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Haskell Guide: Types, Lambda Functions and Type Classes

An introduction to types, lambda functions and type classes in Haskell, the increasingly popular functional programming language.
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© University of Glasgow

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.

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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^2e3 = 4*ae4 = e3*ce5 = e2-e4e6 = sqrt e5e7 = e1+e6
 

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 -> Intf 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 -> Charf i = "abcdefghijklmnopqrstuvwxyz" !! ix :: Intx = 3f :: Char->Stringf x = x:" There is a kind of character in thy life"
 

Polymorphic functions

 
Polymorphic means “having many forms”.
 
 fst :: (a,b) -> afst (x,y) = xsnd :: (a,b) -> bsnd (x,y) = yfst :: (a,b) -> afst (a,b) = asnd :: (a,b) -> bsnd (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 -> Intf 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) 4g :: Int -> Intg = f 3g 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 -> df :: 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]), (Intrightarrow 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
[frac {hbox{assumption — what you’re given}} {hbox{consequence — what you can infer}}]

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

[frac {hbox{$c$ is a constant with a fixed type $T$}} {Gamma vdash c :: T}]
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

[frac {Gamma vdash e_1 :: (alpha rightarrow beta) qquad Gamma vdash e_2 :: alpha } {Gamma vdash (e_1e_2) :: beta}]
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

[frac {Gamma, x :: alpha quad vdash quad e :: beta} {Gamma vdash (lambda x rightarrow e) :: (alpha rightarrow beta)}]
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).
© University of Glasgow
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