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by Joel Kaasinen (Nitor) and John Lång (University of Helsinki)
This is an online course on Functional Programming that uses the Haskell programming language. You can study at your own pace. All the material and exercises are openly available.
This course is aimed at beginners who wish to learn functional programming, but also people who have experience with functional programming and want to learn Haskell in particular. The course assumes no previous knowledge, but knowing at least one programming language beforehand will make the course easier.
Working on the exercises involves knowing how to use the command line, and basic usage of the Git version control system.
This is part 1 of a two-part course. Part 1 covers the basics of Haskell syntax and features. You will learn about recursion, higher-order functions, algebraic data types and some of Haskell’s advanced features. However, part 1 will stick to pure functional programming, without side-effects. I/O and Monads will be introduced in part 2.
The course is split into 8 lectures. They are roughly the same size, but some lectures have more material than others. Each lecture set ends with 10-30 small programming exercises on the topics of the lecture.
The course is worth 5 ECTS credits.
Grading is Fail/Pass/5. To pass the course you must complete at least 50% of each lecture’s exercises. A special grade of 5 is awarded to those who complete 99% of all exercises.
In addition to this course material, the following sources might be useful if you feel like you’re missing examples or explanations.
#haskell
on freenode is a nice place for beginnersHaskell is
Functional – the basic building blocks of programs are functions. Functions can return functions and take functions as arguments. Also, the only looping construct in Haskell is recursion.
Pure - Haskell functions are pure, that is, they don’t have side effects. Side effects mean things like reading a file, printing out text, or changing global variables. All inputs to a function must be in its arguments, and all outputs from a function in its return value. This sounds restricting, but makes reasoning about programs easier, and allows for more optimizations by the compiler.
Lazy - values are only evaluated when they are needed. This makes it possible to work with infinite data structures, and also makes pure programs more efficient.
Strongly typed - every Haskell value and expression has a type. The compiler checks the types at compile-time and guarantees that no type errors can happen at runtime. This means no AttributeErrors (a la Python), ClassCastExceptions (a la Java) or segmentation faults (a la C). The Haskell type system is very powerful and can help you design better programs.
Type inferred - in addition to checking the types, the compiler can deduce the types for most programs. This makes working with a strongly typed language easier. Indeed, most Haskell functions can be written completely without types. However programmers can still give functions and values type annotations to make finding type errors easier. Type annotations also make reading programs easier.
Garbage-collected - like most high-level languages these days, Haskell has automatic memory management via garbage collection. This means that the programmer doesn’t need to worry about allocating or freeing memory, the language runtime handles all of it automatically.
Compiled - even though we mostly use Haskell via the interactive GHCi environment on this course, Haskell is a compiled language. Haskell programs can be compiled to very efficient binaries, and the GHC compiler is very good at optimising functional code into performant machine code.
You’ll learn what these terms mean in practice during this course. Don’t worry if some of them sound abstract right now.
See also: The Haskell Wiki page on Functional programming.
Here’s a showcase of some of the Haskell’s cool features:
Higher-order functions – functions can take functions as arguments:
This results in [3,6]
.
Anonymous functions aka lambdas – you can define single-use helper functions without giving them a name
This results in ["abc","ef"]
.
Partial application – you can define new functions by giving another function only some of the arguments it needs. For example this multiplies all elements in a list by 3:
Algebraic datatypes – a syntax for defining datatypes that can contain a number of different cases:
Now the type Shape
can have values like Point
, Rectangle 3 6
and Circle 5
Pattern matching – defining functions based on cases that correspond to your data definitions:
area Point = 0
area (Rectangle width height) = width * height
area (Circle radius) = 2 * pi * radius
Lists – Unlike many languages, Haskell has a concise built-in syntax for lists. Lists can be built from other lists using list comprehensions. Here’s a snippet that generates names of even length from a set of options for first and last names:
[whole | first <- ["Eva", "Mike"],
last <- ["Smith", "Wood", "Odd"],
let whole = first ++ last,
even (length whole)]
This results in ["EvaSmith","EvaOdd","MikeWood"]
. Thanks to the laziness of Haskell, we can even create so-called infinite lists:
The first ten prime numbers can be then obtained by evaluating
This evaluates to [2,3,5,7,11,13,17,19,23,29]
.
Parameterized types – you can define types that are parameterized by other types. For example [Int]
is a list of Int
s and [Bool]
is a list of booleans. You can define typed functions that work on all kinds of lists, for example reverse
has the type [a] -> [a]
which means it takes a list containing any type a
, and returns a list of the same type.
Type classes – another form of polymorphism where you can give a function a different implementation depending on the arguments’ types. For example the Show
type class defines the function show
that can convert values of various types to strings. The Num
type class defines arithmetic operators like +
that work on all number types (Int
, Double
, Complex
, …).
A brief timeline of Haskell:
The word ‘haskel’ means wisdom in Hebrew, but the name of the Haskell programming language comes from the logician Haskell Curry. The name Haskell comes from the Old Norse words áss (god) and ketill (helmet).
Here are some examples of software projects that were written in Haskell.
See The Haskell Wiki and this blog post for more!
The easiest way to get Haskell is to install the stack
tool, see http://haskellstack.org. The exercises on this course are intended to work with Stack, so you should use it for now.
By the way, if you’re interested in what Stack is, and how it relates to other Haskell tools like Cabal and GHC, read more here or here. We’ll get back to Haskell packages and using them in detail in part 2 of the course.
For now, after installing Stack, just run stack ghci
to get an interactive Haskell environment.
GHCi is the interactive Haskell interpreter. Here’s an example session:
$ stack ghci
GHCi, version 8.0.1: http://www.haskell.org/ghc/ :? for help
Prelude> 1+1
2
Prelude> "asdf"
"asdf"
Prelude> reverse "asdf"
"fdsa"
Prelude> :type "asdf"
"asdf" :: [Char]
Prelude> tail "asdf"
"sdf"
Prelude> :type tail "asdf"
tail "asdf" :: [Char]
Prelude> :type tail
tail :: [a] -> [a]
Prelude> :quit
Leaving GHCi.
By the way, the first time you run stack ghci
it will download GHC and some libraries, so don’t worry if you see some output and have to wait for a while before getting the Prelude>
prompt.
Let’s walk through this. Don’t worry if you don’t understand things yet, this is just a first brush with expressions and types.
Prelude> 1+1
2
The Prelude>
is the GHCi prompt. It indicates we can use the functions from the Haskell base library called Prelude. We evaluate 1 plus 1, and the result is 2.
Prelude> "asdf"
"asdf"
Here we evaluate a string literal, and the result is the same string.
Prelude> reverse "asdf"
"fdsa"
Here we compute the reverse of a string by applying the function reverse
to the value "asdf"
.
Prelude> :type "asdf"
"asdf" :: [Char]
In addition to evaluating expressions we can also ask for their type with the :type
(abbreviated :t
) GHCi command. The type of "asdf"
is a list of characters. Commands that start with :
are part of the user interface of GHCi, not part of the Haskell language.
Prelude> tail "asdf"
"sdf"
Prelude> :t tail "asdf"
tail "asdf" :: [Char]
The tail
function works on lists and returns all except the first element of the list. Here we see tail
applied to "asdf"
. We also check the type of the expression, and it is a list of characters, as expected.
Prelude> :t tail
tail :: [a] -> [a]
Finally, here’s the type of the tail
function. It takes a list of any type as an argument, and returns a list of the same type.
Prelude> :quit
Leaving GHCi.
That’s how you quit GHCi.
Just like we saw in the GHCi example above, expressions and types are the bread and butter of Haskell. In fact, almost everything in a Haskell program is an expression. In particular, there are no statements like in Python, Java or C.
An expression has a value and a type. We write an expression and its type like this: expression :: type
. Here are some examples:
Expression | Type | Value |
---|---|---|
True |
Bool |
True |
not True |
Bool |
False |
"as" ++ "df" |
[Char] |
"asdf" |
Expressions consist of functions applied to arguments. Functions are applied (i.e. called) by placing the arguments after the name of the function – there is no special syntax for a function call.
Haskell | Python, Java or C |
---|---|
f 1 |
f(1) |
f 1 2 |
f(1,2) |
Parentheses can be used to group expressions (just like in math and other languages).
Haskell | Python, Java or C |
---|---|
g h f 1 |
g(h,f,1) |
g h (f 1) |
g(h,f(1)) |
g (h f 1) |
g(h(f,1)) |
g (h (f 1)) |
g(h(f(1))) |
Some function names are made special characters and they are used as operators: between their arguments instead of before them. Function calls bind tighter than operators, just like multiplication binds tighter than addition.
Haskell | Python, Java or C |
---|---|
a + b |
a + b |
f a + g b |
f(a) + g(b) |
f (a + g b) |
f(a+g(b)) |
PS. in Haskell, function application associates left, that is, f g x y
is actually the same as (((f g) x) y)
. We’ll get back to this topic later. For now you can just think that f g x y
is f
applied to the arguments g
, x
and y
.
Here are some basic types of Haskell to get you started.
Type | Literals | Use | Operations |
---|---|---|---|
Int |
1 , 2 , -3 |
Number type (signed, 64bit) | + , - , * , div , mod |
Integer |
1 , -2 , 900000000000000000 |
Unbounded number type | + , - , * , div , mod |
Double |
0.1 , 1.2e5 |
Floating point numbers | + , - , * , / , sqrt |
Bool |
True , False |
Truth values | && , || , not |
String aka [Char] |
"abcd" , "" |
Strings of characters | reverse , ++ |
As you can see, the names of types in Haskell start with a capital letter. Some values like True
also start with a capital letter, but variables and functions start with a lower case letter (reverse
, not
, x
). We’ll get back to the meaning of capital letters in Lecture 2.
Function types are written using the ->
syntax:
argumentType -> returnType
argument1Type -> argument2Type -> returnType
argument1Type -> argument2Type -> argument3Type -> returnType
Looks a bit weird, right? We’ll get back to this as well.
Sometimes, the types you see in GHCi are a bit different than what you’d assume. Here are two common cases.
For now, you should read the type Num a => a
as “any number type”. In Haskell, number literals are overloaded which means that they can be interpreted as any number type (e.g. Int
or Double
). We’ll get back to what Num a
actually means when we talk about type classes later.
The type String
is just an alias for the type [Char]
which means “list of characters”. We’ll get back to lists on the next lecture! In any case, you can use String
and [Char]
interchangeably, but GHCi will mostly use [Char]
when describing types to you.
Here’s a simple Haskell program that does some arithmetic and prints some values.
module Gold where
-- The golden ratio
phi :: Double
phi = (sqrt 5 + 1) / 2
polynomial :: Double -> Double
polynomial x = x^2 - x - 1
f x = polynomial (polynomial x)
main = do
print (polynomial phi)
print (f phi)
If you put this in a file called Gold.hs
and run it with (for example) stack runhaskell Gold.hs
, you should see this output
0.0
-1.0
Let’s walk through the file.
There is one Haskell module per source file. A module consists of definitions.
This is a comment. Comments are not part of the actual program, but text for human readers of the program.
This is a definition of the constant phi
, with an accompanying type annotation (also known as a type signature) phi :: Double
. The type annotation means that phi
has type Double
. The line with a equals sign (=
) is called an equation. The left hand side of the =
is the expression we are defining, and the right hand side of the =
is the definition.
In general a definition (of a function or constant) consists of an optional type annotation and one or more equations
This is the definition of a function called polynomial
. It has a type annotation and an equation. Note how an equation for a function differs from the equation of a constant by the presence of a parameter x
left of the =
sign. Note also that ^
is the power operator in Haskell, not bitwise xor like in many other languages.
This is the definition of a function called f
. Note the lack of type annotation. What is the type of f
?
This is a description of what happens when you run the program. It uses do-syntax and the IO Monad. We’ll get back to those in part 2 of the course.
When you see an example definition like this
you should usually play around with it. Start by running it. There are a couple of ways to do this.
If a definition fits on one line, you can just define it in GHCi using let
:
Prelude> let polynomial x = x^2 - x - 1
Prelude> polynomial 3.0
5.0
For a multi-line definition, you can either use ;
to separate lines, or use the special :{ :}
syntax to paste a block of code into GHCi:
Prelude> :{
Prelude| polynomial :: Double -> Double
Prelude| polynomial x = x^2 - x - 1
Prelude| :}
Prelude> polynomial 3.0
5.0
Finally, you can paste the code into a new or existing .hs
file, and then :load
it into GHCi. If the file has already been loaded, you can also use :reload
.
-- first copy and paste the definition into Example.hs, then run GHCi
Prelude> :load Example.hs
[1 of 1] Compiling Main ( Example.hs, interpreted )
Ok, one module loaded.
*Main> polynomial 3.0
5.0
-- now you can edit the definition
*Main> :reload
[1 of 1] Compiling Main ( Example.hs, interpreted )
Ok, one module loaded.
*Main> polynomial 3
3.0
After you’ve run the example, try modifying it, or making another function that is similar but different. You learn programming by programming, not by reading!
Since Haskell is a typed language, you’ll pretty quickly bump into type errors. Here’s an example of an error during a GHCi session:
Prelude> "string" ++ True
<interactive>:1:13: error:
• Couldn't match expected type ‘[Char]’ with actual type ‘Bool’
• In the second argument of ‘(++)’, namely ‘True’
In the expression: "string" ++ True
In an equation for ‘it’: it = "string" ++ True
This is the most common type error, “Couldn’t match expected type”. Even though the error looks long and scary, it’s pretty simple if you just read through it.
The first line of the error message, <interactive>:1:13: error:
tells us that the error occurred in GHCi. If we had loaded a file, we might instead get something like Sandbox.hs:3:17: error:
, where Sandbox.hs
is the name of the file, 3
is the line number and 17
is the number of a character in the line.
The line • Couldn't match expected type ‘[Char]’ with actual type ‘Bool’
tells us that the immediate cause for the error is that there was an expression of type Bool
, when GHCi was expecting to find an expression of type [Char]
“. The location of this error was indicated in the first line of the error message. Note that the expected type is not always right. Giving type annotations by hand can help debugging typing errors.
The line • In the second argument of ‘(++)’, namely ‘True’
tells that the expression that had the wrong type was the second argument of the operator (++)
. We’ll learn later why it’s surrounded by parentheses.
The full expression with the error was "string" ++ True
. As mentioned above, String
is a type alias for [Char]
, the type of character lists. The first argument to ++
was a list of characters, and since ++
can only combine two lists of the same type, the second argument should’ve been of type [Char]
too.
The line In an equation for ‘it’: it = "string" ++ True
says that the expression occurred in the definition of the variable it
, which is a default variable name that GHCi uses for standalone expressions. If we had a line x = "string" ++ True
in a file, or a declaration let x = "string" ++ True
in GHCi, GHCi would print In an equation for ‘x’: x = "string" ++ True
instead.
There are also others types of errors.
Prelude> True + 1
<interactive>:6:1: error:
• No instance for (Num Bool) arising from a use of ‘+’
• In the expression: True + 1
In an equation for ‘it’: it = True + 1
This is the kind of error you get when you try to use a numeric function like +
on something that’s not a number.
The hardest error to track down is usually this:
Prelude> True +
<interactive>:10:7: error:
parse error (possibly incorrect indentation or mismatched brackets)
There are many ways to cause it. Probably you’re missing some characters somewhere. We’ll get back to indentation later in this lecture.
There’s one thing in Haskell arithmetic that often trips up beginners, and that’s division.
In Haskell there are two division functions, the /
operator and the div
function. The div
function does integer division:
The /
operator performs the usual division:
However, you can only use div
on whole number types like Int
and Integer
, and you can only use /
on decimal types like Double
. Here’s an example of what happens if you try to mix them up:
error:
• No instance for (Fractional Int) arising from a use of ‘/’
• In the expression: x / 2
In an equation for ‘halve’: halve x = x / 2
Just try to keep this in mind for now. We’ll get back to the difference between /
and div
, and what Num
and Fractional
mean when talking about type classes.
So far you’ve seen some arithmetic, reversing a string and so on. How does one write actual programs in Haskell? Many of the usual programming constructs like loops, statements and assignment are missing from Haskell. Next, we’ll go through the basic building blocks of Haskell programs:
In other languages, if
is a statement. It doesn’t have a value, it just conditionally executes other statements.
In Haskell, if
is an expression. It has a value. It selects between two other expressions. It corresponds to the ?:
operator in C or Java.
Python’s conditional expressions are quite close to haskell’s if
:
This is how the same example looks in Haskell:
Because Haskell’s if
returns a value, you always need an else
!
Bool
In order to write if expressions, you need to know how to get values of type Bool
. The most common way is comparisons. The usual ==
, <
, <=
, >
and >=
operators work in Haskell. You can do ordered comparisons (<
, >
) on all sorts of numbers, and equality comparisons (==
) on almost anything:
One oddity of Haskell is that the not-equals operator is written /=
instead of the usual !=
:
Remember that in addition to these comparisons, you can get Bool
values out of other Bool
values by using the &&
(“and”) and ||
(“or”) operators, and the not
function.
login user password = if user == "unicorn73"
then if password == "f4bulous!"
then "unicorn73 logged in"
else "wrong password"
else "unknown user"
Haskell has two different ways for creating local definitions: let...in
and where
.
where
adds local definitions to a definition:
let...in
is an expression:
Local definitions can also be functions:
We’ll get back to the differences between let
and where
, but mostly you can use which ever you like.
