Thoughts on FRP

(N.B. Code presented in this post is theoretical.  I haven't tried to compile it, although I do expect it would work.)

Although various implementations have been under development for some time, FRP has yet to reach a really satisfying breakthrough.  The best semantic model has been that of Conal Elliot's Reactive, but unfortunately it lacks a Smart Enough Compiler and suffers from subtle space leaks.  The most performant is probably Yampa, which is arrow-based.  Arrows turn some people off; I personally find them too verbose and don't like the syntax for recursion.

Apfelmus has recently released a new FRP library, reactive-banana, which shows some early promise towards being a best-of-all-possible-worlds.  In this post I'll examine some of the features of reactive-banana to see why I think this.

Introduction

Reactive-banana is very similar to the semantic model provided by Reactive, although much more minimal.  One fundamental data type is the Event, which can be thought of as a list of values with their time of occurrence, i.e. Event a = [(Time, a)].  The other fundamental type is the Behavior, which can be thought of as a value that changes with time, i.e. Behavior a = \Time -> a.  This is according to the reactive-banana documentation, which also notes that in this implementation Behaviors aren't continuous but are actually step functions.

So how does it look?  Consider a simple system with a button and a toggle: when the button is pressed it generates a 0 if the toggle is off, and 1 when the toggle is on.  We could model it like this:

togEvent :: Event Bool
buttonEvent :: Event ()

-- Upon receiving an Event (), mixEvents generates an Event Int
-- The value of the Int depends upon the most recent Event Bool status
mixEvents :: Event Bool -> Event () -> Event Int

The togEvent and buttonEvent sources are created by connecting to the window framework's event system (see reactive-banana-wx for an example), but mixEvents is more complex.  We could model it with the pure function

mixEventsF :: Bool -> () -> Int
mixEventsF True _ = 1
mixEventsF _    _ = 0

so we just need to find a way to lift this into the FRP system.  Behaviors have an Applicative instance, so let's try it:

mixEvents = apply (mixEventsF <$> behavior False togEvent) buttonEvent

and we've got it.  If we prefer, we could define mixEventsF :: Bool -> Int, but then we'd need to fmap const onto the Behavior to create a Behavior (() -> Int) we could apply to an Event.

So how does this work?  The Behavior functions as an Event Accumulator.  It stores every Bool received from togEvent, and when a button press event is received creates an event from applying the Bool and () to the mixEventsF function.  One clue that this is the proper way to think of Behaviors is by the different ways they can be constructed:

• lifting pure values, and always (these are semantically equivalent, pure is defined as always)
• the behavior smart constructor (only hold the most recent Event)
• various accumulate functions on Events

So if a Behavior is an event accumulator, why do we need it at all?  Why not just use events directly, explicitly storing values when we need them?  In my view, the answer has to do with abstracting implementation from the user.

Push vs Pull

One tension in the FRP design space relates to a push-driven (data-driven) vs pull-driven (demand-driven) architecture.  Push-driven offer theoretically better performance, as data sources typically change relatively slowly (e.g. mouse clicks relative to screen refresh), so this minimizes recomputation.  However, pull-driven seems to fit the functional paradigm better.  Also, if a Behavior models a continuous function, pushing updates at the sample rate could be very wasteful.

The Behavior type (at least in reactive and reactive-banana) moderates between these two models.  We can see how this works by examining the semantics of our mixEvents function.

-- Upon receiving an Event (), mixEvents generates an Event Int
-- The value of the Int depends upon the most recent Event Bool status
mixEvents :: Event Bool -> Event () -> Event Int

Our implementation created a Behavior (() -> Int) by combining a pure function with the Event Bool stream.  This behavior is then sampled whenever an Event () is received.  By converting a discrete Event into something which can be later sampled, the Behavior localizes a change from push-driven to pull-driven.  Thus Event Bools are stored and pulled from as necessary, while Event ()s are pushed through and become Event Ints, which are then pushed until they're also changed into Behaviors (or reactimate'd).

Continuous Time

Another Behavior feature is that semantically (traditionally, but not in reactive-banana) they are continuous functions over time.  There are a lot of reasons to like this feature, which is notably missing from the reactive-banana semantic model.  Originally I was somewhat skeptical, however I've come to the conclusion that the reactive-banana model of Behaviors as step functions is more correct.  Since a Behavior is an accumulator of Events, it makes sense that the Behavior would change only when new Events are received.  As an Event is a datum at a discrete time, the Behavior must therefore be a discrete-valued function.  In particular, there is a conflict inherent in overloading the meaning of a Behavior to both respond to Events and be continuous in time.

Furthermore, I don't think the programmer loses anything by the reactive-banana model.  It's still possible to write functions that are continuous in time and lift them to Behaviors.  This is because the reactive-banana framework has no notion of time.  If you want a time value, you need to provide one yourself, perhaps via a hardware clock.

