Author: Thenaesh Elango

Haskell is a purely functional programming language with strong, static, inferred typing.

While Haskell has its roots in academia, its emphasis on purity and side-effect-free computation makes it a valuable asset in software engineering contexts. Programs written in Haskell tend to be easy to test, refactor and debug, with the compiler usually catching all bugs before the program can even be compiled and run. Consequently, Haskell codebases are extraordinarily stable.

Here's an example of a Haskell program that reads a string of numbers, prints
the sum of the numbers and repeats the process until the string `"quit"`

is
entered. This shall serve as our Hello World.

```
-- the entry point of the program
main :: IO ()
main = do
str <- getLine
if str == "quit"
then return ()
else do
let sumOfNumbers = sumAllNumbersInString str
putStrLn $ show sumOfNumbers
main
sumAllNumbersInString :: String -> Int
sumAllNumbersInString str = sumAll $ extractInts $ tokenize str
-- sums up a list of integers using a higher-order function (the left fold)
sumAll :: [Int] -> Int
sumAll = foldl (+) 0
-- convert each string of digits in a list to an actual integer
extractInts :: [String] -> [Int]
extractInts = fmap read
-- split string by spaces using a built-in function
tokenize :: String -> [String]
tokenize = words
```

Haskell is widely used in a whole range of industries, including banks, financial companies, technology companies and engineering companies use Haskell in a variety of systems. A comprehensive list may be found here.

**This tutorial, in general, assumes a system-wide installation of the Haskell
Platform.** This is primarily for simplicity. It is perfectly acceptable to write
small programs or code not intended for production in this manner.

When using Haskell in an actual project, however, it is **strongly-recommended**
to use Stack. Not doing so may cause dependency management to become a nightmare.

For new users, Haskell may be quickly and easily installed by downloading the Haskell Platform for their respective operating systems. The Haskell Platform contains many common and important Haskell libraries, in addition to the Glasgow Haskell Compiler (GHC).

At the time of writing, the Haskell Platform has binaries available for all common operating systems, and many uncommon ones as well.

For Haskell projects of significant size, it may be necessary to control the exact versions of the compiler and libraries used. For such use cases, the system-wide installation method above may prove unwieldy and inadequate. In cases like these, it may be preferable to have an entire Haskell environment just for that project, together with a curated set of libraries.

In such a scenario, Stack may come in handy. Stack is a package manager of sorts for Haskell, similar to NPM. Installation instructions may be found in the Stack Documentation, and is fairly standard.

A new Stack project may be created and set up with the following:

```
# create the project skeleton
stack new ${PROJECT_NAME}
# go into the project source directory
cd ${PROJECT_NAME}
# install GHC for the project
stack setup
# compile the project
stack build
# run the project executable
stack exec ${PROJECT_NAME}-exe
```

The command `stack new`

is used to create a new project, which contains a
skeleton already set up. This skeleton includes a `${PROJECT_NAME}.cabal`

file,
which contains nearly the entire configuration for the project (compiler/library
versions, modules to be exposed, build targets, etc), and is best thought of as
a sort of `package.json`

or `Gemfile`

for Stack.

The command `stack setup`

downloads and installs GHC. Stack installs GHC
versions into an isolated location in a user's home directory, and does not add
them to the system path. The version used for any particular project depends on
the setting in the project's `${PROJECT_NAME}.cabal`

file.

The commands `stack build`

and `stack exec`

are used to build and run the
project. The executable name for a project named Project is `Project-exe`

. This
name is configurable in `Project.cabal`

.

The rest of the Stack documentation may be found in the official guide.

The command `ghci`

may be used to invoke the GHC interpreter. This launches an
REPL where
Haskell code can be entered and evaluated interactively. This is a very useful
tool when first learning Haskell, and also when debugging code that fails to
compile.

The command `ghc`

may be used to compile Haskell code down to machine code. The
invocation of `ghc`

is very similar to that of `gcc`

.

When using Stack, simply prefix the commands with `stack`

.

Haskell is statically typed, meaning that every variable binds to a value of a specified type. Haskell is also strongly-typed, meaning that every value has a well-defined type.