Even though things like pi
above are often called variables, I’ve chosen to call them definitions here. This is because unlike variables in Python or Java, the values of these definitions can’t be changed. Haskell variables aren’t boxes into which you can put new values, Haskell variables name a value (or rather, an expression) and that’s it.
We’ll talk about immutability again later on this course, but for now it’s enough to know that things like this don’t work.
This is just an infinite loop, because it tries to define a new variable x
with the property x = x+1
. Thus when evaluating x
, Haskell just keeps computing 1+1+1+1+...
indefinitely.
error:
Conflicting definitions for ‘a’
Bound at: <interactive>:14:17
<interactive>:15:17
Here we get a straightforward error when we’re trying to “update” the value of a
.
As a remark, local definitions can shadow the names of variables defined elsewhere. Shadowing is not a side-effect. Instead, shadowing creates a new variable within a more restricted scope that uses the same name as some variable in the outer scope. For example, all of the functions f
, g
, and h
below are legal:
x :: Int
x = 5
f :: Int -> Int
f x = 2 * x
g :: Int -> Int
g y = x where x = 6
h :: Int -> Int
h x = x where x = 3
If we apply them to the global constant x
, we see the effects of shadowing:
It is best to always choose new names for local variables, so that shadowing never happens. That way, the reader of the code will understand where the variables that are used in an expression come from. Note that in the following example, f
and g
don’t shadow each others’ arguments:
A definition (of a function) can consist of multiple equations. The equations are matched in order against the arguments until a suitable one is found. This is called pattern matching.
Pattern matching in Haskell is very powerful, and we’ll keep learning new things about it along this course, but here are a couple of first examples:
greet :: String -> String -> String
greet "Finland" name = "Hei, " ++ name
greet "Italy" name = "Ciao, " ++ name
greet "England" name = "How do you do, " ++ name
greet _ name = "Hello, " ++ name
The function greet
generates a greeting given a country and a name (both String
s). It has special cases for three countries, and a default case. This is how it works:
Prelude> greet "Finland" "Pekka"
"Hei, Pekka"
Prelude> greet "England" "Bob"
"How do you do, Bob"
Prelude> greet "Italy" "Maria"
"Ciao, Maria"
Prelude> greet "Greenland" "Jan"
"Hello, Jan"
The special pattern _
matches anything. It’s usually used for default cases. Because patterns are matched in order, it’s important to (usually) put the _
case last. Consider:
Now the first case gets selected for all inputs.
Prelude> brokenGreet "Finland" "Varpu"
"Hello, Varpu"
Prelude> brokenGreet "Sweden" "Ole"
"Hello, Ole"
GHC even gives you a warning about this code:
<interactive>:1:1: warning: [-Woverlapping-patterns]
Pattern match is redundant
In an equation for ‘brokenGreet’: brokenGreet "Finland" name = ...
Some more examples follow. But first let’s introduce the standard library function show
that can turn (almost!) anything into a string:
So, here’s an example of a function with pattern matching and a default case that actually uses the value (instead of just ignoring it with _
):
describe :: Integer -> String
describe 0 = "zero"
describe 1 = "one"
describe 2 = "an even prime"
describe n = "the number " ++ show n
This is how it works:
You can even pattern match on multiple arguments. Again, the equations are tried in order. Here’s a reimplementation of the login
function from earlier:
login :: String -> String -> String
login "unicorn73" "f4bulous!" = "unicorn73 logged in"
login "unicorn73" _ = "wrong password"
login _ _ = "unknown user"
In Haskell, all sorts of loops are implemented with recursion. Function calls are very efficient, so you don’t need to worry about performance. (We’ll talk about performance later).
Learning how to do simple things with recursion in Haskell will help you use recursion on more complex problems later. Recursion is also often a useful way for thinking about solving harder problems.
Here’s our first recursive function which computes the factorial. In mathematics, factorial is the product of n first positive integers and is written as n!. The definition of factorial is
n! = n * (n-1) * … * 1
For example, 4! = 4*3*2*1 = 24. Well anyway, here’s the Haskell implementation of factorial:
This is how it works. We use ==>
to mean “evaluates to”.
What happens when you evaluate factorial (-1)
?
Here’s another example:
A function can call itself recursively multiple times. As an example let’s consider the Fibonacci sequence from mathematics. The Fibonacci sequence is a sequence of integers with the following definition.
The sequence starts with 1, 1. To get the next element of the sequence, sum the previous two elements of the sequence.
The first elements of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13 and so on. Here’s a function fibonacci
which computes the n
th element in the Fibonacci sequence. Note how it mirrors the mathematical definition.
-- Fibonacci numbers, slow version
fibonacci 1 = 1
fibonacci 2 = 1
fibonacci n = fibonacci (n-2) + fibonacci (n-1)
Here’s how fibonacci 5
evaluates:
fibonacci 5
==> fibonacci 3 + fibonacci 4
==> (fibonacci 1 + fibonacci 2) + fibonacci 4
==> ( 1 + 1 ) + fibonacci 4
==> ( 1 + 1 ) + (fibonacci 2 + fibonacci 3)
==> ( 1 + 1 ) + (fibonacci 2 + (fibonacci 1 + fibonacci 2))
==> ( 1 + 1 ) + ( 1 + ( 1 + 1 ))
==> 5
Note how fibonacci 3
gets evaluated twice and fibonacci 2
three times. This is not the most efficient implementation of the fibonacci
function. We’ll get back to this in the next lecture. Another way to think about the evaluation of the fibonacci function is to visualize it as a tree (we abbreviate fibonacci
as fib
):
This tree then exaclty corresponds with the expression (1 + 1) + (1 + (1 + 1))
. Recursion can often produce chain-like, tree-like, nested, or loopy structures and computations. Recursion is one of the main techniques in functional programming, so it’s worth spending some effort in learning it.
Finally, here’s a complete Haskell module that uses ifs, pattern matching, local defintions and recursion. The module is interested in the Collatz conjecture, a famous open problem in mathematics. It asks:
Does the Collatz sequence eventually reach 1 for all positive integer initial values?
The Collatz sequence is defined by taking any number as a starting value, and then repeatedly performing the following operation:
As an example, the Collatz sequence for 3 is: 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1 … As you can see, once the number reaches 1, it gets caught in a loop.
module Collatz where
-- one step of the Collatz sequence
step :: Integer -> Integer
step x = if even x then down else up
where down = div x 2
up = 3*x+1
-- collatz x computes how many steps it takes for the Collatz sequence
-- to reach 1 when starting from x
collatz :: Integer -> Integer
collatz 1 = 0
collatz x = 1 + collatz (step x)
-- longest finds the number with the longest Collatz sequence for initial values
-- between 0 and upperBound
longest :: Integer -> Integer
longest upperBound = longest' 0 0 upperBound
-- helper function for longest
longest' :: Integer -> Integer -> Integer -> Integer
-- end of recursion, return longest length found
longest' number _ 0 = number
-- recursion step: check if n has a longer Collatz sequence than the current known longest
longest' number maxlength n = if length > maxlength
then longest' n length (n-1)
else longest' number maxlength (n-1)
where length = collatz n
We can load the program in GHCi and play with it.
$ stack ghci
GHCi, version 8.2.2: http://www.haskell.org/ghc/ :? for help
Prelude> :load Collatz.hs
[1 of 1] Compiling Collatz ( Collatz.hs, interpreted )
Ok, one module loaded.
*Collatz>
Let’s verify that our program computes the start of the Collatz sequence for 3 correctly.
*Collatz> step 3
10
*Collatz> step 10
5
*Collatz> step 5
16
How many steps does it take for 3 to reach 1?
*Collatz> collatz 3
7
What’s the longest Collatz sequence for a starting value under 10? What about 100?
*Collatz> longest 10
9
*Collatz> longest 100
97
The lengths of these Collatz sequences are:
*Collatz> collatz 9
19
*Collatz> collatz 97
118
The previous examples have been fancily indented. In Haskell indentation matters, a bit like in Python. The complete set of rules for indentation is hard to describe, but you should get along fine with these rules of thumb:
While you can get away with using tabs, it is highly recommended to use spaces for all indenting.
Some examples are in order.
These all are ok:
i x = let y = x+x+x+x+x+x in div y 5
-- let and in are grouped together, an expression is split
j x = let y = x+x+x
+x+x+x
in div y 5
-- the definitions of a and b are grouped together
k = a + b
where a = 1
b = 1
l = a + b
where
a = 1
b = 1
These are not ok:
-- indentation not increased even though expression split on many lines
i x = let y = x+x+x+x+x+x
in div y 5
-- indentation not increased even though expression is split
j x = let y = x+x+x
+x+x+x
in div y 5
-- grouped things are not aligned
k = a + b
where a = 1
b = 1
-- grouped things are not aligned
l = a + b
where
a = 1
b = 1
-- where is part of the equation, so indentation needs to increase
l = a + b
where
a = 1
b = 1
If you make a mistake with the indentation, you’ll typically get a parse error like this:
Indent.hs:2:1: error: parse error on input ‘where’
The error includes the line number, so just go over that line again. If you can’t seem to get indentation to work, try putting everything on just one long line at first.
At the end of each lecture you’ll find a quiz like this. The quizes aren’t graded, they’re just here to help you check you’ve understood the chapter. You can check your answer by clicking on an option. You’ll see a green background if you were right, a red one if you were wrong. Feel free to guess as many times as you want, just make sure you understand why the right option is right in the end.
What is the Haskell equivalent of the C/Java/Python expression combine(prettify(lawn),construct(house,concrete))
?
combine prettify (lawn) construct (house concerete)
combine (prettify lawn (counstruct house concrete))
combine (prettify lawn) (construct house concrete)
send metric (double population + increase)
?
send(metric(double(population+increase)))
send(metric(double(population)+increase))
send(metric,double(population)+increase)
send(metric,double(population+increase))
Which one of the following claims is true in Haskell?
Which one of the following claims is true in Haskell?
if
always requires both then
and else
What does the function f x = if even (x + 1) then x + 1 else f (x - 1)
do?
x
to the least even number greater than or equal to x
x
to the greatest even number less than or equal to x
Why is 3 * "F00"
not valid Haskell?
3
and "F00"
have different types
"F00"
needs the prefix “0x”
Why does 7.0 `div` 2
give an error?
div
is not defined for the type Double
div
is not defined for the type Int
`...`
is used for delimiting strings.
The course materials, including exercises, are available in a Git repository on GitHub at https://github.com/moocfi/haskell-mooc. If you’re not familiar with Git, see GitHub’s instructions on cloning a repository.
Once you’ve cloned the haskell-mooc
repository, go into the exercises
directory. To download and build dependencies needed for running the exercise tests (such as the correct version of GHC and various libraries), run following command in your terminal:
Do note that the dependencies are multiple gigabytes and it will take a while for the command to finish.
There are primarily two types of files in the exercises
directory: exercise sets named SetNX.hs
and accompanying test program for the exercises named SetNXTest.hs
. Both are Haskell source files, but only the exercise file should to be edited when solving the exercises. Instructions to all individual exercises are embedded in the exercise file as comments.
Use the tests file to check your answers. For example when you have solved some of the exercises in Set1.hs
, run the following command:
The output of the tests looks something like this:
===== EXERCISE 1
+++++ Pass
===== EXERCISE 2
+++++ Pass
===== EXERCISE 3
*** Failed! Falsified (after 2 tests and 1 shrink):
quadruple 1
Expected: 4
Was: 2
----- Fail
===== EXERCISE 4
+++++ Pass
===== EXERCISE 5
+++++ Pass
===== EXERCISE 6
+++++ Pass
===== EXERCISE 7
+++++ Pass
===== EXERCISE 8
+++++ Pass
===== EXERCISE 9
+++++ Pass
===== EXERCISE 10
+++++ Pass
===== EXERCISE 11
+++++ Pass
===== EXERCISE 12
+++++ Pass
===== EXERCISE 13
+++++ Pass
===== EXERCISE 14
+++++ Pass
===== EXERCISE 15
+++++ Pass
===== EXERCISE 16
+++++ Pass
===== EXERCISE 17
+++++ Pass
===== EXERCISE 18
+++++ Pass
===== EXERCISE 19
+++++ Pass
===== TOTAL
1101111111111111111
18 / 19
In the example above, I’ve made a mistake in exercise 3.
To make debugging faster and more straightforward, I can load my exercise file in GHCi, which allows me to evaluate any top-level function manually. For instance I can verify the above mistake by:
$ stack ghci Set1.hs
GHCi, version 8.6.5: http://www.haskell.org/ghc/ :? for help
[1 of 2] Compiling Mooc.Todo ( Mooc/Todo.hs, interpreted )
[2 of 2] Compiling Set1 ( Set1.hs, interpreted )
Ok, two modules loaded.
*Set1> quadruple 1
2
Once you’re done with an exercise set, you can turn it in on the Submit page on the course pages. After that you can see the results of your submission on the Results page and your total score on the My status page.
Note! You may turn in an exercise set as many times as you want.
Note! If you don’t want to use Stack or can’t get it working, you should also be able to run the tests with Cabal like this:
Once you’ve successfully completed all the exercises in a set, you can view the model solutions on the My status page. It’s useful to glance at the model solutions, they might show you a technique you’ve missed!
Maybe
, Either
Often you’ll find you need helper variables in recursion to keep track of things. You can get them by defining a helper function with more arguments. Analogy: arguments of the helper function are variables you update in your loop.
Here’s an example of how you would convert a loop (in Java or Python) into a recursive helper function in Haskell.
Java:
public String repeatString(int n, String str) {
String result = "";
while (n>0) {
result = result+str;
n = n-1;
}
return result;
}
Python:
Haskell:
repeatString n str = repeatHelper n str ""
repeatHelper n str result = if (n==0)
then result
else repeatHelper (n-1) str (result++str)
You might have noticed that the Java and Python implementations look a bit weird since they use while loops instead of for loops. This is because this way the conversion to Haskell is more straightforward.
This can be made a bit tidier by using pattern matching instead of an if
:
repeatString n str = repeatHelper n str ""
repeatHelper 0 _ result = result
repeatHelper n str result = repeatHelper (n-1) str (result++str)
Here’s another example with more variables: computing fibonacci numbers efficiently.
Java:
public int fibonacci(int n) {
int a = 0;
int b = 1;
while (n>1) {
int c = a+b;
a=b;
b=c;
n--;
}
return b;
}
Python:
Haskell:
-- fibonacci numbers, fast version
fibonacci :: Integer -> Integer
fibonacci n = fibonacci' 0 1 n
fibonacci' :: Integer -> Integer -> Integer -> Integer
fibonacci' a b 1 = b
fibonacci' a b n = fibonacci' b (a+b) (n-1)
Take a while to study these and note how the Haskell recursion has the same format as the loop.
Sidenote: Haskell programs often use the apostrophe to name helper functions and alternative versions of functions. Thus the name fibonacci'
for the helper function above. Names like foo'
are usually read foo prime (like in mathematics).
I said earlier that this version of fibonacci is more efficient. Can you see why? The answer is that there are less recursive calls. The expression fibonacci' _ _ n
calls fibonacci' _ _ (n-1)
once, and this means that we can compute fibonacci' _ _ n
in n
steps.
This type of recursion where a function just directly calls itself with different arguments is called tail recursion. As you’ve seen above, tail recursion corresponds to loops. This is why tail recursion is often fast: the compiler can generate a loop in machine code when it sees tail recursion.
Before we move on to new types, let’s go over one more piece of Haskell syntax.
The if then else
is often a bit cumbersome, especially when you have multiple cases. An easier alternative is Haskell’s conditional definition or guarded definition. This is a bit like pattern matching in that you have multiple equations, but you can have arbitrary code deciding which equation to use. Guarded definitions look like this:
A condition can be any expression of type Bool
. The first condition that evaluates to True
is chosen. The word otherwise
is just an alias for True
. It is used to mark the default case.
Here are some examples of using guards. First off, we have a function that describes the given number. Note how it is important to have the "Two"
case before the "Even"
case.
describe :: Int -> String
describe n
| n==2 = "Two"
| even n = "Even"
| n==3 = "Three"
| n>100 = "Big!!"
| otherwise = "The number "++show n
Here is factorial, implemented with guards instead of pattern matching. Unlike the pattern-matching version, this one doesn’t loop forever with negative inputs.
You can even combine guards with pattern matching. Here’s the implementation of a simple age guessing game:
guessAge :: String -> Int -> String
guessAge "Griselda" age
| age < 47 = "Too low!"
| age > 47 = "Too high!"
| otherwise = "Correct!"
guessAge "Hansel" age
| age < 12 = "Too low!"
| age > 12 = "Too high!"
| otherwise = "Correct!"
guessAge name age = "Wrong name!"
Prelude> guessAge "Griselda" 30
"Too low!"
Prelude> guessAge "Griselda" 60
"Too high!"
Prelude> guessAge "Griselda" 47
"Correct!"
Prelude> guessAge "Bob" 30
"Wrong name!"
Prelude> guessAge "Hansel" 10
"Too low!"
So far we’ve always worked with single values like number or booleans. Strings contain multiple characters, but still in some sense a string is just one piece of information. In order to be able to do actual programming, we need to handle variable numbers of items. For this we need data structures.
