Chapter 4: Functional Programming

Thinking in Haskell

Our early learning of Haskell has two distinct aspects. The first is coming to terms with the shift in mindset from imperative programming to functional: we have to replace our programming habits from other languages. We do this not because imperative techniques are bad, but because in a functional language other techniques work better.

Our second challenge is learning our way around the standard Haskell libraries. As in any language, the libraries act as a lever, enabling us to multiply our problem solving power. Haskell libraries tend to operate at a higher level of abstraction than those in many other languages. We’ll need to work a little harder to learn to use the libraries, but in exchange they offer a lot of power.

In this chapter, we’ll introduce a number of common functional programming techniques. We’ll draw upon examples from imperative languages to highlight the shift in thinking that we’ll need to make. As we do so, we’ll walk through some of the fundamentals of Haskell’s standard libraries. We’ll also intermittently cover a few more language features along the way.

A simple command line framework

In most of this chapter, we will concern ourselves with code that has no interaction with the outside world. To maintain our focus on practical code, we will begin by developing a gateway between our “pure” code and the outside world. Our framework simply reads the contents of one file, applies a function to the file, and writes the result to another file.

-- Save this in a source file, e.g. Interact.hs

import System.Environment (getArgs)

interactWith function inputFile outputFile = do
  input <- readFile inputFile
  writeFile outputFile (function input)

main = mainWith myFunction
  where mainWith function = do
          args <- getArgs
          case args of
            [input,output] -> interactWith function input output
            _ -> putStrLn "error: exactly two arguments needed"

        -- replace "id" with the name of our function below
        myFunction = id

This is all we need to write simple, but complete, file processing programs. This is a complete program. We can compile it to an executable named InteractWith as follows.

$ ghc --make InteractWith
[1 of 1] Compiling Main             ( InteractWith.hs, InteractWith.o )
Linking InteractWith ...

If we run this program from the shell or command prompt, it will accept two file names: the name of a file to read, and the name of a file to write.

$ ./Interact
error: exactly two arguments needed
$ ./Interact hello-in.txt hello-out.txt
$ cat hello-in.txt
hello world
$ cat hello-out.txt
hello world

Some of the notation in our source file is new. The do keyword introduces a block of actions that can cause effects in the real world, such as reading or writing a file. The <- operator is the equivalent of assignment inside a do block. This is enough explanation to get us started. We will talk in much more detail about these details of notation, and I/O in general, in Chapter 7, /I/O/.

When we want to test a function that cannot talk to the outside world, we simply replace the name id in the code above with the name of the function we want to test. Whatever our function does, it will need to have the type String -> String: in other words, it must accept a string, and return a string.

Warming up: portably splitting lines of text

Haskell provides a built-in function, lines, that lets us split a text string on line boundaries. It returns a list of strings with line termination characters omitted.

ghci> :type lines
lines :: String -> [String]
ghci> lines "line 1\nline 2"
["line 1","line 2"]
ghci> lines "foo\n\nbar\n"
["foo","","bar"]

While lines looks useful, it relies on us reading a file in “text mode” in order to work. Text mode is a feature common to many programming languages: it provides a special behavior when we read and write files on Windows. When we read a file in text mode, the file I/O library translates the line ending sequence "\r\n" (carriage return followed by newline) to "\n" (newline alone), and it does the reverse when we write a file. On Unix-like systems, text mode does not perform any translation. As a result of this difference, if we read a file on one platform that was written on the other, the line endings are likely to become a mess. (Both readFile and writeFile operate in text mode.)

ghci> lines "a\r\nb"
["a\r","b"]

The lines function only splits on newline characters, leaving carriage returns dangling at the ends of lines. If we read a Windows-generated text file on a Linux or Unix box, we’ll get trailing carriage returns at the end of each line.

We have comfortably used Python’s “universal newline” support for years: this transparently handles Unix and Windows line ending conventions for us. We would like to provide something similar in Haskell.

Since we are still early in our career of reading Haskell code, we will discuss our Haskell implementation in quite some detail.

splitLines :: String -> [String]

Our function’s type signature indicates that it accepts a single string, the contents of a file with some unknown line ending convention. It returns a list of strings, representing each line from the file.

splitLines [] = []
splitLines cs =
    let (pre, suf) = break isLineTerminator cs
    in  pre : case suf of
                ('\r':'\n':rest) -> splitLines rest
                ('\r':rest)      -> splitLines rest
                ('\n':rest)      -> splitLines rest
                _                -> []

isLineTerminator c = c == '\r' || c == '\n'

Before we dive into detail, notice first how we have organized our code. We have presented the important pieces of code first, keeping the definition of isLineTerminator until later. Because we have given the helper function a readable name, we can guess what it does even before we’ve read it, which eases the smooth “flow” of reading the code.

The prelude defines a function named break that we can use to partition a list into two parts. It takes a function as its first parameter. That function must examine an element of the list, and return a Bool to indicate whether to break the list at that point. The break function returns a pair, which consists of the sublist consumed before the predicate returned True (the prefix), and the rest of the list (the suffix).

ghci> break odd [2,4,5,6,8]
([2,4],[5,6,8])
ghci> :module +Data.Char
ghci> break isUpper "isUpper"
("is","Upper")

Since we only need to match a single carriage return or newline at a time, examining one element of the list at a time is good enough for our needs.

The first equation of splitLines indicates that if we match an empty string, we have no further work to do.

In the second equation, we first apply break to our input string. The prefix is the substring before a line terminator, and the suffix is the remainder of the string. The suffix will include the line terminator, if any is present.

The ”pre :” expression tells us that we should add the pre value to the front of the list of lines. We then use a case expression to inspect the suffix, so we can decide what to do next. The result of the case expression will be used as the second argument to the (:) list constructor.

The first pattern matches a string that begins with a carriage return, followed by a newline. The variable rest is bound to the remainder of the string. The other patterns are similar, so they ought to be easy to follow.

A prose description of a Haskell function isn’t necessarily easy to follow. We can gain a better understanding by stepping into ghci, and oberving the behavior of the function in different circumstances.

Let’s start by partitioning a string that doesn’t contain any line terminators.

ghci> splitLines "foo"
["foo"]

Here, our application of break never finds a line terminator, so the suffix it returns is empty.

ghci> break isLineTerminator "foo"
("foo","")

The case expression in splitLines must thus be matching on the fourth branch, and we’re finished. What about a slightly more interesting case?

ghci> splitLines "foo\r\nbar"
["foo","bar"]

Our first application of break gives us a non-empty suffix.

ghci> break isLineTerminator "foo\r\nbar"
("foo","\r\nbar")

Because the suffix begins with a carriage return, followed by a newline, we match on the first branch of the case expression. This gives us pre bound to "foo", and suf bound to "bar". We apply splitLines recursively, this time on "bar" alone.

ghci> splitLines "bar"
["bar"]

The result is that we construct a list whose head is "foo" and whose tail is ["bar"].

ghci> "foo" : ["bar"]
["foo","bar"]

This sort of experimenting with ghci is a helpful way to understand and debug the behavior of a piece of code. It has an even more important benefit that is almost accidental in nature. It can be tricky to test complicated code from ghci, so we will tend to write smaller functions. This can further help the readability of our code.

This style of creating and reusing small, powerful pieces of code is a fundamental part of functional programming.

A line ending conversion program

Let’s hook our splitLines function into the little framework we wrote earlier. Make a copy of the Interact.hs source file; let’s call the new file FixLines.hs. Add the splitLines function to the new source file. Since our function must produce a single string, we must stitch the list of lines back together. The prelude provides an unlines function that concatenates a list of strings, adding a newline to the end of each.

fixLines :: String -> String
fixLines input = unlines (splitLines input)

If we replace the id function with fixLines, we can compile an executable that will convert a text file to our system’s native line ending.

$ ghc --make FixLines
[1 of 1] Compiling Main             ( FixLines.hs, FixLines.o )
Linking FixLines ...

If you are on a Windows system, find and download a text file that was created on a Unix system (for example gpl-3.0.txt). Open it in the standard Notepad text editor. The lines should all run together, making the file almost unreadable. Process the file using the FixLines command you just created, and open the output file in Notepad. The line endings should now be fixed up.

