Haskell/Monad transformers
We have seen how monads can help handling IO
actions, Maybe
, lists, and state.
With monads providing a common way to use such useful generalpurpose tools, a natural thing we might want to do is using the capabilities of several monads at once. For instance, a function could use both I/O and Maybe
exception handling. While a type like IO (Maybe a)
would work just fine, it would force us to do pattern matching within IO
doblocks to extract values, something that the Maybe
monad was meant to spare us from.
Enter monad transformers: special types that allow us to roll two monads into a single one that shares the behavior of both.
Passphrase validation[edit  edit source]
Consider a reallife problem for IT staff worldwide: getting users to create strong passphrases. One approach: force the user to enter a minimum length with various irritating requirements (such as at least one capital letter, one number, one nonalphanumeric character, etc.)
Here's a Haskell function to acquire a passphrase from a user:
getPassphrase :: IO (Maybe String)
getPassphrase = do s < getLine
if isValid s then return $ Just s
else return Nothing
 The validation test could be anything we want it to be.
isValid :: String > Bool
isValid s = length s >= 8
&& any isAlpha s
&& any isNumber s
&& any isPunctuation s
First and foremost, getPassphrase
is an IO
action, as it needs to get input from the user. We also use Maybe
, as we intend to return Nothing
in case the password does not pass the isValid
. Note, however, that we aren't actually using Maybe
as a monad here: the do
block is in the IO
monad, and we just happen to return
a Maybe
value inside it.
Monad transformers not only make it easier to write getPassphrase
but also simplify all the code instances. Our passphrase acquisition program could continue like this:
askPassphrase :: IO ()
askPassphrase = do putStrLn "Insert your new passphrase:"
maybe_value < getPassphrase
case maybe_value of
Just value > do putStrLn "Storing in database..."  do stuff
Nothing > putStrLn "Passphrase invalid."
The code uses one line to generate the maybe_value
variable followed by further validation of the passphrase.
With monad transformers, we will be able to extract the passphrase in one go — without any pattern matching (or equivalent bureaucracy like isJust
). The gains for our simple example might seem small but will scale up for more complex situations.
A simple monad transformer: MaybeT
[edit  edit source]
To simplify getPassphrase
and the code that uses it, we will define a monad transformer that gives the IO
monad some characteristics of the Maybe
monad; we will call it MaybeT
. That follows a convention where monad transformers have a "T
" appended to the name of the monad whose characteristics they provide.
MaybeT
is a wrapper around m (Maybe a)
, where m
can be any monad (IO
in our example):
newtype MaybeT m a = MaybeT { runMaybeT :: m (Maybe a) }
This data type definition specifies a MaybeT
type constructor, parameterized over m
, with a data constructor, also called MaybeT
, and a convenient accessor function runMaybeT
, with which we can access the underlying representation.
The whole point of monad transformers is that they transform monads into monads; and so we need to make MaybeT m
an instance of the Monad
class:
instance Monad m => Monad (MaybeT m) where
return = MaybeT . return . Just
 The signature of (>>=), specialized to MaybeT m:
 (>>=) :: MaybeT m a > (a > MaybeT m b) > MaybeT m b
x >>= f = MaybeT $ do maybe_value < runMaybeT x
case maybe_value of
Nothing > return Nothing
Just value > runMaybeT $ f value
It would also have been possible (though arguably less readable) to write the return function as: return = MaybeT . return . return
.
Starting from the first line of the do
block:
 First, the
runMaybeT
accessor unwrapsx
into anm (Maybe a)
computation. That shows us that the wholedo
block is inm
.  Still in the first line,
<
extracts aMaybe a
value from the unwrapped computation.  The
case
statement testsmaybe_value
: With
Nothing
, we returnNothing
intom
;  With
Just
, we applyf
to thevalue
from theJust
. Sincef
hasMaybeT m b
as result type, we need an extrarunMaybeT
to put the result back into them
monad.
 With
 Finally, the
do
block as a whole hasm (Maybe b)
type; so it is wrapped with theMaybeT
constructor.
It may look a bit complicated, but aside from the copious amounts of wrapping and unwrapping, the implementation of MaybeT
's bind works the same way as the implementation of Maybe
's familiar bind operator:
 (>>=) for the Maybe monad
maybe_value >>= f = case maybe_value of
Nothing > Nothing
Just value > f value
Why use the MaybeT
