"Programming with Effects" by Graham Hutton -- http://www.cs.nott.ac.uk/~gmh/monads

monads.txt

PROGRAMMING WITH EFFECTS
Graham Hutton, January 2015

Shall we be pure or impure?

The functional programming community divides into two camps:

 o "Pure" languages, such as Haskell, are based directly
   upon the mathematical notion of a function as a
   mapping from arguments to results.

 o "Impure" languages, such as ML, are based upon the 
   extension of this notion with a range of possible
   effects, such as exceptions and assignments.

Pure languages are easier to reason about and may benefit
from lazy evaluation, while impure languages may be more
efficient and can lead to shorter programs.

One of the primary developments in the programming language
community in recent years (starting in the early 1990s) has
been an approach to integrating the pure and impure camps,
based upon the notion of a "monad".  This note introduces
the use of monads for programming with effects in Haskell.


Abstracting programming patterns
--------------------------------

Monads are an example of the idea of abstracting out a common
programming pattern as a definition.  Before considering monads,
let us review this idea, by means of two simple functions:

   inc	      :: [Int] -> [Int]
   inc []     =  []
   inc (n:ns) =  n+1 : inc ns

   sqr	      :: [Int] -> [Int]
   sqr []     =  []
   sqr (n:ns) =  n^2 : sqr ns

Both functions are defined using the same programming pattern,
namely mapping the empty list to itself, and a non-empty list
to some function applied to the head of the list and the result
of recursively processing the tail of the list in the same manner.
Abstracting this pattern gives the library function called map

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

using which our two examples can now be defined more compactly:

   inc = map (+1)

   sqr = map (^2)


A simple evaluator
------------------

Consider the following simple language of expressions that are
built up from integer values using a division operator:

   data Expr = Val Int | Div Expr Expr

Such expressions can be evaluated as follows:

   eval           :: Expr -> Int
   eval (Val n)   =  n
   eval (Div x y) =  eval x `div` eval y

However, this function doesn't take account of the possibility of 
division by zero, and will produce an error in this case.  In order
to deal with this explicitly, we can use the Maybe type

   data Maybe a = Nothing | Just a

to define a "safe" version of division

   safediv     :: Int -> Int -> Maybe Int
   safediv n m =  if m == 0 then Nothing else Just (n `div` m)

and then modify our evaluator as follows:

   eval           :: Expr -> Maybe Int
   eval (Val n)   =  Just n
   eval (Div x y) =  case eval x of
		        Nothing -> Nothing
			Just n  -> case eval y of
				      Nothing -> Nothing
				      Just m  -> safediv n m
  
As in the previous section, we can observe a common pattern, namely
performing a case analysis on a value of a Maybe type, mapping Nothing
to itself, and Just x to some result depending upon x.  (Aside: we
could go further and also take account of the fact that the case
analysis is performed on the result of an eval, but this would
lead to the more advanced notion of a monadic fold.)

How should this pattern be abstracted out?  One approach would be
to observe that a key notion in the evaluation of division is the
sequencing of two values of a Maybe type, namely the results of
evaluating the two arguments of the division.  Based upon this
observation, we could define a sequencing function

   seqn                    :: Maybe a -> Maybe b -> Maybe (a,b)
   seqn Nothing   _        =  Nothing
   seqn _         Nothing  =  Nothing
   seqn (Just x)  (Just y) =  Just (x,y)

using which our evaluator can now be defined more compactly:

   eval (Val n)   = Just n
   eval (Div x y) = apply f (eval x `seqn` eval y)
                    where f (n,m) = safediv n m

The auxiliary function apply is an analogue of application for Maybe,
and is used to process the results of the two evaluations:

   apply            :: (a -> Maybe b) -> Maybe a -> Maybe b
   apply f Nothing  =  Nothing
   apply f (Just x) =  f x

In practice, however, using seqn can lead to programs that manipulate
nested tuples, which can be messy.  For example, the evaluation of
an operator Op with three arguments may be defined by:

   eval (Op x y z) = apply f (eval x `seqn` (eval y `seqn` eval z))
                     where f (a,(b,c)) = ...


