Difference between revisions of "Monads as containers"
CaleGibbard (talk  contribs) 
m (Added nondeterminism category) 

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−  A 
+  A [[monad]] is a container type together with a few methods defined on it. 
Monads model different kinds of computations. 
Monads model different kinds of computations. 

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There are a few ways to choose the basic set of functions that one can perform 
There are a few ways to choose the basic set of functions that one can perform 

on these containers to be able to define a monad. Haskell generally uses a 
on these containers to be able to define a monad. Haskell generally uses a 

−  pair of functions called 
+  pair of functions called <code>pure</code> (<code>return = pure</code>) and monadic 
−  sometimes to begin with 
+  bind <code>(>>=)</code>, but it is more natural sometimes to begin with <code>fmap</code>, 
+  <code>pure</code> and <code>join</code>, as these are simpler 

to understand at first. We can later define bind in terms of these. 
to understand at first. We can later define bind in terms of these. 

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function to every object in the container. 
function to every object in the container. 

−  That is, if <code>f</code> is a functor, and we are given a function of type <code>(a > b)</code>, 
+  That is, if <code style="color:darkgreen">f</code> is a functor, and we are given a function of type <code>(<span style="color:firebrick">a</span> > <span style="color:mediumblue">b</span>)</code>, 
−  and a container of type <code>(f a)</code>, we can get a new container of type <code>(f b)</code>. 
+  and a container of type <code>(<span style="color:darkgreen">f</span> <span style="color:firebrick">a</span>)</code>, we can get a new container of type <code>(<span style="color:darkgreen">f</span> <span style="color:mediumblue">b</span>)</code>. 
This is expressed in the type of <code>fmap</code>: 
This is expressed in the type of <code>fmap</code>: 

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fmap :: (Functor f) => (a > b) > f a > f b 
fmap :: (Functor f) => (a > b) > f a > f b 

</haskell> 
</haskell> 

−  ''If you will give me a blueberry for each apple I give you'' <code>(a > b)</code>'', 

+  ''If you will give me a blueberry for each apple I give you'' <code>(<span style="color:firebrick">a</span> > <span style="color:mediumblue">b</span>)</code>'', and I have a box of apples ''<code>(<span style="color:darkgreen">f</span> <span style="color:firebrick">a</span>)</code>'', then I can get a box of blueberries ''<code>(<span style="color:darkgreen">f</span> <span style="color:mediumblue">b</span>)</code>''.'' 

−  and I have a box of apples'' <code>(f a)</code>'', then I can get a box of blueberries'' 

−  <code>(f b)</code>''.'' 

Every monad is a functor. 
Every monad is a functor. 

−  The second method, ''return'', is specific to monads. If <code>m</code> is a monad, then 
+  The second method, ''return'', is specific to monads. If <code style="color:indigo">m</code> is a monad, then 
−  return takes an element of type <code>a</code>, and gives a container of type <code>(m a)</code> 
+  return takes an element of type <code style="color:firebrick">a</code>, and gives a container of type <code>(<span style="color:indigo">m</span> <span style="color:firebrick">a</span>)</code> 
with that element in it. So, its type in Haskell is 
with that element in it. So, its type in Haskell is 

<haskell> 
<haskell> 

return :: (Monad m) => a > m a 
return :: (Monad m) => a > m a 

</haskell> 
</haskell> 

−  ''If I have an apple'' <code>(a)</code> ''then I can put it in a box'' <code>(m a)</code>''.'' 
+  ''If I have an apple'' <code>(<span style="color:firebrick">a</span>)</code> ''then I can put it in a box'' <code>(<span style="color:indigo">m</span> <span style="color:firebrick">a</span>)</code>''.'' 
The third method, ''join'', also specific to monads, takes a container of 
The third method, ''join'', also specific to monads, takes a container of 

−  containers <code>m (m a)</code>, and combines them into one <code>m a</code> in some sensible 
+  containers <code><span style="color:indigo">m</span> (<span style="color:indigo">m</span> <span style="color:firebrick">a</span>)</code>, and combines them into one <code><span style="color:indigo">m</span> <span style="color:firebrick">a</span></code> in some sensible 
fashion. Its Haskell type is 
fashion. Its Haskell type is 

<haskell> 
<haskell> 

join :: (Monad m) => m (m a) > m a 
join :: (Monad m) => m (m a) > m a 

</haskell> 
</haskell> 

−  ''If I have a box of boxes of apples'' <code>(m (m a))</code> ''then I can take the 
+  ''If I have a box of boxes of apples'' <code>(<span style="color:indigo">m</span> (<span style="color:indigo">m</span> <span style="color:firebrick">a</span>))</code> ''then I can take the apples from each, and put them in a new box'' <code>(<span style="color:indigo">m</span> <span style="color:firebrick">a</span>)</code>''.'' 
−  apples from each, and put them in a new box'' <code>(m a)</code>''.'' 

