`
You simply state:
foo [] =
foo l =
But pattern matching go even further.
It is also able to inspect inside datas.
We can replace
foo l = let x = head l
xs = tail l
in if even x
then foo (n+x) xs
else foo n xs
by
foo (x:xs) = if even x
then foo (n+x) xs
else foo n xs
This is a very useful feature.
It makes our code both terse and easier to read.
02_Hard_Part/14_Functions.lhs
We also can currify a bit our definition by removing the `l`:
-- Version 4
evenSum :: Integral a => [a] -> a
evenSum = accumSum 0
where
accumSum n [] = n
accumSum n (x:xs) =
if even x
then accumSum (n+x) xs
else accumSum n xs
02_Hard_Part/15_Functions.lhs
Higher Level Functions
To make things even better we should use higher level functions.
What are these beast?
Higher level functions are functions taking another functions as parameters.
Here are some examples:
import Data.List
filter :: (a -> Bool) -> [a] -> [a]
map :: (a -> b) -> [a] -> [b]
foldl' :: (a -> b -> a) -> a -> [b] -> a
Let's proceed by small steps.
-- Version 5
evenSum l = mysum 0 (filter even l)
where
mysum n [] = n
mysum n (x:xs) = mysum xs (n+x)
filter even [1..10] ⇔ [2,4,6,8,10]
Now you can use the `foldl'` to accumulate a value.
The function `foldl` capture a general coding pattern:
myfunc list = foo initialValue list
foo accumulated [] = accumulated
foo tmpValue (x:xs) = foo (bar tmpValue x) xs
Which can be replaced by:
myfunc list = foldl bar initialValue list
-- Version 6
import Data.List
evenSum l = foldl' mysum 0 (filter even l)
where mysum acc value = acc + value
For each element of the list, `foldl'` will add it to the next.
And finally add 0.
If you really want to know how the magic works.
Here is the definition of `foldl`.
foldl f z [] = z
foldl f z (x:xs) = foldl f (f z x) xs
But as Haskell is lazy, it doesn't evaluate `(f z x)` and push this in the stack.
`foldl'` is a strict version of `foldl`.
If you don't understand what "lazy" and "strict" means,
don't worry, just follow the code as if `fold` and `foldl'` where identical.
Here is what occurs:
~~~
evenSum [1,2,3,4]
⇒ foldl' mysum 0 (filter even [1,2,3,4])
⇒ foldl' mysum 0 [2,4]
⇒ foldl' mysum (mysum 0 2) [4]
⇒ foldl' mysum (0+2) [4]
⇒ foldl' mysum 2 [4]
⇒ foldl' mysum (mysum 2 4) []
⇒ foldl' mysum (2+4) []
⇒ foldl' mysum 6 []
⇒ 6
~~~
Beware!
Most of the time you want to use `foldl'` and not `foldl`.
This is nice, but as `mysum` is a very simple function, giving it a name is a burden.
We can use anonymous functions or lambdas.
-- Version 7
evenSum l = foldl' (\x y -> x+y) (filter even l)
And of course, we remark
(\x y -> x+y) ⇔ (+)
02_Hard_Part/16_Functions.lhs
Finaly
-- Version 8
import Data.List
evenSum :: Integral a => [a] -> a
evenSum l = foldl' (+) 0 (filter even l)
`foldl'` isn't the easiest function to intuit.
If you are not used to it, you should exercise a bit.
I would like to introduce another higher order function: `(.)`.
The `(.)` function correspond to the mathematical composition.
(f . g . h) x ⇔ f ( g (h x))
We can take advantage of this operator to curry a bit more our function:
-- Version 9
import Data.List
evenSum :: Integral a => [a] -> a
evenSum = (foldl' (+) 0) . (filter even)
Also, there already exists a `sum` function.
-- Version 10
import Data.List
evenSum :: Integral a => [a] -> a
evenSum = sum . (filter even)
!!!!!
What power did we gain by using `foldl'`?
You have no more different case to test, it feels more like a mathematical function.
And it become far easier to compose the function with other ones.
