Melding Monads

In May, I gave a brief presentation to the Chicago Haskell User’s Group on modelling shared-state concurrency in a purely functional way without using the IO or STM monads. You can download a copy of the slides and Haskell source files. The slides may be of particular interest to those who are baffled by the continuation monad but are comfortable with the state monad.

The problem motivating the presentation is the Thread Game, attributed to J Strother Moore: given two threads each running the doubler pseudo-code below, with local variables x and y and a single shared global variable a initialized to 1, what values can appear in the global variable? Can you prove your answer?

doubler = loop
            x := a
            y := a
            a := x + y
          end

It should be noted that no synchronization is used, that each line represents an atomic instruction, and that arbitrary schedules are allowed. (e.g. they need not be fair schedules)

The talk covered two implementations of a function from finite schedules to the final value of a, providing a model of the Thread Game. The first implementation used a relatively straightforward technique reminiscent of automata and Turing Machines from theoretical computer science. The local configuration for a thread consists of the local variables plus an instruction pointer. The delta function step_doubler takes a local configuration and the current value of the global variable, and produces a new local configuration and value for the global variable.

type IP     = Int       -- Instruction Pointer
type Local  = Integer   -- Local Variable
type Global = Integer   -- Global Variable

type LocalConf  = (IP, Local, Local)

step_doubler :: (LocalConf,Global) → (LocalConf,Global)
step_doubler ( (ip,x,y) , a )
   = case ip of
      0 → ( (1,a,y) , a   )
      1 → ( (2,x,a) , a   )
      2 → ( (0,x,y) , x+y )

The global configuration consists of the value of the global variable, and one local configuration for each doubler process. The global step function takes the global configuration and a process identifier, and produces another global configuration.

type PID        = Int    -- Process Identifier
type GlobalConf = (LocalConf, LocalConf, Global)

step_global :: GlobalConf → PID → GlobalConf
step_global (d0, d1, a) pid
  = case pid of
     0 → let (d', a') = step_doubler (d0, a) in (d', d1, a')
     1 → let (d', a') = step_doubler (d1, a) in (d0, d', a')

The function from schedules to the final value in a is just a left fold, resulting in the first model of the thread game:

threadGame :: [PID] → Integer
threadGame pids = a 
  where
    ( _, _, a ) = foldl step_global ( (0,0,0), (0,0,0), 1 ) pids

While this technique is generic and relatively straightforward, the code itself is not generic, and is fairly far removed from the original specification. It would seem natural to model doubler using the state monad as shown below. The state monad can handls the local variables and the global variable, but cannot represent the instruction pointer or offer the co-routine facility needed to model the given problem.

doubler = do
    x ← get
    y ← get
    put (x+y)
    doubler

The continuation monad can represent the instruction pointer, but because Haskell is a pure language, further tricks are needed to mediate between the thread scheduler and each doubler process.

Iteratee-based Shared-State Concurrency

The second implementation used an idea derived from Iterators as shown to me by Roshan James. These are a more generic variant of to Oleg’s Iteratees. Iteratees are something of a hot topic these days, being a core feature of the Snap Framework and the subject of assorted blog posts.

data Iterator i o r
   = Done r
   | Result o (i → Iterator i o r)

The idea here is that an iterator is either done, and returns a final result (of type r), or it might want to produce more, so it returns an intermediate result (of type o) along with a continuation to be called if and when further computation is required. Interestingly, Stephen Blackheath also used an equivalent structure in today’s post, Co-routines in Haskell.

Basically, each thread is an Iteratee that accepts the value in the global variable as an input, and produces a new value for the global value and it’s own continuation as output. This continuation represents the local configuration, which consists of the instruction pointer and the two local variables. The thread scheduler manages these continuations and feeds the global variable into the corresponding continuation.

I had (almost) invented iteratees myself a number of years ago, although I didn’t realize the connection until after I had begun to write this blog post. Back in the year 2000, I helped write a paper about implementing finite automata and Turing machines in functional languages by directly using the standard textbook definition. In the process I stumbled on an alternate representation of a DFA that I did not publish, mentioned briefly here.

data DFA st ab = DFA (st → ab → st) st (st → Bool)

data ContDFA ab = ContDFA Bool (ab → ContDFA ab)

The first representation is the standard textbook definition, a automaton consists of a set of states st, an alphabet ab, and a transition function, a start state, and (the characteristic function of) a set of final states. Interestingly, this representation was revisited in the most recent issue of the Monad Reader.

The second representation is a specialized case of Iterator above, without the Done alternative. Basically, the ContDFA represents a single state in the machine, which has a boolean field saying whether or not the state is final, and a method that accepts a letter and returns another state.

When I was first exposed to Iterators, I was confused by them and the way my friend was using them. Funnily enough, I completely missed the connection to my ContDFA invention, even though I was immediately reminded of Mealy Machines and thought of that project! In a way, it was a fortuitous slip of the mind. While thinking about implementing Mealy Machines with continuations in an effort to understand Roshan’s work, I came up with “functional iterators”. (If anybody can suggest a better name, please speak up!)

newtype FunIter i o r
      = FunIter (FunIter o i r → i → r)

Compared to Iterator, FunIter assumes that the consumer of the intermediate values is itself written in the Continuation Passing Style. I don’t understand the exact relationship between Iter and Iterator; they are remarkably similar, but there are trade-offs between what these two structures can represent, trade-offs I don’t understand very well.