From these, we can construct an important operation called ''bind'' or 
From these, we can construct an important operation called ''bind'' or 

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join are in general simpler than bind. 
join are in general simpler than bind. 

−  What bind does is to take a container of type <code>(m a)</code> and a function of type 
+  What bind does is to take a container of type <code>(<span style="color:indigo">m</span> <span style="color:firebrick">a</span>)</code> and a function of type 
−  <code>(a > m b)</code>. It first maps the function over the container, (which would give 
+  <code>(<span style="color:firebrick">a</span> > <span style="color:indigo">m</span> <span style="color:mediumblue">b</span>)</code>. It first maps the function over the container, (which would give 
−  an <code>m (m b)</code>) and then applies join to the result to get a container of type 
+  an <code><span style="color:indigo">m</span> (<span style="color:indigo">m</span> <span style="color:mediumblue">b</span>)</code>) and then applies join to the result to get a container of type 
−  <code>(m b)</code>. Its type and definition in Haskell is then 
+  <code>(<span style="color:indigo">m</span> <span style="color:mediumblue">b</span>)</code>. Its type and definition in Haskell is then 
<haskell> 
<haskell> 

(>>=) :: (Monad m) => m a > (a > m b) > m b 
(>>=) :: (Monad m) => m a > (a > m b) > m b 

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 that were chosen to be primitive 
 that were chosen to be primitive 

</haskell> 
</haskell> 

−  ''If I have a box of apples'' <code>(m a)</code> ''and for each apple, you will give me 

+  ''If I have a box of apples'' <code>(<span style="color:indigo">m</span> <span style="color:firebrick">a</span>)</code> ''and for each apple, you will give me a box of blueberries'' <code>(<span style="color:firebrick">a</span> > <span style="color:indigo">m</span> <span style="color:mediumblue">b</span>)</code> ''then I can get a box with all the blueberries together'' <code>(<span style="color:indigo">m</span> <span style="color:mediumblue">b</span>)</code>''.'' 

−  a box of blueberries'' <code>(a > m b)</code> ''then I can get a box with all the 

−  blueberries together'' <code>(m b)</code>''.'' 

Note that for a given container type, there might be more than one way to 
Note that for a given container type, there might be more than one way to 

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permit one instance of the Monad class per actual type). [Technical side note: 
permit one instance of the Monad class per actual type). [Technical side note: 

The functions return and bind need to satisfy 
The functions return and bind need to satisfy 

−  [ 
+  [[Monad_laws  a few laws]] in order to 
make a monad, but if you define them in a sensible way given what they are 
make a monad, but if you define them in a sensible way given what they are 

supposed to do, the laws will work out. The laws are only a formal way to 
supposed to do, the laws will work out. The laws are only a formal way to 

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<haskell> 
<haskell> 

instance Monad [] where 
instance Monad [] where 

−  return :: a > [a] 
+  return :: a > [a] 
return x = [x]  make a list containing the one element given 
return x = [x]  make a list containing the one element given 

−  (>>=) :: [a] > (a > [b]) > [b] 
+  (>>=) :: [a] > (a > [b]) > [b] 
xs >>= f = concat (map f xs) 
xs >>= f = concat (map f xs) 

 collect up all the results of f (which are lists) 
 collect up all the results of f (which are lists) 

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For more good examples of monads, and lots of explanation see 
For more good examples of monads, and lots of explanation see 

−  http://www. 
+  [http://www.haskell.org/haskellwiki/All_About_Monads All About Monads] which has a catalogue of the commonest 
ones, more explanation as to why you might be interested in monads, and 
ones, more explanation as to why you might be interested in monads, and 

information about how they work. 
information about how they work. 

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Another good example of a monad which perhaps isn't obviously a container at 
Another good example of a monad which perhaps isn't obviously a container at 

−  first, is the Reader monad 
+  first, is the [[Reader monad]]. This monad basically consists 
of functions from a particular type: <code>((>) e)</code>, which might be written 
of functions from a particular type: <code>((>) e)</code>, which might be written 

<code>(e >)</code> if that were supported syntax. These can be viewed as containers 
<code>(e >)</code> if that were supported syntax. These can be viewed as containers 

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this way. 
this way. 

−  The Reader monad models computations which read from (depend on) a shared 
+  The [[Reader monad]] models computations which read from (depend on) a shared 
environment. To clear up the correspondence, the type of the environment is 
environment. To clear up the correspondence, the type of the environment is 

the index type on our indexed containers. 
the index type on our indexed containers. 

