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by Joel Kaasinen (Nitor) and John Lång (University of Helsinki)
This lecture goes over the basic parts of Haskell introduced in part 1 of the course: types, values, pattern matching, functions and recursion.
Remember the primitive types of Haskell? Here they are:
Values | Type | Meaning |
---|---|---|
True , False |
Bool |
Truth values |
0 , 1 , 20 , -37 , … |
Int |
Whole numbers |
'A' , 'a' , '!' , … |
Char |
Characters |
"" , "abcd" , … |
String |
Strings, which are actually just lists of characters, [Char] |
0.0 , -3.2 , 12.3 , … |
Double |
Floating-point numbers |
() |
() |
The so-called unit type with only one value |
It’s possible to combine these primitive types in various ways to form more complex types. Function types, tuple types and list types are examples of types that combine other types.
Values | Type | Meaning |
---|---|---|
(1,2) , (True,'c') , … |
(a, b) |
A pair of a value of type a and a value of type b |
(1,2,3) , (1,2,'c') , … |
(a, b, c) |
A triple of values (of types a , b and c ) |
[] , [1,2,3,4] , … |
[a] |
List of values of type a |
not , reverse , \x -> 1 , \x -> x , … |
a -> b |
Function from type a to type b |
There’s one more powerful mechanism for creating more types: Algebraic datatypes (ADTs). Some examples include:
-- Enumeration types
data Bool = True | False
data Color = Red | Green | Blue
-- Record types that contain fields
data Vector2d = MakeVector Double Double
data Person = Person Int String
-- Parameterized types. Note the type parameter `a`
data PairOf a = TwoValues a a
-- Recursive types
data IntList = Empty | Node Int IntList
-- Complex types which combine many of these features
data Maybe a = Nothing | Just a
data Either a b = Left a | Right b
data List a = Nil | Cons a (List a) -- This is equivalent to the built-in [a] type
data Tree a = Leaf a | Node a (Tree a) (Tree a)
data MultiTree a = MultiTree a [MultiTree a] -- Note the list
Values of these types include:
Values | Type |
---|---|
True , False |
Bool |
Red , Green , Blue |
Color |
MakeVector 1.5 3.2 |
Vector2d |
Person 13 "Bob" |
Person |
TwoValues 1 3 |
PairOf Int |
Empty , Node 3 (Node 4 Empty) |
IntList |
Nothing , Just 3 , Just 4 , … |
Maybe Int |
Nothing , Just 'c' , Just 'd' , … |
Maybe Char |
Left "foo" , Right 13 , … |
Either String Int |
Nil , Cons True Nil , Cons True (Cons False Nil) … |
List Bool |
Leaf 7 , Node 1 (Leaf 0) (Leaf 2) , … |
Tree Int |
MultiTree 'a' [MultiTree 'b' [], MultiTree 'c' []]] , … |
MultiTree Char |
You can combine parameterized types in complex ways, for example with something like Either [String->String] (Maybe String, Int)
.
The names of concrete types start with capital letters. Lowercase letters are used for type variables which indicate parametric polymorphism: functions and values that can have multiple types. Here are some examples of polymorphic function types:
-> [a] -- function from list of any type, to list of the same type
[a] -> a -- function from list of any type, to the element type
[a] -> [a] -- function from tuple to list (a,b)
List literals can be written in using the familiar [x,y,z]
syntax. However, that notation is just a shorthand as lists are actually built up of the list constructors []
and (:)
. These constructors are also used when pattern matching lists. Here are some examples of lists:
Abbreviation | Full list | Type |
---|---|---|
[1,2,3] |
1:2:3:[] |
[Int] |
[[1],[2],[3]] |
(1:[]):(2:[]):(3:[]):[] |
[[Int]] |
"foo" |
'f':'o':'o':[] |
[Char] , also known as String |
There’s also a range syntax for lists:
Range | Result |
---|---|
['a' .. 'z'] |
"abcdefghijklmnopqrstuvwxyz" |
[0 .. 9] |
[0,1,2,3,4,5,6,7,8,9] |
[0, 5 .. 25] |
[0,5,10,15,20,25] |
[x .. y] |
everything from x to y |
[9 .. 3] |
[] |
[y, y-1 .. x] |
everything from y to x in decreasing order |
[9,8 .. 3] |
[9,8,7,6,5,4,3] |
List comprehensions are another powerful way to create lists:
Comprehension | Result |
---|---|
[x^3 | x <- [1..3]] |
[1,8,27] |
[x^2 + y^2 | x <- [1..3], y <- [1..2]] |
[2,5,5,8,10,13] |
[y | x <- [1..10], let y = x^2, even x, y<50] |
[4,16,36] |
[c | c <- "Hello, World!", elem c ['a'..'z']] |
"elloorld" |
In general, [f x | x <- xs, p x]
is the same as map f (filter p xs)
. Also, [y | x <- xs, let y = f x]
is the same as [f x | x <- xs]
. Any combination of <-
, let
and [f x | ...]
is possible.
Just one more note on the syntax. Recall that (:)
associates to the right, e.g. True:False:[]
is the same as True:(False:[])
. (In fact, (True:False):[]
is not even a list, because True:False
attempts to add True
in front of False
which is not a list.)
The basic form of function definition is:
functionName :: argumentType -> returnType
= returnValue functionName argument
For example:
repeatString :: String -> String
= s ++ s repeatString s
Functions taking multiple arguments are defined in a similar manner. Note how the type of a multi-argument function looks like.
surroundString :: String -> String -> String
= around ++ s ++ around surroundString around s
Functions can be polymorphic, can take multiple arguments, and can even take functions as arguments. Here are more examples:
id :: a -> a
id x = x
const :: a -> b -> a
const x y = x
flip :: (a -> b -> c) -> b -> a -> c
flip f x y = f y x
More sophisticated functions can be defined using pattern matching. We can pattern match on the constructors of algebraic datatypes like Maybe
, and also lists with the constructors []
and (:)
.
swap :: (a,b) -> (b,a)
= (y,x)
swap (x,y)
maybe :: b -> (a -> b) -> Maybe a -> b
maybe def _ Nothing = def
maybe _ f (Just x) = Just (f x)
safeHead :: [a] -> Maybe a
= Nothing
safeHead [] :_) = Just x safeHead (x
Yet more sophistication can be achieved with guards. Guards let you define a function case-by-case based on tests of type Bool
. Guards are useful in situations where pattern matching can’t be used. Of course, guards can also be combined with pattern matching:
myAbs :: Int -> Int
myAbs x| x < 0 = -x
| otherwise = x
safeDiv :: Double -> Double -> Maybe Double
safeDiv x y| y == 0 = Nothing
| otherwise = Just (x / y)
buy :: String -> Double -> String
"Banana" money
buy | money < 3.2 = "You don't have enough money for a banana"
| otherwise = "You bought a banana"
product _ = "No such product: " ++ product buy
Case expressions let us pattern match inside functions. They are useful in situations where the result of one function depends on the result of another and we want to match the pattern on the output of the other function:
divDefault :: Double -> Double -> Double -> Double
= case safeDiv x y of
divDefault x y def Nothing -> def
Just w -> w
Let-expressions enable local definitions. Where-clauses work similarly to let
s. For example:
circleArea :: Double -> Double
= let pi = 3.1415926
circleArea r = x * x
square x in pi * square r
circleArea' :: Double -> Double
= pi * square r
circleArea' r where pi = 3.1415926
= x * x square x
Lambda expressions are another occasionally useful syntax for defining functions. Lambda expressions represent anonymous (unnamed) functions. They can be used for defining local functions that are typically used only once.
incrementAll :: [Int] -> [Int]
= map (\x -> x + 1) xs incrementAll xs
Note that f x = y
is the same thing as f = \x -> y
.
Finally, binary operators have sections. Sections are partially applied operators. The section of an operator is obtained by writing the operator and one of its arguments in parentheses. For example, (*2)
multiplies its argument by 2
from the right, e.g. (*2) 5 ==> 5 * 2
. A fractional number (e.g. a Double
) can be inverted with the section (1/)
, e.g. (1/) 2 ==> 0.5
.
incrementAll' :: [Int] -> [Int]
= map (+1) xs incrementAll' xs
Haskell is a functional programming language, which means that functions can be passed in as arguments and returned from functions. As a programming paradigm, functional programming aims to build programs by combining simple functions together to form larger and larger ones.
The most often presented example of functional programming is functional list manipulation using higher-order functions (functions that take functions as arguments) like map
and filter
. Here’s one example from part 1:
-- a predicate that checks if a string is a palindrome
palindrome :: String -> Bool
= str == reverse str
palindrome str
-- palindromes n takes all numbers from 1 to n, converts them to
-- strings using show, and keeps only palindromes
palindromes :: Int -> [String]
= filter palindrome (map show [1..n]) palindromes n
150
palindromes ==> ["1","2","3","4","5","6","7","8","9",
"11","22","33","44","55","66","77","88","99",
"101","111","121","131","141"]
We also encountered other functional programming patterns in part 1, like partial application:
map (take 3) [[1,2,3,4,5],[6,7,8,9,0]]
==> [[1,2,3],[6,7,8]]
Also, function composition:
map reverse . filter (/="Smith")) ["Jones","Smith","White"]
(==> ["senoJ","etihW"]
map (negate . sum) [[1,2,3],[5]]
==> [-6,-5]
And finally, folds:
foldr (*) 1 [2,3,4] ==> 24
foldr max 0 [1,3,7] ==> 7
foldr (++) "" ["abc","de","f"] ==> "abcdef"
To implement a function that uses repetition in Haskell, you need recursion. Haskell has no loops like other programming languages. Here are some simple recursive functions in Haskell:
repeatString :: Int -> String -> String
0 s = ""
repeatString = s ++ repeatString (n-1) s
repeatString n s
times :: Int -> Int -> Int
0 n = 0
times 1 n = n
times = n + times (m - 1) n
times m n
safeLast :: [a] -> Maybe a
= Nothing
safeLast [] = Just x
safeLast [x] :xs) = safeLast xs safeLast (x
To consume or produce a list you often need recursion. Here are the implementations of map
and filter
as examples of recursive list processing:
map :: (a -> b) -> [a] -> [b]
map _ [] = []
map f (x:xs) = f x : map f xs
filter :: (a -> Bool) -> [a] -> [a]
filter _ [] = []
filter pred (x:xs)
| pred x = x : filter pred xs
| otherwise = filter pred xs
Sometimes, a recursive helper function is needed in case you need to keep track of multiple pieces of data.
sumNumbers :: [Int] -> Int
= go 0 xs
sumNumbers xs where go sum [] = sum
sum (x:xs) = go (sum+x) xs go
Here’s a final example utilizing guards, pattern matching, a helper function and recursion:
-- split a string into pieces at the given character
mySplit :: Char -> String -> [String]
= helper [] xs
mySplit c xs where helper piece [] = [piece]
:ys)
helper piece (y| c == y = piece : helper [] ys
| otherwise = helper (piece++[y]) ys
'-' "a-bcd-ef" ==> ["a","bcd","ef"] mySplit
The following functions are parametrically polymorphic:
id :: a -> a
id x = x
head :: [a] -> a
head (x:_) = x
fst :: (a,b) -> a
fst (x,y) = x
Parametrically polymorphic functions always work the same way, no matter what types we’re working with. This means that we can’t define a special implementation of id
just for the Int
type, or fst
for the type (Bool, String)
.
By contrast, ad hoc polymorphism allows different types to have different implementations of the same function. Ad hoc polymorphism in Haskell can be achieved by defining a type class and then declaring instances of that type class for various types. Ad hoc polymorphism is a handy way for expressing a common set of operations, even when the implementation of the operations depends on the type they’re acting on.
Functions that use ad hoc polymorphism have a class constraint in their types. Here are some examples:
negate :: Num a => a -> a
(==) :: Eq a => a -> a -> Bool
sort :: Ord a => [a] -> [a]
A type like Num a => a -> a
means: for any type X
that’s a member of the Num
class, this function has type X -> X
. In other words, we can invoke negate
on any number type, but not on other types:
Prelude> negate 1
-1
Prelude> negate 1.0
-1.0
Prelude> negate True
<interactive>:3:1: error:
No instance for (Num Bool) arising from a use of ‘negate’ •
Here’s a summary of some useful type classes from the standard library.
Eq
is for equality comparison. It contains the ==
operatorOrd
is for order comparison. It contains the ordered comparison operators like <
and =>
, and functions like max
and min
.Num
is for all number types. It contains +
, -
, *
and negate
.Integral
is for whole number types. Most notably, it contains integer division div
.Fractional
is for number types that support division, /
Show
contains the function show :: Show a => a -> String
that converts values to stringsRead
contains the function read :: Read a => String -> a
that is the inverse of show
Sometimes, multiple class constraints are needed. For instance here:
sumTwoSmallest :: (Num a, Ord a) => [a] -> a
= let (a:b:_) = sort xs
sumTwoSmallest xs in a+b
Now that we’ve seen some classes and types, let’s look at the syntax of declaring classes and instances. Here are two class definitions:
class Sized a where
empty :: a -- a thing with size 0
size :: a -> Int
class Eq a where
(==) :: a -> a -> Bool
Consider the following data structures:
data Numbers = None | One Int | Two Int Int
data IntList = Nil | ListNode Int IntList
data Tree a = Leaf | Node a (Tree a) (Tree a)
All of these have sizes that we can count, but we need to perform the operation differently:
instance Sized Numbers where
= None
empty None = 0
size One _) = 1
size (Two _ _) = 2
size (
instance Sized IntList where
= Nil
empty Nil = 0
size ListNode _ list) = 1 + size list
size (
instance Sized (Tree a) where
= Leaf
empty Leaf = 0
size Node _ left right) = 1 + size left + size right size (
We can also easily declare Eq
instances for Numbers
and IntList
:
instance Eq Numbers where
None == None = True
One x) == (One y) = x==y
(Two x y) == (Two z w) = x==z && y==w
(== _ = False -- to handle cases like None == One 1
_
instance Eq IntList where
Nil == Nil = True
ListNode x xs) == (ListNode y ys) = x == y && xs == ys
(== _ = False _
However, since the Tree
datatype is parameterized over the element type a
, we need an Eq a
instance in order to have an Eq (Tree a)
instance. This is achieved by adding a class constraint to the instance declaration. This is called an instance hierarchy.
instance Eq a => Eq (Tree a) where
Leaf == Leaf = True
Node x l r) == (Node x' l' r') = x == x' && l == l' && r == r'
(== _ = False _
Some standard type classes, most notably Show
, Read
, Eq
and Ord
can be derived, that is, you can ask the compiler to generate automatic instances for you. For example we could have derived all of these classes for our earlier Numbers
example.
data Numbers = None | One Int | Two Int Int
deriving (Show, Read, Eq, Ord)
None == One 1 ==> False
Two 1 2 == Two 1 2 ==> True
None < Two 1 2 ==> True
Two 1 3 < Two 1 2 ==> False
show (Two 1 3) ==> "Two 1 3"
What is the type of ('c',not)
[Char]
[Bool]
(Char,Bool -> Bool)
(Char,Bool)
What is the type of ['c',not]
[Char]
[Bool]
(Char,Bool -> Bool)
(Char,Bool)
Which of these is a value of the following type?
data T = X Int | Y String String | Z T
X "foo"
Y "foo"
Z (X 1)
X (Z 1)
What is the type of this function?
:Just x:_) = x
f (_= False f _
Maybe a -> a
[Maybe a] -> a
[Maybe a] -> Bool
[Maybe Bool] -> Bool
What is the type of this function?
= x-y == 0 f x y
(Num a, Eq a) => a -> a -> Bool
Num a => a -> a -> Bool
Eq a => a -> a -> Bool
a -> a -> Bool
Which of the following types can x
have in order for x (&&) y
to not be a type error?
Bool
Bool -> Bool -> Bool
(Bool -> Bool -> Bool) -> Bool -> Bool
Here’s a short recap of how to work on the exercises. The system is the same as for part 1 of the course.
exercises/
directorystack build
to download dependenciesSet9a.hs
stack runhaskell Set9aTest.hs
Purity and laziness were mentioned as key features of Haskell in the beginning of part 1. Let’s take a closer look at them.
Haskell is a pure functional language. This means that the value f x y
is always the same for given x
and y
. In other words, the values of x
and y
uniquely determine the value of f x y
. This property is also called referential transparency.
Purity also means that there are no side effects: you can’t have the evaluation of f x y
read a line from the user - the line would be different on different invocations of f
and would affect the return value, breaking referential transparency! Obviously you need side effects to actually get something done. We’ll get back to how Haskell handles side effects later.
Haskell is a lazy language. This means that a value is not evaluated if it is not needed. An example illustrates this best. Consider these two functions:
= f x -- infinite recursion
f x = x g x y
Evaluating f 1
does not halt due to the infinite recursion. However, this works:
2 (f 1) ==> 2 g
Laziness is not a problem because Haskell is pure. Only the result of the function matters, not the side effects. So if the result of a function is not used, we can simply not evaluate it without changing the meaning (semantics) of a program. Well okay, sometimes we get a terminating program instead of one that goes on for ever, but adding laziness never makes a functioning Haskell program break.
If you’re interested in the theory behind this, check out the Church-Rosser theorem or the Haskell Wiki article Lazy vs. non-strict.
Referential transparency, the feature that an expression always returns the same value for the same inputs, is a very powerful property that we can leverage to reason about programs.
In a C-style language, we might write a procedure that may not always return the same value for the same arguments:
int c = 0;
int funny(int x) {
return x + c++;
}
The expression c++
increments the value of c
and returns the old value of c
. The next time it is evaluated, the value of c
has increased by one. This means that depending on the current value of c
, funny(0)
might return 0
, 1
, 2
, or any other integer value. (It might even return negative values if c
overflows!)
In some situations this kind of behaviour with side-effects may be useful, but there are also times when it is more important to be able to reason about the code easily. The advantage of pure functions is that they can be analyzed using basic mathematical techniques. Sometimes applying math to our functions can even reveal simplifications or optimisations we otherwise wouldn’t have thought about.
Consider the following expression:
map (+1) . reverse . map (-1)
This expression can be simplified to just reverse
. We begin by establishing some helpful facts (or lemmas). First, suppose that we know
map id === id
map f . map g === map (f.g)
reverse . map f === map f . reverse
The fourth fact that we’re going to need is the following:
(+1) . (-1) === id
We can prove fact 4 by reasoning about how (+1) . (-1)
behaves for an arbitrary input x
:
+1) . (-1)) x === ((+1) ((-1) x))
((=== ((+1) (x - 1))
=== (x - 1) + 1
=== x
=== id x
Because we didn’t assume anything about x
, we may conclude that the above chain of equations holds for every x
. Thus,
+1) . (-1) === id (
For those who are familiar with the technique of proof by induction, it is a fun exercise to prove the first three facts also. This course doesn’t discuss induction proofs, though, so don’t sweat if you don’t know induction.
Now, from facts 1-4 it follows that
map (+1) . reverse . map (-1)
=== map (+1) . (reverse . map (-1)) -- By associativity of (.)
=== map (+1) . (map (-1) . reverse) -- By fact 3
=== (map (+1) . map (-1)) . reverse -- By associativity of (.)
=== map ((+1) . (-1)) . reverse -- By fact 2
=== map id . reverse -- By fact 4
=== id . reverse -- By fact 1
=== reverse -- By the definition of id
This course won’t go into details about proving things about programs, but it’s good to know that pure functional programming is very compatible with analysis like this.
The benefits of laziness are best demonstrated with some examples involving infinite lists. Let’s start with repeat 1
, which generates an infinite list of 1
s. If we try to tell GHCi to print the value repeat 1
, it will just keep printing 1
s for ever until we interrupt it using Control-C:
Prelude> repeat 1
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
[^C
However, due to laziness, we can work with infinite lists and write computations that end. We just need to use a finite number of elements from the infinite list. Here are some examples:
Prelude> take 10 $ repeat 1
1,1,1,1,1,1,1,1,1,1]
[Prelude> take 20 $ repeat 1
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
[Prelude> repeat 1 !! 13337
1
An infinite list that just repeats one element can sometimes be necessary, but it’s kind of pointless. Let’s look at some more useful infinite lists next. You can use the [n..]
syntax to generate an infinite list of numbers, starting from n
:
Prelude> take 20 [0..]
0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]
[Prelude> take 10 . map (2^) $ [0..]
1,2,4,8,16,32,64,128,256,512] [
The function cycle
repeats elements from the given list over and over again. It can be useful when dealing with rotations or cycles.
Prelude> take 21 $ cycle "asdf"
"asdfasdfasdfasdfasdfa"
Prelude> take 4 . map (take 4) . tails $ cycle "asdf"
"asdf","sdfa","dfas","fasd"] [
As a more concrete example of how cycle
is useful, let’s look at computing the check digit of Finnish bank transfer transaction numbers (viitenumero). A transaction number consists of any number of digits, followed by a single check digit. The check digit is checked by multiplying the digits (from right to left) with the numbers 7, 3, 1, 7, 3, 1 and so on, and summing the results. If the result of the sum plus the check digit is divisible by 10, the number is valid.
Here’s a concrete example. 116127
is a valid transaction number. The computation goes like this:
digits: 1 1 6 1 2
* * * * *
multipliers: 3 7 1 3 7
3+ 7+ 6+ 3+14 = 33
check digit is 7, 33+7=40 is divisible by 10, valid
Here’s the Haskell code for a transaction number checker. Note how we use use the infinite list cycle [7,3,1]
for the multipliers.
viitenumeroCheck :: [Int] -> Bool
= mod (checksum+checkDigit) 10 == 0
viitenumeroCheck allDigits where (checkDigit:digits) = reverse allDigits
= cycle [7,3,1]
multipliers = sum $ zipWith (*) multipliers digits checksum
1,1,6,1,2,7] ==> True
viitenumeroCheck [1,1,6,1,2,8] ==> False viitenumeroCheck [
Finally, here’s how you would find the first power of 3 that’s larger than 100.
Prelude> head . filter (>100) $ map (3^) [0..]
243
Let’s go through how this works step by step. Note how map and filter are processing the list lazily, one element at a time, as needed. This is similar to how generators or iterators work in languages like Python or Java.
head (filter (>100) (map (3^) [0..]))
==> head (filter (>100) (map (3^) (0:[1..]))) -- evaluate first element of the lazy list
==> head (filter (>100) (1 : map (3^) [1..])) -- map processes the element
==> head (filter (>100) (map (3^) [1..])) -- filter drops the element
==> head (filter (>100) (map (3^) (1:[2..]))) -- evaluate second element of the lazy list
==> head (filter (>100) (3 : map (3^) [2..])) -- map processes the element
==> head (filter (>100) (map (3^) [2..])) -- filter drops the element
-- let's take bigger steps now
==> head (filter (>100) (9 : map (3^) [3..])) -- map processes, filter will drop
==> head (filter (>100) (27 : map (3^) [4..])) -- map processes, filter will drop
==> head (filter (>100) (81 : map (3^) [5..])) -- map processes, filter will drop
==> head (filter (>100) (243 : map (3^) [6..])) -- map processes
==> head (243 : filter (>100) (map (3^) [6..])) -- filter lets the value through
==> 243 -- head returns the result
Laziness will probably feel a bit magical to you right now. You might wonder how it can be implemented. Haskell evaluation is remarkably simple, it’s just different than what you might be used to. Let’s dig in.
In most other programming languages (like Java, C or Python), evaluation proceeds inside-out. Arguments to functions are evaluated before the function.
