Ever since ActiveVRML, the model we’ve been using in functional reactive programming (FRP) for interactive behaviors is (T->a) -> (T->b), for dynamic (time-varying) input of type a and dynamic output of type b (where T is time). In “Classic FRP” formulations (including ActiveVRML, Fran & Reactive), there is a “behavior” abstraction whose denotation is a function of time. Interactive behaviors are then modeled as host language (e.g., Haskell) functions between behaviors. Problems with this formulation are described in Why classic FRP does not fit interactive behavior. These same problems motivated “Arrowized FRP”. In Arrowized FRP, behaviors (renamed “signals”) are purely conceptual. They are part of the semantic model but do not have any realization in the programming interface. Instead, the abstraction is a signal transformer, SF a b, whose semantics is (T->a) -> (T->b). See Genuinely Functional User Interfaces and Functional Reactive Programming, Continued.

Whether in its classic or arrowized embodiment, I’ve been growing uncomfortable with this semantic model of functions between time functions. A few weeks ago, I realized that one source of discomfort is that this model is mostly junk.

This post contains some partially formed thoughts about how to eliminate the junk (“garbage collect the semantics”), and what might remain.

There are two generally desirable properties for a denotational semantics: full abstraction and junk-freeness. Roughly, “full abstraction” means we must not distinguish between what is (operationally) indistinguishable, while “junk-freeness” means that every semantic value must be denotable.

FRP’s semantic model, (T->a) -> (T->b), allows not only arbitrary (computable) transformation of input values, but also of time. The output at some time can depend on the input at any time at all, or even on the input at arbitrarily many different times. Consequently, this model allows respoding to future input, violating a principle sometimes called “causality”, which is that outputs may depend on the past or present but not the future.

In a causal system, the present can reach backward to the past but not forward the future. I’m uneasy about this ability as well. Arbitrary access to the past may be much more powerful than necessary. As evidence, consult the system we call (physical) Reality. As far as I can tell, Reality operates without arbitrary access to the past or to the future, and it does a pretty good job at expressiveness.

Moreover, arbitrary past access is also problematic to implement in its semantically simple generality.

There is a thing we call informally “memory”, which at first blush may look like access to the past, it isn’t really. Rather, memory is access to a present input, which has come into being through a process of filtering, gradual accumulation, and discarding (forgetting). I’m talking about “memory” here in the sense of what our brains do, but also what all the rest of physical reality does. For instance, weather marks on a rock are part of the rock’s (present) memory of the past weather.

A very simple memory-less semantic model of interactive behavior is just a -> b. This model is too restrictive, however, as it cannot support any influence of the past on the present.

Which leaves a question: what is a simple and adequate formal model of interactive behavior that reaches neither into the past nor into the future, and yet still allows the past to influence the present? Inspired in part by a design principle I call “what would reality do?” (WWRD), I’m happy to have some kind of infinitesimal access to the past, but nothing further.

My current intuition is that differentiation/integration plays a crucial role. That information is carried forward moment by moment in time as “momentum” in some sense.

I call intuition cosmic fishing. You feel a nibble, then you’ve got to hook the fish. – Buckminster Fuller

Where to go with these intuitions?

Perhaps interactive behaviors are some sort of function with all of its derivatives. See Beautiful differentiation for an specification and derived implementation of numeric operations, and more generally of Functor and Applicative, on which much of FRP is based.

I suspect the whole event model can be replaced by integration. Integration is the main remaining piece.

How weak a semantic model can let us define integration?

Thanks

My thanks to Luke Palmer and to Noam Lewis for some clarifying chats about these half-baked ideas. And to the folks on #haskell IRC for brainstorming titles for this post. My favorite suggestions were

• luqui: instance HasJunk FRP where
• luqui: Functional reactive programming’s semantic baggage
• sinelaw: FRP, please take out the trash!
• cale: Garbage collecting the semantics of FRP
• BMeph: Take out the FRP-ing Trash

all of which I preferred over my original “FRP is mostly junk”.

I’ve been playing with a simple/general semantics for 3D. In the process, I was surprised to see that a key part of the semantics looks exactly like a key part of the semantics of functional reactivity as embodied in the library Reactive. A closer look revealed a closer connection still, as described in this post.

What is 3D rendering?

Most programmers think of 3D rendering as being about executing sequence of side-effects on frame buffer or some other mutable array of pixels. This way of thinking (sequences of side-effects) comes to us from the design of early sequential computers. Although computer hardware architecture has evolved a great deal, most programming languages, and hence most programming thinking, is still shaped by the this first sequential model. (See John Backus’s Turing Award lecture Can Programming Be Liberated from the von Neumann Style? A functional style and its algebra of programs.) The invention of monadic Imperative functional programming allows Haskellers to think and program within the imperative paradigm as well.

