Combinators.hs

{-----------------------------------------------------------------------------
    reactive-banana
------------------------------------------------------------------------------}
{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE MultiParamTypeClasses #-}

module Reactive.Banana.Combinators (
    -- * Synopsis
    -- | Combinators for building event graphs.
    
    -- * Introduction
    -- $intro1
    Event, Behavior,
    -- $intro2
    interpret,
    
    -- * Core Combinators
    module Control.Applicative,
    module Data.Monoid,
    never, union, unions, filterE, collect, spill, accumE,
    apply, stepper,
    -- $classes
    
    -- * Derived Combinators
    -- ** Filtering
    filterJust, filterApply, whenE, split,
    -- ** Accumulation
    -- $Accumulation.
    accumB, mapAccum,
    -- ** Simultaneous event occurrences
    calm, unionWith,
    -- ** Apply class
    Apply(..),
    ) where

import Control.Applicative
import Control.Monad

import Data.Maybe (isJust, catMaybes)
import Data.Monoid (Monoid(..))


import qualified Reactive.Banana.Internal.EventBehavior1 as Prim
import Reactive.Banana.Internal.Types2 


{-----------------------------------------------------------------------------
    Introduction
------------------------------------------------------------------------------}
{-$intro1

At its core, Functional Reactive Programming (FRP) is about two
data types 'Event' and 'Behavior' and the various ways to combine them.

-}

-- Event
-- Behavior

{-$intro2

As you can see, both types seem to have a superfluous parameter @t@.
The library uses it to rule out certain gross inefficiencies,
in particular in connection with dynamic event switching.
For basic stuff, you can completely ignore it,
except of course for the fact that it will annoy you in your type signatures.

While the type synonyms mentioned above are the way you should think about
'Behavior' and 'Event', they are a bit vague for formal manipulation.
To remedy this, the library provides a very simple but authoritative
model implementation. See "Reactive.Banana.Model" for more.

-}

{-----------------------------------------------------------------------------
    Interpetation
------------------------------------------------------------------------------}
-- | Interpret an event processing function.
-- Useful for testing.
interpret :: (forall t. Event t a -> Event t b) -> [[a]] -> IO [[b]]
interpret f xs =
    map toList <$> Prim.interpret (return . unE . f . E) (map Just xs)


toList :: Maybe [a] -> [a]
toList Nothing   = []
toList (Just xs) = xs

{-----------------------------------------------------------------------------
    Core combinators
------------------------------------------------------------------------------}
singleton :: a -> [a]
singleton x = [x]

-- | Event that never occurs.
-- Think of it as @never = []@.
never    :: Event t a
never = E $ Prim.mapE singleton Prim.never

-- | Merge two event streams of the same type.
-- In case of simultaneous occurrences, the left argument comes first.
-- Think of it as
--
-- > union ((timex,x):xs) ((timey,y):ys)
-- >    | timex <= timey = (timex,x) : union xs ((timey,y):ys)
-- >    | timex >  timey = (timey,y) : union ((timex,x):xs) ys
union    :: Event t a -> Event t a -> Event t a
union e1 e2 = E $ Prim.unionWith (++) (unE e1) (unE e2)

-- | Merge several event streams of the same type.
-- 
-- > unions = foldr union never
unions :: [Event t a] -> Event t a
unions = foldr union never

-- | Allow all event occurrences that are 'Just' values, discard the rest.
-- Variant of 'filterE'.
filterJust :: Event t (Maybe a) -> Event t a
filterJust = E . Prim.filterJust . Prim.mapE (decide . catMaybes) . unE
    where
    decide xs = if null xs then Nothing else Just xs

-- | Allow all events that fulfill the predicate, discard the rest.
-- Think of it as
-- 
-- > filterE p es = [(time,a) | (time,a) <- es, p a]
filterE   :: (a -> Bool) -> Event t a -> Event t a
filterE p = filterJust . fmap (\x -> if p x then Just x else Nothing)

-- | Collect simultaneous event occurences.
-- The result will never contain an empty list.
-- Example:
--
-- > collect [(time1, e1), (time1, e2)] = [(time1, [e1,e2])]
collect   :: Event t a -> Event t [a]
collect e = E $ Prim.mapE singleton (unE e)

-- | Emit simultaneous event occurrences.
-- The first element in the list will be emitted first, and so on.
--
-- Up to strictness, we have
--
-- > spill . collect = id
spill :: Event t [a] -> Event t a
spill e = E $ Prim.mapE concat (unE e)

-- | Construct a time-varying function from an initial value and 
-- a stream of new values. Think of it as
--
-- > stepper x0 ex = \time -> last (x0 : [x | (timex,x) <- ex, timex < time])
-- 
-- Note that the smaller-than-sign in the comparision @timex < time@ means 
-- that the value of the behavior changes \"slightly after\"
-- the event occurrences. This allows for recursive definitions.
-- 
-- Also note that in the case of simultaneous occurrences,
-- only the last one is kept.
stepper :: a -> Event t a -> Behavior t a
stepper x e = B $ Prim.stepperB x $ Prim.mapE last $ unE e

-- | The 'accumE' function accumulates a stream of events.
-- Example:
--
-- > accumE "x" [(time1,(++"y")),(time2,(++"z"))]
-- >    = [(time1,"xy"),(time2,"xyz")]
--
-- Note that the output events are simultaneous with the input events,
-- there is no \"delay\" like in the case of 'accumB'.
accumE   :: a -> Event t (a -> a) -> Event t a
accumE acc = E . mapAccumE acc . Prim.mapE concatenate . unE
    where
    concatenate :: [a -> a] -> a -> ([a],a)
    concatenate fs acc = (tail values, last values)
        where values = scanl' (flip ($)) acc fs

    mapAccumE :: s -> Prim.Event (s -> (a,s)) -> Prim.Event a
    mapAccumE acc =
        Prim.mapE fst . Prim.accumE (undefined,acc) . Prim.mapE (. snd)

