`
You simply state:
foo [] =
foo l =
But pattern matching go even further.
It is also able to inspect inside data.
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 terser and easier to read.
main = print $ evenSum [1..10]
02_Hard_Part/13_Functions.lhs
02_Hard_Part/14_Functions.lhs
In Haskell you can simplify function definition by η-reducing them.
For example, instead of writing:
f x = (some expresion) x
you can simply write
f = some expression
We use this method to remove 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
main = print $ evenSum [1..10]
02_Hard_Part/14_Functions.lhs
02_Hard_Part/15_Functions.lhs
Higher Order Functions
<%= blogimage("escher_polygon.png","Escher") %>
To make things even better we should use higher order functions.
What are these beast?
Higher order functions are functions taking function as parameter.
Here are some examples:
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 (n+x) xs
where
filter even [1..10] ⇔ [2,4,6,8,10]
The function `filter` takes a function of type (`a -> Bool`) and a list of type `[a]`. It returns a list containing only elements for which the function returned `true`.
Our next step is to use another way to simulate loop.
We will 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
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
foldl f z [x1,...xn]
⇔ f (... (f (f z x1) x2) ...) xn
But as Haskell is lazy, it doesn't evaluate `(f z x)` and push this to the stack.
This is why we generally use `foldl'` instead of `foldl`;
`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 `foldl` and `foldl'` where identical.
Now our new version of `evenSum` become:
-- Version 6
-- foldl' isn't accessible by default
-- we need to import it from the module Data.List
import Data.List
evenSum l = foldl' mysum 0 (filter even l)
where mysum acc value = acc + value
Version we can simplify by using directly a lambda notation.
This way we don't have to create the temporary name `mysum`.
-- Version 7
-- Generally it is considered a good practice
-- to import only the necessary function(s)
import Data.List (foldl')
evenSum l = foldl' (\x y -> x+y) 0 (filter even l)
And of course, we remark
(\x y -> x+y) ⇔ (+)
main = print $ evenSum [1..10]
02_Hard_Part/15_Functions.lhs
02_Hard_Part/16_Functions.lhs
Finally
-- Version 8
import Data.List (foldl')
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.
To help you understand what's going on here, a step by step evaluation:
evenSum [1,2,3,4]
⇒ foldl' (+) 0 (filter even [1,2,3,4])
⇒ foldl' (+) 0 [2,4]
⇒ foldl' (+) (0+2) [4]
⇒ foldl' (+) 2 [4]
⇒ foldl' (+) (2+4) []
⇒ foldl' (+) 6 []
⇒ 6
Another useful higher order function is `(.)`.
The `(.)` function correspond to the mathematical composition.
(f . g . h) x ⇔ f ( g (h x))
We can take advantage of this operator to η-reduce our function:
-- Version 9
import Data.List (foldl')
evenSum :: Integral a => [a] -> a
evenSum = (foldl' (+) 0) . (filter even)
Also, we could rename a bit some part to make it clearer:
-- Version 10
import Data.List (foldl')
sum' :: (Num a) => [a] -> a
sum' = foldl' (+) 0
evenSum :: Integral a => [a] -> a
evenSum = sum' . (filter even)
It is time to discuss a bit.
What did we gain by using higher order functions?
At first, you can say it is terseness.
But in fact, it has more to do with better thinking.
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))
squareEvenSum' = evenSum . (map (^2))
squareEvenSum'' = sum' . (map (^2)) . (filter even)
We just had to add another "transformation function"[^0216].
[^0216]: You should remark `squareEvenSum''` is more efficient that the two other versions. The order of `(.)` is important.
~~~
map (^2) [1,2,3,4] ⇔ [1,4,9,16]
~~~
The `map` function simply apply a function to all element of a list.
We didn't had to modify _inside_ the function definition.
It feels more modular.
But there is also you can think more mathematically about your function.
You could then use your function as any other one.
You could compose, map, fold, filter using your new function.
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.
Understanding this was really hard for me when learning Haskell.
A lot of effort is provided to explain you how much functional approach is superior.
Then 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: _Types_.
main = print $ evenSum [1..10]
02_Hard_Part/16_Functions.lhs
Types
<%= blogimage("salvador-dali-the-madonna-of-port-lligat.jpg","Dali, the madonna of port Lligat") %>
> <%=tldr%>
>
> - `type Name = AnotherType` is just an alias and the compiler doesn't do any difference between `Name` and `AnotherType`.
> - `data Name = NameConstructor AnotherType` make a difference.
> - `data` can construct structures which can be recursives.
> - `deriving` is magic and create functions for you.
In Haskell, types are strong and static.
Why is this important? It will help you _a lot_ not to make some mistake.
