{-# LANGUAGE GeneralizedNewtypeDeriving #-} {-| This module provides leak-free and referentially transparent higher-order discrete signals. -} module FRP.Elerea.Simple ( -- * The signal abstraction Signal , SignalGen -- * Embedding into I/O , start , external , externalMulti , debug -- * Basic building blocks , delay , generator , memo , until -- * Derived combinators , stateful , transfer , transfer2 , transfer3 , transfer4 -- * Random sources , noise , getRandom -- * A longer example -- $example ) where import Control.Applicative import Control.Concurrent.MVar import Control.Monad import Control.Monad.Fix import Data.IORef import Data.Maybe import Prelude hiding (until) import System.Mem.Weak import System.Random.Mersenne -- | A signal represents a value changing over time. It can be -- thought of as a function of type @Nat -> a@, where the argument is -- the sampling time, and the 'Monad' instance agrees with the -- intuition (bind corresponds to extracting the current sample). -- Signals and the values they carry are denoted the following way in -- the documentation: -- -- > s = <<s0 s1 s2 ...>> -- -- This says that @s@ is a signal that reads @s0@ during the first -- sampling, @s1@ during the second and so on. You can also think of -- @s@ as the following function: -- -- > s t_sample = [s0,s1,s2,...] !! t_sample -- -- Signals are constrained to be sampled sequentially, there is no -- random access. The only way to observe their output is through -- 'start'. newtype Signal a = S (IO a) deriving (Functor, Applicative, Monad) -- | A dynamic set of actions to update a network without breaking -- consistency. type UpdatePool = [Weak (IO (),IO ())] -- | A signal generator is the only source of stateful signals. It -- can be thought of as a function of type @Nat -> a@, where the -- result is an arbitrary data structure that can potentially contain -- new signals, and the argument is the creation time of these new -- signals. It exposes the 'MonadFix' interface, which makes it -- possible to define signals in terms of each other. The denotation -- of signal generators happens to be the same as that of signals, but -- this partly accidental (it does not hold in the other variants), so -- we will use a separate notation for generators: -- -- > g = <|g0 g1 g2 ...|> -- -- Just like signals, generators behave as functions of time: -- -- > g t_start = [g0,g1,g2,...] !! t_start -- -- The conceptual difference between the two notions is that signals -- are passed a sampling time, while generators expect a start time -- that will be the creation time of all the freshly generated -- signals in the resulting structure. newtype SignalGen a = SG { unSG :: IORef UpdatePool -> IO a } -- | The phases every signal goes through during a superstep. data Phase a = Ready a | Updated a a instance Functor SignalGen where fmap = (<*>).pure instance Applicative SignalGen where pure = return (<*>) = ap instance Monad SignalGen where return = SG . const . return SG g >>= f = SG $ \p -> g p >>= \x -> unSG (f x) p instance MonadFix SignalGen where mfix f = SG $ \p -> mfix (($p).unSG.f) -- | Embedding a signal into an 'IO' environment. Repeated calls to -- the computation returned cause the whole network to be updated, and -- the current sample of the top-level signal is produced as a -- result. This is the only way to extract a signal generator outside -- the network, and it is equivalent to passing zero to the function -- representing the generator. In general: -- -- > replicateM n =<< start <|<<x0 x1 x2 x3 ...>> ...|> == take n [x0,x1,x2,x3,...] -- -- Example: -- -- > do -- > smp <- start (stateful 3 (+2)) -- > res <- replicateM 5 smp -- > print res -- -- Output: -- -- > [3,5,7,9,11] start :: SignalGen (Signal a) -- ^ the generator of the top-level signal -> IO (IO a) -- ^ the computation to sample the signal start (SG gen) = do pool <- newIORef [] S sample <- gen pool return $ do let deref ptr = (fmap.fmap) ((,) ptr) (deRefWeak ptr) res <- sample (ptrs,acts) <- unzip.catMaybes <$> (mapM deref =<< readIORef pool) writeIORef pool ptrs mapM_ fst acts mapM_ snd acts return res -- | Auxiliary function used by all the primitives that create a -- mutable variable. addSignal :: (a -> IO a) -- ^ sampling function -> (a -> IO ()) -- ^ aging