8.7
3 Getting comfortable with first-class functions
3.1 Overview
{- --- fulltitle: Higher-Order Programming Patterns date: Jan 31, 2023 --- -} module HigherOrder where import Data.Char import Test.HUnit import Prelude hiding (filter, foldr, map, pred, product, sum) {- Functions Are Data ================== As in all functional languages, Haskell functions are *first-class* values, meaning that they can be treated just as you would any other data. You can pass functions around in *any* manner that you can pass any other data around. For example, suppose you have the simple functions `plus1` and `minus1` defined via the equations -} plus1 :: Int -> Int plus1 x = x + 1 minus1 :: Int -> Int minus1 x = x - 1 {- Now, you can make a pair containing both the functions -} funp :: (Int -> Int, Int -> Int) funp = (plus1, minus1) {- Or you can make a list containing the functions -} funs :: [Int -> Int] funs = undefined {- Taking Functions as Input ------------------------- This innocent looking feature makes a language surprisingly brawny and flexible, because now, we can write *higher-order* functions that take functions as input and return functions as output! Consider: -} doTwice :: (a -> a) -> a -> a doTwice f x = f (f x) dtTests :: Test dtTests = TestList [ doTwice plus1 4 ~?= 6, doTwice minus1 5 ~?= 3 ] {- Here, `doTwice` takes two inputs: a function `f` and value `x`, and returns the the result of applying `f` to `x`, and feeding that result back into `f` to get the final output. Note how the raw code is clearer to understand than my long-winded English description! Last time we talked about how programs execute in Haskell: we substitute equals-for-equals, just like simplifying equations in math class. Let's think about an example with `doTwice`: ~~~~~ doTwice plus1 10 {- unfold doTwice -} == plus1 (plus1 10) {- unfold first plus1 -} == (plus1 10) + 1 {- unfold other plus1 -} == (10 + 1) + 1 {- old-school arithmetic -} == 12 ~~~~~ What you might infer from this example is that Haskell does not evaluate arguments before calling the functions. In other words, we didn't unfold `plus1 10` (and simplifiy it to `11`) before unfolding the first call to `plus1`. Instead, Haskell waits until we actually need that argument, for example, if we need to add it to something. Executing code in this way is technically called "call-by-name" evaluation. It is a fine way to think about how Haskell evaluates. Under the covers, the compiler is a bit smarter though, and will reuse computations when it can (i.e. "lazy" evaluation). We won't go into the details of lazy evaluation today, but be aware that there is a difference in evaluation order between Haskell and almost every other language that you have seen. Furthermore, what is great is that in Haskell thinking about evaluation in terms of substituting equals-for-equals works anywhere. As a result, we could *also* understand the evaluation of `doTwice` using a more standard order of evaluation. ~~~~~ doTwice plus1 10 {- unfold doTwice -} == plus1 (plus1 10) {- unfold second plus1 -} == plus1 (10 + 1) {- old-school arithmetic -} == plus1 11 {- unfold first plus1 -} == 11 + 1 {- more arithmetic -} == 12 ~~~~~ or even a hybrid of the two versions! ~~~~~ doTwice plus1 10 {- unfold doTwice -} == plus1 (plus1 10) {- unfold second plus1 -} == plus1 (10 + 1) {- unfold first plus1 -} == (10 + 1) + 1 {- old-school arithmetic -} == 12 ~~~~~ Returning Functions as Output ----------------------------- Similarly, it can be useful to write functions that return new functions as output. For example, rather than writing different versions `plus1`, `plus2`, `plus3`, *etc.* we can write a single function `plusn` as -} plusn :: Int -> (Int -> Int) plusn n = f where f x = x + n {- That is, `plusn` returns a function `f` which itself takes as input an integer `x` and adds `n` to it. Lets use it -} plus10 :: Int -> Int plus10 = plusn 10 minus20 :: Int -> Int minus20 = plusn (-20) {- Note the types of the above are `Int -> Int`. That is, `plus10` and `minus20` are functions that take in an integer and return an integer (even though we didn't explicitly give them an argument). -} -- >>> plus10 3 {- -} -- >>> plusn 10 3 {- Partial Application ------------------- In regular arithmetic, the `-` operator is *left-associative*. Hence, 2 - 1 - 1 is equivalent to (2 - 1) - 1 and thus to 0 (and not `2 - (1 - 1) == 2` !). Just like `-` is an arithmetic operator that takes two numbers and returns an number, in Haskell, `->` is a *type operator* that takes two types, the input and output, and returns a new function type. However, `->` is *right-associative*: the type Int -> Int -> Int is equivalent to Int -> (Int -> Int) That is, the first type (a function which takes two Ints) is in reality a function that takes a single Int as input, and returns as *output* a function from `Int` to `Int`! Equipped with this knowledge, consider the function -} plus :: Int -> Int -> Int plus m n = m + n {- Thus, whenever we use `plus` we can either pass in both the inputs at once, as in plus 10 20 or instead, we can *partially* apply the function, by just passing in only one input out of the two that it expects. -} plusfive :: Int -> Int plusfive = plus 5 {- thereby getting as output a function that is *waiting* for the second input (at which point it will produce the final result). -} pfivetest :: Test pfivetest = plusfive 1000 ~?