Combining sequencing and processing
-----------------------------------

The problem of nested tuples can be avoided by returning of our 
original observation of a common pattern: "performing a case analysis
on a value of a Maybe type, mapping Nothing to itself, and Just x to
some result depending upon x".   Abstract this pattern directly gives
a new sequencing operator that we write as >>=, and read as "then":

   (>>=)   :: Maybe a -> (a -> Maybe b) -> Maybe b
   m >>= f =  case m of
                 Nothing -> Nothing
                 Just x  -> f x

Replacing the use of case analysis by pattern matching gives a
more compact definition for this operator:

   (>>=)         :: Maybe a -> (a -> Maybe b) -> Maybe b
   Nothing  >>= _ = Nothing
   (Just x) >>= f = f x

That is, if the first argument is Nothing then the second argument
is ignored and Nothing is returned as the result.  Otherwise, if
the first argument is of the form Just x, then the second argument
is applied to x to give a result of type Maybe b.

The >>= operator avoids the problem of nested tuples of results
because the result of the first argument is made directly available
for processing by the second, rather than being paired up with the
second result to be processed later on.  In this manner, >>= integrates
the sequencing of values of type Maybe with the processing of their
result values.  In the literature, >>= is often called "bind", because
the second argument binds the result of the first.  Note also that
>>= is just apply with the order of its arguments swapped.

Using >>=, our evaluator can now be rewritten as:

   eval (Val n)   = Just n
   eval (Div x y) = eval x >>= (\n ->
                    eval y >>= (\m ->
                    safediv n m))

The case for division can be read as follows: evaluate x and call
its result value n, then evaluate y and call its result value m,
and finally combine the two results by applying safediv.  In
fact, the scoping rules for lambda expressions mean that the
parentheses in the case for division can freely be omitted.

Generalising from this example, a typical expression built using
the >>= operator has the following structure:

   m1 >>= \x1 ->
   m2 >>= \x2 ->
   ...
   mn >>= \xn ->
   f x1 x2 ... xn

That is, evaluate each of the expression m1,m2,...,mn in turn, and
combine their result values x1,x2,...,xn by applying the function f.
The definition of >>= ensures that such an expression only succeeds
(returns a value built using Just) if each mi in the sequence succeeds.
In other words, the programmer does not have to worry about dealing
with the possible failure (returning Nothing) of any of the component
expressions, as this is handled automatically by the >>= operator. 

Haskell provides a special notation for expressions of the above
structure, allowing them to be written in a more appealing form:

   do x1 <- m1
      x2 <- m2
      ...
      xn <- mn
      f x1 x2 ... xn

Hence, for example, our evaluator can be redefined as:

   eval (Val n)   = Just n
   eval (Div x y) = do n <- eval x
                       m <- eval y
                       safediv n m

Exercises:

o Show that the version of eval defined using >>= is equivalent to
  our original version, by expanding the definition of >>=.

o Redefine seqn x y and eval (Op x y z) using the do notation.


Monads in Haskell
-----------------

The do notation for sequencing is not specific to the Maybe type,
but can be used with any type that forms a "monad".  The general
concept comes from a branch of mathematics called category theory.
In Haskell, however, a monad is simply a parameterised type m,
together with two functions of the following types:

   return :: a -> m a

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

(Aside: the two functions are also required to satisfy some simple
properties, but we will return to these later.)  For example, if
we take m as the parameterised type Maybe, return as the function
Just :: a -> Maybe a, and >>= as defined in the previous section,
then we obtain our first example, called the maybe monad.

In fact, we can capture the notion of a monad as a new class
declaration.  In Haskell, a class is a collection of types that
support certain overloaded functions.  For example, the class
Eq of equality types can be declared as follows:

   class Eq a where
      (==) :: a -> a -> Bool
      (/=) :: a -> a -> Bool

      x /= y = not (x == y)

The declaration states that for a type "a" to be an instance of
the class Eq, it must support equality and inequality operators
of the specified types.  In fact, because a default definition
has already been included for /=, declaring an instance of this
class only requires a definition for ==.  For example, the type
Bool can be made into an equality type as follows:

   instance Eq Bool where
      False == False = True
      True  == True  = True
      _     == _     = False

The notion of a monad can now be captured as follows:

   class Monad m where
      return :: a -> m a
      (>>=)  :: m a -> (a -> m b) -> m b

That is, a monad is a parameterised type "m" that supports return
and >>= functions of the specified types.  The fact that m must be
a parameterised type, rather than just a type, is inferred from its
use in the types for the two functions.   Using this declaration,
it is now straightforward to make Maybe into a monadic type:

   instance Monad Maybe where
      -- return      :: a -> Maybe a
      return x       =  Just x

      -- (>>=)       :: Maybe a -> (a -> Maybe b) -> Maybe b
      Nothing  >>= _ =  Nothing
      (Just x) >>= f =  f x

(Aside: types are not permitted in instance declarations, but we
include them as comments for reference.)  It is because of this
declaration that the do notation can be used to sequence Maybe
values.  More generally, Haskell supports the use of this notation
with any monadic type.  In the next few sections we give some 
further examples of types that are monadic, and the benefits
that result from recognising and exploiting this fact.