We specify types explicitly by postfixing the variable names with the type.

```
a :: Int
a = 5
-- unbounded integer type, similar to Java BigInt
b :: Integer
b = product [1..1000] -- this is the factorial of 1000
pi :: Double
pi = 3.141592654
```

Haskell has very powerful type inference engine, so it is possible to omit the type definitions in most cases.

```
a = 5
```

Functions, which are just values, have types too. It is considered good practice in Haskell to specify types for toplevel functions, as a form of documentation, even though the compiler is likely able to infer types.

```
-- input: x of type Double
-- output: x * x of type Double
square :: Double -> Double
square x = x * x
-- computes the hypotenuse of a right triangle given the other two sides
hypotenuse :: Double -> Double -> Double
hypotenuse adj opp = sqrt (square adj + square opp)
```

If the above syntax is confusing and the comments insufficient, the reader may wish to consult the detailed introduction to Haskell syntax here.

The type definition for `square`

is rather obvious. But the type definition of
`hypotenuse`

is a little strange. One would expect `(Double, Double) -> Double)`

instead of `Double -> Double -> Double`

. The reason is that functions in Haskell
are curried, so a two-parameter
function can be called with a single argument, with a one-parameter function
(that takes in the remaining argument and produces the value) being returned.
`(Double, Double) -> Double`

is actually a function that takes in a *single*
2-tuple parameter, which is different from a function that takes in two parameters.

The `->`

binds to the right, so `Double -> Double -> Double`

may be written as
`Double -> (Double -> Double)`

(*a function that takes a double and
returns a function that takes a double and returns a double*).
Calling `hypotenuse 3 4`

is also the same as calling `(hypotenuse 3) 4`

,
as function application binds to the left.

We may go even further with currying, by fixing some parameters in the function:

```
-- hypotenuse of a right triangle whose adjacent side is restricted to 3
hypotenuseWithAdjacent3 :: Double -> Double
hypotenuseWithAdjacent3 = hypotenuse 3
```

This clearly illustrates how currying can be used to reuse and partially specialise code as needed. This idiom comes in handy very often in Haskell, as will be seen later.

It may be of interest to note that **all** functions in Haskell take in at most
one parameter. The illusion of multi-parameter functions is created by currying
and left-binding function calls.

It is possible to create custom data types, either from nothing or from existing types.

```
data TrafficSignal = Red | Amber | Green
-- define some values of type TrafficSignal, all type-inferred
redLight = Red
amberLight = Amber
greenLight = Green
```

The `TrafficSignal`

type is an example of creating data types from nothing.
We call `Red`

, `Amber`

and `Green`

the value constructors and `TrafficSignal`

itself the type constructor. In this case, a `TrafficSignal`

has 3 possible
values, `Red`

, `Amber`

or `Green`

.
**Both type and value constructors must start with a capital letter.**

We make use of types in functions by pattern matching on the value constructors. It is necessary to pattern match on all the value constructors; omitting any will cause the compiler to complain of non-exhaustive pattern matches.

```
makeTrafficDecision :: TrafficSignal -> String
-- leaving any of these out will cause the compiler to complain
makeTrafficDecision Red = "Stop"
makeTrafficDecision Amber = "Carefully Proceed"
makeTrafficDecision Green = "Go"
```

It is also possible to create data types that encapsulate/contain other data types. The value constructors in this case take parameters instead of being bare. Pattern matching is done by "expanding" the value constructor.

```
data HttpRequest = Get String | Post String
handleRequest :: HttpRequest -> String
-- the ++ denotes string concatenation in this context
handleRequest (Get string) = "Get request performed on " ++ string
-- we use _ to denote that we don't care about the actual value
handleRequest (Post _) = "Post request not supported"
```

There is also an additional way to declare data types. Suppose we had a C++ class like so:

```
// we are omitting trivial details like constructors
class Box {
double length;
double breadth;
double height;
double density;
public:
double getVolume() {
return this->length * this->breadth * this->height;
}
double getMass() {
return this->density * this->getVolume();
}
};
```

We could certainly represent a `Box`

as an algebraic data type as follows:

```
-- NOTE: a value constructor can have the same name as the type constructor
data Box = Box Double Double Double Double
```