The basic datastructure in Haskell is the list. Lists are used to store multiple values of the same type (in other words, Haskell lists are homogeneous). This is what a list literal looks like:
A list type is written as [Element]
, where Element
is the type of the lists elements. Here are some more list expressions and their types:
[True,True,False] :: [Bool]
["Moi","Hei"] :: [String]
[] :: [a] -- more about this later
[[1,2],[3,4]] :: [[Int]] -- a list of lists
[1..7] :: [Int] -- range syntax, value [1,2,3,4,5,6,7]
Haskell lists are implemented as singly-linked lists. We’ll return to this later.
The Haskell standard library comes with lots of functions that operate on lists. Here are some of the most important ones, together with their types. We’ll get back to what [a]
actually means in a second, but for now you can imagine it means “any list”.
head :: [a] -> a -- returns the first element
tail :: [a] -> [a] -- returns everything except the first element
init :: [a] -> [a] -- returns everything except the last element
take :: Int -> [a] -> [a] -- returns the n first elements
drop :: Int -> [a] -> [a] -- returns everything except the n first elements
(++) :: [a] -> [a] -> [a] -- lists are catenated with the ++ operator
(!!) :: [a] -> Int -> a -- lists are indexed with the !! operator
reverse :: [a] -> [a] -- reverse a list
null :: [a] -> Bool -- is this list empty?
length :: [a] -> Int -- the length of a list
Sidenote: the last two operations (null
and length
) actually have more generic types, but here I’m pretending that you can only use them on lists.
Lists can be compared with the familiar ==
operator.
Do you remember this from our first GHCi session?
This means that String
is just an alias for [Char]
, which means string is a list of characters. This means you can use all list operations on strings!
Some list operations come from the module Data.List
. You can import a module in code or in GHCi with the import Data.List
syntax. One example is the sort
function which sorts a list:
Note how the set of imported modules shows up in the GHCi prompt.
Here are some examples of working with lists. In this case, instead of showing you output from GHCi, I merely use ==>
to show what an expression evaluates to.
Indexing a list:
Defining a function that discards the 3rd and 4th elements of a list using take
and drop
:
Rotating a list by taking the first element and moving it to the end:
Here’s an example of the range syntax:
Because Haskell is pure, it also means that functions can’t mutate (change) their inputs. Mutation is a side effect, and Haskell functions are only allowed output via their return value. This means that Haskell list functions always return a new list. In practice:
Prelude> let list = [1,2,3,4]
Prelude> reverse list
[4,3,2,1]
Prelude> list
[1,2,3,4]
Prelude> drop 2 list
[3,4]
Prelude> list
[1,2,3,4]
This might seem very inefficient but it turns out it can be both performant and quite useful. We’ll get back to how Haskell datastructures work in a later lecture.
So what does a type like head :: [a] -> a
mean? It means given a list that contains elements of any type a
, the return value will be of the same type a
.
In this type, a
is a type variable. Type variables are types that start with a small letter, e.g. a
, b
, thisIsATypeVariable
. A type variable means a type that is not yet known, or in other words a type that could be anything. Type variables can turn into concrete types (e.g. Bool
) by the process of type inference (also called unification).
Let’s have a look at some examples. If we apply head
to a list of booleans, type inference will compare the type of head’s argument, [a]
, with the type of the actual argument, [Bool]
and deduce that a
must be Bool
. This means that the return type of head
will in this case also be Bool
.
The function tail
takes a list, and returns a list of the same type. If we apply tail
to a list of booleans, the return value will also be a list of booleans.
If types don’t match, we get a type error. Consider the operator ++
which takes two lists of the same type, as we can see from its type [a] -> [a] -> [a]
. If we try to apply ++
to a list of booleans and a list of characters we get an error. This is what happens in GHCi:
Prelude> [True,False] ++ "Moi"
<interactive>:1:16:
Couldn't match expected type `Bool' against inferred type `Char'
Expected type: [Bool]
Inferred type: [Char]
In the second argument of `(++)', namely `"Moi"'
In the expression: [True, False] ++ "Moi"
Type inference is really powerful. It uses the simple process of unification to get us types for practically any Haskell expression. Consider these two functions:
We can ask GHCi for their types, and we will see that type inference has figured out that the two arguments to f
must have the same type, since their heads get put into the same list.
The function g
, which fixed one of the arguments of f
to a string (i.e. [Char]
) gets a narrower type. Type inference has decided that the argument zs
to g
must also have type [Char]
, since otherwise the type of f
would not match the call to f
.
In a type like [Char]
we call Char
a type parameter. A type like the list type that needs a type parameter is called a parameterized type.
The fact that a function like head
can be used with many different types of arguments is called polymorphism. The head
function is said to be polymorphic. There are many forms of polymorphism, and this Haskell form that uses type variables is called parametric polymorphism.
Since Haskell has type inference, you don’t need to give any type annotations. However even though type annotations aren’t required, there are multiple reasons to add them:
A good rule of thumb is to give top-level definitions type annotations.
Maybe
TypeIn addition to the list type, Haskell has other parameterized types too. Let’s look at a very common and useful one: the Maybe
type.
Sometimes an operation doesn’t have a valid return value (E.g. division by zero.). We have a couple of options in this situation. We can use an error value, like -1
. This is a bit ugly, not always possible. We can throw an exception. This is impure. In some other languages we would return a special null value that exists in (almost) all types. However Haskell does not have a null.
The solution Haskell offers us instead is to change our return type to a Maybe
type. This is pure, safe and neat. The type Maybe a
has two constructors: Nothing
and Just
. Nothing
is just a constant, but Just
takes a parameter. More concretely:
Type | Values |
---|---|
Maybe Bool |
Nothing , Just False , Just True |
Maybe Int |
Nothing , Just 0 , Just 1 , … |
Maybe [Int] |
Nothing , Just [] , Just [1,1337] , … |
You can think of Maybe a
as being a bit like [a]
except there can only be 0 or 1 elements, not more. Alternatively, you can think of Maybe a
introducing a null value to the type a
. If you’re familiar with Java, Maybe Integer
is the Haskell equivalent of Java’s Optional<Integer>
.
You can create Maybe
values by either specifying Nothing
or Just someOtherValue
:
Prelude> :t Nothing
Nothing :: Maybe a
Prelude> Just "a camel"
Just "a camel"
Prelude> :t Just "a camel"
Just "a camel" :: Maybe [Char] -- the same as Maybe String
Prelude> Just True
Just True
Prelude> :t Just True
Just True :: Maybe Bool
-- given a password, return (Just username) if login succeeds, Nothing otherwise
login :: String -> Maybe String
login "f4bulous!" = Just "unicorn73"
login "swordfish" = Just "megahacker"
login _ = Nothing
You use a Maybe
value by pattern matching on it. Usually you define patterns for the Nothing
and Just something
cases. Some examples:
-- Multiply an Int with a Maybe Int. Nothing is treated as no multiplication at all.
perhapsMultiply :: Int -> Maybe Int -> Int
perhapsMultiply i Nothing = i
perhapsMultiply i (Just j) = i*j -- Note how j denotes the value inside the Just
intOrZero :: Maybe Int -> Int
intOrZero Nothing = 0
intOrZero (Just i) = i
safeHead :: [a] -> Maybe a
safeHead xs = if null xs then Nothing else Just (head xs)
headOrZero :: [Int] -> Int
headOrZero xs = intOrZero (safeHead xs)
headOrZero [] ==> intOrZero (safeHead []) ==> intOrZero Nothing ==> 0
headOrZero [1] ==> intOrZero (safeHead [1]) ==> intOrZero (Just 1) ==> 1
As you can see above, we can pattern match on the constructors of Maybe
: Just
and Nothing
. We’ll get back to what constructors mean later. For now it’s enough to note that constructors are special values that start with a capital letter that you can pattern match on.
Other constructors that we’ve already seen include the constructors of Bool
– True
and False
. We’ll introduce the constructors of the list type on the next lecture.
Constructors can be used just like Haskell values. Constructors that take no arguments like Nothing
, and False
are just constants. Constructors like Just
that take an argument behave like functions. They even have function types!
Prelude> :t Just
Just :: a -> Maybe a
Either
typeSometimes it would be nice if you could add an error message or something to Nothing
. That’s why we have the Either
type. The Either
type takes two type arguments. The type Either a b
has two constructors: Left
and Right
. Both take an argument, Left
an argument of type a
and Right
an argument of type b
.
Type | Values |
---|---|
Either Int Bool |
Left 0 , Left 1 , Right False , Right True , … |
Either String [Int] |
Left "asdf" , Right [0,1,2] , … |
Either Integer Integer |
Left 0 , Right 0 , Left 1 , Right 1 , … |
Here’s a simple example: a readInt
function that only knows a couple of numbers and returns a descriptive error for the rest. Note the Haskell convention of using Left
for errors and Right
for success.
readInt :: String -> Either String Int
readInt "0" = Right 0
readInt "1" = Right 1
readInt s = Left ("Unsupported string: " ++ s)
Sidenote: the constructors of Either
are called Left
and Right
because they refer to the left and right type arguments of Either
. Note how in Either a b
, a
is the left argument and b
is the right argument. Thus Left
contains a value of type a
and likewise Right
of type b
. The convention of using Right
for success is probably simply because right also means correct. No offense is intended to left-handed people.
Here’s another example: pattern matching an Either
. Just like with Maybe
, there are two patterns for an Either
, one for each constructor.
iWantAString :: Either Int String -> String
iWantAString (Right str) = str
iWantAString (Left number) = show number
As you recall, Haskell lists can only contain elements of the same type. You can’t have a value like [1,"foo",2]
. However, you can use a type like Either
to represent lists that can contain two different types of values. For example we could track the number of people on a lecture, with a possibility of adding an explanation if a value is missing:
lectureParticipants :: [Either String Int]
lectureParticipants = [Right 10, Right 13, Left "easter vacation", Right 17, Left "lecturer was sick", Right 3]
We’ve seen pattern matching in function arguments, but there’s also a way to pattern match in an expression. It looks like this:
As an example let’s rewrite the describe
example from the first lecture using case
:
describe :: Integer -> String
describe 0 = "zero"
describe 1 = "one"
describe 2 = "an even prime"
describe n = "the number " ++ show n
describe :: Integer -> String
describe n = case n of 0 -> "zero"
1 -> "one"
2 -> "an even prime"
n -> "the number " ++ show n
A more interesting example is when the value we’re pattern matching on is not a function argument. For example:
-- parse country code into country name, returns Nothing if code not recognized
parseCountry :: String -> Maybe String
parseCountry "FI" = Just "Finland"
parseCountry "SE" = Just "Sweden"
parseCountry _ = Nothing
flyTo :: String -> String
flyTo countryCode = case parseCountry countryCode of Just country -> "You're flying to " ++ country
Nothing -> "You're not flying anywhere"
We could write the flyTo
function using a helper function for pattern matching instead of using the case-of expression:
flyTo :: String -> String
flyTo countryCode = handleResult (parseCountry countryCode)
where handleResult (Just country) = "You're flying to " ++ country
handleResult Nothing = "You're not flying anywhere"
In fact, a case-of expression can always be replaced with a helper function. Here’s one more example, written in both ways:
-- given a sentence, decide whether it is a statement, question or exclamation
sentenceType :: String -> String
sentenceType sentence = case last sentence of '.' -> "statement"
'?' -> "question"
'!' -> "exclamation"
_ -> "not a sentence"
-- same function, helper function instead of case-of
sentenceType sentence = classify (last sentence)
where classify '.' = "statement"
classify '?' = "question"
classify '!' = "exclamation"
classify _ = "not a sentence"
Prelude> sentenceType "This is Haskell."
"statement"
Prelude> sentenceType "This is Haskell!"
"exclamation"
Things you can use as patterns:
Int
and Integer
constants like (-1)
, 0
, 1
, 2
, …Bool
values True
and False
Char
constants: 'a'
, 'b'
String
constants: "abc"
, ""
Maybe
constructors: Nothing
, (Just x)
Either
constructors: (Left x)
, (Right y)
_
pattern which means “anything, I don’t care”(Just 1)
Places where you can use patterns:
f :: Bool -> Maybe Int -> Int
f False Nothing = 1
f False _ = 2
f True (Just i) = i
f True Nothing = 0
case of
expression:The only thing you really need pattern matching for is getting the value inside a Just
, Left
or Right
constructor. Here are two more examples of this:
-- getElement (Just i) gets the ith element (counting from zero) of a list, getElement Nothing gets the last element
getElement :: Maybe Int -> [a] -> a
getElement (Just i) xs = xs !! i
getElement Nothing xs = last xs
direction :: Either Int Int -> String
direction (Left i) = "you should go left " ++ show i ++ " meters!"
direction (Right i) = "you should go right " ++ show i ++ " meters!"
Prelude> direction (Left 3)
"you should go left 3 meters!"
Prelude> direction (Right 5)
"you should go right 5 meters!"
Other uses (that we’ve seen so far!) of pattern matching can also be accomplished with the ==
operator. However, things like x==Nothing
won’t work in all cases. We’ll find out why when we talk about type classes in lecture 4.
How many values does f x = [x,x]
return?
Why does the expression Nothing 1
cause a type error?
Nothing
takes no arguments
Nothing
returns nothing
Nothing
is a constructor
What is the type of the function f x y = if x && y then Right x else Left "foo"
?
Bool -> Bool -> Either Bool String
String -> String -> Either String String
Bool -> Bool -> Either String Bool
Which of the following functions could have the type Bool -> Int -> [Bool]
f x y = [0, y]
f x y = [x, True]
f x y = [y, True]
What is the type of this function? justBoth a b = [Just a, Just b]
a -> b -> [Maybe a, Maybe b]
a -> a -> [Just a]
a -> b -> [Maybe a]
a -> a -> [Maybe a]
Now with lists and polymorphism in our toolbox, we can finally start to look at functional programming.
In Haskell a function is a value, just like a number or a list is. Functions can be passed as parameters to other functions. Here’s a toy example. The function applyTo1
takes a function of type Int->Int
, applies it to the number 1
, and returns the result.
Let’s define a simple function of type Int->Int
and see applyTo1
in action.
Let’s go back to the type annotation for applyTo1
.
The parentheses are needed because the type Int -> Int -> Int
would be the type of a function taking two Int
arguments. More on this later.
Let’s look at a slightly more interesting example. This time we’ll implement a polymorphic function doTwice
. Note how we can use it with various types of values and functions.
doTwice addThree 1
==> addThree (addThree 1)
==> 7
doTwice tail "abcd"
==> tail (tail "abcd")
==> "cd"
That was a bit boring. Luckily there are many useful list functions that take functions as arguments. By the way, functions that take functions as arguments (or return functions) are often called higher-order functions.
The most famous of these list-processing higher-order functions is map
. It gives you a new list by applying the given function to all elements of a list.
The partner in crime for map
is filter
. Instead of transforming all elements of a list, filter
drops some elements of a list and keeps others. In other words, filter
selects the elements from a list that fulfill a condition.
Here’s an example: selecting the positive elements from a list
Note how both the type signatures of map
and filter
use polymorphism. They work on all kinds of lists. The type of map
even uses two type parameters! Here are some examples of type inference using map
and filter
.
Prelude> :t onlyPositive
onlyPositive :: [Int] -> [Int]
Prelude> :t mapBooleans
mapBooleans :: (Bool -> b) -> [b]
Prelude> :t mapBooleans not
mapBooleans not :: [Bool]
One more thing: remember how constructors were just functions? That means you can pass them as arguments to other functions!
How many “palindrome numbers” are between 1
and n
?
-- a predicate that checks if a string is a palindrome
palindrome :: String -> Bool
palindrome str = str == reverse str
-- palindromes n takes all numbers from 1 to n, converts them to strings using show, and keeps only palindromes
palindromes :: Int -> [String]
palindromes n = filter palindrome (map show [1..n])
palindrome "1331" ==> True
palindromes 150 ==>
["1","2","3","4","5","6","7","8","9",
"11","22","33","44","55","66","77","88","99",
"101","111","121","131","141"]
length (palindromes 9999) ==> 198
How many words in a string start with “a”? This uses the function words
from the module Data.List
that splits a string into words.
countAWords :: String -> Int
countAWords string = length (filter startsWithA (words string))
where startsWithA s = head s == 'a'
The function tails
from Data.List
returns the list of all suffixes (“tails”) of a list. We can use tails
for many string processing tasks. Here’s how tails
works:
Here’s an example where we find what characters come after a given character in a string. First of all, we use tails
, map
and take
to get all substrings of a certain length:
substringsOfLength :: Int -> String -> [String]
substringsOfLength n string = map shorten (tails string)
where shorten s = take n s
There’s some shorter substrings left at the end (can you see why?), but they’re fine for our purposes right now. Now that we have substringsOfLength
, we can implement the function whatFollows c k s
that finds all the occurrences of the character c
in the string s
, and outputs the k
letters that come after these occurrences.
whatFollows :: Char -> Int -> String -> [String]
whatFollows c k string = map tail (filter match (substringsOfLength (k+1) string))
where match sub = take 1 sub == [c]
When using higher-order functions you can find yourself defining lots of small helper functions, like addThree
or shorten
in the previous examples. This is a bit of a chore in the long run, but luckily Haskell’s functions behave a bit weirdly…
Let’s start in GHCi:
So, we’ve defined add
, a function of two arguments, and only given it one argument. The result is not a type error but a new function. The new function just stores (or remembers) the given argument, waits for another argument, and then gives both to add
.