On Unix-like systems, the standard pagers and editors hide Windows line endings. This makes it more difficult to verify that FixLines is actually eliminating them. Here are a few commands that should help.

$ file gpl-3.0.txt
gpl-3.0.txt: ASCII English text
$ unix2dos gpl-3.0.txt
unix2dos: converting file gpl-3.0.txt to DOS format ...
$ file gpl-3.0.txt
gpl-3.0.txt: ASCII English text, with CRLF line terminators

Infix functions

Usually, when we define or apply a function in Haskell, we write the name of the function, followed by its arguments. This notation is referred to as prefix, because the name of the function comes before its arguments.

If a function or constructor takes two or more arguments, we have the option of using it in infix form, where we place it between its first and second arguments. This allows us to use functions as infix operators.

To define or apply a function or value constructor using infix notation, we enclose its name in backtick characters (sometimes known as backquotes). Here are simple infix definitions of a function and a type.

a `plus` b = a + b

data a `Pair` b = a `Pair` b
                  deriving (Show)

-- we can use the constructor either prefix or infix
foo = Pair 1 2
bar = True `Pair` "quux"

Since infix notation is purely a syntactic convenience, it does not change a function’s behavior.

ghci> 1 `plus` 2
3
ghci> plus 1 2
3
ghci> True `Pair` "something"
True `Pair` "something"
ghci> Pair True "something"
True `Pair` "something"

Infix notation can often help readability. For instance, the prelude defines a function, elem, that indicates whether a value is present in a list. If we use elem using prefix notation, it is fairly easy to read.

ghci> elem 'a' "camogie"
True

If we switch to infix notation, the code becomes even easier to understand. It is now clearer that we’re checking to see if the value on the left is present in the list on the right.

ghci> 3 `elem` [1,2,4,8]
False

We see a more pronounced improvement with some useful functions from the Data.List module. The isPrefixOf function tells us if one list matches the beginning of another.

ghci> :module +Data.List
ghci> "foo" `isPrefixOf` "foobar"
True

The isInfixOf and isSuffixOf functions match anywhere in a list and at its end, respectively.

ghci> "needle" `isInfixOf` "haystack full of needle thingies"
True
ghci> "end" `isSuffixOf` "the end"
True

There is no hard-and-fast rule that dictates when you ought to use infix versus prefix notation, although prefix notation is far more common. It’s best to choose whichever makes your code more readable in a specific situation.

Working with lists

As the bread and butter of functional programming, lists deserve some serious attention. The standard prelude defines dozens of functions for dealing with lists. Many of these will be indispensable tools, so it’s important that we learn them early on.

For better or worse, this section is going to read a bit like a “laundry list” of functions. Why present so many functions at once? These functions are both easy to learn and absolutely ubiquitous. If we don’t have this toolbox at our fingertips, we’ll end up wasting time by reinventing simple functions that are already present in the standard libraries. So bear with us as we go through the list; the effort you’ll save will be huge.

The Data.List module is the “real” logical home of all standard list functions. The prelude merely re-exports a large subset of the functions exported by Data.List. Several useful functions in Data.List are not re-exported by the standard prelude. As we walk through list functions in the sections that follow, we will explicitly mention those that are only in Data.List.

ghci> :module +Data.List

Because none of these functions is complex or takes more than about three lines of Haskell to write, we’ll be brief in our descriptions of each. In fact, a quick and useful learning exercise is to write a definition of each function after you’ve read about it.

Basic list manipulation

The length function tells us how many elements are in a list.

ghci> :type length
length :: [a] -> Int
ghci> length []
0
ghci> length [1,2,3]
3
ghci> length "strings are lists, too"
22

If you need to determine whether a list is empty, use the null function.

ghci> :type null
null :: [a] -> Bool
ghci> null []
True
ghci> null "plugh"
False

To access the first element of a list, we use the head function.

ghci> :type head
head :: [a] -> a
ghci> head [1,2,3]
1

The converse, tail, returns all but the head of a list.

ghci> :type tail
tail :: [a] -> [a]
ghci> tail "foo"
"oo"

Another function, last, returns the very last element of a list.

ghci> :type last
last :: [a] -> a
ghci> last "bar"
'r'

The converse of last is init, which returns a list of all but the last element of its input.

ghci> :type init
init :: [a] -> [a]
ghci> init "bar"
"ba"

Several of the functions above behave poorly on empty lists, so be careful if you don’t know whether or not a list is empty. What form does their misbehavior take?

ghci> head []
*** Exception: Prelude.head: empty list

Try each of the above functions in ghci. Which ones crash when given an empty list?

Safely and sanely working with crashy functions

When we want to use a function like head, where we know that it might blow up on us if we pass in an empty list, the temptation might initially be strong to check the length of the list before we call head. Let’s construct an artificial example to illustrate our point.

myDumbExample xs = if length xs > 0
                   then head xs
                   else 'Z'

If we’re coming from a language like Perl or Python, this might seem like a perfectly natural way to write this test. Behind the scenes, Python lists are arrays; and Perl arrays are, well, arrays. So they necessarily know how long they are, and calling len(foo) or scalar(@foo) is a perfectly natural thing to do. But as with many other things, it’s not a good idea to blindly transplant such an assumption into Haskell.

We’ve already seen the definition of the list algebraic data type many times, and know that a list doesn’t store its own length explicitly. Thus, the only way that length can operate is to walk the entire list.

Therefore, when we only care whether or not a list is empty, calling length isn’t a good strategy. It can potentially do a lot more work than we want, if the list we’re working with is finite. Since Haskell lets us easily create infinite lists, a careless use of length may even result in an infinite loop.

A more appropriate function to call here instead is null, which runs in constant time. Better yet, using null makes our code indicate what property of the list we really care about. Here are two improved ways of expressing myDumbExample.

mySmartExample xs = if not (null xs)
                    then head xs
                    else 'Z'

myOtherExample (x:_) = x
myOtherExample [] = 'Z'

Partial and total functions

Functions that only have return values defined for a subset of valid inputs are called partial functions (calling error doesn’t qualify as returning a value!). We call functions that return valid results over their entire input domains total functions.

It’s always a good idea to know whether a function you’re using is partial or total. Calling a partial function with an input that it can’t handle is probably the single biggest source of straightforward, avoidable bugs in Haskell programs.

Some Haskell programmers go so far as to give partial functions names that begin with a prefix such as unsafe, so that they can’t shoot themselves in the foot accidentally.

It’s arguably a deficiency of the standard prelude that it defines quite a few “unsafe” partial functions, like head, without also providing “safe” total equivalents.

More simple list manipulations

Haskell’s name for the “append” function is (++).

ghci> :type (++)
(++) :: [a] -> [a] -> [a]
ghci> "foo" ++ "bar"
"foobar"
ghci> [] ++ [1,2,3]
[1,2,3]
ghci> [True] ++ []
[True]

The concat function takes a list of lists, all of the same type, and concatenates them into a single list.

ghci> :type concat
concat :: [[a]] -> [a]
ghci> concat [[1,2,3], [4,5,6]]
[1,2,3,4,5,6]

It removes one level of nesting.

ghci> concat [[[1,2],[3]], [[4],[5],[6]]]
[[1,2],[3],[4],[5],[6]]
ghci> concat (concat [[[1,2],[3]], [[4],[5],[6]]])
[1,2,3,4,5,6]

The reverse function returns the elements of a list in reverse order.

ghci> :type reverse
reverse :: [a] -> [a]
ghci> reverse "foo"
"oof"

For lists of Bool, the and and or functions generalise their two-argument cousins, (&&) and (||), over lists.

ghci> :type and
and :: [Bool] -> Bool
ghci> and [True,False,True]
False
ghci> and []
True
ghci> :type or
or :: [Bool] -> Bool
ghci> or [False,False,False,True,False]
True
ghci> or []
False

They have more useful cousins, all and any, which operate on lists of any type. Each one takes a predicate as its first argument; all returns True if that predicate succeeds on every element of the list, while any returns True if the predicate succeeds on at least one element of the list.

