{-# LANGUAGE GeneralizedNewtypeDeriving #-} {-| This module provides efficient higher-order discrete signals. For a non 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'|) @ -} module FRP.Elerea.Experimental.Simple ( Signal , SignalGen , start , external , delay , generator , memo , stateful , transfer ) where import Control.Applicative import Control.Monad import Control.Monad.Fix import Data.IORef import Data.Maybe import System.Mem.Weak {-| A signal 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). -} 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. -} 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. -} 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' transfer function emits the value of a signal from the previous superstep, starting with the filler value given in the first argument. 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 ensures that the error can never happen. -} 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 time of generation. It is possible to create new signals in the monad. 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 starting time doesn't matter and can be ignored. -} 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 >> return ())) 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 :: 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 >> return ())) 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. -} 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) {-| 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. It is equivalent to the following expression: @ stateful x0 f = 'mfix' $ \sig -> 'delay' x0 (f '<$>' sig) @ -} 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. It is equivalent to the following expression: @ transfer x0 f s = 'mfix' $ \sig -> 'liftA2' f s '<$>' 'delay' x0 sig @ -} transfer :: a -- ^ initial internal state -> (t -> a -> a) -- ^ state updater function -> Signal t -- ^ input signal -> SignalGen (Signal a) transfer x0 f s = mfix $ \sig -> liftA2 f s <$> delay x0 sig {-| The @Show@ instance is only defined for the sake of 'Num'... -} instance Show (Signal a) where showsPrec _ _ s = "<SIGNAL>" ++ s {-| Equality test is impossible. -} instance Eq (Signal a) where _ == _ = False {-| Error message for unimplemented instance functions. -} unimp :: String -> a unimp = error . ("Signal: "++) instance Ord t => Ord (Signal t) where compare = unimp "compare" min = liftA2 min max = liftA2 max instance Enum t => Enum (Signal t) where succ = fmap succ pred = fmap pred toEnum = pure . toEnum fromEnum = unimp "fromEnum" enumFrom = unimp "enumFrom" enumFromThen = unimp "enumFromThen" enumFromTo = unimp "enumFromTo" enumFromThenTo = unimp "enumFromThenTo" instance Bounded t => Bounded (Signal t) where minBound = pure minBound maxBound = pure maxBound instance Num t => Num (Signal t) where (+) = liftA2 (+) (-) = liftA2 (-) (*) = liftA2 (*) signum = fmap signum abs = fmap abs negate = fmap negate fromInteger = pure . fromInteger instance Real t => Real (Signal t) where toRational = unimp "toRational" instance Integral t => Integral (Signal t) where quot = liftA2 quot rem = liftA2 rem div = liftA2 div mod = liftA2 mod quotRem a b = (fst <$> 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