Prelude> map addThree [1,2,3]
[4,5,6]
Prelude> map (add 3) [1,2,3]
[4,5,6]
Here we can see that we don’t even need to give a name to the function returned by add 3
. We can just use it anywhere where a function of one argument is expected.
This is called partial application. All functions in Haskell behave like this. Let’s have a closer look. Here’s a function that takes many arguments.
We can give between
less arguments and get back new functions, just like we saw with add
:
Prelude> (between 1 5) 2
True
Prelude> let f = between 1 5 in f 2
True
Prelude> map (between 1 3) [1,2,3]
[False,True,False]
Look at the types of partially applying between
. They behave neatly, with arguments disappearing one by one from the type as values are added to the expression.
Prelude> :t between
between :: Integer -> Integer -> Integer -> Bool
Prelude> :t between 1
between 1 :: Integer -> Integer -> Bool
Prelude> :t between 1 2
between 1 2 :: Integer -> Bool
Prelude> :t between 1 2 3
between 1 2 3 :: Bool
Actually, when we write a type like Integer -> Integer -> Integer -> Bool
, it means Integer -> (Integer -> (Integer -> Bool))
. That is, a multi-argument function is just a function that returns a function. Similarly, an expression like between 1 2 3
is the same as ((between 1) 2) 3
, so passing multiple arguments to a function happens via multiple single-argument calls. Representing multi-argument functions like this is called currying (after the logician Haskell Curry). Currying is what makes partial application possible.
Here’s another example of using partial application with map
:
In addition to normal functions, partial application also works with operators. With operators you can choose whether you apply the left or the right argument. (Partially applied operators are also called sections or operator sections). Some examples:
Prelude> map (*2) [1,2,3]
[2,4,6]
Prelude> map (2*) [1,2,3]
[2,4,6]
Prelude> map (1/) [1,2,3,4,5]
[1.0,0.5,0.3333333333333333,0.25,0.2]
Normal Haskell operators are applied with prefix notation, which is just a fancy way to say that the function name comes before the arguments. In contrast, operators are applied with infix notation – the name of the function comes between the arguments.
An infix operator can be converted into a prefix function by adding parentheses around it. For instance,
This is useful especially when an operator needs to be passed as an argument to another function.
As an example, the function zipWith
takes two lists, a binary function, and joins the lists using the function. We can use zipWith (+)
to sum two lists, element-by-element:
Prelude> :t zipWith
zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
Prelude> zipWith (+) [0,2,5] [1,3,3]
[1,5,8]
Without the ability to turn an operator into a function, we’d have to use a helper function – such as add
above.
Note that omitting the parentheses leads into a type error:
Prelude> zipWith + [0,2,5,3] [1,3,3]
<interactive>:1:11: error:
• Couldn't match expected type ‘[Integer]
-> (a -> b -> c) -> [a] -> [b] -> [c]’
with actual type ‘[Integer]’
• The function ‘[0, 2, 5, 3]’ is applied to one argument,
but its type ‘[Integer]’ has none
In the second argument of ‘(+)’, namely ‘[0, 2, 5, 3] [1, 3, 3]’
In the expression: zipWith + [0, 2, 5, 3] [1, 3, 3]
• Relevant bindings include
it :: (a -> b -> c) -> [a] -> [b] -> [c]
(bound at <interactive>:1:1)
The reason for this weird-looking error is that GHCi got confused and thought that we were somehow trying to add zipWith
and [0,2,5,3] [1,3,3]
together. Logically, it deduced that [0,2,5,3]
must be a function since it’s being applied to [1,3,3]
(remember that functions bind tighter than operators).
Unfortunately, error messages can sometimes be obscure, since the compiler cannot always know the “real” cause of the error (which is in this case was omitting the parentheses). Weird error messages are frustrating, but only the programmer knows what was the original intent behind the code.
Another nice feature of Haskell is the syntax for applying a binary function as if it was an infix operator, by surrounding it with backticks (`). For example:
The last spanner we need in our functional programming toolbox is λ (lambda). Lambda expressions are anonymous functions. Consider a situation where you need a function only once, for example in an expression like
A lambda expression allows us to write this directly, without defining a name (big
) for the helper function:
Here are some more examples in GHCi:
Prelude> (\x -> x*x) 3
9
Prelude> (\x -> reverse x == x) "ABBA"
True
Prelude> filter (\x -> reverse x == x) ["ABBA","ACDC","otto","lothar","anna"]
["ABBA","otto","anna"]
Prelude> (\x y -> x^2+y^2) 2 3 -- multiple arguments
13
The Haskell syntax for lambdas is a bit surprising. The backslash character (\
) stands for the greek letter lambda (λ). The Haskell expression \x -> x+1
is trying to mimic the mathematical notation λx. x+1. Other languages use syntax like x => x+1
(JavaScript) or lambda x: x+1
(Python).
Note! You never need to use a lambda expression. You can always instead define the function normally using let
or where
.
By the way, lambda expressions are quite powerful constructs which have a deep theory of their own, known as Lambda calculus. Some even consider purely functional programming languages such as Haskell to be typed extensions of Lambda calculus with extra syntax.
.
and $
OperatorsThe two most common operators in Haskell codebases are probably .
and $
. They are useful when writing code that uses higher-order functions. The first of these, the .
operator, is the function composition operator. Here’s its type
And here’s what it does
(f.g) x ==> f (g x)
You can use function composition to build functions out of other functions, without mentioning any arguments. For example:
double x = 2*x
quadruple = double . double -- computes 2*(2*x) == 4*x
f = quadruple . (+1) -- computes 4*(x+1)
g = (+1) . quadruple -- computes 4*x+1
third = head . tail . tail -- fetches the third element of a list
We can also reimplement doTwice
using (.)
. Note how we can use doTwice
both as applied only to a function, or as applied to a function and a value.
let ttail = doTwice tail
in ttail [1,2,3,4]
==> [3,4]
(doTwice tail) [1,2,3,4] ==> [3,4]
doTwice tail [1,2,3,4] ==> [3,4]
Often function composition is not used when defining a new function, but instead to avoid defining a helper function. For instance, consider the difference between these two expressions:
The other operator, $
is more subtle. Let’s look at its type.
It takes a function of type a -> b
and a value of type a
, and returns a value of type b
. In other words, it’s a function application operator. The expression f $ x
is the same as f x
. This seems pretty useless, but it means that the $
operator can be used to eliminate parentheses! These expressions are the same:
This isn’t that impressive when it’s used to eliminate one pair of parentheses, but together .
and $
can eliminate lots of them! For example, we can rewrite
as
and then
Sometimes the operators .
and $
are useful as functions in their own right. For example, a list of functions can be applied to an argument using map and a section of $
:
map ($"string") [reverse, take 2, drop 2]
==> [reverse $ "string", take 2 $ "string", drop 2 $ "string"]
==> [reverse "string", take 2 "string", drop 2 "string"]
==> ["gnirts", "st", "ring"]
If this seems complicated, don’t worry. You don’t need to use .
and $
in your own code until you’re comfortable with them. However, you’ll bump into .
and $
when reading Haskell examples and code on the internet, so it’s good to know about them. This article might also help.
whatFollows
Now, let’s rewrite the whatFollows
example from earlier using the tools we just saw. Here’s the original version:
substringsOfLength :: Int -> String -> [String]
substringsOfLength n string = map shorten (tails string)
where shorten s = take n s
whatFollows :: Char -> Int -> String -> [String]
whatFollows c k string = map tail (filter match (substringsOfLength (k+1) string))
where match sub = take 1 sub == [c]
To get started, let’s get rid of the helper function substringsOfLength
and move all the code to whatFollows
:
whatFollows c k string = map tail (filter match (map shorten (tails string)))
where shorten s = take (k+1) s
match sub = take 1 sub == [c]
Now let’s use partial application instead of defining shorten
:
whatFollows c k string = map tail (filter match (map (take (k+1)) (tails string)))
where match sub = take 1 sub == [c]
Let’s use .
and $
to eliminate some of those parentheses:
whatFollows c k string = map tail . filter match . map (take (k+1)) $ tails string
where match sub = take 1 sub == [c]
We can also replace match
with a lambda:
Finally, we don’t need to mention the string
parameter at all, since we can just express whatFollows
as a composition of map
, filter
, map
and tails
:
We can even go a bit further by rewriting the lambda using an operator section
\sub -> take 1 sub == [c]
=== \sub -> (==[c]) (take 1 sub)
=== \sub -> (==[c]) ((take 1) sub)
=== \sub -> ((==[c]) . (take 1)) sub
=== ((==[c]) . (take 1))
=== ((==[c]) . take 1)
Now what we have left is:
This is a somewhat extreme version of the function, but when used in moderation the techniques shown here can make code easier to read.
Here are some more examples of functional programming with lists. Let’s start by introducing a couple of new list functions:
takeWhile :: (a -> Bool) -> [a] -> [a] -- take elements from a list as long as they satisfy a predicate
dropWhile :: (a -> Bool) -> [a] -> [a] -- drop elements from a list as long as they satisfy a predicate
There’s also the function elem
, which can be used to check if a list contains an element:
Using these, we can implement a function findSubstring
that finds the earliest and longest substring in a string that consist only of the given characters.
findSubstring :: String -> String -> String
findSubstring chars = takeWhile (\x -> elem x chars)
. dropWhile (\x -> not $ elem x chars)
The function zipWith
lets you combine two lists element-by-element:
zipWith (++) ["John","Mary"] ["Smith","Cooper"]
==> ["JohnSmith","MaryCooper"]
zipWith take [4,3] ["Hello","Warden"]
==> ["Hell","War"]
Sometimes with higher-order functions it’s useful to have a function that does nothing. The function id :: a -> a
is the identity function and just returns its argument.
This seems a bit useless, but you can use it for example with filter
or dropWhile
:
filter id [True,False,True,True] ==> [True,True,True]
dropWhile id [True,True,False,True,False] ==> [False,True,False]
Another very simple but sometimes crucial function is the constant function, const :: a -> b -> a
. It always returns its first argument:
When partially applied it can be used when you need a function that always returns the same value:
Here’s a new operator, :
Prelude> 1:[]
[1]
Prelude> 1:[2,3]
[1,2,3]
Prelude> tail (1:[2,3])
[2,3]
Prelude> head (1:[2,3])
1
Prelude> :t (:)
(:) :: a -> [a] -> [a]
The :
operator builds a list out of a head and a tail. In other words, x : xs
is the same as [x] ++ xs
. Why do we need an operator for this?
Actually, :
is the constructor for lists: it returns a new linked list node. The other list constructor is []
, the empty list. All lists are built using :
and []
. The familiar [x,y,z]
syntax is actually just a nicer way to write x:y:z:[]
, or even more explicitly, x:(y:(z:[]))
. In fact (++)
is defined in terms of :
and recursion in the standard library.
Here’s a picture of how [1,2,3]
is structured in memory:
Using :
we can define recursive functions that build lists. For example here’s a function that builds lists like [3,2,1]
:
Here’s a function that builds a list by iterating a function n
times:
iterate (*2) 4 3 ==> [3,6,12,24,48]
let xs = "terve"
in iterate tail (length xs) xs
==> ["terve","erve","rve","ve","e",""]
Here’s a more complicated example: splitting a string into pieces at a given character:
split :: Char -> String -> [String]
split c [] = []
split c xs = start : split c (drop 1 rest)
where start = takeWhile (/=c) xs
rest = dropWhile (/=c) xs
Last lecture, it was said that constructors are things that can be pattern matched on. Above, it was divulged that the constructors for the list type are :
and []
. We can put one and one together and guess that we can pattern match on :
and []
. This is true! Here’s how you can define your own versions of head
and tail
using pattern matching:
myhead :: [Int] -> Int
myhead [] = -1
myhead (first:rest) = first
mytail :: [Int] -> [Int]
mytail [] = []
mytail (first:rest) = rest
You can nest patterns. That is, you can pattern match more than one element from the start of a list. In this example, we use the pattern (a:b:_)
which is the same as (a:(b:_))
:
sumFirstTwo :: [Integer] -> Integer
-- this equation gets used for lists of length at least two
sumFirstTwo (a:b:_) = a+b
-- this equation gets used for lists of length 0 or 1
sumFirstTwo _ = 0
Another way we can nest patterns is pattern matching on the head while pattern matching on a list. For example this function checks if a list starts with 0
:
startsWithZero :: [Integer] -> Bool
startsWithZero (0:xs) = True
startsWithZero (x:xs) = False
startsWithZero [] = False
Using pattern matching and recursion, we can recursively process a whole list. Here’s how you sum all the numbers in a list:
Here’s how you compute the largest number in a list, this time using a helper function.
myMaximum :: [Int] -> Int
myMaximum [] = 0 -- actually this should be some sort of error...
myMaximum (x:xs) = go x xs
where go biggest [] = biggest
go biggest (x:xs) = go (max biggest x) xs
Note!, “go
” is just a cute name for the helper function here. It’s not special syntax.
It’s often convenient to use nested patterns while consuming a list. Here’s an example that counts how many Nothing
values occur in a list of Maybe
s:
countNothings :: [Maybe a] -> Int
countNothings [] = 0
countNothings (Nothing : xs) = 1 + countNothings xs
countNothings (Just _ : xs) = countNothings xs
Now that we can build and consume lists, let’s do both of them at the same time. This function doubles all elements in a list.
It evaluates like this:
doubleList [1,2,3]
=== doubleList (1:(2:(3:[])))
==> 2*1 : doubleList (2:(3:[]))
==> 2*1 : (2*2 : doubleList (3:[]))
==> 2*1 : (2*2 : (2*3 : doubleList []))
==> 2*1 : (2*2 : (2*3 : []))
=== [2*1, 2*2, 2*3]
==> [2,4,6]
Once you know pattern matching for lists, it’s straightforward to define map
and filter
. Actually, let’s just look at the GHC standard library implementations. Here’s map:
and here’s filter:
filter :: (a -> Bool) -> [a] -> [a]
filter _pred [] = []
filter pred (x:xs)
| pred x = x : filter pred xs
| otherwise = filter pred xs
(Note! Naming the argument _pred
is a way to tell the reader of the code that this argument is unused. It could have been just _
as well.)
When a recursive function evaluates to a new call to that same function with different arguments, it is called tail-recursive. (The recursive call is said to be in tail position.) This is the type of recursion that corresponds to an imperative loop. We’ve already seen many examples of tail-recursive functions, but we haven’t really contrasted the two ways for writing the same function. This is sumNumbers
from earlier in this lecture:
In the second equation the function +
is at the top level, i.e. in tail position. The recursive call to sumNumbers
is an argument of +
. This is sumNumbers
written using a tail recursive helper function:
sumNumbers :: [Int] -> Int
sumNumbers xs = go 0 xs
where go sum [] = sum
go sum (x:xs) = go (sum+x) xs
Note the second equation of go
: it has the recursive call to go
at the top level, i.e. in tail position. The +
is now in an argument to go
.
For a function like sumNumbers
that produces a single value (a number), it doesn’t really matter which form of recursion you choose. The non-tail-recursive function is easier to read, while the tail-recursive one can be easier to come up with. You can try writing a function both ways. The tail-recursive form might be more efficient, but that depends on many details. We’ll talk more about Haskell performance in part 2 of this course.
However, when you’re returning a list there is a big difference between these two forms. Consider the function doubleList
from earlier. Here it is again, implemented first directly, and then via a tail-recursive helper function.
doubleList :: [Int] -> [Int]
doubleList xs = go [] xs
where go result [] = result
go result (x:xs) = go (result++[2*x]) xs
Here the direct version is much more efficient. The (:)
operator works in constant time, whereas the (++)
operator needs to walk the whole list, needing linear time. Thus the direct version uses linear time (O(n)) with respect to the length of the list, while the tail-recursive version is quadratic (O(n²))!
One might be tempted to fix this by using (:)
in the tail-recursive version, but then the list would get generated in the reverse order. This could be fixed with an application of reverse
, but that would make the resulting function quite complicated.
There is another reason to prefer the direct version: laziness. We’ll get back to laziness in part 2 of the course, but for now it’s enough for you to know that the direct way of generating a list is simpler, more efficient and more idiomatic. You should try to practice it in the exercises. Check out the standard library implementations of map
and filter
above, even they produce the list directly without tail recursion!
Haskell has list comprehensions, a nice syntax for defining lists that combines the power of map
and filter
. You might be familiar with Python’s list comprehensions already. Haskell’s work pretty much the same way, but their syntax is a bit different.
Mapping:
Filtering:
In general, these two forms are equivalent:
List comprehensions can do even more. You can iterate over multiple lists:
[ first ++ " " ++ last | first <- ["John", "Mary"], last <- ["Smith","Cooper"] ]
==> ["John Smith","John Cooper","Mary Smith","Mary Cooper"]
You can make local defitions:
[ reversed | word <- ["this","is","a","string"], let reversed = reverse word ]
==> ["siht","si","a","gnirts"]
You can even do pattern matching in list comprehensions!
