Introduction to Algebraic Data Types in Haskell
Most programming languages have a way to make compound data types. In Haskell, we can do that via algebraic data types. Even though the name might sound scary at first, it’s simply a way to construct types.
This article will introduce you to the concept of algebraic data types and show you how to use them.
Read further to learn:
 how to create your own Haskell data types;
 what product and sum types are;
 why algebraic data types are called algebraic;
 how to use common Haskell ADTs such as
Maybe
andEither
;  why functions are called exponential types.
If you want to watch this article as a 10minute video, you can do that on our YouTube channel.
Product types
How to define a new data type in Haskell?
Let’s start by creating a data type for a 2dimensional point.
New data types are created via the data
keyword. To create a Point
data type, we need to provide the name of a type constructor (the name of our type), data constructor (used to construct new instances of the type), followed by the types our type will contain.
 [1] [2] [3]
data Point = Point Double Double
deriving (Show, Eq)
 [1]: Type constructor.
 [2]: Data constructor.
 [3]: Types wrapped.
A few notes on this piece of code.
First of all, there’s a difference between the type constructor and the data constructor. In our example, they are called the same, but they could have been Point
and Point2D
, for example. This frequently confuses beginners.
 Type constructor  Data constructor 
 Name of the type.  Used to construct an instance of a type. 
 Each type can have only one type constructor.  Each type can have multiple data constructors (in case of sum types). 
Second, adding deriving (Show, Eq)
to the type definition above makes it possible to print values of the type and to compare them for equality. You can read more about deriving in this blog post.
Let’s play with our Point
type in GHCi.
We can create new values of this type via the data constructor.
*Main> a = Point 3 4
*Main> a
Point 3.0 4.0
And we can create functions that pattern match on constructors and values inside them.
*Main> distance (Point x1 y1) (Point x2 y2) = sqrt ((x1  x2) ^ 2 + (y1  y2) ^ 2)
*Main> a = Point 3 4
*Main> b = Point 1 2
*Main> distance a b
2.8284271247461903
Definition of a product type
We call Point
(and all types with a similar structure) a product type. All product types combine multiple elements that are in the data structure at the same time. It’s the same as saying that you need this type and that type.
Polymorphic data types
Our previously created Point
data type can contain only doubleprecision floats.
In some cases, we would want it to work with other numbers as well. If so, we need to make it polymorphic (able to work with multiple different data types).
data PPoint a = PPoint a a
deriving (Show, Eq)
Here, we provide the type constructor with a type variable a
, which we can later use in the definition of our type. In contrast to our previous Point
type, PPoint
is a type that needs to be “completed” by providing it with a concrete type.
To better illustrate this fact, we can look at the kinds of both functions. While it is not possible to fully explain kinds in this article, you can think of them as type signatures for types.
If you wish to read more about kinds, I suggest this article by Diogo Castro.
As we can see, Point
is a concrete type.
*Main> :kind Point
Point :: *
In contrast, PPoint
is a function that takes a type and returns a concrete type.
*Main> :kind PPoint
PPoint :: * > *
Another typical example of a polymorphic product type is the tuple type.
*Main> :info (,)
type (,) :: * > * > *
data (,) a b = (,) a b
It takes two types – a
and b
– and returns a type that has a
in the first slot and b
in the second slot.
Records
The individual types of our Point
type are not named. While it doesn’t really add any difficulty right now, working with something like Person String String String String
can be confusing.
An alternative is to use records, which have field labels.
data Point = Point
{ x :: Double
, y :: Double
}
deriving (Show, Eq)
*Main> a = Point 3 4
*Main> a
Point {x = 3.0, y = 4.0}
Records also provide us with getter functions for free. The names of those getters are the same as the field names.
*Main> x a
3.0
*Main> y a
4.0
You can update a record by providing the fields you want to update (rest will stay the same).
*Main> b = a {x = 4}
*Main> b
Point {x = 4.0, y = 4.0}
And you can put these two things together to create functional record updates.
*Main> moveUp point = point {y = y point + 1}
*Main> c = moveUp a
*Main> c
Point {x = 3.0, y = 5.0}
Of course, you can also work with records via pattern matching as with basic product types.
*Main> getX (Point x _) = x
*Main> getX a
3.0
Sum types
There’s another flavor of types – sum types – that lists several possible variants a type could have. You might have encountered something similar under the name of enums or union types.
The most simple example of a sum type is Bool
.
 [1] [2] [3] [4]
data Bool = False  True
 [1]: Type constructor.
 [2, 4]: Data constructors.
 [3]: The pipe operator that separates data constructors.
Bool
can be constructed by either True
or False
.
We can make functions such as a negation that work on the values of Bool
.
neg :: Bool > Bool
neg True = False
neg False = True
There are a lot of sum types in the wild that you wouldn’t even necessarily recognize as such. While it is not defined that way, an Int
can be thought of as the enumeration of all the entries in [2^29 .. 2^291]
, for example.
A more nontrivial example of a sum type would be a data type that fits both a 2dimensional and a 3dimensional point.
data Point = Point2D Double Double  Point3D Double Double Double
deriving (Show, Eq)
Now we can write a function that accepts both types of points by pattern matching on the data constructors.
pointToList :: Point > [Double]
pointToList (Point2D x y) = [x, y]
pointToList (Point3D x y z) = [x, y, z]
Here’s an example of its usage:
*Main> a = Point2D 3 4
*Main> b = Point3D 3 4 5
*Main> pointToList a
[3.0,4.0]
*Main> pointToList b
[3.0,4.0,5.0]
Definition of a sum type
Like product types, sum types are a way of putting together basic types to create a more complex one. But in comparison to product types, only one of those types can be present in any given instance of the type.
In other words, using a sum type is like saying that you need type a or type b: “I need True or False”, “I need a 2D point or a 3D point”, etc.
Product types vs. sum types
Here’s a small table to help you remember the differences between these two groups of types.
  Product types  Sum types 
 Example  data (,) a b = (,) a b
 data Bool = False  True