!!!!!
Suppose we want to modify slightly our function.
We want to get the sum of all even square of element of the list.
~~~
[1,2,3,4] ~> [1,4,9,16] ~> [4,16] ~> 20
~~~
Update the version 10 is extremely easy:
squareEvenSum = sum . (filter even) . (map (^2))
We simply had to add another "transformation function".
~~~
map (^2) [1,2,3,4] ⇔ [1,4,9,16]
~~~
!!!!!
The main advantage is you didn't have to modify _inside_ the function definition, but you just had to use another function.
You encapsulate the function and you could use a "pipe-like" notation and way of thinking.
With the ability of not having to open the pipe to modify the behaviour of your program will fastly become a huge help to think about it.
!!!!!
To modify version 1 is left as an exercise to the reader.
If you believe we reached the end of generalization, then know you are very wrong. For example, there is a way to not only use this function on list but on any recursive type. If you want to know how, I suggest you to read this quite fun article: [Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire by Meijer, Fokkinga and Paterson](http://eprints.eemcs.utwente.nl/7281/0
1/db-utwente-40501F46.pdf).
This example should show you how pure functional programming is
great. Unfortunately, using pure functional programming isn't well
suited for all usages. Or at least it isn't found yet.
One of the great power of Haskell, is the ability to create DSL
(Domain Specific Language)
making it easy to change the programming paradigm.
In fact, Haskell is also great when you want to write imperative style
programming. Understand this was really hard for me when learning Haskell.
Because a lot of effort is provided to explain you how much functional
approach is superior. Than when you attack the imperative style of Haskell, it
is hard to understand why and how.
But before talking about this Haskell super-power, we must talk about another
essential aspect of Haskell; the _types_.
02_Hard_Part/20_Types.lhs
## Types
In Haskell, types are strong and static.
Why is this important? It will help you _a lot_ not to make some mistake.
In fact, most bugs are catched during the compilation of your program.
And the main reason is because of the type inference during compilation.
It will be easy to detect where you used the bad parameter at the wrong place for example.
### Type inference
Static typing is generally essential to reach fast execution time.
But in common languages static typing has the price of bad generalization.
What saves Haskell is that types can be inferred.
Here are some examples on how to simulate a simple square function in Haskell
in other statically typed languages:
square x = x * x
This function can square any Numeral type.
You can provide square an Int, an Integer, a Float a Fractional and even Complex.
square 2
4
square 2.1
4.41
:m Data.Complex
(2 :+ 1) * (2 :+ 1)
3.0 :+ 4.0
`x :+ y` is the notation for the complex (x + ib).
Now compare to the code necessary in C:
int int_square(int x) { return x*x; }
float fl_square(float x) {return x*x; }
complex complex_square (complex z) {
complex tmp;
tmp.real = z.real * z.real - z.img * z.img;
tmp.img = 2 * z.img * z.real;
}
For each type, you need to write a new function.
The only way to work around this problem is to use some meta-programming trick.
For example using the pre-processor.
In C++ there is a better way, the C++ templates:
class Number {
T value;
square() {
value = value*value;
}
}
Number i;
i.square;
Number f;
f.square;
class Complex {
int real;
int img;
Complex operator<*>(Complex z) {
Complex result;
result.real = real*z.real - img*z.img;
result.img = img*z.real + real*z.img;
return res;
}
}
Number z;
z.square
Even with C++ templates you are forced to write a line for each type.
To be fair, there is also a definition of the multiplication of Complex in Haskell.
But it takes only one line.
Somewhere in the source of the module `Data.Complex`:
instance Num (Complex a) where
...
(x:+y) * (x':+y') = (x*x'-y*y') :+ (x*y'+y*x')
...
The inference of type gives Haskell a feeling of the freedom that dynamic
typed languages provide.
02_Hard_Part/21_Types.lhs
Type construction
You can construct you own types.
