Types, lambda functions and type classes

Function types

• Ordinary data types are for primitive data (like $Int$ and $Char$) and basic data structures (like $[Int]$ and $[Char]$).

• Algebraic data types are types that combine other types either as records (‘products’), e.g.

    data Pair = Pair Int Double
  

  or as variants (‘sums’), e.g.

    data Bool = False | True
  

• Functions have types containing an arrow, e.g. $Int \rightarrow String$.
• We now look at function types in more detail.

Lambda expressions

• Lambda expressions (named after the greek letter $\lambda$) play a very important role in functional programming in general and Haskell in particular.

Named and anonymous expressions

• You can give a name $sum$ to an expression $2+2$:

    sum = 2+2

• But you can also write anonymous expressions — expressions that just appear, but are not given names.

    (-b) + sqrt (b^2 - 4*a*c)

• Without anonymous expressions, writing this would almost be like assembly language:

    e1 = (-b)
    e2 = b^2
    e3 = 4*a
    e4 = e3*c
    e5 = e2-e4
    e6 = e1+e5

Some background

• Sometimes in a mathematics or physics book, there are statements like “the function $x^2$ is continuous$\ldots$”

• This is ok when the context makes it clear what $x$ is.

• But it can lead to problems. What does $x*y$ mean?

  • Is it a constant, because both $x$ and $y$ have fixed values?

  • Is it a function of $x$, with a fixed value of $y$?

  • Is it a function of $y$, with a fixed value of $x$?

  • Is it a function of both $x$ and $y$?

• In mathematical logic (and computer programming) we need to be precise about this!

• A lambda expression $\backslash x \rightarrow e$ contains

  • An explicit statement that the formal parameter is $x$, and

  • the expression $e$ that defines the value of the function.

Anonymous functions

• A function can be defined and given a name using an equation:

    f :: Int -> Int
    f x = x+1

• Since functions are “first class”, they are ubiquitous, and it’s often useful to denote a function anonymously.

• This is done using lambda expressions.

    \x -> x+1

Pronounced “lambda x arrow x+1”.

There may be any number of arguments:

    \x y z -> 2*x + y*z

Using a lambda expression

Functions are first class: you can use a lambda expression wherever a function is needed. Thus

    f = \x -> x+1

is equivalent to

    f x = x+1

But lambda expressions are most useful when they appear inside larger expressions.

    map (\x -> 2*x + 1) xs

Monomorphic and polymorphic functions

Monomorphic functions

Monomorphic means “having one form”.

    f :: Int -> Char
    f i = "abcdefghijklmnopqrstuvwxyz" !! i

    x :: Int
    x = 3

    f x :: Char
    f x -- > 'd'

Polymorphic functions

Polymorphic means “having many forms”.

    fst :: (a,b) -> a
    fst (x,y) = x

    snd :: (a,b) -> b
    snd (x,y) = y

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

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

Currying

• Most programming languages allow functions to have any number of arguments.

• But this turns out to be unnecessary: we can restrict all functions to have just one argument, without losing any expressiveness.

• This process is called Currying, in honor of Haskell Curry.

  • The technique makes essential use of higher order functions.

  • It has many advantages, both practical and theoretical.

A function with two arguments

You can write a definition like this, which appears to have two arguments:

    f :: Int -> Int -> Int
    f x y = 2*x + y

But it actually means the following:

    f :: Int -> (Int -> Int)
    f 5 :: Int -> Int

The function takes its arguments one at a time:

    f 3 4 = (f 3) 4

    g :: Int -> Int
    g = f 3
    g 10 -- > (f 3) 10 -- > 2*3 + 10

Grouping: arrow to the right, application left

• The arrow operator takes two types $a \rightarrow b$, and gives the type of a function with argument type $a$ and result type $b$

• An application $e_1\; e_2$ applies a function $e_1$ to an argument $e_2$

• Note that for both types and applications, a function has only one argument

• To make the notation work smoothly, arrows group to the right, and application groups to the left.

    f :: a -> b -> c -> d
    f :: a -> (b -> (c -> d))

    f x y z = ((f x) y) z

Type classes and ad-hoc polymorphism

The type of $(+)$

Note that $fst$ has the following type, and there is no restriction on what types $a$ and $b$ could be.

    fst :: (a,b) -> a

What is the type of $(+)$? Could it be$\ldots$

    (+) :: Int -> Int -> Int
    (+) :: Integer -> Integer -> Integer
    (+) :: Ratio Integer -> Ratio Integer -> Ratio Integer
    (+) :: Double -> Double -> Double

    (+) :: a -> a -> a  -- Wrong! has to be a number

Type classes

Answer: $(+)$ has type $a \rightarrow a \rightarrow a$ for any type $a$ that is a member of the type class $Num$.

    (+) :: Num a => a -> a -> a

• The class $Num$ is a set of types for which $(+)$ is defined

• It includes $Int$, $Integer$, $Double$, and many more.

• But $Num$ does not contain types like $Bool$, $[Char]$, $Int\rightarrow Double$, and many more.

Two kinds of polymorphism

• Parametric polymorphism.

  • A polymorphic type that can be instantiated to any type.

  • Represented by a type variable. It is conventional to use $a$, $b$, $c$, $\ldots$

  • Example: $length :: [a] \rightarrow Int$ can take the length of a list whose elements could have any type.

• Ad hoc polymorphism.

  • A polymorphic type that can be instantiated to any type chosen from a set, called a “type class”

  • Represented by a type variable that is constrained using the $\Rightarrow$ notation.

  • Example: $(+) :: Num\, a \Rightarrow a \rightarrow a \rightarrow a$ says that $(+)$ can add values of any type $a$, provided that $a$ is an element of the type class $Num$.

Type inference

• Type checking takes a type declaration and some code, and determines whether the code actually has the type declared.

• Type inference is the analysis of code in order to infer its type.

• Type inference works by

  • Using a set of type inference rules that generate typings based on the program text

  • Combining all the information obtained from the rules to produce the types.

Type inference rules

The type system contains a number of type inference rules, with the form

Context

• Statements about types are written in the form similar to $\Gamma \vdash e :: \alpha$

• This means ``if you are given a set $\Gamma$ of types, then it is proven that $e$ has type $\alpha$.

Type of constant

If we know the type $T$ of a constant $c$ (for example, we know that $'a' :: Char$), then this is expressed by saying that there is a given theorem that $c :: T$. Furthermore, this holds given any context $\Gamma$.

Type of application

If $e_1$ is a function with type $\alpha \rightarrow \beta$, then the application of $e_1$ to an argument of type $\alpha$ gives a result of type $\beta$.

Type of lambda expression

We have a context $\Gamma$. Suppose that if we’re also given that $x :: \alpha$, then it can be proven that an expression $e :: \beta$. Then we can infer that the function $\lambda x \rightarrow e$ has type $\alpha \rightarrow \beta$.