-- a pure, continuous function
-- using UTCTime to conveniently tie in with getTime.
createShape :: UTCTime -> Shape

-- an event source tied to the window toolkit's redraw event
redrawScreen :: Event ()

-- function to draw a shape to the screen
drawShape :: Shape -> IO ()

getTime :: IO UTCTime
getTime = Data.Time.Clock.getCurrentTime

reactiveSystem :: Prepare ()
reactiveSystem =
 reactimate $ fmap drawShape
                   (apply (pure createShape) (mapIO getTime redrawScreen))

Let's examine how this works.  Fundamentally, when the window system requests a screen redraw, we want to check the current time, generate the correct shape for the time, and finally update the display.  Our system provides the redraw requests via an Event (), which is mapped to the current time, creating an Event UTCTime.  The createShape function is first lifted to a Behavior then applied to the Event UTCTime, creating an Event Shape.  Finally the drawShape function is mapped over this, creating an Event (IO ()) which can be reactimate'd, i.e. each IO () is performed as it becomes available.  In my opinion, by removing time from the reactive-banana framework, both the described systems and the implementation are greatly simplified, with no loss of expressiveness.

Simultaneitys

One aspect that seems to cause a lot of difficulty is simultaneous events, multiple Events which occur at the same time.  There are a few ways these can arise, such as long sampling periods, merging Event streams, and recursive definitions.  reactive-banana specifies that the order of simultaneous events is undefined, providing the orderedDuplicate function to deal with certain cases where timing is important.

I don't like this solution.  Semantically, it would be nice that an event stream were strictly increasing in time, disallowing simultaneous events entirely.  I suspect that this could be made usable by providing the following two primitive functions:

mergeCombine :: (a -> a -> a) -> Event a -> Event a -> Event a

-- When two events arrive simultaneously, create two output events with the right event delayed slightly
mergeOrdered :: Event a -> Event a -> Event a

The mergeOrdered function introduces a slight delay to preserve discrete events, whereas mergeCombine has a function to combine two events to one.  mergeCombine handles the common cases of dropping one event via const and combining with mappend, as well as more specialized functions.  I think these functions, together with the guarantee that an Event stream only produces single events, are sufficient for a workable system.

Conclusion

So that's my initial reaction to reactive-banana.  The library is quite minimal, but so far it's sufficiently expressive for every problem I've thought of.  The new semantics of Behavior seem conducive to developing a correct model of how the implementation works, thereby making it simpler for users to reason about performance and operations.  Requiring users to explicitly manage time may eventually become a burden, however it's a direction worth exploring.  For many problems user code doesn't depend upon time anyway.

enumerator/iteratees and output

I have recently received a few questions about writing output while using enumerator-based I/O. In some cases users have attempted to make enumerators (like enumFd) that will handle output as well, but have difficulty actually making it work.

I think these problems stem from an incorrect application of the enumerator model of I/O. When using enumerators, a file (or other data source) is a resource that can be enumerated over to process data, exactly as a list can be enumerated over in order to access the data contained in the list. Compare the following:

foldl f init xs

enumFd "SomeFile" ==<< stream2list

In the first function, 'xs' is the data to be processed, 'foldl' tells how to access individual items in the data collection, and 'f' and 'init' do the actual processing. In the second, "SomeFile" is the data, 'enumFd' tells how to access the data, and 'stream2list' does the processing. So how does writing fit in? The output file obviously isn't the data source, and it doesn't make sense to enumerate over your output file as there's no data there to process. So it must go within the Iteratee. It turns out that making an iteratee to write data is relatively simple:

> import Data.Iteratee
> import System.IO
> import Control.Monad
>
> writeOut :: FilePath -> IterateeGM [] Char IO ()
> writeOut file = do
>  h <- liftIO $ openFile file WriteMode
>  loop h
>  where
>  loop :: Handle -> IterateeGM [] Char IO ()
>  loop h = do
>   next <- Data.Iteratee.head
>   case next of
>    Just c -> liftIO $ hPutChar h c >> loop
>    Nothing -> liftIO $ hClose h

Add some error handling and you've got a writer. This version could be polymorphic over different StreamChunk instances by generalizing the type (FlexibleContexts may be required as well). Other stream-specific versions could be written that would take advantage of the specific StreamChunk instance (e.g. using Data.ByteString.hPut instead of hPutChar).

I hope this will serve as a very basic introduction to output when using enumerators. In addition to a generic writer like this, it may frequently be beneficial to define special-purpose writers. In a future post I will show a writer that seeks within the output file using a threaded State monad.

Build a better WAVE reader, pt 2

In the last post, I looked at the beginnings of using an Iteratee to create a WAVE file reader. I achieved a runtime of 25 seconds, compared to 1.9 seconds for my benchmark code.

