# Algebraic data type

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This is a [[type]] where we specify the shape of each of the elements. |
This is a [[type]] where we specify the shape of each of the elements. |
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+ | [http://en.wikipedia.org/wiki/Algebraic_data_type Wikipedia has a thorough discussion.] |
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+ | "Algebraic" refers to the property that an Algebraic Data Type is created by "algebraic" operations. The "algebra" here is "sums" and "products": |
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+ | * "sum" is alternation (<hask>A | B</hask>, meaning <hask>A</hask> or <hask>B</hask> but not both) |
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+ | * "product" is combination (<hask>A B</hask>, meaning <hask>A</hask> and <hask>B</hask> together) |
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+ | Examples: |
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+ | * <hask>data Pair = P Int Double</hask> is a pair of numbers, an <hask>Int</hask> and a <hask>Double</hask> together. The tag <hask>P</hask> is used (in constructors and pattern matching) to combine the contained values into a single structure that can be assigned to a variable. |
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+ | * <hask>data Pair = I Int | D Double</hask> is just one number, either an <hask>Int</hask> or else a <hask>Double</hask>. In this case, the tags <hask>I</hask> and <hask>D</hask> are used (in constructors and pattern matching) to distinguish between the two alternatives. |
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+ | Sums and products can be repeatedly combined into an arbitrarily large structures. |
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+ | Algebraic Data Type is not to be confused with [[Abstract_data_type | *Abstract* Data Type]], which (ironically) is its opposite, in some sense. The initialism "ADT" usually means *Abstract* Data Type, but [[GADT]] usually means Generalized *Algebraic* Data Type. |
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==Tree examples== |
==Tree examples== |
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1 4 |
1 4 |
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− | We may actually use a variety of Haskell data declarations that will handle this. |
+ | We may actually use a variety of Haskell data declarations that will handle this. The choice of algebraic data types determines its structural/shape properties. |

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===Binary search tree=== |
===Binary search tree=== |
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===Rose tree=== |
===Rose tree=== |
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− | Alternatatively, it may be represented in what appears to be a totally different stucture. |
+ | Alternatively, it may be represented in what appears to be a totally different stucture. |

<haskell> |
<haskell> |
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data Rose a = Rose a [Rose a] |
data Rose a = Rose a [Rose a] |
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</haskell> |
</haskell> |
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− | In this case, the examlple tree would be: |
+ | In this case, the example tree would be: |

<haskell> |
<haskell> |
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retree = Rose 5 [Rose 3 [Rose 1 [], Rose 4[]], Rose 7 []] |
retree = Rose 5 [Rose 3 [Rose 1 [], Rose 4[]], Rose 7 []] |
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</haskell> |
</haskell> |
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− | The two representations are almost equivalent, with the exception that the |
+ | The differences between the two are that the (empty) binary search tree <hask>Tip</hask> is not representable as a <hask>Rose</hask> tree, and a Rose tree can have an arbitrary and internally varying branching factor (0,1,2, or more). |

− | binary search tree <hask>Tip</hask> is not representable in this <hask>Rose</hask> type declaration. Also, due to laziness, I believe we could represent infinite trees with the above declaration. |
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==See also== |
==See also== |
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*[[Abstract data type]] |
*[[Abstract data type]] |

## Revision as of 06:29, 24 December 2013

This is a type where we specify the shape of each of the elements. Wikipedia has a thorough discussion. "Algebraic" refers to the property that an Algebraic Data Type is created by "algebraic" operations. The "algebra" here is "sums" and "products":

- "sum" is alternation (, meaningA | BorAbut not both)B
- "product" is combination (, meaningA BandAtogether)B

Examples:

- is a pair of numbers, andata Pair = P Int Doubleand aInttogether. The tagDoubleis used (in constructors and pattern matching) to combine the contained values into a single structure that can be assigned to a variable.P
- is just one number, either andata Pair = I Int | D Doubleor else aInt. In this case, the tagsDoubleandIare used (in constructors and pattern matching) to distinguish between the two alternatives.D

Sums and products can be repeatedly combined into an arbitrarily large structures.

Algebraic Data Type is not to be confused with *Abstract* Data Type, which (ironically) is its opposite, in some sense. The initialism "ADT" usually means *Abstract* Data Type, but GADT usually means Generalized *Algebraic* Data Type.

## Contents |

## 1 Tree examples

Suppose we want to represent the following tree:

5 / \ 3 7 / \ 1 4

We may actually use a variety of Haskell data declarations that will handle this. The choice of algebraic data types determines its structural/shape properties.

### 1.1 Binary search tree

In this example, values are stored at each node, with smaller values to the left, greater to the right.

data Stree a = Tip | Node (Stree a) a (Stree a)

and then our example tree would be:

etree = Node (Node (Node Tip 1 Tip) 3 (Node Tip 4 Tip)) 5 (Node Tip 7 Tip)

To maintain the order, such a tree structure is usually paired with a smart constructor.

### 1.2 Rose tree

Alternatively, it may be represented in what appears to be a totally different stucture.

data Rose a = Rose a [Rose a]

In this case, the example tree would be:

retree = Rose 5 [Rose 3 [Rose 1 [], Rose 4[]], Rose 7 []]