Last modified on 9 November 2014, at 16:27

Haskell/Understanding monads

Monads are very useful in Haskell, but the concept is often difficult at first. Since they have so many applications, people often explain them from a particular point of view, and that can confuse your understanding of monads in their full glory.

Historically, monads were introduced into Haskell to perform input/output. A predetermined execution order is crucial for things like reading and writing files, and monadic operations follow an inherent sequence. We discussed sequencing and IO back in Simple input and output using the do notation. Well, do is actually just syntactic sugar over monads.

Monads are by no means limited to input and output. Monads support a whole range of things like exceptions, state, non-determinism, continuations, coroutines, and more. In fact, thanks to the versatility of monads, none of these constructs needed to be built into Haskell as a language; instead, they are defined by the standard libraries.


DefinitionEdit

A monad is defined by three things:

The function and operator are methods of the Monad type class and have types

    return :: a -> M a
    (>>=)  :: M a -> (a -> M b) -> M b

and are required to obey three laws that will be explained later on.

For a concrete example, take the Maybe monad. The type constructor is M = Maybe, while return and (>>=) are defined like this:

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

Maybe is the monad, and return brings a value into it by wrapping it with Just. As for (>>=), it takes a m :: Maybe a value and a g :: a -> Maybe b function. If m is Nothing, there is nothing to do and the result is Nothing. Otherwise, in the Just x case, g is applied to x, the underlying value wrapped in Just, to give a Maybe b result, which might be Nothing, depending on what g does to x. To sum it all up, if there is an underlying value in m, we apply g to it, which brings the underlying value back into the Maybe monad.

The key first step to understand how return and (>>=) work is tracking which values and arguments are monadic and which ones aren't. As in so many other cases, type signatures are our guide to the process.

Motivation: MaybeEdit

To see the usefulness of (>>=) and the Maybe monad, consider the following example: Imagine a family database that provides two functions

    father :: Person -> Maybe Person
    mother :: Person -> Maybe Person

These look up the name of someone's father or mother. In case our database is missing some information, Maybe allows us to return a Nothing value instead of crashing the program.

Let's combine our functions to query various grandparents. For instance, the following function looks up the maternal grandfather:

maternalGrandfather :: Person -> Maybe Person
maternalGrandfather p =
    case mother p of
        Nothing -> Nothing
        Just mom -> father mom

Or consider a function that checks whether both grandfathers are in the database:

    bothGrandfathers :: Person -> Maybe (Person, Person)
    bothGrandfathers p =
        case father p of
            Nothing -> Nothing
            Just dad ->
                case father dad of
                    Nothing -> Nothing
                    Just gf1 ->                          -- found first grandfather
                        case mother p of
                            Nothing -> Nothing
                            Just mom ->
                                case father mom of
                                    Nothing -> Nothing
                                    Just gf2 ->          -- found second grandfather
                                        Just (gf1, gf2)

What a mouthful! Every single query might fail by returning Nothing and the whole function must fail with Nothing if that happens.

Clearly there has to be a better way to write that instead of repeating the case of Nothing again and again! Indeed, that's what the Maybe monad is set out to do. For instance, the function retrieving the maternal grandfather has exactly the same structure as the (>>=) operator, so we can rewrite it as:

    maternalGrandfather p = mother p >>= father

With the help of lambda expressions and return, we can rewrite the two grandfathers function as well:

    bothGrandfathers p =
       father p >>=
           (\dad -> father dad >>=
               (\gf1 -> mother p >>=   -- this line works as "\_ -> mother p", but naming gf1 allows later return
                   (\mom -> father mom >>=
                       (\gf2 -> return (gf1,gf2) ))))

While these nested lambda expressions may look confusing to you, the thing to take away here is that (>>=) releases us from listing all the Nothings, shifting the focus back to the interesting part of the code.

To be a little more precise: The result of father p is a monadic value (in this case, either Just dad or Nothing, depending on whether p's dad is in the database). As the father function takes a regular (non-monadic value), the >>= feeds p's dad to it as a non-monadic value. The result of father dad is then monadic again, and the process continues.

So, >>= helps us pass non-monadic values to functions without leaving a monad. In the case of the Maybe monad, the monadic aspect is the qualifier that we don't know with certainty whether the value will be found.

Type classEdit

In Haskell, the Monad type class is used to implement monads. It is provided by the Control.Monad module and included in the Prelude. The class has the following methods:

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

Aside from return and bind, notice the two additional functions (>>) and fail.

The operator (>>) called "then" is a mere convenience and commonly implemented as

    m >> n = m >>= \_ -> n

>> sequences two monadic actions when the second action does not involve the result of the first, which is common for monads like IO.

    printSomethingTwice :: String -> IO ()
    printSomethingTwice str = putStrLn str >> putStrLn str

The function fail handles pattern match failures in do notation. It's an unfortunate technical necessity and doesn't really have anything to do with monads. You are advised to not call fail directly in your code.

Notions of ComputationEdit

We've seen how (>>=) and return are very handy for removing boilerplate code that crops up when using Maybe. That, however, is not enough to justify why monads matter so much. We will continue our monad studies by rewriting the two-grandfathers function using do notation with explicit braces and semicolons. Depending on your experience with other programming languages, you may find this very suggestive:

    bothGrandfathers p = do {
        dad <- father p;
        gf1 <- father dad;
        mom <- mother p;
        gf2 <- father mom;
        return (gf1, gf2);
      }

If this looks like a code snippet of an imperative programming language to you, that's because it is. In particular, this imperative language supports exceptions : father and mother are functions that might fail to produce results, i.e. raise an exception, and when that happens, the whole do-block will fail, i.e. terminate with an exception.

