Algebraic specifications provide a mathematical framework for describing abstract data types. This framework offers:

- language independence,
- implementation independence,
- specification of required types and operations thereon, and
- specification of semantics of operations.

sort IntSet imports Int, Bool signatures new : -> IntSet insert : IntSet × Int -> IntSet member : IntSet × Int -> Bool remove : IntSet × Int -> IntSetSuch a syntactic specification is sometimes called a

variables i, j : Int; s : IntSet axioms member(new(), i) = false member(insert(s, j), i) = if i = j then true else member(s, i)Is this complete? How do we know?

Classify functions:

- constructors: return IntSet
- inspectors: take IntSet as argument, returning some other value.

Completeness requires (at least): every inspector and auxiliary constructor is defined by one equation for each key constructor. We therefore need equations for remove.

remove(new(), i) = new() remove(insert(s, j), i) = if i = j then remove(s, i) else insert(remove(s, i), j)

Are we done yet? The completeness criterion (an equation defining member and remove for each of the new and insert constructors) is satisfied.

But does this really specify sets? Do the following properties hold?

- Order of insertion is irrelevant.
insert(insert(s, i), j) = insert(insert(s, j), i)

- Multiple insertion is irrelevant.
insert(insert(s, i), i) = insert(s, i)

- The
*final algebra*approach: two values of a sort are equal if they cannot be distinguished by application of inspector functions. That is, the only properties of an ADT should be those that are accessible by the ADT functions. In our example, two sets are equal if they cannot be distinguished by application of`member`

. Both of the required properties can be proven under this assumption. - The
*initial algebra*approach: two values of a sort are equal only if they are provably equal by the axioms. The properties cannot be proven under this assumption.

A good specification should provide equivalent interpretations under both the initial and final algebra approaches. Thus the properties mentioned above should be added in some fashion.

Changes we could make:

- Add the two equations giving a complete set specification.
- Add order-independence, but allow duplicates.
Introduce
`count`

to count occurrences of an element. -
Add neither equation and interpret as lists.
Add
`head`

and`tail`

or`retrieve`

to retrieve by list position.

Algebraic Specifications have a parallel in functional programming. Algebraic data types can be used to directly implement a set.

data IntSet = New | Insert IntSet Int member New i = False member (Insert s j) i | i == j = True | otherwise = member s i remove New i = New remove (Insert s j) i | i == j = remove s i | otherwise = Insert (remove s i) jNote that this implementation has strong parallels to the specification.

One aspect of the above implementation is that there can be different representations of the same set. For example, the following data structures all represent the same set.

`Insert (Insert New 3) 4`

,`Insert (Insert New 4) 3`

and`Insert (Insert (Insert New 4) 3) 3`

An alternative approach is to define `insert`

as a function that converts a set to a canonical form.
It does so by maintaing the set as an ordered list
and avoiding insertion of duplicate elements.

data IntSet = New | Insert IntSet Int new :: IntSet insert :: IntSet -> Int -> IntSet new = New insert New i = Insert New i insert (Insert s j) i | i == j = Insert s j | i < j = Insert (Insert s j) i | otherwise = Insert (insert s i) j member New i = False member (Insert s j) i | i == j = True | i < j = False | otherwise = member s i remove New i = New remove (Insert s j) i | i == j = s | i < j = Insert s j | otherwise = Insert (remove s i) jUsing this implementation the following three expressions have all generate the same representation, namely

` Insert (Insert New 4) 3 `

.
`Insert (Insert New 4) 3`

.`insert (insert new 3) 4`

,`insert (insert new 4) 3`

and`insert (insert (insert new 4) 3) 3`

`member`

and
`remove`

have been written to take advantage
of the ordered representation.