# Monad Transformers

```
{-# LANGUAGE TypeSynonymInstances, FlexibleContexts, NoMonomorphismRestriction, OverlappingInstances, FlexibleInstances, KindSignatures #-}
module Transformers where
import Control.Monad.Error
import Control.Monad.State
import Control.Monad.Writer
```

# Announcements

- HW #5 out after class, due in
*two weeks*

# What monads have we seen so far?

- []
- State
- Maybe
- IO
- Parsing
- AList

# How do we use *multiple* monads at once?

Today, we will see how monads can be used to write (and compose) *evaluators* for programming languages.

Let's look at a simple language of division expressions.

```
data Expr = Val Int
| Div Expr Expr
deriving (Show)
```

Our first evaluator is *unsafe*.

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

Here are two terms that we will use as running examples.

```
ok = Div (Div (Val 1972) (Val 2)) (Val 23)
err = Div (Val 2) (Div (Val 1) (Div (Val 2) (Val 3)))
```

The first evaluates properly and returns a valid answer, and the second fails with a divide-by-zero exception.

```
*Main> eval ok
*Main> eval err
```

We don't like this `eval`

because it can just blow up with a divide by zero error without telling us how it happened. Worse, the error is a *radioactive* value that, spread unchecked through the entire computation.

We can use the `Maybe`

type to capture the failure case: a `Nothing`

result meant that an error happened somewhere, while a `Just n`

result meant that evaluation succeeded yielding `n`

. Morever, the `Maybe`

monad can be used to avoid ugly case-split-staircase-hell.

```
evalMaybe :: Expr -> Maybe Int
evalMaybe (Val n) = return n
evalMaybe (Div x y) = do
n <- evalMaybe x
m <- evalMaybe y
guard (m /= 0)
return (n `div` m)
```

```
{-
case (evalMaybe y) of
Just 0 -> Nothing
Just m -> case evalMaybe x of
Just n -> Just (n `div` m)
Nothing -> Nothing
Nothing -> Nothing
-}
```

which behaves thus

```
*Main> evalMaybe ok
*Main> evalMaybe err
```

## Error Handling Via Exception Monads

The trouble with the above is that it doesn't let us know *where* the divide by zero occurred. It would be nice to have an *exception* mechanism where, when the error occurred, we could just saw `Throw x`

for some value `x`

which would, like an exception go rocketing back to the top and tell us what the problem was.

If you think for a moment, you'll realize this is but a small tweak on the `Maybe`

type; all we need is to jazz up the `Nothing`

constructor so that it carries the exception value.

```
data Exc a = Raise String
| Result a
deriving (Show)
```

Here the `Raise`

is like `Nothing`

but it carries a string denoting what the exception was. We can make the above a `Monad`

much like the `Maybe`

monad.

```
instance Monad Exc where
-- (>>=) :: Exc a -> (a -> Exc b) -> Exc b
(Result x) >>= f = f x
(Raise s) >>= f = Raise s
```

```
-- return :: a -> Exc a
return = Result
```

```
{-
instance MonadPlus Exc where
mzero = Raise "failure?"
mplus :: m a -> m a -> m a
mplus (Result x) y = Result x
mplus (Raise s) y = y
-}
```

and now, we can use our newly minted monad to write a better exception throwing evaluator

```
--- evalExc :: Expr -> Exc Int
evalExc (Val n) = return n
evalExc (Div x y) = do
n <- evalExc x
m <- evalExc y
if (m /= 0)
then return (n `div` m)
else throwErrorExc $ errorS x y
```

where the sidekick `errorS`

generates the error string.

`errorS y m = "Error dividing " ++ show y ++ " by " ++ show m`

Note that this is essentially like the first evaluator; instead of bailing with `Nothing`

we return some (hopefully) helpful message, but the monad takes care of ensuring that the exception is shot back up.

```
*Main> evalExc ok
*Main> evalExc err
```

## Counting Operations Via State Monads

Next, let's stop being so paranoid about errors and instead try to do some *profiling*. Lets imagine that the `div`

operator is very expensive, and that we would like to *count* the number of divisions that are performed while evaluating a particular expression.

As you might imagine, our old friend the state monad is likely to be of service here! The type of store that we'd like to use is just the count of number of division operations, we can store that in an Int.