First you can use aliases or type synonyms.
type Name = String
type Color = String
showInfos :: Name -> Color -> String
showInfos name color = "Name: " ++ name
++ ", Color: " ++ color
name :: Name
name = "Robin"
color :: Color
color = "Blue"
main = putStrLn $ showInfos name color
02_Hard_Part/22_Types.lhs
But it doesn't protect you much.
Try to replace the last line and run the program:
putStrLn $ showInfos color name
In fact you can replace Name, Color and String everywhere.
The compiler will treat them as completely identical.
Another method is to create your own types using the keyword `data`.
data Name = NameConstr String
data Color = ColorConstr String
showInfos :: Name -> Color -> String
showInfos (NameConstr name) (ColorConstr color) =
"Name: " ++ name ++ ", Color: " ++ color
name = NameConstr "Robin"
color = ColorConstr "Blue"
main = putStrLn $ showInfos name color
Now if you switch parameters of `showInfos`, the compiler complains!
A possible mistake you could never do again.
The only price is to be more verbose.
Also remark constructor are functions:
NameConstr :: String -> Name
ColorConstr :: String -> Color
The syntax of `data` is generally:
data TypeName = ConstructorName [types]
| ConstructorName2 [types]
| ...
Generally the usage is to use the same name for the
DataTypeName and DataTypeConstructor.
Example:
data Complex = Num a => Complex a a
Also you can use the record syntax:
data DataTypeName = DataConstrctor {
field1 :: [type of field1]
, field2 :: [type of field2]
...
, fieldn :: [type of fieldn] }
And many accessor are made for you.
Furthermore you can use another order when setting values.
Example:
data Complex = Num a => Complex { real :: a, img :: a}
c = Complex 1.0 2.0
z = Complex { real = 3, img = 4 }
real c ⇒ 1.0
img z ⇒ 4
02_Hard_Part/23_Types.lhs
### Recursive type
You already encountered recursive types.
Typically, you can re-create lists, but with a more verbose syntax:
~~~
data List a = Empty | Cons a (List a)
~~~
If you really want to use an easier syntax you can use infix name for constructors.
~~~
infixr 5 :::
data List a = Nil | a ::: (List a)
~~~
The number after `infixr` is the priority.
If you want to be able to print (`Show`), read (`Read`), test equality (`Eq`) and compare (`Ord`) your new data structure you can tell Haskell to derive the appropriate function for you.
infixr 5 :::
data List a = Nil | a ::: (List a)
deriving (Show,Read,Eq,Ord)
When told to use deriving Show, Haskell create a `show` function for you.
We'll see soon how you could use your own `show` function.
convertList [] = Nil
convertList (x:xs) = x ::: convertList xs
main = do
print (0 ::: 1 ::: Nil)
print (convertList [0,1])
This print:
~~~
0 ::: (1 ::: Nil)
0 ::: (1 ::: Nil)
~~~
02_Hard_Part/30_Trees.lhs
### Trees
Now we'll just give another typical example, binary trees.
import Data.List
data BinTree a = Empty
| Node a (BinTree a) (BinTree a)
deriving (Show)
To generate tree easily, we create a function who add an element to a `BinTree`.
treeInsert :: (Ord a) => BinTree a -> a -> BinTree a
treeInsert Empty x = Node x Empty Empty
treeInsert (Node y left right) x
| x == y = (Node y left right)
| x < y = (Node y (treeInsert left x) right)
| otherwise = (Node y left (treeInsert right x))
Now try this:
main = print $ foldl' treeInsert Empty [7,2,4,8]
You should obtain the following:
~~~
Node 7 (Node 2 Empty (Node 4 Empty Empty)) (Node 8 Empty Empty)
~~~
This is an informative but quite unpleasant representation of our tree.
02_Hard_Part/31_Trees.lhs
Just for fun, let's code a better display for our trees.
I simply had fun into making a nice function to display tree in a general way.
You can safely pass this part if you find it too difficult to follow.
We have few to change to make.
First, as we will play a bit with string, we import the function `replace`
from `Data.String.Utils`.
import Data.List
We remove the `deriving (Show)` in the declaration of our `BinTree` type.
