# Algebraic data type

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In computer programming, especially functional programming and type theory, an algebraic data type is a kind of composite type, i.e., a type formed by combining other types.

## Contents

Two common classes of algebraic types are product types (i.e., tuples and records) and sum types (i.e., tagged or disjoint unions, coproduct types or variant types). [1]

The values of a product type typically contain several values, called fields. All values of that type have the same combination of field types. The set of all possible values of a product type is the set-theoretic product, i.e., the Cartesian product, of the sets of all possible values of its field types.

The values of a sum type are typically grouped into several classes, called variants. A value of a variant type is usually created with a quasi-functional entity called a constructor. Each variant has its own constructor, which takes a specified number of arguments with specified types. The set of all possible values of a sum type is the set-theoretic sum, i.e., the disjoint union, of the sets of all possible values of its variants. Enumerated types are a special case of sum types in which the constructors take no arguments, as exactly one value is defined for each constructor.

Values of algebraic types are analyzed with pattern matching, which identifies a value by its constructor or field names and extracts the data it contains.

Algebraic data types were introduced in Hope, a small functional programming language developed in the 1970s at the University of Edinburgh. [2]

## Examples

One of the most common examples of an algebraic data type is the singly linked list. A list type is a sum type with two variants, Nil for an empty list and Cons xxs for the combination of a new element x with a list xs to create a new list. Here is an example of how a singly linked list would be declared in Haskell:

dataLista=Nil|Consa(Lista)

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

For a slightly more complex example, binary trees may be implemented in Haskell as follows:

dataTree=Empty|LeafInt|NodeTreeTree

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 example, 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->IntdepthEmpty=0depth(Leafn)=1depth(Nodelr)=1+max(depthl)(depthr)

Thus, a Tree given to depth can be constructed using any of Empty, Leaf, or Node and must be matched 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 highly suited to implementing abstract syntax. For example, the following algebraic data type describes a simple language representing numerical expressions:

dataExpression=NumberInt|AddExpressionExpression|MinusExpressionExpression|MultExpressionExpression|DivideExpressionExpression

An element of such a data type would have a form such as Mult (Add (Number 4) (Minus (Number 0) (Number 1))) (Number 2).

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

## Explanation

What is happening is that there is 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 viewed 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 (e.g., "Empty" in the example above), or one piece of data (e.g., “Leaf” has one Int value), or multiple pieces of data (e.g., “Node” has two Tree values).

To do something with a value of this Tree algebraic data type, it is deconstructed using a process termed 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 above has a form that resembles the structure of some possible value of this datatype. The first pattern simply matches values of the constructor Empty. The second pattern 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 evaluate.

The recursion in patterns in this example are trivial, but a possible more complex recursive pattern would be something like Node(Node(Leaf4)x)(Nodey(NodeEmptyz)). 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:

switchon(data.constructor)caseEmpty:return0caseLeaf:letl=data.field1return1caseNode:letl=data.field1letr=data.field2return1+max(depthl)(depthr)

The comparison of this with pattern matching will point out some of the advantages of algebraic data types and pattern matching. The first advantage 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 can be extracted 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 complex recursive patterns, the task becomes difficult for the average human (or compiler, if it must check arbitrary nested if-else constructs) to handle. Similarly, there may be patterns which never match (i.e., are already covered by prior 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. However, 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.

## Theory

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.

dataLista=Nil|Consa(Lista)

is represented in type theory as ${\displaystyle \lambda \alpha .\mu \beta .1+\alpha \times \beta }$ with constructors ${\displaystyle \mathrm {nil} _{\alpha }=\mathrm {roll} \ (\mathrm {inl} \ \langle \rangle )}$ and ${\displaystyle \mathrm {cons} _{\alpha }\ x\ l=\mathrm {roll} \ (\mathrm {inr} \ \langle x,l\rangle )}$.

The Haskell List datatype can also be represented in type theory in a slightly different form, thus: ${\displaystyle \mu \phi .\lambda \alpha .1+\alpha \times \phi \ \alpha }$. (Note how the ${\displaystyle \mu }$ and ${\displaystyle \lambda }$ 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. (The type variable ${\displaystyle \phi }$ is used to suggest a function rather than a base type like ${\displaystyle \beta }$, since ${\displaystyle \phi }$ is like a Greek f.) The function must also now be applied ${\displaystyle \phi }$ to its argument type ${\displaystyle \alpha }$ 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 expressing so-called nested data types, i.e., those where the recursive type differs parametrically from the original. (For more information on nested data types, see the works of Richard Bird, Lambert Meertens, and Ross Paterson.)

In set theory the equivalent of a sum 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).

## Programming languages with algebraic data types

Many programming languages incorporate algebraic data types as a first class notion, including:

## Related Research Articles

In computer science, a binary tree is a tree data structure in which each node has at most two children, which are referred to as the left child and the right child. A recursive definition using just set theory notions is that a (non-empty) binary tree is a tuple, where L and R are binary trees or the empty set and S is a singleton set containing the root. Some authors allow the binary tree to be the empty set as well.

In mathematics and computer science, mutual recursion is a form of recursion where two mathematical or computational objects, such as functions or data types, are defined in terms of each other. Mutual recursion is very common in functional programming and in some problem domains, such as recursive descent parsers, where the data types are naturally mutually recursive.

Standard ML (SML) is a general-purpose, modular, functional programming language with compile-time type checking and type inference. It is popular among compiler writers and programming language researchers, as well as in the development of theorem provers.

In computer science, a list or sequence is an abstract data type that represents a countable number of ordered values, where the same value may occur more than once. An instance of a list is a computer representation of the mathematical concept of a tuple or finite sequence; the (potentially) infinite analog of a list is a stream. Lists are a basic example of containers, as they contain other values. If the same value occurs multiple times, each occurrence is considered a distinct item.

