The reasonable effectiveness of the continuation monad

Posted on October 26, 2019

There are common monads associated with common effects: Maybe for failure, [] (list) for nondeterminism, State for state… What about the continuation monad? We shall see why the answer is all of the above, but better. Indeed, many effects can be understood and implemented in a simple and uniform fashion in terms of first-class continuations.

Extensions and imports for this Literate Haskell file {-# LANGUAGE InstanceSigs, RankNTypes #-} module Continuations where import Control.Applicative ((<|>)) (()) import Control.Monad ( replicateM , when ) import Data.Foldable ( for_ )

A way too short introduction to continuation-passing style

The key insight behind continuations is that producing a result in a function is equivalent to calling another function which does the rest of the computation with that result.

In this small starting example, we apply some function timesThree , and compare the result to 10. We will transform this code in continuation-passing style.

example1 :: Int -> Bool example1 x = 10 < timesThree x where timesThree :: Int -> Int timesThree x = 3 * x

As our first step, following the train of thought above, instead of taking the result of timesThree and doing something ( 10 < _ ) with it, let timesThree do that operation directly.

example2 :: Int -> Bool example2 x = timesThree x where timesThree :: Int -> Bool timesThree x = 10 < 3 * x

Of course, that’s not much of a timesThree function anymore. Moreover, we know how 3 * x is going to be used in this case, but that’s quite counter to modularity. Let us generalize timesThree : instead of hard-coding 10 < _ , we parameterize timesThree by the context in which the result 3 * x will be used. That context is called the continuation k .

example3 :: Int -> Bool example3 x = timesThree x (\ y -> 10 < y ) where (\ timesThree :: Int -> ( Int -> Bool ) -> Bool timesThree x k = k ( 3 * x )

Furthermore, the result of the continuation doesn’t have to be of type Bool ; we can generalize the type of timesThree further to also be parameterized by the result type r of the continuation. In the main body where we apply timesThree , r is specialized to the type of the final result, which is Bool .

example4 :: Int -> Bool example4 x = timesThree x (\ y -> 10 < y ) where (\ timesThree :: Int -> ( Int -> r ) -> r timesThree x k = k ( 3 * x )

That was continuation-passing style (CPS) in a nutshell.

Functions written in CPS can be composed as follows. Let us refactor the comparison 10 < _ into another CPS function greaterThanTen . Once the program is entirely written in CPS, the identity function (here \ z -> z ) is commonly used as the last continuation, which receives the final result.

example5 :: Int -> Bool example5 x = timesThree x (\ y -> (\ greaterThanTen y (\ z -> (\ z )) )) where timesThree :: Int -> ( Int -> r ) -> r timesThree x k = k ( 3 * x ) greaterThanTen :: Int -> ( Bool -> r ) -> r greaterThanTen y k = k ( 10 < y )

Hey, this example looks a lot like do notation… Indeed, note how we changed the result type of timesThree from Int to (Int -> r) -> r ; that mapping between types (_ -> r) -> r defines a monad.

The Cont monad

(The descriptions in this section are principally meant to provide context if you’ve never seen the implementation of Cont before, but they may be quite dense. It’s not necessary to follow every single detail to catch the rest, so skipping forward is an option.)

A function of type ((a -> r) -> r) takes a continuation (a -> r) and is expected to produce a result r . The obvious way to do that is to apply the continuation to a value a , which is exactly the idea behind continuations given at the beginning. In fact that is also what it means to “return” a value in this monad ( pureCont below; the instances are collapsed at the end of this section). As we will soon see, the power of the continuation monad hides in the myriad other ways of using that continuation.

newtype Cont r a = Cont (( a -> r ) -> r ) (( -- Eliminate Cont runCont :: Cont r a -> ( a -> r ) -> r runCont ( Cont m ) = m -- Use the identity continuation to extract the final result. evalCont :: Cont a a -> a evalCont ( Cont m ) = m id pureCont :: a -> Cont r a pureCont a = Cont (\ k -> k a ) (\

The bind (>>=) of the monad captures the pattern in example5 above to compose two CPS functions. We start with a continuation (k :: b -> r) for the whole computation ( Cont r b ). We first apply ma , with a continuation which takes the result a of ma , and passes it to mc , which in turn produces a b that is just what k wants.

bindCont :: Cont r a -> ( a -> Cont r b ) -> Cont r b bindCont ( Cont ma ) mc_ = Cont (\ k -> (\ ma (\ a -> (\ mc a (\ b -> (\ k b ))) ))) where mc = runCont . mc_