`type Store = Int`

We imported the library `State`

above. (This library looks a lot like `State.hs`

from lecture and your homework assignment.)

```
data State a = S (Store -> (a, Store))
instance Monad State where
return x = S $ \s -> (x, s)
(S st) >>= f = S $ \s -> let (x, s') = st s
S st' = f x
in st' s'
runState :: State s a -> s -> (a, s)
runState (S f) x = f x
get :: State s s
get = S (\s -> (s, s))
put :: s -> State s ()
put s' = S (\_ -> ((), s'))
```

For simplicity today, we will only use the State monad with this particular store type. Therefore, we abbreviate the State monad, using this Store type, as 'ST'.

`type ST = State Store`

Armed with the above, we can write a function that increments the number stored in the state.

```
tickDiv :: Int -> Int -> ST Int
tickDiv m n = do
x <- get
put (x + 1)
return (m `div` n)
```

-- put :: Store -> State Store ()

Now, we can write a *profiling* evaluator

```
evalST :: Expr -> ST Int
evalST (Val n) = return n
evalST (Div x y) = do
m <- evalST x
n <- evalST y
tickDiv m n
```

and observe our profiling evaluator at work

```
goST :: Expr -> IO ()
goST e = putStrLn $ "value: " ++ show x ++ ", count: " ++ show s
where (x,s) = runState (evalST e) 0 :: (Int, Int)
```

`*Main> goST ok`

But, alas, to get the profiling we threw out the nifty error handling that we had put in earlier

`*Main> goST err `

# Transformers: Making Monads Multitask

So it looks like Monads can do many thigs, but only *one thing at a time* -- you can either use a monad to do the error management plumbing *OR* to do the state manipulation plumbing, but not at the same time. Is it too much ask for both? I guess we could write a *mega-state-and-exception* monad that supports the operations of both, but that doesn't sound like any fun at all! Worse, if later we decide to add yet another feature, then we would have to make up yet another mega-monad.

We shall take a different approach, where we will keep *wrapping* or decorating monads with extra features, so that we can take a simple monad, and then add the Exception monad's features to it, and then add the State monad's features and so on.

The key to doing this is to not define exception handling, state passing etc as monads, but as **type-level functions from monads to monads.**

This will require a little more work up-front (most of which is done already in well-designed libraries) but after that we can add new features in a modular manner. For example, to get a mega state- and exception- monad, we will start with a dummy `Identity`

monad, apply it to the `StateT`

monad transformer (which yields state-passing monad) and pass the result to the `ExcT`

monad transformer which yields the desired mega monad. Incidentally, the above should remind some of you of the Decorator Design Pattern and others of Python's Decorators.

## Step 1: Describing Monads With Special Features

The first step to being able to compose monads is to define typeclasses that describe monads armed with the special features. For example, the notion of an *exception monad* is captured by the typeclass

```
class Monad m => MonadExc m where
throwErrorExc :: String -> m a
```

which corresponds to monads that are also equipped with an appropriate `throwErrorExc`

function (you can add a `catch`

function too, if you like!) Indeed, we can make `Exc`

an instance of the above by

```
instance MonadExc Exc where
throwErrorExc = Raise
```

See what happens if you change 'Raise' to 'throwErrorExc' in the evaluator 'evalExc' above. What is the new type of the evaluator?

Similarly, we can bottle the notion of a *state monad* in the typeclass

```
class Monad m => MonadST m where
runStateST :: m a -> Store -> m (a, Store)
getST :: m Store
putST :: Store -> m ()
```

which corresponds to monads that are kitted out with the appropriate execution, extraction and modification functions. Using this monad, we can redefine the ticking operation to work for any state monad.

```
tickST :: MonadST m => m ()
tickST = do
x <- getST
putST (x + 1)
```

Needless to say, we can make `ST`

an instance of the above by

```
instance MonadST (State Store) where
-- runStateST :: ST a -> Store -> ST (a, Store)
runStateST m a = return (runState m a)
getST = get
putST = put
```

Now go back and see what happens when you replace `tickStore`

with `tickST`

above.

## Step 2: Using Monads With Special Features

Armed with these two typeclasses, we can write our exception-throwing, step-counting evaluator quite easily

```
evalMega :: (MonadExc m, MonadST m) => Expr -> m Int
evalMega (Val n) = return n
evalMega (Div x y) = do
n <- evalMega x
m <- evalMega y
if (m == 0)
then throwErrorExc $ errorS n m
else do
tickST
return (n `div` m)
```

Note that it is simply the combination of the two evaluators from before -- we use the `throwErrorExc`

from `evalExc`

and the `tickST`

from `evalST`

.

