Z Notation

1
Z Notation

• SWE 623 - Fall 2002
• Covers Ch 3,4 and 5

2
Chapter 3
3
Conventions

• Z has types
• Typed variable
• Input variables end with ?
• Output variables end with !
• Typed constants
• Has notations for
• set definitions
• logical statements and sentences of 1st order
  logic
• Primed variables Unprimed Variable in post
  state

4
Example (PST Page 43)

• AddPair
• Vocab,Vocab set of (Native,Foreign)
• n?Native f?Foreign
• Vocab e WellFormedVocabs /\
• n? e OthoNative /\
• f? e OrthoForegin /\
• Vocab Vocab u (n?,f?)

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Example Cond. (PST Page 40)

• Native,Foreign
• OthoNative set of Native
• OrthoForegin set of Foreign
• WellFormedVocabs
• V set of (Native,Foreign)
• ?n Native f Foreign .(n,f) e V gt
• n e OrthoNative /\
• f e OrthoForeign

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Adding Operations -1

• Message OK,Error
• AddValidPair
• Vocab,Vocab set of (Native,Foreign)
• n?Native, f?Foreign, rep!Message
• Vocab e WellFormedVocabs /\
• n? e OthoNormal /\ f e OrthoForeign /\
• Vocab Vocab U (n?,f?) /\
• rep! OK

7
Adding Operations -2

• Message OK,Error
• AddValidError
• Vocab,Vocab set of (Native,Foreign)
• n?Native, f?Foreign, rep!Message
• Vocab e WellFormedVocabs /\
• n? \e OthoNormal \/ f \e OrthoForeign /\
• Vocab Vocab U (n?,f?) /\
• rep! Error

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Adding Operations - 3

• TotalAddPair AddValidPair U AddPairError
• Go through the example of ToForeign
• Suggested Exercises
• Page 52, 3.3

9
Chapter 4
10
Types and Sets

• Every declared variable must be typed.
• OthoNative set of Native
• x BOOK
• Set Notations
• By Explicitly listing elements
• BOOK,CAR
• Integers Z, Natural Numbers N
• Named Sets
• samllEven 2,4,6,8
• N1 n e N n gt 0

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Sets 1

• Power Set Operator
• P(S) The set of all subsets of S
• P(12) f, 1, 2, 1,2
• Cartesian Product
• A x B (a,b) a e A /\ b e B
• Constructing Subtypes Using Sets
• library P(BOOK x Z)

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Sets 2

• Types as Declarations Abbreviations
• posEven xZ ? yZ x 2y /\ ygt0
• someEven Z and ? yZ someEven 2y /\ ygt0
• Can also write someEven posEven
• Data Types
• POSITION
• on, off POSITION and on / off
• POSITION on off

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(Free) Type Definitions

• Disjoint Union Constructor
• PERSON empltltEMPLOYEEgtgt clientltltCLIENTgt
  gt
• Product Constructor
• NAME (FIRST,LAST)
• Recursive Type Definitions
• FAMTREE nodeltltNAME x FAMTREE x FAMTREEgtgt
  unk

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From the Z Library

• Abbreviations Known to Z
• Range Restrictions in Ordered Types
• 25 2,3,4,5
• 99 9
• 52 f
• Counting Function
• 1,2,3 3
• 51 0

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Extending Quantification Notation

• General Form Declarations Const . Pred
• ?n N n1 gt n
• ?x PERSON ? y Parent (x,y)
• Definite Descriptions m Decs Pred . Expr
• mx N x5 12
• For the expression to be DEFINITE, its value
  should be unique.

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Axiomatic Descriptions

• Amounts to a formal paragraph
• maxlines Z
• maxlines gt 0
• Abbreviations
• isParent (PERSON,PERSON)
• siblings (x,y) xPERSON /\ yPERSON
• ? z PERSON isParent(x,z) /\
• isParent(y,z)

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Generic Definitions(Parameterized Types and
Polymorphism)

• Example
• P(X) \ P(X) f for every set X, for example
• P(N) \ P(N) f and P(Z) \ P(Z) f.
• Such facts are described as formal paragraphs
• X
• f P(X)
• f xX x /x

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Chapter 5

• Z Notation for Functions and Relations

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Relations

• A set of ordered pairs
• Rides (Jane, bus),
• (Tom,car),
• (John,cycle),
• (Jane, car),
• (Tom,cycle)

Jane Tom John
bus car cycle
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Z Notation for Relations - 1

• X lt-gt Y P(X x Y)
• Can be used for declarations
• Rides PEOPLE lt-gt VEHICLES
• Jane -gt Bus, Tom -gt car
• Can write a formal paragraph as
• _ Rides _ PEOPLE lt-gt VEHICLES