The instances of Functor , Applicative , Monad for Cont . instance Functor ( Cont r ) where fmap :: ( a -> b ) -> Cont r a -> Cont r b fmap f ( Cont m ) = Cont (\ k -> m ( k . f )) (\)) instance Applicative ( Cont r ) where pure :: a -> Cont r a pure = pureCont (<*>) :: Cont r ( a -> b ) -> Cont r a -> Cont r b Cont mf <*> Cont ma = Cont (\ k -> (\ mf (\ f -> (\ ma (\ a -> (\ k ( f a )))) )))) instance Monad ( Cont r ) where (>>=) :: Cont r a -> ( a -> Cont r b ) -> Cont r b ( >>= ) = bindCont

We can thus rewrite the example using do -notation for Cont .

example6 :: Int -> Bool example6 x = evalCont $ do y <- timesThree x z <- greaterThanTen y pure z where timesThree :: Int -> Cont r Int timesThree x = Cont (\ k -> k ( 3 * x )) (\)) greaterThanTen :: Int -> Cont r Bool greaterThanTen y = Cont (\ k -> k ( 10 < y )) (\))

Continuation transformers

Here is another way to look at monadic composition of Cont . If we unfold the definition of Cont , a continuation in the continuation monad, a -> Cont r b , is really a function mapping continuations to continuations, we shall call that a continuation transformer: (b -> r) -> (a -> r) . They map “future” continuations to “present” continuations.

This suggests to take a look at the fish operator, which composes monadic continuations.

(>=>) :: Monad m => ( a -> m b ) -> ( b -> m c ) -> ( a -> m c ) (>=>) :: ( a -> Cont r b ) -> ( b -> Cont r c ) -> ( a -> Cont r c )

Looking at the type of (>=>) :

unfold the definition of Cont r b to (b -> r) -> r , swap the arguments of each function a -> (b -> r) -> r to (b -> r) -> (a -> r) .

The result shows that sequencing in the Cont monad (with (>=>) ) is basically function composition. The function f >=> g :: a -> Cont r c takes a continuation c -> r , passes it to the function g to produce a continuation b -> r , which goes into f to produce a continuation a -> r (note that continuations do flow from right to left in f >=> g ).

( >=> ) :: ( a -> Cont r b ) -> ( b -> Cont r c ) -> ( a -> Cont r c ) {- 1 -} ( a -> ( b -> r ) -> r ) -> ( b -> ( c -> r ) -> r ) -> ( a -> ( c -> r ) -> r ) {- 2 -} (( b -> r ) -> a -> r ) -> (( c -> r ) -> b -> r ) -> (( c -> r ) -> a -> r ) (((((( ( y -> x ) -> ( z -> y ) -> ( z -> x ) ( . ) :: ( y -> x ) -> ( z -> y ) -> ( z -> x )

Many monads in one

In spite of (or thanks to) its simplicity, the Cont monad is quite versatile. Many kinds of effects can be represented in Cont , all with only the one Monad instance given above, that knows nothing about effects.

In contrast with free monads, which are just waiting to be interpreted, we can define an effect directly by its operations in Cont .

The main idea is to consider what operations we allow on continuations. Here we describe various restrictions through the result type r in (a -> r) -> r , but there may be other ways.

Identity

Our starting point was that producing a result is equivalent to calling the continuation. If we add the constraint that the result type r is abstract, so that there are no operations possible on it, then calling the continuation with some argument a is the only option, i.e., we must produce a result a , nothing else. In that case, the continuation monad is isomorphic to the identity monad.

To express the restriction that r is abstract, we can use the forall quantifier. If no operations are possible on r , then r could actually be any type. So Cont computations defined under that restriction are polymorphic in r . We name Done the resulting “specialization” of Cont (if we may call it one), with an isomorphism given by pure :: a -> Done a from the Applicative instance above, and runDone :: Done a -> a below.

type Done a = forall r . Cont r a -- forall r. (a -> r) -> r runDone :: Done a -> a runDone ( Cont m ) = m id

Maybe

The next interesting case to consider is that the continuation may be dropped. But, (a -> r) -> r must still somehow produce a result of type r . We thus replace r with Maybe r , so that a computation can produce Nothing instead of calling the continuation ( abort below). As you might expect, the result is a monad which models computations that can exit early with no output, i.e., a variant of the Maybe monad,