But we are missing key information here. Which `Double`

stands for which
attribute? In situations like these, we can use Haskell's record syntax:

```
-- define box as a record type
data Box = Box { length :: Double, breadth :: Double, height :: Double, density :: Double }
getVolume :: Box -> Double
getVolume (Box { length = l, breadth = b, height = h }) = l * b * h
getMass :: Box -> Double
getMass box = getVolume box * density box
silverBox = Box { length = 5, breadth = 10, height = 15, density = 10.5 }
goldBox = Box 5 10 15 19.3 -- we can still use normal construction by parameter order
```

There are a few things to note here, other than the syntax itself. When pattern
matching on a record type, we may omit any parameters we do not need (we do not
even need to specify `_`

). We may also extract values from the record type by
treating the record parameter names as functions from the record type to the
parameter type. For instance, `density`

in the above example is actually a
`Box -> Double`

function. Doing `density silverBox`

will give the value `10.5`

.

Consider the division operator on `Double`

. We may be tempted to define it with
the type `Double -> Double -> Double`

, but the result may be undefined when
dividing by zero. Here's a first stab at a solution to remedy this:

```
-- represents a value that may or may not exist
data MaybeDouble = Undefined | Defined Double
divide :: Double -> Double -> MaybeDouble
divide _ 0 = Undefined
divide x y = Defined (x / y)
```

This ensures that division by zero returns a clearly-defined result instead of something weird.

Now suppose we want to send a HTTP request and retrieve the response data. This response data may not exist as the server may refuse to return the data. We can try to solve the problem in the following manner:

```
data MaybeResponse = NoResponse | GotResponse HttpResponse
makeRequest :: HttpRequest -> MaybeResponse
-- implementation details irrelevant
```

We have `MaybeDouble`

and `MaybeResponse`

, both of which have a common pattern:
they represent possible failure of computation. Naturally, we may wish to abstract
this out. But all the means of abstraction available to us thus far cannot be
used, as we wish to abstract on *types* rather than values.

For this purpose, Haskell supports *type parameters*, much like how C++ has
templates and Java has generics.

We define the following abstraction of failing computations:

```
data Maybe t = Nothing | Just t
divide :: Double -> Double -> Maybe Double
divide _ 0 = Nothing
divide x y = Just (x / y)
makeRequest :: HttpRequest -> Maybe Response
-- implementation details irrelevant
```

Note that we introduce an additional parameter `t`

on the left side of the definition.
This is known as a type parameter, and **must always be lowercase**. This parameter
can then be used in the value constructors as a placeholder for any type that
should be there.

The use of type parameters in this way is similar to the use of generics in Java.
We may think of `Maybe t`

as `Optional<T>`

, if that helps to understand the role
of `t`

.

Note that there can be more than one type parameter. An example is `Either`

,
which represents the result of a computation that returns values of different
types on success or failure:

```
data Either a b = Left a | Right b
divide :: Double -> Double -> Either String Double
divide _ 0 = Left "Attempt to divide by zero!"
divide x y = Right (x / y)
```

Type constructors can be curried just like regular functions or value constructors.
Therefore, `Either String Double`

is a concrete type, while `Either String`

is
a type constructor that takes in the remaining type.

`Maybe`

and `Either`

are both defined in the Haskell prelude library.

We can define a data type in terms of itself. Consider, for instance, a tree. A tree can be thought of as either an empty tree, or a node with a left subtree and right subtree attached. We encode it like so:

```
data Tree t = EmptyTree | Node t (Tree t) (Tree t)
```

Another classic inductive data type is the singly-linked list. The list is either an empty list or the first element together with the rest of the list. While not canonical, this is a very common representation of lists in the functional programming world:

```
data List t = EmptyList | Element t (List t)
```

This representation of lists is actually exactly how traditional lists are defined in Haskell, just with different names and notation as will be seen later.

We are now poised to enter the world of actual functional programming in Haskell.

A function may be defined in one of several ways. We illustrate the various syntaxes for defining a function below, with more details here if needed:

```
sumOfSquares :: Double -> Double -> Double
-- standard definition
sumOfSquares x y = (x * x) + (y * y)
-- lambda function
sumOfSquares = \x y -> (x * x) + (y * y)
fizzBuzz :: Int -> Either String Int
-- the horrible, disgusting, but still perfectly correct way
fizzBuzz x = case x `mod` 15 == 0 of
True -> Left "fizzbuzz"
False -> case x `mod` 3 == 0 of
True -> Left "fizz"
False -> case x `mod` 5 == 0 of
True -> Left "buzz"
False -> Right x
-- far more elegant way using guard patterns
fizzBuzz x
| x `mod` 15 = Left "fizzbuzz"
| x `mod` 3 = Left "fizz"
| x `mod` 5 = Left "buzz"
| otherwise = Right x
```

