{-# LANGUAGE MultiParamTypeClasses, FlexibleInstances #-}
module Temporal.Media (
-- * Time classes
Dur(..), Temporal(..), Stretchable(..),
ToMaybe(..), TemporalFunctor(..),
-- * Transformers
Reversible(..), Sliceable(..), cut,
-- * Structure
Construct(..), Arrangeable(..), Controlable(..),
sequent, parallel, loop, delay, temp,
-- * Media
Media(..), fold, fromMedia,
-- * Simple interperetation
-- ** Event list
Event, EventList(..),
mapEvent, toEvent, toEventList,
-- ** Unit Temporal Media
MediaUnit(..), unMediaUnit, foldU, fromMediaUnit,
Unit(..)
)
where
import Control.Applicative
import Control.Monad
import Data.Function
import Data.Ratio
import Prelude hiding (reverse, take, drop)
import qualified Prelude as P (reverse)
-------------------------------------------------
-- Classes
--
-------------------------------------------------
-- Time classes
-- | time class
class (Num t, Ord t, Fractional t) => Dur t
instance Dur Double
instance Integral a => Dur (Ratio a)
-- | Temporal structures
class Dur t => Temporal t a where
none :: t -> a -- ^ absence of value
dur :: a -> t -- ^ duration of value
-- | Stretching values by given time-factor
class Dur t => Stretchable t a where
stretch :: t -> a -> a
-- | Values covertible to 'Maybe'
--
-- auxiliary class for conversion to 'EventList'
class ToMaybe m where
toMaybe :: m a -> Maybe a
-- | temporal map
class Dur t => TemporalFunctor t f where
tmap :: (t -> a -> b) -> f a -> f b
---------------------------------------------------
-- transformers
--
class Reversible a where
reverse :: a -> a
-- | extracting parts, minimal complete definition: 'slice'.
class Temporal t a => Sliceable t a where
slice :: t -> t -> a -> a -- ^ @slice t0 t1 v@ extracts part of @v@ inside @[t0, t1]@
take :: t -> a -> a
drop :: t -> a -> a
take t = slice 0 t
drop t x = slice t (dur x) x
sliceErrorMessage = error "should be t0 <= t1, for slice t0 t1"
-- | mixing slice and reverse.
--
-- @cut t0 t1 v@ - if @t1 < t0@ reverses result of 'slice'
cut :: (Reversible a, Sliceable t a) => t -> t -> a -> a
cut t0 t1 m
| t0 <= t1 = slice t0 t1 m
| otherwise = slice (tm - t0) (tm - t1) $ reverse m
where tm = dur m
----------------------------------------------------
-- Structure
--
-- | constructor for generic structures
class Construct m where
prim :: a -> m a
-- | composing structures in sequent and parallel ways
class Arrangeable a where
(+:+) :: a -> a -> a
(=:=) :: a -> a -> a
-- | modifer
class Controlable c a where
control :: c -> a -> a
sequent, parallel :: Arrangeable a => [a] -> a
sequent = foldl1 (+:+)
parallel = foldl1 (=:=)
loop :: Arrangeable a => Int -> a -> a
loop n = sequent . replicate n
delay :: (Temporal t a, Arrangeable a) => t -> a -> a
delay t x = none t +:+ x
-- | constructs generic temporal structure @m a@ form time @t@ and initial value @a@
temp :: (Construct m, Temporal t (m a), Stretchable t (m a))
=> t -> a -> m a
temp t = stretch t . prim
----------------------------------------------------
-- Media
-- | Data type to represent temporal media
data Media c a = Prim a -- ^ single value
| Media c a :+: Media c a -- ^ sequential composition
| Media c a :=: Media c a -- ^ parallel composition
| Control c (Media c a) -- ^ specific environment modifier
deriving (Show, Eq)
-- | Folding Media
--
-- > fold prim seq par mod x
--
-- * prim - responds to 'Prim'
--
-- * seq - responds to ':+:'
--
-- * par - responds to ':=:'
--
-- * mod - responds to 'Control'
fold :: (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> (c -> b -> b)
-> Media c a -> b
fold prim seq par mod m =
case m of
Prim a -> prim a
a :+: b -> (seq `on` f) a b
a :=: b -> (par `on` f) a b
Control c a -> mod c $ f a
where f = fold prim seq par mod
instance Functor (Media c) where
fmap f x = case x of
Prim a -> Prim $ f a
a :+: b -> fmap f a :+: fmap f b
a :=: b -> fmap f a :=: fmap f b