In Haskell, most bugs are caught 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 most static typed language are bad to generalize concepts.
What saves Haskell is that it can _infere_ types.
Here is a simple example.
The `square` function in Haskell:
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`. Proof by example:
~~~
% ghci
GHCi, version 7.0.4:
...
Prelude> let square x = x*x
Prelude> square 2
4
Prelude> square 2.1
4.41
Prelude> -- load the Data.Complex module
Prelude> :m Data.Complex
Prelude Data.Complex> square (2 :+ 1)
3.0 :+ 4.0
~~~
`x :+ y` is the notation for the complex (x + ib).
Now compare with the necessary C code:
int int_square(int x) { return x*x; }
float float_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;
}
complex x,y;
y = complex_square(x);
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.
But unlike dynamic typed languages, most error are caught before the execution.
Generally, in Haskell:
> "if it compiles it certainly does what you intended"
02_Hard_Part/21_Types.lhs
Type construction
You can construct your 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/21_Types.lhs
02_Hard_Part/22_Types.lhs
But it doesn't protect you much.
Try to swap the two parameter of `showInfos` and run the program:
putStrLn $ showInfos color name
It will compile and execute.
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 mainly:
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 = DataConstructor {
field1 :: [type of field1]
, field2 :: [type of field2]
...
, fieldn :: [type of fieldn] }
And many accessors 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/22_Types.lhs
02_Hard_Part/23_Types.lhs
Recursive type
You already encountered recursive types: lists.
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 you add `deriving (Show)` to your data declaration, 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/23_Types.lhs
02_Hard_Part/30_Trees.lhs
Trees
<%= blogimage("magritte-l-arbre.jpg","Magritte, l'Arbre") %>
We'll just give another standard example: binary trees.
import Data.List
data BinTree a = Empty
| Node a (BinTree a) (BinTree a)
deriving (Show)
Also we create a function which transform a list into an ordered binary tree.
treeFromList :: (Ord a) => [a] -> BinTree a
treeFromList [] = Empty
treeFromList (x:xs) = Node x (treeFromList (filter (x) xs))
Look at how elegant this function is.
In plain English:
- an empty list will be converted to an empty tree.
- a list `(x:xs)` will be converted to the tree where:
- The root is `x`
- Its left subtree is the tree created from the list of the remaining element of `xs` which are strictly inferior to `x` and
- the right subtree is the tree created from the elements strictly superior to `x` of the list `xs`.
main = print $ treeFromList [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/30_Trees.lhs
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 changes to make.
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`).
We will be able to 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.
We will create our own version of show.
To achieve this, we must declare that our newly created type `BinTree a`
is an instance of the type class `Show`.
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 Tree
-- show a tree and start each line with pref
-- We don't display Empty tree
treeshow pref Empty = ""
-- Leaf
treeshow pref (Node x Empty Empty) =
(pshow pref x)
-- Right branch is empty
treeshow pref (Node x left Empty) =
(pshow pref x) ++ "\n" ++
(showSon pref "`--" " " left)
-- Left branch is empty
treeshow pref (Node x Empty right) =
(pshow pref x) ++ "\n" ++
(showSon pref "`--" " " right)
-- Tree with left and right sons non empty
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 `treeFromList` method remain identical.
treeFromList :: (Ord a) => [a] -> BinTree a
treeFromList [] = Empty
treeFromList (x:xs) = Node x (treeFromList (filter (x) xs))
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))
[ ["YO","DAWG"]
, ["I","HEARD"]
, ["I","HEARD"]
, ["YOU","LIKE","TREES"] ]
Which is equivalent to
print ( treeFromList (
map treeFromList
[ map treeFromList ["YO","DAWG"]
, map treeFromList ["I","HEARD"]
, map treeFromList ["I","HEARD"]
, map treeFromList ["YOU","LIKE","TREES"] ]))
and gives:
~~~
Binary tree of Binary trees of Char binary trees:
< < < 'Y'
: : : `--'O'
: : `--< 'D'
: : : |--'A'
: : : `--'W'
: : : `--'G'
: |--< < 'I'
: | : `--< 'H'
: | : : |--'E'
: | : : | `--'A'
: | : : | `--'D'
: | : : `--'R'
: `--< < 'Y'
: : : `--'O'
: : : `--'U'
: : `--< 'L'
: : : `--'I'
: : : |--'E'
: : : `--'K'
: : `--< 'T'
: : : `--'R'
: : : |--'E'
: : : `--'S'
~~~
Remark how duplicate trees aren't inserted;
there is only one tree corresponding to `"I","HEARD"`.
We have this for (almost) free, because we have declared Tree to be an instance of `Eq`.
See 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!