Recursion is one of the fundamental themes of functional programming. It is the ability of a function to call itself.

```
-- Time: O(n)
-- Space: O(n)
factorial :: Integer -> Integer
factorial 0 = 1
factorial n = n * factorial (n - 1)
-- Time: O(2^n)
-- Space: O(n), may vary due to lazy evaluation
fibonacci :: Integer -> Integer
fibonacci 0 = 0
fibonacci 1 = 1
fibonacci n = fibonacci (n - 2) + fibonacci (n - 1)
```

While Haskell has no primitive loop structures, looping can be simulated by recursion. While attempting this in languages in C may cause a stack overflow, Haskell avoids this via tail-call optimisation, which can be applied to recursive calls that meet certain requirements.

```
-- Time: O(n)
-- Space: O(1)
factorial :: Integer -> Integer
factorial = factorial' 1 where
factorial' p 0 = p
factorial' p n = factorial' (p * n) (n - 1)
-- Time: O(n)
-- Space: O(1)
fibonacci :: Integer -> Integer
fibonacci 0 = 0
fibonacci 1 = 1
fibonacci n = fibonacci' 0 1 n where
fibonacci a _ 0 = a
fibonacci' a b n = fibonacci' b (a + b) (n - 1)
```

We can safely omit the types in the inner function definitions due to type
inference. Also note how we freely use currying in the `factorial`

definition.

As described earlier, a list is an inductive data type, defined as either the empty list or an element concatenated with the rest of the list. The actual list data type is

```
data [] t = [] | (:) t ([] t)
```

where `:`

is an infix value constructor.

IMPORTANT: Infix FunctionsAny function (a value constructor is really just a function) that takes in two parameters whose name consists of nothing but symbols is infix by default.

An infix function like

`+`

may be used in prefix form by enclosing in parentheses. For instance,`1 + 1`

is the same as`(+) 1 1`

.

In the type definition, the prefix form must be used i.e.`(+) :: Int -> Int -> Int`

.

In the function definition, either is acceptable.We will use this concept freely from now on.

Here are several ways to define a list `xs :: [Int]`

containing 2,4,6,8 in that order:

```
-- the crazy way, using prefix notation directly from the list definition
xs = (:) 2 ((:) 4 ((:) 6 ((:) 8 [])))
-- using infix syntax for (:), still annoying to write
xs = 2:(4:(6:(8:[])))
-- taking advantage of binding rules for (:), noiseless and easier to understand at a glance
xs = 2:4:6:8:[]
-- using varying amounts of list syntactic sugar provided by the compiler
xs = 2:4:6:[8]
xs = 2:4:[6,8]
xs = 2:[4,6,8]
xs = [2,4,6,8]
```

The last representation is most commonly used, while the second last is often used when pattern matching on lists. The rest are almost never seen in practice. However, it is hoped that this pedantic exercise helps the reader understand the true nature of lists: an ordinary inductive data type with some compiler syntactic sugar tacked on.

List processing is a very important part of elementary functional programming. This is due to the fact that lists can store large amounts of data, and it is very easy to define powerful abstractions to slice and dice that data in ways typically unknown in imperative programming.

One common idiom is to loop over a list and aggregate their values.

It is possible to run over a list and sum their values recursively like so:

```
sumList :: [Int] -> Int
sumList [] = 0
sumList (x:xs) = x + sumList xs
```

Note the infix pattern match `(x:xs)`

as opposed to `((:) x xs)`

. What if we wish
to take the product of the elements instead of a sum? Then we would write:

```
prodList :: [Int] -> Int
prodList [] = 1
prodList (x:xs) = x * prodList xs
```

It is clear that some abstraction is in order here. The functions are almost identical except for the aggregating function used and the initial value (0 for sum, 1 for product). We can write a generalised aggregating function:

```
-- 1st parameter is the aggregating function (e.g. (+) or (*))
-- 2nd parameter is the initial value
-- 3rd parameter is the list to aggregate
aggregate :: (Int -> Int -> Int) -> Int -> [Int] -> Int
aggregate _ initial [] = initial
aggregate op initial (x:xs) = op x (aggregate op initial xs)
```