Although the type Maybe appears in this definition, the fact that it is a monad is not used anywhere. In fact, whereas monadic composition (>>=) for Maybe is defined with pattern-matching, there is not a single case in the operations for Abortable defined here. Each constructor is only used once, Nothing when aborting the computation, Just as the final continuation to indicate success. So Abortable does not just imitate Maybe , it is even more efficient!

type Abortable a = forall r . Cont ( Maybe r ) a -- forall r. (a -> Maybe r) -> Maybe r abort :: Abortable x abort = Cont (\ _k -> Nothing ) (\ runAbortable :: Abortable a -> Maybe a runAbortable ( Cont m ) = m Just

Contrary to Done and Identity , Abortable is not isomorphic to Maybe . Whereas a Maybe computation must decide to be Just or Nothing on the spot, an Abortable is a function (a -> Maybe r) -> Maybe r , which may inspect the continuation before making a decision, even though “intuitively” it’s not supposed to.

Thus we can construct a computation secondGuess which is expected to return a Bool , calls the continuation with True (like pure True would) but backtracks to False if that fails.

secondGuess :: Abortable Bool secondGuess = Cont (\ k -> k True <|> k False ) (\ pureTrue :: Abortable Bool pureTrue = pure True

runAbortable maps both secondGuess and pureTrue to Just True , but they behave differently with a continuation which fails on True and succeeds on False .

Nevertheless, it is not possible to construct examples such as secondGuess with only the monad operations and abort ; you have to break the Abortable abstraction. In that sense, Abortable is still a practical alternative to the Maybe monad.

Either

Naturally, a slight variant of “exit early” is to “exit early with an explicit error”, obtained by replacing Maybe with Either e .

type Except e a = forall r . Cont ( Either e r ) a -- forall r. (a -> Either e r) -> Either e r throw :: e -> Except e a throw e = Cont (\ _k -> Left e ) (\ runExcept :: Except e a -> Either e a runExcept ( Cont m ) = m Right

State

What if the continuation takes an extra parameter, with result type s -> r ? Then we may want to call it with different parameters, resulting in a notion of stateful computation.

Remember that the result type s -> r is both the result type of the continuation, and of the whole computation ( (a -> s -> r) -> s -> r ). The whole computation can just call the continuation (with some value a ) to produce a result s -> r , or it can first take the parameter s , and obtain an r by calling the continuation with a different state.

Thus get takes that parameter s , and feeds it twice to the continuation k :: s -> s -> r , keeping the state (second argument) unchanged, but also giving it as the main (first) argument of the subsequent computation. The other function, put ignores that parameter, and calls the continuation k :: () -> s -> r with another state given externally.

type State s a = forall r . Cont ( s -> r ) a -- forall r. (a -> s -> r) -> s -> r get :: State s s get = Cont (\ k s -> k s s ) (\ put :: s -> State s () () put s = Cont (\ k _s -> k () s ) (\() runState :: State s a -> s -> ( a , s ) runState ( Cont m ) = m (,) (,)

That State is isomorphic to the standard definition, s -> (s, a) . Indeed, contrary to Abortable , there is no observation to be made about the continuation a -> s -> r when r is abstract.

As with Maybe / Either , there is no pattern-matching on pairs going on. The s and the a are always just two arguments to the continuation, and a pair gets built up only in the final continuation in runState .

Writer

Are we running out of ideas for what to put in Cont _ a ?

Above we tried r , Either e r (sums!), s -> r (exponentials!). Surely we should also try products. The result is not quite nice, because to do anything with the pair we have to break the property that was maintained until now: that the continuation k is the last thing we call.

We can tell an element of a monoid, by appending it in front of whatever the rest of the computation outputs.

type Writer w a = forall r . Cont ( w , r ) a -- forall r. (a -> (w, r)) -> (w, r) tell :: Monoid w => w -> Writer w () () tell w = Cont (\ k -> (\ let ( w0 , r ) = k () in ( w <> w0 , r )) ())) runWriter :: Monoid w => Writer w a -> ( w , a ) runWriter ( Cont m ) = m (\ a -> ( mempty , a )) (\))

State , reversed

Cont (w, r) can also be viewed as a variant of State . Instead of treating w as a monoid, we can let the user update it however they want. However, that update happens after the rest of the computation is done, so the last update (in the order they would appear in a do block for example) is applied first to the initial state. This is the reverse state monad, where modifications map the future state to the past state.

Getting the current state in the RState monad requires recursion: the current state comes from the future (the continuation), which is asking for the current state itself. With this rget operation, you have to be careful not to introduce any causality loop and accidentally tear down the fabric of reality.