ghci> :type all
all :: (a -> Bool) -> [a] -> Bool
ghci> all odd [1,3,5]
True
ghci> all odd [3,1,4,1,5,9,2,6,5]
False
ghci> all odd []
True
ghci> :type any
any :: (a -> Bool) -> [a] -> Bool
ghci> any even [3,1,4,1,5,9,2,6,5]
True
ghci> any even []
False

Working with sublists

The take function, which we already met in the section called “Function application” consisting of the first k elements from a list. Its converse, drop, drops k elements from the start of the list.

ghci> :type take
take :: Int -> [a] -> [a]
ghci> take 3 "foobar"
"foo"
ghci> take 2 [1]
[1]
ghci> :type drop
drop :: Int -> [a] -> [a]
ghci> drop 3 "xyzzy"
"zy"
ghci> drop 1 []
[]

The splitAt function combines the functions of take and drop, returning a pair of the input list, split at the given index.

ghci> :type splitAt
splitAt :: Int -> [a] -> ([a], [a])
ghci> splitAt 3 "foobar"
("foo","bar")

The takeWhile and dropWhile functions take predicates: takeWhile takes elements from the beginning of a list as long as the predicate returns True, while dropWhile drops elements from the list as long as the predicate returns True.

ghci> :type takeWhile
takeWhile :: (a -> Bool) -> [a] -> [a]
ghci> takeWhile odd [1,3,5,6,8,9,11]
[1,3,5]
ghci> :type dropWhile
dropWhile :: (a -> Bool) -> [a] -> [a]
ghci> dropWhile even [2,4,6,7,9,10,12]
[7,9,10,12]

Just as splitAt “tuples up” the results of take and drop, the functions break (which we already saw in the section called “Warming up: portably splitting lines of text” and span tuple up the results of takeWhile and dropWhile.

Each function takes a predicate; break consumes its input while its predicate fails, while span consumes while its predicate succeeds.

ghci> :type span
span :: (a -> Bool) -> [a] -> ([a], [a])
ghci> span even [2,4,6,7,9,10,11]
([2,4,6],[7,9,10,11])
ghci> :type break
break :: (a -> Bool) -> [a] -> ([a], [a])
ghci> break even [1,3,5,6,8,9,10]
([1,3,5],[6,8,9,10])

Searching lists

As we’ve already seen, the elem function indicates whether a value is present in a list. It has a companion function, notElem.

ghci> :type elem
elem :: (Eq a) => a -> [a] -> Bool
ghci> 2 `elem` [5,3,2,1,1]
True
ghci> 2 `notElem` [5,3,2,1,1]
False

For a more general search, filter takes a predicate, and returns every element of the list on which the predicate succeeds.

ghci> :type filter
filter :: (a -> Bool) -> [a] -> [a]
ghci> filter odd [2,4,1,3,6,8,5,7]
[1,3,5,7]

In Data.List, three predicates, isPrefixOf, isInfixOf, and isSuffixOf, let us test for the presence of sublists within a bigger list. The easiest way to use them is using infix notation.

The isPrefixOf function tells us whether its left argument matches the beginning of its right argument.

ghci> :module +Data.List
ghci> :type isPrefixOf
isPrefixOf :: (Eq a) => [a] -> [a] -> Bool
ghci> "foo" `isPrefixOf` "foobar"
True
ghci> [1,2] `isPrefixOf` []
False

The isInfixOf function indicates whether its left argument is a sublist of its right.

ghci> :module +Data.List
ghci> [2,6] `isInfixOf` [3,1,4,1,5,9,2,6,5,3,5,8,9,7,9]
True
ghci> "funk" `isInfixOf` "sonic youth"
False

The operation of isSuffixOf shouldn’t need any explanation.

ghci> :module +Data.List
ghci> ".c" `isSuffixOf` "crashme.c"
True

Working with several lists at once

The zip function takes two lists and “zips” them into a single list of pairs. The resulting list is the same length as the shorter of the two inputs.

ghci> :type zip
zip :: [a] -> [b] -> [(a, b)]
ghci> zip [12,72,93] "zippity"
[(12,'z'),(72,'i'),(93,'p')]

More useful is zipWith, which takes two lists and applies a function to each pair of elements, generating a list that is the same length as the shorter of the two.

ghci> :type zipWith
zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
ghci> zipWith (+) [1,2,3] [4,5,6]
[5,7,9]

Haskell’s type system makes it an interesting challenge to write functions that take variable numbers of arguments[fn:1]. So if we want to zip three lists together, we call zip3 or zipWith3, and so on up to zip7 and zipWith7.

Special string-handling functions

We’ve already encountered the standard lines function in the section called “Warming up: portably splitting lines of text” and its standard counterpart, unlines. Notice that unlines always places a newline on the end of its result.

ghci> lines "foo\nbar"
["foo","bar"]
ghci> unlines ["foo", "bar"]
"foo\nbar\n"

The words function splits an input string on any white space. Its counterpart, unwords, uses a single space to join a list of words.

ghci> words "the  \r  quick \t  brown\n\n\nfox"
["the","quick","brown","fox"]
ghci> unwords ["jumps", "over", "the", "lazy", "dog"]
"jumps over the lazy dog"

Exercises

Write your own “safe” definitions of the standard partial list functions, but make sure that yours never fail. As a hint, you might want to consider using the following types.
```
safeHead :: [a] -> Maybe a
safeTail :: [a] -> Maybe [a]
safeLast :: [a] -> Maybe a
safeInit :: [a] -> Maybe [a]
    
```
Write a function splitWith that acts similarly to words, but takes a predicate and a list of any type, and splits its input list on every element for which the predicate returns False.
```
splitWith :: (a -> Bool) -> [a] -> [[a]]
    
```
Using the command framework from the section called “A simple command line framework” program that prints the first word of each line of its input.
Write a program that transposes the text in a file. For instance, it should convert "hello\nworld\n" to "hw\neo\nlr\nll\nod\n".

How to think about loops

Unlike traditional languages, Haskell has neither a for loop nor a while loop. If we’ve got a lot of data to process, what do we use instead? There are several possible answers to this question.

Explicit recursion

int as_int(char *str)
{
    int acc; /* accumulate the partial result */

    for (acc = 0; isdigit(*str); str++) {
    acc = acc * 10 + (*str - '0');
    }

    return acc;
}

Given that Haskell doesn’t have any looping constructs, how should we think about representing a fairly straightforward piece of code like this?

We don’t have to start off by writing a type signature, but it helps to remind us of what we’re working with.

import Data.Char (digitToInt) -- we'll need ord shortly

asInt :: String -> Int

The C code computes the result incrementally as it traverses the string; the Haskell code can do the same. However, in Haskell, we can express the equivalent of a loop as a function. We’ll call ours loop just to keep things nice and explicit.

loop :: Int -> String -> Int

asInt xs = loop 0 xs

That first parameter to loop is the accumulator variable we’ll be using. Passing zero into it is equivalent to initialising the acc variable in C at the beginning of the loop.

Rather than leap into blazing code, let’s think about the data we have to work with. Our familiar String is just a synonym for [Char], a list of characters. The easiest way for us to get the traversal right is to think about the structure of a list: it’s either empty, or a single element followed by the rest of the list.

We can express this structural thinking directly by pattern matching on the list type’s constructors. It’s often handy to think about the easy cases first: here, that means we will consider the empty-list case.

loop acc [] = acc

An empty list doesn’t just mean “the input string is empty”; it’s also the case we’ll encounter when we traverse all the way to the end of a non-empty list. So we don’t want to “error out” if we see an empty list. Instead, we should do something sensible. Here, the sensible thing is to terminate the loop, and return our accumulated value.

The other case we have to consider arises when the input list is not empty. We need to do something with the current element of the list, and something with the rest of the list.

loop acc (x:xs) = let acc' = acc * 10 + digitToInt x
                  in loop acc' xs

We compute a new value for the accumulator, and give it the name acc'. We then call the loop function again, passing it the updated value acc' and the rest of the input list; this is equivalent to the loop starting another round in C.

Each time the loop function calls itself, it has a new value for the accumulator, and it consumes one element of the input list. Eventually, it’s going to hit the end of the list, at which time the [] pattern will match, and the recursive calls will cease.