= 1005 {- So how does this execute? Again *substitute equals for equals* ~~~~~{.haskell} plusfive 1000 == plus 5 1000 {- definition of plusfive -} == 5 + 1000 {- unfold plus -} == 1005 {- arithmetic -} ~~~~~ Finally, by now it should be pretty clear that `plusn n` is equivalent to the partially applied `plus n`. If you have been following so far, you should know how this behaves. -} doTwicePlus20 :: Int -> Int doTwicePlus20 = doTwice (plus 20) {- First, see if you can figure out the type. Next, see if you can figure out how this evaluates under the "call-by-name" evaluation order. Remember that this means that you should not evaluate arguments before substituting them into the body of a defined function. doTwicePlus20 0 == doTwice (plus 20) 0 {- unfold doTwice -} == (plus 20) ((plus 20) 0) ... undefined (fill this part in) ... == 20 + 20 + 0 == 40 Note that with partial application, that the order of arguments to the function matters. We can partially apply `plus` to its first argument, but there isn't a default way to skip that first argument and apply it to the second. For example, we can easily use partial application to specialize this function to a particular `Int` (don't bother trying to figure out what it does): -} twoArg :: Int -> String -> Bool twoArg i s = length (s ++ show i) >= 2 {- thus -} oneStringArg :: String -> Bool oneStringArg = twoArg 3 {- However, if we wanted to specialize it to a particular `String`, then it is a bit more clumsy. One solution is to use a library function like `flip` (check out its type in ghci!) to swap the order of the arguments. -} oneIntArg :: Int -> Bool oneIntArg = flip twoArg "a" {- Another solution relies on anonymous functions. (See if you can figure it out after reading the section below.) Anonymous Functions ------------------- As we have seen, with Haskell, it is quite easy to create function values that are not bound to any name. For example the expression `plus 1000` yields a function value that doesn't have another way to refer to it. We will see many situations where a particular function is only used once, and hence, there is no need to explicitly name it. Haskell provides a mechanism to create such *anonymous* functions. For example, \x -> x + 1 is an expression that corresponds to a function that takes an argument `x` and returns as output the value `x + 1`. The function has no name, but we can use it in the same place where we would write a function. -} anonTests :: Test anonTests = TestList [ (\x -> x + 1) 100 ~?= (101 :: Int), doTwice (\x -> x + 1) 100 ~?= (102 :: Int) ] {- We call this expression form a "lambda expression", inspired by the [lambda calculus](https://en.wikipedia.org/wiki/Lambda_calculus). The backslash at the beginning of the expression is meant to look a little like the greek letter λ (lambda) and the `->` between the parameter `x` and the body of the function is meant to remind you that this expression form creates a value with a function type. Of course, we could name the function if we wanted to -} plus1' :: Int -> Int plus1' = \x -> x + 1 {- Indeed, in general, a function defining equation f x1 x2 ... xn = e is equivalent to f = \x1 -> \x2 -> ... \xn -> e Furthermore, we can also write nested lambda expressions together, with a single `\` and `->`. f = \x1 x2 ... xn -> e Infix Operations and Sections ----------------------------- In order to improve readability, Haskell allows you to use certain functions as *infix* operators: an infix operator is a function whose name is made of symbols. Wrapping it in parentheses makes it a regular identifier. My personal favorite infix operator is the application function, defined like this: ($) :: (a -> b) -> a -> b f $ x = f x Huh? Doesn't seem so compelling does it? It's just application. Actually, this operator is very handy because infix operators have different precedence than normal application. For example, I can write: minus20 $ plus 30 32 Which means the same as: minus20 (plus 30 32) That is, Haskell interprets everything after the `$` as one argument to `minus20`. I could *not* do this by writing minus20 plus 30 32 --- WRONG! because