function -> IORef (Phase a) -- ^ the mutable variable behind the signal -> IORef UpdatePool -- ^ the pool of update actions -> IO (Signal a) -- ^ the signal created addSignal sample update ref pool = do let upd = readIORef ref >>= \v -> case v of Ready x -> update x _ -> return () fin = readIORef ref >>= \v -> case v of Updated x _ -> writeIORef ref $! Ready x _ -> error "Signal not updated!" sig = S $ readIORef ref >>= \v -> case v of Ready x -> sample x Updated _ x -> return x updateActions <- mkWeak sig (upd,fin) Nothing modifyIORef pool (updateActions:) return sig -- | The 'delay' combinator is the elementary building block for -- adding state to the signal network by constructing delayed versions -- of a signal that emit a given value at creation time and the -- previous output of the signal afterwards (@--@ is undefined): -- -- > delay x0 s = <| <<x0 s0 s1 s2 s3 ...>> -- > <<-- x0 s1 s2 s3 ...>> -- > <<-- -- x0 s2 s3 ...>> -- > <<-- -- -- x0 s3 ...>> -- > ... -- > |> -- -- It can be thought of as the following function (which should also -- make it clear why the return value is 'SignalGen'): -- -- > delay x0 s t_start t_sample -- > | t_start == t_sample = x0 -- > | t_start < t_sample = s (t_sample-1) -- > | otherwise = error \"Premature sample!\" -- -- The way signal generators are extracted by 'generator' ensures that -- the error can never happen. -- -- Example (requires the @DoRec@ extension): -- -- > do -- > smp <- start $ do -- > rec let fib'' = liftA2 (+) fib' fib -- > fib' <- delay 1 fib'' -- > fib <- delay 1 fib' -- > return fib -- > res <- replicateM 7 smp -- > print res -- -- Output: -- -- > [1,1,2,3,5,8,13] delay :: a -- ^ initial output at creation time -> Signal a -- ^ the signal to delay -> SignalGen (Signal a) -- ^ the delayed signal delay x0 (S s) = SG $ \pool -> do ref <- newIORef (Ready x0) let update x = s >>= \x' -> x' `seq` writeIORef ref (Updated x' x) addSignal return update ref pool -- | A reactive signal that takes the value to output from a signal -- generator carried by its input with the sampling time provided as -- the start time for the generated structure. It is possible to -- create new signals in the monad, which is the key to defining -- dynamic data-flow networks. -- -- > generator << <|x00 x01 x02 ...|> -- > <|x10 x11 x12 ...|> -- > <|x20 x21 x22 ...|> -- > ... -- > >> = <| <<x00 x11 x22 ...>> -- > <<x00 x11 x22 ...>> -- > <<x00 x11 x22 ...>> -- > ... -- > |> -- -- It can be thought of as the following function: -- -- > generator g t_start t_sample = g t_sample t_sample -- -- It has to live in the 'SignalGen' monad, because it needs to -- maintain an internal state to be able to cache the current sample -- for efficiency reasons. However, this state is not carried between -- samples, therefore start time doesn't matter and can be ignored. -- -- Refer to the longer example at the bottom to see how it can be -- used. generator :: Signal (SignalGen a) -- ^ the signal of generators to run -> SignalGen (Signal a) -- ^ the signal of generated structures generator (S s) = SG $ \pool -> do ref <- newIORef (Ready undefined) let sample = do SG g <- s x <- g pool writeIORef ref (Updated undefined x) return x addSignal (const sample) (const (() <$ sample)) ref pool -- | Memoising combinator. It can be used to cache results of -- applicative combinators in case they are used in several places. -- It is observationally equivalent to 'return' in the 'SignalGen' -- monad. -- -- > memo s = <|s s s s ...|> -- -- For instance, if @s = f \<$\> s'@, then @f@ will be recalculated -- once for each sampling of @s@. This can be avoided by writing @s -- \<- memo (f \<$\> s')@ instead. However, 'memo' incurs a small -- overhead, therefore it should not be used blindly. -- -- All the functions defined in this module return memoised signals. memo :: Signal a -- ^ the signal to cache -> SignalGen (Signal a) -- ^ a signal observationally equivalent to the argument memo (S s) = SG $ \pool -> do ref <- newIORef (Ready undefined) let sample = s >>= \x -> writeIORef ref (Updated undefined x) >> return x addSignal (const sample) (const (() <$ sample)) ref pool -- | A signal that is true exactly once: the first time the input -- signal is true. Afterwards, it is constantly false, and it holds -- no reference to the input signal. For instance (assuming the rest -- of the input is constantly @False@): -- -- > until <<False False True True False True ...