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in that container at <code>k</code>. 
in that container at <code>k</code>. 

−  The 
+  The [[Monads as computation]] perspective makes the purpose of such a monad perhaps 
more obvious: bind is taking a computation which may read from the environment 
more obvious: bind is taking a computation which may read from the environment 

before producing a value of type <code>a</code>, and a function from values of type <code>a</code> 
before producing a value of type <code>a</code>, and a function from values of type <code>a</code> 

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−  How about the 
+  How about the [[State monad]]? Although I'll admit that 
with State and IO in particular, it is generally more natural to take the 
with State and IO in particular, it is generally more natural to take the 

−  view of 
+  view of [[Monads as computation]], it is good to see that the container analogy 
doesn't break down. The state monad is a particular refinement of the reader 
doesn't break down. The state monad is a particular refinement of the reader 

monad discussed above. 
monad discussed above. 

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I hope that the above was a reasonably clear introduction to what monads are 
I hope that the above was a reasonably clear introduction to what monads are 

−  about. Feel free to make criticisms and ask questions. 
+  about. Feel free to [[Talk:Monads as containersmake criticisms and ask questions]]. 
−  +   [[User:CaleGibbardCaleGibbard]] 

+  
+  ==See also== 

+  
+  * [[Monad]] 

+  * [[Monads as computation]] 

+  
+  
+  [[Category:Tutorials]] 

+  [[Category:Monad]] 

