Learn Haskell Fast and Hard
Blow your mind with Haskell
tl;dr: A very short and dense tutorial for learning Haskell.
Table of Content
I really believe all developer should learn Haskell. I don’t think all should be super Haskell ninjas, but at least, they should discover what Haskell has to offer. Learning Haskell open your mind.
Mainstream languages share the same foundations:
- variables
- loops
- pointers1
- data structures, objects and classes (for most)
Haskell is very different. This language uses a lot of concepts I had never heard about before. Many of those concepts will help you become a better programmer.
But, learning Haskell can be hard. It was for me. In this article I try to provide what I lacked during my learning.
This article will certainly be hard to follow. This is on purpose. There is no shortcut to learning Haskell. It is hard and challenging. But I believe this is a good thing. It is because it is hard that Haskell is interesting.
The conventional method to learning Haskell is to read two books. First “Learn You a Haskell” and just after “Real World Haskell”. I also believe this is the right way to go. But, to learn what Haskell is all about, you’ll have to read them in detail.
On the other hand, this article is a very brief and dense overview of all major aspects of Haskell. I also added some informations I lacked while I learned Haskell.
The article contains five parts:
- Introduction: a short example to show Haskell can be friendly.
- Basic Haskell: Haskell syntax, and some essential notions.
- Hard Difficulty Part:
- Functional style; a progressive example, from imperative to functional style
- Types; types and a standard binary tree example
- Infinite Structure; manipulate an infinite binary tree!
- Hell Difficulty Part:
- Deal with IO; A very minimal example
- IO trick explained; the hidden detail I lacked to understand IO
- Monads; incredible how we can generalize
- Appendix:
- More on infinite tree; a more math oriented discussion about infinite trees
Note: Each time you’ll see a separator with a filename ending in
.lhs
, you could click the filename to get this file. If you save the file asfilename.lhs
, you can run it withrunhaskell filename.lhsSome might not work, but most will. You should see a link just below.
01_basic/10_Introduction/00_hello_world.lhs
Introduction
Install
- Haskell Platform is the standard way to install Haskell.
Tools:
ghc
: Compiler similar to gcc forC
.ghci
: Interactive Haskell (REPL)runhaskell
: Execute a program without compiling it. Convenient but very slow compared to compiled programs.
Don't be afraid
Many book/articles about Haskell start by introducing some esoteric formula (quick sort, Fibonacci, etc…). I will do the exact opposite. At first I won’t show you any Haskell super power. I will start with similarities between Haskell and other programming languages. Let’s jump to the mandatory “Hello World”.
main = putStrLn "Hello World!"
To run it, you can save this code in a hello.hs
and:
~ runhaskell ./hello.hs Hello World!
You could also download the literate Haskell source.
You should see a link just above the introduction title.
Download this file as 00_hello_world.lhs
and:
~ runhaskell 00_hello_world.lhs Hello World!
01_basic/10_Introduction/00_hello_world.lhs
01_basic/10_Introduction/10_hello_you.lhs
Now, a program asking your name and replying “Hello” using the name you entered:
main = do
print "What is your name?"
name <- getLine
print ("Hello " ++ name ++ "!")
First, let us compare with a similar program in some imperative languages:
# Python print "What is your name?" name = raw_input() print "Hello %s!" % name
# Ruby puts "What is your name?" name = gets.chomp puts "Hello #{name}!"
// In C #include <stdio.h> int main (int argc, char **argv) { char name[666]; // <- An Evil Number! // What if my name is more than 665 character long? printf("What is your name?\n"); scanf("%s", name); printf("Hello %s!\n", name); return 0; }
The structure is the same, but there are some syntax differences. A major part of this tutorial will be dedicated to explaining why.
In Haskell, there is a main
function and every object has a type.
The type of main
is IO ()
.
This means, main
will cause side effects.
Just remember that Haskell can look a lot like mainstream imperative languages.
01_basic/10_Introduction/10_hello_you.lhs
01_basic/10_Introduction/20_very_basic.lhs
Very basic Haskell
Before continuing you need to be warned about some essential properties of Haskell.
Functional
Haskell is a functional language. If you have an imperative language background, you’ll have to learn a lot of new things. Hopefully many of these new concepts will help you to program even in imperative languages.
Smart Static Typing
Instead of being in your way like in C
, C++
or Java
, the type system is here to help you.