Haskell evaluation proceeds outside-in instead of inside-out. The definition of the outermost function in an expression is applied without evaluating any arguments. Here’s a concrete example with toy functions f
and g
:
g :: Int -> Int -> Int
= y+1
g x y f :: Int -> Int -> Int -> Int
= g (a*1000) c f a b c
Inside-out (normal) evaluation:
1 (1234*1234) 2
f -- evaluate arguments to f
==> f 1 1522756 2
-- evaluate f
==> g (1*1000) 2
-- evaluate arguments to g
==> g 1000 2
-- evaluate g
==> 2+1
==> 3
Haskell outside-in evaluation:
1 (1234*1234) 2
f -- evaluate f without evaluating arguments
==> g (1*1000) 2
-- evaluate g without evaluating arguments
==> 2+1
==> 3
Note how the unused calculations 1234*1234
and 1*1000
didn’t get evaluated. This is why laziness is often helpful.
Let’s look at a more involved example, with pattern matching and more complex data (lists). Pattern matching drives Haskell evaluation in a very concrete way, as we’ll see. Here are some functions that we’ll use. They’re familiar from the Prelude, but I’ll give them simple definitions.
not True = False
not False = True
map f [] = []
map f (x:xs) = f x : map f xs
length [] = 0
length (x:xs) = 1+length xs
Here’s the inside-out evaluation of an expression:
length (map not (True:False:[]))
==> length (not True : not False : []) -- evaluate call to map
==> length (False:True:[]) -- evaluate calls to not
==> 2
Here’s how the evaluation proceeds in Haskell. Note how it’s not strictly outside-in, since we sometimes need to evaluate inside arguments to be able to know which pattern is matched.
length (map not (True:False:[]))
-- We can't evaluate length since we don't know which equation of length applies,
-- so we look at length's argument. We can apply the second equation of map, so we do.
==> length (not True : map not (False:[]))
-- Now the argument of length has a (:) we can pattern match on, so we apply the
-- second equation of length
==> 1 + length (map not (False:[]))
-- The outermost function is now +, but it can't do anything unless both arguments
-- are numbers. So we need to evaluate length. In order to pick an equation, we need
-- to evaluate the argument of length again. We apply the second equation of map
==> 1 + length (not False : map not ([]))
-- Now we can apply the second equation of length again.
==> 1 + (1 + length (map not []))
-- The outermost + needs a number to be evaluated. The second + also needs a number.
-- We need to evaluate length again, which means we need to pick an equation for length,
-- which means we need to evaluate its argument. This time it is the first equation for
-- map that applies.
==> 1 + (1 + length [])
-- Now we can apply the first equation for length
==> 1 + (1 + 0)
-- The outermost + still can't be evaluated, but the inner one can
==> 1 + 1
-- Finally we evaluate the outer +
==> 2
Note that we didn’t need to evaluate any of the not
applications.
Let’s introduce some terminology. We say that pattern matching forces evaluation. When Haskell evaluates something, it evaluates it to something called weak head normal form (WHNF). WHNF basically means a value that can be pattern matched on. An expression is in WHNF if it can’t be evaluated on its top level. This means it either:
1
False
, Just (1+1)
, 0:filter f xs
(\x -> 1+x)
The most notable class of expressions that is not in WHNF is function applications. If an expression consists of a function (that is not a constructor), applied to some arguments, it is not in WHNF. We must evaluate it in order to get something pattern matchable.
In the previous example we couldn’t pick an equation for length
in length (map not (False:[]))
. The argument (map not ...)
is not in WHNF, so it can’t be pattern matched on. Thus we need to evaluate it. When we apply the second equation of map
, we get length (not False : map not [])
, and now the argument to length is in WHNF since there is a constructor, (:)
, at the top level. This is a bit more evident if we switch from infix to prefix notation and write the argument to length
as (:) (not False) (map not [])
.
In practice, pattern matching is not the only thing that forces evaluation. Primitives like (+)
also force their arguments.
Instead of forcing, some sources talk about strictness, we can say for instance that (+)
is strict in both arguments.
There’s one more thing about Haskell evaluation. Any time you give a value a name, it gets shared. This means that every occurrence of the name points at the same (potentially unevaluated) expression. When the expression gets evaluated, all occurrences of the name see the result.
Let’s look at a very simple example.
= x*x square x
Based on the previous sections, you might imagine evaluation works like the following. The evaluation is first represented textually, and then visually, as an expression tree.
2+2)
square (==> (2+2) * (2+2) -- definition of square
==> 4 * (2+2) -- (*) forces left argument
==> 4 * 4 -- (*) forces right argument
==> 16 -- definition of (*)
However, what really happens is that the expression 2+2
named by the variable x
is only computed once. The result of the evaluation is then shared between the two occurrences of x
inside square
. So here’s the correct evaluation, first textually, and then visually. Note how now instead of an expression tree, we have an expression graph. This is why Haskell evaluation is sometimes called graph reduction.
2+2)
square (==> (2+2) * (2+2)
==> 4 * 4
==> 16
As another example, consider the function f
below and its evaluation.
f :: Int -> Int
= if i>10 then 10 else i f i
_______shared________| |
1+1) ==> if (1+1)>10 then 10 else (1+1)
f (==> if 2>10 then 10 else 2
==> if False then 10 else 2
==> 2
Haskell does not compute 1+1
twice because it was named, and the name was used twice. We can contrast this with another function that takes two arguments:
g :: Int -> Int
= if i>10 then 10 else j g i j
______no sharing_____| |
1+1) (1+1) ==> if (1+1)>10 then 10 else (1+1)
g (==> if 2>10 then 10 else (1+1)
==> if False then 10 else (1+1)
==> (1+1)
==> 2
Here we have two different names for equivalent expressions, and Haskell doesn’t magically share them. Automatically sharing equivalent expressions is an optimization called Common Subexpression Elimination (CSE). You can learn a bit more CSE and Haskell here.
You can name things via
let ... in ...
where
Combined with laziness, sharing means that a name gets evaluated at most once.
You’ll find below a slightly contrived recursive definition of the function even
. It will illustrate the concepts of forcing and sharing.
not :: Bool -> Bool
not True = False
not False = True
(||) :: Bool -> Bool -> Bool
True || _ = True
|| x = x
_
even :: Int -> Bool
even x = x == 0 || not (even (x-1))
Firstly, note that ||
forces its left argument, but not its right argument. (In other words, ||
is strict in its left argument.) This is because we only need to evaluate the left argument of ||
in order to know which equation applies. This means by extension that even
forces its first argument:
||
forces x==0
x==0
forces x
Now let’s evaluate the expression even 2
to WHNF.
even 2
==> 2 == 0 || not (even (2-1)) -- apply definition of even
==> False || not (even (2-1)) -- || forces its first argument
==> not (even (2-1)) -- second equation of ||
==> not ((2-1) == 0 || not (even ((2-1)-1))) -- not forces its argument: apply definition of even
==> not ( 1 == 0 || not (even ( 1 -1))) -- note sharing!
==> not ( False || not (even (1-1)))
==> not (not (even (1-1)))
==> not (not ((1-1) == 0 || not (even ((1-1)-1))))
==> not (not ( 0 == 0 || not (even ( 0 -1)))) -- (sharing)
==> not (not ( True || not (even (0-1))))
==> not (not True)
==> not False
==> True
Note that with this alternate definition even
would not have worked. Can you tell why?
= not (even' (x-1)) || x == 0 even' x
Now we can really understand what’s going on in the infinite list example from earlier. Let’s use these definitions:
head (x:_) = x
head [] = -1
filter p [] = []
filter p (x:xs) = if p x
then x : filter p xs
else filter p xs
map f [] = []
map f (x:xs) = f x : map f xs
-- [0..] is syntax sugar for enumFrom 0
enumFrom n = n : enumFrom (n+1)
And here we go:
head (filter (>100) (map (3^) [0..]))
=== head (filter (>100) (map (3^) (enumFrom 0)))
-- head forces filter, which forces map, which forces enumFrom. We apply the definition of enumFrom.
==> head (filter (>100) (map (3^) (0:[1..])))
-- head forces filter, which forces map. We apply the second equation of map.
==> head (filter (>100) ((3^0) : map (3^) [1..]))
-- head forces filter. We apply the second equation of filter
==> head (if ((3^0)>100)
then (3^0) : filter (>100) (map (3^) [1..])
else filter (>100) (map (3^) [1..]))
-- head forces if, if forces >, > forces ^. Note sharing!
==> head (if (1>100)
then 1 : filter (>100) (map (3^) [1..])
else filter (>100) (map (3^) [1..]))
-- head forces if, if forces >
==> head (if False
then 1 : filter (>100) (map (3^) [1..])
else filter (>100) (map (3^) [1..]))
-- apply definition of if
==> head (filter (>100) (map (3^) [1..]))
-- let's take slightly bigger steps now
==> head (filter (>100) (map (3^) (1:[2..])))
==> head (filter (>100) ((3^1) : map (3^) [2..]))
==> head (filter (>100) (3 : map (3^) [2..]))
==> head (filter (>100) (map (3^) [2..]))
-- and even bigger steps now
==> head (filter (>100) (9 : map (3^) [3..]))
==> head (filter (>100) (27 : map (3^) [4..]))
==> head (filter (>100) (81 : map (3^) [5..]))
==> head (filter (>100) (243 : map (3^) [6..]))
==> head (243 : filter (>100) (map (3^) [6..]))
==> 243
Whew.
Functions that work with lists often have the best performance when they’re written in such a way that they utilize laziness. One way to try to accomplish this is to write list-handling functions that work well with infinite lists.
To write a function that transforms an infinite list, you need to write a function that only looks at a limited prefix of the input list, then outputs a (:)
constructor, and then recurses. Here’s a first example.
everySecond :: [a] -> [a]
= []
everySecond [] :y:xs) = x : everySecond xs everySecond (x
take 10 (everySecond [0..]) ==> [0,2,4,6,8,10,12,14,16,18]
A good heuristic for writing functions that work well with infinite lists is: can the head
of the result be evaluated cheaply? Here are two examples of functions that don’t work with infinite inputs. In the case of mapTailRecursive
, the problem is that it needs to process the whole input before being in WHNF. In the case of myDrop
, the problem is that it uses the function length
, doesn’t work for infinite lists since it tries to iterate until the end of the list.
map :: (a -> b) -> [a] -> [b]
map _ [] = []
map f (x:xs) = f x : map f xs
mapTailRecursive :: (a -> b) -> [a] -> [b]
= go xs []
mapTailRecursive f xs where go (x:xs) res = go xs (res++[f x])
= res go [] res
head (map inc [0..]) ==> head (inc 0 : map inc [1..]) ==> inc 0 ==> 1
head (mapTailRecursive inc [0..])
==> head (go [0..] [])
==> head (go [1..] ([]++[inc 0]))
==> head (go [2..] ([]++[inc 0]++[inc 1]))
==> head (go [3..] ([]++[inc 0]++[inc 1]++[inc 2]))
-- never terminates
drop :: Int -> [a] -> [a]
drop 0 xs = xs
drop _ [] = []
drop n (x:xs) = drop (n-1) xs
myDrop :: Int -> [a] -> [a]
0 xs = xs
myDrop = if n > length xs then [] else myDrop (n-1) (tail xs) myDrop n xs
head (drop 2 [0..]) ==> head (drop 1 [1..]) ==> head (drop 0 [2..]) ==> head [2..] ==> 2
head (myDrop 2 [0..])
==> head (if n > length [0..] then [] else myDrop (n-1) (tail [0..]))
==> head (if n > 1+length [1..] then [] else myDrop (n-1) (tail [0..]))
==> head (if n > 1+1+length [2..] then [] else myDrop (n-1) (tail [0..]))
==> head (if n > 1+1+1+length [3..] then [] else myDrop (n-1) (tail [0..]))
-- never terminates
Pretty much all the list functions in the standard library are written in this form, for example:
head (takeWhile (>=0) [0..]) ==> 0
head (concat (repeat [1,2,3])) ==> 1
head (zip [0..] [2..]) ==> (0,2)
head (filter even [3..]) ==> 4
Remember foldr
from part 1? Let’s have a look at its cousin foldl
. Here’s the definition of foldl
for lists (it’s actually part of the Foldable
typeclass and so works for various other types too). While foldr
processes a list right-to-left, foldl
processes a list left-to-right. To be a bit more exact, foldr
associates to the right while foldl
associates to the left. Note the difference in the next example:
foldr (+) 0 [1,2,3] ==> 1+(2+(3+0))
foldl (+) 0 [1,2,3] ==> ((0+1)+2)+3
Here are the definitions of foldl
and foldr
:
foldl :: (a -> b -> a) -> a -> [b] -> a
foldl f z [] = z
foldl f z (x:xs) = foldl f (f z x) xs
foldr :: (a -> b -> b) -> b -> [a] -> b
foldr f y [] = y
foldr f y (x:xs) = f x (foldr f y xs)
As foldr f y (x:xs) ==> f x (foldr f y xs)
, it enables lazy evaluation to focus on f
on the second step. Hence, foldr
works nicely with lazy or short-circuiting operations:
foldr (&&) True [False,False,False]
==> False && (foldr (&&) True [False,False])
==> False
head (foldr (++) [] ["Hello","World","lorem","ipsum"])
==> head ("Hello" ++ (foldr (++) [] ["World","lorem","ipsum"]))
==> head ('H':("ello" ++ (foldr (++) [] ["World","lorem","ipsum"])))
==> 'H'
However foldl
needs to process the whole list in order to produce a (WHNF) value. The reason is that foldl
remains in the leftmost-outermost position for as long as its list argument remains non-empty. This makes foldl
the priority for lazy evaluation. Only after the list becomes empty does the evaluation proceed into simplifying the folded values.
foldl (&&) True [False,False,False]
==> foldl (&&) (True&&False) [False,False]
==> foldl (&&) ((True&&False)&&False) [False]
==> foldl (&&) (((True&&False)&&False)&&False) []
==> ((True&&False)&&False)&&False
==> ( False &&False)&&False
==> False &&False
==> False
head (foldl (++) [] ["Hello","World","lorem","ipsum"])
==> head (foldl (++) ([]++"Hello") ["World","lorem","ipsum"])
==> head (foldl (++) (([]++"Hello")++"World") ["lorem","ipsum"])
==> head (foldl (++) ((([]++"Hello")++"World")++"lorem") ["ipsum"])
==> head (foldl (++) (((([]++"Hello")++"World")++"lorem")++"ipsum") [])
==> head (((([]++"Hello")++"World")++"lorem")++"ipsum")
-- head forces the last ++, which forces the next-to-last ++, and so on
==> head ((("Hello"++"World")++"lorem")++"ipsum")
-- same happens again
==> head ((('H':("ello"++"World"))++"lorem")++"ipsum")
-- for clarity, let's drop the "ello"++"World" expression which isn't needed
==> head ((('H':__)++"lorem")++"ipsum")
-- now the next-to-last ++ can operate
==> head (('H':(__++"lorem"))++"ipsum")
-- let's drop the __++"lorem" expression
==> head (('H':__)++"ipsum")
-- now the last ++ can operate
==> head ('H':(__++"ipsum"))
==> 'H'
So why use foldl
at all? Let’s return to our first fold example again. Now, since +
is a strict operation, both types of fold need to build up an expression with lots of +
s. The Haskell implementation needs to track this expression in memory, which is why a problem like this is called a space leak.
foldr (+) 0 [1,2,3]
==> 1 + foldr (+) 0 [2,3]
==> 1 + (2 + foldr (+) 0 [3])
==> 1 + (2 + (3 + foldr (+) 0 []))
==> 1 + (2 + (3 + 0))
==> 1 + (2 + 3)
==> 1 + 5
==> 6
foldl (+) 0 [1,2,3]
==> foldl (+) (0+1) [2,3]
==> foldl (+) ((0+1)+2) [3]
==> foldl (+) (((0+1)+2)+3) []
==> ((0+1)+2)+3
==> ( 1 +2)+3
==> 3 +3
==> 6
Now let’s instead look at what happens when we use foldl'
, a version of foldl
that forces its second argument!
+) 0 [1,2,3]
foldl' (==> foldl' (+) (0+1) [2,3]
-- force second argument
==> foldl' (+) 1 [2,3]
==> foldl' (+) (1+2) [3]
-- force second argument
==> foldl' (+) 3 [3]
==> foldl' (+) (3+3) []
-- force second argument
==> foldl' (+) 6 []
==> 6
Now the work is performed incrementally while scanning the list. No space leak! Sometimes too much laziness can cause space leaks, and a bit of strictness can fix them.
You can find foldl'
in the Data.List
module, and it works just like this. But how could one implement foldl'
? We certainly know by now how to do it for a specific type, say Int
. We just add a pattern match on the second argument that doesn’t change the semantics of the function.
foldl'Int :: (Int -> Int -> Int) -> Int -> [Int] -> Int
= z
foldl'Int f z [] 0 (x:xs) = foldl'Int f (f 0 x) xs
foldl'Int f :xs) = foldl'Int f (f z x) xs foldl'Int f z (x
+) 0 [1,2,3]
foldl'Int (==> foldl'Int (+) (0+1) [2,3]
-- to be able to pick between the second and third equations, (0+1) is forced
==> foldl'Int (+) 1 [2,3]
-- the third equation applies
==> foldl'Int (+) (1+2) [3]
-- again, we need to pick between the second and third equations
==> foldl'Int (+) 3 [3]
==> foldl'Int (+) (3+3) []
==> 3+3
==> 6
To write a generic implementation of foldl'
we need to introduce a new built-in function, seq
. The call seq a b
evaluates to b
but forces a
into WHNF. Here are some examples of using seq
in GHCi. To demonstrate what gets evaluated, we use the special value undefined
, which causes an error if something tries to evaluate it into WHNF.
Prelude> seq (not True) 3
3
Prelude> seq undefined 3
*** Exception: Prelude.undefined
Prelude> (seq (not True) 3) + 7
10
Prelude> (seq undefined 3) + 7
*** Exception: Prelude.undefined
Prelude> let f x = f x in seq (f 3) 3
-- ...infinite recursion
As an example of using seq
in a function, here’s a version of head
that doesn’t work for infinite lists (since it evaluates the last element of the list):
strictHead :: [a] -> a
= seq (last xs) (head xs) strictHead xs
Let’s play around with it in GHCi:
Prelude> head [1,2,3]
1
Prelude> strictHead [1,2,3]
1
Prelude> head (1:2:undefined)
1
Prelude> strictHead (1:2:undefined)
*** Exception: Prelude.undefined
Prelude> head [1..]
1
Prelude> strictHead [1..]
-- ...infinite recursion
Finally, here’s a definition for foldl'
. Note how we need to introduce sharing of a new variable, z'
, to be able to make seq
evaluate the new value and then use it in a recursive call. The new definition is also used in a more detailed evaluation of foldl' (+) 0 [1,2,3]
below.
foldl' :: (a -> b -> a) -> a -> [b] -> a
= z
foldl' f z [] :xs) = let z' = f z x
foldl' f z (xin seq z' (foldl f z' xs)
+) 0 [1,2,3]
foldl' (==> seq (0+1) (foldl' (+) (0+1) [2,3]) -- seq forces first argument
| |
+-----sharing-----'
| |
==> seq 1 (foldl' (+) 1 [2,3]) -- first argument to seq in WHNF, seq disappears
==> foldl' (+) 1 [2,3]
==> seq (1+2) (foldl' (+) (1+2) [3])
==> seq 3 (foldl' (+) 3 [3])
==> foldl' (+) 3 [3]
==> seq (3+3) (foldl' (+) (3+3) [])
==> seq 6 (foldl' (+) 6 [])
==> foldl' (+) 6 []
==> 6
We won’t dive deeper into this subject on this course, but it’s important that you’re aware that seq
exists. You can find more about seq
on the Haskell Wiki and learn more about when it is necessary to add strictness in Real World Haskell. Often it’s nicer to use bang patterns instead of seq
, as discussed by FPComplete and Real World Haskell.
Recall lecture 7. Sometimes we need boxed types. There’s a special keyword newtype
that can be used instead of data
when a boxed type is needed. newtype
expects exactly one constructor, with exactly one field. For instance,
newtype Money = Cents Int
However, the following won’t work, you need data
:
-- the compiler won't accept these!
newtype Currency = Dollars Int | Euros Int
newtype Money = Money Int Int
So what’s the difference? In terms of writing code, nothing. You work with a newtype
exactly as you would with a data
. However, the memory layout is different. Using data
introduces an indirection layer (the constructor), but using newtype
doesn’t. The indirection for data
is necessary to support multiple constructors and multiple fields. An illustration:
: memory:
code
data Money = Cents Int x --> Cents --> 100
= Cents 100
x
newtype Money = Cents Int x --> 100
= Cents 100 x
This difference has many repercussions. First of all, newtype
is more efficient: the type can be said to “disappear” when compiling. The type is still checked though, so you get type safety without any performance impact. Secondly, newtypes are strict. Concretely, this means that Money x
is in weak head normal form only if x
is in WHNF. This can be witnessed in GHCi:
-- if we use data, Cents undefined is in WHNF
Prelude> data Money = Cents Int
Prelude> seq (Cents undefined) True
True
-- if we use newtype, Cents undefined isn't in WHNF, and trying
-- to make it so trips up in undefined
Prelude> newtype Money = Cents Int
Prelude> seq (Cents undefined) True
*** Exception: Prelude.undefined
So when should you use newtype
? In general it’s best to use newtype
whenever you have a single-field single-constructor datatype. However nothing will go catastrophically wrong if you always use data
. The newtype
pattern is also often used when you need to define a different type class instance for a type. Here’s an example that defines a number type with an inverted ordering
newtype Inverted = Inverted Int
deriving (Show, Eq)
instance Ord Inverted where
compare (Inverted i) (Inverted j) = compare j i
Prelude Data.List> sort [1,2,3]
1,2,3]
[Prelude Data.List> sort [Inverted 1,Inverted 2,Inverted 3]
Inverted 3,Inverted 2,Inverted 1] [
Now that we know about sharing and path copying, we can make our own cyclic datastructures. Remember the cycle
examples from the list lecture?
Prelude> take 21 $ cycle "asdf"
"asdfasdfasdfasdfasdfa"
This is what it looks like in memory:
Earlier it was said that Haskell data forms directed graphs in memory. This is an example of a directed graph with a cycle.
How can we define structures like this? We just give a value a name, and refer to that name in the value itself. That is, the value is recursive or self-referential. This trick is known as tying the knot. A simple example:
code memory
let xs = 1:2:xs xs -> (1:) -> (2:) -+
in xs ^ |
+------------+
Note how we use the name xs
inside the definition of xs
. When we make a recursive definition like this, sharing causes it to turn into a cyclic structure in memory.
A more fun example: a simple adventure game where the world is a self-referential structure. Note how the cyclic structure is built with local definitions that refer to each other.
data Room = Room String [(String,Room)]
describe :: Room -> String
Room s _) = s
describe (
move :: Room -> String -> Maybe Room
Room _ directions) direction = lookup direction directions
move (
world :: Room
= meadow
world where
= Room "It's a flowery meadow next to a cliff." [("Stay",meadow),("Enter cave",cave)]
meadow = Room "You are in a cave" [("Exit",meadow),("Go deeper",tunnel)]
cave = Room "This is a very dark tunnel. It seems you can either go left or right."
tunnel "Go back",cave),("Go left",pit),("Go right",treasure)]
[(= Room "You fall into a pit. There is no way out." []
pit = Room "A green light from a terminal fills the room. The terminal says <<loop>>."
treasure "Go back",tunnel)]
[(
play :: Room -> [String] -> [String]
= [describe room]
play room [] :ds) = case move room d of Nothing -> [describe room]
play room (dJust r -> describe room : play r ds
Prelude> play world ["Stay","Enter cave","Go deeper","Go back","Go deeper","Go right"]
"It's a flowery meadow next to a cliff.",
["It's a flowery meadow next to a cliff.",
"You are in a cave",
"This is a very dark tunnel. It seems you can either go left or right.",
"You are in a cave",
"This is a very dark tunnel. It seems you can either go left or right.",
"A green light from a computer terminal floods the room. The terminal says <<loop>>."]