The list monad
--------------

The maybe monad provides a simple model of computations that can
fail, in the sense that a value of type Maybe a is either Nothing,
which we can think of as representing failure, or has the form
Just x for some x of type a, which we can think of as success.

The list monad generalises this notion, by permitting multiple
results in the case of success.  More precisely, a value of
[a] is either the empty list [], which we can think of as
failure, or has the form of a non-empty list [x1,x2,...,xn]
for some xi of type a, which we can think of as success.
Making lists into a monadic type is straightforward:

   instance Monad [] where
      -- return :: a -> [a]
      return x  =  [x]

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

(Aside: in this context, [] denotes the list type [a] without
its parameter.)  That is, return simply converts a value into a
successful result containing that value, while >>= provides a
means of sequencing computations that may produce multiple
results: xs >>= f applies the function f to each of the results
in the list xs to give a nested list of results, which is then
concatenated to give a single list of results.

As a simple example of the use of the list monad, a function
that returns all possible ways of pairing elements from two 
lists can be defined using the do notation as follows:

     pairs	 :: [a] -> [b] -> [(a,b)]
     pairs xs ys =  do x <- xs
                       y <- ys
                       return (x,y)

That is, consider each possible value x from the list xs, and 
each value y from the list ys, and return the pair (x,y).  It
is interesting to note the similarity to how this function
would be defined using the list comprehension notation:

     pairs xs ys = [(x,y) | x <- xs, y <- ys]

In fact, there is a formal connection between the do notation
and the comprehension notation.  Both are simply different 
shorthands for repeated use of the >>= operator for lists.
Indeed, the language Gofer that was one of the precursors
to Haskell permitted the comprehension notation to be used 
with any monad.  For simplicity however, Haskell only allows
the comprehension notation to be used with lists.


The state monad
---------------

Now let us consider the problem of writing functions that
manipulate some kind of state, represented by a type whose
internal details are not important for the moment:

   type State = ... 

The most basic form of function on this type is a "state
transformer" (abbreviated by ST), which takes the current
state as its argument, and produces a modified state as
its result, in which the modified state reflects any
side effects performed by the function:

   type ST = State -> State

In general, however, we may wish to return a result value in
addition to updating the state.  For example, a function for
incrementing a counter may wish to return the current value
of the counter.  For this reason, we generalise our type of
state transformers to also return a result value, with the
type of such values being a parameter of the ST type:

   type ST a = State -> (a, State)

Such functions can be depicted as follows, where s is the input
state, s' is the output state, and v is the result value:
                              
                       ^
          +-------+    |  v
     s    |       | ---'
   -----> |       |
          |       | ----->
          +-------+   s'

A state transformer may also wish to take argument values.
However, there is no need to further generalise the ST type
to take account of this, because this behaviour can already
be achieved by exploiting currying.  For example, a state
transformer that takes a character and returns an integer
would have type Char -> ST Int, which abbreviates the curried
function type Char -> State -> (Int, State), depicted by:

     |                 ^
   c |    +-------+    |  n
     `--> |       | ---'
          |       |
   -----> |       | ----->
     s    +-------+   s'

Returning to the subject of monads, it is now straightforward
to make ST into an instance of a monadic type:

   instance Monad ST where
      -- return :: a -> ST a
      return x  =  \s -> (x,s)

      -- (>>=)  :: ST a -> (a -> ST b) -> ST b
      st >>= f  =  \s -> let (x,s') = st s in f x s'

That is, return converts a value into a state transformer that
simply returns that value without modifying the state:

     |                 ^
   x |    +-------+    | x 
     `----|-------|----'
          |       |
   -------|-------|------>
      s   +-------+   s

In turn, >>= provides a means of sequencing state transformers:
st >>= f applies the state transformer st to an initial state
s, then applies the function f to the resulting value x to
give a second state transformer (f x), which is then applied
to the modified state s' to give the final result:

                                        ^ 
          +-------+   x    +-------+    |
     s    |       | -----> |       | ---'
   -----> |  st   |        |   f   |
          |       | -----> |       | ----->
          +-------+   s'   +-------+

Note that return could also be defined by return x s = (x,s).  
However, we prefer the above definition in which the second 
argument s is shunted to the body of the definition using a
lambda abstraction, because it makes explicit that return is
a function that takes a single argument and returns a state
transformer, as expressed by the type a -> ST a:  A similar
comment applies to the above definition for >>=.