Purity
Generally your functions won’t modify anything in the outside world. This means, it can’t modify the value of a variable, can’t get user input, can’t write on the screen, can’t launch a missile. On the other hand, parallelism will be very easy to achieve. Haskell makes it clear where effects occur and where you are pure. Also, it will be far easier to reason about your program. Most bugs will be prevented in the pure parts of your program.
Furthermore pure functions follow a fundamental law in Haskell:
Applying a function with the same parameters always returns the same value.
Laziness
Laziness by default is a very uncommon language design. By default, Haskell evaluates something only when it is needed. In consequence, it provides a very elegant way to manipulate infinite structures for example.
A last warning on how you should read Haskell code.
For me, it is like reading scientific papers.
Some parts are very clear, but when you see a formula, just focus and read slower.
Also, while learning Haskell, it really doesn’t matter much if you don’t understand syntax details.
If you meet a >>=
, <$>
, <-
or any other weird symbol, just ignore them and follows the flow of the code.
Function declaration
You might be used to declare functions like this:
In C
:
int f(int x, int y) { return x*x + y*y; }
In Javascript:
function f(x,y) { return x*x + y*y; }
in Python:
def f(x,y): return x*x + y*y
in Ruby:
def f(x,y) x*x + y*y end
In Scheme:
(define (f x y) (+ (* x x) (* y y)))
Finally, the Haskell way is:
f x y = x*x + y*y
Very clean. No parenthesis, no def
.
Don’t forget, Haskell uses functions and types a lot. It is thus very easy to define them. The syntax was particularly well thought for these objects.
A Type Example
The usual way is to declare the type of your function. This is not mandatory. The compiler is smart enough to discover it for you.
Let’s play a little.
-- We declare the type using ::
f :: Int -> Int -> Int
f x y = x*x + y*y
main = print (f 2 3)
~ runhaskell 20_very_basic.lhs
13
01_basic/10_Introduction/20_very_basic.lhs
01_basic/10_Introduction/21_very_basic.lhs
Now try
f :: Int -> Int -> Int
f x y = x*x + y*y
main = print (f 2.3 4.2)
You get this error:
21_very_basic.lhs:6:23:
No instance for (Fractional Int)
arising from the literal `4.2'
Possible fix: add an instance declaration for (Fractional Int)
In the second argument of `f', namely `4.2'
In the first argument of `print', namely `(f 2.3 4.2)'
In the expression: print (f 2.3 4.2)
The problem: 4.2
isn’t an Int.
01_basic/10_Introduction/21_very_basic.lhs
01_basic/10_Introduction/22_very_basic.lhs
The solution,
don’t declare the type for f
.
Haskell will infer the most general type for us:
f x y = x*x + y*y
main = print (f 2.3 4.2)
It works!
Great, we don’t have to declare a new function for every single type.
For example, in C
, you’ll have to declare a function for int
, for float
, for long
, for double
, etc…
But, what type should we declare? To discover the type Haskell has found for us, just launch ghci:
% ghciGHCi, version 7.0.4: http://www.haskell.org/ghc/ :? for help Loading package ghc-prim ... linking ... done. Loading package integer-gmp ... linking ... done. Loading package base ... linking ... done. Loading package ffi-1.0 ... linking ... done. Prelude>
let f x y = x*x + y*yPrelude>
:type ff :: Num a => a -> a -> a
Uh? What is this strange type?
Num a => a -> a -> a
First, let’s focus on the right part a -> a -> a
.
To understand it, just look at a list of progressive examples:
The written type | Its meaning |
Int |
the type Int |
Int -> Int |
the type function from Int to Int |
Float -> Int |
the type function from Float to Int |
a -> Int |
the type function from any type to Int |
a -> a |
the type function from any type a to the same type a |
a -> a -> a |
the type function of two arguments of any type a to the same type a |
In the type a -> a -> a
, the letter a
is a type variable.
It means f
is a function with two arguments and both arguments and the result have the same type.
The type variable a
could take many different type value.
For example Int
, Integer
, Float
…
So instead of having a forced type like in C
with declaring the function for int
, long
, float
, double
, etc…
We declare only one function like in a dynamically typed language.
Generally a
can be any type.
For example a String
, an Int
, but also more complex types, like Trees
, other functions, etc…
But here our type is prefixed with Num a =>
.
Num
is a type class.
A type class can be understood as a set of types.
Num
contains only types which behave like numbers.
More precisely, Num
is class containing types who implement a specific list of functions, and in particular (+)
and (*)
.
Type classes are a very powerful language construct. We can do some incredibly powerful stuff with this. More on this later.
Finally, Num a => a -> a -> a
means:
Let a
be a type belonging to the Num
type class.
