Haskell/Higher-order functions

At the heart of functional programming is the idea that functions are just like any other value. The power of functional style comes from handling functions themselves as regular values, i.e. by passing functions to other functions and returning them from functions. A function that takes another function (or several functions) as an argument is called a higher-order function. They can be found pretty much anywhere in a Haskell program, and indeed we have already met some of them, such as map and the various folds. We saw commonplace examples of higher-order functions when discussing map in Lists II. Now, we are going to explore some common ways of writing code that manipulates functions.

A sorting algorithm

For a concrete example, we will consider the task of sorting a list. Quicksort is a well-known recursive sorting algorithm. To apply its sorting strategy to a list, we first choose one element and then divide the rest of the list into (A) those elements that should go before the chosen element, (B) those elements equal to the chosen one, and (C) those that should go after. Then, we apply the same algorithm to the unsorted (A) and (C) lists. After enough recursive sorting, we concatenate everything back together and have a final sorted list. That strategy can be translated into a Haskell implementation in a very simple way.

-- Type signature: any list with elements in the Ord class can be sorted.
quickSort :: (Ord a) => [a] -> [a]
-- Base case:
-- If the list is empty, there is nothing to do.
quickSort [] = []

-- The recursive case:
-- We pick the first element as our "pivot", the rest is to be sorted.
-- Note how the pivot itself ends up included in the middle part.
quickSort (x : xs) = (quickSort less) ++ (x : equal) ++ (quickSort more)
    where
        less = filter (< x) xs
        equal = filter (== x) xs
        more = filter (> x) xs

It should be pointed out that our quickSort is rather naïve. A more efficient implementation would avoid the three passes through filter at each recursive step and not use (++) to build the sorted list. Furthermore, unlike our implementation, the original quicksort algorithm does the sorting in-place using mutability. We will ignore such concerns for now, as we are more interested in the usage patterns of sorting functions, rather than in exact implementation.

The Ord class

Almost all the basic data types in Haskell are members of the Ord class, which is for ordering tests what Eq is for equality tests. The Ord class defines which ordering is the "natural" one for a given type. It provides a function called compare, with type:

compare :: (Ord a) => a -> a -> Ordering

compare takes two values and compares them, returning an Ordering value, which is LT if the first value is less than the second, EQ if it is equal and GT if it is greater than. For an Ord type, (<), (==) from Eq and (>) can be seen as shortcuts to compare that check for one of the three possibilities and return a Bool to indicate whether the specified ordering is true according to the Ord specification for that type. Note that each of the tests we use with filter in the definition of quickSort corresponds to one of the possible results of compare, and so we might have written, for instance, less as less = filter (\y -> y `compare` x == LT) xs.

Choosing how to compare

With quickSort, sorting any list with elements in the Ord class is easy. Suppose we have a list of String and we want to sort them; we just apply quickSort to the list. For the rest of this chapter, we will use a pseudo-dictionary of just a few words (but dictionaries with thousands of words would work just as well):

dictionary = ["I", "have", "a", "thing", "for", "Linux"]

quickSort dictionary returns:

["I", "Linux", "a", "for", "have", "thing"]

As you can see, capitalization is considered for sorting by default. Haskell Strings are lists of Unicode characters. Unicode (and almost all other encodings of characters) specifies that the character code for capital letters are less than the lower case letters. So "Z" is less than "a".

To get a proper dictionary-like sorting, we need a case insensitive quickSort. To achieve that, we can take a hint from the discussion of compare just above. The recursive case of quickSort can be rewritten as:

quickSort compare (x : xs) = (quickSort compare less) ++ (x : equal) ++ (quickSort compare more)
    where
        less  = filter (\y -> y `compare` x == LT) xs
        equal = filter (\y -> y `compare` x == EQ) xs
        more  = filter (\y -> y `compare` x == GT) xs

While this version is less tidy than the original one, it makes it obvious that the ordering of the elements hinges entirely on the compare function. That means we only need to replace compare with an (Ord a) => a -> a -> Ordering function of our choice. Therefore, our updated quickSort' is a higher-order function which takes a comparison function along with the list to sort.

quickSort' :: (Ord a) => (a -> a -> Ordering) -> [a] -> [a]
-- No matter how we compare two things the base case doesn't change,
-- so we use the _ "wildcard" to ignore the comparison function.
quickSort' _ [] = []

-- c is our comparison function
quickSort' c (x : xs) = (quickSort' c less) ++ (x : equal) ++ (quickSort' c more)
    where
        less  = filter (\y -> y `c` x == LT) xs
        equal = filter (\y -> y `c` x == EQ) xs
        more  = filter (\y -> y `c` x == GT) xs

We can reuse our quickSort' function to serve many different purposes.

