Stream Fusion For Haskell Arrays

Stream Fusion for Haskell Arrays

Don Stewart Galois Inc

Haskell's Data Types ●

Beautiful algebraic data types: data Set a = Tip | Bin !Int a !(Set a) !(Set a)

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Concise notation, inductive reasoning, type math!

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Polymorphic, strongly typed, side effect free

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Efficient. GCd. Strict, or lazy, or roll your own

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Pointers, pointers...

But for real speed... Sometimes we need unboxed, flat structures:

Arrays in Haskell biodiversity!

Data.Array

Foreign.ForeignPtr

Data.Array.Diff

Data.ByteString

Data.Array.IO

Data.ByteString.Lazy

Data.Array.Storable

Data.PackedString

Data.Array.ST

Data.StorableVector

Data.Array.Unboxed

Data.Vec

Data.Array.CArray

BLAS.Matrix

Data.ArrayBZ

Data.Packed

Foreign.Array

Data.Packed.Vector

Foreign.Ptr

Data.Packed.Matrix

The Perfect Array Type

1.Very, very efficient. Ruthlessly fast. 2.Polymorphic 3.Pure 4.Rich list-like API 5.Compatible with C arrays, other arrays

Data Parallel Haskell ●

Project to target large multicore systems: Chakravarty, Leshchinksiy, Peyton-Jones, Keller, Marlow

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Parallel, distributed arrays, with good interface

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Built from flat, unlifted arrays

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The core of a better array type for mortals

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Built around array fusion “Stream Fusion: From Lists to Streams to Nothing at All” Coutts, Leshchinskiy, Stewar.t 2007.

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Key technique for making arrays flexible and fast

uvector: fast, flat, fused arrays Two data types: mutable arrays and pure arrays data BUArr e = BUArr !Int !Int ByteArray# data MBUArr s e = MBUArr !Int (MutableByteArray# s)

Fill the mutable array, freeze it, and get free substrings, and persistance. ● Low level Haskell ●

Primitive operations length :: BUArr e -> Int length (BUArr _ n _) = n

class UAE e where sizeBU :: Int -> e -> Int indexBU :: BUArr e -> Int -> e readMBU :: MBUArr s e -> Int -> ST s e writeMBU :: MBUArr s e -> Int -> e -> ST s () newMBU :: UAE e => Int -> ST s (MBUArr s e)

Conversions Zero-copying conversion from mutable to pure unsafeFreezeMBU :: MBUArr s e -> Int -> ST s (BUArr e) unsafeFreezeMBU (MBUArr m mba) n = checkLen "unsafeFreezeMBU" m n $ ST $ \s -> (# s, BUArr 0 n (unsafeCoerce# mba) #)

Bounds checking compiled out if -funsafe

Array element instances Simple per-type representation choices instance UAE () where sizeBU _ _ = 0 indexBU (BUArr _ _ _) (I# _) = () readMBU (MBUArr _ _) (I# _) = ST $ \s -> (# s, () #) writeMBU (MBUArr _ _) (I# _) () = ST $ \s -> (# s, () #)

Goal 1: Efficiency Can be a bit fancier... instance UAE Bool where readMBU (MBUArr n mba) i@(I# i#) = ST $ \s -> case readWordArray# mba (bOOL_INDEX i#) s of (# s2, r# #) -> (# s2, (r# `and#` bOOL_BIT i#) `neWord#` int2Word# 0# # bOOL_INDEX :: Int# -> Int# #if SIZEOF_HSWORD == 4 bOOL_INDEX i# = i# `uncheckedIShiftRA#` 5# #elif SIZEOF_HSWORD == 8 bOOL_INDEX i# = i# `uncheckedIShiftRA#` 6# #endif

Relax. Low level stuff done.

Goal 2: polymorphic Abstract over the primitive arrays class UA e where data UArr e data MUArr e :: * -> * lengthU indexU

:: UArr e -> Int :: UArr e -> Int -> e

lengthMU :: MUArr e s -> Int newMU :: Int -> ST s (MUArr e s) freezeMU :: MUArr e s -> Int -> ST s (UArr e) readMU writeMU

:: MUArr e s -> Int -> ST s e :: MUArr e s -> Int -> e -> ST s ()

Goal 3a: Pure Introducing UArr .. purely! newU :: UA => -> ->

e Int (forall s. MUArr e s -> ST s Int) UArr e

newU n init = runST (do ma <- newMU n n' <- init ma freezeMU ma n' )

Mutation encapsulate in ST monad.