Compare our RState with a more conventional definition of it as s -> (a, s) . There, recursion is used in the definition of (>>=) , while get is trivial, which is a situation opposite to RState .

type RState s a = forall r . Cont ( s , r ) a -- forall r. (a -> (s, r)) -> (s, r) rmodify :: ( s -> s ) -> RState s () () rmodify f = Cont (\ k -> (\ let ( s , r ) = k () in ( f s , r )) ())) rget :: RState s s rget = Cont (\ k -> (\ let ( s , r ) = k s in ( s , r )) )) runRState :: RState s a -> s -> ( s , a ) runRState ( Cont m ) s = m (\ a -> ( s , a )) (\))

Tardis

We can combine State , given by s -> r , and RState , given by (s, r) : instead, if we make the continuation result type s -> (s, r) , we obtain a Tardis monad, with one state going forward in time, and one going backwards.

The forward and backward states don’t actually have to be the same, so we can also generalize (s -> (s, r)) into (fw -> (bw, r)) .

type Tardis bw fw a = forall r . Cont ( fw -> ( bw , r )) a )) -- forall r. (a -> fw -> (bw, r)) -> fw -> (bw, r)

List

One last standard type we haven’t tried for r is the type of lists. In our previous examples, computations called the continuation only once (or at least they should, we can exclude secondGuess as a degenerate example). Equipping the result type with the structure of lists, we can call a continuation multiple times, and return a combination of all the results.

This provides a model of nondeterministic computations, keeping track of all possible executions, which is the same interpretation as the standard list [] monad.

decide chooses both True and False , i.e., it calls the continuation on both booleans, and concatenates the results together. vanish chooses nothing, it drops the continuation like abort .

type List a = forall r . Cont [ r ] a -- forall r. (a -> [r]) -> [r] decide :: List Bool decide = Cont (\ k -> k True ++ k False ) (\ vanish :: forall a . List a vanish = Cont (\ _k -> []) (\[]) runList :: List a -> [ a ] runList ( Cont m ) = m (\ a -> [ a ]) (\])

There’s a handful of variations for that one. Use NonEmpty r to rule out vanish ; generalize over an abstract monoid or semigroup r to prevent inspection of the continuation; or use a Tree r to keep track of the order of choices.

type List1 a = forall r . Cont ( NonEmpty r ) a type List' a = forall r . Monoid r => Cont r a type List1' a = forall r . Semigroup r => Cont r a type Tree0 a = forall r . Cont ( Tree r ) a

ContT

There is also a continuation monad transformer, which is simply the continuation monad with a monadic result type m r . The transformers library defines ContT as a newtype mostly so that it has the right kind to be an instance of MonadTrans . All instances stay the same, so here we will prefer a type synonym to keep our Monad instance count at 1. We will refer to ContT and Cont interchangeably, as we’re not too concerned about kinds in this post, whichever looks better in context.

type ContT r m a = Cont ( m r ) a -- (a -> m r) -> m r

What does it mean that ContT is a monad transformer? There is a lift function, which commutes with monadic operations (that’s called a monad morphism). For ContT , lift is simply (>>=) ,

lift :: Monad m => m a -> ContT r m a -- Monad m => m a -> (a -> m r) -> m r lift u = Cont (\ k -> u >>= k ) (\ -- Monad morphism laws: -- lift (pure a) = pure a -- lift (u >>= \ a -> k a) = lift u >>= \ a -> lift (k a)

CodensityT

A closely related sibling is the “codensity” monad transformer, where r is universally quantified, like it is in previous examples. Both ContT and CodensityT can be used to optimize monads that have expensive bind (>>=) operations. We won’t say anything here about the actual differences between ContT and CodensityT .

type CodensityT m a = forall r . Cont ( m r ) a -- forall r. (a -> m r) -> m r

In the examples above, the types we used instead of r happen to be monads, even if we did not rely on that fact. Here’s a quick summary, with the names of the resulting variant of Cont on the left, an equivalent definition in terms of CodensityT in the middle, and their more-or-less standard counterparts on the right as they can be found on Hackage (base, transformers, rev-state and tardis). The words “retracts to” mean that there is a surjective but not injective mapping from the left to the right.

Done = CodensityT Identity isomorphic to Identity Abortable = CodensityT Maybe retracts to Maybe Except e = CodensityT (Either e) retracts to Either e State s = CodensityT (Reader s) isomorphic to State s Writer w = CodensityT (Writer w) retracts to Writer w, or (reverse) State w Tardis s = CodensityT (State s) retracts to Tardis s List = CodensityT [] retracts to []

Many monad transformers in one

The monad transformers corresponding to the above monads also find their equivalent in terms of Cont . They are not exactly isomorphic, but a noteworthy feature, as before, is that they still use the same old Monad instance for Cont . Operations do rely on a Monad constraint for the transformed monad m .