Haskell would think this was the application of `minus20` to the three separate arguments `plus`, `30` and `32`. It is often not a big deal whether one uses the `($)` operator or parentheses and it mostly comes down to a matter of taste. The operator can come in handy in certain situations, but it is always possible to write code without using it. We will see many infix operators in the course of the class; indeed we have already seen some defined in the standard prelude. For example, list `cons`truction (:) :: a -> [a] -> [a] as well as the arithmetic operators `(+)`, `(*)` and `(-)`. Recall also that Haskell allows you to use *any* function as an infix operator, simply by wrapping it inside backticks. -} anotherFive :: Int anotherFive = 2 `plus` 3 {- To further improve readability, Haskell allows you to use *partially applied* infix operators, i.e. infix operators with only a single argument. These are called *sections*. Thus, the section `(+1)` is simply a function that takes as input a number, the argument missing on the left of the `+` and returns that number plus `1`. -} anotherFour :: Int anotherFour = doTwice (+ 2) 0 {- Similarly, the section `(1:)` takes a list of numbers and returns a new list with `1` followed by the input list. So doTwice (1:) [2..5] evaluates to `[1,1,2,3,4,5]`. For practice, define the singleton operation as a section, so that the following test passes. -} singleton :: a -> [a] singleton = undefined singletonTest :: Test singletonTest = singleton True ~?= [True] {- One exception to sections is subtraction. `(-1)` is the integer "minus one", not a section subtracting `1` from its argument. Instead of a section, you can write `\x -> x - 1` or `subtract 1` (`subtract` is a function from the standard library). Polymorphism ============ We used `doTwice` to repeat an arithmetic operation, but the actual body of the function is oblivious to how `f` behaves. We say that `doTwice` is *polymorphic*: it works with different types of values, e.g. functions that increment integers and concatenate strings. This is vital for *abstraction*. The general notion of repeating, i.e. *doing twice* is entirely independent from the types of the operation that is being repeated, and so we shouldn't have to write separate repeaters for integers and strings. Polymorphism allows us to *reuse* the same abstraction `doTwice` in different settings. Of course, with great power, comes great responsibility. The section `(10 <)` takes an integer and returns `True` iff the integer is greater than `10`: -} greaterThan10 :: Int -> Bool greaterThan10 = (10 <) {- However, because the input and output types are different, it doesn't make sense to try `doTwice greaterThan10`. A quick glance at the type of doTwice would tell us this: doTwice :: (a -> a) -> a -> a The `a` above is a *type variable*. The signature above states that the first argument to `doTwice` must be a function that maps values of type `a` to `a`, i.e. it must produce an output that has the same type as its input (so that that output can be fed into the function again!). The second argument must also be an `a` at which point we may are guaranteed that the result from `doTwice` will also be an `a`. The above holds for *any* `a` which allows us to safely re-use `doTwice` in different settings. Of course, if the input and output type of the input function are different, as in `greaterThan10`, then the function is incompatible with `doTwice`. Ok, to make sure you're following, can you figure out what this does? -} ex1 :: (a -> a) -> a -> a ex1 x y = doTwice doTwice x y {- -} ex1Test :: Test ex1Test = undefined {- Polymorphic Data Structures --------------------------- Polymorphic functions that can *operate* on different kinds of values are often associated with polymorphic data structures that can *contain* different kinds of values. The types of such functions and data structures are written with one or more type variables. For example, the list length function: -} len :: [a] -> Int len [] = 0 len (_ : xs) = 1 + len xs {- The function's type states that we can invoke `len` on any kind of list. The type variable `a` is a placeholder that is replaced with the actual type of the list elements at different application sites. Thus, in the following applications of `len`, `a` is replaced with `Double`, `Char` and `[Int]` respectively. len [1.1, 2.2, 3.3, 4.4] :: Int len "mmm donuts!" :: Int len [[], [1], [1,2], [1,2,3]] :: Int Most of the standard list manipulating functions, for example those in the module [`Data.List`][1], have generic types. With a little practice, you'll find that the type signature contains a surprising amount of information about how the function behaves. In particular, note that we cannot "fake" values of generic