And it also might be useful to make our BinTree an instance of (Eq and Ord).
Now we can test equality and compare trees.
data BinTree a = Empty
| Node a (BinTree a) (BinTree a)
deriving (Eq,Ord)
Without the `deriving (Show)`, Haskell doesn't create a show method for us.
Now, we will create our version of show.
For this, we want our newly created type `BinTree a` to be an instance of
the type class `Show`.
To achieve this, the general syntax is:
instance Show (BinTree a) where
show t = ... -- You declare your function here
Here is my version on how to show a binary tree.
Don't worry about the apparent complexity.
I made a lot of improvement in order to display even strange objects.
-- declare BinTree a to be an instance of Show
instance (Show a) => Show (BinTree a) where
-- will start by a '<' before the root
-- and put a : a begining of line
show t = "< " ++ replace '\n' "\n: " (treeshow "" t)
where
treeshow pref Empty = ""
treeshow pref (Node x Empty Empty) =
(pshow pref x)
treeshow pref (Node x left Empty) =
(pshow pref x) ++ "\n" ++
(showSon pref "`--" " " left)
treeshow pref (Node x Empty right) =
(pshow pref x) ++ "\n" ++
(showSon pref "`--" " " right)
treeshow pref (Node x left right) =
(pshow pref x) ++ "\n" ++
(showSon pref "|--" "| " left) ++ "\n" ++
(showSon pref "`--" " " right)
-- show a tree using some prefixes to make it nice
showSon pref before next t =
pref ++ before ++ treeshow (pref ++ next) t
-- pshow replace "\n" by "\n"++pref
pshow pref x = replace '\n' ("\n"++pref) (show x)
-- replace on char by another string
replace c new string =
concatMap (change c new) string
where
change c new x
| x == c = new
| otherwise = x:[] -- "x"
The `treeInsert` method remain identical.
treeInsert :: (Ord a) => BinTree a -> a -> BinTree a
treeInsert Empty x = Node x Empty Empty
treeInsert (Node y left right) x
| x == y = (Node y left right)
| x < y = (Node y (treeInsert left x) right)
| otherwise = (Node y left (treeInsert right x))
To help creating tree, we define:
treeFromList list = foldl' treeInsert Empty list
And now, we can play:
main = do
putStrLn "Int binary tree:"
print $ treeFromList [7,2,4,8,1,3,6,21,12,23]
~~~
Int binary tree:
< 7
: |--2
: | |--1
: | `--4
: | |--3
: | `--6
: `--8
: `--21
: |--12
: `--23
~~~
Now it is far better!
The root is shown by starting by the `<` character.
And each other line start by a `:`.
But we could also use another type.
putStrLn "\nString binary tree:"
print $ treeFromList ["foo","bar","baz","gor","yog"]
~~~
String binary tree:
< "foo"
: |--"bar"
: | `--"baz"
: `--"gor"
: `--"yog"
~~~
As we can test equality and order trees, we can
make tree of trees!
putStrLn "\nBinary tree of Char binary trees:"
print ( treeFromList
(map treeFromList ["baz","zara","bar"]))
~~~
Binary tree of Char binary trees:
< < 'b'
: : |--'a'
: : `--'z'
: |--< 'b'
: | : |--'a'
: | : `--'r'
: `--< 'z'
: : `--'a'
: : `--'r'
~~~
This is why I chosen to prefix each line of tree display by `:` (except for the root).
<%= blogimage("yo_dawg_tree.jpg","Yo Dawg Tree") %>
putStrLn "\nTree of Binary trees of Char binary trees:"
print $ treeFromList
(map treeFromList
[ map treeFromList ["Ia!","Ia!"]
, map treeFromList ["cthul","hu"]
, map treeFromList ["Fhtagn!"] ])
~~~
Binary tree of Binary trees of Char binary trees:
< < < 'I'
: : : |--'!'
: : : `--'a'
: |--< < 'F'
: | : : |--'!'