Oleg's Iteratee code uses a Stream type which internally represents data as Chunk [a]. Haskell lists have many great properties, but they are not particularly efficient for this type of numerical processing. There are many better options, including Arrays, UVectors, and StorableVectors.

Unfortunately all of these options I would want to use introduce a type class restriction on types of elements in the array. This isn't a problem for sound files in particular, although it does greatly limit the utility of the Iteratees in general. One of the Iteratee examples shows how Iteratees can be layered for text processing, using IterateeGM String m a iteratees layered on IterateeGM Char m a iteratees to provide operations on words in a text stream. This is no longer possible, as String is not an instance of the type classes necessary to use one of these more compact data structures. In the end, the performance benefits should be worth it.

I chose StorableVector to replace the basic list, because it seemed handy while I was experimenting. StorableVector turns out to have another good property which is key to achieving the goal of performance close to C.

Adapting the Iteratees to use StorableVector is straightforward. Unfortunately when we've finished, performance is actually worse! In fact, it's as bad as the original RBIO-based version. It turns out that one key function is the culprit: conv_stream.

conv_stream converts between two streams. It is capable of converting multiple elements from one stream to one (or no) elements of the other, using a user-supplied conversion function. The wave file reader uses conv_stream to convert from a stream of Word8's to a stream of Doubles so the user can operate on Doubles, with the library handling all conversions transparently. The original type of conv_stream is:

conv_stream :: Monad m =>
IterateeGM el m (Maybe [el']) -> EnumeratorN el el' m a

Unfortunately, this means our data needs to unpacked from the StorableVector Word8 into separate Word8's, those Word8's assembled into Just [Double], and finally packed into a StorableVector Double for the next Iteratee. This is very inefficient, but easily fixed. Just change the type of conv_stream to

conv_stream :: (Storable el, Storable el', Monad m) =>
IterateeGM el m (Maybe (Vec.Vector el')) -> EnumeratorN el el' m a

and we're done. In fact, the implementation doesn't change at all from the original, just the type. Now no packing or unpacking need take place.

With the new conv_stream, it's necessary to change the conversion function. This is the IterateeGM which converts Word8's into Doubles. I have actually split this code into two parts, a function (called unroller because the original list-based version was a manually-unrolled endian_read) which converts Word8s into Word16 or Word32 as necessary, and a normalization function to normalize the result and convert from Words to Double. The normalization function should be fine as is, all that's necessary to change is the unroller function. Ideally, it would operate on StorableVectors directly. Fortunately this is possible. All that's necessary is to get a ForeignPtr Word8 from a StorableVector, and cast it to ForeignPtr Wordn. A somewhat hackish 16-bit converter can be implemented as follows:

import qualified Foreign.Ptr as FP
import qualified Foreign.ForeignPtr as FFP
import qualified Data.StorableVector as Vec

unroll_16 :: (Monad m) => IterateeGM Word8 m (Maybe (Vec.Vector Word16))
unroll_16 = liftI $ IE_cont step
where
step (Chunk vec)
| Vec.null vec = unroll_16
| Vec.length vec == 1 = liftI $ IE_cont $ step' vec
| Vec.length vec `rem` 2 == 0 = liftI $ IE_done (convert_vec vec) (Chunk $ Vec.empty)
| True = let (h, t) = Vec.splitAt (Vec.length vec - 1) vec
in
liftI $ IE_done (convert_vec h) (Chunk t)
step stream = liftI $ IE_done Nothing stream
step' i (Chunk vec)
| Vec.null vec = liftI $ IE_cont $ step' i
| Vec.length vec `rem` 2 == 1 = let vec' = Vec.append i vec
in
liftI $ IE_done (convert_vec vec') (Chunk $ Vec.empty)
| True = let (h, t) = Vec.splitAt (Vec.length vec - 1) vec
in
liftI $ IE_done (convert_vec $ Vec.append i h) (Chunk t)
step' _i stream = liftI $ IE_done Nothing stream
convert_vec vec = let (fp, off, len) = VB.toForeignPtr vec
f = FP.plusPtr (FFP.unsafeForeignPtrToPtr fp) off
fp' = (FFP.castForeignPtr $ unsafePerformIO $ FFP.newForeignPtr_ f) :: FFP.ForeignPtr Word16
in
Just $ VB.fromForeignPtr fp' (len `div` 2)

and that's it! The big hack is that this only works on little-endian platforms. To be correct, the bytes would need to be swapped on big-endian systems only. Implementing a suitable swap function is currently an excercise for the reader. Note that unsafeForeignPtrToPtr should be safe as there is no finalizer associated with the ForeignPtr, and in any case the memory pointed to won't be GC'd until after the Iteratee is complete.

