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A short introduction to haskell’s type system

import Prelude hiding (id)
import Data.List (delete)

Introduction

In haskell, every expression e has some type(s) T, which we often write like this:

e :: T

Here, an expression can be a value or a function (once again, as a functional programming language, haskell blurs the frontier between what is a function or a value - functions are regarded as values of a particular type)

Here are some examples that we have already seen

True :: Bool
5 :: Int
5 + 5 * 12 :: Int
[1, 2] :: [Int]
["a", "b", "aa"] :: [String]
[["a", "c"], ["aaaa"]] :: [[String]]
length :: [Int] -> Int

Much like other programming types can be regarded as sets of programs with some expected behaviour, which can often be summarized as “does not lead to some crash or very weird behaviour when put together with things of a compatible type”.

In haskell, all types are determined statically, which means at compile-time, much like Java. It means that, unlike in Python which has dynamical typing: * type representations do not pollute the runtime * there can be no type errors while a compiled program runs

Some basic type constructors

• Bool: boolean values, contain True and False
• [Bla]: lists of values of type Bla. The fundamental constructors for this type are the empty list [] and the (:) operator which allows to build a list by defining the element x :: Bla in front (colloquially called the head) and the rest of the list xs :: [Bla] (colloquially called the tail)

-- >>> True : False : []
-- [True,False]

/! the square brackets have a different meaning in code and in type annotations. In type annotation, it means “I want to talk about a type of lists”. In code this means “I am defining/analyzing a list via…”. It is simply a peculiarity of haskell that some notations like square brackets are used both in code and in types for similar things; some other programming languages like OCaml make those two things more distinctive (by calling the type of lists of Bla something like List Bla essentially).

In general the moral of this story is types and code/values live in separate worlds. Never mix types and values in this language.

• Char: characters

-- >>> :t 'c'
-- 'c' :: Char

• Int: fixed-width integers
• Integer: arbitrary sized integers (“bigints”)
• String: string types; in haskell, String = [Char], although there is a special notation for literals

-- >>> 'h' : 'i' : []
-- "hi"

Here might be a good place to mention that one may define constants as well as function in haskell file, using the same kind of notation. This can be useful sometimes with strings if you have e.g. a piece of text/special characters you are going to be using a lot.

upRightCorner :: String
upRightCorner = "┏"

leftRight :: String
leftRight = "━" 

somePath :: String
somePath = upRightCorner ++ concat (replicate 3 leftRight)

hWMsg :: String
hWMsg = "Hello world!"

• (Foo, Bar): pairs of elements of type Foo and Bar

-- >>> :t (True, "aaa")
-- (True, "aaa") :: (Bool, String)

-- >>> :t ("aa", "aa")
-- ("aa", "aa") :: (String, String)

This can be generalized to tuples of arbitrary length

-- >>> :t ()
-- () :: ()

-- >>> :t ("aa", "aa", True)
-- ("aa", "aa", True) :: (String, String, Bool)

Function types

Given types I and O, one can form the function type I -> O using the -> notation, meant to evoque an arrow as in mathematical notation. We have seen how to define function values using definitions in files and how to apply them, so let us not rehash that here; we are going to see fancier ways of defining functions later on when discussing higher-order functions such as these kind of monsters:

functionDefinedInAPointFreeManner =
  (> 56) . (\z -> z `div` 59 + 67 `mod` z)

We have also seen functions which have multiple arguments that can have types I1 -> … -> In -> O. Under the hood, there is something that we will notice again later: this is actually a nesting of the basic binary operator ->, with the rightmost parentheses. That means that implicitly, we have that

Int -> Int -> Int

actually is syntactic sugar for

Int -> (Int -> Int)

and the -> operator is viewed as any other elementary type constructor like tupling as far as the type system is concerned.

