Reactive normal form

The post “Reactive values from the future” presented a simple formulation of functional events and reactive values, built on a functional notion of future values. In the current post, I’ll describe some of the implementation of events and reactive values*.

Reactive values


As mentioned previously, the stepper function corresponds directly to the representation of Reactive.

data Reactive a = Stepper a (Event a)

stepper :: a -> Event a -> Reactive a
stepper = Stepper

The representation of all other functions must therefore produce their result in terms of stepper, which thereby serves as a kind of reactive normal form (RNF). The implementation can be understood as a set of rewrite rules that convert arbitrary reactive expressions into RNF. (These rewrite rules are all provably correct based on a simple semantics of events and reactive values.)

The more general switching form can be expressed in terms of switcher and (monadic) join:

switcher :: Reactive a -> Event (Reactive a) -> Reactive a
r `switcher` er = join (r `stepper` er)

We’ll see below how to reduce join to a stepper.


As a simple example, fmap f applies a function f to a reactive value pointwise, which is equivalent to applying f to the initial value and to each occurrence value.

instance Functor Reactive where
  fmap f (a `Stepper` e) = f a `stepper` fmap f e

The Event functor is also easily defined. Recall that an event is a future reactive value:

newtype Event = Future (Reactive a)

We want to map a given function over that reactive value, which we can do by combining fmap on Future with fmap on Reactive.

instance Functor Event where
  fmap f (Event fut) = Event ((fmap.fmap) f fut)


The Functor definitions for Reactive and Event are straightforward, because the Stepper structure is easily preserved. A more challenging case is Applicative.

instance Applicative Reactive where ... 

First the easy part. A pure value a becomes reactive by using a as the initial value and mempty as the (never-occuring) change event:

  pure a = a `stepper` mempty

Consider next applying a reactive function to a reactive argument:

  rf@(f `Stepper` Event futf) <*> rx@(x `Stepper` Event futx) =
    f x `stepper` Event fut
   where ...

The initial value is f x, and the change event occurs each time either the function or the argument changes. If the function changes first, we will (in the future) apply a new reactive function to an old argument:

           fmap ( rf' -> rf' <*> rx) futf

Similarly, if the argument changes first, we will apply an old reactive function and a new argument:

           fmap ( rx' -> rf <*> rx') futx

Combining these two futures as alternatives:

     fut = fmap ( rf' -> rf' <*> rx ) futf `mappend`
           fmap ( rx' -> rf  <*> rx') futx

A wonderful thing about this (<*>) definition for Reactive is that it automatically caches the previous value of the function or argument when the argument or function changes. This caching property is especially handy in nested applications of (<*>), which can arise either explicitly or through liftA2, liftA3, etc. Consider u == liftA2 f r s or, equivalently, u == pure f <*> r <*> s, where r and s are reactive values, with initial values r0 and s0, respectively. (The (<*>) operator is associates to the left.) The initial value u0 of u is f r0 s0. If r changes from r0 to r1, then the new value of pure f <*> r will be f r1, which then gets applied to s0, i.e., u1 == f r1 s0. However, if instead s changes from s0 to s1, then u1 == f r0 s1. In this latter case, the old value f r0 of pure f <*> r is passed on without having to be recomputed. The savings is significant for functions that do some work based partial applications.


Above we saw a simple definition of switcher:

r `switcher` er = join (r `stepper` er)

How does join work for Reactive? Rather than using the standard definition of join in terms of (>>=), let’s do the reverse:

r >>= h = joinR (fmap h r)

where joinR is join for reactive values:

joinR :: Reactive (Reactive a) -> Reactive a

Defining joinR is similar to (<*>) above. For a reactive-reactive value rr, either rr or the reactive value that is the initial value of rr changes first.

joinR ((a `Stepper` Event fut) `Stepper` e'@(Event fut')) =
  a `stepper` Event fut''
   -- If fut  arrives first, switch and continue waiting for e'.
   -- If fut' arrives first, abandon fut and keep switching with new
   -- reactive values from fut'.
   fut'' = fmap (`switcher` e') fut `mappend` fmap join fut'

Dynamic optimization

Operations on events and reactive values dynamically re-optimize themselves. For instance, the reactive value fmap f r is known to quiesce (stop changing) when r quiesces, which is knowable because futures have a special representation for the future that never comes. (Really, only easy common cases are optimized. An example of a hard-to-detect never-occuring event would be filtering out first the odd numbers and then the even numbers from an Int-valued event.) Similarly, the reactive value liftA2 f r s quiesces once both r and s quiesce. Once reactive values quiesce, no more computational resource is devoted to them.


The definitions above are mainly for reactive values. What about functional events?


Events form a monoid: mempty is the never-occurring event, which is represented by the never-occuring future, while e `mempty` e' is an event consisting of all occurrences from e and e'.

instance Monoid (Event a) where
  mempty  = Event mempty
  Event fut `mappend` Event fut' = Event (fut `merge` fut')

This merge operation combines two Future streams into one.

merge :: Future (Reactive a) -> Future (Reactive a) -> Future (Reactive a)
Never `merge` fut   = fut
fut   `merge` Never = fut
u     `merge` v     =
  (onFut (`merge` v) <$> u) `mappend` (onFut (u `merge`) <$> v)
   onFut f (a `Stepper` Event t') = a `stepper` Event (f t')

The mappend used in the definition of merge is from Future. It picks the earlier future and abandons the other. Thus the first of the two events to occur will contribute an occurrence value (a), leaving a residual event, which is combined with the not-yet-occurring event.

Applicative and Monad

The Applicative and Monad instances for events are like those for lists, and unlike those for reactive values (given above). For instance, the occurrences of ef <*> ex consist of f x for all occurrences f of ef and occurrences x of ex. See the source code for details.


I’m happy with the simplicity of this new functional formulation of events and reactive values. While most previous FRP implementations used polling, Reactive is change-driven and so has the can be much more efficient. It also uses a dynamically self-optimizing representation.


  1. Conal Elliott » Blog Archive » Invasion of the composable Mutant-Bots:

    […] a few recent posts, I’ve been writing about a new basis for functional reactive programming (FRP), […]

  2. Conal Elliott » Blog Archive » Applicative bots:

    […] Instead, I’d like lf <*> lx to have an output whenever either lf or lx has an output. If lf has an output f, I want to output f x, where x is the most recent lx output. Similarly, if lx has an output, I want to output f x, where f is the most recent lf output. This behavior is exactly how Applicative works for Reactive, as described in reactive-normal-form. […]

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