In Haskell an operator is anything built from the characters !#$%&*+./<=>?@\^|-~
. Operators can be defined just like functions (note the slightly different type annotation):
What’s the type of this function? both p q x = p x && q x
a -> Bool -> a -> Bool -> a -> Bool
(a -> Bool) -> (a -> Bool) -> a -> Bool
(a -> Bool) -> (b -> Bool) -> c -> Bool
What’s the type of this function? applyInOut f g x = f (g (f x))
(a -> b) -> (b -> a) -> a -> b
(a -> b) -> (b -> c) -> a -> c
(a -> a) -> (a -> a) -> a -> a
Which one of the following functions adds its first argument to the second?
f x x = x + x
f x = \y -> x + y
f = \x y -> x + x
Which one of the following functions does not satisfy f 1 ==> 1
?
f x = (\y -> y) x
f x = \y -> y
f x = (\y -> x) x
Which one of the following functions is correctly typed?
f x y = not x; f :: (Bool -> Bool) -> Bool
f x = x ++ "a"; f :: Char -> String
f x = 'a' : x; f :: String -> String
How many arguments does drop 2
take?
What does this function do? f (_:x:_) = x
What is the result of reverse $ take 5 . tail $ "This is a test"
?
"i sih"
"set a"
If f :: a -> b
, then what is the type of map (.f)
?
[b -> c] -> [a -> c]
[c -> a] -> [c -> b]
(b -> c) -> [a -> c]
[a] -> [b]
What is the type of the leftmost id
in id id
?
a
a -> a
(a -> a) -> (a -> a)
What is the type of const const
?
(c -> a -> b) -> a
c -> (a -> b -> a)
a -> b -> c -> a
You’ve probably tried to use x==Nothing
to check if a value is Nothing
. Use pattern matching instead. The reason for this error is that values of type Maybe a
can’t be compared because Haskell doesn’t know how to compare values of the polymorphic type a
. You’ll find more about this in the next lecture. Use pattern matching instead of ==
for now.
Before we dive into type classes, let’s introduce the last remaining built-in datatype in Haskell: the tuple. Tuples or pairs (or triples, quadruples, etc) are a way of bundling a couple of values of different types together. You can think of tuples as fixed-length lists (just like Python’s tuples). Unlike lists, each element in the tuple can have a different type. The types of the elements are reflected in the type of the tuple. Here are some examples of tuple types and values:
Type | Example value |
---|---|
(String,String) |
("Hello","World!") |
(Int,Bool) |
(1,True) |
(Int,Int,Int) |
(4,0,3) |
To get values out of tuples, you can use the functions fst
and snd
:
You can also pattern match on tuples. This is often the most convenient way, and also works for tuples of larger sizes. The fst
and snd
functions work only on pairs.
Tuples are very useful in combination with lists. Here are some examples using the zip
, unzip
and partition
functions from the Data.List
module.
zip :: [a] -> [b] -> [(a, b)] -- two lists to list of pairs
unzip :: [(a, b)] -> ([a], [b]) -- list of pairs to pair of lists
partition :: (a -> Bool) -> [a] -> ([a], [a]) -- elements that satisfy and don't satisfy a predicate
zip [1,2,3] [True,False,True]
==> [(1,True),(2,False),(3,True)]
unzip [("Fred",1), ("Jack",10), ("Helen",13)]
==> (["Fred","Jack","Helen"],[1,10,13])
partition (>0) [-1,1,-4,3,2,0]
==> ([1,3,2],[-1,-4,0])
Here’s an example of pattern matching on tuples:
Here’s an example of pattern matching on tuples and lists at the same time:
-- sum all numbers that are paired with True
sumIf :: [(Bool,Int)] -> Int
sumIf [] = 0
sumIf ((True,x):xs) = x + sumIf xs
sumIf ((False,_):xs) = sumIf xs
Consider the functions sumNumbers :: [Int] -> Int
, myMaximum :: [Int] -> Int
, and countNothings :: [Maybe a] -> Int
again.
sumNumbers :: [Int] -> Int
sumNumbers [] = 0
sumNumbers (x:xs) = x + sumNumbers xs
myMaximum :: [Int] -> Int
myMaximum [] = 0
myMaximum (x:xs) = go x xs
where go biggest [] = biggest
go biggest (x:xs) = go (max biggest x) xs
countNothings :: [Maybe a] -> Int
countNothings [] = 0
countNothings (Nothing : xs) = 1 + countNothings xs
countNothings (Just _ : xs) = countNothings xs
They have one common characteristic. They take a list and produce a value that depends on the values of the elements in the given list. They “crunch” or fold a list of many values into a single value.
Prelude has a function called foldr
, which performs a right associative fold over a Foldable
data type. We’ll learn more about Foldable
soon. At this point, it suffices to think of lists, so we define
What this definition says, is that for an empty list [] :: [a]
, foldr
returns the default value y :: b
. For any other list x : xs
, foldr
applies f
to x
and the result of foldr f y xs
(i.e. folding over the rest of the list). It’s a simple definition by recursion.
In other words, foldr
calls its argument function f
repeatedly with two arguments.
f
returned for the rest of the list.Consider the list [1,2,3]
:
The expression foldr (+) 0 [1,2,3]
evaluates as follows:
foldr (+) 0 [1,2,3] ==> foldr (+) 0 (1:2:3:[])
==> 1 + (foldr (+) 0 (2:3:[]))
==> 1 + (2 + (foldr (+) 0 (3:[])))
==> 1 + (2 + (3 + (foldr (+) 0 [])))
==> 1 + (2 + (3 + 0))
The result can be thought of as a tree:
One way to think about foldr f y xs
is that it replaces the (:)
operation with f
and []
with y
. In this case, f
was (+)
and y
was 0
. If you write out how sumNumbers [1,2,3]
behaves, you’ll notice that it performs the same computation as foldr (+) 0 [1,2,3]
does! More generally:
Those more experienced with math may notice that we can prove this claim by induction: Firstly, sumNumbers [] ==> 0
and foldr (+) 0 [] ==> 0
, so in the base case sumNumbers [] == foldr (+) 0 []
. Next, we may assume as our induction hypothesis that sumNumbers xs == foldr (+) 0 xs
for any list xs
. Then, for the list x:xs
, we have sumNumbers (x:xs) ==> x + sumNumbers xs
. Hence, foldr (+) 0 (x:xs) ==> x + foldr (+) 0 xs ==> x + sumNumbers xs
by induction hypothesis. Therefore, by induction, the equation holds.
You don’t need to read, write, or understand induction proofs in this course, but perhaps it is reassuring to know that properties and equalities of functions in Haskell can be (in principle) analysed mathematically, because Haskell is such a nice language. (Equalities and properties can be analysed in any programming language, but for Haskell, this analysis is especially convenient because Haskell is pure.)
Another folding example is the map
function:
To see why this works, consider what foldr helper [] [x1,x2,..,xn]
does:
Now, since helper x xs ==> g x : xs
for every x
and xs
, we get that:
The resulting list, [ g x1, g x2, g x3, ..., g xn ]
, is then exactly what we would have gotten with map g xs
. (This could have been also proved by induction as we did for sumNumbers
.) The lesson to take away is that folding is a particular, yet quite general, way to apply some transformation recursively into some structure (e.g. a list).
How can Haskell’s +
work on both Int
s and Double
s? Why can I compare all sorts of things with ==
? We’ve briefly mentioned constrained types earlier. Let’s see what they really mean. Let’s look at the types of ==
and +
.
The type (Eq a) => a -> a -> Bool
means: for all types a
that belong to the class Eq
, this is a function of type a -> a -> Bool
. That is, if the type a
is a member of the class Eq
, you can give two values of type a
to ==
and get a Bool
result.
Similarly, the type (Num a) => a -> a -> a
means: for all types a
that belong to the class Num
, this is a function of type a -> a -> a
. That is, you can give two values of the same type a
to +
and get out a third value of type a
, as long as a
is a member of Num
.
Num
and Eq
are type classes. A type class is a way to group together types that support similar operations.
Note! A type class is a collection of types. It doesn’t have much to do with the classes of object oriented programming! In some situations, type classes can act like interfaces in object oriented programming. Unfortunately the functions in a type class are often called methods, adding to the confusion.
PS. remember how using type variables for polymorphism was called parametric polymorphism? The fancy word for what type classes achieve is ad-hoc polymorphism. The difference is that with parametric polymorphism the function (e.g. head
) has the same implementation for all types, whereas with ad-hoc polymorphisms there are multiple implementations (consider ==
on numbers and strings).
When you’re working with a concrete type (not a type variable), you can just use type class functions (in this case, (==)
):
Of course, if the type in question isn’t a member of the right class, you get an error. For example:
error:
• No instance for (Num Bool) arising from a use of ‘+’
• In the expression: b + True
In an equation for ‘addTrue’: addTrue b = b + True
However in a polymorphic function, you need to add type constraints. This doesn’t work:
Luckily the error is nice:
error:
• No instance for (Eq a) arising from a use of ‘==’
Possible fix:
add (Eq a) to the context of
the type signature for:
f :: (a -> a) -> a -> Bool
• In the expression: x == g x
In an equation for ‘f’: f g x = x == g x
To signal that f
only works on types that are members of the Eq
class, we add a type constraint (Eq a) =>
to the type annotation.
If you don’t have a type annotation, type inference can provide the constraints!
You can also have multiple constraints:
bothPairsEqual :: (Eq a, Eq b) => a -> a -> b -> b -> Bool
bothPairsEqual left1 left2 right1 right2 = left1 == left2 && right1 == right2
Here are some standard Haskell type classes you should know about.
Eq
We already saw the Eq
class for equality comparisons. Here are the basic operations of the Eq
class and some examples of their use. As you can see pretty much all the types we’ve seen so far, except for functions, are members of Eq
.
Prelude> 1 == 2
False
Prelude> 1 /= 2
True
Prelude> "Foo" == "Bar"
False
Prelude> [[1,2],[3,4]] == [[1,2],[3,4]]
True
Prelude> (\x -> x+1) == (\x -> x+2)
<interactive>:5:1: error:
• No instance for (Eq (Integer -> Integer))
arising from a use of ‘==’
(maybe you haven't applied a function to enough arguments?)
• In the expression: (\ x -> x + 1) == (\ x -> x + 2)
In an equation for ‘it’: it = (\ x -> x + 1) == (\ x -> x + 2)
There are some other useful functions that use the Eq
class, like nub
from the module Data.List
.
Prelude> import Data.List
Prelude Data.List> :t nub
nub :: Eq a => [a] -> [a]
Prelude Data.List> nub [3,5,3,1,1] -- eliminates duplicates
[3,5,1]
Ord
The Ord
class is for ordering (less than, greater than). Again, here are the basic operations and some examples of their use. Note the new Ordering
type. It has values LT
for “less than”, EQ
for “equal” and GT
for “greater than”.
compare :: Ord a => a -> a -> Ordering
(<) :: Ord a => a -> a -> Bool
(>) :: Ord a => a -> a -> Bool
(>=) :: Ord a => a -> a -> Bool
(<=) :: Ord a => a -> a -> Bool
max :: Ord a => a -> a -> a
min :: Ord a => a -> a -> a
Prelude> compare 1 1 -- 1 is EQual to 1
EQ
Prelude> compare 1 3 -- 1 is Less Than 3
LT
Prelude> compare 1 0 -- 1 is Greater Than 0
GT
Prelude> min 5 3
3
Prelude> max 5 3
5
Prelude> "aardvark" < "banana" -- strings are compared alphabetically
True
Prelude> [1,2,3] > [2,5] -- lists are compared like strings
False
Prelude> [1,2,3] > [1,1]
True
When we can compare values, we can also sort lists of them. The function sort
from Data.List
works on all types that belong to the Ord
class.
Prelude> import Data.List
Prelude Data.List> :t sort
sort :: Ord a => [a] -> [a]
Prelude Data.List> sort [6,1,4,8,2]
[1,2,4,6,8]
Prelude Data.List> sort "black sphinx of quartz, judge my vow!" -- remember, strings are lists!
" !,aabcdefghijklmnoopqrstuuvwxyz"
As a last example, let’s sort a list of lists according to length. We’ll need two helper functions:
-- from the module Data.Ord
-- compares two values "through" the function f
comparing :: (Ord a) => (b -> a) -> b -> b -> Ordering
comparing f x y = compare (f x) (f y)
-- from the module Data.List
-- sorts a list using the given comparison function
sortBy :: (a -> a -> Ordering) -> [a] -> [a]
Now the implementation of sortByLength
is straightforward:
-- sorts lists by their length
sortByLength :: [[a]] -> [[a]]
sortByLength = sortBy (comparing length)
Num
, Integral
, Fractional
, Floating
The Num
class contains integer arithmetic:
(+) :: Num a => a -> a -> a
(-) :: Num a => a -> a -> a
(*) :: Num a => a -> a -> a
negate :: Num a => a -> a -- 0-x
abs :: Num a => a -> a -- absolute value
signum :: Num a => a -> a -- -1 for negative values, 0 for 0, +1 for positive values
fromInteger :: Num a => Integer -> a
Num
also shows up in the types of integer literals:
This means that a literal like 12
can be interpreted as a member of any type implementing Num
. When GHC reads a number literal like, 12
it produces code that corresponds to fromIntegral 12
.
Integral
is the class of types that represent whole numbers, like Int
and Integer
. The most interesting functions are div
and mod
for integer division and remainder. All types that belong to Integral
also belong to Num
.
Fractional
is the class for types that have division. All types that belong to Fractional
also belong to Num
.
Floating
contains some additional operations that only make sense for floating point numbers. All types that belong to Floating
also belong to Fractional
(and to Num
).
Read
and Show
The Show
and Read
classes are for the functions show
and read
, that convert values to and from Strings.
As you can see above, you often need to use a type annotation with read
so that the compiler can choose the right implementation.
Foldable
One more thing! You might remember that it was mentioned earlier that the type of length
isn’t [a] -> Int
but something more general. Let’s have a look:
This type looks a bit different than the ones we’ve seen before. The type variable t
has an argument a
. We’ll look at type classes like this in more detail in part 2, but here’s a crash course.
What Foldable
represents is types that you can fold over. The true type of foldr
is:
We’ve succesfully used the fact that lists are Foldable
since we’ve managed to use length
and foldr
on lists. However, Maybe
is also Foldable
! The Foldable
instance for Maybe
just pretends that values of Maybe a
are like lists of length 0 or 1:
We’ll meet some more foldable types next.
Now that we are familiar with the standard type classes, we can look at one of their applications: the Map
and Array
data structures.
Data.Map
The Data.Map
module defines the Map
type. Maps are search trees for key-value pairs. One way to look at this is that a value of type Map k v
is roughly the same as a value of type [(k,v)]
, a list of pairs. However, the operations on a map are more efficient than operations on a list.
Since Data.Map
contains some function with the same names as Prelude
functions, the namespace needs to be imported qualified:
Now we can refer to the map type as Map.Map
, and to various map functions like Map.insert
. Here are the most important functions for maps:
-- Create a Map from a list of key-value pairs
Map.fromList :: Ord k => [(k, a)] -> Map.Map k a
-- Insert a value into a map. Overrides any previous value with the same key.
-- Returns a new map. Does not mutate the given map.
Map.insert :: Ord k => k -> a -> Map.Map k a -> Map.Map k a
-- Get a value from a map using a key. Returns Nothing if the key was not present in the map.
Map.lookup :: Ord k => k -> Map.Map k a -> Maybe a
-- An empty map
Map.empty :: Map.Map k a
The Ord
constraint for the key type of the map is needed because maps are implemented as ordered binary search trees.
Note that like all Haskell values, maps are immutable meaning you can’t change a map once you define it. However, map operations like insert
produce a new map. To perform multiple map operations you need to reuse the return value. Here’s a GHCi session operating on a map.
Prelude> import qualified Data.Map as Map
Prelude Map> let values = Map.fromList [("z",3),("w",4)]
Prelude Map> Map.lookup "z" values
Just 3
Prelude Map> Map.lookup "banana" values
Nothing
Prelude Map> Map.insert "x" 7 values
fromList [("w",4),("x",7),("z",3)]
Prelude Map> values -- note immutability!
fromList [("w",4),("z",3)]
Prelude Map> Map.insert "x" 1 (Map.insert "y" 2 values) -- two insertions
fromList [("w",4),("x",1),("y",2),("z",3)]
Prelude Map>
Here’s an example of representing a bank as a Map String Int
(map from account name to account balance), and withdrawing some money from an account:
withdraw :: String -> Int -> Map.Map String Int -> Map.Map String Int
withdraw account amount bank =
case Map.lookup account bank of
Nothing -> bank -- account not found, no change
Just sum -> Map.insert account (sum-amount) bank -- set new balance
Here’s how you might use the withdraw
function in GHCi
. Note how the maps get printed as fromList
invocations. Also note how calling withdraw ... bank
returns a new bank and doesn’t change the existing bank.
GHCi> let bank = Map.fromList [("Bob",100),("Mike",50)]
GHCi> withdraw "Bob" 80 bank
fromList [("Bob",20),("Mike",50)]
GHCi> bank -- note immutability
fromList [("Bob",100),("Mike",50)]
GHCi> withdraw "Bozo" 1000 bank
fromList [("Bob",100),("Mike",50)]
Data.Map
defines all sorts of useful higher-order functions for updating maps. We can rewrite the withdraw
function using Data.Map.adjust
:
withdraw :: String -> Int -> Map.Map String Int -> Map.Map String Int
withdraw account amount bank = Map.adjust (\x -> x-amount) account bank
Note! There are separate Data.Map.Strict
and Data.Map.Lazy
implementations. When you import Data.Map
you get Data.Map.Lazy
. You can find the documentation for all the Data.Map
functions in the docs for Data.Map.Lazy
. We won’t go into their differences here, but mostly you shold use Data.Map.Strict
in real code.