At this point, the data should be handled entirely as a StorableVector, with no conversions to or from any intermediate lists at any point. Here are results from running the test:

$ time ./test_iter_sb
Just (AudioFormat {numberOfChannels = 2, sampleRate = 44100, bitDepth = 16})
Just 0.977568895535142

real 0m1.661s
user 0m1.335s
sys 0m0.140s

And the proof is in the pudding. This is the fastest Haskell implementation yet, even faster than hsndfile in a lazy stream.

A bit of work remains to be done. It should be cleaned up, with the hack fixed properly. Also I've currently only implemented unroll_16. For 24-bit audio, I expect the creation of a Word24 type and Storable instance will be necessary in order to employ the same technique. However, I think I've proven that a pure-Haskell, high performance audio I/O library is possible.

Build a better WAVE reader, pt 1

I have recently been working on creating an optimal audio I/O library in Haskell. By optimal, I mean the library should do all of the following:

1. Have a clean, functional, usable interface. Operations should be composable, and the user should not have to deal with recursion, folds or the like. Procedural-style interfaces (anything exposing filehandle-type operations to the user) are out too.
   
2. Be space efficient. Some Haskell implementations suffer from space leaks, or can have space leaks if the user isn't careful with data.
3. Have performance comparable to hsndfile, a Haskell binding to libsndfile.
   
4. Be implemented purely in Haskell. This makes it interesting.
   

The standard for audio I/O performance in Haskell has been hsndfile, as was discussed on haskell-cafe not long ago. Testing shows that using hsndfile to read data in a lazy stream fashion (similar to lazy ByteStrings) is about 10 times faster than the next-fastest implementation. Not good for Haskell's claim of having speed comparable to C! Still, I'm convinced it's possible to do better.

My test case consists of finding the peak normalized amplitude of a 16-bit stereo WAVE file. The audio duration is about 6 minutes, with a total file size of about 66 MB. This task involves reading and processing the entire file. The processing is quite minimal, so the speed of an implementation should depend primarily on the efficiency of reading and normalizing audio data.

Using hsndfile in a lazy stream does provide an interface which is familiar to many Haskell users, but I'm not particularly fond of it. If the user is not careful, the entire file can be retained in memory after processing. For a large audio file, this leads to an unacceptable situation. It should be impossible for data to be retained without explicit action from the user. Also, dependencies on foreign libraries can be difficult for some (i.e. Windows) users to resolve.

So having decided to create a native Haskell implementation, the first place to start is with the semantics and interface. I recently read Oleg K.'s presentations on Iteratee processing in Haskell, and I'm convinced this is the best model for functional I/O. It just feels right, what else can I say? Read the paper, see the code.

A preliminary version of Iteratee-based processing was benchmarked as "Enumerator" in the Haskell-art discussion. It was roughly comparable with other Haskell solutions, but has some problems. Notably it doesn't support seeking within the file, which would be nice. After I developed that, Oleg released a new version of Iteratee which does support seek operations via the RBIO monad. So I dropped Enumerator in favor of the new Iteratee + RBIO.

Oleg helpfully provides a TIFF reader with his Iteratee code. I based my wave reader on that, using the same technique of storing Enumerators for each wave sub-chunk in an IntMap. Since they aren't tagged, a list probably would have served just as well, but since most wave files only have 2 or 3 chunks anyway it doesn't seem like it matters much. Changes here aren't going to have a large performance impact.

The first version to use Iteratee + RBIO executes in about 45 seconds. Ouch.

RBIO uses IORef's to communicate seek requests and answers between Iteratees and Enumerators. Unfortunately IORefs are slow, so I'll start by getting rid of them. The data type

data IterateeG el m a = IE_done a (StreamG el)
| IE_cont (StreamG el -> IterateeGM el m a)

can be changed to

data IterateeG el m a = IE_done a (StreamG el)
| IE_cont (StreamG el -> IterateeGM el m a)
| IE_seek FileOffset (StreamG el -> IterateeGM el m a)

This also necessitates adding one more case to (>>=) on Iteratees:

iter_bind m f = m >>== docase
where
docase (IE_done a (Chunk [])) = f a
docase (IE_done a stream) = f a >>== (\r -> case r of
IE_done x _ -> liftI $ IE_done x stream
IE_cont k -> k stream
iter -> liftI iter)
docase (IE_cont k) = liftI $ IE_cont ((>>= f) . k)
docase (IE_seek off k) = liftI $ IE_seek off ((>>= f) . k)

After these changes, the provided Iteratees can be adapted to use the new structure relatively easily. The enumerator that reads from a file, enum_fd_random, must also be updated to understand seek requests. At this point we can remove RBIO entirely, and our execution time is about 33 sec. Adding a few carefully-chosen INLINE's and SPECIALIZATION's gets runtime down to 25 sec. Better, but we're very far from the goal of being comparable with C.

The next step is to change the Iteratee's internal representation of data.