We conclude this section with a technical aside.  In Haskell,
types defined using the "type" mechanism cannot be made into
instances of classes.  Hence, in order to make ST into an
instance of the class of monadic types, in reality it needs
to be redefined using the "data" mechanism, which requires
introducing a dummy constructor (called S for brevity):

   data ST a = S (State -> (a, State))

It is convenient to define our own application function for
this type, which simply removes the dummy constructor:

   apply        :: ST a -> State -> (a,State)
   apply (S f) x = f x

In turn, ST is now defined as a monadic type as follows:

   instance Monad ST where
      -- return :: a -> ST a
      return x   = S (\s -> (x,s))

      -- (>>=)  :: ST a -> (a -> ST b) -> ST b
      st >>= f   = S (\s -> let (x,s') = apply st s in apply (f x) s')

Aside: the runtime overhead of manipulating the dummy constructor
S can be eliminated by defining ST using the "newtype" mechanism
of Haskell, rather than the "data" mechanism.


An example
----------

By way of an example of using the state monad, let us first define
a type of binary trees whose leaves contains values of some type a:

   data Tree a = Leaf a | Node (Tree a) (Tree a)

Here is a simple example:

   tree :: Tree Char
   tree =  Node (Node (Leaf 'a') (Leaf 'b')) (Leaf 'c')

Now consider the problem of defining a function that labels each 
leaf in such a tree with a unique or "fresh" integer.  This can
be achieved by taking the next fresh integer as an additional 
argument to the function, and returning the next fresh integer
as an additional result.  In other words, the function can be 
defined using the notion of a state transformer, in which the
internal state is simply the next fresh integer:

   type State = Int

In order to generate a fresh integer, we define a special
state transformer that simply returns the current state as
its result, and the next integer as the new state:

   fresh :: ST Int
   fresh =  S (\n -> (n, n+1))

Using this, together with the return and >>= primitives that
are provided by virtue of ST being a monadic type, it is now
straightforward to define a function that takes a tree as its
argument, and returns a state transformer that produces the
same tree with each leaf labelled by a fresh integer:

   mlabel            :: Tree a -> ST (Tree (a,Int))
   mlabel (Leaf x)   =  do n <- fresh
                           return (Leaf (x,n))
   mlabel (Node l r) =  do l' <- mlabel l
                           r' <- mlabel r
                           return (Node l' r')

Note that the programmer does not have to worry about the tedious
and error-prone task of dealing with the plumbing of fresh labels,
as this is handled automatically by the state monad.

Finally, we can now define a function that labels a tree by
simply applying the resulting state transformer with zero as
the initial state, and then discarding the final state:

   label  :: Tree a -> Tree (a,Int)
   label t = fst (apply (mlabel t) 0)

For example, label tree gives the following result:

   Node (Node (Leaf ('a',0)) (Leaf ('b',1))) (Leaf ('c',2))

Exercises:

o Define a function app :: (State -> State) -> ST State, such 
  that fresh can be redefined by fresh = app (+1).

o Define a function run :: ST a -> State -> a, such that label
  can be redefined by label t = run (mlabel t) 0.


The IO monad
------------

Recall that interactive programs in Haskell are written using the
type IO a of "actions" that return a result of type a, but may
also perform some input/output.  A number of primitives are
provided for building values of this type, including:

   return  :: a -> IO a
   (>>=)   :: IO a -> (a -> IO b) -> IO b
   getChar :: IO Char
   putChar :: Char -> IO ()

The use of return and >>= means that IO is monadic, and hence
that the do notation can be used to write interactive programs.
For example, the action that reads a string of characters from
the keyboard can be defined as follows:

   getLine :: IO String
   getLine =  do x <- getChar
                 if x == '\n' then
                    return []
                 else
                    do xs <- getLine
                       return (x:xs)

It is interesting to note that the IO monad can be viewed as a
special case of the state monad, in which the internal state is
a suitable representation of the "state of the world":

   type World = ...

   type IO a  = World -> (a,World)

That is, an action can be viewed as a function that takes the
current state of the world as its argument, and produces a value
and a modified world as its result, in which the modified world
reflects any input/output performed by the action.  In reality,
Haskell systems such as Hugs and GHC implement actions in a more
efficient manner, but for the purposes of understanding the
behaviour of actions, the above interpretation can be useful.