+  [[Category:Nondeterminism]] 
Latest revision as of 05:36, 9 March 2021
There now exists a translation of this article into Russian!
A monad is a container type together with a few methods defined on it. Monads model different kinds of computations.
Like Haskell lists, all the elements which a monadic container holds at any one time must be the same type (it is homogeneous).
There are a few ways to choose the basic set of functions that one can perform
on these containers to be able to define a monad. Haskell generally uses a
pair of functions called pure
(return = pure
) and monadic
bind (>>=)
, but it is more natural sometimes to begin with fmap
,
pure
and join
, as these are simpler
to understand at first. We can later define bind in terms of these.
The first of these three, generally called map, (but called fmap
in
Haskell 98) actually comes from the definition of a functor. We can think
of a functor as a type of container where we are permitted to apply a single
function to every object in the container.
That is, if f
is a functor, and we are given a function of type (a > b)
,
and a container of type (f a)
, we can get a new container of type (f b)
.
This is expressed in the type of fmap
:
fmap :: (Functor f) => (a > b) > f a > f b
If you will give me a blueberry for each apple I give you (a > b)
, and I have a box of apples (f a)
, then I can get a box of blueberries (f b)
.
Every monad is a functor.
The second method, return, is specific to monads. If m
is a monad, then
return takes an element of type a
, and gives a container of type (m a)
with that element in it. So, its type in Haskell is
return :: (Monad m) => a > m a
If I have an apple (a)
then I can put it in a box (m a)
.
The third method, join, also specific to monads, takes a container of
containers m (m a)
, and combines them into one m a
in some sensible
fashion. Its Haskell type is
join :: (Monad m) => m (m a) > m a
If I have a box of boxes of apples (m (m a))
then I can take the apples from each, and put them in a new box (m a)
.
From these, we can construct an important operation called bind or
extend, which is commonly given the symbol (>>=)
. When you define your
own monad in Haskell, it will expect you to define just return and bind. It
turns out that mapping and joining come for free from these two. Although only
return and bind are needed to define a monad, it is usually simpler to think
about map, return, and join first, and then get bind from these, as map and
join are in general simpler than bind.
What bind does is to take a container of type (m a)
and a function of type
(a > m b)
. It first maps the function over the container, (which would give
an m (m b)
) and then applies join to the result to get a container of type
(m b)
. Its type and definition in Haskell is then
(>>=) :: (Monad m) => m a > (a > m b) > m b
xs >>= f = join (fmap f xs)
 how we would get bind (>>=) in Haskell if it were join and fmap
 that were chosen to be primitive
If I have a box of apples (m a)
and for each apple, you will give me a box of blueberries (a > m b)
then I can get a box with all the blueberries together (m b)
.
Note that for a given container type, there might be more than one way to define these basic operations (though for obvious reasons, Haskell will only permit one instance of the Monad class per actual type). [Technical side note: The functions return and bind need to satisfy a few laws in order to make a monad, but if you define them in a sensible way given what they are supposed to do, the laws will work out. The laws are only a formal way to give the informal description of the meanings of return and bind I have here.]
It would be good to have a concrete example of a monad at this point, as these functions are not very useful if we cannot find any examples of a type to apply them to.
Lists are most likely the simplest, most illustrative example. Here, fmap
is just the usual map
, return
is just (\x > [x])
and join
is concat
.
instance Monad [] where
return :: a > [a]
return x = [x]  make a list containing the one element given
(>>=) :: [a] > (a > [b]) > [b]
xs >>= f = concat (map f xs)
 collect up all the results of f (which are lists)
 and combine them into a new list
The list monad, in some sense, models computations which could return any number of values. Bind pumps values in, and catches all the values output. Such computations are known in computer science as nondeterministic. That is, a list [x,y,z] represents a value which is all of the values x, y, and z at once.
A couple examples of using this definition of bind:
[10,20,30] >>= \x > [x, x+1]
 a function which takes a number and gives both it and its
 successor at once
= [10,11,20,21,30,31]
[10,20,30] >>= \x > [x, x+1] >>= \y > if y > 20 then [] else [y,y]
= [10,10,11,11,20,20]
And a simple fractal, exploiting the fact that lists are ordered:
f x  x == '#' = "# #"
 otherwise = " "
"#" >>= f >>= f >>= f >>= f
= "# # # # # # # # # # # # # # # #"
You might notice a similarity here between bind and function application or composition, and this is no coincidence. The reason that bind is so important is that it serves to chain computations on monadic containers together.
You might be interested in how, given just bind and return, we can get back to map and join.
Mapping is equivalent to binding to a function which only returns containers
with a single value in them  the value that we get from the function of
type (a > b)
which we are handed.
The function that does this for any monad in Haskell is called liftM
 it
can be written in terms of return and bind as follows:
liftM :: (Monad m) => (a > b) > m a > m b
liftM f xs = xs >>= (return . f)
 take a container full of a's, to each, apply f,
 put the resulting value of type b in a new container,
 and then join all the containers together.
Joining is equivalent to binding a container with the identity map. This is
indeed still called join
in Haskell:
join :: (Monad m) => m (m a) > m a
join xss = xss >>= id
It is common when constructing monadic computations that one ends up with a large chain of binds and lambdas. For this reason, some syntactic sugar called "do notation" was created to simplify this process, and at the same time, make the computations look somewhat like imperative programs.
Note that in what follows, the syntax emphasizes the fact that the list monad
models nondeterminism: the code y < xs
can be thought of as y
taking on
all the values in the list xs
at once.
The above (perhaps somewhat silly) list computations could be written:
do x < [10,20,30]
[x, x+1]
and,
do x < [10,20,30]
y < [x, x+1]
if y > 20 then [] else [y,y]
The code for liftM could be written:
liftM f xs = do a < xs
return (f a)
If you understood the above, then you have a good start on understanding monads in Haskell. Check out Maybe (containers with at most one thing in them, modelling computations that might not return a value) and State (modelling computations that carry around a state parameter using an odd sort of container described below), and you'll start getting the picture as to how things work.
A good exercise is to figure out a definition of bind and return (or fmap, join and return) which make the following tree type a monad. Just keep in mind what they are supposed to do.