: | : : `--'h'
: | : : |--'a'
: | : : | `--'g'
: | : : `--'t'
: | : : `--'n'
: `--< < 'c'
: : : `--'t'
: : : |--'h'
: : : | `--'l'
: : : `--'u'
: : `--< 'h'
: : : `--'u'
~~~
Remark how you can't insert two identical tree;
there is only one tree corresponding to "Ia!".
Note how awesome this structure is.
We can make tree containing not only integer, string and char, but also other trees.
And we can even make a tree containing a tree of trees!
03_monads/00_Introduction.lhs
# Hell Difficulty Part
<%= blogimage("hell_difficulty.jpg","Hell difficulty") %>
Congratulation to get so far!
Now, the real stuff will began.
If you are like me, you should get the functional style.
You should also understand a bit more the advantages of lazyness by default.
But you also don't really understand were to start to make a real program.
And in particular:
- How do you deal with effects?
- Why is there a strange imperative-like notation for dealing with IO?
- How could you deal with mutable data-structures?
- How could you easily parallelize?
I will try to answer each question.
Be prepared, answer might be difficult to get.
But they all be very rewarding.
Be prepared dear adventurer as you will reach the final and thoughest Haskell dungeon!
03_monads/01_IO/01_progressive_io_example.lhs
Deal With IO
Start by resolving a simple example wich deal with user interaction.
> Ask a user to enter a list of numbers.
> Return the sum of the numbers
toList :: String -> [Integer]
toList input = read ("[" ++ input ++ "]")
main = do
putStrLn "Enter a list of numbers (separated by comma):"
input <- getLine
print $ sum (toList input)
It should be straightforward.
03_monads/01_IO/02_progressive_io_example.lhs
But what occurs if the user enter something strange?
Let's try:
~~~
% runghc 02_progressive_io_example.lhs
Enter a list of numbers (separated by comma):
foo
Prelude.read: no parse
~~~
Argh, an evil error message and a crash!
Now we just want to put our own, more Human readable message.
For this, we must detect, something went wrong.
Here is one way to do this.
Use the type `Maybe`. It is a very common type in Haskell.
import Data.Maybe
maybeRead :: Read a => String -> Maybe a
maybeRead s = case reads s of
[(x,"")] -> Just x
_ -> Nothing
What is this thing? Maybe is a type which takes one parameter.
Its definition is:
data Maybe a = Nothing | Just a
This is a nice way to tell there was an error while trying to create/compute
a value.
The `maybeRead` function is a great example of this.
This is a function similar to the function `read`[^1],
but if something goes wrong the returned value is `Nothing`.
If the value is right, it returns `Just `.
[^1]: Which itself is very similar to the javascript `eval` on a string containing JSON).
getListFromString :: String -> Maybe [Integer]
getListFromString str = maybeRead $ "[" ++ str ++ "]"
Now to be a bit more readable, we define a function which goes like this:
If the string has the wrong format, it will return `Nothing`.
Otherwise, for example for "1,2,3", it will return `Just [1,2,3]`.
We simply test the value in our main function.
main = IO ()
main = do
putStrLn "Enter a list of numbers (separated by comma):"
input <- getLine
let maybeList = getListFromString input in
case maybeList of
Just l -> print (sum l)
Nothing -> error "Bad format. Good Bye."
In case of error, we prompt a nice error message.
One very important thing to note is the type of all the function used.
There is only one function which contains `IO` in its type: `main`.
That means, all other functions are pure.
Why purity matters? As a pure function has no side effect, you can for example
evaluate its value in parallel on many different core without any problem.
And you have this by design of Haskell.
03_monads/01_IO/03_progressive_io_example.lhs
Now how could we ask the user again and again util it enters the right answer?
We keep this the first part:
import Data.Maybe
maybeRead :: Read a => String -> Maybe a
maybeRead s = case reads s of
[(x,"")] -> Just x
_ -> Nothing
getListFromString :: String -> Maybe [Integer]
getListFromString str = maybeRead $ "[" ++ str ++ "]"
Now, we create a function which will ask the user for an integer list
until the input is right.
askUser :: IO [Integer]
askUser = do
putStrLn "Enter a list of numbers (separated by comma):"
input <- getLine
let maybeList = getListFromString input in
case maybeList of
Just l -> return l
Nothing -> askUser
This function is of type `IO [Integer]`.