In other words, the expression father p, which has type Maybe Person, is interpreted as a statement of an imperative language that returns a Person as result. This is true for all monads: a value of type M a is interpreted as a statement of an imperative language that returns a value of type a as result; and the semantics of this language are determined by the monad M.[2]

Under this interpretation, the bind operator (>>=) is simply a function version of the semicolon. Just like a let expression can be written as a function application,

   let x = foo in x + 3          corresponds to      (\x -> x + 3) foo

an assignment and semicolon can be written as the bind operator:

   x <- foo; return (x + 3)      corresponds to      foo >>= (\x -> return (x + 3))

The return function lifts a value a to M a, a full-fledged statement of the imperative language corresponding to the monad M.

Different semantics of the imperative language correspond to different monads. The following table shows the classic selection that every Haskell programmer should know. If the idea behind monads is still unclear to you, studying each of the examples in the following chapters will not only give you a well-rounded toolbox but also help you understand the common abstraction behind them.

Monad Imperative Semantics Wikibook chapter
Maybe Exception (anonymous) Haskell/Understanding monads/Maybe
Error Exception (with error description) Haskell/Understanding monads/Error
State Global state Haskell/Understanding monads/State
IO Input/Output Haskell/Understanding monads/IO
[] (lists) Nondeterminism Haskell/Understanding monads/List
Reader Environment Haskell/Understanding monads/Reader
Writer Logger Haskell/Understanding monads/Writer

Furthermore, these different semantics need not occur in isolation. As we will see in a few chapters, it is possible to mix and match them by using monad transformers to combine the semantics of multiple monads in a single monad.

Monad LawsEdit

In Haskell, every instance of the Monad type class (and thus all implementations of (>>=) and return) must obey the following three laws:

    m >>= return     =  m                        -- right unit
    return x >>= f   =  f x                      -- left unit
 
    (m >>= f) >>= g  =  m >>= (\x -> f x >>= g)  -- associativity


Return as neutral elementEdit

The behavior of return is specified by the left and right unit laws. They state that return doesn't perform any computation, it just collects values. For instance,

    maternalGrandfather p = do
            mom <- mother p
            gf  <- father mom
            return gf

is exactly the same as

    maternalGrandfather p = do
            mom  <- mother p
            father mom

by virtue of the right unit law.

Associativity of bindEdit

The law of associativity makes sure that (like the semicolon) the bind operator (>>=) only cares about the order of computations, not about their nesting; e.g. we could have written bothGrandfathers like this (compare with our earliest version without do):

    bothGrandfathers p =
       (father p >>= father) >>=
           (\gf1 -> (mother p >>= father) >>=
               (\gf2 -> return (gf1,gf2) ))

The associativity of the then operator (>>) is a special case:

   (m >> n) >> o  =  m >> (n >> o)

Monadic compositionEdit

It is easier to picture the associativity of bind by recasting the law as

   (f >=> g) >=> h  =  f >=> (g >=> h)

where (>=>) is the monad composition operator, a close analogue of the function composition operator (.), only with flipped arguments. It is defined as:

   (>=>) :: Monad m => (a -> m b) -> (b -> m c) -> a -> m c
   f >=> g = \x -> f x >>= g

We can also flip monad composition to go the other direction using (<=<). The operation order of (f . g) is the same as (f' <=< g').[3]

Monads and Category TheoryEdit

Monads originally come from a branch of mathematics called Category Theory. Fortunately, it is entirely unnecessary to understand category theory in order to understand and use monads in Haskell. The definition of monads in Category Theory actually uses a slightly different presentation. Translated into Haskell, this presentation gives an alternative yet equivalent definition of a monad which can give us some additional insight.[4]

So far, we have defined monads in terms of (>>=) and return. The alternative definition, instead, starts with monads as functors with two additional combinators:

    fmap   :: (a -> b) -> M a -> M b  -- functor
 
    return :: a -> M a
    join   :: M (M a) -> M a

(Reminder: as discussed in the chapter on the functor class, a functor M can be thought of as container, so that M a "contains" values of type a, with a corresponding mapping function, i.e. fmap, that allows functions to be applied to values inside it.)

Under this interpretation, the functions behave as follows:

  • fmap applies a given function to every element in a container
  • return packages an element into a container,
  • join takes a container of containers and flattens it into a single container.

With these functions, the bind combinator can be defined as follows:

    m >>= g = join (fmap g m)

Likewise, we could give a definition of fmap and join in terms of (>>=) and return:

    fmap f x = x >>= (return . f)
    join x   = x >>= id

Is my Monad a Functor?Edit

At this point we might, with good reason, deduce that all monads are by definition functors as well. While according to category theory that is indeed the case, GHC does it differently because of historic accident in the designing of Haskell. While the Monad and Functor classes aren't connected in current versions, GHC 7.10 will finally fix this issue so that every Monad instance will be connected to a matching Functor instance and the corresponding fmap. Meanwhile, Control.Monad defines liftM, a function with a strangely familiar type signature...

    liftM :: (Monad m) => (a1 -> r) -> m a1 -> m r

As you might suspect, liftM is merely fmap implemented with (>>=) and return, just as we have done above. For a properly implemented monad with a matching Functor (that is, any sensible monad) liftM and fmap are interchangeable.


NotesEdit

  1. This return function has nothing to do with the return keyword found in imperative languages like C or Java; don't conflate these two.
  2. By "semantics", we mean what the language allows you to say. In the case of Maybe, the semantics allow us to express failure, as statements may fail to produce a result, leading to the statements that follow it being skipped.
  3. Of course, the functions in regular function composition are non-monadic functions whereas monadic composition takes only monadic functions.
  4. Deep into the Advanced Track, we will cover the theoretical side of the story in the chapter on Category Theory.