How well does this function work? For positive integers, it’s perfectly cromulent.

ghci> asInt "33"
33

But because we were focusing on how to traverse lists, not error handling, our poor function misbehaves if we try to feed it nonsense.

ghci> asInt ""
0
ghci> asInt "potato"
*** Exception: Char.digitToInt: not a digit 'p'

We’ll defer fixing our function’s shortcomings to Q:1.

Because the last thing that loop does is simply call itself, it’s an example of a tail recursive function. There’s another common idiom in this code, too. Thinking about the structure of the list, and handling the empty and non-empty cases separately, is a kind of approach called structural recursion.

We call the non-recursive case (when the list is empty) the base case (sometimes the terminating case). We’ll see people refer to the case where the function calls itself as the recursive case (surprise!), or they might give a nod to mathematical induction and call it the inductive case.

As a useful technique, structural recursion is not confined to lists; we can use it on other algebraic data types, too. We’ll have more to say about it later.

Transforming every piece of input

Consider another C function, square, which squares every element in an array.

void square(double *out, const double *in, size_t length)
{
    for (size_t i = 0; i < length; i++) {
        out[i] = in[i] * in[i];
    }
}

This contains a straightforward and common kind of loop, one that does exactly the same thing to every element of its input array. How might we write this loop in Haskell?

square :: [Double] -> [Double]

square (x:xs) = x*x : square xs
square []     = []

Our square function consists of two pattern matching equations. The first “deconstructs” the beginning of a non-empty list, to get its head and tail. It squares the first element, then puts that on the front of a new list, which is constructed by calling square on the remainder of the empty list. The second equation ensures that square halts when it reaches the end of the input list.

The effect of square is to construct a new list that’s the same length as its input list, with every element in the input list substituted with its square in the output list.

Here’s another such C loop, one that ensures that every letter in a string is converted to uppercase.

#include <ctype.h>

char *uppercase(const char *in)
{
    char *out = strdup(in);

    if (out != NULL) {
        for (size_t i = 0; out[i] != '\0'; i++) {
            out[i] = toupper(out[i]);
        }
    }

    return out;
}

Let’s look at a Haskell equivalent.

import Data.Char (toUpper)

upperCase :: String -> String

upperCase (x:xs) = toUpper x : upperCase xs
upperCase []     = []

Here, we’re importing the toUpper function from the standard Data.Char module, which contains lots of useful functions for working with Char data.

Our upperCase function follows a similar pattern to our earlier square function. It terminates with an empty list when the input list is empty; and when the input isn’t empty, it calls toUpper on the first element, then constructs a new list cell from that and the result of calling itself on the rest of the input list.

These examples follow a common pattern for writing recursive functions over lists in Haskell. The base case handles the situation where our input list is empty. The recursive case deals with a non-empty list; it does something with the head of the list, and calls itself recursively on the tail.

Mapping over a list

The square and upperCase functions that we just defined produce new lists that are the same lengths as their input lists, and do only one piece of work per element. This is such a common pattern that Haskell’s prelude defines a function, map, to make it easier. map takes a function, and applies it to every element of a list, returning a new list constructed from the results of these applications.

Here are our square and upperCase functions rewritten to use map.

square2 xs = map squareOne xs
    where squareOne x = x * x

upperCase2 xs = map toUpper xs

This is our first close look at a function that takes another function as its argument. We can learn a lot about what map does by simply inspecting its type.

ghci> :type map
map :: (a -> b) -> [a] -> [b]

The signature tells us that map takes two arguments. The first is a function that takes a value of one type, a, and returns a value of another type, b.

Since map takes a function as argument, we refer to it as a higher-order function. (In spite of the name, there’s nothing mysterious about higher-order functions; it’s just a term for functions that take other functions as arguments, or return functions.)

Since map abstracts out the pattern common to our square and upperCase functions so that we can reuse it with less boilerplate, we can look at what those functions have in common and figure out how to implement it ourselves.

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

myMap f (x:xs) = f x : myMap f xs
myMap _ _      = []

We try out our myMap function to give ourselves some assurance that it behaves similarly to the standard map.

ghci> :module +Data.Char
ghci> map toLower "SHOUTING"
"shouting"
ghci> myMap toUpper "whispering"
"WHISPERING"
ghci> map negate [1,2,3]
[-1,-2,-3]

This pattern of spotting a repeated idiom, then abstracting it so we can reuse (and write less!) code, is a common aspect of Haskell programming. While abstraction isn’t unique to Haskell, higher order functions make it remarkably easy.

Selecting pieces of input

Another common operation on a sequence of data is to comb through it for elements that satisfy some criterion. Here’s a function that walks a list of numbers and returns those that are odd. Our code has a recursive case that’s a bit more complex than our earlier functions: it only puts a number in the list it returns if the number is odd. Using a guard expresses this nicely.

oddList :: [Int] -> [Int]

oddList (x:xs) | odd x     = x : oddList xs
               | otherwise = oddList xs
oddList _                  = []

Let’s see that in action.

ghci> oddList [1,1,2,3,5,8,13,21,34]
[1,1,3,5,13,21]

Once again, this idiom is so common that the prelude defines a function, filter, which we have already introduced. It removes the need for boilerplate code to recurse over the list.

ghci> :type filter
filter :: (a -> Bool) -> [a] -> [a]
ghci> filter odd [3,1,4,1,5,9,2,6,5]
[3,1,1,5,9,5]

The filter function takes a predicate and applies it to every element in its input list, returning a list of only those for which the predicate evaluates to True. We’ll revisit filter again soon, in the section called “Folding from the right”

Computing one answer over a collection

Another common thing to do with a collection is reduce it to a single value. A simple example of this is summing the values of a list.

mySum xs = helper 0 xs
    where helper acc (x:xs) = helper (acc + x) xs
          helper acc _      = acc

Our helper function is tail recursive, and uses an accumulator parameter, acc, to hold the current partial sum of the list. As we already saw with asInt, this is a “natural” way to represent a loop in a pure functional language.

For something a little more complicated, let’s take a look at the Adler-32 checksum. This is a popular checksum algorithm; it concatenates two 16-bit checksums into a single 32-bit checksum. The first checksum is the sum of all input bytes, plus one. The second is the sum of all intermediate values of the first checksum. In each case, the sums are computed modulo 65521. Here’s a straightforward, unoptimised Java implementation. (It’s safe to skip it if you don’t read Java.)

public class Adler32
{
    private static final int base = 65521;

    public static int compute(byte[] data, int offset, int length)
    {
        int a = 1, b = 0;

        for (int i = offset; i < offset + length; i++) {
            a = (a + (data[i] & 0xff)) % base;
            b = (a + b) % base;
        }

        return (b << 16) | a;
    }
}

Although Adler-32 is a simple checksum, this code isn’t particularly easy to read on account of the bit-twiddling involved. Can we do any better with a Haskell implementation?

import Data.Char (ord)
import Data.Bits (shiftL, (.&.), (.|.))

base = 65521

adler32 xs = helper 1 0 xs
    where helper a b (x:xs) = let a' = (a + (ord x .&. 0xff)) `mod` base
                                  b' = (a' + b) `mod` base
                              in helper a' b' xs
          helper a b _      = (b `shiftL` 16) .|. a

This code isn’t exactly easier to follow than the Java code, but let’s look at what’s going on. First of all, we’ve introduced some new functions. The shiftL function implements a logical shift left; (.&.) provides bitwise “and”; and (.|.) provides bitwise “or”.

Once again, our helper function is tail recursive. We’ve turned the two variables we updated on every loop iteration in Java into accumulator parameters. When our recursion terminates on the end of the input list, we compute our checksum and return it.

If we take a step back, we can restructure our Haskell adler32 to more closely resemble our earlier mySum function. Instead of two accumulator parameters, we can use a pair as the accumulator.

adler32_try2 xs = helper (1,0) xs
    where helper (a,b) (x:xs) =
              let a' = (a + (ord x .&. 0xff)) `mod` base
                  b' = (a' + b) `mod` base
              in helper (a',b') xs
          helper (a,b) _      = (b `shiftL` 16) .|. a

Why would we want to make this seemingly meaningless structural change? Because as we’ve already seen with map and filter, we can extract the common behavior shared by mySum and adler32_try2 into a higher-order function. We can describe this behavior as “do something to every element of a list, updating an accumulator as we go, and returning the accumulator when we’re done”.

