**Question :**

**Haskell’s algebraic data types,**

**Answer :**

I’m trying to fully understand all of Haskell’s concepts.

In what ways are algebraic data types similar to generic types, e.g., in C# and Java? And how are they different? What’s so algebraic about them anyway?

I’m familiar with universal algebra and its rings and fields, but I only have a vague idea of how Haskell’s types work.

,

Haskell’s *algebraic data types* are named such since they correspond to an *initial algebra* in category theory, giving us some laws, some operations and some symbols to manipulate. We may even use algebraic notation for describing regular data structures, where:

`+`

represents sum types (disjoint unions, e.g.`Either`

).`â€¢`

represents product types (e.g. structs or tuples)`X`

for the singleton type (e.g.`data X a = X a`

)`1`

for the unit type`()`

- and
for the least fixed point (e.g. recursive types), usually implicit.`Î¼`

with some additional notation:

`XÂ²`

for`Xâ€¢X`

In fact, you might say (following Brent Yorgey) that a Haskell data type is regular if it can be expressed in terms of `1`

, `X`

, `+`

, `â€¢`

, and a least ï¬xed point.

With this notation, we can concisely describe many regular data structures:

- Units:
`data () = ()`

`1`

- Options:
`data Maybe a = Nothing | Just a`

`1 + X`

- Lists:
`data [a] = [] | a : [a]`

`L = 1+Xâ€¢L`

- Binary trees:
`data BTree a = Empty | Node a (BTree a) (BTree a)`

`B = 1 + Xâ€¢BÂ²`

Other operations hold (taken from Brent Yorgey’s paper, listed in the references):

- Expansion: unfolding the fix point can be helpful for thinking about lists.
`L = 1 + X + XÂ² + XÂ³ + ...`

(that is, lists are either empty, or they have one element, or two elements, or three, or …) - Composition,
`â—¦`

, given types`F`

and`G`

, the composition`F â—¦ G`

is a type which builds â€œF-structures made out of G-structuresâ€ (e.g.`R = X â€¢ (L â—¦ R)`

,where`L`

is lists, is a rose tree. - Differentiation, the derivative of a data type D (given as D’) is the type of D-structures with a single â€œholeâ€, that is, a distinguished location not containing any data. That amazingly satisfy the same rules as for differentiation in calculus:
`1â€² = 0`

`Xâ€² = 1`

`(F + G)â€² = F' + Gâ€²`

`(F â€¢ G)â€² = F â€¢ Gâ€² + Fâ€² â€¢ G`

`(F â—¦ G)â€² = (Fâ€² â—¦ G) â€¢ Gâ€²`

*References:*

- Oh My!, Brent A. Yorgey, Haskellâ€™10, September 30, 2010, Baltimore, Maryland, USA
- Conor McBride POPL 2008

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