This is a function from type a
to (a -> a
).
Yes, strange. In fact, in Haskell no function really has two arguments. Instead all functions have only one argument. But we will note that taking two arguments is equivalent to taking one argument and returning a function taking the second argument as parameter.
More precisely f 3 4
is equivalent to (f 3) 4
.
Note f 3
is a function:
f :: Num a :: a -> a -> a
g :: Num a :: a -> a
g = f 3
g y ⇔ 3*3 + y*y
Another notation exists for functions. The lambda notation allows us to create functions without assigning them a name. We call them anonymous function. We could have written:
g = \y -> 3*3 + y*y
The \
is used because it looks like λ
and is ASCII.
If you are not used to functional programming your brain should start to heat up. It is time to make a real application.
01_basic/10_Introduction/22_very_basic.lhs
01_basic/10_Introduction/23_very_basic.lhs
But just before that, we should verify the type system works as expected:
f :: Num a => a -> a -> a
f x y = x*x + y*y
main = print (f 3 2.4)
It works, because, 3
is a valid representation both for Fractional numbers like Float and for Integer.
As 2.4
is a Fractional number, 3
is then interpreted as being also a Fractional number.
01_basic/10_Introduction/23_very_basic.lhs
01_basic/10_Introduction/24_very_basic.lhs
If we force our function to work with different types, it will fail:
f :: Num a => a -> a -> a
f x y = x*x + y*y
x :: Int
x = 3
y :: Float
y = 2.4
main = print (f x y) -- won't work because type x ≠ type y
The compiler complains. The two parameters must have the same type.
If you believe it is a bad idea, and the compiler should make the transformation from a type to another for you, you should really watch this great (and funny) video: WAT
01_basic/10_Introduction/24_very_basic.lhs
Essential Haskell
I suggest you to skim this part. Think of it like a reference. Haskell has a lot of features. Many informations are missing here. Get back here if notation feels strange.
I use the ⇔
symbol to state that two expression are equivalent.
It is a meta notation, ⇔
does not exists in Haskell.
I will also use ⇒
to show what is the return of an expression.
Notations
Arithmetic
3 + 2 * 6 / 3 ⇔ 3 + ((2*6)/3)
Logic
True || False ⇒ True
True && False ⇒ False
True == False ⇒ False
True /= False ⇒ True (/=) is the operator for different
Powers
x^n for n an integral (understand Int or Integer)
x**y for y any kind of number (Float for example)
Integer
have no limit except the capacity of your machine:
4^103
102844034832575377634685573909834406561420991602098741459288064
Yeah!
And also rational numbers FTW!
But you need to import the module Data.Ratio
:
$ ghci
....
Prelude> :m Data.Ratio
Data.Ratio> (11 % 15) * (5 % 3)
11 % 9
Lists
[] ⇔ empty list
[1,2,3] ⇔ List of integral
["foo","bar","baz"] ⇔ List of String
1:[2,3] ⇔ [1,2,3], (:) prepend one element
1:2:[] ⇔ [1,2]
[1,2] ++ [3,4] ⇔ [1,2,3,4], (++) concatenate
[1,2,3] ++ ["foo"] ⇔ ERROR String ≠ Integral
[1..4] ⇔ [1,2,3,4]
[1,3..10] ⇔ [1,3,5,7,9]
[2,3,5,7,11..100] ⇔ ERROR! I am not so smart!
[10,9..1] ⇔ [10,9,8,7,6,5,4,3,2,1]
Strings
In Haskell strings are list of Char
.
'a' :: Char
"a" :: [Char]
"" ⇔ []
"ab" ⇔ ['a','b'] ⇔ 'a':"b" ⇔ 'a':['b'] ⇔ 'a':'b':[]
"abc" ⇔ "ab"++"c"
Remark: In real code you shouldn’t use list of char to represent text. You should mostly use
Data.Text
instead. If you want to represent stream of ASCII char, you should useData.ByteString
.
Tuples
The type of couple is (a,b)
.
Elements in a tuple can have different type.
-- All these tuple are valid
(2,"foo")
(3,'a',[2,3])
((2,"a"),"c",3)
fst (x,y) ⇒ x
snd (x,y) ⇒ y
fst (x,y,z) ⇒ ERROR: fst :: (a,b) -> a
snd (x,y,z) ⇒ ERROR: snd :: (a,b) -> b
Deal with parentheses
To remove some parentheses you can use two functions: ($)
and (.)
.