This is better, but perhaps we could generalise this even further beyond `Int`

.
We then arrive at the following, by simply changing the type signature:

```
-- 1st parameter is the aggregating function (e.g. (+) or (*))
-- 2nd parameter is the initial value
-- 3rd parameter is the list to aggregate
aggregate :: (a -> b -> b) -> b -> [a] -> b
aggregate _ initial [] = initial
aggregate op initial (x:xs) = op x (aggregate op initial xs)
```

This function is known as `foldl`

in the Haskell prelude library, and there is
also a variant called `foldr`

that does the aggregation from the right instead.

One may wish to take in a list, transform every element in the list, and output the resulting list. This is known as a map, and may be defined as:

```
map :: (a -> b) -> [a] -> [b]
map _ [] = []
map f (x:xs) = (f x):(map f xs)
```

The type definition itself contains a wealth of information. The `map`

function
takes in a "transformer", the list to be transformed, and return the transformed
list. An example of its usage would be:

```
-- xs is [1,4,9,16]
xs = map (\x -> x * x) [1,2,3,4]
```

One may also wish to remove certain elements, that fail some predicate, from a given list. This is known as a filter:

```
filter :: (t -> Bool) -> [t] -> [t]
filter _ [] = []
filter predicate (x:xs)
| predicate x == True = x:xs
| otherwise = xs
```

This example uses guard patterns. An example of using filter would be:

```
-- xs is [2,4]
xs = filter (\x -> x `mod` 2 == 0) [1,2,3,4]
```

It is left as an exercise for the reader to implement `map`

and `filter`

in terms of `foldl`

(or `aggregate`

as defined above, which is the same).

Recursion is a natural fit with inductive data types other than lists. One example would be finding an element in a binary tree:

```
find :: Tree Int -> Int -> Bool
find EmptyTree _ = False
find (Node x left right) target
| x == target = True
| x < target = find left target
| x > target = find right target
```

The above runs in O(log n) as long as the tree is balanced.

Typeclasses are essentially contracts/constraints imposed on types. They are similar to how Java interfaces are constraints imposed on Java classes. When used properly, they are an extremely powerful tool in helping to structure code.

```
-- "class" here has nothing to do with OOP
class Eq t where
(==) :: t -> t -> Bool
(!==) :: t -> t -> Bool
-- this ensures that we don't have to define (!==) separately
a != b = not (a == b)
```

We have just defined a typeclass called `Eq`

. As its name probably suggests,
this typeclass is used when we wish to define the meaning of equality on types.
We then *instantiate* the typeclass with the `TrafficSignal`

type, like so:

```
instance Eq TrafficSignal where
-- note that these are infix function DEFINITIONS
-- we can define infix operators directly in infix notation
Red == Red
Amber == Amber
Green == Green
```

We have thus defined `(==)`

completely for `TrafficSignal`

. Note that `(!=)`

now comes for free, since we have defined it in terms of `(==)`

in the typeclass
itself.

Here's another example:

```
instance Eq (List t) where
EmptyList == EmptyList
(Element x xs) == (Element y ys) = (x == y) && (xs == ys)
```

Here, we define the equality of a list in terms of its underlying elements.
This seems reasonable. However, running this program will give an error. This
is because we are attempting to compare the underlying elements (of type `t`

)
using `(==)`

, which is not guaranteed to be defined on `t`

.

The solution, in this case, is to enforce a typeclass constraint prerequisite
on `t`

by writing:

```
instance (Eq t) => Eq (List t) where
-- as before
```

Here is an example of `Eq`

being defined on `Tree`

s:

```
instance (Eq t) => Eq (Tree t) where
EmptyTree == EmptyTree
(Node x left right) == (Node x' left' right') = (x == x') && (left == left') && (right == right')
tree1 = Node 1 (Node 2 EmptyTree) (Node 3 EmptyTree)
tree2 = Node 1 (Node 2 EmptyTree) EmptyTree
tree3 = Node 1 (Node 2 EmptyTree) EmptyTree
-- some experiments
tree1 == tree2 -- False
tree2 == tree3 -- True
tree3 != tree1 -- True
```