This kind of function is called a fold, because it “folds up” a list. There are two kinds of fold over lists, foldl for folding from the left (the start) and foldr for folding from the right (the end).

The left fold

Here is the definition of foldl.

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

foldl step zero (x:xs) = foldl step (step zero x) xs
foldl _    zero []     = zero

The foldl function takes a “step” function, an initial value for its accumulator, and a list. The “step” takes an accumulator and an element from the list, and returns a new accumulator value. All foldl does is call the “stepper” on the current accumulator and an element of the list, and passes the new accumulator value to itself recursively to consume the rest of the list.

We refer to foldl as a “left fold” because it consumes the list from left (the head) to right.

Here’s a rewrite of mySum using foldl.

foldlSum xs = foldl step 0 xs
    where step acc x = acc + x

That local function step just adds two numbers, so let’s simply use the addition operator instead, and eliminate the unnecessary where clause.

niceSum :: [Integer] -> Integer
niceSum xs = foldl (+) 0 xs

Notice how much simpler this code is than our original mySum? We’re no longer using explicit recursion, because foldl takes care of that for us. We’ve simplified our problem down to two things: what the initial value of the accumulator should be (the second parameter to foldl), and how to update the accumulator (the (+) function). As an added bonus, our code is now shorter, too, which makes it easier to understand.

Let’s take a deeper look at what foldl is doing here, by manually writing out each step in its evaluation when we call niceSum [1,2,3].

foldl (+) 0 (1:2:3:[])
          == foldl (+) (0 + 1)             (2:3:[])
          == foldl (+) ((0 + 1) + 2)       (3:[])
          == foldl (+) (((0 + 1) + 2) + 3) []
          ==           (((0 + 1) + 2) + 3)

We can rewrite adler32_try2 using foldl to let us focus on the details that are important.

adler32_foldl xs = let (a, b) = foldl step (1, 0) xs
                   in (b `shiftL` 16) .|. a
    where step (a, b) x = let a' = a + (ord x .&. 0xff)
                          in (a' `mod` base, (a' + b) `mod` base)

Here, our accumulator is a pair, so the result of foldl will be, too. We pull the final accumulator apart when foldl returns, and bit-twiddle it into a “proper” checksum.

Why use folds, maps, and filters?

A quick glance reveals that adler32_foldl isn’t really any shorter than adler32_try2. Why should we use a fold in this case? The advantage here lies in the fact that folds are extremely common in Haskell, and they have regular, predictable behavior.

This means that a reader with a little experience will have an easier time understanding a use of a fold than code that uses explicit recursion. A fold isn’t going to produce any surprises, but the behavior of a function that recurses explicitly isn’t immediately obvious. Explicit recursion requires us to read closely to understand exactly what’s going on.

This line of reasoning applies to other higher-order library functions, including those we’ve already seen, map and filter. Because they’re library functions with well-defined behavior, we only need to learn what they do once, and we’ll have an advantage when we need to understand any code that uses them. These improvements in readability also carry over to writing code. Once we start to think with higher order functions in mind, we’ll produce concise code more quickly.

Folding from the right

The counterpart to foldl is foldr, which folds from the right of a list.

foldr :: (a -> b -> b) -> b -> [a] -> b
foldr step zero (x:xs) = step x (foldr step zero xs)
foldr _    zero []     = zero

Let’s follow the same manual evaluation process with foldr (+) 0 [1,2,3] as we did with niceSum in the section called “The left fold”

foldr (+) 0 (1:2:3:[])
          == 1 +           foldr (+) 0 (2:3:[])
          == 1 + (2 +      foldr (+) 0 (3:[])
          == 1 + (2 + (3 + foldr (+) 0 []))
          == 1 + (2 + (3 + 0))

The difference between foldl and foldr should be clear from looking at where the parentheses and the “empty list” elements show up. With foldl, the empty list element is on the left, and all the parentheses group to the left. With foldr, the zero value is on the right, and the parentheses group to the right.

There is a lovely intuitive explanation of how foldr works: it replaces the empty list with the zero value, and every constructor in the list with an application of the step function.

1 : (2 : (3 : []))
1 + (2 + (3 + 0 ))

At first glance, foldr might seem less useful than foldl: what use is a function that folds from the right? But consider the prelude’s filter function, which we last encountered in the section called “Selecting pieces of input” filter using explicit recursion, it will look something like this.

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

Perhaps surprisingly, though, we can write filter as a fold, using foldr.

myFilter p xs = foldr step [] xs
    where step x ys | p x       = x : ys
                    | otherwise = ys

This is the sort of definition that could cause us a headache, so let’s examine it in a little depth. Like foldl, foldr takes a function and a base case (what to do when the input list is empty) as arguments. From reading the type of filter, we know that our myFilter function must return a list of the same type as it consumes, so the base case should be a list of this type, and the step helper function must return a list.

Since we know that foldr calls step on one element of the input list at a time, with the accumulator as its second argument, what step does must be quite simple. If the predicate returns True, it pushes that element onto the accumulated list; otherwise, it leaves the list untouched.

The class of functions that we can express using foldr is called primitive recursive. A surprisingly large number of list manipulation functions are primitive recursive. For example, here’s map written in terms of foldr.

myMap :: (a -> b) -> [a] -> [b]
myMap f xs = foldr step [] xs
    where step x ys = f x : ys

In fact, we can even write foldl using foldr!

myFoldl :: (a -> b -> a) -> a -> [b] -> a
myFoldl f z xs = foldr step id xs z
    where step x g a = g (f a x)

Returning to our earlier intuitive explanation of what foldr does, another useful way to think about it is that it transforms its input list. Its first two arguments are “what to do with each head/tail element of the list”, and “what to substitute for the end of the list”.

The “identity” transformation with foldr thus replaces the empty list with itself, and applies the list constructor to each head/tail pair:

identity :: [a] -> [a]
identity xs = foldr (:) [] xs

It transforms a list into a copy of itself.

ghci> identity [1,2,3]
[1,2,3]

If foldr replaces the end of a list with some other value, this gives us another way to look at Haskell’s list append function, (++).

ghci> [1,2,3] ++ [4,5,6]
[1,2,3,4,5,6]

All we have to do to append a list onto another is substitute that second list for the end of our first list.

append :: [a] -> [a] -> [a]
append xs ys = foldr (:) ys xs

Let’s try this out.

ghci> append [1,2,3] [4,5,6]
[1,2,3,4,5,6]

Here, we replace each list constructor with another list constructor, but we replace the empty list with the list we want to append onto the end of our first list.

As our extended treatment of folds should indicate, the foldr function is nearly as important a member of our list-programming toolbox as the more basic list functions we saw in the section called “Working with lists” produce a list incrementally, which makes it useful for writing lazy data processing code.

Left folds, laziness, and space leaks

To keep our initial discussion simple, we used foldl throughout most of this section. This is convenient for testing, but we will never use foldl in practice.

The reason has to do with Haskell’s non-strict evaluation. If we apply foldl (+) [1,2,3], it evaluates to the expression (((0 + 1) + 2) + 3). We can see this occur if we revisit the way in which the function gets expanded.

foldl (+) 0 (1:2:3:[])
          == foldl (+) (0 + 1)             (2:3:[])
          == foldl (+) ((0 + 1) + 2)       (3:[])
          == foldl (+) (((0 + 1) + 2) + 3) []
          ==           (((0 + 1) + 2) + 3)

The final expression will not be evaluated to 6 until its value is demanded. Before it is evaluated, it must be stored as a thunk. Not surprisingly, a thunk is more expensive to store than a single number, and the more complex the thunked expression, the more space it needs. For something cheap like arithmetic, thunking an expresion is more computationally expensive than evaluating it immediately. We thus end up paying both in space and in time.

