain what is meant by the composition of the relations

studies ; teaches~

• The composition of relates students to the teachers who teach modules the students study.

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• Give an expression for the set of persons who teach person p.

78


• Give an expression for the set of persons who teach person p.

(studies ; teaches~) {p}

79


• Give an expression for the number of persons who teach both person p and person q.

80


• Give an expression for the number of persons who teach both person p and person q.

#((studies ; teaches~) {p} (studies ; teaches~) {q} )

81


• Give an expression for that part of the relation studies that pertains to international students(inter)

•

82


• Give an expression for that part of the relation studies that pertains to international students(inter)

•

• inter studies

83


• Give an expression for that states that p and q teach some of the same international students.

84


• Give an expression for that states that p and q teach some of the same international students.

((teaches; studies ~) {p} (teaches; studies ~) {q} ) inter

85


• The following declarations are part of the description of an international conference.


[LANGUAGE] the set of world languages

official: LANGUAGE

86


• The following declarations are part of the description of an international conference.



official: LANGUAGE

87




official: LANGUAGECONFERENCE PERSONLANGUAGE

_speaks_ : PERSON LANGUAGE

88

Chapter 9 Question 15A

Write expression that states every delegate speaks at least one language

delagates dom speaks

89

Chapter 9 Question 15B

Write expression that states every delegate speaks at least one official language of the conference

delagates ran speaks

90

Chapter 9 Question 15C

Write expression that states every delegate speaks at least one official language of the conference

lang:LANGUAGE

(del:PERSON | del delagates del speaks lang)

91

Chapter 9 Question 15E

• Write an operation schema to register a new delegate and the set of languages he or she speaks.

Registerdel?:PERSON

lang?: LANGUAGE

CONFERENCEdel? delegates

delegates = delegates {del?}

speaks =

speaks {lan:LANGUAGE | lan lang? del? lan}

official = official’

92

Labs

• Using pen and paper do Chapter 2 Exercises 5.

• Using pen and paper do exam example Edward, Fleur, and Gareth.

• Type both of the above into Word using symbols or Z fonts.

93

Labs

• Calculate the following unions.

• S1 = [1,2], S2 = [2, 3, 4, 5], S3 = [1, 2, [2, 3, 4, 5]], union(S1,S2,W).

• S1 = [1,2], S2 = [2, 3, 4, 5], S3 = [1, 2, [2, 3, 4, 5]], union(S1,S3,W).

• S1 = [1,2], S2 = [2, 3, 4, 5], S3 = [1, 2, [2, 3, 4, 5]], union(S3,S2,W).

94

Labs

• Start Prolog

• Copy sets.pro from: X:\distrib\wmt2\maths\prolog

• Consult sets.pro

• Use the sets programs to solve the previous union of S1,S2 and S3.

95

Labs

• Using prolog do exam example Edward, Fleur, and Gareth.

96

Labs

• 1)Using pen and paper do Chapter 2 Exercises 5.

• 2)Using pen and paper do exam example Edward, Fleur, and Gareth.

• 3)Calculate the following unions.

• S1 = [1,2], S2 = [2, 3, 4, 5], S3 = [1, 2, [2, 3, 4, 5]], union(S1,S2,W).

• S1 = [1,2], S2 = [2, 3, 4, 5], S3 = [1, 2, [2, 3, 4, 5]], union(S1,S3,W).

• S1 = [1,2], S2 = [2, 3, 4, 5], S3 = [1, 2, [2, 3, 4, 5]], union(S3,S2,W).

• 4)Type both 1 and 2 above into Word using symbols or Z fonts.

• 5)Start Prolog

• 6)Copy sets.pro from: X:\distrib\wmt2\maths\prolog

• 7)Consult sets.pro

• 8)Use the sets programs to solve the previous union of S1,S2,S3

• 9)Using prolog do exam example Edward, Fleur, and Gareth.

1 relations chapter 9 formal specification using z

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