We present another common typeclass called `Ord`

, which defines order for a type:

```
-- anything that instantiates Ord must also instantiate Eq
-- this makes the typeclass definitions simpler as (==) is already provided and can be used
class (Eq t) => Ord t where
-- the only one we actually need to implement when instantiating
(<) :: t -> t -> Bool
-- we predefine these and can then get them all for free
(>) :: t -> t -> Bool
a > b = not ((a < b) || (a == b))
(<=) :: t -> t -> Bool
a <= b == (a < b) || (a == b)
(>=) :: t -> t -> Bool
a >= b = not (a < b)
```

Consider the following function to check if the elements in the following list are all in ascending order:

```
isAscending :: [t] -> Bool
isAscending [] = True -- handle 0-element lists
isAscending (x:[]) = True -- handle 1-element lists
isAscending (x:y:xs) = (x < y) && isAscending (y:xs) -- recursive case
```

This function seems reasonable, except for one minor detail: we (and the compiler)
are not sure if `t`

can be compared using `(<)`

!. To remedy this, we need to
explicitly state that `t`

instantiates `Ord`

, thereby allowing the use of `(<)`

.
We do this by adding the constraint in the function type definition:

```
isAscending :: (Ord t) => [t] -> Bool
```

We can now try out the `isAscending`

function:

```
isAscending [1,2,4,3] -- False
isAscending [1,2,3,4,5] -- True
isAscending [] -- True
```

Up to this point, we have been instantiating typeclasses with concrete types,
such as `TrafficSignal`

and `Tree t`

. It is also possible to instantiate
typeclasses with **parameterized** type constructors like `Tree`

and `List`

.

Consider the following typeclass `Container`

that is instantiated by types that
have some notion of constituent elements and size. For instance, a `List`

has a
length and contains elements of some type. A `Tree`

has nodes and a size (number
of nodes). The length is independent of type of element contained within.

```
class Container s where
-- t is an arbitrary unconstrained type
size :: s t -> Int
```

We can instantiate `Container`

with `Tree`

and `List`

. These are parameterized
type constructors, not concrete types. We can even instantiate with `Maybe`

.

```
instance Container Tree where
size EmptyTree = 0
size (Node _ left right) = 1 + size left + size right
instance Container List where
size EmptyList = 0
size (Element _ restOfList) = 1 + size restOfList
instance Container Maybe where
size Nothing = 0
size (Just _) = 1
```

As an exercise, the reader may wish to redefine the size of a `Tree`

to mean
"height of tree" rather than "number of nodes". It is necessary to instantiate
`Container`

with `Tree`

differently to achieve this. The function
`max :: (Ord a) => a -> a -> a`

may come in handy (`Int`

is an instance of `Ord`

).

Consider the `map`

function previously defined. The type
of `map`

is `(a -> b) -> [a] -> [b]`

, which means that it operates only on lists.
We may imagine extending maps to `Tree`

s and `Maybe`

s in the following manner:

```
map :: (a -> b) -> Tree a -> Tree b
map :: (a -> b) -> Maybe a -> Maybe b
```

The two type definitions above look very similar and suggest a generalization:
types that can be mapped over. We call such types *functors*, and can represent
their behaviour with a typeclass.

```
class Functor f where
-- f is a type constructor that takes in one type parameter
fmap :: (a -> b) -> f a -> f b
instance Functor Tree where
fmap _ EmptyTree = EmptyTree
fmap f (Node x left right) = Node (f x) (fmap f left) (fmap f right)
instance Functor Maybe where
fmap _ Nothing = Nothing
fmap f (Just x) = Just (f x)
```

We can then map over values of any functor:

```
sq x = x * x
fmap sq (Just 5) -- returns Just 25
fmap sq Nothing -- returns Nothing
fmap sq (Node 1 (Node 2 EmptyTree) (Node 3 (Node 4 EmptyTree) EmptyTree))
-- returns (Node 1 (Node 4 EmptyTree) (Node 9 (Node 16 EmptyTree) EmptyTree))
```

More information about functors, including the functor laws, may be found here.