types. For example, try to replace the `undefined` below with a result that doesn't throw an exception (like `undefined` does) or go into an infinite loop. (N.B.: Using a function that starts with `unsafe` doesn't count.) -} impossible :: a impossible = undefined {- Because `impossible` has to have *any* type, there is no real value that we can provide for it. This type says that 'impossible' has *whatever* type you want it to have---i.e. the type system will allow you do do anything with 'impossible', such as add it to another number -} ok1 :: Int ok1 = impossible + 1 {- concatenate it to a `String` -} ok2 :: String ok2 = "Hello" ++ impossible {- or test it like a boolean -} ok3 :: String ok3 = if impossible then "a" else "b" {- This reasoning extends to other types too. For example, the generic type of the const function const :: a -> b -> a tells us that the output of this function (if there is any) must be the first argument. There is no other way to produce a generic result of type 'a'. (And the second argument must be completely ignored, there is no way to use it in a generic way.) "Bottling" Computation Patterns With Polymorphic Higher-Order Functions ======================================================================= The combination of polymorphism and higher-order functions is the secret sauce that makes FP so tasty. It allows us to take *patterns of computation* that reappear in different guises in different places, and crisply specify them as reusable strategies. Let's look at some concrete examples... Computation Pattern: Iteration ------------------------------ Let's write a function that converts a string to uppercase. Recall that, in Haskell, a `String` is nothing but a list of `Char`s. So we must start with a function that will convert an individual `Char` to its uppercase version. Once we find this function, we will simply *walk over the list*, and apply the function to each `Char`. How might we find such a transformer? Lets query [Hoogle][2] for a function of the appropriate type! Ah, we see that the module [`Data.Char`][3] contains such a function: toUpper :: Char -> Char Using this, we can write a simple recursive function that does what we need: -} toUpperString :: String -> String toUpperString [] = [] toUpperString (x : xs) = toUpper x : toUpperString xs {- This pattern of recursion appears all over the place. For example, suppose we represent a location on the plane using a pair of `Double`s (for the x- and y- coordinates) and we have a list of points that represent a polygon. -} type XY = (Double, Double) type Polygon = [XY] {- It's easy to write a function that *shifts* a point by a specific amount: -} shiftXY :: XY -> XY -> XY shiftXY (dx, dy) (x, y) = (x + dx, y + dy) {- How would we translate a polygon? Just walk over all the points in the polygon and translate them individually. -} shiftPoly :: XY -> Polygon -> Polygon shiftPoly _ [] = [] shiftPoly d (xy : xys) = shiftXY d xy : shiftPoly d xys {- Now, some people (using some languages) might be quite happy with the above code. But what separates a good programmer from a great one is the ability to *abstract*. The functions `toUpperString` and `shiftPoly` share the same computational structure: they walk over a list and apply a function to each element. We can abstract this common pattern out as a higher-order function, `map`. Since the two functions we're abstracting differ only in what they do to each list element, so we'll just take that as an input! -} map :: (a -> b) -> [a] -> [b] map _ [] = [] map f (x : xs) = f x : map f xs {- The type of `map` tells us exactly what it does: it takes an `a -> b` transformer and list of `a` values, and transforms each `a` value to return a list of `b` values. We can now safely reuse the pattern, by *instantiating* the transformer with different specific operations. -} toUpperString' :: String -> String toUpperString' xs = map toUpper xs shiftPoly' :: XY -> Polygon -> Polygon shiftPoly' d = \x -> map (shiftXY d) x {- Much better. But let's make sure our refactoring didn't break anything! -} testMap :: Test testMap = TestList [ toUpperString' "abc" ~?= toUpperString "abc", shiftPoly' (0.5, 0.5) [(1, 1), (2, 2), (3, 3)] ~?