Haskell Programming Tutorial: Recursive Functions on Lists
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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
.
- 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
-
- Every list must be either
-
- ([]) or
-
- ((x : xs)) for some (x) (the head of the list) and (xs) (the tail)
-
- Every list must be either
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)).
-
- The recursive definition follows the structure of the data:
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)
-
- There are some common cases:
-
- 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
-
- Start with an argument of type (a)
-
- Apply a function (g :: a to b) to it, getting an intermediate result of type (b)
-
- 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)
- Haskell uses
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]
- map applies a function to every element of a list
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
- A common style is to define a set of simple computations using map, and to compose them.
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
- The function (f) takes the current value of the accumulator and a list element, and gives the new value of the accumulator.
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), 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
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.)
Functional Programming in Haskell: Supercharge Your Coding
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Functional Programming in Haskell: Supercharge Your Coding
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