02_Hard_Part/31_Trees.lhs
02_Hard_Part/40_Infinites_Structures.lhs
Infinite Structures
<%= blogimage("escher_infinite_lizards.jpg","Escher") %>
It is often stated that Haskell is _lazy_.
In fact, if you are a bit pedantic, you should state that [Haskell is _non-strict_](http://www.haskell.org/haskellwiki/Lazy_vs._non-strict).
Laziness is just a common implementation for non-strict languages.
Then what does not-strict means? From the Haskell wiki:
> Reduction (the mathematical term for evaluation) proceeds from the outside in.
>
> so if you have `(a+(b*c))` then you first reduce `+` first, then you reduce the inner `(b*c)`
For example in Haskell you can do:
-- numbers = [1,2,..]
numbers :: [Integer]
numbers = 0:map (1+) numbers
take' n [] = []
take' 0 l = []
take' n (x:xs) = x:take' (n-1) xs
main = print $ take' 10 numbers
And it stops.
How?
Instead of trying to evaluate `numbers` entirely,
it evaluates elements only when needed.
Also, note in Haskell there is a notation for infinite lists
~~~
[1..] ⇔ [1,2,3,4...]
[1,3..] ⇔ [1,3,5,7,9,11...]
~~~
And most function will work with them.
Also there exists the function `take` equivalent to our `take'`.
02_Hard_Part/40_Infinites_Structures.lhs
02_Hard_Part/41_Infinites_Structures.lhs
This code is mostly the same as the preceding one.
import Debug.Trace (trace)
import Data.List
data BinTree a = Empty
| Node a (BinTree a) (BinTree a)
deriving (Eq,Ord)
-- 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"
nullTree = Node 0 nullTree nullTree
A complete binary tree were each node is equal to 0.
Now I will prove you can manipulate this object using the following function:
-- take all element of a BinTree
-- up to some depth
treeTakeDepth _ Empty = Empty
treeTakeDepth 0 _ = Empty
treeTakeDepth n (Node x left right) = let
nl = treeTakeDepth (n-1) left
nr = treeTakeDepth (n-1) right
in
Node x nl nr
See what occurs for this program:
main = print $ treeTakeDepth 4 nullTree
This code compile, run and stop giving the following result:
~~~
< 0
: |-- 0
: | |-- 0
: | | |-- 0
: | | `-- 0
: | `-- 0
: | |-- 0
: | `-- 0
: `-- 0
: |-- 0
: | |-- 0
: | `-- 0
: `-- 0
: |-- 0
: `-- 0
~~~
Just to heat your neurones a bit more,
let's make a slightly more interesting tree:
iTree = Node 0 (dec iTree) (inc iTree)
where
dec (Node x l r) = Node (x-1) (dec l) (dec r)
inc (Node x l r) = Node (x+1) (inc l) (inc r)
Another way to create this tree is to use an higher order function.
This function should be similar to `map`, but should work on `BinTree` instead of list.
Here is such a function:
-- apply a function to each node of Tree
treeMap :: (a -> b) -> BinTree a -> BinTree b
treeMap f Empty = Empty
treeMap f (Node x left right) = Node (f x)
(treeMap f left)
(treeMap f right)
_Hint_: I won't talk more about this here.
If you are interested of the generalization of `map` to other data structure,
search for functor and `fmap`.
Our definition is now:
infTreeTwo :: BinTree Int
infTreeTwo = Node 0 (treeMap (\x -> x-1) infTreeTwo)
(treeMap (\x -> x+1) infTreeTwo)
Look at the result for
main = print $ treeTakeDepth 4 infTreeTwo
~~~
< 0
: |-- -1
: | |-- -2
: | | |-- -3
: | | `-- -1
: | `-- 0
: | |-- -1
: | `-- 1
: `-- 1
: |-- 0
: | |-- -1
: | `-- 1
: `-- 2
: |-- 1
: `-- 3
~~~
main = do
print $ treeTakeDepth 4 nullTree
print $ treeTakeDepth 4 infTreeTwo
02_Hard_Part/41_Infinites_Structures.lhs
Hell Difficulty Part
Congratulation to get so far!
Now, some of the really hardcore stuff could start.
If you are like me, you should get the functional style.
You should also understand a bit more the advantages of laziness 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?
Be prepared, answer might be difficult to get.
But they all be very rewarding.
03_Hell/01_IO/01_progressive_io_example.lhs
Deal With IO
<%= blogimage("magritte_carte_blanche.jpg","Magritte, Carte blanche") %>
> <%=tldr%>
>
> A typical function doing `IO` look a lot like an imperative language:
>
> ~~~
> f :: IO a
> f = do
> x <- action1
> action2 x
> y <- action3
> action4 x y
> ~~~
>
> - To set a value to an object we use `<-` .