Here’s what the world
of the game looks like in memory:
,-----------------,
v |
+-----------------------|-----------------+
meadow-->|Room "It's..." ["Stay" o, "Enter cave" o]|
+---------------------------------------|-+
^ v
+--------------------------|----------------+
cave---->|Room "You are..." ["Exit" o, "Go deeper" o]|
+-----------------------------------------|-+
^ v
+-----------------------------|----------------------------+
tunnel-->|Room "This is..." ["Go back" o, "Go left" o, "Go right" o]|<--------,
+------------------------------------------|-------------|-+ |
| | |
,------------------------------' | |
v v |
+---------------------+ +-----------------------------|-+
pit----->|Room "You fall..." []| treasure--->|Room "A green..." ["Go back" o]|
+---------------------+ +-------------------------------+
We’ve now seen three types of recursion. Recursive functions call themselves. Recursive types allow us to express arbitarily large structures. Recursive values are one way to implement infinite structures.
Even though Haskell is a pure programming language, we can sometimes gain insights by sprinkling in a bit of impurity.
We can use the function trace :: String -> a -> a
from the module Debug.Trace
to peek into Haskell evaluation. The expression trace "message" x
is the same as x
, but prints message
when it is evaluated (forced). We can use trace
to witness the laziness of the ||
operator:
Prelude> import Debug.Trace
Prelude Debug.Trace> trace "a" True
aTrue
Prelude Debug.Trace> trace "a" False || trace "b" True
a
bTrue
Prelude Debug.Trace> trace "a" True || trace "b" True
aTrue
We can also have a look at when list elements are evaluated. Note how length
doesn’t need to evaluate the elements of the list, and sum
needs to evaluate all of them. (To be precise, head xs
doesn’t actually evaluate the first element of xs
, but returns it to GHCi, which evaluates it in order to show it.)
Prelude Debug.Trace> head [trace "first" 1, trace "second" 2, trace "third" 3]
first1
Prelude Debug.Trace> last [trace "first" 1, trace "second" 2, trace "third" 3]
third3
Prelude Debug.Trace> length [trace "first" 1, trace "second" 2, trace "third" 3]
3
Prelude Debug.Trace> sum [trace "first" 1, trace "second" 2, trace "third" 3]
third
second
first6
Debug.Trace
also offers useful variants of trace
. A notable one is traceShowId x
which prints show x
and evaluates to x
. Let’s verify the evaluation of our previous head-filter-map example using traceShowId
. Note how even though we map traceShowId
over the infinite list [0..]
, only 6 values are actually evaluated. The last 243 is the returned value, not a trace print.
Prelude Debug.Trace> head (filter (>100) (map (\x -> traceShowId (3^x)) [0..]))
1
3
9
27
81
243
243
Debug.Trace
is especially useful when you have an infinite recursion bug. Here’s an example:
-- computes sums like 7+5+3+1
sumEverySecond :: Int -> Int
0 = 0
sumEverySecond = n + sumEverySecond (n-2) sumEverySecond n
6 ==> 12
sumEverySecond 7 ==> doesn't terminate sumEverySecond
We can debug this by adding a trace
to wrap the whole recursive case.
sumEverySecond :: Int -> Int
0 = 0
sumEverySecond = trace ("sumEverySecond "++show n) (n + sumEverySecond (n-2)) sumEverySecond n
Prelude Debug.Trace> sumEverySecond 6
6
sumEverySecond 4
sumEverySecond 2
sumEverySecond 12
Prelude Debug.Trace> sumEverySecond 7
7
sumEverySecond 5
sumEverySecond 3
sumEverySecond 1
sumEverySecond -1
sumEverySecond -3
sumEverySecond -5
sumEverySecond -- and so on
A ha! The problem is that our recursion base case of sumEverySecond 0
is not enough to stop the recursion.
Finally, a word of caution. Using trace
, and especially traceShowId
, can cause things that would not otherwise get evaluated to get evaluated. For example:
Prelude Debug.Trace> traceHead xs = head (traceShowId xs)
Prelude Debug.Trace> traceHead [0..]
-- never terminates since it's trying to show an infinite list
So feel free to use Debug.Trace
when working on the exercises, but try to leave trace
calls out of your final answers. Some exercise sets check your imports and disallow Debug.Trace
.
We’ll see a more principled way of dealing with side effects in the next lecture!
Which of these statements is true?
reverse . reverse . reverse === reverse
reverse . reverse === reverse
reverse . id === id
Which of these is an infinite list that starts with [0,1,2,1,2,1,2...]
?
cycle [0,1,2]
0:repeat [1,2]
0:cycle [1,2]
0:[1,2..]
What’s the next step when evaluating this expression?
head (map not (True:False:[]))
head (False : True : [])
head (not True)
head (False : map not (False:[]))
head (not True : map not (False:[]))
Which of these values is not in weak head normal form?
map
f 1 : map f (2 : [])
Just (not False)
(\x -> x) True
Which of these statements about the following function is true?
0 x = 1+x
f = 2+x f _ x
f
is strict in its left argument
f
is strict in its right argument
f
forces both of its arguments
Does this function work with infinite lists as input? Why?
= []
f [] :xs) = x : map not xs f (x
[]
case, which is never reached.
map
, which evaluates the whole list.
map
, which works with infinite lists.
What about this one?
= map (+(sum xs)) xs f xs
map
, which evaluates the whole list.
head
of the result needs the whole input list.
[]
case
map
, which works with infinite lists.
RealWorld -> (a,RealWorld)
Forget what we talked about functional programming and purity. Actually, Haskell is the world’s best imperative programming language! Let’s start:
= do
questionnaire putStrLn "Write something!"
<- getLine
s putStrLn ("You wrote: "++s)
Prelude> questionnaire
Write something!
Haskell!
You wrote: Haskell!
Reading input and writing output was easy enough. We can also read stuff over the network. Here’s a complete Haskell program that fetches a some words from a URL using HTTP and prints them.
import Network.HTTP
import Control.Monad
= do
main <- simpleHTTP (getRequest "http://httpbin.org/base64/aGFza2VsbCBmb3IgZXZlcgo=")
rsp <- getResponseBody rsp
body words body) $ \w -> do
forM_ (putStr "word: "
putStrLn w
You can find this program in the course repository as exercises/Examples/FetchWords.hs
, and you can run it like this:
$ cd exercises/Examples
$ stack runhaskell FetchWords.hs
word: haskell
word: for
word: ever
What’s going on here? Let’s look at the types:
Prelude> :t putStrLn
putStrLn :: String -> IO ()
Prelude> :t getLine
getLine :: IO String
A value of type IO a
is an operation that produces a value of type a
. So getLine
is an IO operation that produces a string. The ()
type is the so called unit type, its only value is ()
. It’s mostly used when an IO operation doesn’t return anything (but rather just has side effects).
A comparison with Java (method) types might help:
Haskell type | Java type |
---|---|
doIt :: IO () |
void doIt() |
getSomething :: IO Int |
int getSomething() |
force :: a -> b -> IO () |
void force(a arg0, b arg1) |
mogrify :: c -> IO d |
d mogrify(c arg) |
IO operations can be combined into bigger operations using do-notation.
do operation
operation arg<- operationThatReturnsStuff
variable let var2 = expression
operationThatProducesTheResult var2
You can find useful IO operations in the standard library modules Prelude and System.IO
Here’s an IO operation that asks the user for a string, and prints out the length of the string.
query :: IO ()
= do
query putStrLn "Write something!" -- run an operation, ignore produced value
<- getLine -- run an operation, capture produced value
s let n = length s -- run a pure function
putStrLn ("You wrote "++show n++" characters") -- run an operation, passing on the produced value
Prelude> query
Write something!
lorem ipsumYou wrote 11 characters
The value produced by the last line of a do
block is the value produced by the whole block. Note how askForALine
has the same type as getLine
, IO String
:
askForALine :: IO String
= do
askForALine putStrLn "Please give me a line"
getLine
In addition to IO operations like query
you can also run IO operations that produce values, like askForALine
, in GHCi. You can use <-
to capture the result of the operation into a variable if you want.
Prelude> askForALine
Please give me a line
this is a line"this is a line"
Prelude> line <- askForALine
Please give me a line
this is a linePrelude> :t line
line :: String
Prelude> line
"this is a line"
If you need to give your operation parameters, you can just make a function that returns an operation. Note how ask
has a function type with a ->
, just like a normal function. We also use normal function definition syntax to give the parameter the name question
.
ask :: String -> IO String
= do
ask question putStrLn question
getLine
Prelude> ask "What is love?"
What is love?
Baby don't hurt me!
"Baby don't hurt me!"
Prelude> response <- ask "Who are you?"
Who are you?
The programmer.
Prelude> response
"The programmer."
Prelude> :t response
response :: String
Prelude> :t ask
ask :: String -> IO String
Prelude> :t ask "Who are you?"
"Who are you?" :: IO String ask
return
The Haskell function return
is named a bit misleadingly. In other languages return
is a built-in keyword, but in Haskell it’s just a function. The return :: a -> IO a
function takes a value and turns it into an operation, that produces the value.
produceThree :: IO Int
= return 3
produceThree
printThree :: IO ()
= do
printThree <- produceThree
three putStrLn (show three)
That doesn’t sound very useful does it? Combined with do-notation it is. Here we return a boolean according to whether the user answered Y
or N
:
yesNoQuestion :: String -> IO Bool
= do
yesNoQuestion question putStrLn question
<- getLine
s return (s == "Y")
Prelude> yesNoQuestion "Fire the missiles?"
Fire the missiles?
Y
True
Prelude> answer <- yesNoQuestion "Are you sure?"
Are you sure?
N
Prelude> :t answer
answer :: Bool
Prelude> answer
False
Note! This means that return does not stop execution of an operation (unlike return in Java or C). Remember that in do-blocks, the last line decides which value to produce. This means that this operation produces 2
:
produceTwo :: IO Int
= do return 1
produceTwo return 2
Prelude> produceTwo
2
Let’s look at this another way. The do
notation allows us to cause a sequence of side-effects, and finally to produce a value.
produceThree = do putStrLn "1" -- side effect, produces (), which is ignored
return 2 -- no side effect, produces 2, which is ignored
getLine -- side effect, produces a String, which is ignored
return 3 -- no side effect, produces 3, which is passed on
Prelude> final <- produceThree
1
this line is ignored
Prelude> final
3
Also note that these are the same operation:
do ...
<- op
x return x
do ...
op
Since return
is a function, you should remember to parenthesize any complex expressions:
return (f x : xs)
-- alternatively:
return $ f x : xs
do
and TypesLet’s look at the typing of do-notation in more detail. A do-block builds a value of type IO <something>
. For example in
= do
foo ...
lastOp
The lastOp
must be of type IO X
(for some X
). The type of foo
will also be IO X
. Let’s look at an example with parameters next:
= do
bar x y ...
lastOp arg
The lastOp
must be of type Y -> IO X
(so that lastOp arg
has type IO X
). The type of bar
will be A -> B -> IO X
(and inside bar
we’ll have x :: A
and y :: B
).
If we use return
:
= do
quux x ...
return value
The function quux
will have type A -> IO B
, where x :: A
and value :: B
.
Let’s look at the typing of <-
next. If op :: IO X
and you have var <- op
, var
will have type X
. We’ve seen this in many GHCi examples.
The last line of a do
cannot be foo <- bar
. It can’t be let foo = bar
either. The last line determines what the whole operation produces, so it must be an operation (for example, return something
).
Here’s a worked example:
alwaysFine :: IO Bool
= do
alwaysFine putStrLn "What?" -- :: IO ()
return 2 -- :: IO Int, produced value is discarded
<- getLine -- getLine :: IO String, thus s :: String
s putStrLn s -- putStrLn :: String -> IO (), thus putStrLn s :: IO ()
let b = True -- b :: Bool
return b -- :: IO Bool
-- Thus, alwaysFine :: IO Bool
The typing rules guarantee that you can not “escape” IO
. Even though <-
gives you an X
from an IO X
, you can only use <-
inside do
. However a do
always means a value of type IO Y
. In other words: you can temporarily open the IO
box, but you must return into it. “What happens in IO, stays in IO.”
We’ll talk more about what this means later. For now, it’s enough to know that if you have a function with a non-IO type, like for example myFunction :: Int -> [String] -> String
, the function can not have IO happening inside it. It is a pure function.
For the following examples, we’ll need two new operations.
print :: Show a => a -> IO () -- print a value using the show function
readLn :: Read a => IO a -- get a line and convert it to a value using the read function
The usual tools of recursion, guards and if-then-else also work in the IO
world. Here’s an IO operation that’s defined using a guard:
printDescription :: Int -> IO ()
printDescription n| even n = putStrLn "even"
| n==3 = putStrLn "three"
| otherwise = print n
Prelude> printDescription 2
even
Prelude> printDescription 3
threePrelude> printDescription 5
5
Here’s an operation that prints all numbers in a list using recursion and pattern matching:
printList :: [Int] -> IO ()
= return () -- do nothing
printList [] :xs) = do print x
printList (x-- recursion printList xs
Prelude> printList [1,2,3]
1
2
3
Here are two slightly more complicated examples of recursive IO operations. They use the value produced by the recursive call. The operation readAndSum n
reads n
numbers from the user and prints their sum. The operation ask questions
shows each string in questions
to the user, reads a response, and returns a list of all the responses.
readAndSum :: Int -> IO Int
0 = return 0
readAndSum = do
readAndSum n <- readLn -- read one number
i <- readAndSum (n-1) -- recursion: read and sum rest of numbers
s return (i+s) -- produce result
Prelude> s <- readAndSum 3
2
4
5
Prelude> s
11
ask :: [String] -> IO [String]
= return []
ask [] :questions) = do
ask (questionputStr question
putStrLn "?"
<- getLine -- get one answer
answer <- ask questions -- recursion: get rest of answers
answers return (answer:answers) -- produce result
Prelude> replies <- ask ["What is your name","How old are you"]
What is your name?
Yog-Sothoth
How old are you?
The question is meaningless
Prelude> replies
"Yog-Sothoth","The question is meaningless"] [
Additionally, we have some IO
-specific control structures, or rather, functions. These come from the module Control.Monad
.
-- when b op performs op if b is true
when :: Bool -> IO () -> IO ()
-- unless b op performs op if b is false
unless :: Bool -> IO () -> IO ()
-- do something many times, collect results
replicateM :: Int -> IO a -> IO [a]
-- do something many times, throw away the results
replicateM_ :: Int -> IO a -> IO ()
-- do something for every list element
mapM :: (a -> IO b) -> [a] -> IO [b]
-- do something for every list element, throw away the results
mapM_ :: (a -> IO b) -> [a] -> IO ()
-- the same, but arguments flipped
forM :: [a] -> (a -> IO b) -> IO [b]
forM_ :: [a] -> (a -> IO b) -> IO ()
Using these, we can rewrite our earlier examples:
printList :: [Int] -> IO ()
= mapM_ print xs printList xs
= do
readAndSum n <- replicateM n readLn
numbers return (sum numbers)
ask :: [String] -> IO [String]
= do
ask questions
forM questions askOne
askOne :: String -> IO String
= do
askOne question putStr question
putStrLn "?"
getLine
do
and IndentationIt’s easy to run into weird indentation problems when using do-notation. Here are some rules of thumb to help you get it right.
The most important rule of do and indentation is all operations in a do-block must start in the same column.
Some examples of this rule:
-- This is not OK, putStrLn is way too left
= do y <- getLine
foo putStrLn y
-- This is not OK either
= do y <- getLine
foo putStrLn y
-- This is OK
= do y <- getLine
foo putStrLn y
-- This is also OK: putting a line break after do
= do
foo <- getLine
y putStrLn y
A related rule is when an operation goes over multiple lines, indent the follow-up lines. If you don’t indent, it’ll look like a new operation!
-- This is not OK, the string starts a new operation
= do putStrLn
quux "this long string"
print 1
-- This is OK
= do putStrLn
quux "this long string"
print 1
Here’s one more example, with nested do-blocks, and two different valid indentations.
-- This is OK
= do quux
foo x <- blorg
y do thing
when y (
otherThing)return 3
-- This is also OK: starting putting a line break after do, using $
= do
foo x
quux<- blorg
y $ do
when y
thing
otherThingreturn 3
After all these short one-off examples, let’s turn to something a bit longer. Let’s write a program to fetch all type annotations from all .hs
files. We use IO operations like readFile
and listDirectory
to read and find files, but also pure code like map
and filter
to do the actual processing. First off, here’s a recap of the library operations we’re using:
-- split string into lines
lines :: String -> [String]
-- `isSuffixOf suf list` is true if list ends in suf
:: Eq a => [a] -> [a] -> Bool
Data.List.isSuffixOf-- `isInfixOf inf list` is true if inf occurs inside list
:: Eq a => [a] -> [a] -> Bool
Data.List.isInfixOf-- FilePath is just an alias for String
type FilePath = String
-- get entire contents of file
readFile :: FilePath -> IO String
-- list files in directory
:: FilePath -> IO [FilePath]
System.Directory.listDirectory-- is the given file a directory?
:: FilePath -> IO Bool System.Directory.doesDirectoryExist
And here’s the program itself. You can also find it in the course repository as exercises/Examples/ReadTypes.hs
.
module Examples.ReadTypes where
import Control.Monad (forM)
import Data.List (isInfixOf, isSuffixOf)
import System.Directory (listDirectory, doesDirectoryExist)
-- a line is a type signature if it contains :: but does not contain =
isTypeSignature :: String -> Bool
= not (isInfixOf "=" s) && isInfixOf "::" s
isTypeSignature s
-- return list of types for a .hs file
readTypesFile :: FilePath -> IO [String]
readTypesFile file| isSuffixOf ".hs" file = do content <- readFile file
let ls = lines content
return (filter isTypeSignature ls)
| otherwise = return []
-- list children of directory, prepend directory name
qualifiedChildren :: String -> IO [String]
= do childs <- listDirectory path
qualifiedChildren path return (map (\name -> path++"/"++name) childs)
-- get type signatures for all entries in given directory
-- note mutual recursion with readTypes
readTypesDir :: String -> IO [String]
= do childs <- qualifiedChildren path
readTypesDir path <- forM childs readTypes
typess return (concat typess)
-- recursively read types contained in a file or directory
-- note mutual recursion with readTypesDir
readTypes :: String -> IO [String]
= do isDir <- doesDirectoryExist path
readTypes path if isDir then readTypesDir path else readTypesFile path
-- main is the IO action that gets run when you run the program
main :: IO ()
= do ts <- readTypes "."
main mapM_ putStrLn ts
We can run this program by going to the directory exercises/Examples
and running:
$ stack runhaskell ReadTypes.hs
deposit :: String -> Int -> Bank -> Bank
withdraw :: String -> Int -> Bank -> (Int,Bank)
runBankOp :: BankOp a -> Bank -> (a,Bank)
... and so on
The exact output will vary according to the contents of the directory, of course.
Let’s return to the functional world. How can we reconcile IO operations with Haskell being a pure and lazy language? Something like putStrLn :: String -> IO ()
is a pure function that returns an operation. How is it pure? putStrLn x
is the same when x
is the same. In other words: an operation is a pure description of a chain of side effects. Only executing the operation causes those side effects. When a Haskell program is run, only one operation is executed - it’s called main :: IO ()
. Other operations can be run only by linking them up to main
.
When in GHCi, if an expression you type in evaluates to an operation, GHCi runs that operation for you. Here’s a demonstration of the purity of print
:
Prelude> x = print 1 -- creates operation, doesn't run it
Prelude> x -- runs the operation
1
Prelude> x -- runs it again!
1
Operations are values just like numbers, lists and functions. We can write code that operates on operations. This function takes two operations, a
and b
, and returns an operation that asks the user which one he’d like to run.
choice :: IO x -> IO x -> IO x
=
choice a b do putStr "a or b? "
<- getLine
x case x of "a" -> a
"b" -> b
-> do putStrLn "Wrong!"
_ choice a b
Prelude> choice (putStrLn "A!!!!") (putStrLn "B!!!!")
or b? z
a Wrong!
or b? a
a A!!!!
Using operations specified as parameters lets us write functions like mapM_
, which we met earlier. The implementation is a recursive IO operation that takes another IO operation as a parameter. Conceptually complicated, but simple when you read the code:
mapM_ :: (a -> IO b) -> [a] -> IO ()
mapM_ op [] = return () -- do nothing for an empty list
mapM_ op (x:xs) = do op x -- run operation on first element
mapM_ op xs -- run operation on rest of list, recursively
Prelude> mapM_ print [1,2,3]
1
2
3
So far the only side-effects we’ve been able to produce in IO have been terminal (getLine
, print
) and file (readFile
, listDirectory
) IO. Imperative programs written in Java, Python or C have other types of side effects too that we can’t express in pure Haskell. One of these is mutable (i.e. changeable) state. A pure function can not read mutable state, because otherwise two invocations of the same function might not return the same value.
The Haskell type IORef a
from the module Data.IORef
is a mutable reference to a value of type a
newIORef :: a -> IO (IORef a) -- create a new IORef containing a value
readIORef :: IORef a -> IO a -- produce value contained in IORef
writeIORef :: IORef a -> a -> IO () -- set value in IORef
modifyIORef :: IORef a -> (a -> a) -> IO () -- modify value contained in IORef with a pure function
Here are some examples of using an IORef in GHCi:
Prelude> :m +Data.IORef
Prelude Data.IORef> myRef <- newIORef "banana"
Prelude Data.IORef> readIORef myRef
"banana"
Prelude Data.IORef> writeIORef myRef "apple"
Prelude Data.IORef> readIORef myRef
"apple"
Prelude Data.IORef> modifyIORef myRef reverse
Prelude Data.IORef> readIORef myRef
"elppa"
Here’s an example of using an IORef
to sum the values in a list. Note the similarity with an imperative loop.
sumList :: [Int] -> IO Int
= do r <- newIORef 0 -- initialize r to 0
sumList xs -> modifyIORef r (x+)) -- for every xs, add it to r
forM_ xs (\x -- get last value of r readIORef r
Using IORef
isn’t necessary most of the time. Haskell style prefers recursion, arguments and return values. However real world programs might need one or two IORefs occasionally.
A value of type IO X
is an IO operation that produces a value of type X when run. Operations are pure values. Only running the operation causes the side effects.
IO operations can combined together using do
-notation:
op :: X -> IO Y
= do operation -- run operation
op arg -- run operation with argument
operation2 arg <- operation3 arg -- run operation with argument, store result
result let something = f result -- run a pure function f, store result
-- last operation produces the the return value finalOperation
The return x
operation is an operation that always produces value x
. When x :: a
, return x :: IO a
.