The type

(Int, Int) -> Int

is also a type of functions with two arguments, but coded with tuples and is not the same type, although it is morally the same. Going between the two types is called curryfication and uncurryfication, the arrowful type being the curryfied form. In functional style, we tend to like better function written in a currified form, for reasons that we will explain later.

Since -> is a basic type constructor, types such as [(Bool -> Bool, Bool)] are perfectly legal, and may even correspond to real things

>>> :t [(not, True)]
[(not, True)] :: [(Bool -> Bool, Bool)]

We will have further occasions to appreciate this in the future, with some actual consequences for programming in a functional style. But it is worth mentioning that part of the style of functional programming come from the fact that function types can be freely mixed with the other type constructors in a principled way, and a lot of the conventions such as the one for functions with multiple arguments being encoded with multiple arrows instead of tuple types also stem from this.

Parametric polymorphism

What makes haskell type system powerful is that expressions may actually have several types, including user-defined expressions.

take :: Int -> [Int] -> [Int]
length :: [[Int]] -> Int
length :: [Bool] -> Int
5 :: Float
[] :: [Int]
[] :: [Bool]

This relies not on some ad-hock compiler definitions, a type hierarchy or some kind of template programming, but rather on parametric polymorphism based on the Hindley-Milner type system. The idea behind this is that we can have function types that quantify over types using type variables.

Consider for instance the identity function (which returns its input).

id x = x

The identity can operate on any kind of data, be it numbers, lists, etc.

-- >>> id 5
-- 5

-- >>> id True
-- True

-- >>> id [["abc"], ["hh","jj"]]
-- [["abc"],["hh","jj"]]

We can check its type via the :t or :type directive in ghci/hugs:

-- >>> :t id
-- id :: p -> p

In a type, a lowercase identifier is called a type variable. They are supposed to denote any types. Here p -> p should be read as “for every element with given type p, this is a function which returns something of type p”.

Here are other examples:

-- >>> :t take
-- take :: Int -> [a] -> [a]

-- >>> :t fst
-- fst :: (a, b) -> a

-- >>> :t snd
-- snd :: (a, b) -> b

Below is an example of a custom polymorphic function definition, along with its type signature.

delta :: b -> (b, b)
delta x = (x, x)

The type should be read as “for every b, this function takes a b and returns a pair of bs”. We can check it works out of the box with all datatypes.

-- >>> delta 3
-- (3,3)

-- >>> delta "Hello"
-- ("Hello","Hello")

In all scenarios, the exact same piece of code is ran.

Note that there is almost a most general type associated to a function, and unless you are doing some rather unorthodox things with typeclasses, haskell interpreters/compilers will be able to guess them. You can try it out for yourself by typing definitions without the accompanying type signature and see what haskell tells you.

guessMyPolymorphicType x y = (x,x,y)

-- >>> :t guessMyPolymorphicType
-- guessMyPolymorphicType :: b -> c -> (b, b, c)

It is also possible to give a type annotation to a polymorphic function which is not the most general possible. If you do so, the typechecker won’t complain as long as you are consistent with the most general type, and will restrict your function to the types you specified.

fstVectorInt :: (Int, Int) -> Int
fstVectorInt x = fst x


-- >>> fstVectorInt (2,2)
-- 2

-- >>> fstVectorInt (True, False)
-- Couldn't match expected type `Int' with actual type `Bool'
-- In the expression: False
-- In the first argument of `fstVectorInt', namely `(True, False)'
-- In the expression: fstVectorInt (True, False)
-- Couldn't match expected type `Int' with actual type `Bool'
-- In the expression: True
-- In the first argument of `fstVectorInt', namely `(True, False)'
-- In the expression: fstVectorInt (True, False)

Type classes

Sometimes, it is useful to have a common syntax for operations that have a similar meaning but operate on different datatypes. For instance, one may want to be able to use the same symbol for addition of integer numbers (be they fixed-width or bigints) and floating point numbers.