>> = -- > <| <<False False True False False False False False False False ...>> -- > << --- False True False False False False False False False ...>> -- > << --- --- True False False False False False False False ...>> -- > << --- --- --- True False False False False False False ...>> -- > << --- --- --- --- False True False False False False ...>> -- > << --- --- --- --- --- True False False False False ...>> -- > << --- --- --- --- --- --- False False False False ...>> -- > ... -- > |> -- -- It is observationally equivalent to the following expression (which -- would hold onto @s@ forever): -- -- > until s = do -- > step <- transfer False (||) s -- > dstep <- delay False step -- > memo (liftA2 (/=) step dstep) -- -- Example: -- -- > do -- > smp <- start $ do -- > cnt <- stateful 0 (+1) -- > tick <- until ((>=3) <$> cnt) -- > return $ liftA2 (,) cnt tick -- > res <- replicateM 6 smp -- > print res -- -- Output: -- -- > [(0,False),(1,False),(2,False),(3,True),(4,False),(5,False)] until :: Signal Bool -- ^ the boolean input signal -> SignalGen (Signal Bool) -- ^ a one-shot signal true only the first time the input is true until (S s) = SG $ \pool -> do ref <- newIORef (Ready undefined) rsmp <- mfix $ \rs -> newIORef $ do x <- s writeIORef ref (Updated undefined x) when x $ writeIORef rs $ do writeIORef ref (Updated undefined False) return False return x let sample = join (readIORef rsmp) addSignal (const sample) (const (() <$ sample)) ref pool -- | A signal that can be directly fed through the sink function -- returned. This can be used to attach the network to the outer -- world. The signal always yields the value last written to the -- sink. In other words, if the sink is written less frequently than -- the network sampled, the output remains the same during several -- samples. If values are pushed in the sink more frequently, only -- the last one before sampling is visible on the output. -- -- Example: -- -- > do -- > (sig,snk) <- external 4 -- > smp <- start (return sig) -- > r1 <- smp -- > r2 <- smp -- > snk 7 -- > r3 <- smp -- > snk 9 -- > snk 2 -- > r4 <- smp -- > print [r1,r2,r3,r4] -- -- Output: -- -- > [4,4,7,2] external :: a -- ^ initial value -> IO (Signal a, a -> IO ()) -- ^ the signal and an IO function to feed it external x = do ref <- newIORef x return (S (readIORef ref), writeIORef ref) -- | An event-like signal that can be fed through the sink function -- returned. The signal carries a list of values fed in since the -- last sampling, i.e. it is constantly @[]@ if the sink is never -- invoked. The order of elements is reversed, so the last value -- passed to the sink is the head of the list. Note that unlike -- 'external' this function only returns a generator to be used within -- the expression constructing the top-level stream, and this -- generator can only be used once. -- -- Example: -- -- > do -- > (gen,snk) <- externalMulti -- > smp <- start gen -- > r1 <- smp -- > snk 7 -- > r2 <- smp -- > r3 <- smp -- > snk 9 -- > snk 2 -- > r4 <- smp -- > print [r1,r2,r3,r4] -- -- Output: -- -- > [[],[7],[],[2,9]] externalMulti :: IO (SignalGen (Signal [a]), a -> IO ()) -- ^ a generator for the event signal and the associated sink externalMulti = do var <- newMVar [] return (SG $ \pool -> do let sig = S $ readMVar var update <- mkWeak sig (return (),takeMVar var >> putMVar var []) Nothing modifyIORef pool (update:) return sig ,\val -> do vals <- takeMVar var putMVar var (val:vals) ) -- | A pure stateful signal. The initial state is the first output, -- and every subsequent state is derived from the preceding one by -- applying a pure transformation. -- -- Example: -- -- > do -- > smp <- start (stateful "x" ('x':)) -- > res <- replicateM 5 smp -- > print res -- -- Output: -- -- > ["x","xx","xxx","xxxx","xxxxx"] stateful :: a -- ^ initial state -> (a -> a) -- ^ state transformation -> SignalGen (Signal a) stateful x0 f = mfix $ \sig -> delay x0 (f <$> sig) -- | A stateful transfer function. The current input affects the -- current output, i.e. the initial state given in the first argument -- is considered to appear before the first output, and