What’s a functional alternative? Rendering is a function from something to something else. Let’s call these somethings (3D) “Geometry” and (2D) “Image”, where Geometry and Image are types of functional (immutable) values.

type Rendering = Image Color

render :: Geometry -> Rendering

To simplify, I’m assuming a fixed view. What remains is to define what these two types mean and, secondarily, how to represent and implement them.

An upcoming post will suggest an answer for the meaning of Geometry. For now, think of it as a collection of curved and polygonal surfaces, i.e., the outsides (boundaries) of solid shapes. Each point on these surfaces has a location, a normal (perpendicular direction), and material properties (determining how light is reflected by and transmitted through the surface at the point). The geometry will contain light sources.

Next, what is the meaning of Image? A popular answer is that an image is a rectangular array of finite-precision encodings of color (e.g., with eight bits for each of red, blue, green and possibly opacity). This answer is leads to poor compositionality and complex meanings for operations like scaling and rotation, so I prefer another model. As in Pan, an image (the meaning of the type Image Color) is a function from infinite continuous 2D space to colors, where the Color type includes partial opacity. For motivation of this model and examples of its use, see Functional images and the corresponding Pan gallery of functional images. Composition occurs on infinite & continuous images.

After all composition is done, the resulting image can be sampled into a finite, rectangular array of finite precision color encodings. I’m talking about a conceptual/semantic pipeline. The implementation computes the finite sampling without having to compute the values for entire infinite image.

Rendering has several components. I’ll just address one and show how it relates to functional reactive programming (FRP).

Visual occlusion

One aspect of 3D rendering is hidden surface determination. Relative to the viewer’s position and orientation, some 3D objects may fully or partially occluded by nearer objects.

An image is a function of (infinite and continuous) 2D space, so specifying that function is determining its value at every sample point. Each point can correspond to a number of geometric objects, some closer and some further. If we assume for now that our colors are fully opaque, then we’ll need to know the color (after transformation and lighting) of the nearest surface point that is projected onto the sample point. (We’ll remove this opacity assumption later.)

Let’s consider how we’ll combine two Geometry values into one:

union :: Geometry -> Geometry -> Geometry

Because of occlusion, the render function cannot be compositional with respect to union. If it were, then there would exist a functions unionR such that

forall ga gb. render (ga `union` gb) == render ga `unionR` render gb

In other words, to render a union of two geometries, we can render each and combine the results.

The reason we can’t find such a unionR is that render doesn’t let unionR know how close each colored point is. A solution then is simple: add in the missing depth information:

type RenderingD = Image (Depth, Color)  -- first try

renderD :: Geometry -> RenderingD

Now we have enough information for compositional rendering, i.e., we can define unionR such that

forall ga gb. renderD (ga `union` gb) == renderD ga `unionR` renderD gb

where

unionR :: RenderingD -> RenderingD -> RenderingD

unionR im im' p = if d <= d' then (d,c) else (d',c')
 where
   (d ,c ) = im  p
   (d',c') = im' p

When we’re done composing, we can discard the depths:

render g = snd . renderD g

or, with Semantic editor combinators:

render = (result.result) snd renderD

Simpler, prettier

The unionR is not very complicated, but still, I like to tease out common structure and reuse definitions wherever I can. The first thing I notice about unionR is that it works pointwise. That is, the value at a point is a function of the values of two other images at the same point. The pattern is captured by liftA2 on functions, thanks to the Applicative instance for functions.

liftA2 :: (b -> c -> d) -> (a -> b) -> (a -> c) -> (a -> d)

So that

unionR = liftA2 closer

closer (d,c) (d',c') = if d <= d' then (d,c) else (d',c')

Or

closer dc@(d,_) dc'@(d',_) = if d <= d' then dc else dc'

Or even

closer = minBy fst

where

minBy f u v = if f u <= f v then u else v

This definition of unionR is not only simpler, it’s quite a bit more general, as type inference reveals:

unionR :: (Ord a, Applicative f) => f (a,b) -> f (a,b) -> f (a,b)

closer :: Ord a => (a,b) -> (a,b) -> (a,b)

Once again, simplicity and generality go hand-in-hand.