In computer science, pattern matching is the act of checking a given sequence of tokens for the presence of the constituents of some pattern. In contrast to pattern recognition, the match usually has to be exact: "either it will or will not be a match." The patterns generally have the form of either sequences or tree structures. Uses of pattern matching include outputting the locations of a pattern within a token sequence, to output some component of the matched pattern, and to substitute the matching pattern with some other token sequence.

Curry is an experimental functional logic programming language, based on the Haskell language. It merges elements of functional and logic programming, including constraint programming integration.

In computer science, a tagged union, also called a variant, variant record, choice type, discriminated union, disjoint union, sum type or coproduct, is a data structure used to hold a value that could take on several different, but fixed, types. Only one of the types can be in use at any one time, and a tag field explicitly indicates which one is in use. It can be thought of as a type that has several "cases", each of which should be handled correctly when that type is manipulated. This is critical in defining recursive datatypes, in which some component of a value may have the same type as the value itself, for example in defining a type for representing trees, where it is necessary to distinguish multi-node subtrees and leaves. Like ordinary unions, tagged unions can save storage by overlapping storage areas for each type, since only one is in use at a time.

The Glasgow Haskell Compiler (GHC) is an open-source native code compiler for the functional programming language Haskell. It provides a cross-platform environment for the writing and testing of Haskell code and it supports numerous extensions, libraries, and optimisations that streamline the process of generating and executing code. GHC is the most commonly used Haskell compiler. The lead developers are Simon Peyton Jones and Simon Marlow.

In computer science, tree traversal is a form of graph traversal and refers to the process of visiting each node in a tree data structure, exactly once. Such traversals are classified by the order in which the nodes are visited. The following algorithms are described for a binary tree, but they may be generalized to other trees as well.

System F, also known as the (Girard–Reynolds) polymorphic lambda calculus or the second-order lambda calculus, is a typed lambda calculus that differs from the simply typed lambda calculus by the introduction of a mechanism of universal quantification over types. System F thus formalizes the notion of parametric polymorphism in programming languages, and forms a theoretical basis for languages such as Haskell and ML. System F was discovered independently by logician Jean-Yves Girard (1972) and computer scientist John C. Reynolds (1974).

In computer science, corecursion is a type of operation that is dual to recursion. Whereas recursion works analytically, starting on data further from a base case and breaking it down into smaller data and repeating until one reaches a base case, corecursion works synthetically, starting from a base case and building it up, iteratively producing data further removed from a base case. Put simply, corecursive algorithms use the data that they themselves produce, bit by bit, as they become available, and needed, to produce further bits of data. A similar but distinct concept is generative recursion which may lack a definite "direction" inherent in corecursion and recursion.

In computer programming languages, a recursive data type is a data type for values that may contain other values of the same type. Data of recursive types are usually viewed as directed graphs.

In computer science, recursion is a method of solving a problem where the solution depends on solutions to smaller instances of the same problem. Such problems can generally be solved by iteration, but this needs to identify and index the smaller instances at programming time. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science.

The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite statement. In the same manner, an infinite number of computations can be described by a finite recursive program, even if this program contains no explicit repetitions.

In mathematics, an initial algebra is an initial object in the category of F-algebras for a given endofunctor F. This initiality provides a general framework for induction and recursion.

In category theory, the concept of catamorphism denotes the unique homomorphism from an initial algebra into some other algebra.

In functional programming, fold refers to a family of higher-order functions that analyze a recursive data structure and through use of a given combining operation, recombine the results of recursively processing its constituent parts, building up a return value. Typically, a fold is presented with a combining function, a top node of a data structure, and possibly some default values to be used under certain conditions. The fold then proceeds to combine elements of the data structure's hierarchy, using the function in a systematic way.

In functional programming, a generalized algebraic data type is a generalization of parametric algebraic data types.

In computer programming, an anamorphism is a function that generates a sequence by repeated application of the function to its previous result. You begin with some value A and apply a function f to it to get B. Then you apply f to B to get C, and so on until some terminating condition is reached. The anamorphism is the function that generates the list of A, B, C, etc. You can think of the anamorphism as unfolding the initial value into a sequence.

A zipper is a technique of representing an aggregate data structure so that it is convenient for writing programs that traverse the structure arbitrarily and update its contents, especially in purely functional programming languages. The zipper was described by Gérard Huet in 1997. It includes and generalizes the gap buffer technique sometimes used with arrays.

In computer science, polymorphic recursion refers to a recursive parametrically polymorphic function where the type parameter changes with each recursive invocation made, instead of staying constant. Type inference for polymorphic recursion is equivalent to semi-unification and therefore undecidable and requires the use of a semi-algorithm or programmer supplied type annotations.

## References

1. Records and variants- OCaml manual section 1.4 Archived 2020-04-28 at the Wayback Machine
2. Paul Hudak; John Hughes; Simon Peyton Jones; Philip Wadler. "A history of Haskell: being lazy with class". Proceedings of the third ACM SIGPLAN conference on History of programming languages. Presentations included Rod Burstall, Dave MacQueen, and Don Sannella on Hope, the language that introduced algebraic data types
3. Calculus of Inductive Constructions, and basic standard libraries : Datatypes and Logic.
4. "CppCon 2016: Ben Deane "Using Types Effectively"" via www.youtube.com.
5. "Enum Instance". Haxe - The Cross-platform Toolkit.
6. "JEP 360: Sealed Classes (Preview)". OpenJDK.

This article is based on material taken from algebraic data type at the Free On-line Dictionary of Computing prior to 1 November 2008 and incorporated under the "relicensing" terms of the GFDL, version 1.3 or later.