-- By default:
f g h x ⇔ (((f g) h) x)
-- the $ replace parenthesis from the $
-- to the end of the expression
f g $ h x ⇔ f g (h x) ⇔ (f g) (h x)
f $ g h x ⇔ f (g h x) ⇔ f ((g h) x)
f $ g $ h x ⇔ f (g (h x))
-- (.) the composition function
(f . g) x ⇔ f (g x)
(f . g . h) x ⇔ f (g (h x))
01_basic/20_Essential_Haskell/10a_Functions.lhs
Useful notations for functions
Just a reminder:
x :: Int ⇔ x is of type Int
x :: a ⇔ x can be of any type
x :: Num a => a ⇔ x can be any type a
such that a belongs to Num type class
f :: a -> b ⇔ f is a function from a to b
f :: a -> b -> c ⇔ f is a function from a to (b→c)
f :: (a -> b) -> c ⇔ f is a function from (a→b) to c
Defining the type of a function before its declaration isn’t mandatory. Haskell infers the most general type for you. But it is considered a good practice to do so.
Infix notation
square :: Num a => a -> a
square x = x^2
Note ^
use infix notation.
For each infix operator there its associated prefix notation.
You just have to put it inside parenthesis.
square' x = (^) x 2
square'' x = (^2) x
We can remove x
in the left and right side!
It’s called η-reduction.
square''' = (^2)
Note we can declare function with '
in their name.
Here:
square
⇔square'
⇔square''
⇔square '''
Tests
An implementation of the absolute function.
absolute :: (Ord a, Num a) => a -> a
absolute x = if x >= 0 then x else -x
Note: the if .. then .. else
Haskell notation is more like the
¤?¤:¤
C operator. You cannot forget the else
.
Another equivalent version:
absolute' x
| x >= 0 = x
| otherwise = -x
Notation warning: indentation is important in Haskell. Like in Python, a bad indentation could break your code!
01_basic/20_Essential_Haskell/10a_Functions.lhs
Hard Part
The hard part can now begin.
Functional style
In this section, I will give a short example of the impressive refactoring ability provided by Haskell. We will select a problem and solve it using a standard imperative way. Then I will make the code evolve. The end result will be both more elegant and easier to adapt.
Let’s solve the following problem:
Given a list of integers, return the sum of the even numbers in the list.
example:
[1,2,3,4,5] ⇒ 2 + 4 ⇒ 6
To show differences between the functional and imperative approach, I’ll start by providing an imperative solution (in Javascript):
function evenSum(list) { var result = 0; for (var i=0; i< list.length ; i++) { if (list[i] % 2 ==0) { result += list[i]; } } return result; }
But, in Haskell we don’t have variables, nor for loop. One solution to achieve the same result without loops is to use recursion.
Remark:
Recursion is generally perceived as slow in imperative languages. But it is generally not the case in functional programming. Most of the time Haskell will handle recursive functions efficiently.
Here is a C
version of the recursive function.
Note that for simplicity, I assume the int list ends with the first 0
value.
int evenSum(int *list) { return accumSum(0,list); } int accumSum(int n, int *list) { int x; int *xs; if (*list == 0) { // if the list is empty return n; } else { x = list[0]; // let x be the first element of the list xs = list+1; // let xs be the list without x if ( 0 == (x%2) ) { // if x is even return accumSum(n+x, xs); } else { return accumSum(n, xs); } } }
Keep this code in mind. We will translate it into Haskell. But before, I need to introduce three simple but useful functions we will use:
even :: Integral a => a -> Bool head :: [a] -> a tail :: [a] -> [a]
even
verifies if a number is even.
even :: Integral a => a -> Bool even 3 ⇒ False even 2 ⇒ True
head
returns the first element of a list:
head :: [a] -> a head [1,2,3] ⇒ 1 head [] ⇒ ERROR
tail
returns all elements of a list, except the first:
tail :: [a] -> [a] tail [1,2,3] ⇒ [2,3] tail [3] ⇒ [] tail [] ⇒ ERROR
Note that for any non empty list l
,
l ⇔ (head l):(tail l)
The first Haskell solution.