Data.Array
Another type that works kind of like a list but is more efficient for some operations is the array. Arrays are familiar from many other programming languages, but Haskell arrays are a bit different.
Unlike the Data.Map
module, the Data.Array
can just be imported normally:
Now we can look at the type of the array
function that constructs an array.
There are a couple of things to notice here. First of all, the Array
type is parameterized by two types: the index type and the element type. Most other programming languages only parameterize arrays with the element type, but the index type is always int
. In Haskell, we can have, for example, an Array Char Int
: an array indexed by characters, or Array Bool String
, an array indexed by booleans, or even Array (Int,Int) Int
, a two-dimensional array of ints.
Not all types can be index types. Only types that are similar to integers are suitable. That is the reason for the Ix i
class constraint. The Ix
class collects all the types that can be used as array indexes.
Secondly, the array
function takes an extra (i,i)
parameter. These are the minimun and maximum indexes of the array. Unlike some other languages, where arrays always start at index 0 or 1, in Haskell you can define an array that starts from 7 and goes to 11. So here’s that array:
myArray :: Array Int String
myArray = array (7,11) [(7,"seven"), (8,"eight"), (9,"nine"), (10,"ten"), (11,"ELEVEN")]
Listing all the indices and elements in order can be a bit cumbersome, so there’s also the listArray
constructor that just takes a list of elements in order:
Arrays are used with two new operators:
-- Array lookup
(!) :: Ix i => Array i e -> i -> e
-- Array update
(//) :: Ix i => Array i e -> [(i, e)] -> Array i e
Here’s an example GHCi
session:
Prelude> import Data.Array
Prelude Data.Array> let myArray = listArray (7,11) ["seven", "eight", "nine", "ten", "ELEVEN"]
Prelude Data.Array> myArray
array (7,11) [(7,"seven"),(8,"eight"),(9,"nine"),(10,"ten"),(11,"ELEVEN")]
Prelude Data.Array> myArray ! 8
"eight"
Prelude Data.Array> myArray // [(8,"ocho"),(9,"nueve")]
array (7,11) [(7,"seven"),(8,"ocho"),(9,"nueve"),(10,"ten"),(11,"ELEVEN")]
You might be wondering why the (//)
operator does multiple updates at once. The reason is the main weakness of Haskell arrays: immutability. Since arrays can’t be changed in place, (//)
must copy the whole array. This is why in Haskell it’s often preferable to use lists or maps to store data that needs to be updated. However, arrays may still be useful when constructed once and then used for a large number of lookups. We’ll get back to how Haskell data structures work in the next lecture.
Note! In this course we’ll use only Array
, a simple array type that’s specified in the Haskell standard. There are many other array types like the mutable IOArray
and the somewhat obscure DiffArray
. There are also type classes for arrays like IArray
and MArray
. In addition to arrays there is a wide family of Vector
types that can be more practical than Array
for real programs.
The Map
and Array
type are instances of Foldable
just like lists are! This means you can use functions like length
and foldr
on them:
length (array (7,11) [(7,"seven"),(8,"eight"),(9,"nine"),(10,"ten"),(11,"ELEVEN")])
==> 5
foldr (+) 0 (Map.fromList [("banana",3),("egg",7)])
==> 10
Haskell libraries tend to have pretty good docs. We’ve linked to docs via Hackage (https://hackage.haskell.org) previously, but it’s important to know how to find the docs by your self too. The tool for generating Haskell documentation is called Haddock so sometimes Haskell docs are referred to as haddocks.
Hackage is the Haskell package repository (just like PyPI for Python, Maven Central for Java or NPM for Javascript). In addition to the actual packages, it hosts documentation for them. Most of the modules that we use on this course are in the package called base
. You can browse the docs for the base package at http://hackage.haskell.org/package/base.
When you’re not quite sure where the function you’re looking for is, Hoogle (https://hoogle.haskell.org/) can help. Hoogle is a search engine for Haskell documentation. It is a great resource when you need to check what was the type of foldr
or which packages contain a function named reverse
.
Finally, since this course is using the stack
tool, you can also browse the documentation for the libraries stack has installed for you with the commands
stack haddock --open
stack haddock --open <package>
This has the added benefit of getting exactly the right version of the documentation.
In summary, here are the main ways of reading Haskell library documentation:
stack
you can use stack haddock --open
or stack haddock --open <package>
to open docs in your browser.What is the type of swap . swap
?
(a, b) -> (a, b)
(a, b) -> (b, a)
a -> a
What is the type of \f g x -> (f x, g x)
?
(a -> b) -> (c -> d) -> (a,c) -> (b, d)
(a -> b) -> (a -> c) -> a -> (b, c)
(a -> b) -> (b -> a) -> a -> (b, a)
What is the type of \t -> (fst . fst $ t, (snd . fst $ t, snd t))
?
(a, (b, c)) -> (a, (b, c))
(a, (b, c)) -> ((a, b), c)
((a, b), c) -> (a, (b, c))
What does the function foldr (\x xs -> xs ++ [x]) []
do?
What does the function foldr (\(x, y) zs -> x : y : zs) []
do?
What is the type of foldr (\n b -> n == 3 && b)
?
(Foldable t, Eq a, Num a) => Bool -> t a -> Bool
(Foldable t, Eq a, Num a, Bool b) => b -> t a -> b
(Foldable t, Eq a, Num a) => Bool -> [ a ] -> Bool
What is the type of \x -> case x of (True, "Foo") -> show True ++ "Foo"
?
Either Bool String -> String
(Bool, String) -> String
Show a => (Bool, String) -> a
Haskell has a system called algebraic datatypes for defining new types. This sounds fancy, but is rather simple. Let’s dive in by looking at the standard library definitions of some familiar types:
With this syntax you too can define types:
-- definition of a type with three values
data Color = Red | Green | Blue
-- a function that uses pattern matching on our new type
rgb :: Color -> [Double]
rgb Red = [1,0,0]
rgb Green = [0,1,0]
rgb Blue = [0,0,1]
Prelude> :t Red
Red :: Color
Prelude> :t [Red,Blue,Green]
[Red,Blue,Green] :: [Color]
Prelude> rgb Red
[1.0,0.0,0.0]
Types like Bool
, Ordering
and Color
that just list a bunch of constants are called enumerations or enums in Haskell and other languages. Enums are useful, but you need other types as well. Here we define a type for reports containing an id number, a title, and a body:
This is how you create a report:
Prelude> :t ConstructReport 1 "Title" "This is the body."
ConstructReport 1 "Title" "This is the body." :: Report
You can access the fields with pattern matching:
reportContents :: Report -> String
reportContents (ConstructReport id title contents) = contents
setReportContents :: String -> Report -> Report
setReportContents contents (ConstructReport id title _contents) = ConstructReport id title contents
The things on the right hand side of a data
declaration are called constructors. True
, False
, Red
and ConstructReport
are all examples of constructors. A type can have multiple constructors, and a constructor can have zero or more fields.
Here is a datatype for a standard playing card. It has five constructors, of which Joker
has zero fields and the others have one field.
Constructors with fields have function type and can be used wherever functions can:
Prelude> :t Heart
Heart :: Int -> Card
Prelude> :t Club
Club :: Int -> Card
Prelude> map Heart [1,2,3]
[Heart 1,Heart 2,Heart 3]
Prelude> (Heart . (\x -> x+1)) 3
Heart 4
By the way, there’s something missing from our Card
type. Look at how it behaves compared to Ordering
and Bool
:
Prelude> EQ
EQ
Prelude> True
True
Prelude> Joker
<interactive>:1:0:
No instance for (Show Card)
arising from a use of `print' at <interactive>:1:0-4
Possible fix: add an instance declaration for (Show Card)
In a stmt of a 'do' expression: print it
The problem is that Haskell does not know how to print the types we defined. As the error says, they are not part of the Show
class. The easy solution is to just add a deriving Show
after the type definition:
The deriving
syntax is a way to automatically make your class a member of certain basic type classes, most notably Read
, Show
and Eq
. We’ll talk more about what this means later.
So why are these datatypes called algebraic? This is because, theoretically speaking, each datatype can be a sum of constructors, and each constructor is a product of fields. It makes sense to think of these as sums and products for many reasons, one being that we can count the possible values of each type this way:
data Bool = True | False -- corresponds to 1+1. Has 2 possible values.
data TwoBools = TwoBools Bool Bool -- corresponds to Bool*Bool, i.e. 2*2. Has 4 possible values.
data Complex = Two Bool Bool | One Bool | None
-- corresponds to Bool*Bool+Bool+1 = 2*2+2+1 = 7. Has 7 possible values.
There is a rich theory of algebraic datatypes. If you’re interested, you might find more info here or here.
We introduced type parameters and parametric polymorphism when introducing lists in Lecture 2. Since then, we’ve seen other parameterized types like Maybe
and Either
. Now we’ll learn how we can define our own parameterized types.
The definition for Maybe
is:
What’s a
? We define a parameterized type by mentioning a type variable (a
in this case) on the left side of the =
sign. We can then use the same type variable in fields for our constructors. This is analogous to polymorphic functions. Instead of defining separate functions
and so on, we define one function head :: [a] -> a
that works for all types a
. Similarly, instead of defining multiple types
we define one type Maybe a
that works for all types a
.
Here’s our first own parameterized type Described
. The values of type Described a
contain a value of type a
and a String
description.
data Described a = Describe a String
getValue :: Described a -> a
getValue (Describe x _) = x
getDescription :: Described a -> String
getDescription (Describe _ desc) = desc
Prelude> :t Describe
Describe :: a -> String -> Described a
Prelude> :t Describe True "This is true"
Describe True "This is true" :: Described Bool
Prelude> getValue (Describe 3 "a number")
3
Prelude> getDescription (Describe 3 "a number")
"a number"
In the above definitions, we’ve used a
as a type variable. However any word that starts with a lower case letter is fine. We could have defined Maybe
like this:
The rules for Haskell identifiers are:
a
, map
, xs
)Maybe
, Just
, Card
, Heart
)Note that a type and its constructor can have the same name. This is very common in Haskell code for types that only have one constructor. In this material we try to avoid it to avoid confusion. Here are some examples:
Beware of mixing up types and constructors. Luckily types and constructors can never occur in the same context, so you get a nice error:
Prelude> Maybe -- trying to use a type name as a value
<interactive>:1:1: error:
• Data constructor not in scope: Maybe
Prelude> undefined :: Nothing -- trying to use a constructor as a type
<interactive>:2:14: error:
Not in scope: type constructor or class ‘Nothing’
Types can have multiple type parameters. The syntax is similar to defining functions with many arguments. Here’s the definition of the standard Either
type:
So far, all of the types we’ve defined have been of constant size. We can represent one report or one colour, but how could we represent a collection of things? We could use lists of course, but could we define a list type ourselves?
Just like Haskell functions, Haskell data types can be recursive. This is no weirder than having an object in Java or Python that refers to another object of the same class. This is how you define a list of integers:
data IntList = Empty | Node Int IntList
deriving Show
ihead :: IntList -> Int
ihead (Node i _) = i
itail :: IntList -> IntList
itail (Node _ t) = t
ilength :: IntList -> Int
ilength Empty = 0
ilength (Node _ t) = 1 + ilength t
We can use the functions defined above to work with lists of integers:
Prelude> ihead (Node 3 (Node 5 (Node 4 Empty)))
3
Prelude> itail (Node 3 (Node 5 (Node 4 Empty)))
Node 5 (Node 4 Empty)
Prelude> ilength (Node 3 (Node 5 (Node 4 Empty)))
3
Note that we can’t put values other than Int
s inside our IntList
:
Prelude> Node False Empty
<interactive>:3:6: error:
• Couldn't match expected type ‘Int’ with actual type ‘Bool’
• In the first argument of ‘Node’, namely ‘False’
In the expression: Node False Empty
In an equation for ‘it’: it = Node False Empty
To be able to put any type of element in our list, let’s do the same thing with a type parameter. This is the same as the built in type [a]
, but with slightly clunkier syntax:
Note how we need to pass the the type parameter a
onwards in the recursion. We need to write Node a (List a)
instead of Node a List
. The Node
constructor has two arguments. The first has type a
, and the second has type List a
. Here are the reimplementations of some standard list functions for our List
type:
lhead :: List a -> a
lhead (Node h _) = h
ltail :: List a -> List a
ltail (Node _ t) = t
lnull :: List a -> Bool
lnull Empty = True
lnull _ = False
llength :: List a -> Int
llength Empty = 0
llength (Node _ t) = 1 + llength t
Prelude> lhead (Node True Empty)
True
Prelude> ltail (Node True (Node False Empty))
Node False Empty
Prelude> lnull Empty
True
Note that just like with normal Haskell lists, we can’t have elements of different types in the same list:
Prelude> Node True (Node "foo" Empty)
<interactive>:5:12: error:
• Couldn't match type ‘[Char]’ with ‘Bool’
Expected type: List Bool
Actual type: List [Char]
• In the second argument of ‘Node’, namely ‘(Node "foo" Empty)’
In the expression: Node True (Node "foo" Empty)
In an equation for ‘it’: it = Node True (Node "foo" Empty)
Just like a list, we can also represent a binary tree:
Our tree contains nodes, which contain a value of type a
and two child trees, and empty trees.
In case you’re not familiar with binary trees, they’re a data structure that’s often used as the basis for other data structures (Data.Map
is based on trees!). Binary trees are often drawn as (upside-down) pictures, like this:
The highest node in the tree is called the root (0
in this case), and the nodes with no children are called leaves
(2
, 3
and 4
in this case). We can define this tree using our Tree
type like this:
example :: Tree Int
example = (Node 0 (Node 1 (Node 2 Empty Empty)
(Node 3 Empty Empty))
(Node 4 Empty Empty))
The height of a binary tree is length of the longest path from the root to a leaf. In Haskell terms, it’s how many nested levels of Node
constructors you need to build the tree. The height of our example tree is 3. Here’s a function that computes the height of a tree:
treeHeight :: Tree a -> Int
treeHeight Empty = 0
treeHeight (Node _ l r) = 1 + max (treeHeight l) (treeHeight r)
treeHeight Empty ==> 0
treeHeight (Node 2 Empty Empty)
==> 1 + max (treeHeight Empty) (treeHeight Empty)
==> 1 + max 0 0
==> 1
treeHeight (Node 1 Empty (Node 2 Empty Empty))
==> 1 + max (treeHeight Empty) (treeHeight (Node 2 Empty Empty))
==> 1 + max 0 1
==> 2
treeHeight (Node 0 (Node 1 Empty (Node 2 Empty Empty)) Empty)
==> 1 + max (treeHeight (Node 1 Empty (Node 2 Empty Empty))) (treeHeight Empty)
==> 1 + max 2 0
==> 3
In case you’re familiar with binary search trees, here are the definitions of the lookup and insert opertions for a binary search tree. If you don’t know what I’m talking about, you don’t need to understand this.
lookup :: Int -> Tree Int -> Bool
lookup x Empty = False
lookup x (Node y l r)
| x < y = lookup x l
| x > y = lookup x r
| otherwise = True
insert :: Int -> Tree Int -> Tree Int
insert x Empty = Node x Empty Empty
insert x (Node y l r)
| x < y = Node y (insert x l) r
| x > y = Node y l (insert x r)
| otherwise = Node y l r
If some fields need to be accessed often, it can be convenient to have helper functions for reading those fields. For instance, the type Person
might have multiple fields:
A list of persons might look like the following:
people :: [Person]
people = [ MkPerson "Jane Doe" 21 "Houston" "Texas" "Engineer"
, MkPerson "Maija Meikäläinen" 35 "Rovaniemi" "Finland" "Engineer"
, MkPerson "Mauno Mutikainen" 27 "Turku" "Finland" "Mathematician"
]
Suppose that we need to find all engineers from Finland:
query :: [Person] -> [Person]
query [] = []
query (MkPerson name age town state profession):xs
| state == "Finland" && profession == "Engineer" =
(MkPerson name age town state profession) : query xs
| otherwise = query xs
Thus,
Note that the types of the fields give little information on what is the intended content in those fields. We need to remember in all places in the code that town
goes before state
and not vice versa.
Haskell has a feature called record syntax that is helpful in these kinds of cases. The datatype Person
can be defined as a record:
data Person = MkPerson { name :: String, age :: Int, town :: String, state :: String, profession :: String}
deriving Show
We can still define values of Person
normally, but the Show
instance prints the field names for us:
Prelude> MkPerson "Jane Doe" 21 "Houston" "Texas" "Engineer"
MkPerson {name = "Jane Doe", age = 21, town = "Houston", state = "Texas", profession = "Engineer"}
However, we can also define values using record syntax. Note how the fields don’t need to be in any specific order now that they have names.
Prelude> MkPerson {name = "Jane Doe", town = "Houston", profession = "Engineer", state = "Texas", age = 21}
MkPerson {name = "Jane Doe", age = 21, town = "Houston", state = "Texas", profession = "Engineer"}
Most importantly, We get accessor functions for the fields for free:
Prelude> :t profession
profession :: Person -> String
Prelude> profession (MkPerson "Jane Doe" 21 "Houston" "Texas" "Engineer")
"Engineer"
We can now rewrite the query function using these accessor functions:
query :: [Person] -> [Person]
query [] = []
query (x:xs)
| state x == "Finland" && profession x == "Engineer" =
x : query xs
| otherwise = query xs
You’ll probably agree that the code looks more pleasant now.