>>> :t  (5 :: Int) + 6
(5 :: Int) + 6 :: Int

>>> :t (5.2 :: Double) + 2
(5.2 :: Double) + 2 :: Double

On the other hand, we cannot add lists of numbers together

>>> ([8] + [])
No instance for (Num [Integer]) arising from a use of `it_a2BVU'
In the first argument of `evalPrint', namely `it_a2BVU'
In a stmt of an interactive GHCi command: evalPrint it_a2BVU

Another example is the (==) operator, that can be used on a variety of datatypes, but certainly not all.

>>> 1 == 2
False

>>> [[1,1]] == [[1],[1]]
False

>>> take == drop
No instance for (Eq
                   (Int -> [a0_a2C4S[tau:1]] -> [a0_a2C4S[tau:1]]))
  arising from a use of `=='
  (maybe you haven't applied a function to enough arguments?)
In the expression: take == drop
In an equation for `it_a2C3T': it_a2C3T = take == drop

These are instances of what is called ad-hoc polymorphism and is similar in flavour with the way operator overloading is done with languages like java or python, but with a more flexible mechanism than messing with a rigid object-oriented hierarchy. The way haskell does it is that it allows to define so called type classes which are essentially a signature for a bunch of operation parameterized by a bunch of type variables.

Let us walk through the example of the type class that provides the ability to use equality. It is called Eq and is defined as follows in haskell (this is commented not to shadow the official definition, and actually the actual definition is a bit fancier, but we can pretend it is the definition for now):

class Eq a where
  (==) :: a -> a -> Bool

These two lines say: a type Bla belongs to the typeclass Eq if someone provides an operation (==) :: Bla -> Bla -> Bool and explicitly says this is tied to the typeclass Eq. Note that this does not create any function or any type; it simply declares that the typeclass exists and allows to then provide implementation. The way to do so for, say, a = Bool is to write

instance Eq Bool where
  (==) b c = b && c || not b && not c

This is called a typeclass instance. After this declaration, if the haskell compiler sees a statement like True == False, it would use the above function for the implementation of (==). For this to make sense, haskell also ensures each type can have at most one typeclass instance per typeclass.

Here is another example: the Show typeclass that is about types that can be converted to strings (note in passing that the difference between operators and functions is only one of notation; for haskell these are the same kind of object, and one can convert between the two viewpoints using either parentheses or backticks):

class Show a where
  show :: a -> String

instance Show Bool where
  show b = if b then
              "True"
           else
              "False"

There are fancier ways of defining typeclasses and instances tha make the system very powerful and used throughout all haskell libraries; you should not have to explore how to define custom classes and instances for this module, but since haskell provides a lot of default typeclasses and instances, you should know how to use the provided instances and to type functions that use them.

Let us consider an example: the function delete from the Data.List module shipped with your haskell installation. It takes two inputs, which may be for instance an integer and a list of integers, and returns a list of integers where the first occurence of the integer is deleted.

>>> delete 3 [2,3,4,3]
[2,4,3]

Without going into details, it should be fairly clear that * the implementation of this can be generalized to types other than integers, such as Char. Which is nice for string operations for instance

>>> delete 'e' "Hello"
"Hllo"

• however, one cannot do this for all datatypes; we need the ability to compare the first argument for equality with the elements within the second argument. Hence, the type of these arguments should have a typeclass for Eq. We can check we may get into issues if they don’t, like in the following example (OK the example is a bit contrived; a lot of types in haskell have an instance for Eq :))

>>> head (delete take [drop]) 1 []
No instance for (Eq (Int -> [()] -> [()]))
  arising from a use of `it_a2CaO'
  (maybe you haven't applied a function to enough arguments?)
In the first argument of `evalPrint', namely `it_a2CaO'
In a stmt of an interactive GHCi command: evalPrint it_a2CaO

So to capture these two things, the fact that the input type could be anything, but that the type of delete requires an instance of Eq, it is written this way:

>>> :t delete
delete :: Eq a => a -> [a] -> [a]

Before the part of the polymorphic type a -> [a] -> [a] that says what the function does, we add a typeclass constraint Eq a => that says that there should be an instance of Eq for a. So the informal way to read such a type would be to say: this is a function such that * for every a (quantify over the type variables) * that have an instance for Eq (relativize wrt the typeclass constraints) * this takes an argument of type a, an argument of type [a] and returns an value of type [a]

In particular, => is not the same thing as ->!! Count only the -> arrows if you want to figure out the number of arguments to a function.