Meditate for a moment on the type of above evaluator; note that it works with *any monad* that is both a exception- and a state- monad! Indeed, if, as I exhorted you to, you had gone back and studied the types of `evalST`

and `evalExc`

you would find that each of those functions required the underlying monad to be a state-manipulating and exception-handling monad respectively. In contrast, the above evaluator simply demands both features.

But where do we get monads that are both state-manipulating and exception-handling?

Answer: we build them piece-by-piece. We're going to define some type level functions that will *add* state manipulation and exception handling to any pre-existing monad.

## Step 3: Injecting Special Features into Monads

To *add* special features to existing monads, we use *monad transformers*, which are type operators `t`

that map a monad `m`

to a monad `t m`

. The key ingredient of a transformer is that it must have a function `promote`

that can take an `m`

value (ie action) and turn it into a `t m`

value (ie action):

```
class Transformer (t :: (* -> *) -> * -> *) where
promote :: Monad m => m a -> t m a
```

Now, that just defines the *type* of a transformer, let's see some real transformers!

**A Transformer For Exceptions**

Consider the following datatype declaration, which uses record syntax.

`data ExcT (m :: * -> *) a = MkExc { unMkExc :: m (Exc a) }`

-- MkExc :: m (Exc a) -> ExcT m a -- unMkExc :: ExcT m a -> m (Exc a)

It is simply a type with two parameters -- the first is a monad `m`

inside which we will put the exception monad `Exc a`

. In other words, the `ExcT m a`

simply *injects* the `Exc a`

monad *into* the value slot of the `m`

monad. By convention, the names of monad transformers end with 'T'.

It is easy to formally state that the above is a bonafide transformer

```
instance Transformer ExcT where
-- promote :: Monad m => m a -> ExcT m a
promote = MkExc . promote_ where
-- promote_ :: (Monad m) => m a -> m (Exc a)
-- promote_ :: (Monad m1, Monad m2) => m1 t -> m1 (m2 t)
promote_ mt = do
t <- mt
return (return t)
```

where the generic `promote_`

function simply injects the value from the outer monad `m`

into the inner monad `m1`

.

Consequently, any operation on the input monad `m`

can be directly promoted into an action on the transformed monad, and so the transformation *preserves* all the operations on the original monad.

Now, the real trick is twofold, we ensure that if `m`

is a monad, then transformed `ExcT m`

is an *exception monad*, that is an `MonadExc`

.

First, we show the transformer output is a monad:

```
instance Monad m => Monad (ExcT m) where
-- return :: a -> ExcT m a
return x = MkExc (return (return x))
```

```
-- (>>=) :: ExcT m a -> (a -> ExcT m b) -> ExcT m b
-- p :: m (Exc a)
(MkExc p) >>= f = MkExc $ do x <- p
case x of
Raise s -> return (Raise s)
Result a -> unMkExc (f a)
```

and next we ensure that the transformer is an exception monad by equipping it with `throwErrorExc`

```
instance Monad m => MonadExc (ExcT m) where
-- throwErrorExc :: String -> ExcT m a
throwErrorExc = MkExc . return . Raise
```

**A Transformer For State**

Next, we will build a transformer for the state monad, following, more or less, the recipe for exceptions. Here is the type for the transformer

newtype ST a = MkST { unMkST :: Store -> (a, Store) }

`newtype STT m a = MkSTT { unMkSTT :: Store -> m (a, Store) }`

MkSTT :: (Store -> m (a,Store)) -> STT m a unMkSTT :: STT m a -> (Store -> m (a,Store))

Thus, in effect, the enhanced monad is a state-update where the output is the original monad as we do the state-update and return as output the new state wrapped inside the parameter monad.

Note that the monad transformer is *not*:

`newtype STT m a = m (Store -> (a, Store))`

We'd like the inner monad to apply to the result of the state transformation.

```
instance Transformer STT where
-- Monad m => m a -> STT m a
-- m :: m a
-- MkSTT :: (Store -> m (a,Store)) -> STT m a
promote m = MkSTT $ \s -> do a <- m
return (a,s)
```