Control c a -> Control c $ fmap f a
instance Monad (Media c) where
return = Prim
ma >>= f = case ma of
Prim a -> f a
a :+: b -> (a >>= f) :+: (b >>= f)
a :=: b -> (a >>= f) :=: (b >>= f)
Control c a -> Control c $ a >>= f
instance Applicative (Media c) where
pure = return
(<*>) = ap
-- time
instance Temporal t a => Temporal t (Media c a) where
none = Prim . none
dur = fold dur (+) max (const id)
instance Stretchable t a => Stretchable t (Media c a) where
stretch d = fmap $ stretch d
-- transformers
instance Reversible a => Reversible (Media c a) where
reverse x = case x of
Prim a -> Prim $ reverse a
a :+: b -> ((:+:) `on` reverse) b a
a :=: b -> ((:=:) `on` reverse) a b
Control c a -> Control c $ reverse a
instance Sliceable t a => Sliceable t (Media c a) where
slice t0 t1 m
| t1 < t0 = sliceErrorMessage
| t0 < 0 = none (abs t0) :+: slice 0 t1 m
| t1 > tm = slice t0 tm m :+: none (t1 - tm)
| otherwise = case m of
Prim a -> Prim $ slice t0 t1 a
a :+: b -> sliceSeq t0 t1 a b
a :=: b -> ((:=:) `on` slice t0 t1) a b
Control c a -> Control c $ slice t0 t1 a
where tm = dur m
sliceSeq :: Sliceable t a => t -> t -> Media c a -> Media c a -> Media c a
sliceSeq t0 t1 a b
| t1 <= ta = slice t0 t1 a
| t0 >= ta = slice (t0 - ta) (t1 - ta) b
| otherwise = slice t0 ta a :+: slice 0 (t1 - ta) b
where ta = dur a
-- Structure
instance Construct (Media c) where
prim = Prim
instance Arrangeable (Media c a) where
(+:+) = (:+:)
(=:=) = (:=:)
instance Controlable c (Media c a) where
control = Control
----------------------------------------------------
-- Meaning
-- | Media imterpretation
--
-- given two functions (to convert elementary value @(a -> b)@,
-- and to convert modifiers @(c -> b -> b)@) `fromMedia` interprets 'Media' structure
fromMedia :: Arrangeable b => (a -> b) -> (c -> b -> b) -> Media c a -> b
fromMedia prim mod = fold prim (+:+) (=:=) mod
--------------------------------------------------------------------------------------
-- | Event type
--
-- @(t, dt, a)@ - value @a@ starts at @t@ and lasts for @dt@
type Event t a = (t, t, a)
-- | list of events with given total time
data EventList t a = EventList t [Event t a]
deriving (Show, Eq)
toEvent :: (Temporal t (m a), ToMaybe m) => m a -> EventList t a
toEvent a = EventList (dur a) $ maybe [] (return . singleEvent) $ toMaybe a
where singleEvent x = (0, dur a, x)
mapEvent :: Dur t => (a -> b) -> Event t a -> Event t b
mapEvent f (t, dt, a) = (t, dt, f a)
instance Dur t => Functor (EventList t) where
fmap f (EventList t es) = EventList t $ fmap (mapEvent f) es
-- time
instance Dur t => Temporal t (EventList t a) where
none t = EventList t []
dur (EventList t _) = t
instance (Dur t, Stretchable t a) => Stretchable t (EventList t a) where
stretch d (EventList t es) = EventList (d * t) $ map (stretchEvent d) es
where stretchEvent d (t, dt, a) = (d * t, d * dt, stretch d a)
instance Dur t => TemporalFunctor t (EventList t) where
tmap f (EventList t es) = EventList t $ map (tmapEvent f) es
where tmapEvent f (t, dt, a) = (t, dt, f dt a)
-- structure
instance Dur t => Construct (EventList t) where
prim a = EventList 1 [(0, 1, a)]
instance Dur t => Arrangeable (EventList t a) where
(EventList t es) +:+ (EventList t' es') =
EventList (t + t') (es ++ map (delayEvent t) es')
where delayEvent d (t, dt, a) = (t + d, dt, a)
(EventList t es) =:= (EventList t' es') =
EventList (max t t') $ merge es es'
where merge [] x = x
merge x [] = x
merge (a@(ta, _, _):as) (b@(tb, _, _):bs)
| ta < tb = a : merge as (b:bs)
| otherwise = b : merge (a:as) bs
instance Dur t => Controlable () (EventList t a) where
control = const id
-- meaning
-- | converting to 'EventList'
--
-- 'toEventList' mapps generic temporal value @(m a)@ that can be
-- represented with @(t, Maybe a)@ to 'EventList'
toEventList :: (Temporal t (m a), ToMaybe m)
=> (c -> EventList t a -> EventList t a)