 Intuition  Give me a and b  Give me a or b 
Algebraic data types
So why are these types called product and sum types? Let’s get into it.
If you remember your school math lessons, you worked with numbers (111, 222, 333, etc.), variables (xxx, yyy, zzz, etc.), and operators (+++, −−, ∗*∗, etc.). In algebraic data types, our numbers are the number of possible values a type can have and our operators are 
and data constructors.
Summing types
If we use 
in the definition of a type, the type can have a value from the values of types on either side of the operator. As such, the amount of possible values it has is the sum of the amount of values those types have.
For example, False
contains only one value. True
also contains only one value. Bool = False  True
contains 1+11 + 11+1 values. If we add an Unknown
value to Bool
, we will have a type with three possible values, and so on.
Multiplying types
If we use a data constructor, our type can have all the possible combinations of the sets of values we provide. As such, the amount of possible values it has is the product of the amount of values those types have.
For example, if our type consists of two booleans, such as whether a person checked in for both parts of a return flight, it will have 2∗2=42 * 2 = 42∗2=4 possible values.
data CheckedInStatus = CheckedInStatus Bool Bool
Possible values of CheckedInStatus
:
True True
True False
False True
False False
Definition of an algebraic data type
By putting together sum and product types, we can construct elaborate types out of simple building blocks.
And this is what algebraic data types work with. They are a collection of one or more data constructors, composed with the 
operator. In other words, they are a sum of products.
 [1] [2] [3]
data Point = Point2D Double Double  Point3D Double Double Double
 The Point data type is a sum ([2]) of products ([1], [3]).
Common ADTs
Now, let’s look at two commonly used ADTs in Haskell: Maybe
and Either
.
Maybe
First ADT we’ll cover is Maybe
, which you might have encountered in other languages as Optional
.
*Main> :info Maybe
type Maybe :: * > *
data Maybe a = Nothing  Just a
Sometimes, a function might not be able to return a value for a certain input. In that case, we can use the Maybe
type. It has two possible data constructors: Just
or Nothing
. If the function succeeds, we wrap the result in Just
. Otherwise, we return Nothing
, which symbolizes something similar to null.
For example, Prelude
has a scary function called head
, which works on lists but not all of them.
In case we call it with an empty list, we’ll get an exception:
*Main> head []
*** Exception: Prelude.head: empty list
We can make it give a result for each input by pattern matching on the contents of the list and returning Nothing
in the case of an empty list.
safeHead :: [a] > Maybe a
safeHead [] = Nothing
safeHead (x : _) = Just x
*Main> safeHead []
Nothing
*Main> safeHead [1, 2, 3]
Just 1
All in all, you can think of Maybe
as a safer alternative to null.
Either
Now, what if you want to know what made the function fail?
In that case, there is another data type that we can use – Either
. It functions similarly to what is called Result
in other languages.
*Main> :info Either
type Either :: * > * > *
data Either a b = Left a  Right b
In contrast to Maybe
, it can store something on the left side, such as an error message.
Let’s rewrite our safeHead
function with Either
.
safeHead :: [a] > Either String a
safeHead [] = Left "I have no head."
