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In computer programming, particularly functional programming and type theory, an algebraic data type (sometimes also called a variant type) is a datatype each of whose values is data from other datatypes wrapped in one of the constructors of the datatype. Any wrapped datum is an argument to the constructor. In contrast to other datatypes, the constructor is not executed and the only way to operate on the data is to unwrap the constructor using pattern matching.

The most common algebraic data type is a list with two constructors: Nil or [] for an empty list, and Cons (an abbreviation of construct), ::, or : for the combination of a new element with a shorter list (for example Cons 1 [2, 3, 4] or 1:[2,3,4]).

Special cases of algebraic types are product types i.e. tuples and records (only one constructor), sum types or tagged unions (many constructors with a single argument) and enumerated types (many constructors with no arguments). Algebraic types are one kind of composite type (i.e. a type formed by combining other types).

An algebraic data type may also be an abstract data type (ADT) if it is exported from a module without its constructors. Values of such a type can only be manipulated using functions defined in the same module as the type itself.

In set theory the equivalent of an algebraic data type is a disjoint union – a set whose elements are pairs consisting of a tag (equivalent to a constructor) and an object of a type corresponding to the tag (equivalent to the constructor arguments).

## ExamplesEdit

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

data Tree = Empty

| Leaf Int

| Node Tree Tree

Here, Empty, Leaf and Node are called data constructors. Tree is a type constructor (in this case a nullary one). In the rest of this article constructor shall mean data constructor. Similarly, in OCaml syntax the above example may be written:

type tree = Empty

| Leaf of int

| Node of tree * tree

In most languages that support algebraic data types, it's possible to define polymorphic 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

| 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.

## ExplanationEdit

What is happening is that we have a datatype, which can be “one of several types of things.” Each “type of thing” is associated with an identifier called a constructor, which can be thought of as a kind of tag for that kind of data. Each constructor can carry with it a different type of data. A constructor could carry no data at all (e.g. "Empty" in the example above), carry one piece of data (e.g. “Leaf” has one Int value), or multiple pieces of data (e.g. “Node” has two Tree values).

When we want to do something with a value of this Tree algebraic data type, we deconstruct it using a process known as pattern matching. It involves matching the data with a series of patterns. The example function "depth" above pattern-matches its argument with three patterns. When the function is called, it finds the first pattern that matches its argument, performs any variable bindings that are found in the pattern, and evaluates the expression corresponding to the pattern.

Each pattern has a form that resembles the structure of some possible value of this datatype. The first pattern above simply matches values of the constructor Empty. The second pattern above matches values of the constructor Leaf. Patterns are recursive, so then the data that is associated with that constructor is matched with the pattern "n". In this case, a lowercase identifier represents a pattern that matches any value, which then is bound to a variable of that name — in this case, a variable “n” is bound to the integer value stored in the data type — to be used in the expression to be evaluated.

The recursion in patterns in this example are trivial, but a possible more complex recursive pattern would be something like Node (Node (Leaf 4) x) (Node y (Node Empty z)). Recursive patterns several layers deep are used for example in balancing red-black trees, which involve cases that require looking at colors several layers deep.

The example above is operationally equivalent to the following pseudocode:

switch on (data.constructor)

case Empty:

  return 0

case Leaf:

  let n = data.field1

  return 1

case Node:

  let l = data.field1

  let r = data.field2

  return 1 + max (depth l) (depth r)


The comparison of this with pattern matching will point out some of the advantages of algebraic data types and pattern matching. First is type safety. The pseudocode above relies on the diligence of the programmer to not access field2 when the constructor is a Leaf, for example. Also, the type of field1 is different for Leaf and Node (for Leaf it is Int; for Node it is Tree), so the type system would have difficulties assigning a static type to it in a safe way in a traditional record data structure. However, in pattern matching, the type of each extracted value is checked based on the types declared by the relevant constructor, and how many values you can extract is known based on the constructor, so it does not face these problems.

Second, in pattern matching, the compiler statically checks that all cases are handled. If one of the cases of the “depth” function above were missing, the compiler would issue a warning, indicating that a case is not handled. This task may seem easy for the simple patterns above, but with many complicated recursive patterns, the task becomes difficult for the average human (or compiler, if it has to check arbitrary nested if-else constructs) to handle. Similarly, there may be patterns which never match (i.e. it is already covered by previous patterns), and the compiler can also check and issue warnings for these, as they may indicate an error in reasoning.

Do not confuse these patterns with regular expression patterns used in string pattern matching. The purpose is similar — to check whether a piece of data matches certain constraints, and if so, extract relevant parts of it for processing — but the mechanism is very different. This kind of pattern matching on algebraic data types matches on the structural properties of an object rather than on the character sequence of strings.

## TheoryEdit

A general algebraic data type is a possibly recursive sum type of product types. Each constructor tags a product type to separate it from others, or if there is only one constructor, the data type is a product type. Further, the parameter types of a constructor are the factors of the product type. A parameterless constructor corresponds to the empty product. If a datatype is recursive, the entire sum of products is wrapped in a recursive type, and each constructor also rolls the datatype into the recursive type.

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

is represented in type theory as

$\lambda \alpha. \mu \beta. 1 + \alpha \times \beta$

with constructors $\mathrm{nil}_\alpha = \mathrm{roll}\ (\mathrm{inl}\ \langle\rangle)$ and $\mathrm{cons}_\alpha\ x\ l = \mathrm{roll}\ (\mathrm{inr}\ \langle x, l\rangle)$.

The Haskell List datatype can also be represented in type theory in a slightly different form, as follows:

$\mu \phi. \lambda \alpha. 1 + \alpha \times \phi\ \alpha$.

(Note how the $\mu$ and $\lambda$ constructs are reversed relative to the original.) The original formation specified a type function whose body was a recursive type; the revised version specifies a recursive function on types. (We use the type variable $\phi$ to suggest a function rather than a "base type" like $\beta$, since $\phi$ is like a Greek "f".) Note that we must also now apply the function $\phi$ to its argument type $\alpha$ in the body of the type.

For the purposes of the List example, these two formulations are not significantly different; but the second form allows one to express so-called nested data types, i.e., those where the recursive type differs parametrically from the original.

## Programming languages with algebraic data typesEdit

The following programming languages have algebraic data types as a first class notion:

• Clean
• F#