The function evenSum
returns the sum of all even numbers in a list:
-- Version 1
evenSum :: [Integer] -> Integer
evenSum l = accumSum 0 l
accumSum n l = if l == []
then n
else let x = head l
xs = tail l
in if even x
then accumSum (n+x) xs
else accumSum n xs
To test a function you can use ghci
:
% ghci GHCi, version 7.0.3: http://www.haskell.org/ghc/ :? for help Loading package ghc-prim ... linking ... done. Loading package integer-gmp ... linking ... done. Loading package base ... linking ... done. Prelude> :load 11_Functions.lhs [1 of 1] Compiling Main ( 11_Functions.lhs, interpreted ) Ok, modules loaded: Main. *Main> evenSum [1..5] 6
Here is an example of execution2:
*Main> evenSum [1..5] accumSum 0 [1,2,3,4,5] 1 is odd accumSum 0 [2,3,4,5] 2 is even accumSum (0+2) [3,4,5] 3 is odd accumSum (0+2) [4,5] 4 is even accumSum (0+2+4) [5] 5 is odd accumSum (0+2+4) [] l == [] 0+2+4 0+6 6
Coming from an imperative language all should seem right. In reality many things can be improved. First, we can generalize the type.
evenSum :: Integral a => [a] -> a
Next, we can use sub functions using where
or let
.
This way our accumSum
function won’t pollute the global namespace.
-- Version 2
evenSum :: Integral a => [a] -> a
evenSum l = accumSum 0 l
where accumSum n l =
if l == []
then n
else let x = head l
xs = tail l
in if even x
then accumSum (n+x) xs
else accumSum n xs
Next, we can use pattern matching.
-- Version 3
evenSum l = accumSum 0 l
where
accumSum n [] = n
accumSum n (x:xs) =
if even x
then accumSum (n+x) xs
else accumSum n xs
What is pattern matching? Use values instead of general parameter names3.
Instead of saying: foo l = if l == [] then <x> else <y>
You simply state:
foo [] = <x> foo l = <y>
But pattern matching goes even further. It is also able to inspect the inner data of a complex value. 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
with
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.
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
Higher Order Functions
To make things even better we should use higher order functions. What are these beasts? Higher order functions are functions taking functions 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 a loop.
We will use the foldl
function to accumulate a value.
The function foldl
captures 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 pushes it 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
becomes:
-- 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 note that
(\x y -> x+y) ⇔ (+)
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 study it 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 corresponds 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 some parts 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”4.
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 anything inside the function definition. It feels more modular. But in addition you can think more mathematically about your function. You can then use your function as any other one. You can 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 lists 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.
This example should show you how great pure functional programming is. Unfortunately, using pure functional programming isn’t well suited to all usages. Or at least such a language hasn’t been found yet.
One of the great powers of Haskell is the ability to create DSLs (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 has been done to explain to you how much functional approach is superior. Then when you start 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.
Types
tl;dr:
type Name = AnotherType
is just an alias and the compiler doesn’t do any difference betweenName
andAnotherType
.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 greatly to avoid mistakes. 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 wrong parameter at the wrong place for example.
Type inference
Static typing is generally essential to reach fast execution time. But most statically typed languages are bad at generalizing concepts. Haskell’s saving grace is that it can infer 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
with 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 amount of code necessary in C:
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:
#include <iostream> #include <complex> using namespace std; template<typename T> T square(T x) { return x*x; } int main() { // int int sqr_of_five = square(5); cout << sqr_of_five << endl; // double cout << (double)square(5.3) << endl; // complex cout << square( complex<double>(5,3) ) << endl; return 0; }
C++ does a far better job than C. For more complex function the syntax can be hard to follow: look at this article for example.
In C++ you must declare that a function can work with different types. In Haskell this is the opposite. The function will be as general as possible by default.
Type inference gives Haskell the feeling of freedom that dynamically typed languages provide. But unlike dynamically typed languages, most errors are caught before the execution. Generally, in Haskell:
“if it compiles it certainly does what you intended”
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
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
Recursive type
You already encountered a recursive type: 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 an 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 functions 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 can use your own show
function.
convertList [] = Nil
convertList (x:xs) = x ::: convertList xs
main = do
print (0 ::: 1 ::: Nil)
print (convertList [0,1])
This prints:
0 ::: (1 ::: Nil)
0 ::: (1 ::: Nil)
Trees
We’ll just give another standard example: binary trees.
import Data.List
data BinTree a = Empty
| Node a (BinTree a) (BinTree a)
deriving (Show)
We will also create a function which turns a list into an ordered binary tree.
treeFromList :: (Ord a) => [a] -> BinTree a
treeFromList [] = Empty
treeFromList (x:xs) = Node x (treeFromList (filter (<x) xs))
(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 a tree where:- The root is
x
- Its left subtree is the tree created from members of the list
xs
which are strictly inferior tox
and - the right subtree is the tree created from members of the list
xs
which are strictly superior tox
.