If we wanted a descending order, we could just reverse our original sorted list with reverse (quickSort dictionary). Yet to actually do the initial sort descending, we could supply quickSort' with a comparison function that returns the opposite of the usual Ordering.

-- the usual ordering uses the compare function from the Ord class
usual = compare

-- the descending ordering, note we flip the order of the arguments to compare
descending x y = compare y x

-- the case-insensitive version is left as an exercise!
insensitive = ... 
-- How can we do case-insensitive comparisons without making a big list of all possible cases?

Note

Data.List offers a sort function for sorting lists. It does not use quicksort; rather, it uses an efficient implementation of an algorithm called mergesort. Data.List also includes sortBy, which takes a custom comparison function just like our quickSort'

Exercises
Write insensitive, such that quickSort' insensitive dictionary gives ["a", "for", "have", "I", "Linux", "thing"].

Higher-Order Functions and Types

The concept of currying (the generating of intermediate functions on the way toward a final result) was first introduced in the earlier chapter "Lists II". This is a good place to revisit how currying works.

Our quickSort' has type (a -> a -> Ordering) -> [a] -> [a].

Most of the time, the type of a higher-order function provides a guideline about how to use it. A straightforward way of reading the type signature would be "quickSort' takes, as its first argument, a function that gives an ordering of two as. Its second argument is a list of as. Finally, it returns a new list of as". This is enough to correctly guess that it uses the given ordering function to sort the list.

Note that the parentheses surrounding a -> a -> Ordering are mandatory. They specify that a -> a -> Ordering forms a single argument that happens to be a function.

Without the parentheses, we would get a -> a -> Ordering -> [a] -> [a] which accepts four arguments (none of which are themselves functions) instead of the desired two, and that wouldn't work as desired.

Remember that the -> operator is right-associative. Thus, our erroneous type signature a -> a -> Ordering -> [a] -> [a] means the same thing as a -> (a -> (Ordering -> ([a] -> [a]))).

Given that -> is right-associative, the explicitly grouped version of the correct quickSort' signature is actually (a -> a -> Ordering) -> ([a] -> [a]). This makes perfect sense. Our original quickSort lacking the adjustable comparison function argument was of type [a] -> [a]. It took a list and sorted it. Our new quickSort' is simply a function that generates quickSort style functions! If we plug in compare for the (a -> a -> Ordering) part, then we just return our original simple quickSort function. If we use a different comparison function for the argument, we generate a different variety of a quickSort function.

Of course, if we not only give a comparison function as an argument but also feed in an actual list to sort, then the final result is not the new quickSort-style function; instead, it continues on and passes the list to the new function and returns the sorted list as our final result.

Exercises

(Challenging) The following exercise combines what you have learned about higher order functions, recursion and I/O. We are going to recreate what is known in imperative languages as a for loop. Implement a function for :: a -> (a -> Bool) -> (a -> a) -> (a -> IO ()) -> IO () for i p f job = -- ??? An example of how this function would be used might be for 1 (<10) (+1) print which prints the numbers 1 to 9 on the screen. The desired behaviour of for is: starting from an initial value i, for first checks p i, if it evaluates to true it then executes job i, else it stops and does nothing. If job i was executed, it then uses f to modify this value and checks to see if the modified value f i satisfies some condition p. If it doesn't, it stops; otherwise, the for loop continues, using the modified f i in place of i. Implement the for loop in Haskell. The paragraph just above gives an imperative description of the for loop. Describe your implementation in more functional terms. Some more challenging exercises you could try Consider a task like "print the list of numbers from 1 to 10". Given that print is a function, and we want to apply it to a list of numbers, using map sounds like the natural thing to do. But would it actually work? Implement a function sequenceIO :: [IO a] -> IO [a]. Given a list of actions, this function runs each of the actions in order and returns all their results as a list. Implement a function mapIO :: (a -> IO b) -> [a] -> IO [b] which given a function of type a -> IO b and a list of type [a], runs that action on each item in the list, and returns the results. This exercise was inspired from a blog post by osfameron. No peeking!

Function manipulation

We will close the chapter by discussing a few examples of common and useful general-purpose higher-order functions. Familiarity with these will greatly enhance your skill at both writing and reading Haskell code.

