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Haskell Programming Tutorial: Recursive Functions on Lists

This article provides a Haskell programming guide on recursive functions on lists.
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© University of Glasgow

Computing with lists

  • There are two approaches to working with lists:
    • Write functions to do what you want, using recursive definitions that traverse the list structure.
    • Write combinations of the standard list processing functions.
  • The second approach is preferred, but the standard list processing functions do need to be defined, and those definitions use the first approach (recursive definitions).
  • We’ll cover both methods.

Recursion on lists

    • A list is built from the empty list ([]) and the function (cons; :: ; arightarrow [a] rightarrow [a]). In Haskell, the function (cons) is actually written as the operator ((:)) , in other words : is pronounced as cons.


    • Every list must be either
        • ([]) or


        • ((x : xs)) for some (x) (the head of the list) and (xs) (the tail)





where ((x : xs)) is pronounced as (x, mathit{cons}, xs)



    • The recursive definition follows the structure of the data:
        • Base case of the recursion is ([]).


        • Recursion (or induction) case is ((x : xs)).





Some examples of recursion on lists


Recursive definition of length


The length of a list can be computed recursively as follows:


length :: [a] -> Int -- function type
length [] = 0 -- base case
length (x:xs) = 1 + length xs -- recursion case


Recursive definition of filter



    • filter is given a predicate (a function that gives a Boolean result) and a list, and returns a list of the elements that satisfy the predicate.



filter :: (a->Bool) -> [a] -> [a]


Filtering is useful for the “generate and test” programming paradigm.


filter (<5) [3,9,2,12,6,4] -- > [3,2,4]


The library definition for filter is shown below. This relies on guards.


filter :: (a -> Bool) -> [a] -> [a]
filter pred [] = []
filter pred (x:xs)
 | pred x = x : filter pred xs
 | otherwise = filter pred xs


Computations over lists



    • Many computations that would be for/while loops in an imperative language are naturally expressed as list computations in a functional language.


    • There are some common cases: 
        • Perform a computation on each element of a list: (map)


        • Iterate over a list, from left to right: (foldl)


        • Iterate over a list, from right to left: (foldr)




    • It’s good practice to use these three functions when applicable


    • And there are some related functions that we’ll see later



Function composition



    • We can express a large computation by “chaining together” a sequence of functions that perform smaller computations




    1. Start with an argument of type (a)


    1. Apply a function (g :: a to b) to it, getting an intermediate result of type (b)


    1. Then apply a function (f :: b to c) to the intermediate result, getting the final result of type (c)




    • The entire computation (first (g), then (f)) is written as (f circ g).


    • This is traditional mathematical notation; just remember that in (f circ g), the functions are used in right to left order.


    • Haskell uses . as the function composition operator 
      (.) :: (b->c) -> (a->b) -> a -> c
      (f . g) x = f (g x)




Performing an operation on every element of a list: map



    • map applies a function to every element of a list 
      map f [x0,x1,x2] -- > [f x0, f x1, f x2]




Composition of maps



    • map is one of the most commonly used tools in your functional toolkit


    • A common style is to define a set of simple computations using map, and to compose them. 
      map f (map g xs) = map (f . g) xs




This theorem is frequently used, in both directions.


Recursive definition of map


map :: (a -> b) -> [a] -> [b]
map _ [] = []
map f (x:xs) = f x : map f xs


Folding a list (reduction)



    • An iteration over a list to produce a singleton value is called a fold


    • There are several variations: folding from the left, folding from the right, several variations having to do with “initialisation”, and some more advanced variations.


    • Folds may look tricky at first, but they are extremely powerful, and they are used a lot! And they aren’t actually very complicated.



Left fold: foldl



    • foldl is fold from the left


    • Think of it as an iteration across a list, going left to right.


    • A typical application is (foldl, f, z, xs)


    • The (z :: b) is an initial value


    • The (xs :: [a]) argument is a list of values which we combine systematically using the supplied function (f)


    • A useful intuition: think of the (z :: b) argument as an “accumulator”.


    • The function (f) takes the current value of the accumulator and a list element, and gives the new value of the accumulator. 
      foldl :: (b->a->b) -> b -> [a] -> b




Examples of foldl with function notation

mathtt{foldl,f,z,[]} &rightsquigarrow & z
mathtt{foldl,f,z,[x0]} & rightsquigarrow & f,z,x0
mathtt{foldl,f,z,[x0,x1]} & rightsquigarrow & f,(f,z,x0),x1
mathtt{foldl,f,z,[x0,x1,x2]} & rightsquigarrow & f,(f,(f,z,x0),x1), x2end{aligned}]

Examples of foldl with infix notation


In this example, + denotes an arbitrary operator for f; it isn’t supposed to mean specifically addition.


foldl (+) z [] -- > z
foldl (+) z [x0] -- > z + x0
foldl (+) z [x0,x1] -- > (z + x0) + x1
foldl (+) z [x0,x1,x2] -- > ((z + x0) + x1) + x2


Recursive definition of foldl


foldl :: (b -> a -> b) -> b -> [a] -> b
foldl f z0 xs0 = lgo z0 xs0
 lgo z [] = z
 lgo z (x:xs) = lgo (f z x) xs


Right fold: foldr



    • Similar to (foldl), but it works from right to left



foldr :: (a -> b -> b) -> b -> [a] -> b


Examples of foldr with function notation

mathtt{foldr,f, z, [] } & rightsquigarrow & z
mathtt{foldr, f, z, [x0] } & rightsquigarrow & f, x0, z
mathtt{foldr, f, z, [x0,x1] } & rightsquigarrow & f, x0, (f, x1, z)
mathtt{foldr, f, z, [x0,x1,x2] } & rightsquigarrow & f, x0, (f, x1, (f, x2, z))end{aligned}]

Examples of foldr with operator notation


foldr (+) z [] -- > z
foldr (+) z [x0] -- > x0 + z
foldr (+) z [x0,x1] -- > x0 + (x1 + z)
foldr (+) z [x0,x1,x2] -- > x0 + (x1 + (x2 + z))


Recursive definition of foldr


foldr :: (a -> b -> b) -> b -> [a] -> b
foldr k z = go
 go [] = z
 go (y:ys) = y `k` go ys


Relationship between foldr and list structure


We have seen that a list [x0,x1,x2] can also be written as


 x0 : x1 : x2 : []


Folding (cons) (:) over a list using the empty list [] as accumulator gives:


foldr (:) [] [x0,x1,x2]
 -- >
 x0 : x1 : x2 : []


This is identical to constructing the list using (:) and [] ! We can formalise this relationship as follows:

[foldr ; cons ; [] ; xs ; = ; xs]

Some applications of folds


sum xs = foldr (+) 0 xs
product xs = foldr (*) 1 xs


We can actually “factor out” the (xs) that appears at the right of each side of the equation, and write:


sum = foldr (+) 0
product = foldr (*) 1

(This is sometimes called “point free” style because you’re programming solely with the functions; the data isn’t mentioned directly.)

© University of Glasgow
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