This gives you accessor functions like field1 :: TypeName -> Field1Type
for free.
In addition to the data
keyword, there are two additional ways of defining types in Haskell.
The newtype
keyword works like data
, but you can only have a single constructor with a single field. It’s sometimes wise to use newtype
for performance resons, but we’ll get back to those in part 2.
The type
keyword introduces a type alias. Type aliases don’t affect type checking, they just offer a shorthand for writing types. For example the familiar String
type is an alias for [Char]
:
This means that whenever the compiler reads String
, it just immediately replaces it with [Char]
. Type aliases seem useful, but they can easily make reading type errors harder.
Remember how lists were represented in memory as linked lists? Let’s look in more detail at what algebraic datatypes look like in memory.
Haskell data forms directed graphs in memory. Every constructor is a node, every field is an edge. Names (of variables) are pointers into this graph. Different names can share parts of the structure. Here’s an example with lists. Note how the last two elements of x
are shared with y
and z
.
What happens when you make a new version of a datastructure is called path copying. Since Haskell data is immutable, the changed parts of the datastructure get copied, while the unchanged parts can be shared between the old and new versions.
Consider the definition of ++
:
We are making a copy of the first argument while we walk it. For every :
constructor in the first input list, we are creating a new :
constructor in the output list. The second argument can be shared. It is not used at all in the recursion. Visually:
One more way to think about it is this: we want to change the tail
pointer of the list element (3:)
. That means we need to make a new (3:)
. However the (2:)
points to the (3:)
so we need a new copy of the (2:)
as well. Likewise for (1:)
.
The graphs that we get when working with lists are fairly simple. As a more involved example, here is what happens in memory when we run the binary tree insertion example from earlier in this lecture.
insert :: Int -> Tree Int -> Tree Int
insert x Empty = Node x Empty Empty
insert x (Node y l r)
| x < y = Node y (insert x l) r
| x > y = Node y l (insert x r)
| otherwise = Node y l r
Note how the old and the new tree share the subtree with 3 and 4 since it wasn’t changed, but the node 7 that was “changed” and all nodes above it get copied.
Why can’t we map Nothing
?
Nothing
doesn’t take arguments
Nothing
returns nothing
Nothing
is a constructor.
If we define data Boing = Frick String Boing (Int -> Bool)
, what is the type of Frick
?
Boing
String -> Boing -> Int -> Bool -> Boing
String -> Boing -> (Int -> Bool) -> Boing
If we define data ThreeLists a b c = ThreeLists [a] [b] [c]
, what is the type of the constructor ThreeLists
?
[a] -> [b] -> [c] -> ThreeLists
a -> b -> c -> ThreeLists a b c
[a] -> [b] -> [c] -> ThreeLists a b c
[a] -> [b] -> [c] -> ThreeLists [a] [b] [c]
If we define data TwoLists a b = TwoList {aList :: [a], bList :: [b]}
, what is the type of the function aList
?
aList
is not a function, it is a field
TwoLists a b -> [a]
[a] -> TwoLists a b
[a]
We’ve seen class constraints like Eq a =>
in types. We know how to use existing classes with existing types. But how do we use existing classes with our own types? How can we define our own classes?
Here’s how to make your own type a member of the Eq
class:
data Color = Black | White
instance Eq Color where
Black == Black = True
White == White = True
_ == _ = False
A class instance is an instance
block that contains definitions for the functions in that class. Here we define how ==
works on Color
.
A type class is defined using class
syntax. The functions in the class are given types. Here’s a class Size
that contains one function, size
:
Instances of a class are defined with instance
syntax we’ve just seen. Here is how we make Int
and [a]
members of the Size
class:
Our class Size
behaves just like existing type classes. We can use size
anywhere where a function can be used, and Haskell can infer types with Size
constraints for us:
Prelude> :t size
size :: Size a => a -> Int
Prelude> size [True,False]
2
Prelude> let sizeBoth a b = [size a, size b]
Prelude> :t sizeBoth
sizeBoth :: (Size a1, Size a2) => a1 -> a2 -> [Int]
A class can contain multiple functions, and even constants. Here we define a new version of the Size
class with more content.
class Size a where
empty :: a
size :: a -> Int
sameSize :: a -> a -> Bool
instance Size (Maybe a) where
empty = Nothing
size Nothing = 0
size (Just a) = 1
sameSize x y = size x == size y
instance Size [a] where
empty = []
size xs = length xs
sameSize x y = size x == size y
The Haskell 2010 allows only a very specific type of class instance. Let’s look at some instances that aren’t allowed. The examples use the Size
class:
We saw a Size [a]
instance above. Why not define an instance just for lists of booleans?
error:
• Illegal instance declaration for ‘Size [Bool]’
(All instance types must be of the form (T a1 ... an)
where a1 ... an are *distinct type variables*,
and each type variable appears at most once in the instance head.
Use FlexibleInstances if you want to disable this.)
• In the instance declaration for ‘Size [Bool]’
Dang. As the error tries to tell us, we can only define instances where all type parameters are different type variables. That is, we can define instance Size (Either a b)
but we can’t define:
instance Size (Either String a)
– since String
is not a type variableinstance Size (Either a a)
– since the type variables aren’t differentinstance Size [[a]]
– since [a]
is not a type variableWhy is this? This rule guarantees that it’s simple for the compiler to look up the correct type class instance. It can just look at what the top-level type constructor is, and then pick an instance.
The GHC Haskell implementation has many extensions to the type class system – we’ll get back to some of them in part 2 of this course.
Did you notice how in the previous example we gave sameSize
the same definition in both instances? This is a very common occurrence, and it’s why Haskell classes can have default implementations. As a first example, here’s an Example
type class for giving example values of types.
class Example a where
example :: a -- the main example for the type `a`
examples :: [a] -- a short list of examples
examples = [example] -- ...defaulting to just the main example
instance Example Int where
example = 1
examples = [0,1,2]
instance Example Bool where
example = True
Here’s how Example
works. Note how the default implementation of examples
got used in the Bool
case but not in the Int
case. Also note the need for explicit type signatures to tell GHCi which instance we’re interested in. Without them, we would get an “Ambiguous type variable” error.
Prelude> example :: Bool
True
Prelude> example :: Int
1
Prelude> examples :: [Bool]
[True]
Prelude> examples :: [Int]
[0,1,2]
The standard type classes use lots of default implementations to make implementing the classes easy. Here is the standard definitions for Eq
(formatted for readability).
class Eq a where
(==) :: a -> a -> Bool
x == y = not (x /= y)
(/=) :: a -> a -> Bool
x /= y = not (x == y)
Note how both operations have a default implementation in terms of the other. This means we could define an Eq
instance with no content at all, but the resulting functions would just recurse forever. In practice, we want to define at least one of ==
and /=
.
When there are lots of default implementations, it can be hard to know which functions you need to implement yourself. For this reason class documentation usually mentions the minimal complete definition. For Eq
, the docs say “Minimal complete definition: either == or /=.”
Let’s look at Ord
next. Ord
has 7 operations, all with default implementations in terms of each other. By the way, note the quirky way of defining multiple type signatures at once. It’s okay, it’s a feature of Haskell, this is how Ord
is defined in the standard. (We’ll get back to what the (Eq a) =>
part means soon.)
class (Eq a) => Ord a where
compare :: a -> a -> Ordering
(<), (<=), (>=), (>) :: a -> a -> Bool
max, min :: a -> a -> a
compare x y | x == y = EQ
| x <= y = LT
| otherwise = GT
x <= y = compare x y /= GT
x < y = compare x y == LT
x >= y = compare x y /= LT
x > y = compare x y == GT
max x y | x <= y = y
| otherwise = x
min x y | x <= y = x
| otherwise = y
With this definition it’s really hard to know what the minimal complete definition is. Luckily the docs tell us “Minimal complete definition: either compare or <=.”
As a final word on default implementations, if there is never a need to override the default definition, the function can be moved out of the class for simplicity. Consider a class like Combine
below:
class Combine a where
combine :: a -> a -> a
combine3 :: a -> a -> a -> a
combine3 x y z = combine x (combine y z)
It’s hard to think of a case where combine3
would be given any other definition, so why not move it out of the class:
class Combine a where
combine :: a -> a -> a
combine3 :: Combine a => a -> a -> a -> a
combine3 x y z = combine x (combine y z)
As an example, here are the Eq
and Ord
instances for a simple pair type. Note how the definition uses the minimal complete definition rules by only defining ==
and <=
.
data IntPair = IntPair Int Int
deriving Show
instance Eq IntPair where
IntPair a b == IntPair c d = a==b && c==d
instance Ord IntPair where
IntPair a b <= IntPair c d
| a<c = True
| a>c = False
| otherwise = b<=d
*Main> (IntPair 1 2) < (IntPair 2 3)
True
*Main> (IntPair 1 2) > (IntPair 2 3)
False
*Main> compare (IntPair 1 2) (IntPair 2 3)
LT
As we’ve seen many times already, deriving
is a way to get automatically generated class instances. The Read
and Show
classes should pretty much always be derived to get the standard behaviour. The derived instance for Eq
is typically what you want. It requires constructors and fields to match.
The derived Ord
instance might not be what you want. It orders constructors left-to-right, and then compares fields inside constructors left-to-right. An example:
Prelude> Dead < Alive "Bob" 35 -- constructors are ordered left-to-right
True
Prelude> Alive "Barbara" 35 < Alive "Clive" 17 -- names are compared before ages
True
Prelude> Alive "Clive" 17 < Alive "Clive" 30 -- finally, ages are compared if names match
True
You can use the :info
command in GHCi to get the contents and instances of a class. These days the info even includes the minimal complete definition (see the MINIMAL pragma). For example:
Prelude> :info Num
class Num a where
(+) :: a -> a -> a
(-) :: a -> a -> a
(*) :: a -> a -> a
negate :: a -> a
abs :: a -> a
signum :: a -> a
fromInteger :: Integer -> a
{-# MINIMAL (+), (*), abs, signum, fromInteger, (negate | (-)) #-}
-- Defined in ‘GHC.Num’
instance Num Word -- Defined in ‘GHC.Num’
instance Num Integer -- Defined in ‘GHC.Num’
instance Num Int -- Defined in ‘GHC.Num’
instance Num Float -- Defined in ‘GHC.Float’
instance Num Double -- Defined in ‘GHC.Float’
Both classes and instances can form hierarchies. This means that a class or instance depends on another class or instance.
Let’s try to define an Eq
instance for a simple pair type:
data Pair a = MakePair a a
deriving Show
instance Eq (Pair a) where
(MakePair x y) == (MakePair a b) = x==a && y==b
error:
• No instance for (Eq a) arising from a use of ‘==’
Possible fix: add (Eq a) to the context of the instance declaration
• In the first argument of ‘(&&)’, namely ‘x == a’
In the expression: x == a && y == b
In an equation for ‘==’:
(MakePair x y) == (MakePair a b) = x == a && y == b
The compiler is trying to tell us that our Eq (Pair a)
instance needs an Eq a
instance to work. How can we compare pairs of values of type a
if we can’t compare values of type a
? To solve this we need to add a type constraint to the instance declaration, just like we’ve added type constraints to function definitions.
Now we can compare pairs, as long as the element type is comparable. However, we can’t compare, say, pairs of functions, since functions don’t have an Eq instance.
Prelude> MakePair 1 1 == MakePair 1 1
True
Prelude> MakePair reverse reverse == MakePair reverse reverse
<interactive>:15:1: error:
• No instance for (Eq ([a0] -> [a0])) arising from a use of ‘==’
(maybe you haven't applied a function to enough arguments?)
• In the expression:
MakePair reverse reverse == MakePair reverse reverse
In an equation for ‘it’:
it = MakePair reverse reverse == MakePair reverse reverse
Let’s continue with another example. Here’s a simple type class and an instance
Now we can write a function that checks a list:
In order to turn this into a Check [a]
instance, we need to add a constraint to the instance declaration. Our Check [a]
is based on the Check a
instance.
This means that our Check [a]
instance is only valid when there is a corresponding Check a
instance. For example, if we try to invoke a Check [Bool]
instance, we get an error about the missing Check Bool
instance:
Prelude> check [True,False]
<interactive>:1:1: error:
• No instance for (Check Bool) arising from a use of ‘check’
• In the expression: check [True, False]
In an equation for ‘it’: it = check [True, False]
Also, if we try to define Check [a]
instance without the constraint, we get an error (with a pretty good suggestion!)
• No instance for (Check a) arising from a use of ‘check’
Possible fix:
add (Check a) to the context of the instance declaration
If you think about it, this instance hierarchy allows us to circumvent the limitation that we can’t make a Check [Int]
instance.
Finally, sometimes multiple constraints are needed. Consider for example the Eq
instance for Either
:
instance (Eq a, Eq b) => Eq (Either a b) where
Left x == Left y = x==y
Right x == Right y = x==y
_ == _ = False
Respectively, a class can depend on another class. This is useful for instance when you want to use functions from another class in your default implementations:
class Size a where
size :: a -> Int
class Size a => SizeBoth a where
sizeBoth :: a -> a -> Int
sizeBoth x y = size x + size y
In cases like this we say SizeBoth
is a subclass of Size
. Note again the confusion with object oriented programming. Examples of subclasses in the standard library include:
Another way to look at subclasses is that if you have a class Main a => Sub a
, you must provide an instance Main MyType
in order to be able to declare instance Sub MyType
.
What are the functions in the Eq
class?
(==), (/=)
(==)
(==)
, (<)
, (>)
For which of the following classes can we get automatic instances with deriving
?
Num
Ord
Size
Which of the following instance declarations is legal?
instance Eq Maybe
instance Eq (a,a)
instance Eq (Maybe Int)
instance Eq (a,b)
Given the following definition of the class BitOperations
class BitOperations a where
bitNot :: a -> a
bitNot x = bitNand bitTrue x
bitAnd :: a -> a -> a
bitAnd x y = bitNot (bitOr (bitNot x) (bitNot y))
bitOr :: a -> a -> a
bitOr x y = bitNot (bitAnd (bitNot x) (bitNot y))
bitNand :: a -> a -> a
bitNand x y = bitNot (bitAnd x y)
which set of operations is not a minimal complete definition of BitOperations
?
bitNand, bitAnd
bitAnd, bitOr
bitAnd, bitNot
bitNot, bitOr
The declaration instance Num a => Eq (Pair a)
tells me that
Num
are instances of Eq
Pair a
is an instance of Eq
if a
is an instance of Num
Eq (Pair a)
inherits the instance Num a
The declaration class Num a => Fractional a
tells me that
Fractional
must be instances of Num
Num
must be instances of Fractional
Fractional
, I also get an instance for Num
Num
, I also get an instance for Fractional
This lecture offers an introduction to design patterns for typed functional programming. These patterns are both useful when writing Haskell programs, and offer a nice arena for practicing skills from the previous lectures.
Sometimes you don’t need a new type, but instead can just reuse a standard type. For example, repesenting car register plate numbers with String
. However, if your code is full of String
s, it can be easy to accidentally mix up e.g. a car’s model and registration in a function like registerCar :: String -> String -> CarRegistry -> CarRegistry
.
For situations like this it’s common to create a new type that just contains a String
(a “boxed” string):
We can now give registerCar
a slightly nicer type, String -> Plate -> CarRegistry -> CarRegistry
. Additionally, we can restrict the operations that are possible on Plate
s to a subset of those that are possible on strings. For example, there is no need to combine the register plate numbers of two cars. Thus we don’t need to offer a function concatPlates :: Plate -> Plate -> Plate
. We can also define a smart constructor for Plate
that checks that the register number is in the correct format:
parsePlate :: String -> Maybe Plate
parsePlate string
| correctPlateNumber string = Just (Plate string)
| otherwise = Nothing
Here’s another example: representing money. If we just store money as Int
s, the compiler won’t protect us from mistakes like multiplying money with money. If instead we implement our own Money
type that wraps Int
, we get type safety. Additionally, we can encapsulate the fact that money is represented as an integer amount of cents.
data Money = Money Int
deriving Show
renderMoney :: Money -> String
renderMoney (Money cents) = show (fromIntegral cents / 100)
(+!) :: Money -> Money -> Money
(Money a) +! (Money b) = Money (a+b)
scale :: Money -> Double -> Money
scale (Money a) x = Money (round (fromIntegral a * x))
addVat :: Money -> Money
addVat m = m +! scale m 0.24
renderMoney (Money 100 +! Money 150)
==> "2.5"
scale (Money 299) 0.24
==> Money 72
addVat (Money 299)
==> Money 371
Note! If you’re familiar with Object-Oriented Programming, this is a bit like encapsulation.
Haskell’s algebraic datatypes are really powerful at modeling things based on cases. It’s often useful to think of types as defining the set of possible cases, and functions handling those cases (often via pattern matching). Let’s look at two examples.