Useful IO operations:
-- printing & reading
putStr :: String -> IO ()
putStrLn :: String -> IO ()
print :: Show a => a -> IO ()
getLine :: IO String
readLn :: Read a => IO a
-- control structures from Control.Monad
when :: Bool -> IO () -> IO () -- when b op performs op if b is true
unless :: Bool -> IO () -> IO () -- unless b op performs op if b is false
replicateM :: Int -> IO a -> IO [a] -- do something many times, collect results
replicateM_ :: Int -> IO a -> IO () -- do something many times, throw away the results
mapM :: (a -> IO b) -> [a] -> IO [b] -- do something for every list element
mapM_ :: (a -> IO b) -> [a] -> IO () -- do something for every list element, throw away the results
forM :: [a] -> (a -> IO b) -> IO [b] -- the same, but arguments flipped
forM_ :: [a] -> (a -> IO b) -> IO ()
-- files
readFile :: FilePath -> IO String
What is the type of this IO operation?
= do putStrLn x
foo x <- getLine
y return (length y)
String -> IO String
IO Int
String -> IO Int
IO String -> IO Int
Which of these lines could be used in place of ????
quux :: String -> IO [String]
= do y <- getLine
quux q <- getLine
z putStrLn (y++z)
????
q <- getLine
return (y++z)
return [q]
ans <- return [y,z]
What values does blorg [1,2,3]
print? That is, what values x
does it call print x
for. The value produced by blorg
doesn’t count.
= return 0
blorg [] :xs) = do m <- blorg xs
blorg (xprint x
return (m+x)
1
, 2
, 3
1
, 2
, 3
, 6
3
, 2
, 1
3
, 2
, 1
, 6
Which of these can a function of type Int -> IO Int
do?
Which of these can a function of type IO Int -> Int
do?
Remember the map
function for lists? Here’s the definition again:
map :: (a -> b) -> [a] -> [b]
map _ [] = []
map g (x:xs) = g x : map g xs
It applies a function g :: a -> b
to each element of a list of type [a]
, returning a list of type [b]
. Another way to express the type of map
would be (a -> b) -> ([a] -> [b])
. This is the same type because ->
associates to right. The extra parentheses emphasize the fact that map
converts the function g :: a -> b
into a function map g :: [a] -> [b]
. This means that map
is a higher-order function that transforms functions to functions.
As map
is parametrically polymorphic, its definition doesn’t depend on the type of values stored in the list. Thus, every function of type a -> b
is converted into a function of type [a] -> [b]
using exactly the same logic. Using the definition above, we can see that:
map (|| True) [True, True, False]
==> [True || True, True || True, False || True]
==> [True, True, True]
map (+1) [1,2,3]
==> [1 + 1, 2 + 1, 3 + 1]
==> [2, 3, 4]
map (++"1") ["1", "2", "3"]
==> ["1" ++ "1", "2" ++ "1", "3" ++ "1"]
==> ["11", "21", "31"]
What’s notable here is that map
preserves the structure of a list. The length of the list and the relative positions of the elements are the same. The general idea is demonstrated in the picture below.
Let’s see if we can find other similar functions. A value of type Maybe a
is kind of like a list of length at most 1. Let’s map over a Maybe
! Can you see the similarity with the definition of map
?
mapMaybe :: (a -> b) -> Maybe a -> Maybe b
Nothing = Nothing
mapMaybe f Just x) = Just (f x) mapMaybe f (
Here too, the structure of the value is preserved. A Nothing
turns into a Nothing
, and a Just
turns into a Just
. Here too, we can think of the type as (a -> b) -> (Maybe a -> Maybe b)
, converting (or “lifting”) a normal function into a function that works on Maybes.
One more example: consider binary trees.
data Tree a = Leaf | Node a (Tree a) (Tree a)
mapTree :: (a -> b) -> Tree a -> Tree b
Leaf = Leaf
mapTree f Node val left right) = Node (f val) (mapTree f left) (mapTree f right) mapTree f (
A binary tree might look like this:
After mapTree g
the tree would looke like this:
Functor
ClassNow we have three different structure-preserving mapping functions. Three similar operations over different types. Could we write a type class to capture this similarity?
map :: (a -> b) -> [a] -> [b]
mapMaybe :: (a -> b) -> Maybe a -> Maybe b
mapTree :: (a -> b) -> Tree a -> Tree b
A naive attempt at writing a type class runs into problems. If we try to abstract over Maybe c
, it seems we can’t write the right type for the map operation. We’d need to be able to change the type parameter c
somehow.
class Mappable m where
mapThing :: (a -> b) -> m -> m
instance Mappable (Maybe c) where
mapThing :: (a -> b) -> Maybe c -> Maybe c
= ... mapThing
Luckily Haskell type classes have a feature we haven’t covered before. You can write classes for type constructors in addition to types. What does this mean? Let’s just have a look at the standard type class Functor
that does what we tried to do with our Mappable
.
class Functor f where
fmap :: (a -> b) -> f a -> f b
Note how the type parameter f
is a type constructor: it’s being passed a
and b
arguments in different parts of the type of fmap
. Now let’s see the instance for Maybe
.
instance Functor Maybe where
-- In this instance, the type of fmap is:
-- fmap :: (a -> b) -> Maybe a -> Maybe b
fmap f Nothing = Nothing
fmap f (Just x) = Just (f x)
Now fmap
has the right type and we can implement it like mapMaybe
! Note how we’ve declared instance Functor Maybe
instead of instance Functor (Maybe a)
. The type Maybe a
isn’t a functor, the type constructor Maybe
is.
The type constructor for lists is written []
. It’s special syntax, just like other list syntax. However if the type [a]
was written List a
, the type constructor []
would mean List
.
instance Functor [] where
fmap = map
Here’s the final of our examples, as a Functor
instance.
data Tree a = Leaf | Node a (Tree a) (Tree a)
instance Functor Tree where
fmap _ Leaf = Leaf
fmap f (Node val left right) = Node (f val) (fmap f left) (fmap f right)
Sidenote: the term functor comes originally from a branch of mathematics called category theory. However, to work with Haskell you don’t need to know any category theory. As you progress in learning Haskell, you might get interested in category theory, and it can be a valuable source of new ideas for programming. Category theory can feel intimidating, so it’s good to know that you can get on fine without it. For now, when you see functor, you can just think “something I can map over”, or perhaps “a container”.
Let’s zoom out a bit. When we have an instance Functor MyFun
we know that we can map a type X
into a new type MyFun X
(since MyFun
is a type constructor), but also that we can lift a function f
that takes an X
argument into a function fmap f
that takes a MyFun X
argument! So you could say we’re mapping both on the type level and the value level.
Oh right, one more thing. Once you’ve gotten the hang of fmap
you might find yourself using it quite a bit. For code that uses fmap
heavily it can be nice to use its infix alias, <$>
. Consider the symmetry between $
and <$>
in these examples:
+1) <$> [1,2,3] ==> [2,3,4]
(not <$> Just False ==> Just True
reverse . tail $ "hello" ==> "olle"
reverse . tail <$> Just "hello" ==> Just "olle"
-- which is the same as
fmap (reverse . tail) (Just "hello") ==> Just "olle"
What is this “preserving of the structure” that was mentioned above exactly? The following two functor laws are expected to hold for any Functor
instance f
(though unfortunately Haskell compilers can’t enforce them):
fmap id === id
fmap (f . g) === fmap f . fmap g
Don’t worry if that sounded abstract! The first law says that a functor maps id :: a -> a
into id :: f a -> f a
. (id
is the identity function, meaning that id x = x
.) Let’s be concrete and see how it works for the list [1,2,3]
:
fmap id [1,2,3] ==> map id [1,2,3]
==> map id (1:[2,3])
==> id 1 : map id [2,3]
==> 1 : map id [2,3]
==> 1 : id 2 : map id [3]
==> 1 : 2 : id 3 : map id []
==> 1 : 2 : 3 : []
=== [1,2,3]
On the other hand,
id [1,2,3] ==> [1,2,3]
Hence, the result of fmap id [1,2,3]
was the same as the result of id [1,2,3]
, so the first functor law holds in this case. It’s not hard to show that the first functor law holds for any list whatsoever.
The first functor law is really a very simple proposition if you think about it. It just says that if we apply fmap
to a function that changes nothing (id
) the resulting function (fmap id
) again changes nothing. Thus, the act of applying fmap
itself preserves the structure of the functor.
How about the second functor law? For lists, consider what happens if we fmap
the function negate.(*2)
(remember, negate
maps x
to -x
and (*2)
multiplies its argument by 2
):
fmap (negate.(*2)) [1,2,3] ==> map (negate.(*2)) [1,2,3]
==> (negate.(*2)) 1 : map (negate.(*2)) [2,3]
==> negate (1 * 2) : map (negate.(*2)) [2,3]
==> -2 : map (negate.(*2)) [2,3]
==> -2 : (negate.(*2)) 2 : map (negate.(*2)) [3]
==> -2 : -4 : map (negate.(*2)) [3]
==> -2 : -4 : (negate.(*2)) 3 : map (negate.(*2)) []
==> -2 : -4 : -6 : []
==> [-2,-4,-6]
Let’s consider the right-hand side of the second functor law in this case:
fmap negate . fmap (*2)) [1,2,3] ==> (map negate . map (*2)) [1,2,3]
(==> map negate (map (*2) [1,2,3])
==> map negate [2,4,6]
==> [-2,-4,-6]
The second functor law turns out to hold in this particular case. In fact, it holds in all cases (exercise!).
In general, the second functor law says that composing two functions first and then applying fmap
must yield the same result as performing fmap
on those functions and then composing the resulting function. In other words, the order of applying fmap
and composing doesn’t matter. (These two operations are said to commute.)
There are also higher-order functions that fail to satisfy functor laws. Consider the function badMap
:
badMap :: (a -> b) -> [a] -> [b]
= []
badMap f [] :y:xs) = f x : badMap f xs
badMap f (x:xs) = f x : badMap f xs badMap f (x
This function violates the first functor law. For instance:
id [1,2,3] ==> badMap id (1:2:[3])
badMap ==> id 1 : badMap id [3]
==> 1 : badMap id [3]
==> 1 : badMap id (3:[])
==> 1 : id 3 : badMap id []
==> 1 : 3 : []
==> [1,3]
Applying badMap id
to the list [1,2,3]
changed the list as the element 2
was dropped.
As mentioned, Haskell compilers can’t detect if a functor obeys its laws or not. A Haskell compiler would happily accept an instance Functor []
that used badMap
instead of map
as the fmap
implementation. This is a limitation of the type system of Haskell. There are technologies such as LiquidHaskell or dependently typed languages such as Agda, Idris, Coq, or Lean that could actually enforce functor laws so that unlawful functor instances wouldn’t compile. These technologies are outside the scope of this course, however.
Remember that Functor
was a class for type constructors. If we try to define an instance of Functor
for a type, we get an error:
Prelude> instance Functor Int where
<interactive>:1:18: error:
Expected kind ‘* -> *’, but ‘Int’ has kind ‘*’
• In the first argument of ‘Functor’, namely ‘Int’
• In the instance declaration for ‘Functor Int’
The error message talks about kinds. Kinds are types of types. A type like Int
, Bool
or Maybe Int
that can contain values has kind *
. A type constructor has a kind that looks like a function, for example, Maybe
has kind * -> *
. This means that the Maybe
type constructor must be applied to a type of kind *
to get a type of kind *
.
We can ask GHCi for the kinds of types:
Prelude> :kind Int
Int :: *
Prelude> :kind Maybe
Maybe :: * -> *
Prelude> :kind Maybe Int
Maybe Int :: *
If we ask GHCi for info about the Functor
class, it tells us that instances of Functor
must have kind * -> *
:
Prelude> :info Functor
class Functor (f :: * -> *) where
fmap :: (a -> b) -> f a -> f b
...
Here are some examples of even more complex kinds.
-- multiple type parameters
Prelude> :kind Either
Either :: * -> * -> *
Prelude> data Either3 a b c = Left a | Middle b | Right c
Prelude> :kind Either3
Either3 :: * -> * -> * -> *
-- a type parameter of kind *->*
Prelude> data IntInside f = IntInside (f Int)
Prelude> :kind IntInside
IntInside :: (* -> *) -> *
You won’t bump into kinds that much in Haskell programming, but sometimes you’ll see error messages that talk about kinds, so it’s good to know what they are.
Foldable
, AgainWe briefly covered the class Foldable
, which occurs in many type signatures of basic functions, in part 1. For example:
length :: Foldable t => t a -> Int
sum :: (Foldable t, Num a) => t a -> a
minimum :: (Foldable t, Ord a) => t a -> a
foldMap :: (Foldable t, Monoid m) => (a -> m) -> t a -> m
From these type signatures we can see that Foldable
, just like Functor
, is a class for type constructors (things of kind * -> *
). The essence of Foldable
is to be a class for things you can fold over. The class definition could be as simple as
class Foldable (t :: *->*) where
foldr :: (a -> b -> b) -> b -> t a -> b
However, for performance reasons, the class contains many methods (you can see them yourself by checking :info Foldable
in GHCi!), but when we’re defining an instance for Foldable
it’s enought to define just foldr
.
Another way of thinking of the Foldable
class is processing elements left-to-right, in other words, if Functor
was the class for containers, Foldable
is the class for ordered containers.
As an example, let’s implement Functor
and Foldable
for our own pair type.
data Pair a = Pair a a
deriving Show
instance Functor Pair where
-- fmap f applies f to all values
fmap f (Pair x y) = Pair (f x) (f y)
instance Foldable Pair where
-- just like applying foldr over a list of length 2
foldr f initialValue (Pair x y) = f x (f y initialValue)
-- an example function that uses both instances
doubleAndCount :: (Functor f, Foldable f) => f Int -> Int
= sum . fmap (*2) doubleAndCount
Now, we can use Pair
almost wherever we can use a list:
fmap (+1) (Pair 3 6) ==> Pair 4 7
fmap (+1) [3,6] ==> [4,7]
foldr (*) 1 (Pair 3 6) ==> 18
foldr (*) 1 [3,6] ==> 18
length (Pair 3 6) ==> 2
length [3,6] ==> 2
minimum (Pair 3 6) ==> 3
minimum [3,6] ==> 3
Pair 3 6) ==> 18
doubleAndCount (3,6] ==> 18 doubleAndCount [
Other types we’ve met that are Foldable
include Data.Map
and Data.Array
.
So, to summarize, a functor is a type constructor f
and the corresponding Functor f
instance such that fmap
satisfies the two functor laws. These laws assert that fmap
must preserve the identity function and distribute over function composition. More informally, fmap
lifts a function g :: a -> b
operating on values to one operating on containers: fmap g :: f a -> f b
. Basically all well-behaving data structures in Haskell are functors.
What’s the type of fmap
?
a -> b -> f a -> f b
(a -> b) -> f a -> f b
Functor f => a -> b -> f a -> f b
Functor f => (a -> b) -> f a -> f b
Which code snippet completes the next Functor
instance?
data Container x = Things x [x]
instance Functor Container where
????
fmap f (Things x ys) = Things (f x) [f x]
fmap f (Things x ys) = Things (f x) (map f ys)
fmap f (Things x ys) = Things (f x) ys
fmap f (Things x ys) = f (Things x ys)
What’s the kind of [a]
?
*
* -> *
[a]
What’s the kind of Foo
?
data Foo x = FooConst
*
* -> *
Foo
What’s the kind of Bar
?
data Bar = Baz | Qux Int
*
* -> *
Bar
What is the value of foldr (-) 1 (Just 2)
?
Just -1
Just 1
Which code snippet completes the next Foldable
instance?
data Container x = Things x [x]
instance Foldable Container where
????
foldr f z (Things x ys) = f x z
foldr f z (Things x ys) = foldr f x ys
foldr f z (Things x ys) = f x (foldr f z ys)
foldr f z (Things x ys) = foldr f z (x:ys)
In this lecture we’ll build up to the concept of a monad using a number of examples. By now you should be familiar with all the Haskell features needed for understanding monads.
Monads are a famously hard topic in programming, which is partly due to weird terminology, partly due to bad tutorials, and partly due to trying to understand monads too early when learning Haskell. Monads are introduced this late in the course in an attempt to make understanding them easier.
If you find this lecture hard, don’t despair, many others have found the topic hard as well. There are many many productive Haskell programmers who have managed to understand monads, so the task is not hopeless.
One final word of caution: monads, like functors, are a concept originally from a branch of mathematics called Category Theory. However, and I can’t stress this enough, you do not need to know anything or even care about category theory to understand monads in Haskell programming. Just like one can work with object-oriented programming or functional programming without knowing the theory of objects or functions, one can work with monads without understanding the math associated with them. Category theory can be a rewarding topic for a functional programmer, but it’s not a mandatory one.
When working with many Maybe
values, the code tends to become a bit messy. Let’s look at some examples. First, we combine some functions returning Maybe String
. Note the nested case
we need in stealSecret
: It’s not fun to write.
-- Try to login with a password.
-- `Just username` on success, `Nothing` otherwise.
login :: String -> Maybe String
"f4bulous!" = Just "unicorn73"
login "swordfish" = Just "megahacker"
login = Nothing
login _
-- Get a secret associated with a user.
-- Not all users have secrets.
secret :: String -> Maybe String
"megahacker" = Just "I like roses"
secret = Nothing
secret _
-- Login and return the user's secret, if any
stealSecret :: String -> Maybe String
=
stealSecret password case login password of
Nothing -> Nothing
Just user -> case secret user of
Nothing -> Nothing
Just s -> Just ("Stole secret: "++s)
"swordfish" ==> Just "Stole secret: I like roses"
stealSecret "f4bulous!" ==> Nothing
stealSecret "wrong_password" ==> Nothing stealSecret
Next up, we modify a list of pairs. We use the Maybe
-returning function lookup
from the Prelude. Here we have an if inside a case instead of a nested case.
-- Get the value corresponding to a key from a key-value list.
lookup :: (Eq a) => a -> [(a, b)] -> Maybe b
-- Set the value of key to val in the given key-value list,
-- but only if val is larger than the current value!
increase :: Eq a => a -> Int -> [(a,Int)] -> Maybe [(a,Int)]
=
increase key val assocs case lookup key assocs
of Nothing -> Nothing
Just x -> if (val < x)
then Nothing
else Just ((key,val) : delete (key,x) assocs)
This type of code is pretty common, and usually repeats the same pattern: if any intermediate result is Nothing
, the whole result is Nothing
. Let’s try to make writing code like this easier by defining a chaining operator ?>
. The chaining operator takes a result and the next step of computation, and only runs the next step if the result was a Just
value.
(?>) :: Maybe a -> (a -> Maybe b) -> Maybe b
-- if we failed, don't even bother running the next step:
Nothing ?> _ = Nothing
-- otherwise run the next step:
Just x ?> f = f x
The chaining operator streamlines our examples nicely. Note how we can define simple helper functions that take care of one step of the computation instead of writing one big expression.
stealSecret :: String -> Maybe String
=
stealSecret password ?>
login password ?>
secret
decoratewhere decorate s = Just ("Stole secret: "++s)
increase :: Eq a => a -> Int -> [(a,Int)] -> Maybe [(a,Int)]
=
increase key val assocs lookup key assocs ?>
?>
check
buildResultwhere check x
| val < x = Nothing
| otherwise = Just x
= Just ((key,val) : delete (key,x) assocs) buildResult x
Here’s another example: safe list indexing built from safeHead
and safeTail
:
safeHead :: [a] -> Maybe a
= Nothing
safeHead [] :xs) = Just x
safeHead (x
safeTail :: [a] -> Maybe [a]
= Nothing
safeTail [] :xs) = Just xs
safeTail (x
safeThird :: [a] -> Maybe a
= safeTail xs ?> safeTail ?> safeHead
safeThird xs
safeNth :: Int -> [a] -> Maybe a
0 xs = safeHead xs
safeNth = safeTail xs ?> safeNth (n-1) safeNth n xs
1,2,3,4]
safeThird [==> Just 3
1,2]
safeThird [==> Nothing
5 [1..10]
safeNth ==> Just 6
11 [1..10]
safeNth ==> Nothing
PS. note that ?>
associates to the left as is the default in Haskell. That means that op ?> f ?> g
means (op ?> f) ?> g
. The alternative, op ?> (f ?> g)
would not even type check!
Sidenote: this ?>
operator expresses the if-result pattern that’s very common in other languages. Here is how one would write op val ?> f
in Python and Java.
# Python
= op(val)
x if x:
f(x)
// Java
Object x = op(val);
if (x != null) {
f(x);
}
The difference between the if-result pattern and our ?>
is that we use the Nothing
value to explicitly signal failure, instead of relying on the fact that any variable can be None
(or False
) in Python, or that any Object
reference can be null
in Java.
Let’s explore the concept of chaining with another example: logging. The type Logger
represents a value plus a list of log messages (produced by the computation that produced the value).
-- Logger definition
data Logger a = Logger [String] a deriving Show
getVal :: Logger a -> a
Logger _ a) = a
getVal (getLog :: Logger a -> [String]
Logger s _) = s
getLog (
-- Primitive operations:
nomsg :: a -> Logger a
= Logger [] x -- a value, no message
nomsg x
annotate :: String -> a -> Logger a
= Logger [s] x -- a value and a message
annotate s x
msg :: String -> Logger ()
= Logger [s] () -- just a message msg s
Here’s a login
function that logs some details about the usernames and passwords it processes. Note how we run into complicated code in login
when we need to handle multiple Logger
values.
validateUser :: String -> Logger Bool
"paul.atreides" = annotate "Valid user" True
validateUser "ninja" = nomsg True
validateUser = annotate ("Invalid user: "++u) False
validateUser u
checkPassword :: String -> String -> Logger Bool
"paul.atreides" "muad'dib" = annotate "Password ok" True
checkPassword "ninja" "" = annotate "Password ok" True
checkPassword = annotate ("Password wrong: "++pass) False
checkPassword _ pass
login :: String -> String -> Logger Bool
=
login user password let validation = validateUser user
in if (getVal validation)
then let check = checkPassword user password
in Logger (getLog validation ++ getLog check) (getVal check)
else validation
"paul.atreides" "muad'dib"
login ==> Logger ["Valid user","Password ok"] True
"paul.atreides" "arrakis"
login ==> Logger ["Valid user","Password wrong: arrakis"] False
"ninja" ""
login ==> Logger ["Password ok"] True
"leto.atreides" "paul"
login ==> Logger ["Invalid user: leto.atreides"] False
Let’s try to streamline this code by defining a chaining operator for Logger
. The important thing when doing multiple Logger
operations is to preserve all the logs. Here’s a chaining operator, #>
, and an example of how it can be used to log some arithmetic computations.
(#>) :: Logger a -> (a -> Logger b) -> Logger b
Logger la a #> f = let Logger lb b = f a -- feed value to next step
in Logger (la++lb) b -- bundle result with all messages
-- square a number and log a message about it
square :: Int -> Logger Int
= annotate (show val ++ "^2") (val^2)
square val
-- add 1 to a number and log a message about it
add :: Int -> Logger Int
= annotate (show val ++ "+1") (val+1)
add val
-- double a number and log a message about it
double :: Int -> Logger Int
= annotate (show val ++ "*2") (val*2)
double val
-- compute the expression 2*(x^2+1) with logging
compute :: Int -> Logger Int
=
compute x
square x#> add
#> double
3
compute ==> Logger ["3^2","9+1","10*2"] 20
We can streamline login
quite a bit by using #>
. Note how we don’t need to worry about combining logs together. Also note how we use a lambda expression instead of defining a helper function.
login :: String -> String -> Logger Bool
=
login user password
validateUser user#>
-> if valid then checkPassword user password
\valid else nomsg False
To ramp things up a bit, let’s use Logger
in a recursive list processing function. Here’s a logging version of filter
. Note how the code chains a log message before the recursive call in order to keep the order of log entries nice.