constructor before the do
block while we have the accessor runMaybeT
within do
? Well, the do
block must be in the m
monad, not in MaybeT m
(which lacks a defined bind operator at this point).
As usual, we also have to provide instances for the superclasses of Monad
, Applicative
and Functor
:
instance Monad m => Applicative (MaybeT m) where
pure = return
(<*>) = ap
instance Monad m => Functor (MaybeT m) where
fmap = liftM
In addition, it is convenient to make MaybeT m
an instance of a few other classes:
instance Monad m => Alternative (MaybeT m) where
empty = MaybeT $ return Nothing
x <> y = MaybeT $ do maybe_value < runMaybeT x
case maybe_value of
Nothing > runMaybeT y
Just _ > return maybe_value
instance Monad m => MonadPlus (MaybeT m) where
mzero = empty
mplus = (<>)
instance MonadTrans MaybeT where
lift = MaybeT . (liftM Just)
MonadTrans
implements the lift
function, so we can take functions from the m
monad and bring them into the MaybeT m
monad in order to use them in do
blocks. As for Alternative
and MonadPlus
, since Maybe
is an instance of those classes it makes sense to make the MaybeT m
an instance too.
Passphrase validation, simplified[edit  edit source]
The above passphrase validation example can now be simplified using the MaybeT
monad transformer as follows:
getPassphrase :: MaybeT IO String
getPassphrase = do s < lift getLine
guard (isValid s)  Alternative provides guard.
return s
askPassphrase :: MaybeT IO ()
askPassphrase = do lift $ putStrLn "Insert your new passphrase:"
value < getPassphrase
lift $ putStrLn "Storing in database..."
The code is now simpler, especially in the user function askPassphrase
. Most importantly, we do not have to manually check whether the result is Nothing
or Just
: the bind operator takes care of that for us.
Note how we use lift
to bring the functions getLine
and putStrLn
into the MaybeT IO
monad. Also, since MaybeT IO
is an instance of Alternative
, checking for passphrase validity can be taken care of by a guard
statement, which will return empty
(i.e. IO Nothing
) in case of a bad passphrase.
Incidentally, with the help of MonadPlus
it also becomes very easy to ask the user ad infinitum for a valid passphrase:
askPassphrase :: MaybeT IO ()
askPassphrase = do lift $ putStrLn "Insert your new passphrase:"
value < msum $ repeat getPassphrase
lift $ putStrLn "Storing in database..."
A plethora of transformers[edit  edit source]
The transformers package provides modules with transformers for many common monads (MaybeT
, for instance, can be found in Control.Monad.Trans.Maybe). These are defined consistently with their nontransformer versions; that is, the implementation is basically the same except with the extra wrapping and unwrapping needed to thread the other monad. From this point on, we will use precursor monad to refer to the nontransformer monad (e.g. Maybe in MaybeT) on which a transformer is based and base monad to refer to the other monad (e.g. IO in MaybeT IO) on which the transformer is applied.
To pick an arbitrary example, ReaderT Env IO String
is a computation which involves reading values from some environment of type Env
(the semantics of Reader
, the precursor monad) and performing some IO
in order to give a value of type String
. Since the bind operator and return
for the transformer mirror the semantics of the precursor monad, a do
block of type ReaderT Env IO String
will, from the outside, look a lot like a do
block of the Reader
monad, except that IO
actions become trivial to embed by using lift
.
Type juggling[edit  edit source]
We have seen that the type constructor for MaybeT
is a wrapper for a Maybe
value in the base monad. So, the corresponding accessor runMaybeT
gives us a value of type m (Maybe a)
 i.e. a value of the precursor monad returned in the base monad. Similarly, for the ListT
and ExceptT
transformers, which are built around lists and Either
respectively:
runListT :: ListT m a > m [a]
and
runExceptT :: ExceptT e m a > m (Either e a)
Not all transformers are related to their precursor monads in this way, however. Unlike the precursor monads in the two examples above, the Writer
, Reader
, State
, and Cont
monads have neither multiple constructors nor constructors with multiple arguments. For that reason, they have run... functions which act as simple unwrappers, analogous to the run...T of the transformer versions. The table below shows the result types of the run... and run...T functions in each case, which may be thought of as the types wrapped by the base and transformed monads respectively.^{[1]}
Precursor  Transformer  Original Type ("wrapped" by precursor) 
Combined Type ("wrapped" by transformer) 