Since it’s so easy to define custom types in Haskell, it’s quite convenient to use more descriptive types instead of booleans or strings. Consider a list of persons. In some other language if you wanted to sort the persons into ascending order by name you might use a call like sortPersons(persons, "name", true)
. In Haskell you can do this instead:
data Person = Person {name :: String, age :: Int}
deriving Show
data SortOrder = Ascending | Descending
data SortField = Name | Age
sortByField :: SortField -> [Person] -> [Person]
sortByField Name ps = sortBy (comparing name) ps
sortByField Age ps = sortBy (comparing age) ps
sortPersons :: SortField -> SortOrder -> [Person] -> [Person]
sortPersons field Ascending ps = sortByField field ps
sortPersons field Descending ps = reverse (sortByField field ps)
persons = [Person "Fridolf" 73, Person "Greta" 60, Person "Hans" 65]
sortPersons Name Ascending persons
==> [Person {name = "Fridolf", age = 73},Person {name = "Greta", age = 60},Person {name = "Hans", age = 65}]
sortPersons Age Descending persons
==> [Person {name = "Fridolf", age = 73},Person {name = "Hans", age = 65},Person {name = "Greta", age = 60}]
Note how you can’t accidentally typo the field name (unlike with strings), and how you don’t need to remember whether true
refers to ascending or descending order.
Let’s move on to the next example. Many Haskell functions don’t work with empty lists (consider head []
). If you’re writing code that needs to track whether lists are possibly empty or guaranteed to not be empty, you can use the NonEmpty
type from the Data.List.NonEmpty module.
Consider the definition of NonEmpty
:
Here the type represents a lack of cases. The type NonEmpty a
will always consist of a value of type a
, and some further a
s, collected in a list. Here are some example values of NonEmpty Int
:
By the way, this is also an example of an infix constructor. We’ve already met another infix constructor earlier, the list constructor (:)
. Any operator that begins with a colon (the :
character) can be used as an infix constructor. We can pattern match on (:|)
just like on (:)
, as you’ll see in the examples below.
Here are the functions that convert between normal lists and nonempty lists. Note how we can’t have a function [a] -> NonEmpty a
, but must instead use Maybe
to represent the possibility that the list was, indeed, empty. Note also how toList
has only one equation, we can’t have a toList []
situation due to the type NonEmpty
.
nonEmpty :: [a] -> Maybe (NonEmpty a)
nonEmpty [] = Nothing
nonEmpty (x:xs) = Just (x :| xs)
toList :: NonEmpty a -> [a]
toList (x :| xs) = x : xs
nonEmpty [1,2,3] ==> Just (1 :| [2,3])
nonEmpty [1] ==> Just (1 :| [])
nonEmpty [] ==> Nothing
toList (1 :| [2,3]) ==> [1,2,3]
Here are head
and last
implemented for NonEmpty
:
By the way, these functions are available as Data.List.NonEmpty.head
and Data.List.NonEmpty.last
along with many other useful functions.
In summary, if you write types that represent all possible cases for your values, and then write functions that handle those cases, your code will be simple and correct.
A pattern that comes up surprisingly often in functional programming is the monoid (not to be confused with a monad!). Explanations of monoids are often very mathematical, but the idea is simple: combining things.
Many functions and operators we use are associative. This is just a fancy way of saying they don’t need parentheses. For example, all of these expressions have the value 16 because addition is associative:
Examples of associative operations are easy to come by in Haskell. For example the ++
operator for catenating lists is associative: it doesn’t matter whether you do ([1] ++ [2,3]) ++ [4]
or [1] ++ ([2,3] ++ 4)
– the result is [1,2,3,4]
.
Another great example is the function composition operator. Both (head . tail) . tail
and head . (tail . tail)
compute the third element of a list.
However not all operators are associative. The most familiar examples are subtraction and exponentiation. (1-2)-3
is -4
but 1-(2-3)
is 2
. Similarly, (2^3)^2
is 64
while 2^(3^2)
is 512. One needs to be careful with parentheses when using operators that are not associative.
Another operator that’s not associative is the list constructor, (:)
. This time the reason is even more fundamental: while True:(False:[])
is ok, (True:False):[]
does not even type! In order for an operation to be associative, it needs to take two arguments of the same type.
In addition to operators, functions can also be associative. The syntax looks a bit different, but a function f
is associative if these are the same:
Two widely-used associative functions are the min
and max
functions:
Mathematically speaking, an associative function (or operator) forms a semigroup. Haskell has a type class Semigroup
(defined in the module Data.Semigroup
) that can be used when a type has one clear associative operation.
Lists are an instance of Semigroup
with (++)
as (<>)
:
Types that have multiple different associative operators usually aren’t made an instance of Semigroup. An example is Int
, which has many associative functions like +
, *
and max
. Instead, the Haskell standard library uses boxing (see earlier in this lecture). Here are the definitions for Sum
and Product
:
data Sum a = Sum a
instance Num a => Semigroup (Sum a) where
Sum a <> Sum b = Sum (a+b)
data Product a = Product a
instance Num a => Semigroup (Product a) where
Product a <> Product b = Product (a*b)
By the way, this is another benefit of boxing things: being able to declare different type class instances!
Similarly, we have box types Min
and Max
. Let’s play around in GHCi a bit:
Prelude> import Data.Semigroup
Prelude Data.Semigroup> Product (2::Int) <> Product 3 <> Product 1
Product {getProduct = 6}
Prelude Data.Semigroup> Sum 3 <> Sum 5 <> Sum 7
Sum {getSum = 15}
Prelude Data.Semigroup> Product 2 <> Product 3 <> Product 1
Product {getProduct = 6}
Prelude Data.Semigroup> Min 4 <> Min 3 <> Min 5
Min {getMin = 3}
Prelude Data.Semigroup> Max 4 <> Max 3 <> Max 5
Max {getMax = 5}
If we listen to the mathematicians for a moment again, a monoid is a semigroup with a neutral element. A neutral element is a zero: an element that does nothing when combined with other elements. Here are some examples:
-- 0 is the neutral element of (+)
3 + 0 ==> 3
0 + 3 ==> 3
-- 1 is the neutral element of (*)
1 * 5 ==> 5
5 * 1 ==> 5
-- [] is the neutral element of (++)
[] ++ [1,2] ==> [1,2]
[1,2] ++ [] ==> [1,2]
The Haskell type class Monoid
(from the module Data.Monoid
) represents monoids.
Here are the Monoid
instances corresponding to our three examples of neutral elements:
instance Monoid (Sum a) where
mempty = Sum 0
instance Monoid (Product a) where
mempty = Product 1
instance Monoid [] where
mempty = []
So, what is a monoid for a programmer? A type forms a monoid if there’s a way of combining two elements of the type together so that parenthesis don’t matter, and there’s a also an “empty element” that can be combined with things without changing them. When thought of like this, monoids come up in programming quite often!
What use is this Monoid
class? Can’t we just write 1 + 2
instead of Sum 1 <> Sum 2
? We can, yes, but some library functions work on all Monoid
types.
The reason we want both a neutral element and an associative binary operator is that those are the exact two things we need in order to reduce or fold multiple elements into one value. This is the job of:
Sidenote: one way to define mconcat
is foldr (<>) mempty
. Do you remember foldr
?
Let’s look at why we need the properties of Monoid
to implement mconcat
. Firstly, we need mempty
to handle empty lists:
Secondly, we need associativity to be able to reduce a list [x,y,z]
to a unique value. If <>
were not associative, we would have two possible values for mconcat [x,y,z]
, namely (x<>y)<>z
and x<>(y<>z)
.
The most useful Monoid
function is foldMap
:
That type signature looks scary, but concrete cases are simpler:
foldMap Max [1::Int,4,2] ==> Max 4
foldMap Product [1::Int,4,2] ==> Product 8
-- We need the ::Int to avoid an "Ambiguous type variable" error when printing the result
Let’s break down that type. We know that an example of a Foldable t => t a
type is [a]
, so we can rewrite the type as
We can build this function out of functions we already know:
Oh, by the way, thanks to the (Monoid a, Monoid b) => Monoid (a,b)
instance we can even compute the maximum and product in one pass:
Note, you don’t need to use monoids in your own code, but you’ll eventually bump into them when using Haskell libraries so it’s good to know what they are.
Due to various historical and performance reasons, the definition of the Monoid
and Semigroup
classes aren’t just
Although you can mostly pretend they are. The actual definitions are:
class Semigroup a where
-- | An associative operation.
(<>) :: a -> a -> a
-- Combine elements of a nonempty list with <>
sconcat :: NonEmpty a -> a
sconcat as = ... -- default implementation omitted
-- Combine a value with itself using <>, n times
stimes :: Integral b => b -> a -> a
stimes n x = ... -- default implementation omitted
class Semigroup a => Monoid a where
mempty :: a
mappend :: a -> a -> a
mappend = (<>)
-- Combine elements of a list with <>
mconcat :: [a] -> a
mconcat = ... -- default implementation omitted
As you can see, all the operations except <>
and mempty
have default definitions, so a normal Monoid
instance declaration looks just like this:
A question novice Haskell programmers often ask (or at least should ask!) is: when should I use type classes? This section offers one answer.
Let’s look at a concrete example. A vehicle can be either a car or an airplane. We can model this with algebraic datatypes (as we’ve seen earlier in this chapter), but also with type classes. Here’s the datatype version:
data Vehicle = Car String | Airplane String
sound :: Vehicle -> String
sound (Car _) = "brum brum"
sound (Airplane _) = "zooooom"
Here’s the class version. Note how each case gets its own datatype, which are collected together in a type class.
data Car = Car String
data Airplane = Airplane String
class VehicleClass a where
sound :: a -> String
instance VehicleClass Car where
sound (Car _) = "brum brum"
instance VehicleClass Airplane where
sound (Airplane _) = "zooooom"
What is the difference between these solutions? The data-based solution is closed, meaning the set of cases is fixed and we can handle all of them in one place. The class-based solution is open, meaning we can add new cases, even in other modules.
An open abstraction is nice when we want extensibility. In the class-based solution, another module could define a bike:
A closed abstraction is good when we want to know that we’ve handled all cases, consider for example the function canCollide
which checks whether two vehicles can collide:
canCollide :: Vehicle -> Vehicle -> Bool
canCollide (Car _) (Car _) = True
canCollide (Airplane _) (Airplane _) = True
canCollide _ _ = False
This would be very hard to implement reliably in the class-based solution. Consider for example how collision checks between Bike
s and Car
s would get handled.
Sometimes it’s useful to implement a mini programming language for describing parts of your software. The fancy term for these is an Embedded Domain-Specific Language (EDSL). Haskell is well suited to modeling and interpreting languages. The expressions of the language are represented using (often recursive) algebraic data types. The language can be interpreted (that is, evaluated or run) by a recursive function.
Here’s an example of a language for describing price computations for products in a web shop.
data Discount = DiscountPercent Int -- A percentage discount
| DiscountConstant Int -- A constant discount
| MinimumPrice Int -- Set a minimum price
| ForCustomer String Discount -- Discounts can be conditional
| Many [Discount] -- Apply a number of discounts in row
The language is interpreted by the function applyDiscount
that takes a customer name, a price, a discount, and returns a price.
applyDiscount :: String -> Int -> Discount -> Int
applyDiscount _ price (DiscountPercent percent) = price - (price * percent) `div` 100
applyDiscount _ price (DiscountConstant discount) = price - discount
applyDiscount _ price (MinimumPrice minPrice) = max price minPrice
applyDiscount customer price (ForCustomer target discount)
| customer == target = applyDiscount customer price discount
| otherwise = price
applyDiscount customer price (Many discounts) = go price discounts
where go p [] = p
go p (d:ds) = go (applyDiscount customer p d) ds
Here we apply a discount chain of -50%, -$30 with a minimum price of $35:
applyDiscount "Bob" 120 (DiscountPercent 50)
==> 60
applyDiscount "Bob" 60 (DiscountConstant 30)
==> 30
applyDiscount "Bob" 30 (MinimumPrice 35)
==> 35
applyDiscount "Bob" 120 (Many [DiscountPercent 50, DiscountConstant 30, MinimumPrice 35])
==> 35
Here we have different discounts for Ssarah and Yvonne:
applyDiscount "Yvonne" 100 (Many [ForCustomer "Yvonne" (DiscountConstant 10), ForCustomer "Ssarah" (DiscountConstant 20)])
==> 90
applyDiscount "Ssarah" 100 (Many [ForCustomer "Yvonne" (DiscountConstant 10), ForCustomer "Ssarah" (DiscountConstant 20)])
==> 80
As you can see, even a simple Discount
type can generate complex behaviours because it is self-referential (recursive). Using Discount
we are able to represent the discount logic of our webshop as data instead of writing code.
There are multiple reasons for representing logic as data instead of code. Unlike code, data can easily be stored in a file or database, or even transmitted over the network. We can also use the same data for multiple purposes, for exaple we could visualize the discount chains in an administration user interface.
This course has been centered around pure functional programming. We’ve done lots of arithmetic, reversed lists, worked with binary trees, but so far we haven’t been able to affect the world outside our GHCi.
Things like reading input, writing to a file, or talking over the network are side effects. Side effects can’t be represented with pure functional code. A function like
can’t be pure, because if it were, readInputFromUser "What is your name?"
would always have to return the same result. However, representing side-effects and impurity in a pure language is possible. There are many ways of doing it, and the Haskell way is to use Monads.
Monads are reputedly difficult to understand. That is probably because they are so abstract. I think it’s best to focus on practical and concrete cases first. Here’s a taste of the IO
Monad, which you can use for all sorts of side effects in Haskell.
Let’s start!
another line
What we’ve seen here is the IO action getLine
. It has type IO String
. This means that GHCi can execute the action to produce a value of type String
. When we enter line <- getLine
into GHCi, we mean:
Execute the IO action
getLine
, and give the result the nameline
.
After we’ve received line
, it’s a pure String
value and we can work with it normally.
Some IO actions take parameters. For example putStrLn :: String -> IO ()
takes a String
and returns an IO
action that prints that string. The ()
type is a special type that only has one value, ()
. In this case IO ()
means that this IO always produces the same empty value ()
. You can run IO actions by just
Prelude> :t putStrLn
putStrLn :: String -> IO ()
Prelude> :t putStrLn "hello"
putStrLn "hello" :: IO ()
Prelude> val <- putStrLn "hello"
hello
Prelude> val
()
If you don’t need the return value of an IO action, you can run it in GHCi without the <-
:
You can build your own IO actions by combining other actions with do-notation. A do
block lists IO actions that are executed in order.
printTwoThings :: IO ()
printTwoThings = do
putStrLn "Hello!"
putStrLn "How are you?"
greet :: IO ()
greet = do
putStrLn "What's your name?"
name <- getLine
putStrLn ("Hello, " ++ name)
Prelude> printTwoThings
Hello!
How are you?
Prelude> greet
What's your name?
Seraphim
Hello, Seraphim
It feels as if we can just do side effects where ever we want with these IO actions. However, it’s important to remember the distinction between defining an IO action and executing it.
Let’s try to print while mapping over a list
This didn’t print anything, because even though we defined our action
, it wasn’t given to GHCi for execution. Because printAndIncrement
returns an Int
, it can’t return an action. Ok, let’s try another approach:
That didn’t print anything either! Let’s see why:
Prelude> :t map putStrLn ["string1","string2"]
map putStrLn ["string1","string2"] :: [IO ()]
Prelude> :t length (map putStrLn ["string1","string2"])
length (map putStrLn ["string1","string2"]) :: Int
We generated a list of IO actions and computed the value of the list. Defining IO actions is pure, it’s running them that causes side effects. Since the type of our expression was Int
, no IO
actions could land in GHCi and be executed.
If we instead return an IO action, it does get run:
Prelude> :t head (map putStrLn ["string1","string2"])
head (map putStrLn ["string1","string2"]) :: IO ()
Prelude> head (map putStrLn ["string1","string2"])
string1
Here too, the code that produces the action putStrLn "string1"
is pure, it’s only after the IO action is executed by GHCi that we see the printed string. And as you can see, the other IO action, putStrLn "string2"
, never got run.
If this feels complicated, don’t worry. We’ll get back to this on part 2 of the course.
We know that GHCi can run IO actions. What about actual Haskell programs? The way Haskell programs work is that the IO action called main
gets executed when the program is run. Recall our example program from Lecture 1.
module Gold where
-- The golden ratio
phi :: Double
phi = (sqrt 5 + 1) / 2
polynomial :: Double -> Double
polynomial x = x^2 - x - 1
f x = polynomial (polynomial x)
main = do
print (polynomial phi)
print (f phi)
Here we see some pure code and a main
IO action that prints two things (print
is just putStrLn
combined with show
).
We can place this code in a file called Gold.hs
, compile it into an executable, and run it:
$ ghc -main-is Gold Gold.hs
[1 of 1] Compiling Gold ( Gold.hs, Gold.o )
Linking Gold ...
$ ./Gold
0.0
-1.0
So far, we’ve learned about Haskell’s syntax and types, quite a bit of functional programming and about some language features like type classes.
We’ve also seen some type-oriented programming, and even gotten a taste of I/O in Haskell.
Now you know how to write a real computer program in Haskell, but there’s still much to learn.
We’re hoping to have Part 2 of the course ready soon. Part 2 will cover topics like Monads, IO and how Haskell works under the hood. We’ll also get to do some real world programming with networks and databases. Oh and testing in Haskell is also covered.
While you’re waiting for that, here are some Haskell resources you can study:
I also recommend working on some programming problems in Haskell, like the ones from:
You can also keep extending your final project and maybe generate some cool art in Haskell.
In any case – thank you so much for tagging along, and we hope you have a great rest of the year!
Open up the exercise file Set8.hs
and follow the instructions there. Have fun!
This course was made possible by Nitor who donated hours and hours of Joel’s working time for this project. Thank you! Check out our open positions if you’re interested in working somewhere that values continuous learning.
Thanks to the whole Haskell Mooc team, especially