-- sometimes you don't need the previous value:
(##>) :: Logger a -> Logger b -> Logger b
Logger la _ ##> Logger lb b = Logger (la++lb) b
filterLog :: (Eq a, Show a) => (a -> Bool) -> [a] -> Logger [a]
= nomsg []
filterLog f [] :xs)
filterLog f (x| f x = msg ("keeping "++show x) ##> filterLog f xs #> (\xs' -> nomsg (x:xs'))
| otherwise = msg ("dropping "++show x) ##> filterLog f xs
>0) [1,-2,3,-4,0]
filterLog (==> Logger ["keeping 1","dropping -2","keeping 3","dropping -4","dropping 0"] [1,3]
In the previous example we just wrote some state (the log). Sometimes we need computations that change some sort of shared state. Let’s look at accounts in a small bank. We’ll first define a datatype for the state of the bank: the balances of all accounts, as a map from account name to balance.
import qualified Data.Map as Map
data Bank = Bank (Map.Map String Int)
deriving Show
Here’s how we can deposit some money to an account. We use the function adjust
from Data.Map
to modify the map.
-- Apply a function to one value in a map
:: Ord k => (a -> a) -> k -> Map.Map k a -> Map.Map k a Map.adjust
deposit :: String -> Int -> Bank -> Bank
Bank accounts) =
deposit accountName amount (Bank (Map.adjust (\x -> x+amount) accountName accounts)
Withdrawing money is a bit more complicated, since we want to handle some special cases like the account not existing, or the account not having enough money. We use the library function findWithDefault
to help us along.
-- Fetch the value corresponding to a key from a map,
-- or a default value in case the key does not exist
:: Ord k => a -> k -> Map.Map k a -> a Map.findWithDefault
withdraw :: String -> Int -> Bank -> (Int,Bank)
Bank accounts) =
withdraw accountName amount (let -- balance is 0 for a nonexistant account
= Map.findWithDefault 0 accountName accounts
balance -- can't withdraw over balance
= min amount balance
withdrawal = Map.adjust (\x -> x-withdrawal) accountName accounts
newAccounts in (withdrawal, Bank newAccounts)
Finally, let’s write a function that takes at most 100 money from one account, splits the money in half, and deposits it in two accounts. Pay attention to how we need to carefully thread the different versions of the bank, bank
, bank1
, bank2
and bank3
to make sure all transactions happen in the right order.
share :: String -> String -> String -> Bank -> Bank
=
share from to1 to2 bank let (amount,bank1) = withdraw from 100 bank
= div amount 2
half -- carefully preserve all money, even if amount was an odd number
= amount-half
rest = deposit to1 half bank1
bank2 = deposit to2 rest bank2
bank3 in bank3
"wotan" "siegfried" "brunhilde"
share Bank (Map.fromList [("brunhilde",0),("siegfried",0),("wotan",1000)]))
(==> Bank (Map.fromList [("brunhilde",50),("siegfried",50),("wotan",900)])
"wotan" "siegfried" "brunhilde"
share Bank (Map.fromList [("brunhilde",0),("siegfried",0),("wotan",91)]))
(==> Bank (Map.fromList [("brunhilde",46),("siegfried",45),("wotan",0)])
Code like this turns up often in Haskell when you’re doing serial updates to one value, while also performing some other computations on the side. It’s easy to make a mistake, and the type system won’t help you if you e.g. reuse the bank1
value. Let’s rewrite share
so that we don’t need to refer to the bank itself. We can again use the same chaining idea to accomplish this.
-- `BankOp a` is an operation that transforms a Bank value,
-- while returning a value of type `a`
data BankOp a = BankOp (Bank -> (a,Bank))
-- running a BankOp on a Bank
runBankOp :: BankOp a -> Bank -> (a,Bank)
BankOp f) bank = f bank
runBankOp (
-- Running one BankOp after another
(+>>) :: BankOp a -> BankOp b -> BankOp b
+>> op2 = BankOp combined
op1 where combined bank = let (_,bank1) = runBankOp op1 bank
in runBankOp op2 bank1
-- Running a parameterized BankOp, using the value returned
-- by a previous BankOp. The implementation is a bit tricky
-- but it's enough to understand how +> is used for now.
(+>) :: BankOp a -> (a -> BankOp b) -> BankOp b
+> parameterized = BankOp combined
op where combined bank = let (a,bank1) = runBankOp op bank
in runBankOp (parameterized a) bank1
-- Make a BankOp out of deposit.
-- There is no return value so we use ().
depositOp :: String -> Int -> BankOp ()
= BankOp depositHelper
depositOp accountName amount where depositHelper bank = ((), deposit accountName amount bank)
-- Make a BankOp out of withdraw. Note how
-- withdraw accountName amount :: Bank -> (Int,Bank)
-- is almost a BankOp already!
withdrawOp :: String -> Int -> BankOp Int
= BankOp (withdraw accountName amount) withdrawOp accountName amount
Let’s see how chaining works with these bank operations.
Prelude> bank = Bank (Map.fromList [("edsger",10),("grace",50)])
-- Running a number of operations using +>>
Prelude> runBankOp (depositOp "edsger" 1) bank
Bank (fromList [("edsger",11),("grace",50)]))
((),
Prelude> runBankOp (depositOp "edsger" 1 +>> depositOp "grace" 1) bank
Bank (fromList [("edsger",11),("grace",51)]))
((),
Prelude> runBankOp (depositOp "edsger" 1 +>> depositOp "grace" 1 +>> withdrawOp "edsger" 11) bank
11,Bank (fromList [("edsger",0),("grace",51)]))
(
-- Using +> to implement a transfer from one account to the other:
Prelude> runBankOp (withdrawOp "edsger" 5 +> depositOp "grace") bank
Bank (fromList [("edsger",5),("grace",55)]))
((),
Prelude> runBankOp (withdrawOp "edsger" 100 +> depositOp "grace") bank
Bank (fromList [("edsger",0),("grace",60)])) ((),
Note how a value of type BankOp
represents a process that transforms the bank. The initial state of the bank must be supplied using runBankOp
. This makes sense because BankOp
transformations can be composed, unlike Bank
states. Having to use runBankOp
makes the distinction between defining operations and executing them clearer.
Now that we’re familiar with manipulating BankOp
values, we can implement share
as a BankOp
. We implement a helper distributeOp
to make the code a bit neater.
-- distribute amount to two accounts
distributeOp :: String -> String -> Int -> BankOp ()
=
distributeOp to1 to2 amount
depositOp to1 half+>>
depositOp to2 restwhere half = div amount 2
= amount - half
rest
shareOp :: String -> String -> String -> BankOp ()
=
shareOp from to1 to2 100
withdrawOp from +>
distributeOp to1 to2
"wotan" "siegfried" "brunhilde")
runBankOp (shareOp Bank (Map.fromList [("brunhilde",0),("siegfried",0),("wotan",1000)]))
(==> ((),Bank (Map.fromList [("brunhilde",50),("siegfried",50),("wotan",900)]))
"wotan" "siegfried" "brunhilde")
runBankOp (shareOp Bank (Map.fromList [("brunhilde",0),("siegfried",0),("wotan",91)]))
(==> ((),Bank (Map.fromList [("brunhilde",46),("siegfried",45),("wotan",0)]))
That was pretty clean wasn’t it? We don’t need to mention the bank at all, we can almost program as if in an imperative language while staying completely pure.
You can find all of this code in the course repository under exercises/Examples/Bank.hs
.
We’ve now seen three different types with a chaining operation:
(?>) :: Maybe a -> (a -> Maybe b) -> Maybe b
(#>) :: Logger a -> (a -> Logger b) -> Logger b
(+>) :: BankOp a -> (a -> BankOp b) -> BankOp b
Just like previously with map
and Functor
, there is a type class that captures this pattern. Note that Monad
is a class for type constructors, just like Functor
.
class Monad m where
(>>=) :: m a -> (a -> m b) -> m b
There are some additional operations in Monad
too:
-- lift a normal value into the monad
return :: a -> m a
-- simpler chaining (like our ##>)
(>>) :: m a -> m b -> m b
>> b = a >>= \_ -> b -- remember: _ means ignored argument a
Recall that the Functor
class was about a generic map
operation. Similarly, the Monad
class is just about a generic chaining operation.
fmap :: Functor f => (a->b) -> f a -> f b
(>>=) :: Monad m => m a -> (a -> m b) -> m b
The expression operation >>= next
takes a monadic operation operation :: m a
, and does some further computation with the value that it produces using next :: a -> m b
. If this feels too abstract, just recall how chaining works for Maybe
:
(>>=) :: Maybe a -> (a -> Maybe b) -> Maybe b
-- if we failed, don't even bother running the next step
Nothing >>= _ = Nothing
-- otherwise run the next step
Just x >>= f = f x
Here’s the full Monad
instance for Maybe
and some examples.
instance Monad Maybe where
Just x) >>= k = k x
(Nothing >>= _ = Nothing
Just _) >> k = k
(Nothing >> _ = Nothing
return x = Just x
Just 1 >>= \x -> return (x+1)
==> Just 2
Just "HELLO" >>= (\x -> return (length x)) >>= (\x -> return (x+1))
==> Just 6
Just "HELLO" >>= \x -> Nothing
==> Nothing
Just "HELLO" >> Just 2
==> Just 2
Just 2 >> Nothing
==> Nothing
Here are the stealSecret
and increase
examples rewritten with monad operations. The changes are ?>
to >>=
and Just
to return
.
stealSecret :: String -> Maybe String
=
stealSecret password >>=
login password >>=
secret
decoratewhere decorate s = return ("Stole secret: "++s)
-- Set the value of key to val in the given key-value list,
-- but only if val is larger than the current value!
increase :: Eq a => a -> Int -> [(a,Int)] -> Maybe [(a,Int)]
=
increase key val assocs lookup key assocs >>=
>>=
check
buildResultwhere check x
| val < x = Nothing
| otherwise = return x
= return ((key,val) : delete (key,x) assocs) buildResult x
do
Here’s an example of what a complex monad operation might look like.
= op1 >>= continue
f where continue x = op2 >> op3 >>= continue2 x
= op4 >> op5 x y continue2 x y
Let’s see what happens when we transform this code a bit. First off, let’s inline the definitions.
= op1 >>= (\x ->
f >>
op2 >>= (\y ->
op3 >>
op4 op5 x y))
Due to lambda expressions continuing to the end of the expression, we can omit the parentheses. Let’s also indent differently.
= op1 >>= \x ->
f >>
op2 >>= \y ->
op3 >>
op4 op5 x y
Now we can notice the similarity with do
notation. The do
block below is actually the same code!
= do x <- op1
f
op2<- op3
y
op4 op5 x y
To clarify, do
notation is just a nicer syntax for the monad operations (>>=
and >>
) and lambdas. Here’s how do notation gets transformed into monad operations. Note! the definition is recursive.
do x <- op a ~~~> op a >>= \x -> do ...
...
do op a ~~~> op a >> do ...
...
do let x = expr ~~~> let x = expr in do ...
...
do finalOp ~~~> finalOp
Here’s safeNth
using do notation:
safeHead :: [a] -> Maybe a
= Nothing
safeHead [] :xs) = Just x
safeHead (x
safeTail :: [a] -> Maybe [a]
= Nothing
safeTail [] :xs) = Just xs
safeTail (x
safeNth :: Int -> [a] -> Maybe a
0 xs = safeHead xs
safeNth = do t <- safeTail xs
safeNth n xs -1) t safeNth (n
Here is increase
one last time, now with do notation
-- Set the value of key to val in the given key-value list,
-- but only if val is larger than the current value!
increase :: Eq a => a -> Int -> [(a,Int)] -> Maybe [(a,Int)]
=
increase key val assocs do oldVal <- lookup key assocs
check oldValreturn ((key,val) : delete (key,oldVal) assocs)
where check x
| val < x = Nothing
| otherwise = return x
We should be able to write a Monad
instance for Logger
ourselves, by setting >>=
to #>
. However, due to recent changes in the Haskell language we must implement Functor
and Applicative
instances to be allowed to implement the Monad
instance. Functor
we’ve already met, but what’s Applicative
? We’ll find out later. Let’s implement the instances:
import Control.Monad
data Logger a = Logger [String] a deriving Show
msg :: String -> Logger ()
= Logger [s] ()
msg s
-- The Functor instance just maps over the stored value
instance Functor Logger where
fmap f (Logger log x) = Logger log (f x)
-- This is an Applicative instance that works for any
-- monad, you can just ignore it for now. We'll get back
-- to Applicative later.
instance Applicative Logger where
pure = return
<*>) = ap
(
-- Finally, the Monad instance
instance Monad Logger where
return x = Logger [] x
Logger la a >>= f = Logger (la++lb) b
where Logger lb b = f a
We don’t need the nomsg
operation any more since it’s just return
. We can also reimplement the annotate
operation using monad operations.
nomsg :: a -> Logger a
= return x
nomsg x
annotate :: String -> a -> Logger a
= msg s >> return x annotate s x
Here are the compute
and filterLog
examples rewritten using do-notation. Note how nice filterLog
is with do-notation.
= do
compute x <- annotate "^2" (x*x)
a <- annotate "+1" (a+1)
b "*2" (b*2)
annotate
filterLog :: (Show a) => (a -> Bool) -> [a] -> Logger [a]
= return []
filterLog f [] :xs)
filterLog f (x| f x = do msg ("keeping "++show x)
<- filterLog f xs
xs' return (x:xs')
| otherwise = do msg ("dropping "++show x)
filterLog f xs
3
compute ==> Logger ["^2","+1","*2"] 20
>0) [1,-2,3,-4,0]
filterLog (==> Logger ["keeping 1","dropping -2","keeping 3","dropping -4","dropping 0"] [1,3]
Haskell’s State
monad is a generalized version of our BankOp
type. The State
type is parameterized by two types, the first being the type of the state, and the second the type of the value produced. State Bank a
would be equivalent to our BankOp a
. You can find the State
monad in the module Control.Monad.Trans.State
of the transformers
package. Here’s a simplified implementation of State
.
data State s a = State (s -> (a,s))
State f) s = f s
runState (
-- operation that overwrites the state (and produces ())
put :: s -> State s ()
= State (\oldState -> ((),state))
put state
-- operation that produces the current state
get :: State s s
= State (\state -> (state,state))
get
-- operation that modifies the current state with a function (and produces ())
modify :: (s -> s) -> State s ()
= State (\state -> ((), f state))
modify f
-- Functor and Applicative instances skipped
instance Monad (State s) where
return x = State (\s -> (x,s))
>>= f = State h
op where h state0 = let (val,state1) = runState op state0
= f val
op2 in runState op2 state1
Note how we declare an instance Monad (State s)
. We’re using a partially-applied type constructor because instances of Monad
can only be declared for type constructors that take one more type parameter. This might be a bit clearer if you look at how m
, Maybe
and State
occur in the type of >>=
below.
class Monad m where
(>>=) :: m a -> (a -> m b) -> m b
instance Monad Maybe where
(>>=) :: Maybe a -> (a -> Maybe b) -> Maybe b
instance Monad (State s) where
(>>=) :: State s a -> (a -> State s b) -> State s b
Let’s look at some examples of working with State
. To start off, let’s consider computations of type State Int a
, which represent working with a simple counter.
-- adds i to the value of the counter
add :: Int -> State Int ()
= do old <- get
add i +i) put (old
1 >> add 3 >> add 5 >> add 6) 0
runState (add ==> ((),15)
example :: State Int Int
= do add 3 -- increment state by 3
example <- get -- value is current state, i.e. initial+3
value 1000 -- increment state by 1000
add + 1) -- overwrite state with value+1, i.e. initial+4
put (value return value -- produce value, i.e. intial+3
1
runState example ==> (4,5) -- initial is 1, state is initial+4=5, produces initial+3=4
Note how a value of type State s a
represents a process that transforms the state (just like BankOp
). The initial state must be supplied using runState
. Again, having to use runState
makes the distinction between defining operations and executing them clearer.
A state can replace an accumulator parameter when processing a list. Here are two examples: finding the largest element of a list, and finding values in a list that occur directly after a 0
.
findLargest :: Ord a => [a] -> State a ()
= return ()
findLargest [] :xs) = do
findLargest (x-> max x y) -- update state with max of current value and previous largest value
modify (\y -- process rest of list findLargest xs
1,2,7,3]) 0 ==> ((),7) runState (findLargest [
-- store the given value in the state list
remember :: a -> State [a] ()
= modify (x:)
remember x
valuesAfterZero :: [Int] -> ((),[Int])
= runState (go xs) []
valuesAfterZero xs where go :: [Int] -> State [Int] ()
0:y:xs) = do remember y
go (:xs)
go (y:xs) = go xs
go (x= return () go []
0,1,2,3,0,4,0,5,0,0,6]
valuesAfterZero [==> ((),[6,0,5,4,1])
mapM
The control structures from the IO lecture work in all monads. Here are their real types.
when :: Monad m => Bool -> m () -> m () -- conditional operation
unless :: Monad m => Bool -> m () -> m () -- same, but condition is flipped
replicateM :: Monad m => Int -> m a -> m [a] -- do something many times
replicateM_ :: Monad m => Int -> m a -> m () -- same, but ignore the results
mapM :: Monad m => (a -> m b) -> [a] -> m [b] -- do something on a list's elements
mapM_ :: Monad m => (a -> m b) -> [a] -> m () -- same, but ignore the results
forM :: Monad m => [a] -> (a -> m b) -> m [b] -- mapM but arguments reversed
forM_ :: Monad m => [a] -> (a -> m b) -> m () -- same, but ignore the results
As we can see here, we can use mapM
over all of the monads we’ve met so far:
mapM (\x -> if (x>0) then Just (x-1) else Nothing) [1,2,3] ==> Just [0,1,2]
mapM (\x -> if (x>0) then Just (x-1) else Nothing) [1,0,3] ==> Nothing
mapM (\x -> msg "increment" >> msg (show x) >> return (x+1)) [1,2,3]
==> Logger ["increment","1","increment","2","increment","3"] [2,3,4]
mapM (\x -> modify (x+) >> return (x+1)) [1,2,3]) 0
runState (==> ([2,3,4],6)
Some more examples:
safeHead :: [a] -> Maybe a
= Nothing
safeHead [] :xs) = Just x
safeHead (xfirsts :: [[a]] -> Maybe [a]
= forM xs safeHead firsts xs
1,2,3],[4,5],[6]] ==> Just [1,4,6]
firsts [[1,2,3],[],[6]] ==> Nothing firsts [[
-- an abbreviated version of an example from the last section
findLargest :: Ord a => [a] -> State a ()
= mapM_ update xs
findLargest xs where update x = modify (\y -> max x y)
1,2,7,3]) 0 ==> ((),7) runState (findLargest [
let increment = modify (+1) >> get
= replicateM 4 increment
ops in runState ops 0
==> ([1,2,3,4],4)
Here’s filter
reimplemented using the State
monad:
rememberElements :: (a -> Bool) -> [a] -> State [a] ()
= mapM_ maybePut xs
rememberElements f xs where maybePut x = when (f x) (modify (++[x]))
sfilter :: (a -> Bool) -> [a] -> [a]
= finalState
sfilter f xs where (_, finalState) = runState (rememberElements f xs) []
even [1,2,3,4,5]
sfilter ==> [2,4]
We can write our own operations that work for all monads. This is made possible by type classes, as we’ve seen before. If you only use monad operations like return
and do-notation, the type system will infer a generic type for your function.
= if b then op else return ()
mywhen b op
= return ()
mymapM_ op [] :xs) = do op x
mymapM_ op (x mymapM_ op xs
*Main> :t mywhen
mywhen :: (Monad m) => Bool -> m () -> m ()
*Main> :t mymapM_
mymapM_ :: (Monad m) => (t -> m a) -> [t] -> m ()
We can use these generic operations in each of our example monads:
perhapsDecrease :: Int -> Maybe Int
= do
perhapsDecrease x <=0) Nothing
mywhen (xreturn (x-1)
2 ==> Just 1
perhapsDecrease 0 ==> Nothing perhapsDecrease
search :: (Show a, Eq a) => a -> [a] -> Logger ()
= mymapM_ look ys
search x ys where look y = mywhen (x==y) (msg ("Found "++show y))
3 [1,2,3,4,3,2] ==> Logger ["Found 3","Found 3"] () search
sumPositive :: [Int] -> State Int ()
= mymapM_ f xs
sumPositive xs where f x = when (x>0) (modify (x+))
1,-4,2,3]) 0 ==> ((),6) runState (sumPositive [
One useful operation hasn’t yet been introduced: liftM
.
liftM :: Monad m => (a->b) -> m a -> m b
= do x <- op
liftM f op return (f x)
The liftM
operation makes it easy to write code with pure and monadic parts.
negate (Just 3)
liftM ==> Just (-3)
sort $ firsts [[4,6],[2,1,0],[3,3,3]]
liftM ==> Just [2,3,4]
negate get) 3
runState (liftM ==> (-3,3)
Does the type of liftM
look familiar? It’s just like the type of fmap
! In fact, it’s easy to define a functor instance for a monad: just set fmap = liftM
. Since every Monad
needs to be a Functor
these days, modern Haskell style prefers fmap
over liftM
.
fmap :: Functor f => (a->b) -> f a -> f b
fmap negate (Just 3)
==> Just (-3)
fmap sort $ firsts [[4,6],[2,1,0],[3,3,3]]
==> Just [2,3,4]
fmap negate get) 3
runState (==> (-3,3)
The list monad (that is, the Monad
instance for []
) represents computations with multiple return values. It’s useful for searching through alternatives. Here’s a first example. For every x
we produce both x
and -x
:
1,2,3] >>= \x -> [-x,x]
[==> [-1,1,-2,2,-3,3]
We can filter out unsuitable values by produing an empty list:
1,2,3] >>= \x -> if x>1 then [x] else []
[==> [2,3]
If we’re using do-notation the list monad starts to look more like a looping construct:
do word <- ["Blue", "Green"]
<- [1,2,3]
number return (word ++ show number)
==> ["Blue1","Blue2","Blue3","Green1","Green2","Green3"]
More interesting example: find all the pairs in a list that sum to k
. (The same element twice counts as a pair.)
findSum :: [Int] -> Int -> [(Int,Int)]
= do a <- xs
findSum xs k <- xs
b if (a+b==k) then [(a,b)] else []
1,2,3,4,5] 5
findSum [==> [(1,4),(2,3),(3,2),(4,1)]
A final, more complex example. We find all palindromes from a string using the list monad, and then find the longest one.
import Data.List (sortBy)
substrings :: String -> [String]
= do start <- [0..length xs - 1]
substrings xs <- [start+1..length xs - 1]
end return $ drop start $ take end $ xs
palindromesIn :: String -> [String]
= do s <- substrings xs
palindromesIn xs if (s==reverse s) then return s else []
= head . sortBy f $ palindromesIn xs
longestPalindrome xs where f s s' = compare (length s') (length s) -- longer is smaller
"aabbacddcaca"
palindromesIn ==> ["a","aa","a","abba","b","bb","b","a","acddca","c","cddc","d","dd","d","c","cac","a","c"]
"aabbacddcaca"
longestPalindrome ==> "acddca"
Here’s the surprisingly simple implementation of the list monad:
instance Monad [] where
return x = [x] -- an operation that produces one value
>>= f = concat (map f lis) -- compute f for all values, combine the results lis
We’ve actually seen the list monad previously in the guise of list comprehensions. Compare this reimplementation of findSum
to the earlier one that uses do
-notation.
findSum :: [Int] -> Int -> [(Int,Int)]
= [(a,b) | a <- xs, b <- xs, a+b==k ] findSum xs k
As you’ve probably guessed by now, IO
is a monad. However the implementations of the IO
type and instance Monad IO
are compiler built-ins. You couldn’t implement the IO monad just using standard Haskell, unlike Maybe
monad, State
monad and other monads we’ve seen.
