Z:%20Operations%20on%20Schemas

1
Z Operations on Schemas

• David Lightfoot
• based on work of
• Andrew Simpson

2
Reference

• Using Z Specification, Refinement, and Proof,
• Jim Woodcock and Jim Davies,
• Prentice-Hall,
• 1996
• (Chapter 11)

3
Agenda

• Operation schemas
• Input and output
• Initialisation schemas
• Schema disjunction
• Schema conjunction
• Schema negation

4
The schema language

• The schema language is used to structure and
  compose mathematical descriptions of systems
• It collates pieces of information, encapsulates
  them, and names them for re-use
• It is the second component of the Z notation the
  first is the mathematical language

5
Change of state

• To describe the effect of an operation, we
  consider two copies of the state schema one
  describing the state before, the other describing
  the state afterwards
• An operation schema describes the relationship
  between the two states
• The inclusion of two copies of the state ensures
  that the constraint part of the state schema (the
  state invariant) is preserved

6
Operation schemas

• An operation schema includes two copies of the
  corresponding state schema
• ? Operation ________
• State
• State?
• ?___________
• ?
• ?_____________
• We use the undecorated state to represent the
  state before the operation and the primed state
  to represent the state afterwards

7
Example

• ? Purchase0 ________
• BoxOffice
• BoxOffice?
• ?___________
• s? ? seating \ dom sold
• sold? sold ? s? ? c?
• seating? seating
• ?_____________

8
Input and output

• An operation may involve inputs and outputs
• These are declared in the normal way, although
  there is a convention regarding their names
• the name of an input must end in a question mark
  ?
• the name of an output must end in an exclamation
  mark !

9
Example

• ? Operation ________
• State
• State?
• i? I
• o! O
• ?___________
• ?_____________

10
Example

• ? Purchase0 ________
• BoxOffice
• BoxOffice?
• s? Seat
• c? Customer
• ?___________
• s? ? seating \ dom sold
• sold? sold ? s? ? c?
• seating? seating
• ?_____________

11
? (Delta) and ? (Xi)

• There is another convention regarding operation
  schemas
• if a schema describes an operation upon a state
  described by S, we include ?S (delta S)in its
  declaration (in place of S and S?)
• if, in addition, the operation leaves the state
  unchanged we include ?S (in place of ?S)

12
Example

• ? Purchase0 ________
• ?BoxOffice
• s? Seat
• c? Customer
• ?___________
• s? ? seating \ dom sold
• sold? sold ? s? ? c?
• seating? seating
• ?_____________

13
Example

• An operation that leaves the state of the box
  office unchanged
• ? QueryAvailability ________
• ?BoxOffice
• available! ?
• ?___________
• available! (seating \ dom sold)
• ?_____________

14
Initialisation

• An initialisation is a special operation for
  which the before state is unimportant
• Such an operation can be modelled by an operation
  schema that contains only a decorated copy of the
  state
• ? StateInit ________
• State?
• ?___________
• ?_____________

15
Question

• How might we complete the following?
• ? BoxOfficeInit ________
• BoxOffice?
• allocation? ?Seat
• ?___________
• ?_____________

16
BoxOfficeInit

• ? BoxOfficeInit ________
• BoxOffice?
• allocation? ?Seat
• ?___________
• seating? allocation?
• sold? ?
• ?_____________

17
Schema disjunction

• If S and T are two schemas then their
  disjunction, S ? T is also a schema
• in which the declaration is a merging of the two
  declarations
• in which the constraint is a disjunction
  (oring) of the two constraints

18
Schema conjunction

• If S and T are two schemas then their
  conjunction, S ? T is also a schema
• in which the declaration is a merging of the two
  declarations
• in which the constraint is a disjunction
  (anding) of the two constraints

19
Schema negation

• If S is a schema then its negation, ? Sis also
  a schema
• in which the declaration the same as that of S
• in which the constraint is the negation
  (noting) of the constraint of S

20
Examples

• ? S ________
• a A
• b B
• ?________
• P
• ?__________

• ? T ________
• b B
• c C
• ?________
• Q
• ?__________

• S ? T is equivalent to
• ?________
• a A
• b B
• c C
• ?________
• P ? Q
• ?__________

• S ? T is equivalent to
• ?________
• a A
• b B
• c C
• ?________
• P ? Q
• ?__________

21
Examples

• ? S ________
• a A
• b B
• ?________
• P
• ?__________

• ? S is equivalent to
• ?________
• a A
• b B
• ?________
• ? P
• ?__________

22
Example

• If we define
• ? NotAvailable ________
• ?BoxOffice
• s? Seat
• ?___________
• s? ? seating \ dom sold
• ?_____________
• then the schema disjunction
• Purchase0 ? NotAvailable
• describes a total operation

23
Constructing operations

• Although disjunction is the obvious operator for
  constructing operation schemas, conjunction can
  also be useful
• Response okay sorry

• ? Success ________
• r! Response
• ?___________
• r! okay
• ?____________

• ? Failure ________
• r! Response
• ?___________
• r! sorry
• ?_____________

24
Total operation

• The operation of purchasing a seat may be
  described by
• Purchase ?
• (Purchase0 ? Success)
• ?
• (NotAvailable ? Failure)

25
Question

• How might we complete the following operation?
• ? ReturnTicket0 ________
• ?BoxOffice
• s? Seat
• c? Customer
• ?___________
• ?_____________

26
ReturnTicket0

• ? ReturnTicket0 ________
• ?BoxOffice
• s? Seat
• c? Customer
• ?___________
• s? ? c? ? sold
• sold? sold \ s? ? c?
• seating? seating
• ?_____________

27
Question

• How might we complete the following operation?
• ? ReturnNotPossible ________
• ?BoxOffice
• s? Seat
• c? Customer
• ?___________
• ?_____________

28
ReturnNotPossible

• ? ReturnNotPossible ________
• ?BoxOffice
• s? Seat
• c? Customer
• ?___________
• s? ? c? ? sold
• ?_____________

29
Summary

• We may use operation schemas to describe the
  effect of operations on the state of our system
• The inclusion of ?S in an operation schema
  denotes that the operation will change the state
  of S
• The inclusion of ?S indicates that the operation
  is concerned with the state of S, but will leave
  it unchanged
• Operation schemas may have input and output
• An initialisation schema describes the state of
  our system at the beginning of its life
• Disjunction and conjunction may be used to
  combine operation schemas