Such a type means, that we retrieved a value of this type through some IO actions.
Sometimes you could see someone telling you while waving their hands:
"This is an `[Integer]` but inside an IO"
If you want to understand the details behind all of this, you have to continue.
Finally our main function is quite simpler:
main :: IO ()
main = do
list <- askUser
print $ sum list
03_monads/01_IO/04_progressive_io_example.lhs
## `IO` trick explained.
Why did we used some strange syntax, and what exactly is this `IO` type.
It looks a bit like magic.
For now let's just forget about all the pure part of our program, and focus
on the impure part:
askUser :: IO [Integer]
askUser = do
putStrLn "Enter a list of numbers (separated by comma):"
input <- getLine
let maybeList = getListFromString input in
case maybeList of
Just l -> return l
Nothing -> askUser
main :: IO ()
main = do
list <- askUser
print $ sum list
First noticiable thing:
the structure of these function is very similar to the one of an imperative language.
The fact is, Haskell is powerful enough to recreate function to help code look like in an imperative language.
For example, if you wish you could create a `while` in Haskell.
In fact, for dealing with `IO`, imperative style is generally more appropriate.
But, you also see there are some light differences.
The notation is a bit strange.
This is here that reside the beauty of how Haskell handle IOs.
Imagine you want to write a pure language.
But, a completely pure language will have few utility in real life.
Wihout effect, you couldn't print anything on a screen, read the user input, etc...
You can imagine, in standard impure language, there is a hidden global variable.
For example, you could write something in a file.
Somebody else could modify this file.
And you could later read the content of the file.
Each time something changed in the external world, it was like a global variable had changed its value.
This global variable can be represented as a World state.
Now, to have a pure language with some utility you could simply state the execution of your program will be an evaluation of the main function with the following type.
~~~
main :: World -> World
~~~
Which means, main instead of having a global variable accessible by all functions of you program.
Main will be given as parameter an id representing the state of the World on which you can access.
And it will certainly make some changes to it.
In reality, the real type is closer to
~~~
main :: World -> ((),World)
~~~
The `()` type is the null type.
Nothing to see here.
Now let's write our main function:
~~~
main w0 =
let (list,w1) = askUser w0 in
let (x,w2) = print (sum list,w1) in
x
~~~
Also remember, the order of evaluation is generally not fixed in Haskell.
For example in general to evaluate `f a b`, you have many choices:
- first eval `a` then `b` then `f a b`
- first eval `b` then `a` then `f a b`.
- eval `a` and `b` in parallel then `f a b`
This is true, because we should work in a pure language.
Now, if you look at the main function, it is clear you must eval the first
line before the second one since, to evaluate the second line you have
to get a parameter given by the evaluation of the first line.
Such trick works nicely.
The compiler will at each step provide a pointer to a new real world id.
Under the hood, `print` will evaluate as:
- print something on the screen
- modify the id of the world
- evaluate as `((),new world id)`.
Now, if you look at the style of the main function, it is clearly awkward.
Let's try to make the same to the askUser function:
~~~
askUser :: World -> ([Integer],World)
~~~
The type has changed as we will modify the "World" we simulate this by
returning a world value different than the input "World" value.
This way we remain "pure" in the language.
You could write a completely pure implementation and it will works.
In the real world, the evaluation will have some side effect each time a function
return another value of the world input.
Before:
askUser :: IO [Integer]
askUser = do
putStrLn "Enter a list of numbers:"
input <- getLine
let maybeList = getListFromString input in
case maybeList of
Just l -> return l
Nothing -> askUser
After:
askUser w0 =
let (_,w1) = putStrLn "Enter a list of numbers:"
(input,w2) = getLine w1
(l,w3) = case getListFromString input of
Just l -> (l,w2)
Nothing -> askUser w2
in
(l,w3)
This is similar, but awkward. All these `let ... in`. Even if with Haskell
you could remove most, it's still awkard.