-- strict version of scanl
scanl' :: (a -> b -> a) -> a -> [b] -> [a]
scanl' f x ys = x : case ys of
    []   -> []
    y:ys -> let z = f x y in z `seq` scanl' f z ys

-- | Apply a time-varying function to a stream of events.
-- Think of it as
-- 
-- > apply bf ex = [(time, bf time x) | (time, x) <- ex]
apply    :: Behavior t (a -> b) -> Event t a -> Event t b
apply bf ex = E $ Prim.applyE (Prim.mapB map $ unB bf) (unE ex)

{-$classes

/Further combinators that Haddock can't document properly./

> instance Monoid (Event t (a -> a))

This monoid instance is /not/ the straightforward instance
that you would obtain from 'never' and 'union'.
Instead of just merging event streams, we use 'unionWith' to compose
the functions. This is very useful in the context of 'accumE' and 'accumB'
where simultaneous event occurrences are best avoided.

> instance Applicative (Behavior t)

'Behavior' is an applicative functor. In particular, we have the following functions.

> pure :: a -> Behavior t a

The constant time-varying value. Think of it as @pure x = \\time -> x@.

> (<*>) :: Behavior t (a -> b) -> Behavior t a -> Behavior t b

Combine behaviors in applicative style.
Think of it as @bf \<*\> bx = \\time -> bf time $ bx time@.

-}

{- No monoid instance, sorry.

instance Monoid (Event t (a -> a)) where
    mempty  = never
    mappend = unionWith (flip (.))
-}

instance Functor (Event t) where
    fmap f e = E $ Prim.mapE (map f) (unE e)

instance Applicative (Behavior t) where
    pure x    = B $ Prim.pureB x
    bf <*> bx = B $ Prim.applyB (unB bf) (unB bx)

instance Functor (Behavior t) where
    fmap = liftA

{-----------------------------------------------------------------------------
    Derived Combinators
------------------------------------------------------------------------------}
{-

Unfortunately, we can't make a  Num  instance because that would
require  Eq  and  Show .

instance Num a => Num (Behavior t a) where
    (+) = liftA2 (+)
    (-) = liftA2 (-)
    (*) = liftA2 (*)
    negate = fmap negate
    abs    = fmap abs
    signum = fmap signum
    fromInteger = pure . fromInteger
-}

-- | Allow all events that fulfill the time-varying predicate, discard the rest.
-- Generalization of 'filterE'.
filterApply :: Behavior t (a -> Bool) -> Event t a -> Event t a
filterApply bp = fmap snd . filterE fst . apply ((\p a-> (p a,a)) <$> bp)

-- | Allow events only when the behavior is 'True'.
-- Variant of 'filterApply'.
whenE :: Behavior t Bool -> Event t a -> Event t a
whenE bf = filterApply (const <$> bf)

-- | Split event occurrences according to a tag.
-- The 'Left' values go into the left component while the 'Right' values
-- go into the right component of the result.
split :: Event t (Either a b) -> (Event t a, Event t b)
split e = (filterJust $ fromLeft <$> e, filterJust $ fromRight <$> e)
    where
    fromLeft  (Left  a) = Just a
    fromLeft  (Right b) = Nothing
    fromRight (Left  a) = Nothing
    fromRight (Right b) = Just b


-- | Combine simultaneous event occurrences into a single occurrence.
--
-- > unionWith f e1 e2 = fmap (foldr1 f) <$> collect (e1 `union` e2)
unionWith :: (a -> a -> a) -> Event t a -> Event t a -> Event t a
unionWith f e1 e2 = E $ Prim.unionWith g (unE e1) (unE e2)
    where g xs ys = singleton $ foldr1 f (xs ++ ys)

-- | Keep only the last occurrence when simultaneous occurrences happen.
calm :: Event t a -> Event t a
calm = fmap last . collect



-- $Accumulation.
-- Note: all accumulation functions are strict in the accumulated value!
-- acc -> (x,acc) is the order used by 'unfoldr' and 'State'.

-- | The 'accumB' function is similar to a /strict/ left fold, 'foldl''.
-- It starts with an initial value and combines it with incoming events.
-- For example, think
--
-- > accumB "x" [(time1,(++"y")),(time2,(++"z"))]
-- >    = stepper "x" [(time1,"xy"),(time2,"xyz")]
-- 
-- Note that the value of the behavior changes \"slightly after\"
-- the events occur. This allows for recursive definitions.
accumB   :: a -> Event t (a -> a) -> Behavior t a
-- accumB x (Event e) = behavior $ AccumB x e
accumB  acc = stepper acc . accumE acc

-- | Efficient combination of 'accumE' and 'accumB'.
mapAccum :: acc -> Event t (acc -> (x,acc)) -> (Event t x, Behavior t acc)
mapAccum acc ef = (fst <$> e, stepper acc (snd <$> e))
    where e = accumE (undefined,acc) ((. snd) <$> ef)


infixl 4 <@>, <@

-- | Class for overloading the 'apply' function.
class (Functor f, Functor g) => Apply f g where
    -- | Infix operation for the 'apply' function, similar to '<*>'
    (<@>) :: f (a -> b) -> g a -> g b
    -- | Convenience function, similar to '<*'
    (<@)  :: f a -> g b -> g a
    
    f <@ g = (const <$> f) <@> g 

instance Apply (Behavior t) (Event t) where
    (<@>) = apply