Derived primitives
------------------

An important benefit of abstracting out the notion of a monad
is that it then becomes possible to define a number of useful 
functions that work in an arbitrary monad.  For example, the
"map" function on lists can be generalised as follows:

   liftM     :: Monad m => (a -> b) -> m a -> m b
   liftM f mx = do x <- mx
                   return (f x)

Similarly, "concat" on lists generalises to:

   join     :: Monad m => m (m a) -> m a
   join mmx =  do mx <- mmx
                  x  <- mx
                  return x

It is sometimes useful to sequence two monadic expressions,
but discard the result value produced by the first:

   (>>)     :: Monad m => m a -> m b -> m b
   mx >> my =  do _ <- mx
                  y <- my
                  return y

For example, in the state monad the >> operator is just normal
sequential composition, written as ; in most languages.

As a final example, we can define a function that transforms
a list of monadic expressions into a single such expression that
returns a list of results, by performing each of the argument
expressions in sequence and collecting their results:

   sequence          :: Monad m => [m a] -> m [a]
   sequence []       =  return []
   sequence (mx:mxs) =  do x  <- mx
                           xs <- sequence mxs
                           return (x:xs)

Exercises:

o Define liftM and join more compactly by using >>=.

o Explain the behaviour of sequence for the maybe monad.

o Define another monadic generalisation of map:

     mapM :: Monad m => (a -> m b) -> [a] -> m [b]

o Define a monadic generalisation of foldr:

     foldM :: Monad m => (a -> b -> m a) -> a -> [b] -> m a


The monad laws
--------------

Earlier we mentioned that the notion of a monad requires that the
return and >>= functions satisfy some simple properties.    The
first two properties concern the link between return and >>=:

   return x >>= f  =  f x		(1)

   mx >>= return  =  mx			(2)

Intuitively, equation (1) states that if we return a value x and
then feed this value into a function f, this should give the same
result as simply applying f to x.  Dually, equation (2) states
that if we feed the results of a computation mx into the function
return, this should give the same result as simply performing mx.
Together, these equations express --- modulo the fact that the
second argument to >>= involves a binding operation --- that
return is the left and right identity for >>=.

The third property concerns the link between >>= and itself, and
expresses (again modulo binding) that >>= is associative:

     (mx >>= f) >>= g
   =					(3)
     mx >>= (\x -> (f x >>= g))

Note that we cannot simply write mx >>= (f >>= g) on the right 
hand side of this equation, as this would not be type correct.

As an example of the utility of the monad laws, let us see how
they can be used to prove a useful property of the liftM function
from the previous section, namely that it distributes over the
composition operator for functions, in the sense that:

   liftM (f . g)  =  liftM f . liftM g

This equation generalises the familiar distribution property of
map from lists to an arbitrary monad.  In order to verify this
equation, we first rewrite the definition of liftM using >>=:

   liftM f mx  =  mx >>= \x -> return (f x)

Now the distribution property can be verified as follows:

     (liftM f . liftM g) mx
   =    applying .
     liftM f (liftM g mx)
   =    applying the second liftM
     liftM f (mx >>= \x -> return (g x))
   =    applying liftM 
     (mx >>= \x -> return (g x)) >>= \y -> return (f y)
   =    equation (3)
     mx >>= (\z -> (return (g z) >>= \y -> return (f y)))
   =    equation (1)
     mx >>= (\z -> return (f (g z)))
   =    unapplying . 
     mx >>= (\z -> return ((f . g) z)))
   =    unapplying liftM
     liftM (f . g) mx

Exercise:

o Show that the maybe monad satisfies equations (1), (2) and (3).


An exercise
-----------

Given the type

   data Expr a = Var a | Val Int | Add (Expr a) (Expr a)

of expressions built from variables of type "a", show that this
type is monadic by completing the following declaration:

   instance Monad Expr where
      -- return       :: a -> Expr a
      return x         = ...

      -- (>>=)        :: Expr a -> (a -> Expr b) -> Expr b
      (Var a)   >>= f  = ...
      (Val n)   >>= f  = ...
      (Add x y) >>= f  = ...

Hint: think carefully about the types involved.  With the aid of an 
example, explain what the >>= operator for this type does.


Other topics
------------

The subject of monads is a large one, and we have only scratched
the surface here.  If you are interested in finding out more, 
two suggestions for further reading would be to look at "monads
with a zero a plus" (which extend the basic notion with two 
extra primitives that are supported by some monads), and "monad
transformers" (which provide a means to combine monads.)  For
example, see sections 3 and 7 of the following article, which
concerns the monadic nature of functional parsers:

   http://www.cs.nott.ac.uk/~gmh/monparsing.pdf

For a more in-depth exploration of the IO monad, see Simon Peyton
Jones' excellent article on the "awkward squad":

   http://research.microsoft.com/Users/simonpj/papers/marktoberdorf/