> - The type of each line is `IO *`;
> in this example:
> - `action1 :: IO b`
> - `action2 x :: IO ()`
> - `action3 :: IO c`
> - `action4 x y :: IO a`
> - `x :: b`, `y :: c`
> - Few objects have the type `IO a`, this should help you to choose.
> In particular you cannot use pure function directly here.
> To use pure function you could do `action2 (purefunction x)` for example.
In this section, I will explain how to use IO, not how they work.
You'll see how Haskell separate pure from impure part of the program.
Don't stop because you're trying to understand the details of the syntax.
Answer will come in the next section.
What to achieve?
> Ask a user to enter a list of numbers.
> Print 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 to understand the behavior of this program.
Let's analyze the types in more detail.
~~~
putStrLn :: String -> IO ()
getLine :: IO String
print :: Show a => a -> IO ()
~~~
Or more interestingly, we remark each expression in the `do` block has a type of `IO a`.
main = do
putStrLn "Enter ... " :: IO ()
getLine :: IO String
print Something :: IO ()
We should also remark the effect of the `<-` symbol.
~~~
do
x <- something
~~~
If `something :: IO a` then `x :: a`.
Another important remark to use `IO`.
All line in a do block must have one of the two forms:
~~~
action1 :: IO a
-- in this case, generally a = ()
~~~
or
~~~
value <- action2 -- where
-- bar z t :: IO b
-- value :: b
~~~
These two kind of line will correspond to two different way of sequencing actions.
The meaning of this sentence should be clearer at the end of the next section.
03_Hell/01_IO/01_progressive_io_example.lhs
03_Hell/01_IO/02_progressive_io_example.lhs
Now let's see how this behave.
For example, what occur 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!
The first evolution will be to answer with a more friendly 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
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 `.
Don't try to understand too much of this function.
I use a lower level function than `read`; `reads`.
[^1]: Which itself is very similar to the javascript `eval` on a string containing JSON).
maybeRead :: Read a => String -> Maybe a
maybeRead s = case reads s of
[(x,"")] -> Just x
_ -> Nothing
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]`.
getListFromString :: String -> Maybe [Integer]
getListFromString str = maybeRead $ "[" ++ str ++ "]"
We simply have to 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.
Remark the type of each expression in the main's do block remains of the form `IO a`.
The only strange construction is `error`.
I'll say `error msg` will simply take the needed type (here `IO ()`).
One very important thing to note is the type of all the defined function.
There is only one function which contains `IO` in its type: `main`.
That means main is impure.
But main use `getListFromString` which is pure.
It is then clear just by looking at declared types where are pure and impure functions.
Why purity matters?
I certainly forget many advantages, but the three main reason are:
- It is far easier to think about pure code than impure one.
- Purity protect you from all hard to reproduce bugs due to border effects.
- You can evaluate pure functions in any order or in parallel without risk.
This is why, you should generally put as most code as possible in pure functions.
03_Hell/01_IO/02_progressive_io_example.lhs
03_Hell/01_IO/03_progressive_io_example.lhs
Our next evolution will be to ask the user again and again until it enters a valid answer.
We keep 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 type `[Integer]` through some IO actions.
Some people might explain while waving their hands:
> «This is an `[Integer]` inside an `IO`»
If you want to understand the details behind all of this, you'll have to read the next section.
But sincerely, if you just want to _use_ IO.
Just exercise a little and remember to think about the type.
Finally our main function is quite simpler:
main :: IO ()
main = do
list <- askUser
print $ sum list
We have finished with our introduction to `IO`.
This was quite a fast. Here are the main things to remind:
- in the `do` bloc, each expression must have the type `IO a`.
You are then limited in the number of expression you could use.
For example, `getLine`, `print`, `putStrLn`, etc...
- Try to externalize the pure function as much as possible.
- the `IO a` type means: an IO _action_ which return an element of type `a`.
`IO` represent action; under the hood, `IO a` is the type of a function.
Read the next section if you are curious.
If you exercise a bit, you should be able to _use_ `IO`.
> _Exercises_:
>
> - Make a program that sum all its argument. Hint: use the function `getArgs`.
03_Hell/01_IO/03_progressive_io_example.lhs
IO trick explained
<%= blogimage("magritte_pipe.jpg","Magritte, ceci n'est pas une pipe") %>
> Here is a <%=tldr%> for this section.
>
> To separate pure from impure part,
> the main is defined as a function
> which modify the state of the world
>
> ~~~
> main :: World -> World
> ~~~
>
> A function is granted to have side effect only if it gets this value.