However, true side effects fit the monad pattern just like State
and Maybe
. Just like with other monads, we’re separating the pure definitions of operations from the process of running the operations. As a bonus, you can use all the generic monad operations (mapM
and friends) with IO.
Here are some examples of writing IO using monad operations.
printTwoThings :: IO ()
= putStrLn "One!" >> print 2
printTwoThings
echo :: IO ()
= getLine >>= putStrLn
echo
verboseEcho :: IO ()
= getLine >>= \s -> putStrLn ("You wrote: " ++ s)
verboseEcho
query :: String -> IO String
= putStrLn question >> getLine
query question
confirm :: String -> IO Bool
= putStrLn question >> fmap interpret getLine
confirm question where interpret "Y" = True
= False interpret _
Prelude> printTwoThings
One!
2
Prelude> verboseEcho
The Iliad
You wrote: The Iliad
Prelude> answer <- query "Why am I here?"
Why am I here?
Good question!
Prelude> answer
"Good question!"
Prelude> b <- confirm "Fire warheads?"
Fire warheads?
no no no noPrelude> b
False
Prelude> b <- confirm "Make love, not war?"
Make love, not war?
Y
Prelude> b
True
Once you’ve gotten familiar with the concept of a monad, you’ll start seeing monadlike things in other languages too. The most well-known examples of this are Option types, Java Streams and JavaScript promises . If you know these languages or concepts from before, you might find this section illuminating. If you don’t, feel free to skip this.
Many langages have an option type. This type is called Optional<T>
in Java, std::optional<T>
in C++, Nullable<T>
in C#, and so on. These types often have behaviour resembling the Haskell Maybe
monad, for example:
Optional.flatMap
corresponds to >>=
: it lets you apply a Function<T,<Optional<U>>
to an Optional<T>
and get an Optional<U>
.Nullable
types. For example, a + null
becomes null
.Java Streams have a monadlike API too. Streams are about producing many values incrementally. Just like with Optional, the method Stream.flatMap
lets us take a Stream<T>
, combine it with a Function<T,Stream<U>>
and get a Stream<U>
.
As an example, if lines
is a Stream<String>
, words
takes a String
and returns a Stream<String>
and readInt
takes a String
and returns an Integer
, we can write:
<Integer> parseNumbers(Stream<String> lines) {
Streamreturn lines.flatMap(words).map(read);
}
This corresponds to the following Haskell list monad code:
parseNumbers :: [String] -> [Int]
= fmap read (strings >>= words) parseNumbers strings
"123 456","7 89"] ==> [123,456,7,89] parseNumbers [
There is much disagreement about whether Promises in JavaScript really are monads or not. However, some similarities are obvious.
First, consider the similarities between Promise.then
and >>=
. Both take an operation (promise or monadic operation), and combine it with a function that returns a new operation.
function concatPromises(promise1, promise2) {
return promise1.then(value1 => promise2.then(value2 => value1+value2));
}
>> concatPromises(Promise.resolve("abc"), Promise.resolve("def")).then(console.log)
abcdef
concatMonadic :: Monad m => m String -> m String -> m String
= op1 >>= (\value1 -> op2 >>= (\value2 -> return (value1++value2))) concatMonadic op1 op2
Prelude> concatMonadic (Just "abc") (Just "def")
Just "abcdef"
Next, let’s consider the similarities between async/await and do-notation. Both are nicer syntaxes for working with the raw Promise.then
or >>=
mechanisms. We reimplement concatPromises
using async/await, and concatMonadic
using do-notation. Their behaviour stays the same.
async function concatPromises(promise1, promise2) {
let value1 = await promise1;
let value2 = await promise2;
return value1+value2;
}
concatMonadic :: Monad m => m String -> m String -> m String
= do
concatMonadic op1 op2 <- op1
value1 <- op2
value2 return (value1++value2)
Monad
type class is a way to represent different ways of executing recipes
Maybe
)Monad
class operations (>>=
, >>
) directlydo
-notationM
is a monad, values of type M a
are operations that produce a result of type a
mapM
etc)
State
operation is easier than deciphering a complicated recursion with stateThis and the previous lecture have covered many parts where the GHC version of Haskell differs from standard Haskell 2010. Here’s a short list of the changes GHC has made, just so you know:
length
, sum
, foldr
etc. generalized to work on Foldable
instead of just listsFunctor
and Applicative
are superclasses of Monad
fail
method has been moved from the Monad
type class to its own MonadFail
classWhat is the expression equivalent to the following do block?
do y <- z
s yreturn (f y)
z >> \y -> s y >> return (f y)
z >>= \y -> s y >> return (f y)
z >> \y -> s y >>= return (f y)
What is the type of \x xs -> return (x : xs)
?
Monad m => a -> [a] -> m [a]
Monad m => a -> [m a] -> [m a]
a -> [a] -> Monad [a]
What is the type of \x xs -> return x : xs
?
Monad m => a -> [a] -> m [a]
Monad m => a -> [m a] -> [m a]
a -> [a] -> Monad [a]
What is the type of (\x xs -> return x) : xs
?
Monad m => a -> [a] -> m [a]
Monad m => a -> [m a] -> [m a]
a -> [a] -> Monad [a]
Now that you know monads, you pretty much know everything about Haskell to start writing real programs that use libraries to do useful things. This lecture will go over some examples of libraries that are commonly used in such real programs. Using these libraries is also a good way to practice using monads, reading docs, and understanding type errors.
Note! When reading the documentation for the libraries, remember to pay attention to the library version. You can see the versions used on the course in the tests.cabal
file. The links in the course material always take you to the right version, as does the stack haddock --open <package>
command. See also Reading Docs in Part 1.
Text
and ByteString
So far, we’ve been using the Haskell String
type to work with strings. However String
is just [Char]
, a linked list of characters. This is horribly inefficient, both in terms of memory, and in terms of time. Once we move beyond processing short strings and start processing whole files or network requests, a more time efficient string type becomes a must.
There are two types that are used as replacements for String
, with slightly different semantics:
Data.Text
represents a sequence of Unicode characters, just like String
, only more efficient. Used when dealing with text.Data.ByteString
represents a sequence of bytes. Used when dealing with binary data.Additionally, both of these types come in lazy and strict variants. The docs for Data.Text
summarize the difference well:
The strict
Text
type requires that an entire string fit into memory at once. The lazyText
type is capable of streaming strings that are larger than memory using a small memory footprint… Each module provides an almost identical API…
All of these types (Text
and ByteString
, strict and lazy) offer pack
and unpack
functions for converting from and to plain String
s. The types also come with specialized versions of familiar list functions like reverse
, take
, map
and so on.
Text
Let’s go through a short GHCi session demonstrating the use of Data.Text
. As the documentation says, the Data.Text
module is designed to be imported qualified. We can convert a String
into a Text
with the function T.pack
. Note how a value of type Text
gets printed just like a String
.
Prelude> import qualified Data.Text as T
Prelude T> :t T.pack
:: String -> T.Text
T.packPrelude T> phrase = T.pack "brevity is the soul of wit"
Prelude T> :t phrase
phrase :: T.Text
Prelude T> phrase
"brevity is the soul of wit"
We can use the functions from Data.Text
to operate on values of Text
. Many of these are named like their counterparts for String
s or lists from Prelude
.
Prelude T> :t T.length
:: T.Text -> Int
T.lengthPrelude T> T.length phrase
26
Prelude T> T.head phrase
'b'
Prelude T> T.take 4 phrase
"brev"
Prelude T> :t T.words
:: T.Text -> [T.Text]
T.wordsPrelude T> T.words phrase
"brevity","is","the","soul","of","wit"]
[Prelude T> :t T.map
:: (Char -> Char) -> T.Text -> T.Text
T.mapPrelude T> T.map (\c -> if c=='o' then '0' else c) phrase
"brevity is the s0ul 0f wit"
A useful detail is that Text
has a Monoid
instance that glues Text
values together. You can also use the functions T.append
and T.concat
.
Prelude T> phrase <> phrase
"brevity is the soul of witbrevity is the soul of wit"
Prelude T> T.append phrase phrase
"brevity is the soul of witbrevity is the soul of wit"
Prelude T> T.concat [phrase,phrase,phrase]
"brevity is the soul of witbrevity is the soul of witbrevity is the soul of wit"
If you want to write a recursive function that pattern matches on a Text
like you would on a String
, you can use the function T.uncons :: T.Text -> Maybe (Char, T.Text)
to split a Text
into a head and a tail. Here’s a simple example:
countLetter :: Char -> T.Text -> Int
=
countLetter c t case T.uncons t of
Nothing -> 0
Just (x,rest) -> (if x == c then 1 else 0) + countLetter c rest
Prelude T> countLetter 't' phrase
3
Note that Data.Text
implements the strict Text
type. You need to use Data.Text.Lazy
for the lazy variant. As mentioned earlier, one difference between these two types is that the strict type does not work for infinite strings:
Prelude T> T.head (T.pack (repeat 'x'))
-- never returns
Prelude T> import qualified Data.Text.Lazy as TL
Prelude T TL> TL.head (TL.pack (repeat 'x'))
'x'
Another practical problem is that you can end up with a mismatch between strict and lazy Text
s when using libraries. You can usually fix this by using toStrict
or fromStrict
as needed.
Prelude T TL> lazyPhrase = TL.pack "brevity is the soul of wit"
Prelude T TL> :t lazyPhrase
lazyPhrase :: TL.Text
Prelude T TL> :t phrase
phrase :: T.Text
Prelude T TL> lazyPhrase == phrase
<interactive>: error:
Couldn't match expected type ‘TL.Text’
• type ‘T.Text’
with actual NB: ‘T.Text’ is defined in ‘Data.Text.Internal’
TL.Text’ is defined in ‘Data.Text.Internal.Lazy’
‘In the second argument of ‘(==)’, namely ‘phrase’
• In the expression: lazyPhrase == phrase
In an equation for ‘it’: it = lazyPhrase == phrase
Prelude T TL> :t TL.toStrict
:: TL.Text -> T.Text
TL.toStrictPrelude T TL> :t TL.fromStrict
:: T.Text -> TL.Text
TL.fromStrictPrelude T TL> TL.toStrict lazyPhrase == phrase
True
ByteString
We can walk through pretty much the same GHCi session using ByteString
instead of Text
. However, note how the ByteString
is built up from Word8
values and not Char
values. A Char
can represent an arbitrary unicode codepoint like for a character like 'Å'
, but a Word8
represents a byte: a number from 0 to 255. Unfortunately and somewhat confusingly, ByteString
values get printed like String
s.
Prelude> import Data.Word
Prelude Data.Word> import qualified Data.ByteString as B
Prelude Data.Word B> binary = B.pack [99,111,102,102,101,101]
Prelude Data.Word B> :t binary
binary :: B.ByteString
Prelude Data.Word B> :t B.pack
:: [Word8] -> B.ByteString
B.packPrelude Data.Word B> binary
"coffee"
Prelude Data.Word B> :t B.length
:: B.ByteString -> Int
B.lengthPrelude Data.Word B> B.length binary
6
Prelude Data.Word B> :t B.head
:: B.ByteString -> Word8
B.headPrelude Data.Word B> B.head binary
99
Prelude Data.Word B> B.take 4 binary
"coff"
Prelude Data.Word B> :t B.map
:: (Word8 -> Word8) -> B.ByteString -> B.ByteString
B.mapPrelude Data.Word B> B.map (+1) binary
"dpggff"
The same caveats apply to the differences between strict and lazy ByteString
as for Text
:
Prelude B Data.Char> B.head (B.pack (repeat 99))
-- never returns
Prelude Data.Word B> import qualified Data.ByteString.Lazy as BL
Prelude Data.Word B BL> BL.head (BL.pack (repeat 99))
99
Prelude Data.Word B BL> binary == BL.pack [99]
<interactive>: error:
Couldn't match expected type ‘B.ByteString’
• type ‘BL.ByteString’
with actual NB: ‘BL.ByteString’ is defined in ‘Data.ByteString.Lazy.Internal’
B.ByteString’ is defined in ‘Data.ByteString.Internal’
‘In the second argument of ‘(==)’, namely ‘BL.pack [99]’
• In the expression: binary == BL.pack [99]
In an equation for ‘it’: it = binary == BL.pack [99]
Prelude Data.Word B BL> :t BL.toStrict
:: BL.ByteString -> B.ByteString
BL.toStrictPrelude Data.Word B BL> :t BL.fromStrict
:: B.ByteString -> BL.ByteString
BL.fromStrictPrelude Data.Word B BL> binary == BL.toStrict (BL.pack [99])
False
You might be wondering why on earth do we have both Text and ByteString. The difference is subtle but real. When we operate on Text
we operate character by character, regardless of what those characters are and how they are encoded. When we operate on ByteString
we operate on bytes, regardless of what those bytes represent.
Characters, numbers, and data structures are abstractions that help us humans to deal with complex programming tasks. The computer memory is essentially just an enormous sequence of bytes. The machine doesn’t care how we interpret those bytes. The essential difference between Text
and ByteString
is the way how the bytes are grouped and interpreted.
To illustrate this difference, we’ll look at the UTF-8 text encoding. A text encoding is a method for representing characters as bytes. UTF-8 can represent all the millions of characters defined by Unicode. Because bytes can only store values from 0 to 255, this means that a character can get encoded into multiple bytes. The bits and bytes of the UTF-8 string “Ha∫keλ!” can be interpreted in various ways:
(If you see different characters in the picture and in “Ha∫keλ!”, it means that your browser is either interpreting the encoding incorrectly or the font you’re using doesn’t support all of the characters.)
If we read the same stream of bits using a different encoding, we’ll see other characters. For example the string above would be interpreted as “Haâ«keÎ»!” using the Latin-1 text encoding.
By the way, when dealing with raw binary data, it’s often convenient to use the hexadecimal number system which uses a single symbol 0
, 1
, …, 9
, A
, B
, …, F
to represent all of the sixteen possible combinations of four bits. We don’t need hexadecimal in this course but you can check out Wikipedia if you’re interested in learning more about hexadecimal.
We can explore the same example using code. The function Data.Text.Encoding.encodeUtf8 :: Text -> ByteString
encodes the characters in a Text into bytes in a ByteString using UTF-8.
Prelude> import qualified Data.Text as T
Prelude T> import qualified Data.ByteString as B
Prelude T B> T.length (T.pack "haskell")
7
Prelude T B> T.length (T.pack "Ha∫keλ!")
7
Prelude T B> import Data.Text.Encoding
Prelude T B Data.Text.Encoding> encodeUtf8 (T.pack "haskell")
"haskell"
Prelude T B Data.Text.Encoding> encodeUtf8 (T.pack "Ha∫keλ!")
"Ha\226\136\171ke\206\187!"
Prelude T B Data.Text.Encoding> B.length (encodeUtf8 (T.pack "haskell"))
7
Prelude T B Data.Text.Encoding> B.length (encodeUtf8 (T.pack "Ha∫keλ!"))
10
If we are processing ASCII text, that is, characters that can be represented using single bytes, we can use Text
and ByteString
interchangeably. There are namespaces Data.ByteString.Char8
and Data.ByteString.Lazy.Char8
that offer functions that operate on ByteString
s using Char
values instead of Word8
. However, care must be taken to make sure all the characters really are plain ASCII characters, or surprising things will happen.
Prelude T B> import qualified Data.ByteString.Char8 as B8
Prelude T B B8> B8.pack "abc"
"abc"
Prelude T B B8> :t B8.pack
:: String -> B.ByteString
B8.packPrelude T B B8> :t B.pack
:: [Word8] -> B.ByteString
B.packPrelude T B B8> B8.cons 'a' (B8.pack "bc")
"abc"
Prelude T B B8> putStrLn (B8.unpack (B8.pack "€λ훈")) -- non-ASCII characters get truncated
¬»ÈPrelude T B B8> putStrLn (T.unpack (T.pack "€λ훈"))
€λ훈
The next libraries we’ll look at work inside the IO monad. Here’s a short recap of what we learned about monads in the last lecture.
M
is a monad, values of type M X
are operations that can be executed to produce a value of type X
.Monad
type class (return
, >>=
, >>
),do
-notation,mapM
.return
is not a keyword and does not cause an operation to stop executing. Instead, return x
is the operation that always produces x and does nothing else.do
-notation looks like:= do
foo y -- run an operation
operation1 <- operation2 -- run an operation and keep the produced value
val -- run an operation with parameters
operation3 val y mapM_ (\x -> operation4 val x) things -- use a generic monad operation and a lambda
-- the final line of the do decides which value the whole block produces operation5 val
Sometimes it feels like everything in the world happens over HTTP and Web Apis. Your web browser, your smartphone apps, your bank, your coffee pot, and even your doorbell, all talk to servers using the HTTP protocol.
Let’s look at how we can set up a simple HTTP server in Haskell. The standard low level components for this are called WAI and Warp. WAI, the Web Application Interface gives us a way to define how HTTP requests are handled. Warp is a simple HTTP server that runs the logic we have defined using WAI. That probably sounds a bit abstract right now, but a simple example will help.
The file exercises/Examples/HelloServer.hs
implements a HTTP server that always responds with “Hello World!”. You can try it out by going to the exercises/Examples
directory and running it with stack runhaskell HelloServer.hs
. After that you can visit http://localhost:3421 in your browser to see the response from the server.
module Examples.HelloServer where
import qualified Data.ByteString.Lazy.Char8 as BL
import Network.HTTP.Types.Status (status200)
import Network.Wai (Application, responseLBS)
import Network.Wai.Handler.Warp (run)
port :: Int
= 3421
port
main :: IO ()
= run port application
main
-- type Application = Request -> (Response -> IO ResponseReceived) -> IO ResponseReceived
application :: Application
=
application request respond "Hello World!")) respond (responseLBS status200 [] (BL.pack
Let’s look at the types in this example. There’s a lot going on here. First off, Application
is a type alias for a thing that implements the logic of a web server. The run
function from Warp can run Application
s:
run :: Port -> Application -> IO ()
type Application = Request -> (Response -> IO ResponseReceived) -> IO ResponseReceived
We’ll talk about what Request
and Response
are soon, but from this type we can see that an Application
is an IO operation that takes as arguments a request of type Request
, and an IO operation respond :: Response -> IO ResponseReceived
. Arguments like respond
are called callbacks in many contexts. They allow us to call back to the library who called the application. The Application
operation has to produce the same special ResponseReceived
type that respond
. You can think of this type as a token proving that respond
was called by the Application
.
That might sound intimidating: but looking at the code things are relatively simple: our server
is an Application
and takes two parameters: request
and respond
.
WAI uses lots of types like Port
, Request
, Response
, Status
to represent HTTP concepts. It’s useful to look these up in the documentation when you bump into them. For example Port
is just an alias for Int
. As another example, we can see the responseLBS
function has the type
responseLBS :: Status -> ResponseHeaders -> ByteString -> Response
where Status
is defined in Network.HTTP.Types.Status
, ResponseHeaders
is a type alias for [Header]
from Network.HTTP.Types.Header
, ByteString
is a lazy ByteString
, and the result type Response
is defined by Network.WAI
.
Finally, note that we take a shortcut and use the function Data.ByteString.Lazy.Char8.pack
to convert a String
into a ByteString
. This only works for ASCII text.
A web server that always responds with the same text isn’t that interesting. Let’s look at how we can give different responses to different requests next. There are many parts in a HTTP request, but for this lecture we’re going to focus on the path. In a URL like http://example.com/abcd/ef/file
, the /abcd/ef/file
part is the path. WAI has the function
pathInfo :: Request -> [Text]
This gives us the path of the requested URL, split at the /
characters.
The file exercises/Examples/PathServer.hs
implements a web server that has three different pages:
As before, you can run the server by going into the exercises/Examples
directory, and running stack runhaskell PathServer.hs
.
After implementing a HTTP server we can participate in the global graph of applications talking to each other that’s called the internet. But what use is talking if we can’t remember? A real application needs to be able to persist data even when it is restarted. A common way to accomplish this is to use a database.
There are many different kinds of databases, but arguably the most widely used simple database is SQLite. SQLite is a library that lets you store data in a file and process it using SQL, the Structured Query Language. With SQLite there is no need to run a separate database server a la PostgreSQL or MySQL.
In case you’re not familiar with SQL, don’t worry, you won’t need to write any queries of your own in the exercises. If you’d like to learn a bit of SQL now, there are lots of tutorials on the web. See W3Schools, SQL Zoo or Codecademy.
There are many libraries for Haskell for using SQLite, but we’ll look at one called sqlite-simple here. Let’s explore the library in GHCi for a bit.
All the functions live inside Database.SQLite.Simple
. You can open a database by giving a filename to open
, which is an IO operation that produces a Connection
.
Prelude> import Database.SQLite.Simple
Prelude Database.SQLite.Simple> :t open
open :: String -> IO Connection
Prelude Database.SQLite.Simple> db <- open "example.sqlite"
To run an SQL query you can use IO operation query_
, which takes a Connection
and a Query
, and produces a list of results. The Query
type is just a simple newtype
around Text
. The result type of query_
is polymorphic: you can read any type that satisfies the FromRow
type class from the database. If this feels confusing, compare it to the type of read
: Read a => String -> a
. The FromRow
class is like Read
for this database: it represents types that can be read out of the database. Anyway, let’s read the number 1
from the database:
Prelude Database.SQLite.Simple> :t query_
query_ :: FromRow r => Connection -> Query -> IO [r]
Prelude Database.SQLite.Simple> :info Query
newtype Query = Query {fromQuery :: Data.Text.Internal.Text}
-- Defined in ‘Database.SQLite.Simple.Types’
-- ... rest of output omitted
Prelude Database.SQLite.Simple> import qualified Data.Text as T
Prelude Database.SQLite.Simple T> q = Query (T.pack "SELECT 1;")
Prelude Database.SQLite.Simple T> res <- query_ db q :: IO [[Int]]
Prelude Database.SQLite.Simple T> res
1]] [[
By the way, all of these initial examples use simple SELECT x, y, z;
queries that just return constant data. We’ll worry about actual tables in the database later.
Without the type signature, we get an error from GHCi, which can’t decide which type we want to read out from the database:
Prelude Database.SQLite.Simple T> res <- query_ db q
<interactive>:17:8: error:
Ambiguous type variable ‘r0’ arising from a use of ‘query_’
• FromRow r0)’ from being solved.
prevents the constraint ‘(Probable fix: use a type annotation to specify what ‘r0’ should be.
-- rest of error omitted
Before we go on, let’s have a closer look at FromRow
. If you’ve bumped into SQL before, you know that an SQL query returns a number of rows, and each row consists of a number of values (also called columns). To be able to interpret the result of an SQL query into Haskell data, we need a way to interpret these values and rows. Thus sqlite-simple defines two classes, FromField
and FromRow
, and a bunch of instances like the following. (You can find these instances from the docs or by asking GHCi with :info FromRow
etc.)
instance FromField Int
instance FromField Bool
instance FromField String
instance FromField Text
instance FromField a => FromRow [a]
instance (FromField a, FromField b) => FromRow (a,b)
instance (FromField a, FromField b, FromField c) => FromRow (a,b,c)
In essence, basic Haskell datatypes fulfill the FromField
class, and various Haskell collections fulfill the FromRow
class. Our earlier example was using the FromRow [a]
and FromField Int
instances to get a [[Int]]
out of query_
. Here’s a simple query that uses some other datatypes:
Prelude Database.SQLite.Simple T> q = Query (T.pack "SELECT 1, true, 'string';")
Prelude Database.SQLite.Simple T> query_ db q :: IO [(Int,Bool,String)]
1,True,"string")] [(
What happens if the SQL and Haskell types don’t match? Well, you get a runtime error, just like if you try to invoke read "True" :: Int
.