-> Media c (m a) -> EventList t a
toEventList = fromMedia toEvent
-------------------------------------------------------------------------
-------------------------------------------------------------------
-- Special case : Media with explicit time. Value is unit
--
-- | Media with explicit time
--
-- Value is unit (undividable, invariant to reverse and time stretching)
-- O(1) 'dur'
data Dur t => MediaUnit t c a = MediaUnit t (Media c (Unit t a))
unMediaUnit :: Dur t => MediaUnit t c a -> Media c (Unit t a)
unMediaUnit (MediaUnit _ m) = m
instance Dur t => Functor (MediaUnit t c) where
fmap f (MediaUnit t m) = MediaUnit t $ fmap (fmap f) m
instance Dur t => Monad (MediaUnit t c) where
return = MediaUnit 1 . return . return
(MediaUnit t ma) >>= f = MediaUnit (dur ma') ma'
where ma' = ft =<< ma
ft ta = case unMediaUnit . f <$> ta of
(Unit t (Just a)) -> stretch t a
(Unit t Nothing) -> none t
instance Dur t => Applicative (MediaUnit t c) where
pure = return
(<*>) = ap
-- time
instance Dur t => Temporal t (MediaUnit t c a) where
none t = MediaUnit t $ none t
dur (MediaUnit t _) = t
instance Dur t => Stretchable t (MediaUnit t c a) where
stretch d (MediaUnit t m) = MediaUnit (t * d) $ stretch d m
instance Dur t => TemporalFunctor t (MediaUnit t c) where
tmap f (MediaUnit t m) = MediaUnit t $ fmap (tmap f) m
-- transformers
-- | 'fold' replica for MediaUnit
foldU :: Dur t => (t -> a -> b) -> (b -> b -> b) -> (b -> b -> b) -> (c -> b -> b)
-> MediaUnit t c a -> Maybe b
foldU prim seq par mod = fold prim' (liftA2 seq) (liftA2 par) mod' . unMediaUnit
where prim' (Unit t a) = prim t <$> a
mod' c = fmap (mod c)
instance Dur t => Reversible (MediaUnit t c a) where
reverse (MediaUnit t m) = MediaUnit t $ reverse $ m
instance Dur t => Sliceable t (MediaUnit t c a) where
slice t0 t1 (MediaUnit t a) = MediaUnit (t1 - t0) $ slice t0 t1 a
-- Structure
instance Dur t => Construct (MediaUnit t c) where
prim = MediaUnit 1 . prim . prim
instance Dur t => Arrangeable (MediaUnit t c a) where
a +:+ b = MediaUnit (dur a + dur b) $ (unMediaUnit a) +:+ (unMediaUnit b)
a =:= b
| ta < tb = f tb (a' +:+ none (tb - ta)) b'
| ta > tb = f ta (b' +:+ none (ta - tb)) a'
| otherwise = f ta a' b'
where ta = dur a
tb = dur b
a' = unMediaUnit a
b' = unMediaUnit b
f t a b = MediaUnit t $ a =:= b
instance Dur t => Controlable c (MediaUnit t c a) where
control c (MediaUnit t a) = MediaUnit t $ control c a
-- meaning
-- | Interpretation of 'MediaUnit'
--
-- it relies on properties of 'Unit' (it's temporal and covertible to 'Maybe')
fromMediaUnit :: Dur t => (c -> EventList t a -> EventList t a)
-> MediaUnit t c a -> EventList t a
fromMediaUnit f = toEventList f . unMediaUnit
------------------------------------------------------------
-- Unit
-- | unit values that can happen and lasts for some time
data Dur t => Unit t a = Unit t (Maybe a)
deriving (Show, Eq)
instance Dur t => Functor (Unit t) where
fmap f (Unit t a) = Unit t $ fmap f a
instance Dur t => Monad (Unit t) where
return = prim
(Unit t a) >>= f = case fmap f a of
Nothing -> none t
Just (Unit t' b) -> Unit (t * t') b
instance Dur t => Applicative (Unit t) where
pure = return
(<*>) = ap
-- time
instance Dur t => Temporal t (Unit t a) where
none t = Unit t Nothing
dur (Unit t _) = t
instance Dur t => Stretchable t (Unit t a) where
stretch d (Unit t a) = Unit (d * t) a
instance Dur t => ToMaybe (Unit t) where
toMaybe (Unit _ a) = a
instance Dur t => TemporalFunctor t (Unit t) where
tmap f (Unit t a) = Unit t $ fmap (f t) a
-- Unit transformers
instance Dur t => Reversible (Unit t a) where
reverse = id
instance Dur t => Sliceable t (Unit t a) where
slice t0 t1 u@(Unit t a)
| t1 < t0 = sliceErrorMessage
| t1 < (t - eps) || t0 > eps = none $ t1 - t0
| otherwise = u
where eps = 1e-6
-- structure
instance Dur t => Construct (Unit t) where
prim a = Unit 1 $ Just a