Schemas-Formal Specification Methods in Software Development-Lecture Slides, Slides of Software Development Methodologies

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Chapter No. 11

Schemas

Introduction

• In the Z notation there are two languages
  • the mathematical language and
  • the schema language.
• The mathematical language is used to describe various aspects of a design - objects, and - the relationships between them.
• The schema language is used to structure and compose descriptions - collating pieces of information, encapsulating them, and naming them for re-use. docsity.com

Introduction

• We believe that the use of schemas helps to promote a good specification style.
• However, as with any notation, the language of schemas requires careful and judicious application if it is not to be abused.
• We should take care to develop simple theories and to use schemas to present them in an elegant and comprehensible fashion.

Introduction

• This chapter is an introduction to schemas their appearance, and information it contain
• We see how they may be used as types, as declarations, and as predicates.
• In subsequent chapters, we present a language of schema operators, and show how schemas may be used in reasoning about formal descriptions.

The Schema

• One of the most basic things that we can do to help the reader or indeed the writer of a specification is to identify commonly used concepts and factor them out from the mathematical description of a system.
• In this way, we can encapsulate an important concept and give it a name, thus increasing our vocabulary and our mental power.

The Schema

• In formal specifications, we see a pattern occurring over and over again: a piece of mathematical text which is a structure describing some variables whose values are constrained in some way.
• We call this introduction of variables under some constraint a schema.

Example: Concert Hall

• The box office maintains a record of which seats have been sold, and to whom.
• This relationship is a functional because no seat can be sold to two different customers: sold ³ Seat © Customer
• To allow for the possibility that seats may be added to or removed from, we introduce a set seating, a subset of Seat, to represent the seating allocated for the performance.

Example: Concert Hall

• It should not be possible to book seating that has not been allocated; the following predicate should be true at all times dom sold º seating
• That is, the domain of sold should be a subset of seating.
• This property, together with the declarations of sold and seating, forms a schema which we shall call BoxOffice.

Formalizing Box Office

Example 11.3 We can write the box office schema in horizontal form, as

[Seating : ¡ Seat; sold : Seat © Customer | dom sold º seating]

Or in vertical form as

[Seating : ¡ Seat; sold : Seat © Customer

dom sold º seating]

Associating Names with Schema

• The schema language includes a special operator for associating names with it.
• Horizontal

Name á [declaration | predicate

• Vertical

Name declaration

predicate

Constraint as a Part of Predicate

• Declaration part of box office includes constraint that relation sold between Seat and Customer must be functional.
• The following schema, in which this constraint appears as part of the predicate, is entirely equivalent:

BoxOffice [Seating : ¡ Seat; sold : Seat ¨ Customer

dom sold º seating]^ sold ³ Seat © Customer

Another Way of Defining Schema

Example : 11.6 The schema of Example 11.5 could be written in the following form:

BoxOffice Seating : ¡ Seat sold : Seat ¨ Customer

dom sold º seating sold ³ Seat © Customer

Schemas as Type

• Schema below corresponds to a composite data type with two components: an integer called a, and a set of integers called c.

SchemaOne a : ¤ c : ¡ ¤

• We may introduce elements of this type in the usual way: the declaration s : SchemaOne introduces an object s of schema type SchemaOne.

Binding in a Schema

• To write an object of schema type in extension, we list the component names and the values to which they are bound. This requires a new piece of notation:

Ýa â 2, c â {1, 2, 3}Þ

is a binding in which a is bound to 2 and c is bound to the set {1, 2, 3}