safeHead (x : _) = Right x
*Main> safeHead []
Left "I have no head."
*Main> safeHead [1, 2, 3]
Right 1
All in all, you can think of Either
as a safer alternative to exceptions.
Exponential types (functions)
To blow your mind a little bit in the end: functions can also add to our “type algebra” since they also have types.
Imagine we have a data type for traffic lights.
data Light = Green  Yellow  Red
How many possible values are in the type Light > Bool
? (One can imagine that they encode all possible rules for when is it legal to cross the street.)
Let’s try to write them all out.
 True if it is Green, False if it is Yellow or Red.
 True if it is Green or Yellow, False if it is Red.
 True if it is Green, Yellow, or Red.
 True if it is Yellow or Red, False if it is Green.
 True if it is Red, False if it is Green or Yellow.
 False if it is Green, Yellow, or Red.
 True if it is Green or Red, False if it is Yellow.
 True if it is Yellow, False if it is Green or Red.
The final number is 888, or 232^323.
Turns out, if we have two types aaa and bbb with the amount of values inside those types being ∣a∣a∣a∣ and ∣b∣b∣b∣, respectively, then there are ∣b∣∣a∣b^ {a}∣b∣∣a∣ functions in the set of possible functions from aaa to bbb.
Conclusion
In this article, we explored common ways of defining your own data types in Haskell. We looked at product types and sum types and how they work together to create algebraic data types. We also looked at common data types such as Maybe
and Either
, and saw how functions are exponential data types.
Haskell’s type system is large and enables you to be very expressive, so there are a lot of things that we didn’t cover in our blog post, such as newtypes.
If you want to read more of our Haskell articles, follow us on Twitter or Dev.
Exercises
In case you want some practice with algebraic data types, here are a couple of quick exercises.
Create a data type called
Person
that stores a person’s full name, address, and phone number. Create a function for getting a person’s name and a function for changing their phone number.Convert the data type created in exercise 1 to a record.
Given a data type for days of the week:
data Day = Monday  Tuesday  Wednesday  Thursday  Friday  Saturday  Sunday
, write two functions:Recall the
Maybe
data type we covered earlier. Write a ‘tail’ function for a list with the type signature ofsafeTail :: [a] > Maybe [a]
. It should take a list and return the list without the first element, wrapped inJust
. In case that is not possible, it should returnNothing
.
Some examples of its behavior:
Appendix
Here’s a handy table of some of the types we have covered in this article and how to compute the cardinality (how many members the type has) of those types, assuming that the cardinalities of their components are known.
Note: ∣a∣a∣a∣ notes the cardinality of the type aaa in the table.
 Name  Haskell  Cardinality 
 Bool  data Bool = False  True

1+11 + 11+1

 Maybe  data Maybe a = Nothing  Just a

1+∣a∣1 + a1+∣a∣

 Either  data Either a b = Left a  Right b

∣a∣+∣b∣a + b∣a∣+∣b∣

 Tuple  (a, b)

∣a∣∗∣b∣a * b∣a∣∗∣b∣

 Function  a > b

∣b∣∣a∣b ^{a}∣b∣∣a∣

 2D point  data Point = Point Double Double

∣Double∣∗∣Double∣Double * Double∣Double∣∗∣Double∣

 2D or 3D point  data Point = Point2D Double Double  Point3D Double Double Double

∣Double∣∗∣Double∣+∣Double∣∗∣Double∣∗∣Double∣Double * Double + Double * Double * Double∣Double∣∗∣Double∣+∣Double∣∗∣Double∣∗∣Double∣

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