data Tree a = Leaf a  Branch (Tree a) (Tree a)
For more good examples of monads, and lots of explanation see All About Monads which has a catalogue of the commonest ones, more explanation as to why you might be interested in monads, and information about how they work.
The question that many people will be asking at this point is "What does this all have to do with IO?".
Well, in Haskell, IO is a monad. How does this mesh with the notion of a container?
Consider getChar :: IO Char
 that is, an IO container with a character
value in it. The exact character that the container holds is determined when
the program runs by which keys the user presses. Trying to get the character
out of the box will cause a side effect: the program stops and waits for the
user to press a key. This is generally true of IO values  when you get a
value out with bind, side effects can occur. Many IO containers don't
actually contain interesting values. For example,
putStrLn "Hello, World!" :: IO ()
That is, the value returned by putStrLn "Hello, World!"
is an IO container
filled with a value of type ()
, a not so interesting type. However, when
you pull this value out during a bind operation, the string Hello, World!
is printed on the screen. So another way to think about values of type IO t
is as computations which when executed, may have side effects before
returning a value of type t.
One thing that you might notice as well, is that there is no ordinary Haskell
function you can call (at least not in standard Haskell) to actually get a
value out of an IO container/computation, other than bind, which puts it right
back in. Such a function of type IO a > a
would be very unsafe in the pure
Haskell world, because the value produced could be different each time it was
called, and the IO computation could have side effects, and there would be no
way to control when it was executed (Haskell is lazy after all). So how do IO
actions ever get run? The IO action called main
runs when the program is
executed. It can make use of other IO actions in the process, and everything
starts from there.
When doing IO, a handy special form of bind when you just want the side effects and don't care about the values returned by the container on the left is this:
(>>) :: Monad m => m a > m b > m b
m >> k = m >>= \_ > k
An example of doing some IO in do notation:
main = do putStrLn "Hello, what is your name?"
name < getLine
putStrLn ("Hello " ++ name ++ "!")
or in terms of bind, making use of the special form:
main = putStrLn "Hello, what is your name?" >>
getLine >>= \name >
putStrLn ("Hello " ++ name ++ "!")
or, very primitive, without the special form for bind:
main = putStrLn "Hello, what is your name?" >>= \x >
getLine >>= \name >
putStrLn ("Hello " ++ name ++ "!")
Another good example of a monad which perhaps isn't obviously a container at
first, is the Reader monad. This monad basically consists
of functions from a particular type: ((>) e)
, which might be written
(e >)
if that were supported syntax. These can be viewed as containers
indexed by values of type e
, having one spot for each and every value of
type e
. The primitive operations on them follow naturally from thinking
this way.
The Reader monad models computations which read from (depend on) a shared environment. To clear up the correspondence, the type of the environment is the index type on our indexed containers.
type Reader e = (>) e  our monad
Return simply produces the container having a given value at every spot.
return :: a > (Reader e a)
return x = (\k > x)
Mapping a function over such a container turns out to be nothing more than what composition does.
fmap :: (a > b) > Reader e a > Reader e b
= (a > b) > (e > a) > (e > b)
 by definition, (Reader a b) = (a > b)
fmap f xs = f . xs
How about join? Well, let's have a look at the types.
join :: (Reader e) (Reader e a) > (Reader e a)
= (e > e > a) > (e > a)  by definition of (Reader a)
There's only one thing the function of type (e > a)
constructed could
really be doing:
join xss = (\k > xss k k)
From the container perspective, we are taking an indexed container of indexed
containers and producing a new one which at index k
, has the value at index
k
in the container at index k
.
So we can derive what we want bind to do based on this:
(>>=) :: (Reader e a) > (a > Reader e b) > (Reader e b)
= (e > a) > (a > (e > b)) > (e > b)  by definition
xs >>= f = join (fmap f xs)
= join (f . xs)
= (\k > (f . xs) k k)
= (\k > f (xs k) k)
Which is exactly what you'll find in other definitions of the Reader monad.
What is it doing? Well, it's taking a container xs
, and a function f
from
the values in it to new containers, and producing a new container which at
index k
, holds the result of looking up the value at k
in xs
, and then
applying f
to it to get a new container, and finally looking up the value
in that container at k
.
The Monads as computation perspective makes the purpose of such a monad perhaps
more obvious: bind is taking a computation which may read from the environment
before producing a value of type a
, and a function from values of type a
to computations which may read from the environment before returning a value
of type b
, and composing these together, to get a computation which might
read from the (shared) environment, before returning a value of type b
.
How about the State monad? Although I'll admit that with State and IO in particular, it is generally more natural to take the view of Monads as computation, it is good to see that the container analogy doesn't break down. The state monad is a particular refinement of the reader monad discussed above.
I won't go into huge detail about the state monad here, so if you don't already know what it's for, what follows may seem a bit unnatural. It's perhaps better taken as a secondary way to look at the structure.
For reference to the analogy, a value of type (State s a)
is like a
container indexed by values of type s
, and at each index, it has a value of
type a
and another, new value of type s
. The function runState
does
this "lookup".
newtype State s a = State { runState :: (s > (a,s)) }
What does return do? It gives a State
container with the given element at
every index, and with the "address" (a.k.a. state parameter) unchanged.
return :: a > State s a
return x = State (\s > (x,s))
Mapping does the natural thing, applying a function to each of the values of
type a
, throughout the structure.
fmap :: (a > b) > (State s a) > (State s b)
fmap f (State m) = State (onVal f . m)
where onVal f (x, s) = (f x, s)
Joining needs a bit more thought. We want to take a value of type
(State s (State s a))
and turn it into a (State s a)
in a natural way.
This is essentially removal of indirection. We take the new address and new
box that we get from looking up a given address in the box, and we do another
lookup  note that this is almost the same as what we did with the reader
monad, only we use the new address that we get at the location, rather than
the same address as for the first lookup.
So we get:
join :: (State s (State s a)) > (State s a)
join xss = State (\s > uncurry runState (runState xss s))
I hope that the above was a reasonably clear introduction to what monads are about. Feel free to make criticisms and ask questions.