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3.2

## The University of Glasgow # 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)

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, which we will investigate properly next week.

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 $$a$$
2. Apply a function $$g :: a \to b$$ to it, getting an intermediate result of type $$b$$
3. 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

\begin{aligned} \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)\, x2\end{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
where
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

\begin{aligned} \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
where
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.)