> But look at a typical main function:
>
> ~~~
> main w0 =
> let (v1,w1) = action1 w0 in
> let (v2,w2) = action2 v1 w1 in
> let (v3,w3) = action3 v2 w2 in
> action4 v3 w3
> ~~~
>
> We have a lot of temporary elements (here `w1`, `w2` and `w3`)
> which must be passed to the next action.
>
> We create a function `bind` or `(>>=)`.
> With `bind` we need no more temporary name.
>
> ~~~
> main =
> action1 >>= action2 >>= action3 >>= action4
> ~~~
>
> Bonus: Haskell has a syntactical sugar for us:
>
> ~~~
> main = do
> v1 <- action1
> v2 <- action2 v1
> v3 <- action3 v2
> action4 v3
> ~~~
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 commas):"
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 remark; it looks like an imperative structure.
Haskell is powerful enough to make some pure code to look imperative.
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 should had remarked the notation is a bit unusual.
Here is why, in detail.
In an impure language, the state of the world can be seen as a huge hidden global variable.
This hidden variable is accessible by all function of your language.
For example, you can read and write a file in any function.
The fact a file exists or not, can be seen as different state of the world.
For Haskell this state is not hidden.
It is explicitly said `main` is a function that _potentially_ change the state of the world.
It's type is then something like:
main :: World -> World
Not all function could have access to this variable.
Those who have access to this variable can potentially be impure.
Functions whose the world variable isn't provided to should be pure[^032001].
[^032001]: There are some _unsafe_ exception to this rule. But you shouldn't see such usage on a real application except might be for some debugging purpose.
Haskell consider the state of the world is an input variable for `main`.
But the real type of main is closer to this one[^032002]:
[^032002]: For the curious the real type is `data IO a = IO {unIO :: State# RealWorld -> (# State# RealWorld, a #)}`. All the `#` as to do with optimisation and I swapped the fields in my example. But mostly, the idea is exactly the same.
main :: World -> ((),World)
The `()` type is the null type.
Nothing to see here.
Now let's rewrite our main function with this in mind:
main w0 =
let (list,w1) = askUser w0 in
let (x,w2) = print (sum list,w1) in
x
First, we remark, that all function which have side effect must have the type:
World -> (a,World)
Where `a` is the type of result.
For example, a `getChar` function should have the type `World -> (Char,World)`.
Another thing to remark is the trick to fix the order of evaluation.
In Haskell to evaluate `f a b`, you generally 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)
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:" in
let (input,w2) = getLine w1 in
let (l,w3) = case getListFromString input of
Just l -> (l,w2)
Nothing -> askUser w2
in
(l,w3)
This is similar, but awkward.
Look at all these temporary `w?` names.
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 similar to:
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 w0 in
let (z,w2) = action2 w1 in
let (t,w3) = action3 w2 in
...
Each action can take 0 to some parameters.
And in particular, each action can take a parameter from the result of a line above.
For example, we could also have:
let (_,w1) = action1 x w0 in
let (z,w2) = action2 w1 in
let (_,w3) = action3 x z w2 in
...
And of course `actionN w :: (World) -> (a,World)`.
> IMPORTANT, there are only two important pattern for us:
>
> ~~~
> let (x,w1) = action1 w0 in
> let (y,w2) = action2 w1 in
> ~~~
>
> and
>
> ~~~
> let (_,w1) = action1 w0 in
> let (y,w2) = action2 w1 in
> ~~~
<%= leftblogimage("jocker_pencil_trick.jpg","Jocker pencil trick") %>
Now, we will make a magic trick.
We will make the temporary world symbol "disappear".
We will `bind` the two lines.
Let's define the `bind` function.
Its type is quite intimidating at first:
bind :: (World -> (a,World))
-> (a -> (World -> (b,World)))
-> (World -> (b,World))
But remember that `(World -> (a,World))` is the type for an IO action.
Now let's rename it for 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 and 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.
This type help us simplify the type of `bind`:
bind :: IO a
-> (a -> IO b)
-> IO b
It says that `bind` takes two IO actions as parameter and return another IO action.
Now, remember the _important_ patterns. The first was:
let (x,w1) = action1 w0 in
let (y,w2) = action2 x w1 in
(y,w2)
Look at the types:
action1 :: IO a
action2 :: a -> IO b
(y,w2) :: IO b
Doesn't seem familiar?
(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 use a better notation.
Let's use `(>>=)` instead of `bind`.
`(>>=)` is an infix function like
`(+)`; reminder `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
x <- action1
y <- action2
z <- action3
...
Is replaced by:
action1 >>= \x ->
action2 >>= \y ->
action3 >>= \z ->
...
Note you can use `x` in `action2` and `x` and `y` in `action3`.