Let us look at how it is done in a custom example which takes three arguments

inOrder :: Ord a => a -> a -> a -> Bool
inOrder x y z = x < y && y < z

Here the Ord typeclass shows up to allow the use of the (<) operator.

In some scenarios, one may need several typeclass constraints; let us give some example of the syntax needed in type annotations:

exampleWithNumAndOrd :: (Ord a, Num a) => a -> Bool
exampleWithNumAndOrd x = x * x > x

exampleWithALotOfTypeclassConstraints :: (Show a, Ord a, Num a, Read a) => a -> (String, Bool)
exampleWithALotOfTypeclassConstraints x = (show x, read "3" * x > x)

Regarding the typeclasses that are provided with haskell, you should probably get familiar with the following during the term:

• Num is the typeclass that provides basic numerical operations like (+), (-) (*), as well as conversion functions from numerals. That is the reason why numerals in ghci have type Num a => a: they are of any type a that can be converted to a Num (much like the empty list is a list of any type [a])

• Eq is the typeclass that provide operators (==) and (/=) that stand for equality and inequality

• Ord is the typeclass that provides comparison operators (>), (>=), …

• Show provides a conversion function from the parameter type to strings. In particular it is used in the interactive prompt to display values! That is why you may have errors mentioning Show if you try to ask ghci about values that have no obvious representations as strings (such as function types): it is not that the value does not exist or is illegal, but simply that ghci does not know how to print it because there is no instance of Show for the type.

• Read provides the converse functionality to Show, i.e. the ability to parse a string representation into your favorite value.

• Enum provides facilities to enumerate all values of a type. This is what work under the hood of the .. notation for lists.

Here is one of the main function, which gives the “next element” in a list

>>> :t succ
succ :: Enum a => a -> a

enumFromTo :: Enum a => a -> a -> [a]
enumFromTo x y = [x .. y]

Type annotations

In some rare cases, the typechecker will be unable to infer a type for a haskell expression due to either very tricky code with typeclasses, or because of some language extensions (not applicable to the module). In those case, it is possible to help the typechecker by asking that a given subexpression have a particular type using the :: notation within a definition. Here is a (bogus) example:

aConstant = (4 :: Int) + 6

>>> :t aConstant
aConstant :: Int

Without the type annotation in the function, this would have turned into a Num a => a.

Practice: in ghci/VSCode/Hoogle

The type system helps writting haskell programs in the sense that it filters out programs that carry dreadful runtime errors in other languages while also offering enough expressiveness to not crush users under bureaucracy (hopefully).

Now understanding the type system helps you: * understand the errors the compilers/interpreters print out. All errors that are not runtime or syntax errors are type errors, and typically the haskell tool do there best to assume your programs can typecheck, and try to provide an explanation when they can’t figure out a valid type for all expressions. Sometimes these explanations are a bit complex and will mention type variables and typeclasses, because the tools try really hard! Do try to look first at the line number and location of the error, and then the reasoning offered by the tools to try to understand what might be wrong

• you can then ask what is the type of an expression expr by typing :t expr in ghci, or hovering over it in VSCode if you have the haskell plugin and HLS installed and and gain useful information from the type

• on Hoogle, you can search for haskell functions and their documentations by providing either their name or their types. This is very useful e.g. if you think the base install should provide some basic function that you don’t know the name but can guess the interface of!

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