Flexible array representations instance UA () where newtype UArr () = UAUnit Int newtype MUArr () s = MUAUnit Int lengthU (UAUnit n) = n indexU (UAUnit _) _ = () sliceU (UAUnit _) _ n = UAUnit n lengthMU (MUAUnit n) = newMU n = readMU (MUAUnit _) _ = writeMU (MUAUnit _) _ _= freezeMU (MUAUnit _) n

n return $ MUAUnit n return () return () = return $ UAUnit n

Goal 4: list-like operations data (:*:) a b = !a :*: !b instance (UA a, UA b) => UA (a :*: b) where data UArr (a :*: b) = UAProd !(UArr a)

!(UArr b)

data MUArr (a :*: b) s = MUAProd !(MUArr a s) !(MUArr b s) indexU (UAProd l r) i = indexU l i :*: indexU r i

Support for numeric stuff instance (RealFloat a, UA a) => UA (Complex a) where newtype UArr (Complex a) = UAComplex (UArr (a :*: a)) newtype MUArr (Complex a) s = MUAComplex (MUArr (a :*: a) s) indexU (UAComplex arr) i = case indexU arr i of (a :*: b) -> a :+ b

But that's not the end •

Strict, pure arrays are a bit too inefficient

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Too much copying, not enough sharing

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Impure languages would just mutate inplace

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But we need to find some other way to deforest.

Goal 1&2: Efficiency Stream Fusion data Step s a = Done | Skip !s | Yield !a !s data Stream a = exists s. Stream (s -> Step s a) !s Int

Abstract sequence transformers ● Non-recursive ● General fusion rule for removing intermediates ● We'll convert arrays into abstract sequences ● Non-recursive things we can optimise ruthlessly ●

Conversion to and from arrays streamU :: UA a => UArr a -> Stream a streamU arr = Stream next 0 n where n = lengthU arr next i | i == n = Done | otherwise = Yield (arr `indexU` i) (i+1) unstreamU :: UA a => Stream a -> UArr a unstreamU st@(Stream next s n) = newDynU n (\marr -> unstreamMU marr st)

Convert recursive array loops to non-recursive streams mapU :: (UA e, UA e') => (e -> e') -> UArr e -> UArr e' mapU f = unstreamU . mapS f . streamU headU :: UA e => UArr e -> e headU = headS . StreamU lastU :: UA e => UArr e -> e lastU = foldlU (flip const)

The fusion rule ●

Use rules to remove redundant conversions "streamU/unstreamU" forall s. streamU (unstreamU s) = s

Compositions of non-recursive functions left over ● Then combine streams using general optimisations ● Arrays at the end will be fused from the combined stream pipeline ●

Filling a mutable array unstreamMU :: UA a => MUArr a s -> Stream a -> ST s Int unstreamMU marr (Stream next s n) = fill s 0 where fill s !i = case next s of Done -> return i Skip s' -> s' `seq` fill s' i Yield x s' -> s' `seq` do writeMU marr i x fill s' (i+1)

New streams emptyS :: Stream a emptyS = Stream (const Done) () 0 replicateS :: Int -> a -> Stream a replicateS n x = Stream next 0 n where next i | i == n = Done | otherwise = Yield x (i+1) enumFromToS :: (Ord a, RealFrac a) => a -> a -> Stream a enumFromToS n m = Stream next n (truncate (m - n)) where lim = m + 1/2 next s | s > lim = Done | otherwise = Yield s (s+1)

Transforming streams mapS :: (a -> b) -> Stream a -> Stream b mapS f (Stream next s n) = Stream next' s n where next' s = case next s of Done -> Done Skip s' -> Skip s' Yield x s' -> Yield (f x) s' foldS :: (b -> a -> b) -> b -> Stream a -> b foldS f z (Stream next s _) = fold z s where fold !z s = case next s of Yield x !s' -> fold (f z x) s' Skip !s' -> fold z s' Done -> z

Zipping streams zipWithS :: (a -> b -> c) -> Stream a -> Stream b -> Stream c zipWithS f (Stream next1 s m) (Stream next2 t n) = Stream next (s :*: t) m where next (s :*: t) = case next1 s of Done -> Done Skip s' -> Skip (s' :*: t) Yield x s' -> case next2 t of Done -> Done Skip t' -> Skip (s :*: t') Yield y t' -> Yield (f x y) (s' :*: t')

Arrays to streams to nothing at all ...

Future ●

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Allow users to pick and choose between fused or direct implementations Write some big programs in this style Goal 4: more conversions from other array types (e.g. ByteStrings, Ptr a) Conversions to and from other sequence types via streams – no overhead for the conversion DPH's goals: parallel nested arrays, fusible mutable arrays.

OM NOM NOM NOM

It's on hackage.haskell.org