Writer  WriterT  (a, w) 
m (a, w)

Reader  ReaderT  r > a 
r > m a

State  StateT  s > (a, s) 
s > m (a, s)

Cont  ContT  (a > r) > r 
(a > m r) > m r

Notice that the precursor monad type constructor is absent in the combined types. Without interesting data constructors (of the sort that Maybe
and lists have), there is no reason to retain the precursor monad type after unwrapping the transformed monad. It is also worth noting that in the latter three cases we have function types being wrapped. StateT
, for instance, turns statetransforming functions of the form s > (a, s)
into statetransforming functions of the form s > m (a, s)
; only the result type of the wrapped function goes into the base monad. ReaderT
is analogous.ContT
is different because of the semantics of Cont
(the continuation monad): the result types of both the wrapped function and its function argument must be the same, and so the transformer puts both into the base monad. In general, there is no magic formula to create a transformer version of a monad; the form of each transformer depends on what makes sense in the context of its nontransformer type.
Lifting[edit  edit source]
We will now have a more detailed look at the lift
function, which is critical in daytoday use of monad transformers. The first thing to clarify is the name "lift". One function with a similar name that we already know is liftM
. As we have seen in Understanding monads, it is a monadspecific version of fmap
:
liftM :: Monad m => (a > b) > m a > m b
liftM
applies a function (a > b)
to a value within a monad m
. We can also look at it as a function of just one argument:
liftM :: Monad m => (a > b) > (m a > m b)
liftM
converts a plain function into one that acts within m
. By "lifting", we refer to bringing something into something else — in this case, a function into a monad.
liftM
allows us to apply a plain function to a monadic value without needing doblocks or other such tricks:
bind notation  do notation  liftM 

monadicValue >>=
\x > return (f x)

do x < monadicValue
return (f x)

liftM f monadicValue

The lift
function plays an analogous role when working with monad transformers. It brings (or, to use another common word for that, promotes) base monad computations to the combined monad. By doing so, it allows us to easily insert base monad computations as part of a larger computation in the combined monad.
lift
is the single method of the MonadTrans
class, found in Control.Monad.Trans.Class. All monad transformers are instances of MonadTrans
, and so lift
is available for them all.
class MonadTrans t where
lift :: (Monad m) => m a > t m a
There is a variant of lift
specific to IO
operations, called liftIO
, which is the single method of the MonadIO
class in Control.Monad.IO.Class.
class (Monad m) => MonadIO m where
liftIO :: IO a > m a
liftIO
can be convenient when multiple transformers are stacked into a single combined monad. In such cases, IO
is always the innermost monad, and so we typically need more than one lift to bring IO
values to the top of the stack. liftIO
is defined for the instances in a way that allows us to bring an IO
value from any depth while writing the function a single time.
Implementing lift
[edit  edit source]
Implementing lift
is usually pretty straightforward. Consider the MaybeT
transformer:
instance MonadTrans MaybeT where
lift m = MaybeT (liftM Just m)
We begin with a monadic value of the base monad. With liftM
(fmap
would have worked just as fine), we slip the precursor monad (through the Just
constructor) underneath, so that we go from m a
to m (Maybe a)
). Finally, we wrap things up with the MaybeT
constructor. Note that the liftM
here works in the base monad, just like the doblock wrapped by MaybeT
in the implementation of (>>=)
we saw early on was in the base monad.
Exercises 


