Haskell Type vs Data Constructor

 

I am learning Haskell from learnyouahaskell.com. I am having trouble understanding type constructors and data constructors. For example, I don't really understand the difference between this: 

data Car = Car { company :: String  
               , model :: String  
               , year :: Int  
               } deriving (Show)

and this: 

data Car a b c = Car { company :: a  
                     , model :: b  
                     , year :: c   
                     } deriving (Show)

I understand that the first is simply using one constructor ( Car ) to built data of type Car . I don't really understand the second one.

Also, how do data types defined like this: 

data Color = Blue | Green | Red

fit into all of this?

From what I understand, the third example ( Color ) is a type which can be in three states: Blue , Green or Red . But that conflicts with how I understand the first two examples: is it that the type Car can only be in one state, Car , which can take various parameters to build? If so, how does the second example fit in?

Essentially, I am looking for an explanation that unifies the above three code examples/constructs.

In a data declaration, a type constructor is the thing on the left hand side of the equals sign. The data constructor(s) are the things on the right hand side of the equals sign. You use type constructors where a type is expected, and you use data constructors where a value is expected.

Data constructors

To make things simple, we can start with an example of a type that represents a colour. 

data Colour = Red | Green | Blue

Here, we have three data constructors. Colour is a type, and Green is a constructor that contains a value of type Colour . Similarly, Red and Blue are both constructors that construct values of type Colour . We could imagine spicing it up though! 

data Colour = RGB Int Int Int

We still have just the type Colour , but RGB is not a value – it's a function taking three Ints and returning a value! RGB has the type 

RGB :: Int -> Int -> Int -> Colour

RGB is a data constructor that is a function taking some values as its arguments, and then uses those to construct a new value. If you have done any object-oriented programming, you should recognise this. In OOP, constructors also take some values as arguments and return a new value!

In this case, if we apply RGB to three values, we get a colour value! 

Prelude> RGB 12 92 27
#0c5c1b

We have constructed a value of type Colour by applying the data constructor. A data constructor either contains a value like a variable would, or takes other values as its argument and creates a new value. If you have done previous programming, this concept shouldn't be very strange to you.

Intermission

If you'd want to construct a binary tree to store String s, you could imagine doing something like 

data SBTree = Leaf String
            | Branch String SBTree SBTree

What we see here is a type SBTree that contains two data constructors. In other words, there are two functions (namely Leaf and Branch ) that will construct values of the SBTree type. If you're not familiar with how binary trees work, just hang in there. You don't actually need to know how binary trees work, only that this one stores String s in some way.

We also see that both data constructors take a String argument – this is the String they are going to store in the tree.

But! What if we also wanted to be able to store Bool , we'd have to create a new binary tree. It could look something like this: 

data BBTree = Leaf Bool
            | Branch Bool BBTree BBTree

Type constructors

Both SBTree and BBTree are type constructors. But there's a glaring problem. Do you see how similar they are? That's a sign that you really want a parameter somewhere.

So we can do this: 

data BTree a = Leaf a
             | Branch a (BTree a) (BTree a)

Now we introduce a type variable a as a parameter to the type constructor. In this declaration, BTree has become a function. It takes a type as its argument and it returns a new type.

It is important here to consider the difference between a concrete type (examples include Int , [Char] and Maybe Bool ) which is a type that can be assigned to a value in your program, and a type constructor function which you need to feed a type to be able to be assigned to a value. A value can never be of type "list", because it needs to be a "list of something". In the same spirit, a value can never be of type "binary tree", because it needs to be a "binary tree storing something".

If we pass in, say, Bool as an argument to BTree , it returns the type BTree Bool , which is a binary tree that stores Bool s. Replace every occurrence of the type variable a with the type Bool , and you can see for yourself how it's true.

If you want to, you can view BTree as a function with the kind 

BTree :: * -> *

Kinds are somewhat like types – the * indicates a concrete type, so we say BTree is from a concrete type to a concrete type.

Wrapping up

Step back here a moment and take note of the similarities.

  • A data constructor is a "function" that takes 0 or more values and gives you back a new value.

  • A type constructor is a "function" that takes 0 or more types and gives you back a new type.

    • Data constructors with parameters are cool if we want slight variations in our values – we put those variations in parameters and let the guy who creates the value decide what arguments they are going to put in. In the same sense, type constructors with parameters are cool if we want slight variations in our types! We put those variations as parameters and let the guy who creates the type decide what arguments they are going to put in.