Flipping arguments

flip is a handy little Prelude function. It takes a function of two arguments and returns a version of the same function with the arguments swapped.

flip :: (a -> b -> c) -> b -> a -> c

flip in use:

Prelude> (flip (/)) 3 1
0.3333333333333333
Prelude> (flip map) [1,2,3] (*2)
[2,4,6]

We could have used flip to write a point-free version of the descending comparing function from the quickSort example:

descending = flip compare

flip is particularly useful when we want to pass a function with two arguments of different types to another function and the arguments are in the wrong order with respect to the signature of the higher-order function.

Composition

The (.) composition operator is another higher-order function. It has the signature:

(.) :: (b -> c) -> (a -> b) -> a -> c

(.) takes two functions as arguments and returns a new function which applies the second function to the argument and then the first. (Writing the type of (.) as (b -> c) -> (a -> b) -> (a -> c) can make that easier to see.)

Composition and higher-order functions provide a range of powerful tricks. For a tiny sample, first consider the inits function, defined in the module Data.List. Quoting the documentation, it "returns all initial segments of the argument, shortest first", so that:

Prelude Data.List> inits [1,2,3]
[[],[1],[1,2],[1,2,3]]

We can provide a one-line implementation for inits (written point-free for extra dramatic effect) using only the following higher-order functions from Prelude: flip, scanl, (.) and map:

myInits :: [a] -> [[a]]
myInits = map reverse . scanl (flip (:)) []

Swallowing a definition so condensed may look daunting at first, so analyze it slowly, bit by bit, recalling what each function does and using the type signatures as a guide.

The definition of myInits is super concise and clean with use of parentheses kept to a bare minimum. Naturally, if one goes overboard with composition by writing mile-long (.) chains, things will get confusing; but, when deployed reasonably, these point-free styles shine. Furthermore, the implementation is quite "high level": we do not deal explicitly with details like pattern matching or recursion; the functions we deployed — both the higher-order ones and their functional arguments — take care of such plumbing.

Application

($) is a curious higher-order operator. Its type is:

($) :: (a -> b) -> a -> b

It takes a function as its first argument, and all it does is to apply the function to the second argument, so that, for instance, (head $ "abc") == (head "abc").

You might think that ($) is completely useless! However, there are two interesting points about it. First, ($) has very low precedence, unlike regular function application which has the highest precedence. In effect, that means we can avoid confusing nesting of parentheses by breaking precedence with $. We write a non-point-free version of myInits without adding new parentheses:

myInits :: [a] -> [[a]]
myInits xs = map reverse . scanl (flip (:)) [] $ xs

Furthermore, as ($) is just a function which happens to apply functions, and functions are just values, we can write intriguing expressions such as:

map ($ 2) [(2*), (4*), (8*)]

(Yes, that is a list of functions, and it is perfectly legal.)

uncurry and curry

As the name suggests, uncurry is a function that undoes currying; that is, it converts a function of two arguments into a function that takes a pair as its only argument.

uncurry :: (a -> b -> c) -> (a, b) -> c

Prelude> let addPair = uncurry (+)
Prelude> addPair (2, 3)
5

One interesting use of uncurry occasionally seen in the wild is in combination with ($), so that the first element of a pair is applied to the second.

Prelude> uncurry ($) (reverse, "stressed")
"desserts"

There is also curry, which is the opposite of uncurry.

curry :: ((a, b) -> c) -> a -> b -> c

Prelude> curry addPair 2 3 -- addPair as in the earlier example.
5

Because most Haskell functions are already curried, curry is nowhere near as common as uncurry.

id and const

Finally, we should mention two functions which, while not higher-order functions themselves, are most often used as arguments to higher-order functions. id, the identity function, is a function with type a -> a that returns its argument unchanged.

Prelude> id "Hello"
"Hello"

Similar in spirit to id, const is an a -> b -> a function that works like this:

Prelude> const "Hello" "world"
"Hello"

const takes two arguments, discards the second and returns the first. Seen as a function of one argument, a -> (b -> a), it returns a constant function, which always returns the same value no matter what argument it is given.

id and const might appear worthless at first. However, when dealing with higher-order functions it is sometimes necessary to pass a dummy function, be it one that does nothing with its argument or one that always returns the same value. id and const give us convenient dummy functions for such cases.

Exercises

Write implementations for curry, uncurry and const. Describe what the following functions do without testing them: uncurry const curry fst curry swap, where swap :: (a, b) -> (b, a) swaps the elements of a pair. (swap can be found in Data.Tuple.) (Very hard) Use foldr to implement foldl. Hint: begin by reviewing the sections about foldr and foldl in Lists III. There are two solutions; one is easier but relatively boring and the other is truly interesting. For the interesting one, think carefully about how you would go about composing all functions in a list.