can never be -- observed, and subsequent states are determined by combining the -- preceding state with the current output of the input signal using -- the function supplied. -- -- Example: -- -- > do -- > smp <- start $ do -- > cnt <- stateful 1 (+1) -- > transfer 10 (+) cnt -- > res <- replicateM 5 smp -- > print res -- -- Output: -- -- > [11,13,16,20,25] transfer :: a -- ^ initial internal state -> (t -> a -> a) -- ^ state updater function -> Signal t -- ^ input signal -> SignalGen (Signal a) transfer x0 f s = mfix $ \sig -> do sig' <- delay x0 sig memo (liftA2 f s sig') -- | A variation of 'transfer' with two input signals. transfer2 :: a -- ^ initial internal state -> (t1 -> t2 -> a -> a) -- ^ state updater function -> Signal t1 -- ^ input signal 1 -> Signal t2 -- ^ input signal 2 -> SignalGen (Signal a) transfer2 x0 f s1 s2 = mfix $ \sig -> do sig' <- delay x0 sig memo (liftA3 f s1 s2 sig') -- | A variation of 'transfer' with three input signals. transfer3 :: a -- ^ initial internal state -> (t1 -> t2 -> t3 -> a -> a) -- ^ state updater function -> Signal t1 -- ^ input signal 1 -> Signal t2 -- ^ input signal 2 -> Signal t3 -- ^ input signal 3 -> SignalGen (Signal a) transfer3 x0 f s1 s2 s3 = mfix $ \sig -> do sig' <- delay x0 sig memo (liftM4 f s1 s2 s3 sig') -- | A variation of 'transfer' with four input signals. transfer4 :: a -- ^ initial internal state -> (t1 -> t2 -> t3 -> t4 -> a -> a) -- ^ state updater function -> Signal t1 -- ^ input signal 1 -> Signal t2 -- ^ input signal 2 -> Signal t3 -- ^ input signal 3 -> Signal t4 -- ^ input signal 4 -> SignalGen (Signal a) transfer4 x0 f s1 s2 s3 s4 = mfix $ \sig -> do sig' <- delay x0 sig memo (liftM5 f s1 s2 s3 s4 sig') -- | A random signal. -- -- Example: -- -- > do -- > smp <- start noise :: IO (IO Double) -- > res <- replicateM 5 smp -- > print res -- -- Output: -- -- > [0.12067753390401374,0.8658877349182655,0.7159264443196786,0.1756941896012891,0.9513646060896676] noise :: MTRandom a => SignalGen (Signal a) noise = memo (S randomIO) -- | A random source within the 'SignalGen' monad. getRandom :: MTRandom a => SignalGen a getRandom = SG (const randomIO) -- | A printing action within the 'SignalGen' monad. debug :: String -> SignalGen () debug = SG . const . putStrLn -- | The Show instance is only defined for the sake of Num... instance Show (Signal a) where showsPrec _ _ s = "<SIGNAL>" ++ s -- | Equality test is impossible. instance Eq (Signal a) where _ == _ = False -- | Error message for unimplemented instance functions. unimp :: String -> a unimp = error . ("Signal: "++) instance Ord t => Ord (Signal t) where compare = unimp "compare" min = liftA2 min max = liftA2 max instance Enum t => Enum (Signal t) where succ = fmap succ pred = fmap pred toEnum = pure . toEnum fromEnum = unimp "fromEnum" enumFrom = unimp "enumFrom" enumFromThen = unimp "enumFromThen" enumFromTo = unimp "enumFromTo" enumFromThenTo = unimp "enumFromThenTo" instance Bounded t => Bounded (Signal t) where minBound = pure minBound maxBound = pure maxBound instance Num t => Num (Signal t) where (+) = liftA2 (+) (-) = liftA2 (-) (*) = liftA2 (*) signum = fmap signum abs = fmap abs negate = fmap negate fromInteger = pure . fromInteger instance Real t => Real (Signal t) where toRational = unimp "toRational" instance Integral t => Integral (Signal t) where quot = liftA2 quot rem = liftA2 rem div = liftA2 div mod = liftA2 mod quotRem a b = (fst <$> qrab,snd <$> qrab) where qrab = quotRem <$> a <*> b divMod a b = (fst <$> dmab,snd <$> dmab) where dmab = divMod <$> a <*> b toInteger = unimp "toInteger" instance Fractional t => Fractional (Signal t) where (/) = liftA2 (/) recip = fmap recip fromRational = pure . fromRational instance Floating t => Floating (Signal t) where pi = pure pi exp = fmap exp sqrt = fmap sqrt log = fmap log (**) = liftA2 (**) logBase = liftA2 logBase sin = fmap sin tan = fmap tan cos = fmap cos asin = fmap asin atan = fmap atan acos = fmap acos sinh = fmap sinh tanh = fmap tanh cosh = fmap cosh asinh = fmap asinh atanh = fmap atanh acosh = fmap acosh {- $example For a not entirely trivial example, let's create a dynamic collection of countdown timers, where each expired timer is removed from the collection. First of all, we'll need a simple tester function: @ sigtest gen = 'replicateM' 15 '=<<' 'start' gen @ We can try it with a trivial example: @ \> sigtest $ 'stateful' 2 (+3) [2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47] @ Our first definition will be a signal representing a simple named timer: @ countdown :: String -\> Int -\> SignalGen (Signal (String,Maybe Int)) countdown name t = do let tick prev = do { t \<- prev ; 'guard' (t \> 0) ; 'return' (t-1) } timer \<- 'stateful' (Just t) tick 'return' ((,) name '<$>' timer) @ Let's see if it works: @ \> sigtest $ countdown \"foo\" 4 [(\"foo\",Just 4),(\"foo\",Just 3),(\"foo\",Just 2),(\"foo\",Just 1),(\"foo\",Just 0), (\"foo\",Nothing),(\"foo\",Nothing),(\"foo\",Nothing),...] @ Next, we will define a timer source that takes a list of timer names, starting values and start times and creates a signal that delivers the list of new timers at every point: @ timerSource :: [(String, Int, Int)] -\> SignalGen (Signal [Signal (String, Maybe Int)]) timerSource ts = do let gen t = 'mapM' ('uncurry' countdown) newTimers where newTimers = [(n,v) | (n,v,st) \<- ts, st == t] cnt \<- 'stateful' 0 (+1) 'generator' (gen '<$>' cnt) @ Now we need to encapsulate the timer source signal in another signal expression that takes care of maintaining the list of live timers. Since working with dynamic collections is a recurring task, let's define a generic combinator that maintains a dynamic list of signals given a source and a test that tells from the output of each signal whether it should be kept. We can use @mdo@ expressions (a variant of @do@ expressions allowing forward references) as syntactic sugar for 'mfix' to make life easier: @ collection :: Signal [Signal a] -\> (a -\> Bool) -\> SignalGen (Signal [a]) collection source isAlive = mdo sig \<- 'delay' [] ('map' 'snd' '<$>' collWithVals') coll \<- 'memo' ('liftA2' (++) source sig) let collWithVals = 'zip' '<$>' ('sequence' '=<<' coll) '<*>' coll collWithVals' \<- 'memo' ('filter' (isAlive . 'fst') '<$>' collWithVals) 'return' $ 'map' 'fst' '<$>' collWithVals' @ We need recursion to define the @coll@ signal as a delayed version of its continuation, which does not contain signals that need to be removed in the current sample. At every point of time the running collection is concatenated with the source. We define @collWithVals@, which simply pairs up every signal with its current output. The output is obtained by extracting the current value of the signal container and sampling each element with 'sequence'. We can then derive @collWithVals'@, which contains only the signals that must be kept for the next round along with their output. Both @coll@ and @collWithVals'@ have to be memoised, because they are used more than once (the program would work without that, but it would recalculate both signals each time they are used). By throwing out the respective parts, we can get both the final output and the collection for the next step (@coll'@). Now we can easily finish the original task: @ timers :: [(String, Int, Int)] -\> SignalGen (Signal [(String, Int)]) timers timerData = do src \<- timerSource timerData getOutput '<$>' collection src ('isJust' . 'snd') where getOutput = 'fmap' ('map' (\\(name,Just val) -> (name,val))) @ As a test, we can start four timers: /a/ at t=0 with value 3, /b/ and /c/ at t=1 with values 5 and 3, and /d/ at t=3 with value 4: @ \> sigtest $ timers [(\"a\",3,0),(\"b\",5,1),(\"c\",3,1),(\"d\",4,3)] [[(\"a\",3)],[(\"b\",5),(\"c\",3),(\"a\",2)],[(\"b\",4),(\"c\",2),(\"a\",1)], [(\"d\",4),(\"b\",3),(\"c\",1),(\"a\",0)],[(\"d\",3),(\"b\",2),(\"c\",0)], [(\"d\",2),(\"b\",1)],[(\"d\",1),(\"b\",0)],[(\"d\",0)],[],[],[],[],[],[],[]] @ If the noise of the applicative lifting operators feels annoying, she (<http://personal.cis.strath.ac.uk/~conor/pub/she/>) comes to the save. Among other features it provides idiom brackets, which can substitute the explicit lifting. For instance, it allows us to define @collection@ this way: @ collection :: Stream [Stream a] -> (a -> Bool) -> StreamGen (Stream [a]) collection source isAlive = mdo sig \<- 'delay' [] (|'map' ~'snd' collWithVals'|) coll \<- 'memo' (|source ++ sig|) collWithVals' \<- 'memo' (|'filter' ~(isAlive . 'fst') (|'zip' ('sequence' '=<<' coll) coll|)|) 'return' (|'map' ~'fst' collWithVals'|) @ -}