But what for line not using the `<-`?
Easy another function `blindBind`:
blindBind :: IO a -> IO b -> IO b
blindBind action1 action2 w0 =
bind action (\_ -> action2) w0
I didn't simplified this definition for clarity purpose.
Of course we can use a better notation, we'll use the `(>>)` operator.
And
do
action1
action2
action3
Is transformed into
action1 >>
action2 >>
action3
Also, another function is quite useful.
putInIO :: a -> IO a
putInIO x = IO (\w -> (x,w))
This is the general way to put pure value inside the "IO context".
The general name for `putInIO` is `return`.
This is quite a bad name when you learn Haskell. `return` is very different from what you might be used to.
03_Hell/01_IO/21_Detailled_IO.lhs
To finish, let's translate our example:
askUser :: IO [Integer]
askUser = do
putStrLn "Enter a list of numbers (separated by commas):"
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
Is translated into:
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 ++ "]"
askUser :: IO [Integer]
askUser =
putStrLn "Enter a list of numbers (sep. by commas):" >>
getLine >>= \input ->
let maybeList = getListFromString input in
case maybeList of
Just l -> return l
Nothing -> askUser
main :: IO ()
main = askUser >>=
\list -> print $ sum list
You can compile this code to verify it continues to work.
Imagine what it would look like without the `(>>)` and `(>>=)`.
03_Hell/01_IO/21_Detailled_IO.lhs
03_Hell/02_Monads/10_Monads.lhs
Monads
<%= blogimage("dali_reve.jpg","Dali, reve. It represent a weapon out of the mouth of a tiger, itself out of the mouth of another tiger, itself out of the mouth of a fish itsleft out of a grenade. I could have choosen a picture of the Human centipede as it is a very good representation of what a monad really is. But just to thing about it, I find this disgusting and that wasn't the purpose of this document.") %>
Now the secret can be revealed: `IO` is a _monad_.
Being a monad means you have access to some syntactical sugar with the `do` notation.
But mainly, you have access to some coding pattern which will ease the flow of your code.
> **Important remarks**:
>
> - Monad are not necessarily about effects!
> There are a lot of _pure_ monads.
> - Monad are more about sequencing
For the Haskell language `Monad` is a type class.
To be an instance of this type class, you must provide the functions `(>>=)` and `return`.
The function `(>>)` will be derived from `(>>=)`.
Here is how the type class `Monad` is declared (mostly):
class Monad m where
(>>=) :: m a -> (a -> m b) -> m b
return :: a -> m a
(>>) :: m a -> m b -> m b
f >> g = f >>= \_ -> g
-- You should generally safely ignore this function
-- which I believe exists for historical reason
fail :: String -> m a
fail = error
> Remarks:
>
> - the keyword `class` is not your friend.
> A Haskell class is _not_ a class like in object model.
> A Haskell class has a lot similarities with Java interfaces.
> A better word should have been `typeclass`.
> That means a set of types.
> For a type to belong to a class, all function of the class must be provided for this type.
> - In this particular example of type class, the type `m` must be a type that take an argument.
> for example `IO a`, but also `Maybe a`, `[a]`, etc...
> - To be a useful monad, your function must obey some rule.
> If your construction does not obey these rules strange things might happens:
>
> ~~~
> return a >>= k == k a
> m >>= return == m
> m >>= (\x -> k x >>= h) == (m >>= k) >>= h
> ~~~
Maybe is a monad
There exists a lot of different type that are instance of `Monad`.
One of the easiest to describe is `Maybe`.
If you have a sequence of `Maybe` values, you could use monad to manipulate them.
It is particularly useful to remove very deep `if..then..else..` constructions.
Imagine a complex bank operation. You are eligible to gain about 700€ only
if you can afford to follow a list of operation without being negative.
