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Z: Operations on Schemas

David Lightfoot

based on work of

Andrew Simpson

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Reference

Using Z: Specification, Refinement, and Proof,

Jim Woodcock and Jim Davies,

Prentice-Hall,

1996

(Chapter 11)

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Agenda

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

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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

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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

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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

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Example

Purchase0 BoxOffice BoxOffice … s? seating \ dom sold sold = sold {s? c?} seating = seating

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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 !

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Example

Operation State State i?: I o!: O …

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Example

Purchase0 BoxOffice BoxOffice s?: Seat c?: Customer s? seating \ dom sold sold = sold {s? c?} seating = seating

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(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)

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Example

Purchase0 BoxOffice s?: Seat c?: Customer s? seating \ dom sold sold = sold {s? c?} seating = seating

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Example

• An operation that leaves the state of the box office unchanged:

QueryAvailability BoxOffice available!: available! = # (seating \ dom sold)

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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 …

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Question

• How might we complete the following?

BoxOfficeInit BoxOffice allocation?: Seat …

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BoxOfficeInit

BoxOfficeInit BoxOffice allocation?: Seat seating = allocation? sold =

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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

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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

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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

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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

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Examples

S a: A b: B P S is equivalent to:

a: A b: B P

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Example

If we define:

NotAvailable BoxOffice s?: Seat s? seating \ dom soldthen the schema disjunction

Purchase0 NotAvailable

describes a total operation

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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

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Total operation

The operation of purchasing a seat may be described by:

Purchase (Purchase0 Success)

(NotAvailable Failure)

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Question

• How might we complete the following operation?

ReturnTicket0 BoxOffice s?: Seat c?: Customer …

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ReturnTicket0

ReturnTicket0 BoxOffice s?: Seat c?: Customer s? c? sold sold = sold \ {s? c?} seating = seating

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Question

• How might we complete the following operation?

ReturnNotPossible BoxOffice s?: Seat c?: Customer …

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ReturnNotPossible

ReturnNotPossible BoxOffice s?: Seat c?: Customer s? c? sold

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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