When GHC is evaluating a thunked expression, it uses an internal stack to do so. Because a thunked expression could potentially be infinitely large, GHC places a fixed limit on the maximum size of this stack. Thanks to this limit, we can try a large thunked expression in ghci without needing to worry that it might consume all of memory.

ghci> foldl (+) 0 [1..1000]
500500

From looking at the expansion above, we can surmise that this creates a thunk that consists of 1000 integers and 999 applications of (+). That’s a lot of memory and effort to represent a single number! With a larger expression, although the size is still modest, the results are more dramatic.

ghci> foldl (+) 0 [1..1000000]
*** Exception: stack overflow

On small expressions, foldl will work correctly but slowly, due to the thunking overhead that it incurs. We refer to this invisible thunking as a space leak, because our code is operating normally, but using far more memory than it should.

On larger expressions, code with a space leak will simply fail, as above. A space leak with foldl is a classic roadblock for new Haskell programmers. Fortunately, this is easy to avoid.

The Data.List module defines a function named foldl' that is similar to foldl, but does not build up thunks. The difference in behavior between the two is immediately obvious.

ghci> foldl  (+) 0 [1..1000000]
*** Exception: stack overflow
ghci> :module +Data.List
ghci> foldl' (+) 0 [1..1000000]
500000500000

Due to the thunking behavior of foldl, it is wise to avoid this function in real programs: even if it doesn’t fail outright, it will be unnecessarily inefficient. Instead, import Data.List and use foldl'.

Exercises

Use a fold (choosing the appropriate fold will make your code much simpler) to rewrite and improve upon the asInt function from the section called “Explicit recursion”

asInt_fold :: String -> Int

Your function should behave as follows.

ghci> asInt_fold "101"
ghci> asInt_fold "-31337"
-31337
ghci> asInt_fold "1798"
1798

Extend your function to handle the following kinds of exceptional conditions by calling error.

ghci> asInt_fold ""
0
ghci> asInt_fold "-"
0
ghci> asInt_fold "-3"
-3
ghci> asInt_fold "2.7"
*** Exception: Char.digitToInt: not a digit '.'
ghci> asInt_fold "314159265358979323846"
564616105916946374

The asInt_fold function uses error, so its callers cannot handle errors. Rewrite it to fix this problem.

type ErrorMessage = String
asInt_either :: String -> Either ErrorMessage Int

ghci> asInt_either "33"
Right 33
ghci> asInt_either "foo"
Left "non-digit 'o'"

The Prelude function concat concatenates a list of lists into a single list, and has the following type.
```
concat :: [[a]] -> [a]
    
```
Write your own definition of concat using foldr.
Write your own definition of the standard takeWhile function, first using explicit recursion, then foldr.
The Data.List module defines a function, groupBy, which has the following type.
```
groupBy :: (a -> a -> Bool) -> [a] -> [[a]]
    
```
Use ghci to load the Data.List module and figure out what groupBy does, then write your own implementation using a fold.
How many of the following prelude functions can you rewrite using list folds?
- any
- cycle
- words
- unlines
For those functions where you can use either foldl' or foldr, which is more appropriate in each case?

Anonymous (lambda) functions

In many of the function definitions we’ve seen so far, we’ve written short helper functions.

isInAny needle haystack = any inSequence haystack
    where inSequence s = needle `isInfixOf` s

Haskell lets us write completely anonymous functions, which we can use to avoid the need to give names to our helper functions. Anonymous functions are often called “lambda” functions, in a nod to their heritage in the lambda calculus. We introduce an anonymous function with a backslash character, \, pronounced /lambda/[fn:2]. This is followed by the function’s arguments (which can include patterns), then an arrow -> to introduce the function’s body.

Lambdas are most easily illustrated by example. Here’s a rewrite of isInAny using an anonymous function.

isInAny2 needle haystack = any (\s -> needle `isInfixOf` s) haystack

We’ve wrapped the lambda in parentheses here so that Haskell can tell where the function body ends.

Anonymous functions behave in every respect identically to functions that have names, but Haskell places a few important restrictions on how we can define them. Most importantly, while we can write a normal function using multiple clauses containing different patterns and guards, a lambda can only have a single clause in its definition.

The limitation to a single clause restricts how we can use patterns in the definition of a lambda. We’ll usually write a normal function with several clauses to cover different pattern matching possibilities.

safeHead (x:_) = Just x
safeHead _ = Nothing

But as we can’t write multiple clauses to define a lambda, we must be certain that any patterns we use will match.

unsafeHead = \(x:_) -> x

This definition of unsafeHead will explode in our faces if we call it with a value on which pattern matching fails.

ghci> :type unsafeHead
unsafeHead :: [t] -> t
ghci> unsafeHead [1]
1
ghci> unsafeHead []
*** Exception: Lambda.hs:7:13-23: Non-exhaustive patterns in lambda

The definition type-checks, so it will compile, so the error will occur at runtime. The moral of this story is to be careful in how you use patterns when defining an anonymous function: make sure your patterns can’t fail!

Another thing to notice about the isInAny and isInAny2 functions we showed above is that the first version, using a helper function that has a name, is a little easier to read than the version that plops an anonymous function into the middle. The named helper function doesn’t disrupt the “flow” of the function in which it’s used, and the judiciously chosen name gives us a little bit of information about what the function is expected to do.

In contrast, when we run across a lambda in the middle of a function body, we have to switch gears and read its definition fairly carefully to understand what it does. To help with readability and maintainability, then, we tend to avoid lambdas in many situations where we could use them to trim a few characters from a function definition. Very often, we’ll use a partially applied function instead, resulting in clearer and more readable code than either a lambda or an explicit function. Don’t know what a partially applied function is yet? Read on!

We don’t intend these caveats to suggest that lambdas are useless, merely that we ought to be mindful of the potential pitfalls when we’re thinking of using them. In later chapters, we will see that they are often invaluable as “glue”.

Partial function application and currying

You may wonder why the -> arrow is used for what seems to be two purposes in the type signature of a function.

ghci> :type dropWhile
dropWhile :: (a -> Bool) -> [a] -> [a]

It looks like the -> is separating the arguments to dropWhile from each other, but that it also separates the arguments from the return type. But in fact -> has only one meaning: it denotes a function that takes an argument of the type on the left, and returns a value of the type on the right.

The implication here is very important: in Haskell, all functions take only one argument. While dropWhile looks like a function that takes two arguments, it is actually a function of one argument, which returns a function that takes one argument. Here’s a perfectly valid Haskell expression.

ghci> :module +Data.Char
ghci> :type dropWhile isSpace
dropWhile isSpace :: [Char] -> [Char]

Well, that looks useful. The value dropWhile isSpace is a function that strips leading white space from a string. How is this useful? As one example, we can use it as an argument to a higher order function.

ghci> map (dropWhile isSpace) [" a","f","   e"]
["a","f","e"]

Every time we supply an argument to a function, we can “chop” an element off the front of its type signature. Let’s take zip3 as an example to see what we mean; this is a function that zips three lists into a list of three-tuples.

ghci> :type zip3
zip3 :: [a] -> [b] -> [c] -> [(a, b, c)]
ghci> zip3 "foo" "bar" "quux"
[('f','b','q'),('o','a','u'),('o','r','u')]

If we apply zip3 with just one argument, we get a function that accepts two arguments. No matter what arguments we supply to this compound function, its first argument will always be the fixed value we specified.

ghci> :type zip3 "foo"
zip3 "foo" :: [b] -> [c] -> [(Char, b, c)]
ghci> let zip3foo = zip3 "foo"
ghci> :type zip3foo
zip3foo :: [b] -> [c] -> [(Char, b, c)]
ghci> (zip3 "foo") "aaa" "bbb"
[('f','a','b'),('o','a','b'),('o','a','b')]
ghci> zip3foo "aaa" "bbb"
[('f','a','b'),('o','a','b'),('o','a','b')]
ghci> zip3foo [1,2,3] [True,False,True]
[('f',1,True),('o',2,False),('o',3,True)]

When we pass fewer arguments to a function than the function can accept, we call this partial application of the function: we’re applying the function to only some of its arguments.