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Z Notation for Relations - 2

• _ lt _ Z lt-gt Z
• ? x,y Z . ( x lt y ? ? z N . y xz)
• Domains and Ranges
• dom R xeX ? yY x -gt y e R
• ran R yeY ? xX x -gt y e R

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An Example in Z 1

• Consider a stock database.
• initialStock Stock
• initialStock lt maxlines
• ? i ITEM n N (i,n)einitalStock /\ nlt100
• stockItem P ITEM
• stockItems dom initialStock

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An Example in Z - Cont

• Hence if initialStock is
• widget1 -gt1, widget2 -gt12, widget3 -gt 8
• Then stockItems is the set
• widget1, widget2, widget3

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Operations on Relations

• Because relations are sets, set operations such
  as union, intersection, difference can be applied
  to them.

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Example

• A set of ordered pairs
• Rides (Jane, bus),
• (Tom,car),
• (John,cycle),
• (Jane, car),
• (Tom,cycle)
• Rides U (Ben,tricycle)

Jane Tom John
bus car cycle
tricycle
Ben
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Other Relational Operators

• Inverse
• (x,y)eR-1 ? (y,x)eR
• (tricycle,ben)eRides-1 ?(ben,tricycle)eRides
• Domain Restrictions
• R some domain eltslt R
• S lt R xXyY xeS /\ x -gty e R
• femaleRiders jane lt Riders
• (jane,bus),(jane,car)

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Other Relational Operators - 2

• Range Restrictions
• R R gtsome domain elts
• R gt T xXyY yeT /\ x -gty e R
• carRiders Riders gt car
• (jane,bus),(tom,car)

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Removing Elements

• From the Domain
• S lt R xXyY xeSc /\ x -gt y e R
• Men lt Riders (jane,bus),(jane,car)
• From the Range
• R gt T xXyY yeTc /\ x -gt y e R
• Riders gt cyclist (jane,bus),(jane,car)
  ,(tom,car)

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Composing Relations

• riders PERSON ? vehicles
• hasWheels Vehicles ? N
• ridersOnofWheels PERSON ? N
• R S (x,y) ? z (x,y)eR /\ (y,z)e S

1 2 3 4 5 6
Jane John Tom ben
Car Bus Cycle tricycle
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Relational Image

• If R X ? Y and S P(X), then
• R(S) yY ? xX x e S /\ x -gt y e R
• R(S) ran( S lt R)
• Example
• R(women) car,bus
• Look at exercises on Page 100

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Functions

• Functions
• Relations where every element in its domain is
  related to at most one element in the range.
• Total function
• A function in which every element in the domain
  is mapped to some element in the range
• Z Notation
• isDaughterOf PEOPLE PEOPLE

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Function Definitions

• Example Function
• square Z -gt N
• ? x Z . Square x x.x
• Finite Functions
• SSN PEOPLE N

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Functions - 3

• Injective Functions
• Functions where an element in the range is
  related to at most one element in the domain.
• Z Notation
• countryOfCapitol CITIES COUNTRY
• Surjective Functions
• A function in which every element in the range is
  related to some element in the domain
• SSN N PEOPLE

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Operations on Functions

• Because functions are relations, all relational
  operations are valid for functions.
• An Example
• X,Y
• first (X x Y) -gt X
• second (X x Y) -gt Y
• ? x XyY first(x,y) x /\ second(x,y) y

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Lambda Notation

• General Form
• l declarations predicate term
• Examples
• 1. l n N n lt5 n2
• 2. l n, m N m gt n (m-n)

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Sequences

• Sets where order matters
• Represented as functions in Z
• Example
• lt Alice, Bob, Cindygt 1 -gt Alice,2 -gt Bob
• 3-gt Cindy

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Operations on Sequences - 1

• Head, tail front and last
• Let seq1 X seq x \ltgt
• Then head, tail can be defined as follows
• X
• head seq1 X -gt X
• tail seq1 X -gt X
• ? s seq1 X
• head s s1 /\
• tail s l n1 .. s 1. S(n1)

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Concatenating Sequences

• Concatenation
• Appending a second list to the back of the first
• Z notation
• Example
• lt1,2,3gtlt4,5gt lt1,2,3,4,5gt
• X
• __ seq X x seq X -gt seq X
• ? s,t seq X st sUn1..t (ns) -gt tn

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Representing Bags in Z

• Records the name of the element and how many
  times it occurs in the bag.
• Hence Bag X X N
• X
• count bag X (X N)
• _ U _ (bag X x bag X) bag X
• ? x X bag Bag count B (lx X.0) () B
• ? B1,B2 bag X xX
• count (B1 U B2 ) x count B1 x count B2 x