    A case study

    As the home stretch here, we can consider the Maybe a type. Its definition is 

    data Maybe a = Nothing
             | Just a

    Here, Maybe is a type constructor that returns a concrete type. Just is a data constructor that returns a value. Nothing is a data constructor that contains a value. If we look at the type of Just , we see that 

    Just :: a -> Maybe a

    In other words, Just takes a value of type a and returns a value of type Maybe a . If we look at the kind of Maybe , we see that 

    Maybe :: * -> *

    In other words, Maybe takes a concrete type and returns a concrete type.

    Once again! The difference between a concrete type and a type constructor function. You cannot create a list of Maybe s - if you try to execute 

    [] :: [Maybe]

    you'll get an error. You can however create a list of Maybe Int , or Maybe a . That's because Maybe is a type constructor function, but a list needs to contain values of a concrete type. Maybe Int and Maybe a are concrete types (or if you want, calls to type constructor functions that return concrete types.)

    Haskell has algebraic data types, which very few other languages have. This is perhaps what's confusing you.

    In other languages, you can usually make a "record", "struct" or similar, which has a bunch of named fields that hold various different types of data. You can also sometimes make an "enumeration", which has a (small) set of fixed possible values (eg, your Red , Green and Blue ).

    In Haskell, you can combine both of these at the same time. Weird, but true!

    Why is it called "algebraic"? Well, the nerds talk about "sum types" and "product types". For example: 

    data Eg1 = One Int | Two String

    An Eg1 value is basically either an integer or a string. So the set of all possible Eg1 values is the "sum" of the set of all possible integer values and all possible string values. Thus, nerds refer to Eg1 as a "sum type". On the other hand: 

    data Eg2 = Pair Int String

    Every Eg2 value consists of both an integer and a string. So the set of all possible Eg2 values is the Cartesian product of the set of all integers and the set of all strings. The two sets are "multiplied" together, so this is a "product type".

    Haskell's algebraic types are sum types of product types. You give a constructor multiple fields to make a product type, and you have multiple constructors to make a sum (of products).

    As an example of why that might be useful, suppose you have something that outputs data as either XML or JSON, and it takes a configuration record - but obviously, the configuration settings for XML and for JSON are totally different. So you might do something like this: 

    data Config = XML_Config {...} | JSON_Config {...}

    (With some suitable fields in there, obviously.) You can't do stuff like this in normal programming languages, which is why most people aren't used to it.

    Start with the simplest case: 

    data Color = Blue | Green | Red

    This defines a "type constructor" Color which takes no arguments - and it has three "data constructors", Blue , Green and Red . None of the data constructors takes any arguments. This means that there are three of type Color : Blue , Green and Red .

    A data constructor is used when you need to create a value of some sort. Like: 

    myFavoriteColor :: Color
myFavoriteColor = Green

    creates a value myFavoriteColor using the Green data constructor - and myFavoriteColor will be of type Color since that's the type of values produced by the data constructor.

    A type constructor is used when you need to create a type of some sort. This is usually the case when writing signatures: 

    isFavoriteColor :: Color -> Bool

    In this case, you are calling the Color type constructor (which takes no arguments).

    Still with me?

    Now, imagine you not only wanted to create red/green/blue values but you also wanted to specify an "intensity". Like, a value between 0 and 256. You could do that by adding an argument to each of the data constructors, so you end up with: 

    data Color = Blue Int | Green Int | Red Int

    Now, each of the three data constructors takes an argument of type Int . The type constructor ( Color ) still doesn't take any arguments. So, my favorite color being a darkish green, I could write 

        myFavoriteColor :: Color
    myFavoriteColor = Green 50

    And again, it calls the Green data constructor and I get a value of type Color .

    Imagine if you don't want to dictate how people express the intensity of a color. Some might want a numeric value like we just did. Others may be fine with just a boolean indicating "bright" or "not so bright". The solution to this is to not hardcode Int in the data constructors but rather use a type variable: 

    data Color a = Blue a | Green a | Red a

    Now, our type constructor takes one argument (another type which we just call a !) and all of the data constructors will take one argument (a value!) of that type a . So you could have 

    myFavoriteColor :: Color Bool
myFavoriteColor = Green False

    or 

    myFavoriteColor :: Color Int
myFavoriteColor = Green 50

    Notice how we call the Color type constructor with an argument (another type) to get the "effective" type which will be returned by the data constructors. This touches the concept of kinds which you may want to read about over a cup of coffee or two.

    Now we figured out what data constructors and type constructors are, and how data constructors can take other values as arguments and type constructors can take other types as arguments. HTH.

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