Both Iterator and FunIter are capable of expressing the coroutine facility needed to model the thread game, however, my Iterator solution has a few vestigial features, and I like my FunIter solution a bit better.

newtype FunIter i o r
      = FunIter {runIter :: FunIter o i r → i → r}

type Thread r st a = Cont (FunIter st st r) a

runThread :: Thread r st r → FunIter st st r
runThread m = runCont m (\r → FunIter (\_ _ → r))

I used a continuation monad to capture the local continuations, while FunIter mediates the transition between the local continuations and the scheduler continuation. We can then define get and put, which yield to the scheduler continuation:

get    = Cont (\k → FunIter (\(FunIter sk) st →  sk (k st) st))
put st = Cont (\k → FunIter (\(FunIter sk) _  →  sk (k ()) st))

Here is a more generic thread scheduler:

update :: Int → a → [a] → [a]
update n x [] = []
update n x (y:ys) 
  | n <= 0    =  x : ys 
  | otherwise =  y : update (n-1) x ys

runSchedule :: [FunIter st st st] → [PID] → st → st
runSchedule threads []          st = st
runSchedule threads (pid:pids)  st = runIter (threads !! pid) sched st
   where
     sched = FunIter $ \iter st' → 
               runSchedule (update pid iter threads) pids st'

And finally, we can bring everything together into an alternate model of the thread game:

threadGame' :: [PID] → Integer
threadGame' sched = runSchedule [runThread doubler, runThread doubler] sched 1

I don’t know who first invented Iteratees, but the idea has been lurking in a variety of different contexts for some time. The functional iterators presented here might be original. The second solution is a bit longer and more esoteric, but it’s more generic and should scale better in terms of source code and maintenance.

I have four draft posts started that I don’t have finished. But I won’t be finishing any of them today, instead I’ll issue an erratum for my paper in Issue 14 of the Monad Reader. On page 53 I wrote:

Writer monads partially enforce the efficient use of list concatenation. In the MTL, Writer [e] a is just a newtype isomorphism for ([e], a). It provides a function tell that takes a list and concatenates the remainder of the computation onto the end of the list. This naturally associates to the right, and thus avoids quadratic behavior. Of course, tell accepts arbitrary length lists, and one could inefficiently produce a long argument to tell.

The writer monad does no such thing. In fact, it’s quite easy to write a function using the writer monad that exhibits quadratic behavior. For example here is a function that uses the writer monad to produce the list [1..n] in quadratic time:

enumTo n
  | n <= 0    = return ()
  | otherwise = enumTo (n-1) >> tell [n]

By unfolding the definitions of return and (>>=) for Control.Monad.Writer, and the Monoid instance for List, we can find this is equivalent to:

enumToWriter n
  | n <= 0    = ((), [])
  | otherwise = let (a,w ) = enumToWriter (n-1) 
                    (b,w') = (\_ -> ((),[n])) ()
                 in (b,w ++ w')

By beta-reducing the let bindings and making use of snd, we get

enumToWriter'2 n
  | n <= 0    = ((), [])
  | otherwise = ((), snd (enumToWriter'2 (n-1)) ++ [n])

And finally, we can hoist the pairs out of the loop, if we want:

enumToWriter'3 n = ((), loop n)
  where
    loop n
      | n <= 0    = []
      | otherwise = loop (n-1) ++ [n]

It should now be clear that this is quadratic, as the first element of the result gets copied (n-1) times, the second element gets copied (n-2) times, and so on. However, there is a solution. By adjusting the monad, the original definition can be executed in linear time. My erroneous quote is actually an accurate description the continuation-based writer monad.

 
instance (Monoid w) => MonadWriter w (Cont w) where
   tell xs = Cont (\k -> xs `mappend` k ())
   -- or,  in a direct style:
   -- tell xs = mapCont (xs `mappend`) (return ())

The expression runCont (enumTo n) (const []) produces the list [1..n] in linear time. The continuation-based writer monad provides the linear-time guarantee that I described but erroneously attributed to the regular writer monad. By unfolding the definition of return and (>>=) for Control.Monad.Cont and our definition for tell above, we can find out what this algorithm actually does:

-- unfold return and (>>=)
enumToCont n 
  | n <= 0    = k ()
  | otherwise = \k -> enumToCont (n-1) (\a  -> (\_ -> tell [n]) a k) 
-- beta-reduction
--            = \k -> enumToCont (n-1) (\_a -> tell [n] k)
-- unfold tell
--            = \k -> enumToCont (n-1) (\_  -> (\k -> [n] ++ k ()) k)
-- beta-reduction
--            = \k -> enumToCont (n-1) (\_  -> [n] ++ k ())
-- unfold (++)
--            = \k -> enumToCont (n-1) (\_  -> n : k ())