- The root is
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.
Just for fun, let’s code a better display for our trees. I simply had fun making a nice function to display trees in a general way. You can safely skip this part if you find it too difficult to follow.
We have a few changes to make.
We remove the deriving (Show)
from the declaration of our BinTree
type.
And it might also 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 of how to show a binary tree. Don’t worry about the apparent complexity. I made a lot of improvements in order to display even stranger 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
-- shows a tree and starts each line with pref
-- We don't display the 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 children non empty
treeshow pref (Node x left right) =
(pshow pref x) ++ "\n" ++
(showSon pref "|--" "| " left) ++ "\n" ++
(showSon pref "`--" " " right)
-- shows a tree using some prefixes to make it nice
showSon pref before next t =
pref ++ before ++ treeshow (pref ++ next) t
-- pshow replaces "\n" by "\n"++pref
pshow pref x = replace '\n' ("\n"++pref) (show x)
-- replaces one 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 remains identical.
treeFromList :: (Ord a) => [a] -> BinTree a
treeFromList [] = Empty
treeFromList (x:xs) = Node x (treeFromList (filter (<x) xs))
(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 the line with the <
character.
And each following line starts with 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 chose to prefix each line of tree display by :
(except for the root).
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'
Notice 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 trees containing not only integers, strings and chars, but also other trees. And we can even make a tree containing a tree of trees!
02_Hard_Part/40_Infinites_Structures.lhs
Infinite Structures
It is often stated that Haskell is lazy.
In fact, if you are a bit pedantic, you should state that Haskell is 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 functions will work with them.
Also, there is a built-in function take
which is equivalent to our take'
.
02_Hard_Part/40_Infinites_Structures.lhs
02_Hard_Part/41_Infinites_Structures.lhs
Suppose we don’t mind having an ordered binary tree. Here is an infinite binary tree:
nullTree = Node 0 nullTree nullTree
A complete binary tree where 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 compiles, runs and stops giving the following result:
< 0
: |-- 0
: | |-- 0
: | | |-- 0
: | | `-- 0
: | `-- 0
: | |-- 0
: | `-- 0
: `-- 0
: |-- 0
: | |-- 0
: | `-- 0
: `-- 0
: |-- 0
: `-- 0
Just to heat up 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 a 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 by the generalization of map
to other data structures,
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
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
tl;dr:
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 doaction2 (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
5,
but if something goes wrong the returned value is Nothing
.
If the value is right, it returns Just <the value>
.
Don’t try to understand too much of this function.
I use a lower level function than read
; reads
.
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 anIO
»
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 typeIO 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 typea
.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
Here is a tl;dr: 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
andw3
) which must be passed to the next action.We create a function
bind
or(>>=)
. Withbind
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 pure6.
Haskell consider the state of the world is an input variable for main
.
But the real type of main is closer to this one7:
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
thenb
thenf a b
- first eval
b
thena
thenf a b
. - eval
a
andb
in parallel thenf 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
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
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 beentypeclass
. 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 exampleIO a
, but alsoMaybe 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
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 monads8!
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 we saw some simple construction. Unfortunately we removed two properties of our tree:
- no duplicate node value
- 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.
Our first step is to create some pseudo-random number list:
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))
(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 thattreeTakeDepth 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
In order to resolve these problem we will modify slightly our
treeFromList
and shuffle
function.
A first problem, is the lack of infinite different number in our implementation of shuffle
.
We generated only 4331
different numbers.
To resolve this we make a slightly better shuffle
function.
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 (<x) xs
is empty.
Then to resolve this problem, I’ll authorize some error in the creation of our binary tree.
This new version of code can create binary tree which don’t have the following property for some of its nodes:
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
right = 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
andnbTry
, it seems to work nicely. But in the worst case, it can be exponential. Create a worst case list to give as parameter totreeFromList
.
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 usesafefilter'
directly as if was not in the where of safefilter) find a definition off
such that with probability1
, 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) right = treeFromList' (safefilter' (>x) xs (f n) f = ???
04_Appendice/01_More_on_infinite_trees/11_Infinite_Trees.lhs
-
Even if most recent languages try to hide them, they are present.↩
-
I know I’m cheating. But I will talk about non-strict later.↩
-
For the brave, a more complete explanation of pattern matching can be found here.↩
-
You should remark
squareEvenSum''
is more efficient that the two other versions. The order of(.)
is important.↩ -
Which itself is very similar to the javascript
eval
on a string containing JSON).↩ -
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.↩
-
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.↩ -
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.↩