In the example above, we have a partially applied function, zip3 "foo", and a new function, zip3foo. We can see that the type signatures of the two and their behavior are identical.

This applies just as well if we fix two arguments, giving us a function of just one argument.

ghci> let zip3foobar = zip3 "foo" "bar"
ghci> :type zip3foobar
zip3foobar :: [c] -> [(Char, Char, c)]
ghci> zip3foobar "quux"
[('f','b','q'),('o','a','u'),('o','r','u')]
ghci> zip3foobar [1,2]
[('f','b',1),('o','a',2)]

Partial function application lets us avoid writing tiresome throwaway functions. It’s often more useful for this purpose than the anonymous functions we introduced in the section called “Anonymous (lambda) functions” the isInAny function we defined there, here’s how we’d use a partially applied function instead of a named helper function or a lambda.

isInAny3 needle haystack = any (isInfixOf needle) haystack

Here, the expression isInfixOf needle is the partially applied function. We’re taking the function isInfixOf, and “fixing” its first argument to be the needle variable from our parameter list. This gives us a partially applied function that has exactly the same type and behavior as the helper and lambda in our earlier definitions.

Partial function application is named currying, after the logician Haskell Curry (for whom the Haskell language is named).

As another example of currying in use, let’s return to the list-summing function we wrote in the section called “The left fold”

niceSum :: [Integer] -> Integer
niceSum xs = foldl (+) 0 xs

We don’t need to fully apply foldl; we can omit the list xs from both the parameter list and the parameters to foldl, and we’ll end up with a more compact function that has the same type.

nicerSum :: [Integer] -> Integer
nicerSum = foldl (+) 0

Sections

Haskell provides a handy notational shortcut to let us write a partially applied function in infix style. If we enclose an operator in parentheses, we can supply its left or right argument inside the parentheses to get a partially applied function. This kind of partial application is called a section.

ghci> (1+) 2
3
ghci> map (*3) [24,36]
[72,108]
ghci> map (2^) [3,5,7,9]
[8,32,128,512]

If we provide the left argument inside the section, then calling the resulting function with one argument supplies the operator’s right argument. And vice versa.

Recall that we can wrap a function name in backquotes to use it as an infix operator. This lets us use sections with functions.

ghci> :type (`elem` ['a'..'z'])
(`elem` ['a'..'z']) :: Char -> Bool

The above definition fixes elem’s second argument, giving us a function that checks to see whether its argument is a lowercase letter.

ghci> (`elem` ['a'..'z']) 'f'
True

Using this as an argument to all, we get a function that checks an entire string to see if it’s all lowercase.

ghci> all (`elem` ['a'..'z']) "Frobozz"
False

If we use this style, we can further improve the readability of our earlier isInAny3 function.

isInAny4 needle haystack = any (needle `isInfixOf`) haystack

As-patterns

Haskell’s tails function, in the Data.List module, generalises the tail function we introduced earlier. Instead of returning one “tail” of a list, it returns all of them.

ghci> :m +Data.List
ghci> tail "foobar"
"oobar"
ghci> tail (tail "foobar")
"obar"
ghci> tails "foobar"
["foobar","oobar","obar","bar","ar","r",""]

Each of these strings is a suffix of the initial string, so tails produces a list of all suffixes, plus an extra empty list at the end. It always produces that extra empty list, even when its input list is empty.

ghci> tails []

What if we want a function that behaves like tails, but which only returns the non-empty suffixes? One possibility would be for us to write our own version by hand. We’ll use a new piece of notation, the @ symbol.

suffixes :: [a] -> [[a]]
suffixes xs@(_:xs') = xs : suffixes xs'
suffixes _ = []

The pattern xs @ (_ : xs') is called an as-pattern, and it means “bind the variable xs to the value that matches the right side of the @ symbol”.

In our example, if the pattern after the “@” matches, xs will be bound to the entire list that matched, and xs' to all but the head of the list (we used the wild card _ pattern to indicate that we’re not interested in the value of the head of the list).

ghci> tails "foo"
["foo","oo","o",""]
ghci> suffixes "foo"
["foo","oo","o"]

The as-pattern makes our code more readable. To see how it helps, let us compare a definition that lacks an as-pattern.

noAsPattern :: [a] -> [[a]]
noAsPattern (x:xs) = (x:xs) : noAsPattern xs
noAsPattern _ = []

Here, the list that we’ve deconstructed in the pattern match just gets put right back together in the body of the function.

As-patterns have a more practical use than simple readability: they can help us to share data instead of copying it. In our definition of noAsPattern, when we match (x : xs), we construct a new copy of it in the body of our function. This causes us to allocate a new list node at run time. That may be cheap, but it isn’t free. In contrast, when we defined suffixes, we reused the value xs that we matched with our as-pattern. Since we reuse an existing value, we avoid a little allocation.

Code reuse through composition

It seems a shame to introduce a new function, suffixes, that does almost the same thing as the existing tails function. Surely we can do better?

Recall the init function we introduced in the section called “Working with lists” last element of a list.

suffixes2 xs = init (tails xs)

This suffixes2 function behaves identically to suffixes, but it’s a single line of code.

ghci> suffixes2 "foo"
["foo","oo","o"]

If we take a step back, we see the glimmer of a pattern here: we’re applying a function, then applying another function to its result. Let’s turn that pattern into a function definition.

compose :: (b -> c) -> (a -> b) -> a -> c
compose f g x = f (g x)

We now have a function, compose, that we can use to “glue” two other functions together.

suffixes3 xs = compose init tails xs

Haskell’s automatic currying lets us drop the xs variable, so we can make our definition even shorter.

suffixes4 = compose init tails

Fortunately, we don’t need to write our own compose function. Plugging functions into each other like this is so common that the Prelude provides function composition via the (.) operator.

suffixes5 = init . tails

The (.) operator isn’t a special piece of language syntax; it’s just a normal operator.

ghci> :type (.)
(.) :: (b -> c) -> (a -> b) -> a -> c
ghci> :type suffixes
suffixes :: [a] -> [[a]]
ghci> :type suffixes5
suffixes5 :: [a] -> [[a]]
ghci> suffixes5 "foo"
["foo","oo","o"]

We can create new functions at any time by writing chains of composed functions, stitched together with (.), so long (of course) as the result type of the function on the right of each (.) matches the type of parameter that the function on the left can accept.

As an example, let’s solve a simple puzzle: counting the number of words in a string that begin with a capital letter.

ghci> :module +Data.Char
ghci> let capCount = length . filter (isUpper . head) . words
ghci> capCount "Hello there, Mom!"
2

We can understand what this composed function does by examining its pieces. The (.) function is right associative, so we will proceed from right to left.

ghci> :type words
words :: String -> [String]

The words function has a result type of [String], so whatever is on the left side of (.) must accept a compatible argument.

ghci> :type isUpper . head
isUpper . head :: [Char] -> Bool

This function returns True if a word begins with a capital letter (try it in ghci), so filter (isUpper . head) returns a list of Strings containing only words that begin with capital letters.

ghci> :type filter (isUpper . head)
filter (isUpper . head) :: [[Char]] -> [[Char]]

Since this expression returns a list, all that remains is calculate the length of the list, which we do with another composition.

Here’s another example, drawn from a real application. We want to extract a list of macro names from a C header file shipped with libpcap, a popular network packet filtering library. The header file contains a large number definitions of the following form.

#define DLT_EN10MB      1       /* Ethernet (10Mb) */
#define DLT_EN3MB       2       /* Experimental Ethernet (3Mb) */
#define DLT_AX25        3       /* Amateur Radio AX.25 */

Our goal is to extract names such as DLT_EN10MB and DLT_AX25.

import Data.List (isPrefixOf)

dlts :: String -> [String]

dlts = foldr step [] . lines

We treat an entire file as a string, split it up with lines, then apply foldr step [] to the resulting list of lines. The step helper function operates on a single line.

where step l ds
        | "#define DLT_" `isPrefixOf` l = secondWord l : ds
        | otherwise                     = ds
      secondWord = head . tail . words

If we match a macro definition with our guard expression, we cons the name of the macro onto the head of the list we’re returning; otherwise, we leave the list untouched.