The lesson, is, naive IO implementation in Pure functional language is awkward!
Fortunately, some have found a better way to handle this problem.
We see a pattern.
Each line is of the form:
~~~
let (y,w') = action x w in
~~~
Even if for some line the first `x` argument isn't needed.
The output type is a couple, `(answer, newWorldValue)`.
Each function `f` must have a type of kind:
~~~
f :: World -> (a,World)
~~~
Not only this, but we can also remark we use them always
with the following general pattern:
~~~
let (y,w1) = action1 x w0 in
let (z,w2) = action2 y w1 in
...
~~~
Now, we will make a magic trick. We will make the world variable "disappear".
We will `bind` the two lines. Let's define the `bind` function.
~~~
bind :: (World -> (a,World))
-> (a -> (World -> (b,World)))
-> (World -> (b,World))
~~~
(World -> (a,World)) is the type for an IO action. Like getLine, printing something, etc... Now let's rename it for more clarity.
~~~
type IO a = World -> (a, World)
~~~
Some example of functions:
~~~
getLine :: IO String
print :: Show a => a -> IO ()
~~~
`getLine` is an IO action which take a world as parameter, then return a couple (String,World).
Which can be said as: `getLine` is of type IO String.
Which we also see as, an IO action which will return a String "embeded inside an IO".
The function `print` is also interresting.
It takes on argument which can be shown.
In fact it takes two arguments.
The first is the value to print and the other is the state of world.
It then return a couple of type `((),World)`.
This means it changes the world state, but don't give anymore data.
We simplify the bind type:
~~~
bind :: IO a
-> (b -> IO b)
-> IO b
~~~
The function bind take two actions.
The type is quite intimidating. But stay with me here.
On a line like
~~~
let (x,w1) = action1 w0 in
let (y,w2) = action2 x w1 in
(y,w2)
~~~
On the first line, action1 is of type `(World -> (a,World))`.
On the second line, action2 is of type `(a -> (World -> (b,World))`.
`bind`:
- take a function similar to all lines as first argument wich returns a `(a,World)`
- take a function with take an `a` as argument and returns a line wich return a `(b,World)`
- return a line wich returns a `(b,World)`.
~~~
(bind action1 action2) w0 =
let (x, w1) = action1 w0
(y, w2) = action2 x w1
in (y, w2)
~~~
The idea is to hide the World argument with this function. Let's go:
As example imagine if we wanted to simulate:
~~~
let (line1,w1) = getLine w0 in
let ((),w2) = print line1 in
((),w2)
~~~
Now, using the bind function:
~~~
(res,w2) = (bind getLine (\l -> print l)) w0
~~~
As print is of type (World -> ((),World)), we know res = () (null type).
If you didn't saw what was magic here, let's try with three lines this time.
~~~
let (line1,w1) = getLine w0 in
let (line2,w2) = getLine w1 in
let ((),w3) = print (line1 ++ line2) in
((),w3)
~~~
Which is equivalent to:
~~~
(res,w3) = bind getLine (\line1 ->
bind getLine (\line2 ->
print (line1 ++ line2)))
~~~
Didn't you remark something?
Yes, there isn't anymore temporary World variable used anywhere!
This is _MA_. _GIC_.
We can make thinks look better. Let's call bind (>>=) which is an infix function.
Infix is like (+), 3 + 4 <=> "(+) 3 4"
~~~
(res,w3) = getLine >>=
(\line1 -> getLine >>=
(\line2 -> print (line1 ++ line2)))
~~~
Ho Ho Ho! Happy Christmas Everyone!
Haskell has made a syntactical sugar for us:
~~~
do
y <- f x
z <- g y
t <- h y z
~~~
Is replaced by:
~~~
(f >>= (\y ->
g y >>= (\z ->
h y z >>= (\t ->
...
))))
~~~
Which is perfect for IO.
Now we also just need a way to remove the last statement containing a World value.
Easy, just write a simple function return