`> import Data.Char`

In all functional languages, functions are *first-class* values, meaning, that they can be treated just as you would any other data. That is, you can pass functions around to in *any* manner that you can pass any other data around in. For example, suppose you have a 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 two instances of the function

`> funp = (plus1, minus1)`

Lets see what the type of the pair is

`ghc> :type funp `

funp :: (Int -> Int, Int -> Int)

or you can make a list containing five copies

`> funs = [plus1, minus1, plus1]`

Again, note that the type is most descriptive

`ghc> :type funs `

funs :: [Int -> Int]

This innocent looking feature makes a langage surprisingly brawny and flexible, because now, we can write *higher-order* functions that other functions as input and return functions as output! Consider

`> doTwice f x = f (f x)`

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, thereby twice applying `f`

to `x`

in a manner of speaking. Note how the raw code is clearer to understand than my long-winded English description!

Lets run it,

`ghc> doTwice plus1 10`

12

ghc> doTwice minus1 100

98

Did I mention that the execution model is just *substitute equals for equals*

`doTwice plus1 10 == {- unfold doTwice -} `

plus1 (plus1 10)

== {- unfold plus1 -}

plus1 (10 + 1)

== {- unfold plus1 -}

(10 + 1) + 1

== {- old-school arithmetic -}

12

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 just write a single function `plusn`

as

`> plusn :: Int -> (Int -> Int)`

> plusn n = f

> where f x = x + n

That is, `plusn`

returns as output a function `f`

which itself takes as input an integer `x`

and adds `n`

to it. Lets use it

`> plus10 = plusn 10`

> minus20 = plusn (-20)

Note the types of the above

`plus10 :: Int -> Int`

That is, `plus10`

will take an input integer and return the incremented-by–10 integer as output.

`ghc> plus10 100`

110

ghc> minus20 1000

980

You can also directly use `plusn`

`ghc> (plusn 25) 100`

125

What happened? You do the math

`(plusn 25) 100 == {- unfold plusn -}`

f 100

where f x = x + 25

== {- unfold f -}

100 + 25

== {- arithmetic -}

125

Super easy!

In regular arithmetic, the `-`

operator is *left-associative*. Hence,

` 2 - 1 - 1 == (2 - 1) - 1 == 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 of function, which takes two integers, is in reality a function that takes a single integer as input, and returns as *output* a function from integers to integers! Equipped with this knowledge, consider the function

`> plus :: Int -> Int -> Int`

> plus n x = n + x

Thus, whenever we use `plus`

we can either pass in both the inputs at once

`ghci> plus 10 20`

30

or instead, we can *partially* apply the function, by just passing in only one input

`> 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.)

`ghci> plus5 1000`

1005

Again, how ? (Cough) *Substitute equals for equals*

`plusfive 1000 == {- definition of plusfive -}`

plus 5 1000

== {- unfold plus -}

5 + 1000

== 1005

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 = doTwice (plus 20)`

First, see if you can figure out the type.

`doTwicePlus20 :: Int -> Int`

Next, see if you can figure out how this evaluates.

`ghci> doTwicePlus20 0`

40

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 is not bound to any name.

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 + 2`

. The function has no name, but we can use it in the same place where we would write a function.

`ghci> (\x -> x + 1) 100`

101

ghci> doTwice (\x -> x + 1) 100

102

Of course, we could name the function if we wanted to

`> plus1' = \x -> x + 1`

Indeed, in general, a function defining equation

`f x1 x2 ... xn = e `

is equivalent to

`f = \x1 x2 ... xn -> e `

In order to improve readability, Haskell allows you to use certain functions as *infix* operations: a function whose name appears in parentheses can be used as an infix operation. My personal favorite infix operator is the *pipeline* function defined thus

`> (|>) x f = f x`

Huh? Doesn’t seem so compelling does it? Actually, its very handy because I can now write deeply nested function applications in an easy to read unix-style pipes manner. For example

`ghci> 0 |> plus 1 `

3

ghci> 0 |> plus 1 |> plus5

6

We can get a triple-repeating version of `doTwice`

as

`> doThrice f x = x |> f |> f |> f`

It is easy to check that

`doThrice f x == f (f (f x))`

We will see many other such operators in the course of the class; indeed many standard operators including that we have used already are defined in this manner in the standard library.