While the individual functions in the body of secondWord are by now familiar to us, it can take a little practice to piece together a chain of compositions like this. Let’s walk through the procedure.

Once again, we proceed from right to left. The first function is words.

ghci> :type words
words :: String -> [String]
ghci> words "#define DLT_CHAOS    5"
["#define","DLT_CHAOS","5"]

We then apply tail to the result of words.

ghci> :type tail
tail :: [a] -> [a]
ghci> tail ["#define","DLT_CHAOS","5"]
["DLT_CHAOS","5"]
ghci> :type tail . words
tail . words :: String -> [String]
ghci> (tail . words) "#define DLT_CHAOS    5"
["DLT_CHAOS","5"]

Finally, applying head to the result of drop 1 . words will give us the name of our macro.

ghci> :type head . tail . words
head . tail . words :: String -> String
ghci> (head . tail . words) "#define DLT_CHAOS    5"
"DLT_CHAOS"

Use your head wisely

After warning against unsafe list functions in the section called “Safely and sanely working with crashy functions” here we are calling both head and tail, two of those unsafe list functions. What gives?

In this case, we can assure ourselves by inspection that we’re safe from a runtime failure. The pattern guard in the definition of step contains two words, so when we apply words to any string that makes it past the guard, we’ll have a list of at least two elements, "#define" and some macro beginning with "DLT_".

This the kind of reasoning we ought to do to convince ourselves that our code won’t explode when we call partial functions. Don’t forget our earlier admonition: calling unsafe functions like this requires care, and can often make our code more fragile in subtle ways. If we for some reason modified the pattern guard to only contain one word, we could expose ourselves to the possibility of a crash, as the body of the function assumes that it will receive two words.

Tips for writing readable code

So far in this chapter, we’ve come across two tempting looking features of Haskell: tail recursion and anonymous functions. As nice as these are, we don’t often want to use them.

Many list manipulation operations can be most easily expressed using combinations of library functions such as map, take, and filter. Without a doubt, it takes some practice to get used to using these. In return for our initial investment, we can write and read code more quickly, and with fewer bugs.

The reason for this is simple. A tail recursive function definition has the same problem as a loop in an imperative language: it’s completely general. It might perform some filtering, some mapping, or who knows what else. We are forced to look in detail at the entire definition of the function to see what it’s really doing. In contrast, map and most other list manipulation functions do only one thing. We can take for granted what these simple building blocks do, and focus on the idea the code is trying to express, not the minute details of how it’s manipulating its inputs.

In the middle ground between tail recursive functions (with complete generality) and our toolbox of list manipulation functions (each of which does one thing) lie the folds. A fold takes more effort to understand than, say, a composition of map and filter that does the same thing, but it behaves more regularly and predictably than a tail recursive function. As a general rule, don’t use a fold if you can compose some library functions, but otherwise try to use a fold in preference to a hand-rolled a tail recursive loop.

As for anonymous functions, they tend to interrupt the “flow” of reading a piece of code. It is very often as easy to write a local function definition in a let or where clause, and use that, as it is to put an anonymous function into place. The relative advantages of a named function are twofold: we don’t need to understand the function’s definition when we’re reading the code that uses it; and a well chosen function name acts as a tiny piece of local documentation.

Space leaks and strict evaluation

The foldl function that we discussed earlier is not the only place where space leaks can arise in Haskell code. We will use it to illustrate how non-strict evaluation can sometimes be problematic, and how to solve the difficulties that can arise.

Avoiding space leaks with seq

We refer to an expression that is not evaluated lazily as strict, so foldl' is a strict left fold. It bypasses Haskell’s usual non-strict evaluation through the use of a special function named seq.

foldl' _    zero []     = zero
foldl' step zero (x:xs) =
    let new = step zero x
    in  new `seq` foldl' step new xs

This seq function has a peculiar type, hinting that it is not playing by the usual rules.

ghci> :type seq
seq :: a -> t -> t

It operates as follows: when a seq expression is evaluated, it forces its first argument to be evaluated, then returns its second argument. It doesn’t actually do anything with the first argument: seq exists solely as a way to force that value to be evaluated. Let’s walk through a brief application to see what happens.

foldl' (+) 1 (2:[])

This expands as follows.

let new = 1 + 2
in new `seq` foldl' (+) new []

The use of seq forcibly evaluates new to 3, and returns its second argument.

foldl' (+) 3 []

We end up with the following result.

Thanks to seq, there are no thunks in sight.

Learning to use seq

Without some direction, there is an element of mystery to using seq effectively. Here are some useful rules for using it well.

To have any effect, a seq expression must be the first thing evaluated in an expression.

-- incorrect: seq is hidden by the application of someFunc
-- since someFunc will be evaluated first, seq may occur too late
hiddenInside x y = someFunc (x `seq` y)

-- incorrect: a variation of the above mistake
hiddenByLet x y z = let a = x `seq` someFunc y
                    in anotherFunc a z

-- correct: seq will be evaluated first, forcing evaluation of x
onTheOutside x y = x `seq` someFunc y

To strictly evaluate several values, chain applications of seq together.

chained x y z = x `seq` y `seq` someFunc z

A common mistake is to try to use seq with two unrelated expressions.

badExpression step zero (x:xs) =
    seq (step zero x)
        (badExpression step (step zero x) xs)

Here, the apparent intention is to evaluate step zero x strictly. Since the expression is duplicated in the body of the function, strictly evaluating the first instance of it will have no effect on the second. The use of let from the definition of foldl' above shows how to achieve this effect correctly.

When evaluating an expression, seq stops as soon as it reaches a constructor. For simple types like numbers, this means that it will evaluate them completely. Algebraic data types are a different story. Consider the value (1 + 2) : (3 + 4) : []. If we apply seq to this, it will evaluate the (1 + 2) thunk. Since it will stop when it reaches the first (:) constructor, it will have no effect on the second thunk. The same is true for tuples: seq ((1 + 2), (3 + 4)) True will do nothing to the thunks inside the pair, since it immediately hits the pair’s constructor.

If necessary, we can use normal functional programming techniques to work around these limitations.

strictPair (a,b) = a `seq` b `seq` (a,b)
strictList (x:xs) = x `seq` x : strictList xs
strictList []     = []

It is important to understand that seq isn’t free: it has to perform a check at runtime to see if an expression has been evaluated. Use it sparingly. For instance, while our strictPair function evaluates the contents of a pair up to the first constructor, it adds the overheads of pattern matching, two applications of seq, and the construction of a new tuple. If we were to measure its performance in the inner loop of a benchmark, we might find it to slow the program down.

Aside from its performance cost if overused, seq is not a miracle cure-all for memory consumption problems. Just because you can evaluate something strictly doesn’t mean you should. Careless use of seq may do nothing at all; move existing space leaks around; or introduce new leaks.

The best guides to whether seq is necessary, and how well it is working, are performance measurement and profiling, which we will cover in Chapter 25, /Profiling and optimization/. From a base of empirical measurement, you will develop a reliable sense of when seq is most useful.

Footnotes

[fn:1] Unfortunately, we do not have room to address that challenge in this book.

[fn:2] The backslash was chosen for its visual resemblance to the Greek letter lambda, λ. Although GHC can accept Unicode input, it correctly treats λ as a letter, not as a synonym for \.

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Chapter 4: Functional Programming

Thinking in Haskell

A simple command line framework

Warming up: portably splitting lines of text

A line ending conversion program

Infix functions

Working with lists

Basic list manipulation

Safely and sanely working with crashy functions

Partial and total functions

More simple list manipulations

Working with sublists

Searching lists

Working with several lists at once

Special string-handling functions

Exercises

How to think about loops

Explicit recursion

Transforming every piece of input

Mapping over a list

Selecting pieces of input

Computing one answer over a collection

The left fold

Why use folds, maps, and filters?

Folding from the right

Left folds, laziness, and space leaks

Exercises

Further reading

Anonymous (lambda) functions

Partial function application and currying

Sections

As-patterns

Code reuse through composition

Use your head wisely

Tips for writing readable code

Space leaks and strict evaluation

Avoiding space leaks with seq

Learning to use seq

Footnotes