deposit value account = account + value
withdraw value account = account - value
eligible :: (Num a,Ord a) => a -> Bool
eligible account =
let account1 = deposit 100 account in
if (account1 < 0)
then False
else
let account2 = withdraw 200 account1 in
if (account2 < 0)
then False
else
let account3 = deposit 100 account2 in
if (account3 < 0)
then False
else
let account4 = withdraw 300 account3 in
if (account4 < 0)
then False
else
let account5 = deposit 1000 account4 in
if (account5 < 0)
then False
else
True
main = do
print $ eligible 300 -- True
print $ eligible 299 -- False
03_Hell/02_Monads/10_Monads.lhs
03_Hell/02_Monads/11_Monads.lhs
Now, let's make it better using Maybe and the fact it is a Monad
deposit :: (Num a) => a -> a -> Maybe a
deposit value account = Just (account + value)
withdraw :: (Num a,Ord a) => a -> a -> Maybe a
withdraw value account = if (account < value)
then Nothing
else Just (account - value)
eligible :: (Num a, Ord a) => a -> Maybe Bool
eligible account = do
account1 <- deposit 100 account
account2 <- withdraw 200 account1
account3 <- deposit 100 account2
account4 <- withdraw 300 account3
account5 <- deposit 1000 account4
Just True
main = do
print $ eligible 300 -- Just True
print $ eligible 299 -- Nothing
03_Hell/02_Monads/11_Monads.lhs
03_Hell/02_Monads/12_Monads.lhs
Not bad, but we can make it even better:
deposit :: (Num a) => a -> a -> Maybe a
deposit value account = Just (account + value)
withdraw :: (Num a,Ord a) => a -> a -> Maybe a
withdraw value account = if (account < value)
then Nothing
else Just (account - value)
eligible :: (Num a, Ord a) => a -> Maybe Bool
eligible account =
deposit 100 account >>=
withdraw 200 >>=
deposit 100 >>=
withdraw 300 >>=
deposit 1000 >>
return True
main = do
print $ eligible 300 -- Just True
print $ eligible 299 -- Nothing
We have proved Monad are nice to make our code more elegant.
Note this idea of code organization, in particular for `Maybe` can be used
in most imperative language.
In fact, this is the kind of construction we make naturally.
> An important remark:
>
> The first element in the sequence being evaluated to `Nothing` will stop
> the complete evaluation.
> That means, you don't execute all lines.
> You have this for free, thanks to laziness.
The `Maybe` monad proved to be useful while being a very simple example.
We saw the utility of the `IO` monad.
But now a cooler example, lists.
03_Hell/02_Monads/12_Monads.lhs
03_Hell/02_Monads/13_Monads.lhs
The list monad
<%= blogimage("golconde.jpg","Golconde de Magritte") %>
The list monad help us to simulate non deterministic computation.
Here we go:
import Control.Monad (guard)
allCases = [1..10]
resolve :: [(Int,Int,Int)]
resolve = do
x <- allCases
y <- allCases
z <- allCases
guard $ 4*x + 2*y < z
return (x,y,z)
main = do
print resolve
MA. GIC. :
~~~
[(1,1,7),(1,1,8),(1,1,9),(1,1,10),(1,2,9),(1,2,10)]
~~~
For the list monad, there is also a syntactical sugar:
print $ [ (x,y,z) | x <- allCases,
y <- allCases,
z <- allCases,
4*x + 2*y < z ]
I won't list all the monads, but there is a lot of monads.
The usage of monad simplify the manipulation of some notion in pure languages.
In particular, monad are very useful for:
- IO,
- non deterministic computation,
- generating pseudo random numbers,
- keeping configuration state,
- writing state,
- ...
If you have followed me until here, then you've done it!
You know monads[^03021301]!
[^03021301]: Well, you'll certainly need to exercise a bit to be used to them and to understand when you can use them and create your own. But you already made a big step further.
03_Hell/02_Monads/13_Monads.lhs
Appendix
This section is not so much about learning Haskell.
It is just here to discuss some details further.
04_Appendice/01_More_on_infinite_trees/10_Infinite_Trees.lhs
More on Infinite Tree
In the section [Infinite Structures](#infinite-structures) we saw some simple construction.
Unfortunately we removed two properties of our tree:
1. no duplicate node value
2. well ordered tree
In this section we will try to keep the first property.
Concerning the second one, we must relax this one but we'll discuss on how to
keep it as much as possible.
This code is mostly the same as the one in the [tree section](#trees).
import Data.List
data BinTree a = Empty
| Node a (BinTree a) (BinTree a)
deriving (Eq,Ord)
-- 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"
shuffle = map (\x -> (x*3123) `mod` 4331) [1..]
Just as reminder here are the definition of `treeFromList`
treeFromList :: (Ord a) => [a] -> BinTree a
treeFromList [] = Empty
treeFromList (x:xs) = Node x (treeFromList (filter (x) xs))
and `treeTakeDepth`:
treeTakeDepth _ Empty = Empty
treeTakeDepth 0 _ = Empty
treeTakeDepth n (Node x left right) = let
nl = treeTakeDepth (n-1) left
nr = treeTakeDepth (n-1) right
in
Node x nl nr
See the result of:
main = do
putStrLn "take 10 shuffle"
print $ take 10 shuffle
putStrLn "\ntreeTakeDepth 4 (treeFromList shuffle)"
print $ treeTakeDepth 4 (treeFromList shuffle)
~~~
% runghc 02_Hard_Part/41_Infinites_Structures.lhs
take 10 shuffle
[3123,1915,707,3830,2622,1414,206,3329,2121,913]
treeTakeDepth 4 (treeFromList shuffle)
< 3123
: |--1915
: | |--707
: | | |--206
: | | `--1414
: | `--2622
: | |--2121
: | `--2828
: `--3830
: |--3329
: | |--3240
: | `--3535
: `--4036
: |--3947
: `--4242
~~~
Yay! It ends!
