» CS 330 Lecture 23 – Ways of Making Functions: Composition

CS 330 Lecture 23 – Ways of Making Functions: Composition

March 31, 2017 by Chris Johnson. Filed under cs330, lectures, spring 2017.

Dear students,

There are at least four ways of defining functions in Haskell:

by defining a named function
by partially applying parameters to get a new function expecting only the remaining parameters
by composing a gauntlet of functions together
by defining unnamed lambdas

We’ve seen the first two. Now let’s turn to composition. You’ve almost certainly worked with composition in your math classes. What is meant by (f . g)(x)? It means g is evaluated at x, and its result is used to further evaluate f.

Composing means to chain operations together, much like pipelining at the shell. The difference between composition and pipelining or Haskell’s $ operator is a matter of precedence. Haskell’s compose operator lets us chain functions together before we evaluate any of them. We can produce new functions by composing old ones together:

h = f . g
h 5      -- these two calls will
f (g 5)  --   produce the same result

That means we can eliminate a lot of clutter from our code! Those gauntlets of function calls that our data must pass through can be shrunk down to a single meaningful name.

Time for some examples:

Let’s write rsort which sorts a list in reverse:

rsort = Ord a => [a] -> [a]
rsort = reverse . sort

Let’s write scalebias to map a value using a linear function:

scalebias :: Double -> Double -> Double -> Double
scalebias scale bias x = scale * x + bias     -- without composition
scalebias' scale bias = (+ bias) . (* scale)  -- with composition

Now we write calc2fahren like we should have back in high school:

c2f :: Double -> Double
c2f = scalebias' 1.8 32

Let’s write a function to compute the number of negatives in a list:
```
nnegatives :: [Integer] -> Int
nnegatives = length . filter (< 0)
```
Let’s write a function that gives us the second element of a list, the neck:
```
neck :: [a] -> a
neck = head . tail
```

Let’s write a function that gives us a list of the unique items within another list:

onlies :: Ord a => [a] -> [a]
onlies = concat . filter single . group . sort
  where single list = length list == 1

Let’s write a function to create a spiral. As a first stop, let’s write some utility type synonyms and the (big three) functions to transform a 2D point:
```
type Point = (Double, Double)

scale :: Double -> Point -> Point
scale factor (x, y) = (factor * x, factor * y)

translate :: (Double, Double) -> Point -> Point
translate (dx, dy) (x, y) = (x + dx, y + dy)

rotate :: Double -> Point -> Point
rotate degrees (x, y) = (x', y')
  where radians = degrees * pi / 180
        x' = x * cos radians - y * sin radians
        y' = x * sin radians + y * cos radians
```
We can make a spiral by repeatedly scaling and rotating a point. Each step will be this compound operation:
```
spiral degrees growthRate = rotate degrees . scale growthRate
```
Okay, now it’s time to write a standalone Haskell program that does its own IO instead of operating in the REPL. Sadly, IO is one of those things that doesn’t fit well in a pure functional language. IO is stateful. It embeds a notion of time: first this line, then this line, and then that one… Pure function languages forbid the notion of time and state. They are about mathematical truth, which is timeless. We must leave the world of getters (expressions) and enter the world of doers (statements). Let’s start with a simple function that prints a Point:
```
print2 (x, y) = printf "%.5f,%.5f\n" x y
```
Now let’s write our first main. It is different:
```
main = do
  let p = (1, 0)
  print2 p
  let p' = spiral 1.05 20 p
  print2 p'
  return ()
```
We’ll talk about Haskell’s IO more another day. Right now let’s just remember these things: we must return () at the end, time sequences must be embedded in a do block, and let bind values to identifiers with let. What if we want to execute multiple steps? We can write a recursive method. main will kick it off with the initial point and number of iterations:
```
main = step (1, 0) 100
```
Since there’s only one line, we don’t need do. The step function might look this this:
```
step p n = do
  let p' = spiral 1.05 20 p
  print2 p'
  if n > 0 then
    step p' (n - 1)
  else
    return ()
```
Now we can execute this script from the command-line:
```
runhaskell foo.hs
```
To make this a little more flexible, let’s accept some command-line parameters:
```
step :: (Double, Double) -> Double -> Double -> Int -> IO ()
step p rate degrees n = do
  let p' = spiral rate degrees p
  print2 p'
  if n > 0 then
    step p' rate degrees (n - 1)
  else
    return ()

main = do
  args <- getArgs
  let x = read (head args) :: Double
  let y = read (neck args) :: Double
  let rate = read (args !! 2) :: Double
  let degrees = read (args !! 3) :: Double
  let n = read (args !! 4) :: Int
  step (x, y) rate degrees n
```

Let’s write a function to transform a rectangle around an arbitrary pivot point. Let’s represent with rectangle with four numbers: the x- and y-coordinates of the bottom-left corner, a width, and a height. Let’s create some helper functions to determine the coordinates of the points:

type Rectangle = (Point, Double, Double)

bl (p, _, _) = p
br ((left, bottom), width, _) = (left + width, bottom)
tl ((left, bottom), _, height) = (left, bottom + height)
tr ((left, bottom), width, height) = (left + width, bottom + height)

To rotate around an arbitrary pivot, we pass a point through this gauntlet:

rotateAround :: Point -> Double -> Point -> Point
rotateAround pivot@(dx, dy) degrees = translate pivot . rotate degrees . translate (-dx, -dy)

Finally, we can write a main to rotate a rectangle by the number of degrees provided at the command-line:

main = do
  args <- getArgs
  let degrees = read (head args) :: Double
  let r = (0, 0, 1, 1)
  print2 $ rotateAround (1, 1) degrees (bl r)
  print2 $ rotateAround (1, 1) degrees (br r)
  print2 $ rotateAround (1, 1) degrees (tr r)
  print2 $ rotateAround (1, 1) degrees (tl r)
  print2 $ rotateAround (1, 1) degrees (bl r)
  return ()

Here’s your TODO list:

Read Execution in the Kingdom of Nouns. On a quarter sheet, write down 2-3 questions, observations, or thoughtful agreements or disagreements.

See you next time, when we discuss lambdas and higher-order functions!

Sincerely,

P.S. It’s Haiku Friday!

scramble = fry . whisk . crack
hardboil = trash . peel . boil
vandalize = freeze . throw