`ghci> :type (+)`

(+) :: (Num a) => a -> a -> a

ghci> :type (++)

(++) :: [a] -> [a] -> [a]

ghci> :type (:)

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

Furthermore, Haskell allows you to use *any* function as an infix operator, simply by wrapping it inside backticks.

`ghci> 2 `plus` 3`

5

Recall the function from the last lecture

`> clone x n | n <= 0 = []`

> | otherwise = x : clone x (n-1)

We invoke it in an infix-style like so

`ghci> 30 `clone` 3`

[30,30,30]

To further improve readability, Haskell allows you to use *partially applied* infix operators, ie 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`

.

`ghci> doThrice (+1) 0`

3

Similarly, the section `(1:)`

takes a list of numbers and returns a new list with `1`

followed by the input list. Consequently,

`ghci> doTwice (1:) [2..5]`

[1,1,3,4,5]

We used to `doTwice`

to repeat an arithmetic operation, but the actual body of the function is oblivious to how `f`

behaves. For example,

`ghci> doTwice (+1) 0`

2

ghci> doTwice (++ "I don't care! ") "Isn't this neat ?"

"Isn't this neat ? I don't care! I don't care! "

Thus, `doTwice`

is *polymorphic* in that it works with different kinds of values, eg functions that increment integers and concatenate strings. This is vital for *abstraction*. The general notion of repeating, ie *doing twice* is entirely independent of the *specific* 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`

`ghci> doTwice greaterThan10 100`

*Main> doTwice

<interactive>:1:0:

No instance for (Show ((t -> t) -> t -> t))

arising from a use of `print' at <interactive>:1:0-6

Possible fix:

add an instance declaration for (Show ((t -> t) -> t -> t))

In a stmt of a 'do' expression: print it

Urgh!!! However, a quick glance at the type of doTwice would have spared us this grief.

`ghci> :type doTwice`

doTwice :: (t -> t) -> t -> t

The `t`

above is a *type variable*. The signature above states that the first argument to `doTwice`

must be a function that maps values of type `t`

to `t`

, ie 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 a `t`

at which point we may are guaranteed that the result from `doTwice`

will also be a `t`

. The above holds *for any* `t`

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 = doTwice doTwice`

Polymorphic functions which can *operate* on different kinds values are often associated with polymorphic data structures which can *contain* different kinds of values. These are also represented by types containing type variables.

For example, the list length function

`> len :: [a] -> Int`

> len [] = 0

> len (x:xs) = 1 + len xs

doesn’t peep inside the actual contents of the list; it only counts how many there are. This property is crisply specified in the function’s signature, which 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 at different application sites. Thus, in the below instances, `a`

is replaced with `Double`

, `Char`

and `[Int]`

respectively.

`ghci> len [1.1, 2.2, 3.3, 4.4]`

4

ghci> len "mmm donuts!"

11

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

4

Most standard list manipulating functions, for example those in the standard library Data.List have generic types. You’ll find that the type signature contains a surprising amount of information about how the function behaves.

`(++) :: [a] -> [a] -> [a]`

head :: [a] -> a

tail :: [a] -> [a]

The tag-team of polymorphism and higher-order functions is the secret sauce that makes FP so tasty. It allows us to take arbitrary, *patterns of computation* that reappear in different guises in different places, and crisply specify them as safely reusable strategies. That sounds very woolly (I hope), lets look at some concrete examples.

Lets write a function that converts a string to uppercase. Recall that in Haskell, a `String`

is just a list of `Char`

. We must start with a function that will convert an individual `Char`

to its uppercase version. Once we find this function, we will simply *jog over the list*, and apply the function to each `Char`

.

How might we find such a transformer? Lets query Hoogle for a function of the appropriate type! Ah, we see that the module `Data.Char`

contains a function.

`toUpper :: Char -> Char`

and so now, we can write the simple recursive function

`toUpperString [] = []`

toUpperString (c:cs) = toUpper c : toUpperString cs

As you might imagine, this sort of recursion appears all over the place. For example, suppose I represent a using a pair of `Double`

(for the x- and y- coordinates.) and I have a list of points that represent a polygon.