Next, we ensure that the transformer output is a monad:

```
instance Monad m => Monad (STT m) where
-- return :: a -> STT m a
return x = promote $ return x
-- (>>=) :: STT m a -> (a -> STT m b) -> STT m b
-- m :: Store -> m (a,Store)
-- f :: a -> STT m b
MkSTT m >>= f = MkSTT $ \s0 -> do -- m
(a,s1) <- m s0
unMkSTT (f a) s1
```

and next we ensure that the transformer is a state monad by equipping it with the operations from `MonadST`

```
instance Monad m => MonadST (STT m) where
--runStateST :: STT m a -> Store -> STT m (a, Store)
runStateST (MkSTT f) s = MkSTT $ \s' -> do
result <- f s
return (result,s')
```

```
--getST :: STT m Store
getST = MkSTT getSTT where
--getSTT :: Store -> m (Store, Store)
getSTT s = return (s,s)
```

```
--putST :: Store -> STT m ()
putST s = MkSTT putSTT where
--putSTT :: Store -> m ((), Store)
putSTT _ = return ((),s)
```

## Step 4: Preserving Old Features of Monads

Of course, we must make sure that the original features of the monads are not lost in the transformed monads. For this purpose, we will just use the `promote`

operation to directly transfer operations from the old monad into the transformed monad.

Thus, we can ensure that if a monad was already a state-manipulating monad, then the result of the exception-transformer is *also* a state-manipulating monad.

```
instance MonadExc m => MonadExc (STT m) where
-- throwErrorExc :: String -> STT m a
throwErrorExc = promote . throwErrorExc
-- :: String -> m a
```

```
instance MonadST m => MonadST (ExcT m) where
getST = promote getST
putST = promote . putST
-- runStateST :: ExcT m a -> Store -> ExcT m (a, Store)
-- m :: m (Exc a)
-- s :: Store
-- runStateST :: m a -> Store -> m (a,Store)
runStateST (MkExc m) s = MkExc $ do
(exc,s) <- runStateST m s
-- result :: m (Exc a,Store)
-- want :: m (Exc (a,Store))
return $ case exc of
Raise err -> Raise err
Result x -> Result (x,s)
```

## Step 5: Whew! Put together and Run

Finally, we can put all the pieces together and run the transformers. We could *order* the transformations differently (and that can have different consequences on the output as we will see.)

```
evalExSt :: Expr -> STT Exc Int
evalExSt = evalMega
evalStEx :: Expr -> ExcT ST Int
evalStEx = evalMega
```

which we can run as

```
goExSt :: Expr -> IO ()
goExSt e = putStr $ pr (evalExSt e) where
pr :: STT Exc Int -> String
pr (MkSTT f) = case (f 0) of
Raise s -> "Raise: " ++ s ++ "\n"
Result (v, cnt) -> "Count: " ++ show cnt ++ "\n" ++
"Result: " ++ show v ++ "\n"
```

```
goStEx :: Expr -> IO ()
goStEx e = putStr $ pr (evalStEx e) where
pr :: ExcT ST Int -> String
pr (MkExc f) = "Count: " ++ show cnt ++ "\n" ++ show r ++ "\n" where
(r, cnt) = runState f 0
```

```
*Main> goStEx ok
*Main> goStEx err
*Main> goExSt ok
*Main> goExSt err
```

# The Monad Transformer Library

While it is often *instructive* to roll your own versions of code, as we did above, in practice you should reuse as much as you can from standard libraries.

## Error Monads and Transformers

The above sauced-up exception-tracking version of `Maybe`

already exists in the standard type Either.

```
*Main> :info Either
data Either a b = Left a | Right b -- Defined in Data.Either
```

The `Either`

type is a generalization of our `Exc`

type, where the exception is polymorphic, rather than just being a `String`

. In other words the hand-rolled `Exc a`

corresponds to the standard `Either String a`

type.

The standard MonadError typeclass corresponds directly with `MonadExc`

developed above.

```
*Main> :info MonadError
class (Monad m) => MonadError e m | m -> e where
throwError :: e -> m a
catchError :: m a -> (e -> m a) -> m a
-- Defined in Control.Monad.Error.Class
instance (Monad m, Error e) => MonadError e (ErrorT e m)
-- Defined in Control.Monad.Error
instance (Error e) => MonadError e (Either e)
-- Defined in Control.Monad.Error
instance MonadError IOError IO -- Defined in Control.Monad.Error
```

Note that `Either String`

is an instance of `MonadError`

much like `Exc`

is an instance of `MonadExc`

. Finally, the `ErrorT`

transformer corresponds to the `ExcT`

transformer developed above and its output is guaranteed to be an instance of `MonadError`

.