Implementing transformers[edit  edit source]
The State transformer[edit  edit source]
As an additional example, we will now have a detailed look at the implementation of StateT
. You might want to review the section on the State monad before continuing.
Just as the State monad might have been built upon the definition newtype State s a = State { runState :: (s > (a,s)) }
, the StateT transformer is built upon the definition:
newtype StateT s m a = StateT { runStateT :: (s > m (a,s)) }
StateT s m
will have the following Monad
instance, here shown alongside the one for the precursor state monad:
State  StateT 

newtype State s a =
State { runState :: (s > (a,s)) }
instance Monad (State s) where
return a = State $ \s > (a,s)
(State x) >>= f = State $ \s >
let (v,s') = x s
in runState (f v) s'

newtype StateT s m a =
StateT { runStateT :: (s > m (a,s)) }
instance (Monad m) => Monad (StateT s m) where
return a = StateT $ \s > return (a,s)
(StateT x) >>= f = StateT $ \s > do
(v,s') < x s  get new value and state
runStateT (f v) s'  pass them to f

Our definition of return
makes use of the return
function of the base monad. (>>=)
uses a doblock to perform a computation in the base monad.
Note
Incidentally, we can now finally explain why, back in the chapter about State
, there was a state
function instead of a State
constructor. In the transformers and mtl packages, State s
is implemented as a type synonym for StateT s Identity
, with Identity
being the dummy monad introduced in an exercise of the previous section. The resulting monad is equivalent to the one defined using newtype
that we have used up to now.
If the combined monads StateT s m
are to be used as state monads, we will certainly want the allimportant get
and put
operations. Here, we will show definitions in the style of the mtl package. In addition to the monad transformers themselves, mtl provides type classes for the essential operations of common monads. For instance, the MonadState
class, found in Control.Monad.State, has get
and put
as methods:
instance (Monad m) => MonadState s (StateT s m) where
get = StateT $ \s > return (s,s)
put s = StateT $ \_ > return ((),s)
Note
instance (Monad m) => MonadState s (StateT s m)
should be read as: "For any type s
and any instance of Monad
m
, s
and StateT s m
together form an instance of MonadState
". s
and m
correspond to the state and the base monad, respectively. s
is an independent part of the instance specification so that the methods can refer to it − for instance, the type of put
is s > StateT s m ()
.
There are MonadState
instances for state monads wrapped by other transformers, such as MonadState s m => MonadState s (MaybeT m)
. They bring us extra convenience by making it unnecessary to lift uses of get
and put
explicitly, as the MonadState
instance for the combined monads handles the lifting for us.
It can also be useful to lift instances that might be available for the base monad to the combined monad. For instance, all combined monads in which StateT
is used with an instance of MonadPlus
can be made instances of MonadPlus
:
instance (MonadPlus m) => MonadPlus (StateT s m) where
mzero = StateT $ \_ > mzero
(StateT x1) `mplus` (StateT x2) = StateT $ \s > (x1 s) `mplus` (x2 s)
The implementations of mzero
and mplus
do the obvious thing; that is, delegating the actual work to the instance of the base monad.
Lest we forget, the monad transformer must have a MonadTrans
, so that we can use lift
:
instance MonadTrans (StateT s) where
lift c = StateT $ \s > c >>= (\x > return (x,s))
The lift
function creates a StateT
state transformation function that binds the computation in the base monad to a function that packages the result with the input state. If, for instance, we apply StateT to the List monad, a function that returns a list (i.e., a computation in the List monad) can be lifted
into StateT s []
where it becomes a function that returns a StateT (s > [(a,s)])
. I.e. the lifted computation produces multiple (value,state) pairs from its input state. This "forks" the computation in StateT, creating a different branch of the computation for each value in the list returned by the lifted function. Of course, applying StateT
to a different monad will produce different semantics for the lift
function.
Exercises 


Acknowledgements[edit  edit source]
This module uses a number of excerpts from All About Monads, with permission from its author Jeff Newbern.
Notes
 ↑ The wrapping interpretation is only literally true for versions of the mtl package older than 2.0.0.0 .