Schemas in Z

1
Schemas in Z

• Chapter 6 in Potter Sinclair and Till

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Schemas

• Zs main representational mechanism of structure
  and functions.
• Example
• initialStock Stock
• initialStock lt maxlines
• ? i ITEM n N (i,n)einitalStock /\ nlt100
• stockItem P ITEM
• stockItems dom initialStock

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Schemas Describing Abstract States
Account Opened
Regular Withdrawal
AuthorizedWithdrawal
Account in Good Standing
AccountOverdrawn
Overdraft Paid Off
Deposit
Account closed
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Schemas Describing Abstract States

• Example 1
• BankAccount 1
• total, minBalance R
• total gt minBalance
• Example 2
• BankAccount 2
• total, minBalance R
• inGoodStanding BOOLEAN
• inGoodStanding ?(total gt minBalance)

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Schemas Describing Operations

• BankAccount with Operations
• total, minBalance R
• withdraw R BOOL
• overDraft R x BOOL R
• total gt minBalance
• ?x?Rout!Bool. total x? lt minBalance gt
  out! withdraw(x?)/\out! TRUE /\ total
  total-x
• \/total x? gt minBalance gt out! FALSE /\
• out! withdraw(x?)/\ total total

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Schema Calculus Operations on Schemas

• Schemas can be built from other schemas.
• Schema calculus has some operations on schemas to
  build new schemas from the old.
• Operations
• Inclusion
• Decoration
• Disjunction
• Conjunction
• Negation
• composition

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Inclusion - 1

• BankAccount with Operations
• Bank Account 1
• Bank Overdraft
• Where Bank Overdraft can be defined separately

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Inclusion - 2

• Bank Regular Withdraw
• total, minBalance R
• withdraw R BOOL
• overDraft R x BOOL R
• total gt minBalance
• ?x?Rout!Bool. total x? lt minBalance gt
  out! withdraw(x)/\out! TRUE /\
• minBalance minBalance-x?
• \/total x? gt minBalance gt out! FALSE /\
• out! withdraw(x?)/\ total total

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Decoration

• Adding ?, ! And to variables is referred to as
  decoration.
• Can use for schemas themselves.
• Example
• Double Min Balance
• Bank Account, Bank Account
• doubleMinBalance R R
• monBalance 2minBalance

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Full Expansion of a Decorated Schema

• Replace primed entities by their definitions
• Double Min Balance
• total, total, minBalance, minBalance R
• doubleMinBalance R R
• total gt minBalance /\ total gt minBalance

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Schema Disjunction

• Want to specify define two cases separately and
  state that they are two cases of the same
  thing.
• Example
• Bank WithdrawCash Bank RegularWithdraw \/ Bank
  Overdraft
• Bank Regular Withdraw defined earlier

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Schema Disjunction - 2

• Bank Overdraft
• Bank Account 1, Bank Account 1
• overdraft R x BOOL BOOL
• x? R, approve? BOOL
• ?x?Rout!Bool. approve? gt (out! /\
  totaltotal-x?)
• total total-x?
• \/not approve? gt (not out! /\ totaltotal)

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Schema Conjunction

• When one schema can be written specified as two
  specifications
• Results in taking the conjunction of both
  constraints
• Example
• Bank Overdraft sendMessage/\adjustBalance

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Schema Conjunction - 2

• send Message
• Bank Account 1, Bank Account 1
• overdraft R x BOOL BOOL
• x? R, approve? BOOL
• ?x?Rout!Bool. approve? gt out!
  /\totaltotal-x?)
• \/? approve? gt (? out! /\ totaltotal)

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Schema Conjunction - 3

• Adjust Total
• Bank Account 1, Bank Account 1
• overdraft R x BOOL BOOL
• x? R, approve? BOOL
• ?x?Rout!Bool. approve? gt out! /\ total
  ttotal-x?
• \/? approve? gt (? out! /\ totaltotal)

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Schema Negation

• Example 1
• BankAccount 1
• total, minBalance R
• total gt minBalance
• Bad bank Account ? Bank Account 1
• Bad BankAccount
• total, minBalance R
• ?(total gt minBalance)

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Schema Hiding Operations

• Used to hide some variables and declarations of
  already known schemas.
• Use ? to hide variables.
• Format
• Decs Pred \ Hidden variables
• Reduced Decs ? Hidden Decs . Predicate

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Schemas Hiding Example -1

• BankAccount with Operations
• total, minBalance R
• withdraw R BOOL
• overDraft R x BOOL R
• total gt minBalance
• ?x?R ?app?out!Bool. total x? lt
  minBalancegtout! overDraft x?, app? /\out! /\
• total total-x
• \/total x? lt minBalance gt out! FALSE /\
• out! overDraft(x?,app?)/\ totaltotal

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Schemas Hiding Example -2

• BankAccount with Operations
• total, minBalance R
• withdraw R BOOL
• overDraft R x BOOL R
• total gt minBalance
• ?x?R ? app?out!Bool. total x? lt
  minBalancegtout! overDraft x?, app? /\out! /\
• total total-x
• \/total x? lt minBalance gt out! FALSE /\
• out! overDraft(x?,app?)/\ totaltotal

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Schema Composition

• Recall that function composition
• If f S1 -gt S2 and g S2 -gt S3 then
• fog S1 -gt S3
• Schema Composition imports this idea to schemas.
• Example Consider overdrawn bank account
• To close account
• Must pay up to minimum balance
• Then close account from account in good standing

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Schema Composition Method

• Replace primed variables ()in first schema with
  double primed variables ()
• S/
• Replace un-primed variables in second schema with
  double primed variables ()
• T/
• Existentially quantify variables in double primed
  state ()
• ? State . S//\T/

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Schema Composition Example -1

• toGoodStanding
• D Bank Account 1
• toGoodStanding R -gt NULL
• total lt minBalance
• ? x?R ( xtotal gt minBalance) gt
• total total x?

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Schema Composition Example -2

• closeInGoodStanding
• D Bank Account 1
• close BOOL -gt R
• total gt minBalance
• ? x?BOOL, return!R ( totalgtminBalance) gt
  return!total /\ total0
• closeFromDefault toGoodStanding o
• closeFromGoodStanding

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Schema Composition Example -3

• closeFromDefault
• D Bank Account 1
• close BOOL -gt R
• ? Bank Account
• ? x? R toGoodStanding x /\ close true return

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Schema Preconditions

• Applicable only to schemas representing
  operations
• Pre Op ? State Out! . Op
• Means that the schema should be applied to those
  state that result in the given predicate
  satisfies in the post state.
• Here Op is the operation

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Schema Precondition Example

• Simple
• x,y N
• x lt y
• NonEndPoint
• D Simple Z!N
• x lt x lt z! lt y lt y

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Constructing pre NonEndPoint

• preNonEndPoint
• Simple
• ? x,y,z! N x lt x lt z lt y lt y
• Can simplify the statement
• ? x,y,z! N x lt x lt z lt y lt y to
• x3 lt y

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Schema Types

• A way to record data types by specifying filed
  and their types of a structure.
• Example
• Book
• title, author, isbn, publisher String
• Pubdate, pages Z
• slength title lt 80, slength author lt 80,
• (checksum isbn) mode 11 0
• O lt pages /\ 1800 lt pubdate lt 2050