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Chapter No. 10

Free Types

Introduction

• In the course of a specification we may define a variety of data structures: lists, arrays, or trees of elements drawn from one or more basic types.
• These structures could be modeled using a combination of sets and relations, but resulting definitions would be verbose.
• A more elegant, concise alternative is offered by free types: sets with explicit structuring information.

The Natural Numbers

• The set of all natural numbers, N, is already part of our mathematical language
• It has been defined as a subset of the built- in type Z.
• Here, the construction of a set similar to N will prove a useful illustration of free type.
• Thus we attempt to define such a set, beginning with a basic type nat, a zero element, and a partial function called succ

First Attempt Defining Natural No.

• Every element n is either constant zero or result of applying the succ to an element m.

zero : nat succ : nat © nat

Õ n : nat × n = zerp Ú Ö m : nat × n = succ m

• Unfortunately, this is not enough to capture the essence of N. docsity.com

3 rd^ Attempt Defining Natural No.

• Having realized that the function used to construct naturals must be total, try again:

zero : nat succ : nat ª nat

Õ n : nat × n = zerp Ú Ö m : nat × n = succ m {zero} » ran succ = ¸

• We failed to exclude possibility that some element is successor of two or more othersdocsity.com

Last Attempt Defining Natural No.

• We must add a fourth requirement that the function used must be injective.

zero : nat succ : nat ¬ nat

{zero} » ran succ = ¸ {zero} ¼ ran succ = nat

• The final requirement is that nat should be
• the smallest set that meets the conditions

Free Type Definition

• The above abbreviation not only fails to introduce names, also fails making distinct
• No guarantee, red is different from green.
• Consider free type definition

Colors :: = red | orange | yellow | green | blue | indigo | violet

• It introduces a set Colors, and seven distinct constants

Free Type Definition

• In this definition, we may infer that Colors is the smallest set containing the seven distinct elements red, orange, yellow, green, blue, indigo, and violet.
• The order is unimportant i.e. the definition

Colors :: = indigo | red | orange | yellow | green | blue | violet, same as

Colors :: = red | orange | yellow | green | blue | indigo | violet

Example 10.

• Suppose The University of Oxford awards BA, MSc., D. Phil, MA. For ceremonial purposes, degrees are ordered as follows:
• MA is the highest ranking
• D. Phil takes second place,
• MSc and BA third and fourth respectively.
• Suppose that we wish to represent this ordered collection of degrees as a free type.
• Ordering is similar to first four naturals

Degree ::= status Ý 0..3 Þ

First Attempt Defining Natural No.

• Every element n is either constant zero or result of applying the succ to an element m.

ba, msc, dphil, ma : Degree

ba = status 0 msc = status 1 dphil = status 2 ma = status 3

Partial Functions

Example 8.1 An organization has a system for keeping track of its employees while they are on the premises. Each employee is issued with an active badge which reports their current position to a central database.

• If set of all people is Person , and set of all locations is Location , then information may be described by a relation where_is of type Person  Locations
• This relation is a partial function, i.e., where_is  Person © Location

Total Functions

Definition: if each element of X is related to unique element of Y then partial function is a total function denoted by X ª Y.

X ª Y == { f : X©Y | dom f = X × f }

b c

d

1

2 3

X f Y

Example : Total Functions

Example 8.3 If Rachel is an employee,

then we may write where_is rachel to

denote her current location. This makes

sense only if the database has a unique

record of Rachel’s whereabouts. If

where_is is the function

{otto § lobby, peter § meeting,

quentin § meeting, rachel § meeting}

Then we may observe that

where_is rachel = meeting

Lambda Functions

• Suppose that f is a function whose domain is precisely those elements of X that satisfy a constraint p. If the result of applying f to an arbitrary element x can be written as the expression e, then f can be described as

f = {x :X | p  x § e}

• Lambda notation offers more conciseness ( declaration | constraint  result)
• Using lambda notation, our function f is f = (x :X | p  e)