## State Monads and Transformers

Similarly, the `ST`

monad that we wrote above is but a pale reflection of the more general State monad.

```
*Main> :info State
type State s = StateT s Data.Functor.Identity.Identity
-- Defined in Control.Monad.Trans.State.Lazy
```

The `MonadExc`

typeclass corresponds directly with the standard MonadState typeclass is the proper version

of our `MonadST`

rendered above.

```
*Main> :info MonadState
class (Monad m) => MonadState s m | m -> s where
get :: m s
put :: s -> m ()
-- Defined in Control.Monad.State.Class
instance (Monad m) => MonadState s (StateT s m)
-- Defined in Control.Monad.State.Lazy
instance MonadState s (State s)
-- Defined in Control.Monad.State.Lazy
```

Note that `State s`

is already an instance of `MonadState`

much like `ST`

is an instance of `MonadST`

. Finally, the `StateT`

transformer corresponds to the `STT`

transformer developed above and its output is guaranteed to be an instance of `MonadState`

.

Thus, if we stick with the standard libraries, we can simply write

```
tick :: (MonadState Int m) => m ()
tick = do { n <- get; put (n+1) }
```

```
eval1 :: (MonadError String m, MonadState Int m) =>
Expr -> m Int
eval1 (Val n) = return n
eval1 (Div x y) = do n <- eval1 x
m <- eval1 y
if m == 0
then throwError $ errorS y m
else do tick
return $ n `div` m
```

```
evalSE :: Expr -> StateT Int (Either String) Int
evalSE = eval1
```

```
*Main> runStateT (evalSE ok) 0
*Main> runStateT (evalSE err) 0
```

You can stack them in the other order if you prefer

```
evalES :: Expr -> ErrorT String (State Int) Int
evalES = eval1
```

which will yield a different result

```
*Main> runState (runErrorT (evalES ok)) 0
*Main> runState (runErrorT (evalES err)) 0
```

see that we actually get the division-count (upto the point of failure) even when the computation bails.

## Tracing Operations Via Logger Monads

Next, we will spice up our computations to also *log* messages (a *pure* variant of the usual method where we just *print* the messages to the screen.) This can be done with the standard Writer monad, which supports a `tell`

action that logs the string you want (and allows you to later view the entire log of the computation).

To accomodate logging, we juice up our evaluator directly as

```
eval2 :: (MonadError String m, MonadState Int m,
MonadWriter String m) =>
Expr -> m Int
eval2 v =
case v of
Val n -> do tell $ msg v n
return n
Div x y -> do n <- eval2 x
m <- eval2 y
if m == 0
then throwError $ errorS y m
else do tick
tell $ msg v (n `div` m)
return $ n `div` m
```

where the `msg`

function is simply

```
msg :: (Show a, Show b) => a -> b -> String
msg t r = "term: " ++ show t ++ ", yields " ++ show r ++ "\n"
```

Note that the only addition to the previous evaluator is the `tell`

operations! We can run the above using

```
evalWSE :: Expr -> WSE Int
evalWSE = eval2
```

where `WSE`

is a type abbreviation

`type WSE a = WriterT String (StateT Int (Either String)) a `

That is, we simply use the `WriterT`

transformer to decorate the underlying monad that carries the state and exception information.

```
*Main> runStateT (runWriterT (evalWSE ok)) 0
*Main> runStateT (runWriterT (evalWSE err)) 0
```

That looks a bit ugly, so we can write our own pretty-printer

```
instance Show a => Show (WSE a) where
show m = case runStateT (runWriterT m) 0 of
Left s -> "Error: " ++ s
Right ((v, w), s) -> "Log:\n" ++ w ++ "\n" ++
"Count: " ++ show s ++ "\n" ++
"Value: " ++ show v ++ "\n"
```

after which we get

```
*Main> print $ evalWSE ok
*Main> print $ evalWSE err
```

*How come we didn't get any log in the error case?*

The answer lies in the *order* in which we compose the transformers; since the error wraps the log, if the computation fails, the log gets thrown away. Instead, we can just wrap the other way around

```
type ESW a = ErrorT String (WriterT String (State Int)) a
evalESW :: Expr -> ESW Int
evalESW = eval2
```

after which, everything works just fine!

`*Main> evalESW err`

```
instance Show a => Show (ESW a) where
show m = "Log:\n" ++ log ++ "\n" ++
"Count: " ++ show cnt ++ "\n" ++
result
where ((res, log), cnt) = runState (runWriterT (runErrorT m)) 0
result = case res of
Left s -> "Error: " ++ s
Right v -> "Value: " ++ show v
```

## News :

Welcome to CIS 552!

See the home page for basic
information about the course, the schedule for the lecture notes
and assignments, the resources for links to the required software
and online references, and the syllabus for detailed information about
the course policies.