= shiftPoly (0.5, 0.5) [(1, 1), (2, 2), (3, 3)] ] {- By the way, what happened to the list parameters of `toUpperString` and `shiftPoly`? Two words: *partial application*. In general, in Haskell, a function definition equation f x = e x is identical to f = e as long as `x` isn't used in `e`. Thus, to save ourselves the trouble of typing, and the blight of seeing the vestigial `x`, we often prefer to just leave it out altogether. (As an exercise, you may like to prove to yourself using just equational reasoning, using the equality laws we have seen, that the above versions of `toUpperString` and `shiftPoly` are equivalent.) We've already seen a few other examples of the map pattern. Recall the `listIncr` function, which added 1 to each element of a list: -} listIncr :: [Int] -> [Int] listIncr [] = [] listIncr (x : xs) = (x + 1) : listIncr xs {- We can write this more cleanly with map, of course: -} listIncr' :: [Int] -> [Int] listIncr' = undefined {- Computation Pattern: Folding ---------------------------- Once you've put on the FP goggles, you start seeing a handful of computation patterns popping up everywhere. Here's another... Lets write a function that *adds* all the elements of a list. -} sum :: [Int] -> Int sum [] = 0 sum (x : xs) = x + sum xs {- Next, a function that *multiplies* the elements of a list. -} product :: [Int] -> Int product [] = 1 product (x : xs) = x * product xs {- Can you see the pattern? Again, the only bits that are different are the `base` case value, and the function being used to combine the list element with the recursive result at each step. We'll just turn those into parameters, and lo! -} foldr :: (a -> b -> b) -> b -> [a] -> b foldr _f base [] = base foldr f base (x : xs) = x `f` foldr f base xs {- Now, each of the individual functions are just specific instances of the general `foldr` pattern. -} sum', product' :: [Int] -> Int sum' = foldr (+) 0 product' = foldr (*) 1 testFoldr :: Test testFoldr = TestList [ sum' [1, 2, 3] ~?= sum [1, 2, 3], product' [1, 2, 3] ~?= product [1, 2, 3] ] {- To develop some intuition about `foldr` let's unfold an example a few times by hand. In Haskell, we can substitute equals-for-equals anywhere so we can unfold definitions eagerly if we want. ~~~~~ foldr f base [x1,x2,...,xn] == f x1 (foldr f base [x2,...,xn]) {- unfold foldr -} == f x1 (f x2 (foldr f base [...,xn])) {- unfold foldr -} == x1 `f` (x2 `f` ... (xn `f` base)) ~~~~~ Aha! It has a rather pleasing structure that mirrors that of lists; the `:` is replaced by the `f` and the `[]` is replaced by `base`. So can you see how to use it to eliminate recursion from the recursion from our list-length function? len :: [a] -> Int len [] = 0 len (x:xs) = 1 + len xs -} len' :: [a] -> Int len' = undefined {- Once you have defined `len` in this way, see if you can trace how it works on a small example: ~~~~~~~~~ len' (1:2:[]) == foldr (\x y -> 1 + y) 0 (1:2:[]) == (\x y -> 1 + y) 1 (foldr (\x y -> 1 + y) 0 (2:[])) ... == 2 ~~~~~~~~~ Or, how would you use foldr to eliminate the recursion from this? -} factorial :: Int -> Int factorial 0 = 1 factorial n = n * factorial (n -1) factorial' :: Int -> Int factorial' n = undefined {- OK, one more. The standard list library function `filter` has this type: -} filter :: (a -> Bool) -> [a] -> [a] {- The idea is that it the output list should contain only the elements of the first list for which the input function returns `True`. So: -} testFilter :: Test testFilter = TestList [ filter (> 10) [1 .. 20] ~?= ([11 .. 20] :: [Int]), filter (\l -> sum l <= 42) [[10, 20], [50, 50], [1 .. 5]] ~?= [[10, 20], [1 .. 5]] ] {- Can we implement filter using foldr? Sure! -} filter pred = undefined runTests :: IO Counts runTests = runTestTT $ TestList [testMap, testFoldr, testFilter] {- Which is more readable? HOFs or Recursion ----------------------------------------- As a beginner, you might find the explicitly recursive versions of some of these functions easier to follow than the `map` and `foldr` versions. However, as you write more Haskell, you will probably start to find the latter are far easier, because `map` and `foldr` encapsulate such common patterns that you'll become completely accustomed to thinking in terms of them and other similar abstractions. In contrast, explicitly writing out the recursive pattern matching should start to feel needlessly low-level. Every time you see a recursive function, you have to understand how the knots are tied -- and worse, there is potential for making silly off-by-one type errors if you re-jigger the basic strategy every time. As an added bonus, it can be quite useful and profitable to *parallelize* and *distribute* the computation patterns (like `map` and `foldr`) in just one place, thereby allowing arbitrary hundreds or thousands of instances to benefit in a single shot! Haskell doesn't do this out of the box, but these ideas readily translate to languages designed for [parallel computation][4]. We'll see some other similar patterns later on. [1]: http://hackage.haskell.org/packages/archive/base/latest/doc/html/Data-List.html "Data.List" [2]: http://haskell.org/hoogle "Hoogle Query: Char -> Char" [3]: http://hackage.haskell.org/packages/archive/base/latest/doc/html/Data-Char.html "Data.Char" [4]: http://en.wikipedia.org/wiki/MapReduce "MapReduce" -}