Beware though, it will only work if you always have something to put into a branch.
For example
treeTakeDepth 4 (treeFromList [1..])
will loop forever.
Simply because, it will try to access the head of `filter (<1) [2..]`.
But filter is not smart enought to understand that the result is the empty list.
Nonetheless, it is still a very cool example of what non strict program has to offer.
Left as an exercise to the reader:
- Could you prove that there exists some number `n` such that `treeTakeDepth n (treeFromList shuffle)` will enter in an infinite loop.
- Find an upper bound for `n`.
- Prove there is no `shuffle` list such that, for any depth, the program ends.
04_Appendice/01_More_on_infinite_trees/10_Infinite_Trees.lhs
04_Appendice/01_More_on_infinite_trees/11_Infinite_Trees.lhs
This code is mostly the same as the preceding one.
import Debug.Trace (trace)
import Data.List
data BinTree a = Empty
| Node a (BinTree a) (BinTree a)
deriving (Eq,Ord)
-- 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"
treeTakeDepth _ Empty = Empty
treeTakeDepth 0 _ = Empty
treeTakeDepth n (Node x left right) = let
nl = treeTakeDepth (n-1) left
nr = treeTakeDepth (n-1) right
in
Node x nl nr
shuffle = map rand [1..]
where
rand x = ((p x) `mod` (x+c)) - ((x+c) `div` 2)
p x = m*x^2 + n*x + o -- some polynome
m = 3123
n = 31
o = 7641
c = 1237
This shuffle function has the property (hopefully) not to have an upper nor lower bound.
But having a better shuffle list isn't enough not to enter an infinite loop.
Generally, we cannot decide whether `filter ( Any element of the left (resp. right) branch must all be strictly inferior (resp. superior) to the label of the root.
Remark it will remains _mostly_ an ordered binary tree.
Furthermore, by construction, each node value is unique in the tree.
Here is our new version of `treeFromList`. We simply have replaced `filter` by `safefilter`.
treeFromList :: (Ord a, Show a) => [a] -> BinTree a
treeFromList [] = Empty
treeFromList (x:xs) = Node x left right
where
left = treeFromList $ safefilter (x) xs
This new function `safefilter` is almost equivalent to `filter` but don't enter infinite loop if the result is a finite list.
If it cannot find an element for which the test is true after 10000 consecutive steps, then it considers to be the end of the search.
safefilter :: (a -> Bool) -> [a] -> [a]
safefilter f l = safefilter' f l nbTry
where
nbTry = 10000
safefilter' _ _ 0 = []
safefilter' _ [] _ = []
safefilter' f (x:xs) n =
if f x
then x : safefilter' f xs nbTry
else safefilter' f xs (n-1)
Now run the program and be happy:
main = do
putStrLn "take 10 shuffle"
print $ take 10 shuffle
putStrLn "\ntreeTakeDepth 8 (treeFromList shuffle)"
print $ treeTakeDepth 8 (treeFromList $ shuffle)
You should realize the time to print each value is different.
This is because Haskell compute each value when it needs it.
And in this case, this is when asked to print it on the screen.
Impressively enough, try to replace the depth from `8` to `100`.
It will work without killing your RAM!
The flow and the memory management is done naturally by Haskell.
Left as an exercise to the reader:
- Even with large constant value for `deep` and `nbTry`, it seems to work nicely. But in the worst case, it can be exponential.
Create a worst case list to give as parameter to `treeFromList`.
_hint_: think about (`[0,-1,-1,....,-1,1,-1,...,-1,1,...]`).
- I first tried to implement `safefilter` as follow:
safefilter' f l = if filter f (take 10000 l) == []
then []
else filter f l
Explain why it doesn't work and can enter into an infinite loop.
- Suppose that `shuffle` is real random list with growing bounds.
If you study a bit this structure, you'll discover that with probability 1,
this structure is finite.
Using the following code
(suppose we could use `safefilter'` directly as if was not in the where of safefilter)
find a definition of `f` such that with probability `1`,
treeFromList' shuffle is infinite. And prove it.
Disclaimer, this is only a conjecture.
treeFromList' [] n = Empty
treeFromList' (x:xs) n = Node x left right
where
left = treeFromList' (safefilter' (x) xs (f n)
f = ???
04_Appendice/01_More_on_infinite_trees/11_Infinite_Trees.lhs