`> type XY = (Double, Double)`

> type Polygon = [XY]

Now, its 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 jog over all the points in the polygon and translate them individually

`shiftPoly :: XY -> Polygon -> Polygon`

shiftPoly d [] = []

shiftPoly d (xy:xys) = shiftXY d xy : shiftPoly d xys

Now, in a lesser language, you might be quite happy with the above code. But what separates a good programmer from a great one, is the ability to *abstract*.

Like humans and monkeys, the functions `toUpperString`

and `shiftPoly`

share 93% of their DNA — the notion of *jogging* over the list. The common pattern is described by the polymorphic higher-order function `map`

`map f [] = []`

map f (x:xs) = (f x) : (map f xs)

How did we arrive at this? Well, you find what is enshrine in the function’s body that which is *common* to the different instances, namely the recursive jogging strategy; and the bits that are different, simply become the function’s parameters! Thus, the `map`

function abstracts, or if you have a vivid imagination, locks up in a bottle, the extremely common pattern of jogging over the list.

Verily, the type of `map`

tells us exactly what it does

`ghci> :type map`

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

That is, it takes an `a -> b`

transformer and list of `a`

values, and transforms each value to return a list of `b`

values. We can now safely reuse the pattern, by *instantiating* the transformer with different specific operations.

`> toUpperString = map toUpper`

> shiftPoly = map shiftXY

Much better.

By the way, what happened to the 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`

doesn’t appear 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, to prove to yourself using just equational reasoning (using the different equality laws we have seen) that the above versions of `toUpperString`

and `shiftPoly`

are equivalent.

Once you’ve put on the FP goggles, you start seeing computation patterns everywhere.

Lets write a function that *adds* all the elements of a list.

`listAdd [] = 0`

listAdd (x:xs) = x + (listAdd xs)

Next, a function that *multiplies* the elements of a list.

`listMul [] = 1`

listMul (x:xs) = x * (listMul xs)

Can you see the pattern? Again, the only bits that are different are the `base`

case value, and the `op`

being performed at each step. We’ll just turn those into parameters, and lo!

`foldr op base [] = base`

foldr op base (x:xs) = x `op` (foldr op base xs)

Now, each of the individual functions are just specific instances of the general `foldr`

pattern.

`> listAdd = foldr (+) 0`

> listMul = foldr (*) 1

To develop some intuition about `foldr`

lets “run” it a few times by hand.

`foldr op base [x1,x2,...,xn] `

== {- unfold -}

x1 `op` (foldr op base [x2,...,xn])

== {- unfold -}

x1 `op` (x2 `op` (foldr op base [...,xn]))

== {- unfold -}

x1 `op` (x2 `op` (... `op` (xn `op` base)))

Aha! It has a rather pleasing structure that mirrors that of lists; the `:`

is replaced by the `op`

and the `[]`

is replaced by `base`

. Thus, can you see how to use it to eliminate recursion from the recursion from

`listLen [] = 0`

listLen (x:xs) = 1 + (listLen xs)

`> listLen = foldr (\_ tailLen -> 1 + tailLen) 0`

How would you use it to eliminate the recursion from

`factorial 0 = 1`

factorial n = n * factorial (n-1)

`> factorial n = foldr (*) 1 [1..n]`

One last pattern exercise. How about this fellow from last lecture:

`fuseActions :: [IO ()] -> IO () `

fuseActions [] = return ()

fuseActions (a1:acts) = do a1

fuseActions acts

Can you spot the pattern?

`> fuseActions :: [IO ()] -> IO () `

> fuseActions = foldr (>>) (return ())

As a beginner, you might think that the recursive versions of `toUpperString`

and `shiftPoly`

are easier to follow than the `map`

versions. Certainly, `fold`

takes a bit of getting used to.

However, as you get used to the light, you will find the latter is infact far easier to follow, because once abstraction gets into your bones you’ll know exactly what an instance does at a single glance.

In contrast, recursion is lower-level, so 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 `fold`

) in just one place, thereby allowing arbitrary hundreds or thousands of instances to benefit in a single shot!