An *applicative functor* is a functor that allows for a more advanced type of
mapping. We shall jump straight into the (abridged) typeclass definition and the
example of `Maybe`

as an applicative functor:

```
class (Functor f) => Applicative f where
pure :: a -> f a
(<*>) :: f (a -> b) -> f a -> f b -- generalized map function
instance Applicative Maybe where
pure x = Just x
Nothing <*> _ = Nothing
_ <*> Nothing = Nothing
Just f <*> Just x = Just (f x)
```

Applicative functors have the concept of *lifting*, embodied in `pure`

, where
a value is taken and placed in the context of a functor. For instance, in the
context of `Maybe`

, `pure 5`

returns the value
`Just 5`

.

Applicative functors allow a more general form of mapping, where it is possible to use an N-parameter function to map over N functors. To understand the value of this, consider the following code:

```
euclideanDistance :: Double -> Double -> Double -> Double
euclideanDistance x y z = sqrt ((x * x) + (y * y) + (z * z))
(pure euclideanDistance) <*> Just 1 <*> Just 2 <*> Just 3
-- returns Just 3.7416573867739413
```

The above code can be written with just `fmap`

in the ordinary `Functor`

class,
but will involve incredible contortions.

More information about applicative functors can be found
here. There is a
lot of additional functionality available in the `Applicative`

typeclass. We
have barely scratched the surface.

No Haskell tutorial will be complete without an introduction to the fabled monad. Monads have been described with various analogies, as well as with notorious phrases from category theory like "a monad is a monoid in the category of endofunctors".

None of these are useful for the software engineer, so we dispense with them and opt for just showing the code:

```
class (Applicative m) => Monad m where
-- this function, called "bind", is at the heart of the monad
(>>=) :: m a -> (a -> m b) -> m b
-- we could actually just use pure, but return is here for historical reasons
return :: a -> m a
return = pure
```

A monad is essentially an applicative functor that allows for operations to be
chained together with a value carried in the background. To consider this, let
us consider the familiar case of `Maybe`

, which is a monad.

```
instance Monad Maybe where
Nothing >>= _ = Nothing
(Just x) >>= f = Just (f x)
-- maybeSomeValue is Just 50
maybeSomeValue = Just 5 >>= (\x -> Just (x * x) >>= (\x -> Just (x + x)))
```

Essentially, the bind function allows for values carried inside the monad (which
is ultimately just a functor) to be extracted and passed into another computation.
This explanation may seem obtuse, but consider the same code, with some extracted
whitespace added and the `return`

function used:

```
maybeSomeValue = Just 5 >>= (\x ->
Just (x * x) >>= (\x ->
return (x + x)))
```

If the reader squints hard enough, this looks like an imperative program! It looks like the following is being done:

```
maybeSomeValue = do
x <- Just 5
x <- Just (x * x)
return (x + x)
```

The result of the imperative-looking code is exactly the same as that of the
original computation, if traced through. Using monads to provide an imperative
interface in a functional program is such a common pattern that the `do`

notation
was conceived as syntactic sugar to make writing such a pattern easier. That means
that the imperative-looking code is actually valid Haskell!

In addition to `Maybe`

, there are several other monads. A major example is the
IO monad, which allows external state to be encapsulated in the monad an interfaced
with in a manner familiar to imperative programmers.

Monads are a big topic, and additional resources are available:

- Monads
- Monad Laws that every monad should obey
- IO Monad
- State Monad, allows state to be carried in a monadic context, allowing imperative-style computation
- ST Monad, allows mutable state to be carried in a monadic context, useful for implementing inherently destructive algorithms
- Arrays, allows constant-time access to elements like a C array, with mutable variants in the
`IO`

and`ST`

monads provided

- Learn You a Haskell is a good beginner text
for learning Haskell. It does not have much real-world examples, but does quite
a good job in explaining difficult theoretical concepts (e.g. functors,
applicative functors and monads) well. It is recommended to read the material
in 4 chunks:
- 1-6: basic functional programming
- 7: modules (very important for large codebases)
- 8-10: details of the type-system
- 11-14: monads

- Real World Haskell is a rather old text which is possibly outdated. Still, it shows plenty of examples of how Haskell may be used in actual real-life scenarios (databases, web programming, etc).
- Haskell Wikibook is a comprehensive Haskell reference. This resource really shines when it comes to aggregating and covering advanced Haskell topics that typically appear elsewhere in a very ad-hoc fashion.
- Typeclassopedia is a reference for the major typeclasses contained in the Haskell hierarchical libraries. Use it to determine which typeclasses are related to which (e.g. every monad is an applicative functor, which is in turn a functor).