And, if we like, we can avoid passing () to the continuation argument, resulting in a normal accumulator argument:

enumToCont' n k
  | n <= 0    = k
  | otherwise = enumToCont' (n-1) (n:k)

By writing it with (:), we know it has to associate to the right and thus operates in linear time. In the quadratic-time solution, the (++) cannot be unfolded. However, this solution isn’t incremental: trying to evaluate take 5 (enumToCont' (10^12) []) will more than likely run your computer completely out of virtual memory long before it finishes. It’s not to difficult to see why: enumFromCont' produces a list element equal to it’s input, and it has to count down from a trillion before it produces [1,2,3,4,5, ...]. So it really is better to write it as:

enumTo' lim = loop 1
  where
     loop n 
       | n <= lim  = tell [n] >> loop (n+1)
       | otherwise = return ()

This definition is both linear and incremental under both the continuation-based writer and the normal writer monad, but it isn’t a generic transformation that applies to any algorithm using the writer monad.

These two writer monads have a number tradeoffs; for example, the continuation-based writer monad cannot be used with the corecursive queue transformer in the paper. An efficient alternative is the regular writer monad with a list representation that offers efficient concatenation, such as difference lists.

For further reading, you might enjoy Asymptotic Improvement of Computations over Free Monads by Janis Voigtländer, which explores this phenomenon further.

So Issue 14 of the Monad Reader is out, which includes my paper “Lloyd Allison’s Corecursive Queues” with a few significant revisions. Compared to earlier versions mentioned on this blog, I moved a few sections around to improve the flow: the first 8 pages now contain an uninterrupted informal derivation of the queue operators. I also demonstrated a new fixpoint operator on continuations, an implementation of the standard mfix fixpoint on the codensity monad, and argued that mapCont cannot be implemented in terms of callCC.

However, this post focuses on the section that I moved, entitled “Monad Transformers” from pages 51 to 56. It’s basically a short rant about why I don’t like monad transformers, an argument that later culminates in a mostly broken corecursive queue transformer. In retrospect, it’s somewhat unfortunate that I did not reread the classic paper Monad Transformers and Modular Interpreters sometime before Issue 14 came out. I did read that paper approximately nine or ten years ago. Although the paper was helpful in understanding how monads could be shaped to my own ends, now that I actually understand the contents of the paper, it feels rather crufty.

Section 8.1 defines correctness criteria for a monad transformer and associated lifted operations, which I quote as follows:

The basic requirement of lifting is that any program which does not use the added features should behave in the same way after a monad transformer is applied.

The thrust of my argument is that this requirement is indeed very basic; one would hope that certain properties useful for reasoning inside a given monadic language would also be preserved. This additional requirement seems rather hopeless. However, the pièce de résistance of the argument is that the continuation transformer is incorrect by their own criteria, at least in the context of a lazy language such as Haskell or Gofer.

I demonstrate an expression that depends on the laziness of the lazy state monad, and fails to terminate after a continuation transformer is applied. (As an aside, it doesn’t matter if this is a ContT r (StateT st Identity) monad or StateT st (ContT r Identity), they are the same monad with the same operations.) In retrospect, this seems obvious: something written in the continuation passing style specifies an evaluation order independent of the host language, and applying the continuation transformer corresponds to a call-by-value CPS transformation of part of the program.

The example involves a definition that computes a right fold over a level-order traversal of a tree:

foldrByLevel :: (MonadState (Queue a) m)
             => (a -> [a])
             -> (a -> b -> b) -> b -> [a] -> m b 
foldrByLevel childrenOf f b as = fold as 
  where
    fold []     = deQ >>= maybe (return b)
                          (\a -> fold (childrenOf a))
    fold (a:as) = do 
                     enQ a 
                     b <- fold as
                     return (f a b)

If we use this definition to traverse an infinite tree, it will be productive when run with Control.Monad.State.Lazy, but will get stuck in an infinite non-productive loop when combined with Control.Monad.Cont. You can download a complete file that demonstrates this phenomenon. There are “simpler” expressions that demonstrate this, but the example I gave is itself interesting because it is useful in other contexts.

As demonstrated in my paper, the laziness can be restored by tweaking the definition of foldrByLevel to use mapCont. However, this is a leaky abstraction: in order to maintain the same behavior, we must modify existing code to make use of the “added” features in ways that are not backwards compatible. I do not know how to write a lazy state monad using continuations, or a sensible way of writing a single definition of foldrByLevel that behaves the same on both the lazy state monad and the continuation state monad.

(I use the word “sensible” because one could provide a unfaithful definition of mapCont for the lazy state monad that happens to work in the case of foldrByLevel, but fails in other contexts.)

What impact does this have on the notion that continuations are the mother of all monads? Is there a monad that corresponds to a call-by-name CPS transformation? Is it even possible to express the lazy state monad using continuations?

I do not yet have answers to these questions. One thing is clear, however: its tricky to generalize Filinski’s Representing Monads to lazy evaluation, if indeed it is possible to do so fully.