Prelude Database.SQLite.Simple T> query_ db q :: IO [(Int,Int,Int)]
*** Exception: ConversionFailed {errSQLType = "TEXT", errHaskellType = "Int", errMessage = "need an int"}
To mirror the FromRow
and FromField
classes, sqlite-simple also defines the ToRow
and ToField
classes for writing into the database. Here’s the type of the query
function, which allows us to use parameterized queries.
query :: (ToRow q, FromRow r) => Connection -> Query -> q -> IO [r]
And here are some instances for ToRow
and ToField
:
instance ToField Int
instance ToField Bool
instance ToField String
instance ToField Text
instance ToField Int
instance ToField a => ToRow [a]
instance (ToField a, ToField b) => ToRow (a, b)
instance (ToField a, ToField b, ToField c) => ToRow (a, b, c)
instance ToField a => ToRow (Only a)
Parameterized queries use a ?
character to denote slots where parameters can be passed in. Here’s a simple example:
Prelude Database.SQLite.Simple T> input = (1,"hello") :: (Int,String)
Prelude Database.SQLite.Simple T> parameterized = Query (T.pack "SELECT ?+1, true, ?;")
Prelude Database.SQLite.Simple T> query db parameterized input :: IO [(Int,Bool,String)]
2,True,"hello")] [(
Note! When performing a query with only one parameter, there are two ToRow
instances you can use: ToField a => ToRow [a]
and ToField a => Only a
. The Only
datatype is defined in Data.Tuple.Only
and is kind of a workaround for the fact that Haskell doesn’t have one-element tuples. Alternatively, a list of size 1 works as well. The same applies for queries that return rows with only one column: you can use either [[X]]
or [Only X]
as the return type. Here’s an example:
Prelude Database.SQLite.Simple T> q = Query (T.pack "SELECT lower(?);")
Prelude Database.SQLite.Simple T> query db q (Only "HELLO") :: IO [Only String]
Only {fromOnly = "hello"}]
[Prelude Database.SQLite.Simple T> query db q ["HELLO"] :: IO [[String]]
"hello"]] [[
That’s pretty much all you need to know about sqlite-simple: open
, query_
, query
, FromRow
, ToRow
. Oh right, one more thing. There are the functions execute
and execute_
if you don’t need the result of the query. They’re useful for inserting things into the database, for example.
execute_ :: Connection -> Query -> IO ()
execute :: ToRow q => Connection -> Query -> q -> IO ()
You’ll find an example program that uses sqlite-simple to maintain a phone book under exercises/Examples/Phonebook.hs
. The program keeps the phonebook in a file called phonebook.db
and works like this (run from the exercises/Examples
directory in the course repository):
$ stack runhaskell Phonebook.hs
(a)dd or (q)uery?
a
Name?
bob
Phone?
1234
$ stack runhaskell Phonebook.hs
(a)dd or (q)uery?
a
Name?
bob
Phone?
5678
$ stack runhaskell Phonebook.hs
(a)dd or (q)uery?
a
Name?
samantha
Phone?
1357
$ stack runhaskell Phonebook.hs
(a)dd or (q)uery?
q
Name?
bob
2 numbers:
["1234"]
["5678"]
PS. If you’re devastated by the lack of compile-time type checking for SQL queries, you can have a look at some of the more advanced SQL libraries for Haskell like Beam or Opaleye. This course is using sqlite-simple for simplicity and to avoid dwelling on the details of SQL too much.
The Applicative
type class is a middle ground between Functor
(which you can’t do that much with) and Monad
(which pretty much allows you to write arbitrary programs). Reasons to use Applicative
instead of Monad
include:
Applicative
allows less operations, it can be optimized better than Monad
.Applicative
interface is easier to reason about.Monad
instance for your type, but there is an Applicative
instance. This is rare.So what is an Applicative
? Let’s look at a definition.
class Functor f => Applicative f where
pure :: a -> f a
liftA2 :: (a -> b -> c) -> f a -> f b -> f c
-- other operations omitted for now
So an Applicative
is a Functor
that allows us to build singleton values via pure
, and combine two values into one using liftA2
. This adds a lot of power compared to a bare functor. Computations using functors are necessarily linear: fmap :: (a -> b) -> f a -> f b
takes in one functorial value, and outputs another. By contrast, pure
takes in no functorial value and outputs one, and liftA2
takes in two and returns one.
Sidenote: the term Applicative comes from the term Applicative Functor, which sounds like it comes from Category Theory, but was actually introduced in a programming paper.
That’s enough abstract mumbo jumbo for now. Let’s look at what sort of computations we can express using Applicative operations (and fmap
). We’ll start with the Maybe
applicative. Here’s a streamlined definition:
instance Applicative Maybe where
pure x = Just x
Just x) (Just y) = Just (f x y)
liftA2 f (= Nothing liftA2 f _ _
You’ll see the definition uses the same type of failure propagation as the Monad Maybe
instance. Let’s use this when parsing monetary values:
data Currency = EUR | USD
deriving (Show, Eq)
data Money = Money Int Currency
deriving (Show, Eq)
parseCurrency :: String -> Maybe Currency
"e" = pure EUR
parseCurrency "€" = pure EUR
parseCurrency "$" = pure USD
parseCurrency = Nothing
parseCurrency _
parseAmount :: String -> Maybe Int
= readMaybe
parseAmount
parseMoney :: String -> String -> Maybe Money
=
parseMoney amountString currencyString Money (parseAmount amountString) (parseCurrency currencyString) liftA2
"123" "€" ==> Just (Money 123 EUR)
parseMoney "45" "$" ==> Just (Money 45 USD)
parseMoney "4x" "€" ==> Nothing
parseMoney "45" "£" ==> Nothing parseMoney
That worked out nicely. However, if we try to expand on this we soon run into the limits of Applicative
. For example, consider this sumMoney
function that sums Money
values but fails if they are not in the same currency:
sumMoney :: Money -> Money -> Maybe Money
Money a c) (Money b c')
sumMoney (| c == c' = Just (Money (a+b) c)
| otherwise = Nothing
We can’t apply it to two Maybe Money
values using Applicative
operations. We need the Maybe
monad for that:
example :: Maybe Money
= do x <- parseMoney "123" "€"
example <- parseMoney "45" "$"
y sumMoney x y
If we try to use liftA2
, we’re stuck with a Maybe (Maybe Money)
type. In addition, we can now get two different types of failures: Nothing
and Just Nothing
, depending on what level the error happens on. This would be a clear case for switching to a Monad
instance.
liftA2 sumMoney (parseMoney "123" "e") (parseMoney "45" "€")
==> Just (Just (Money 168 EUR))
liftA2 sumMoney (parseMoney "123" "e") (parseMoney "45" "$")
==> Just Nothing
liftA2 sumMoney (parseMoney "123" "e") (parseMoney "xxx" "e")
==> Nothing
Sidenote: the name liftA2
sounds a bit cumbersome, but it’s an analogy with the liftM
, liftM2
and so on functions for monads. Recall that liftM
was just fmap
, so perhaps liftA2
should’ve been called fmap2
.
Let’s have a look at the applicative instances for another Functor
we’ve seen. The Applicative
instance for the list functor goes through all possible combinations of values (just like the list monad). Here’s the instance:
instance Applicative [] where
pure x = [x]
= [f x y | x <- xs, y <- ys] liftA2 f xs ys
And here’s an example: generating some phrases.
things :: [String]
= ["tangerine","bandit","diamond"]
things
fruits :: [String]
= ["apple", "tangerine"]
fruits
phrases :: [String]
= liftA2 combine things fruits
phrases where combine t f = "a " ++ t ++ " the size of a " ++ f
= liftA2 copy [1,2,3] fruits
bunches where copy n f = unwords (replicate n f)
==> ["a tangerine the size of a apple",
phrases "a tangerine the size of a tangerine",
"a bandit the size of a apple",
"a bandit the size of a tangerine",
"a diamond the size of a apple",
"a diamond the size of a tangerine"]
==> ["apple","tangerine",
bunches "apple apple","tangerine tangerine",
"apple apple apple","tangerine tangerine tangerine"]
There are a handful of operators for Applicatives that are quite handy. They are <$>
, <*>
, <*
and *>
.
Let’s start with <$>
, which is just an infix version of fmap
:
(<$>) :: Functor f => (a -> b) -> f a -> f b
<$> x = fmap f x f
not <$> Just True ==> Just False
not <$> Nothing ==> Nothing
negate <$> [1,2,3] ==> [-1,-2,-3]
That’s kinda nice on its own, but it really gets to shine when combined with this Applicative operator:
(<*>) :: Applicative f => f (a -> b) -> f a -> f b
The type tells you what <*>
does: it’s function application lifted to an applicative. Here are some standalone examples:
Just not <*> Just True ==> Just False
Nothing <*> Just True ==> Nothing
Just not <*> Nothing ==> Nothing
+1),(*2)] <*> [10,100] ==> [11,101,20,200] [(
The real magic happens when we combine <$>
and <*>
: we can then lift functions of arbitrarily many arguments to an Applicative!
say :: String -> Int -> String -> String
= x ++ " has " ++ show i ++ " " ++ y say x i y
<$> Just "haskell" <*> Just 99 <*> Just "operators"
say ==> Just "haskell has 99 operators"
<$> Nothing <*> Just 99 <*> Just "operators"
say ==> Nothing
<$> ["bob","jake"] <*> [2,3] <*> ["bananas","cars"]
say ==> ["bob has 2 bananas",
"bob has 2 cars",
"bob has 3 bananas",
"bob has 3 cars",
"jake has 2 bananas",
"jake has 2 cars",
"jake has 3 bananas",
"jake has 3 cars"]
What’s going on here? Let’s step through the evaluation. The key is that each <*>
partially applies one more argument to the function.
<$> Just "haskell" <*> Just 99 <*> Just "operators"
say === ((say <$> Just "haskell") <*> Just 99) <*> Just "operators"
=== (fmap say (Just "haskell") <*> Just 99) <*> Just "operators"
==> (Just (say "haskell") <*> Just 99) <*> Just "operators"
==> Just (say "haskell" 99) <*> Just "operators"
==> Just (say "haskell" 99 "operators")
==> Just "haskell has 99 operators"
Perhaps looking at the types will make it clearer:
<$> Just "haskell" :: Maybe (Int -> String -> String)
say <$> Just "haskell" <*> Just 99 :: Maybe ( String -> String)
say <$> Just "haskell" <*> Just 99 <*> Just "operators" :: Maybe ( String) say
The next two operators are a bit simpler:
(*>) :: Applicative f => f a -> f b -> f b
*> y = liftA2 (\a b -> b) x y
x
(<*) :: Applicative f => f a -> f b -> f a
<* y = liftA2 (\a b -> a) x y x
You can compare the types to a more familiar operator:
(>>) :: Monad m => m a -> m b -> m b
What the operators <*
and *>
mean is: run both of these operations, but only keep one result. The arrow points to the result that’s kept:
Just 1 *> Just 2 ==> Just 2
Just 1 <* Just 2 ==> Just 1
Just 1 <* Nothing ==> Nothing
Nothing <* Just 2 ==> Nothing
These operators might seem trivial, but they’re useful when combining checks. For example:
decrease :: Int -> Maybe Int
= if i>0 then Just (i-1) else Nothing
decrease i
small :: Int -> Maybe Int
= if i<10 then Just i else Nothing
small i
decreaseSmall :: Int -> Maybe Int
-- do what decrease does, but fail if small fails
= decrease i <* small i decreaseSmall i
4 ==> Just 3
decreaseSmall 0 ==> Nothing
decreaseSmall 11 ==> Nothing decreaseSmall
Now that we’ve seen all these operators, we can understand the full definition of Applicative
. All the operators have definitions in terms of liftA2
, so it’s enought to define liftA2
and pure
when implementing an Applicative
instance.
class Functor f => Applicative f where
pure :: a -> f a
liftA2 :: (a -> b -> c) -> f a -> f b -> f c
(<*>) :: f (a -> b) -> f a -> f b
(*>) :: f a -> f b -> f b
(<*) :: f a -> f b -> f a
Let’s look at an Applicative that’s a bit more interesting than Maybe or lists. Often in programming we need to validate some inputs from the user. In these cases it’s useful to gather together all the errors that the input might have. The file exercises/Examples/Validation.hs
implements the Validation
datatype:
data Validation a = Ok a | Errors [String]
deriving (Show,Eq)
The Applicative
instance for Validation
works like this:
+) (Ok 1) (Ok 2)
liftA2 (==> Ok 3
+) (Errors ["oh no"]) (Errors ["boom"])
liftA2 (==> Errors ["oh no","boom"]
Note how in contrast to the Maybe
Applicative, we have many different kinds of failures.
Here’s a worked example that introduces some helpers and then uses them to congratulate somebody on their birthday:
invalid :: String -> Validation a
= Errors [err]
invalid err
check :: Bool -> String -> a -> Validation a
check b err x| b = pure x
| otherwise = invalid err
birthday :: String -> Int -> Validation String
= liftA2 congratulate checkedName checkedAge
birthday name age where checkedName = check (length name < 10) "Name too long" name
= check (age < 99) "Too old" age
checkedAge = "Happy "++show a++"th birthday "++n++"!" congratulate n a
"Guy" 31
birthday ==> Ok "Happy 31th birthday Guy!"
"Guybrush Threepwood" 31
birthday ==> Errors ["Name too long"]
"Yog-sothoth" 10000
birthday ==> Errors ["Name too long","Too old"]
Oh right, here are the Functor
and Applicative
instances for Validation
:
instance Functor Validation where
fmap f (Ok x) = Ok (f x)
fmap _ (Errors e) = Errors e
instance Applicative Validation where
pure x = Ok x
Ok x) (Ok y) = Ok (f x y)
liftA2 f (Errors e1) (Ok y) = Errors e1
liftA2 f (Ok x) (Errors e2) = Errors e2
liftA2 f (Errors e1) (Errors e2) = Errors (e1++e2) liftA2 f (
The definition of liftA2
for Validation
shows that the errors are collected together left-to-right. This can be seen in the example above where the expression liftA2 congratulate checkedName checkedAge
outputs the error ("Name too long"
) from checkedName
first, and the error ("Too old"
) from checkedAge
last.
traverse
So far we’ve dealt with fixed-size things and Applicatives: we’ve applied a function of two or three arguments to some things. What if we have an arbitrary amount of inputs? What if we need to validate a list?
Let’s look at some ways to implement a function like this:
1,2,3]
allPositive [==> Ok [1,2,3]
1,2,3,-4]
allPositive [==> Errors ["Not positive: -4"]
1,-2,3,-4]
allPositive [==> Errors ["Not positive: -2","Not positive: -4"]
As always, when working with lists, pattern matching and recursion are usually the way to go. Here’s a recursive solution:
allPositive :: [Int] -> Validation [Int]
= Ok []
allPositive [] :xs) = liftA2 (:) checkThis checkRest
allPositive (xwhere checkThis = check (x>=0) ("Not positive: "++show x) x
= allPositive xs checkRest
It’s a bit of a chore to always spell out a recursion like this. If we were working in a Monad
we could just use a helper like mapM
:
mapM (\x -> if x>=0 then Just x else Nothing) [1,2,3]
==> Just [1,2,3]
mapM (\x -> if x>=0 then Just x else Nothing) [1,2,3,-4]
==> Nothing
The equivalent of mapM
for Applicative
is called traverse
. It’s a member of the type class Traversable
:
traverse :: (Traversable t, Applicative f) => (a -> f b) -> t a -> f (t b)
That’s one heck of a type signature, so let’s simplify it a bit. Lists are Traversable
, so we can specialize this type into:
traverse :: Applicative f => (a -> f b) -> [a] -> f [b]
That looks like what we need! For Applicatives that are also Monads, traverse
is just another name for mapM
:
traverse (\x -> if x>=0 then Just x else Nothing) [1,2,3]
==> Just [1,2,3]
traverse (\x -> if x>=0 then Just x else Nothing) [1,2,3,-4]
==> Nothing
But for our Validation
, which isn’t a Monad
, traverse
is exactly what we want:
allPositive :: [Int] -> Validation [Int]
= traverse checkNumber xs
allPositive xs where checkNumber x = check (x>=0) ("Not positive: "++show x) x
1,2,3]
allPositive [==> Ok [1,2,3]
1,2,3,-4]
allPositive [==> Errors ["Not positive: -4"]
1,-2,3,-4]
allPositive [==> Errors ["Not positive: -2","Not positive: -4"]
Note how traverse
for Validation
collects all the errors together, in the order they occur in the original list.
PS. In fact, Validation
is one of the few examples of an Applicative
that can’t be a Monad
. Can you figure out why?
Traversable
So what things are Traversable
? Many familiar structures. Here are some examples:
decrease :: Int -> Maybe Int
= if i>0 then Just (i-1) else Nothing decrease i
-- Lists are Traversable
traverse decrease [1,2,3] ==> Just [0,1,2]
traverse decrease [1,0,3] ==> Nothing
-- Arrays are Traversable
traverse decrease (array (1,3) [(1,10),(2,11),(3,12)])
==> Just (array (1,3) [(1,9),(2,10),(3,11)])
-- Maps are Traversable
traverse decrease (M.fromList [("a",1),("b",2)])
==> Just (M.fromList [("a",0),("b",1)])
traverse decrease (M.fromList [("a",1),("b",0)])
==> Nothing
-- Either is Traversable
traverse decrease (Left "abc") ==> Just (Left "abc")
traverse decrease (Right 3) ==> Just (Right 2)
traverse decrease (Right 0) ==> Nothing
So Traversable
is a type class for all sorts of containers, kind of like Foldable
. Indeed, if you look at the definition, Traversable
is a subclass of Foldable
. It also turns out that traverse
and mapM
are methods of the class!
class (Functor t, Foldable t) => Traversable t where
traverse :: Applicative f => (a -> f b) -> t a -> f (t b)
mapM :: Monad m => (a -> m b) -> t a -> m (t b)
It can be hard to keep the types straight here. Let’s go back to the type of traverse
:
traverse :: (Traversable t, Applicative f) => (a -> f b) -> t a -> f (t b)
Here we have two functors: t
and f
. The t
functor is also a Foldable
and a Traversable
and the f
functor is also an Applicative
. The traverse
function lets us run f
-operations inside a t
-container.
Don’t worry if that feels abstract. In practice, you pretty much always use traverse
on lists.
Alternative
If you play around with applicatives a bit you’ll start to notice some limits to their power. For example, when writing parsers like we did in the parseMoney
example, it would be nice to be able to try a couple of different parsers and take any non-failure result. This is easy enough to write for a concrete Applicative like Maybe
, as can be seen below.
data Answer = Yes | No
deriving (Show, Eq)
parseYes :: String -> Maybe Answer
"y" = Just Yes
parseYes "yes" = Just Yes
parseYes "maybe" = Just Yes
parseYes = Nothing
parseYes _
parseNo :: String -> Maybe Answer
"n" = Just No
parseNo "no" = Just No
parseNo "maybe" = Just No
parseNo = Nothing
parseNo _
eitherOf :: Maybe x -> Maybe x -> Maybe x
Just x) _ = Just x
eitherOf (Nothing mx = mx
eitherOf
parseAnswer :: String -> Maybe Answer
-- prefer positive answers!
= eitherOf (parseYes s) (parseNo s) parseAnswer s
"yes" ==> Just Yes
parseAnswer "y" ==> Just Yes
parseAnswer "n" ==> Just No
parseAnswer "maybe" ==> Just Yes
parseAnswer "x" ==> Nothing parseAnswer
How could we generalize eitherOf
? We can’t give it the type Applicative f => f x -> f x -> f x
because then the implementation would need to be effectively something like eitherOf a b = liftA2 something a b
, but then eitherOf Nothing (Just x)
would be Nothing
(since that’s how the Applicative instance works)!
It turns out we need a new type class: Alternative
. An Alternative adds two operations to Applicative: empty
means no results, and <|>
means combining results.
class Applicative f => Alternative f where
empty :: f a
(<|>) :: f a -> f a -> f a
-- some other operations omitted
Now we can rewrite our parsing code using general operations:
data Answer = Yes | No
deriving (Show, Eq)
parseYes :: Alternative f => String -> f Answer
"y" = pure Yes
parseYes "yes" = pure Yes
parseYes "maybe" = pure Yes
parseYes = empty
parseYes _
parseNo :: Alternative f => String -> f Answer
"n" = pure No
parseNo "no" = pure No
parseNo "maybe" = pure No
parseNo = empty
parseNo _
parseAnswer :: Alternative f => String -> f Answer
= parseYes s <|> parseNo s parseAnswer s
We can also pick which Alternative
we run our parser in to get different behaviours. Maybe
gives us only one result, while []
gives us all possible results.
> parseAnswer "yes" :: Maybe Answer
Just Yes
> parseAnswer "maybe" :: Maybe Answer
Just Yes
> parseAnswer "yes" :: [Answer]
Yes]
[> parseAnswer "maybe" :: [Answer]
Yes,No] [
The Alternative
instances for []
and Maybe
are unsurprising:
instance Alternative [] where
= []
empty <|>) = (++)
(
instance Alternative Maybe where
= Nothing
empty Just x <|> _ = Just x
Nothing <|> mx = mx
The Validation
type is also an Alternative
. The instance collects together all error messages, just like the Applicative
instance did.
instance Alternative Validation where
= Errors []
empty Ok x <|> _ = Ok x
Errors e1 <|> Ok y = Ok y
Errors e1 <|> Errors e2 = Errors (e1++e2)
Here’s a final example: validating contact information, which is either a phone number or an email address.
data ContactInfo = Email String | Phone String
deriving Show
validateEmail :: String -> Validation ContactInfo
= check (elem '@' s) "Not an email: should contain a @" (Email s)
validateEmail s
checkLength :: String -> Validation ContactInfo
= check (length s <= 10) "Not a phone number: should be at most 10 digits" (Phone s)
checkLength s
checkDigits :: String -> Validation ContactInfo
= check (all isDigit s) "Not a phone number: should be all numbers" (Phone s)
checkDigits s
validatePhone :: String -> Validation ContactInfo
= checkDigits s *> checkLength s
validatePhone s
validateContactInfo :: String -> Validation ContactInfo
= validateEmail s <|> validatePhone s validateContactInfo s
"user@example.com"
validateContactInfo ==> Ok (Email "user@example.com")
"01234"
validateContactInfo ==> Ok (Phone "01234")
"01234567890"
validateContactInfo ==> Errors ["Not an email: should contain a @","Not a phone number: should be at most 10 digits"]
"01234567890x"
validateContactInfo ==> Errors ["Not an email: should contain a @",
"Not a phone number: should be all numbers",
"Not a phone number: should be at most 10 digits"]
"x"
validateContactInfo ==> Errors ["Not an email: should contain a @",
"Not a phone number: should be all numbers"]
Note how here, just like in previous examples, the errors are collected left-to-right: the errors from validateEmail
come before the errors from validatePhone
. The error from checkDigits
comes before the error from checkLength
.
There are multiple reasons for learning Applicatives, even if they do not provide any additional power over Monads. First off, as discussed in Lecture 13, the GHC standard library nowadays makes Applicative instances for all Monads compulsory. Thus a working Haskell programmer is bound to see lots of Applicative instances.