ListT

Turning the previous examples into monad transformers is left as an exercise for the reader.

Here we will focus on List ; it is an interesting case because a monad transformer corresponding to lists is notoriously non-obvious. The obvious candidate m [a] is not a monad (unless m is commutative).

Curiously, we have the “monad” part down for free, and we only need to solve “list” and “transformer”.

We briefly saw earlier that we can get a “list” monad by using any monoid instead of [r] as the result type. We also saw that a monadic result type m r makes a monad transformer. In addition, any monad defines a monoid m () if we ignore the result (we can also use a different monoid instead of () but that doesn’t seem as interesting), with pure () as the unit and (*>) (or (>>) ) for composition. In fact, we only need an Applicative constraint for the “list” operations, but lift still requires Monad .

We already had all the ingredients to make a list monad transformer!

Reading the definition of ListT slowly, it takes a continuation (a -> m ()) , and produces a computation m () . What can it actually do? Mostly, call the continuation with various values of a in some order.

type ListT m a = Cont ( m ()) a ()) -- (a -> m ()) -> m () decideM :: Applicative m => ListT m Bool decideM = Cont (\ k -> k True *> k False ) (\ vanishM :: Applicative m => ListT m a vanishM = Cont (\ _k -> pure ()) (\()) runListT :: Applicative m => ( a -> m ()) -> ListT m a -> m () ())() runListT k ( Cont m ) = m k

The list transformer is a nice pattern for deeply nested loops common in enumeration/search algorithms.

Here are three nested for_ loops:

-- All 3 bit patterns threebit :: IO () () threebit = for_ [ 0 , 1 ] $ \ i -> for_ [ 0 , 1 ] $ \ j -> for_ [ 0 , 1 ] $ \ k -> printDigits [ i , j , k ] printDigits :: [ Int ] -> IO () () printDigits ds = do for_ ds (\ i -> putStr ( show i )) (\)) putStrLn ""

Here they are again, where each value is bound using do notation thanks to the list transformer (this combination is really neat: Cont $ for_ [ ... ] ).

-- All 3 bit patterns threebit' :: IO () () threebit' = runListT printDigits $ do i <- Cont $ for_ [ 0 , 1 ] j <- Cont $ for_ [ 0 , 1 ] k <- Cont $ for_ [ 0 , 1 ] pure [ i , j , k ]

Once iteration is captured in a monad, we can iterate across dimensions:

-- All 8 bit patterns eightbit :: IO () () eightbit = runListT printDigits $ replicateM 8 ( Cont ( for_ [ 0 , 1 ])) ])) -- 00000000 -- 00000001 -- 00000010 -- 00000011 -- 00000100 -- 00000101 -- 00000110 -- ...

All of that is technically possible with just the list monad. The transformer really adds the ability to interleave enumeration and computation.

-- All 8 bit patterns, but show only the suffix that changed at every step. eightbit' :: IO () () eightbit' = runListT pure $ do for_ [ 0 .. 7 ] $ \ n -> do i <- Cont $ for_ [ 0 , 1 ] lift $ when ( i == 1 ) $ putStr ( replicate n ' ' ) lift $ putStr ( show ( i :: Int )) )) lift $ putStrLn "" -- 00000000 -- 1 -- 10 -- 1 -- 100 -- 1 -- 10 -- ...

Other list monad transformers

This “list monad transformer” is actually different from another incarnation which may be found on Hackage. The more common version of a “list monad transformer” is an “effectful list”, where the list constructors are interleaved with computations.

newtype ListT m a = ListT ( m ( Maybe ( a , ListT m a ))) )))

The biggest difference is that the “effectful list” transformer naturally supports an uncons operation, which evaluates the effectful list and pauses after producing the first element (or the empty list).

The trade-off is that uncons has a cost in usability. The paused computation must be resumed explicitly: it may be dropped, or resumed more than once. The continuation transformer, by not allowing such “interruptions”, may offer stronger guarantees for resource management.

Conclusion

The continuation monad can thus serve as a uniform foundation for many kinds of monadic effects, and is even often a more efficient replacement of “standard” monads.

“Control operations” might cause some difficulties; those are operations parameterized by computations, such as catch and bracket ; they weren’t discussed here, but I think the problems can be overcome.

See also