Haskell Programming Tutorial: Recursive Functions on Lists

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 \(cons\; :: \; a\rightarrow [a] \rightarrow [a]\). 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)
     
     

• • 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

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]

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

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. 1. Start with an argument of type \(a\)

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

1. 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
    

     

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

Examples of foldl with infix notation

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 xs0wherelgo z [] = zlgo 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

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 = gowherego [] = zgo (y:ys) = y `k` go ys

Relationship between foldr and list structure

 x0 : x1 : x2 : []

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

Some applications of folds

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

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