Algebraic Data Type - Examples

Examples

One of the most common uses of algebraic data types is to represent singly linked lists. A list type is a sum type with two variants, Nil for an empty list and Cons x xs for the combination of a new element x with a list xs to create a new list:

data List a = Nil | Cons a (List a)

Cons is an abbreviation of construct. Many languages have special syntax for lists. For example, Haskell and ML use for Nil, :: or : for Cons, and square brackets for entire lists. So Cons 1 (Cons 2 (Cons 3 Nil)) would normally be written as 1:2:3: or in Haskell, or as 1::2::3:: or in ML.

For another example, in Haskell we can define a new algebraic data type, Tree:

data Tree = Empty | Leaf Int | Node Tree Tree

Here, Empty represents an empty tree, Leaf contains a piece of data, and Node organizes the data into branches.

In most languages that support algebraic data types, it is possible to define parametric types. Examples are given later in this article.

Somewhat similar to a function, a data constructor is applied to arguments of an appropriate type, yielding an instance of the data type to which the type constructor belongs. For instance, the data constructor Leaf is logically a function Int -> Tree, meaning that giving an integer as an argument to Leaf produces a value of the type Tree. As Node takes two arguments of the type Tree itself, the datatype is recursive.

Operations on algebraic data types can be defined by using pattern matching to retrieve the arguments. For example, consider a function to find the depth of a Tree, given here in Haskell:

depth :: Tree -> Int depth Empty = 0 depth (Leaf n) = 1 depth (Node l r) = 1 + max (depth l) (depth r)

Thus, a Tree given to depth can be constructed using any of Empty, Leaf or Node and we must match for any of them respectively to deal with all cases. In case of Node, the pattern extracts the subtrees l and r for further processing.

Algebraic data types are particularly well-suited to the implementation of abstract syntax. For instance, the following algebraic data type describes a simple language representing numerical expressions:

data Expression = Number Int | Add Expression Expression | Minus Expression | Mult Expression Expression | Divide Expression Expression

An element of such a data type would have a form such as Mult (Add (Number 4) (Minus (Number 1))) (Number 2).

Writing an evaluation function for this language is a simple exercise; however, more complex transformations also become feasible. For instance, an optimization pass in a compiler might be written as a function taking an abstract expression as input and returning an optimized form.