Secondly, you’ll bump into Applicative operators a lot even in Monadic code. Expressions like f <$> x <*> y <*> z
can be useful in many Monadic contexts. Also, since the Traversable
type class is built in terms of Applicative
, you’ll often use applicative operations with it.
Thirdly, Applicatives are an excellent exercise in understanding functional design patterns. They marry the Functor pattern with the Monoid pattern, and Alternative brings in yet another Monoid-like dimension. Being able to efficiently work with Applicatives will make working with further abstractions like monad transformers or lenses easier.
Lastly, there are a couple of types that are Applicatives but not Monads. Validation
is one example, and a very practical one at that. Without understanding of Applicative, we’d have no way to recognize and generalize operations over types like this. Another such type is ZipList
.
Even though we only looked at some very simple and concrete Applicatives in this lecture, there are plenty of Haskell libraries that use Applicatives for big things. Here are some examples.
The same sort of idea as our Validation
Applicative has been implemented in the validation and either libraries.
There are multiple parser libraries that use Applicatives. For example, regex-applicative, optparse-applicative, yamlparse-applicative, json-stream, and so on.
So what’s the relationship between an Monad and an Applicative? If an Applicative is also a Monad, the following laws hold:
pure === return
fmap === liftM
fmap f op === do x <- op
return (f x)
=== liftM2
liftA2
=== do x <- op1
liftA2 f op1 op2 <- op2
y return (f x y)
*> op2 === op1 >> op2
op1
<*> op2 === do f <- op1
op1 <- op2
x return (f x)
When working in a Monad
, you can freely mix Applicative
and Functor
with Monad
operations. As an example, let’s rewrite mapM
until it only uses applicative operations, and thus get an implementation for traverse
. This is our starting point:
= return []
myMapM op [] :xs) = do y <- op x
myMapM op (x<- myMapM op xs
ys return (y:ys)
GHCi tells us it only works for Monads:
Prelude> :t myMapM
myMapM :: Monad m => (a -> m b) -> [a] -> m [b]
Let’s apply the pure === return
and the liftA2
laws from above:
= pure []
myMapM op [] :xs) = liftA2 (:) (op x) (myMapM op xs) myMapM op (x
Ta-da! Now myMapM
works for any Applicative
:
Prelude> :t myMapM
myMapM :: Applicative f => (a -> f b) -> [a] -> f [b]
What’s the type of x
in liftA2 (&&) Nothing x
?
Applicative f => f Bool
Applicative Bool
Maybe Bool
How many elements does liftA2 f xs ys
have when xs
and ys
are lists?
length xs + length ys
length xs * length ys
min (length xs) (length ys)
Which of these expressions is equivalent to liftA2 f x y
? There might be multiple correct answers.
f <$> x <*> y
f <*> x <*> y
f <*> x <$> y
fmap f x <*> y
pure f <*> x <*> y
If f :: a -> Maybe b
and xs :: [a]
, which of these expressions has a type different from the others?
fmap f xs
traverse f xs
map f xs
For which Applicative
do the expressions pure x <* pure y
and pure x <|> pure y
produce different results?
Maybe
[]
Validation
This final lecture will go over some minor topics that didn’t fit in anywhere else. You’ve finished all the hard parts of the course. Now it’s time to sit back, relax, and enjoy some cool Haskell!
One of the benefits of purity is that pure functions are easy to test: you don’t need to set up any global state, you can just pass in arguments and check that the result is ok. In this section, we’ll take a quick tour of the property-based testing library QuickCheck, which has also been used for checking that your answers to exercises are right on this course!
Let’s look at testing a (faulty) implementation of reverse
. You can find this and the following examples in the file exercises/Examples/QuickCheck.hs
.
rev :: [a] -> [a]
= []
rev [] :xs) = xs ++ [x] rev (x
We can write an individual test case using the ===
operator from QuickCheck:
(===) :: (Eq a, Show a) => a -> a -> Property
propRevSmall :: Property
= rev [1,2] === [2,1] propRevSmall
We can ask QuickCheck to run them in GHCi:
*Examples.QuickCheck> quickCheck propRevSmall
+++ OK, passed 1 test.
So far so good. However this isn’t really what QuickCheck was made for. QuickCheck was designed for property-based testing where you can state a property your code should have, and QuickCheck runs your code with randomized inputs, checking the property every time. What would be a simple property that a correct implementation of reverse
has? Reversing a list twice should certainly give us back the same list. Let’s write it out:
propRevTwice :: [Int] -> Property
= rev (rev xs) === xs propRevTwice xs
Our Property
has an argument, which means that QuickCheck will generate random values and run the test. We can use the verboseCheck
function to see which values are run. We can also give a parameter to the test ourselves if we wish to check a specific value.
*Examples.QuickCheck> quickCheck propRevTwice
+++ OK, passed 100 tests.
*Examples.QuickCheck> verboseCheck propRevTwice
Passed:
[]== []
[]
Passed:
1]
[1] == [1]
[
Passed:
-2,1,-1]
[-2,1,-1] == [-2,1,-1]
[-- lots of output
+++ OK, passed 100 tests.
*Examples.QuickCheck> quickCheck (propRevTwice [1,2,3])
+++ OK, passed 1 test.
Even this property didn’t catch the bug in our implementation. Let’s try another one. Here’s a property about how rev (xs ++ ys)
behaves. You might want to take a moment to convince yourself that it should hold for a correct rev
function.
propRevTwo :: [Int] -> [Int] -> Property
= rev (xs ++ ys) === rev ys ++ rev xs propRevTwo xs ys
Let’s see if it holds for our implementation:
*Examples.QuickCheck> quickCheck propRevTwo
*** Failed! Falsified (after 5 tests and 3 shrinks):
0,0]
[1]
[0,1,0] /= [1,0,0] [
Finally, a failure! There’s a bit to unpack here. First off, QuickCheck tells us the arguments with which the property failed: they are [0,0]
and [1]
. We can check this ourselves:
*Examples.QuickCheck> quickCheck (propRevTwo [0,0] [1])
*** Failed! Falsified (after 1 test):
0,1,0] /= [1,0,0] [
Next up, what does “after 5 tests and 3 shrinks” mean? One of the cool things about QuickCheck is that when it finds a failure, it tries some related values in order to find a nicer, smaller failure. We can see this in action with verboseShrinking
, which prints out all the failures QuickCheck goes through:
*Examples.QuickCheck> quickCheck (verboseShrinking propRevTwo)
Failed:
4,-1,-4]
[1,4]
[-1,-4,1,4,4] /= [4,1,-1,-4,4]
[
Failed:
-1,-4]
[1,4]
[-4,1,4,-1] /= [4,1,-4,-1]
[
Failed:
-4]
[1,4]
[1,4,-4] /= [4,1,-4]
[
Failed:
4]
[1,4]
[1,4,4] /= [4,1,4]
[
Failed:
0]
[1,4]
[1,4,0] /= [4,1,0]
[
Failed:
0]
[0,4]
[0,4,0] /= [4,0,0]
[
Failed:
0]
[0,2]
[0,2,0] /= [2,0,0]
[
Failed:
0]
[0,1]
[0,1,0] /= [1,0,0] [
QuickCheck went down from a counterexample of [4,1,-1,4,4]
all the way to [1,0,0]
. Pretty sweet!
Sometimes you need to limit the values QuickCheck generates. Your function might not work on all inputs, for instance? Let’s try writing a test for last
.
propLast :: [Int] -> Property
= last xs === head (reverse xs) propLast xs
*Examples.QuickCheck> quickCheck propLast
*** Failed! Exception: 'Prelude.last: empty list' (after 1 test):
[]
In this case, we can fix the test just by switching to another input type. QuickCheck defines the NonEmptyList
type (not to be confused with Data.List.NonEmpty
!), which is just a wrapper for a normal list. However, when generating values of NonEmptyList
, QuickCheck won’t generate empty lists.
newtype NonEmptyList a = NonEmpty [a]
propLastFixed :: NonEmptyList Int -> Property
NonEmpty xs) = last xs === head (reverse xs) propLastFixed (
*Examples.QuickCheck> quickCheck propLastFixed
+++ OK, passed 100 tests.
There are other modifiers like this, for example Positive
for positive numbers, NonNegative
for non-negative numbers, or SortedList
for a sorted list. Here’s an example of a more complex test. We check that the nth element of cycle xs
is correct. Both of the modifiers are needed, since !!
doesn’t work with negative inputs, and cycle []
is an error.
propCycle :: NonEmptyList Int -> NonNegative Int -> Property
NonEmpty xs) (NonNegative n) =
propCycle (cycle xs !! n === xs !! (mod n (length xs))
forAll
Sometimes we need to limit the range of inputs to a test even further. As a simple example, here’s a test that Data.Char.toUpper
changes the character passed to it:
propToUpperChanges :: Char -> Property
= toUpper c =/= c propToUpperChanges c
quickCheck propToUpperChanges*** Failed! Falsified (after 1 test and 1 shrink):
'A'
'A' == 'A'
Of course, it only changes lowercase letters. How can we write a test for that? There is no Lowercase
modifier available that would work like Positive
or NonEmptyList
. We need to fall back to explicit generation of values using forAll
:
propToUpperChangesLetter :: Property
= forAll (elements ['a'..'z']) propToUpperChanges propToUpperChangesLetter
*Examples.QuickCheck> verboseCheck propToUpperChangesLetter
Passed:
's'
'S' /= 's'
Passed:
'z'
'Z' /= 'z'
-- lots of output omitted
+++ OK, passed 100 tests.
Perfect! Let’s look at the types to see what’s going on here.
elements :: [a] -> Gen a
'a'..'z'] :: Gen Char
elements [forAll :: (Show a, Testable prop) => Gen a -> (a -> prop) -> Property
'a'..'z']) :: Testable prop => (Char -> prop) -> Property forAll (elements [
There are some new types here. A value of type Gen a
is a generator for values of type a
. We’ll talk a bit more about Gen
in the next section, but in this section you’ll see a couple of functions that return Gen
s so that we can use them with forAll
. The elements
function is a generator that returns one of the elements of the given list at random, as you might have guessed.
The Testable
type class is the same that the quickCheck
function uses. It exists so that quickCheck
can test types like [Int] -> Bool -> Property
in addition to just plain Property
values.
quickCheck :: Testable prop => prop -> IO ()
In addition, some simple types like Bool
have a Testable
instance, so that you can write tests using normal Haskell predicates instead of ===
:
listHasZero :: [Int] -> Bool
= elem 0 xs listHasZero xs
*Examples.QuickCheck> quickCheck (listHasZero [1,0,2])
+++ OK, passed 1 test.
*Examples.QuickCheck> quickCheck listHasZero
*** Failed! Falsified (after 1 test):
[]
Getting back to forAll
, we can use forAll
to write more complex tests. Here’s a test that checks that sort xs
has the same elements as xs
. Note how we use NonEmptyList
to guarantee that the forAll
has some elements to pick from.
propSort :: NonEmptyList Int -> Property
NonEmpty xs) =
propSort (-> elem x (sort xs)) forAll (elements xs) (\x
We’ve only just scratched the surface of QuickCheck. Here are some pointers to some things you’ll find useful when you start writing larger QuickCheck tests.
Sometimes the output from QuickCheck isn’t quite verbose enough. You can add your own lines to the output by using the counterexample
combinator:
counterexample :: Testable prop => String -> prop -> Property
As an example, let’s add logging of the input to rev
to propRevTwo
:
propRevTwo' :: [Int] -> [Int] -> Property
=
propRevTwo' xs ys let input = xs ++ ys
in counterexample ("Input: " ++ show input) $
=== rev ys ++ rev xs rev input
*Examples.QuickCheck> quickCheck propRevTwo'
*** Failed! Falsified (after 4 tests and 5 shrinks):
0]
[0,1]
[Input: [0,0,1]
0,1,0] /= [1,0,0] [
As you might have guessed, Gen
is a Monad
. You can write your own generators by combining the generators defined by QuickCheck. You can check what your generators output using sample
.
someLetters :: Gen String
= do
someLetters <- elements "xyzw"
c <- choose (1,10)
n return (replicate n c)
*Examples.QuickCheck> sample someLetters
"yyyyyyyy"
"zzzzzzzzz"
"xxxxxxxxx"
"yyyyyyy"
"yyy"
"ww"
"xxxxxx"
"yyy"
"yyyyyyy"
"xxxxxxxxxx"
"y"
Closely related to generators is the Arbitrary
type class. Arbitrary
is how QuickCheck generates all those inputs automatically.
class Arbitrary a where
arbitrary :: Gen a
shrink :: a -> [a]
If you’re writing tests for custom types, you either need to use forAll
, or implement an Arbitrary
instance. Here’s what happens if you’re missing an instance:
data Switch = On | Off
deriving (Show, Eq)
toggle :: Switch -> Switch
On = Off
toggle Off = On
toggle
propToggleTwice :: Switch -> Property
= s === toggle (toggle s) propToggleTwice s
*Examples.QuickCheck> quickCheck propToggleTwice
error:
No instance for (Arbitrary Switch)
• of ‘quickCheck’
arising from a use In the expression: quickCheck propToggleTwice •
Here are the two ways to fix it:
*Examples.QuickCheck> quickCheck (forAll (elements [On,Off]) propToggleTwice)
+++ OK, passed 100 tests.
instance Arbitrary Switch where
= elements [On,Off] arbitrary
Boo! There’s a ghost in the type system! Let’s have a look at what phantom types can do for you.
Phantom types are types that don’t take any values. They are related to newtypes (see lecture 10) in that both are a way to add additional type checking without affecting the evaluation of the program at all.
Let’s use phantom types to track which currency an amount of money is in. We define the phantom types EUR
and USD
(note how they don’t have any constructors!), and the parameterized type Money a
that doesn’t use the type parameter a
for anything. Then we can define two constants, one in euros and the other in dollars. You can find all of the code from this section in the file exercises/Examples/Phantom.hs
.
data EUR
data USD
data Money currency = Money Double
deriving Show
dollar :: Money USD
= Money 1
dollar
twoEuros :: Money EUR
= Money 2 twoEuros
Note how the type signatures for dollar
and twoEuros
are doing all the work here. An expression like Money 1
has the polymorphic type Money currency
if we don’t restrict it to a more specific type. We give dollar
and twoEuros
more limited types explicitly. This is analogous to defining something like one :: Int; one = 1
, since the constant 1
has the polymorphic type Num p => p
but we give it a more restricted type.
*Examples.Phantom> :t Money
Money :: Double -> Money currency
*Examples.Phantom> :t Money 1
Money 1 :: Money currency
Now that we have some constants, we can write functions that operate on them. Let’s start with a function scaleMoney
that multiplies an amount of money by a number. The currency stays constant. Here too, the type signature is doing the work: without the type signature, Haskell would infer a type of Double -> Money a -> Money b
.
scaleMoney :: Double -> Money currency -> Money currency
Money a) = Money (factor * a) scaleMoney factor (
*Examples.Phantom> :t scaleMoney 3 twoEuros
3 twoEuros :: Money EUR
scaleMoney *Examples.Phantom> :t scaleMoney 3 dollar
3 dollar :: Money USD scaleMoney
Next up: adding two amounts that are in the same currency. We get a nice type error if we try to add values in two different currencies.
addMoney :: Money currency -> Money currency -> Money currency
Money a) (Money b) = Money (a+b) addMoney (
*Examples.Phantom> :t addMoney dollar dollar
dollar :: Money USD
addMoney dollar*Examples.Phantom> :t addMoney twoEuros twoEuros
twoEuros :: Money EUR
addMoney twoEuros*Examples.Phantom> :t addMoney twoEuros dollar
error:
Couldn't match type ‘USD’ with ‘EUR’
• Expected type: Money EUR
Actual type: Money USD
In the second argument of ‘addMoney’, namely ‘dollar’
• In the expression: addMoney twoEuros dollar
As before, the type signature is crucial. Here’s the same implementation with an unrestricted type. Now we can add anything to anything!
addMoneyUnsafe :: Money x -> Money y -> Money z
Money a) (Money b) = Money (a+b) addMoneyUnsafe (
*Examples.Phantom> addMoneyUnsafe twoEuros dollar
Money 3.0
We can keep going with this approach, and define currency conversions. We define the type Rate
that uses phantom types to track the currencies it’s translating between. The types of convert
and invert
are restricted to have the properties we want. There’s also an unrestricted version of the convert function to let you compare the types.
data Rate from to = Rate Double
deriving Show
eurToUsd :: Rate EUR USD
= Rate 1.22
eurToUsd
convert :: Rate from to -> Money from -> Money to
Rate r) (Money a) = Money (r*a)
convert (
invert :: Rate from to -> Rate to from
Rate r) = Rate (1/r)
invert (
convertUnsafe :: Rate from to -> Money x -> Money y
Rate r) (Money a) = Money (r*a) convertUnsafe (
*Examples.Phantom> convert eurToUsd twoEuros
Money 2.44
*Examples.Phantom> convert eurToUsd dollar
error:
Couldn't match type ‘USD’ with ‘EUR’
• Expected type: Money EUR
Actual type: Money USD
In the second argument of ‘convert’, namely ‘dollar’
• In the expression: convert eurToUsd dollar
In an equation for ‘it’: it = convert eurToUsd dollar
*Examples.Phantom> convert (invert eurToUsd) dollar
Money 0.819672131147541
*Examples.Phantom> convertUnsafe eurToUsd dollar
Money 1.22
Note! The words currency
, from
, to
and so forth in the previous examples are just type variables. There’s nothing special going on with them. We could’ve as well given invert
a type like Rate a b -> Rate b a
without any change in type safety.
This approach that uses phantom types has clear benefits: giving us type errors for invalid code. Also, compared to defining lots of concrete types like data MoneyEur = MoneyEur Double
, with phantom types we need to implement functions like scaleMoney
and addMoney
only once. Also, we’re able to define polymorphic and reusable concepts like Rate
. You can contrast this approach with the Boxing section of Lecture 7.
However, phantom types also have downsides. Without advanced tricks we can’t really handle currencies that are defined at runtime (for example: reading an amount from a user). It’s also easy to end up in a place where you start needing language extensions like Generalized Algebraic Datatypes, type families and other type-level programming constructs. Eventually you’re all the way in the world of dependent typing.
So what are good applications for phantom types? When you need to track some simple, but crucial information, that is known at compile-time. An example that’s somewhat better than currencies is tracking whether inputs from the user have been sanitated to prevent attacks like SQL injection or cross-site scripting.
We can use the types Input Safe
and Input Unsafe
to track whether strings are safe for passing into the database or not. If our module only exports the makeInput
function, and not the Input
constructor, the type system ensures that any inputs must pass through the escapeInput
function at some point before going into a database function like addForumComment
.
data Safe
data Unsafe
data Input a = Input String
-- Public constructor function for Input, only allows constructing
-- Unsafe Inputs from Strings.
makeInput :: String -> Input Unsafe
= Input xs
makeInput xs
-- Adds comment to the database.
addForumComment :: Input Safe -> IO Result
= ...
addForumComment
-- We can combine inputs, but that won't change their safety
concatInputs :: Input a -> Input a -> Input a
Input xs) (Input ys) = Input (xs++ys)
concatInputs (
-- Strip bad characters to turn an unsafe input safe
escapeInput :: Input Unsafe -> Input Safe
Input xs) = Input (filter (\c -> isAlpha c || isSpace c) xs) escapeInput (
One of the great things about purity is that it makes parallelism, computing many things at the same time, very easy. Let’s have a look at how we can do this in Haskell. First off, we need to start a new GHCi that has parallel execution enabled. The simplest way is:
$ stack ghci --ghci-options "+RTS -N"
Next up, let’s define a very naive version of the Fibonacci function (remember Lecture 1?) , enable performance statistics with :set +s
, and see how long it takes to compute five values of the function:
Prelude> fib 0 = 1; fib 1 = 1; fib n = fib (n-1) + fib (n-2)
Prelude> :set +s
Prelude> map fib [29,29,29,29,29]
832040,832040,832040,832040,832040]
[7.54 secs, 2,440,860,632 bytes) (
Now let’s bring in the module Control.Parallel.Strategies that defines ways to evaluate values in parallel. We’ll use the parList rseq
strategy to evaluate all the elements of the list to WHNF, in parallel.
Prelude> import Control.Parallel.Strategies
Prelude Control.Parallel.Strategies> withStrategy (parList rseq) (map fib [29,29,29,29,29])
832040,832040,832040,832040,832040]
[4.80 secs, 488,531,384 bytes) (
That’s almost 2 times as fast, on the 2-core machine that this example is being run on. Pretty nice. The coolest thing here is that we were able to define the computation (map fib ...
) completely separately from the evaluation strategy (parList rseq
), separating the what to compute from the how to compute.
Computer science makes the distinction between parallel and concurrent computations. Parallel computations are those that just run separate independent computations in parallel (in other words, parallelism is pure). Concurrent computations are those where there are multiple interacting threads of computation. Concurrency usually involves threads, locks, messages and deadlocks.
In addition to great tooling for parallelism, Haskell also has good tooling for concurrency via threads. Since concurrency is all about side-effects, concurrent computations happen in the IO
Monad. The classic example of threading is two threads, one printing a stream of As and the other a stream of Bs. Here it is in Haskell:
printA :: IO ()
= putStrLn (replicate 40 'A')
printA
printB :: IO ()
= putStrLn (replicate 40 'B')
printB
concurrency :: IO ()
= do
concurrency
forkIO printA
forkIO printBreturn ()
Prelude Control.Concurrent> concurrency
AABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABB
The operation forkIO :: IO () -> IO ThreadId
takes an IO operation and starts running it in the backgroud. It produces a ThreadId
that can be used to e.g. terminate the thread.
If we want to add actual communication between threads, we can use abstractions like MVar
(a mutable thread-safe variable) or Chan
(a queue).
Here’s a simple example where one thread writes a value to an MVar
and another one waits for them and prints them. An MVar
works like a mailbox: it is either empty or full. Calling takeMVar
on an empty box waits for the box to get filled (with a putMVar
). Symmetrically, trying to putMVar
into a full box waits until the box is empty.
takeMVar :: MVar a -> IO a
putMVar :: MVar a -> a -> IO ()
newEmptyMVar :: IO (MVar a)
send :: [String] -> MVar String -> IO ()
= mapM_ (putMVar var) values
send values var
receive :: MVar String -> IO ()
= do val <- takeMVar var
receive var print val
-- loop unless at last value
/="end") (receive var)
when (val
concurrency2 :: IO ()
= do
concurrency2 <- newEmptyMVar
var "hello","world","and","goodbye","end"] var)
forkIO (send [
forkIO (receive var)return ()
Prelude Control.Concurrent Control.Monad> concurrency2
"hello"
"world"
"and"
"goodbye"
"end"
Congratulations! You’ve reached the end of this two part course on Functional Programming in Haskell. What next? You definitely know enough Haskell to keep learning on your own. The online Haskell community is very friendly and there are lots of blog posts and other content explaining advanced techniques and features. You can find lots of interesting stuff by following for example:
#haskell
on libera.chatJust keep writing Haskell, studying things (like libraries and tools) when you bump into them, and slowly accumulate experience. Lots of the things you learn with Haskell will be transferable to other languages like TypeScript, Elm, Rust or F#.
Finally, here is an incomplete list of things that got left out of this course, but are worth looking into:
~
patterns) and @
patterns. See e.g. A Gentle Introduction to Haskell.MultiParamTypeClasses
, ViewPatterns
etc. This is one good guidefix
functionThis course was made possible by Nitor who donated hours and hours of Joel’s working time for this project. Thank you!
Thanks to the whole Haskell Mooc team, especially
Thanks to all the students who patiently waited for part 2 and reported errors in the material & exercises!