Another type class morphism

Let’s see if we can make union rendering simpler and more inevitable. Rendering is nearly a homomorphism. That is, render nearly distributes over union, but we have to replace union by unionR. I’d rather eliminate this discrepancy, ending up with

forall ga gb. renderD (ga `op` gb) == renderD ga `op` renderD gb

for some op that is equal to union on the left and unionR on the right. Since union and unionR have different types (with neither being a polymorphic instance of the other), op will have to be a method of some type class.

My favorite binary method is mappend, from Monoid, so let’s give it a try. Monoid requires there also to be an identity element mempty and that mappend be associative. For Geometry, we can define

instance Monoid Geometry where
  mempty  = emptyGeometry
  mappend = union

Images with depth are a little trickier. Image already has a Monoid instance, whose semantics is determined by the principle of type class morphisms, namely

The meaning of an instance is the instance of the meaning

The meaning of an image is a function, and that functions have a Monoid instance:

instance Monoid b => Monoid (a -> b) where
  mempty = const mempty
  f `mappend` g =  a -> f a `mappend` g a

which simplifies nicely to a standard form, by using the Applicative instance for functions.

instance Applicative ((->) a) where
  pure      = const
  hf <*> xf =  a -> (hf a) (xf a)

instance Monoid b => Monoid (a -> b) where
  mempty  = pure   mempty
  mappend = liftA2 mappend

We’re in luck. Since we’ve defined unionR as liftA2 closer, so we just need it to turn out that closer == mappend and that closer is associative and has an identity element.

However, closer is defined on pairs, and the standard Monoid instance on pairs doesn’t fit.

instance (Monoid a, Monoid b) => Monoid (a,b) where
  mempty = (mempty,mempty)
  (a,b) `mappend` (a',b') = (a `mappend` a', b `mappend` b')

To avoid this conflict, define a new data type to be used in place of pairs.

data DepthG d a = Depth d a  -- first try

Alternatively,

newtype DepthG d a = Depth { unDepth :: (d,a) }

I’ll go with this latter version, as it turns out to be more convenient.

Then we can define our monoid:

instance Monoid (DepthG d a) where
  mempty  = Depth (maxBound,undefined)
  Depth p `mappend` Depth p' = Depth (p `closer` p')

The second method definition can be simplified nicely

  mappend = inDepth2 closer

where

  inDepth2 = unDepth ~> unDepth ~> Depth

using the ideas from Prettier functions for wrapping and wrapping and the notational improvement from Matt Hellige’s Pointless fun.

FRP — Future values

The Monoid instance for Depth may look familiar to you if you’ve been following along with my future values or have read the paper Simply efficient functional reactivity. A future value has a time and a value. Usually, the value cannot be known until its time arrives.

newtype FutureG t a = Future (Time t, a)

instance (Ord t, Bounded t) => Monoid (FutureG t a) where
  mempty = Future (maxBound, undefined)
  Future (s,a) `mappend` Future (t,b) =
    Future (s `min` t, if s <= t then a else b)

When we’re using a non-lazy (flat) representation of time, this mappend definition can be written more simply:

  mappend = minBy futTime

  futTime (Future (t,_)) = t

Equivalently,

  mappend = inFuture2 (minBy fst)

The Time type is really nothing special about time. It is just a synonym for the Max monoid, as needed for the Applicative and Monad instances.

This connection with future values means we can discard more code.

type RenderingD d = Image (FutureG d Color)
renderD :: (Ord d, Bounded d) => Geometry -> RenderingD d

Now we have our monoid (homo)morphism properties:

renderD mempty == mempty

renderD (ga `mappend` gb) == renderD ga `mappend` renderD gb

And we’ve eliminated the code for renderR by reusing and existing type (future values).

Future values?

What does it mean to think about depth/color pairs as being “future” colors? If we were to probe outward along a ray, say at the speed of light, we would bump into some number of 3D objects. The one we hit earliest is the nearest, so in this sense, mappend on futures (choosing the earlier one) is the right tool for the job.

I once read that a popular belief in the past was that vision (light) reaches outward to strike objects, as I’ve just described. I’ve forgotten where I read about that belief, though I think in a book about perspective, and I’d appreciate a pointer from someone else who might have a reference.

We moderns believe that light travels to us from the objects we see. What we see of nearby objects comes from the very recent past, while of further objects we see the more remote past. From this modern perspective, therefore, the connection I’ve made with future values is exactly backward. Now that I think about it in this way, of course it’s backward, because we see (slightly) into the past rather than the future.

Fixing this conceptual flaw is simple: define a type of “past values”. Give them exactly the same representation as future values, and deriving its class instances entirely.

newtype PastG t a = Past (FutureG t a)
  deriving (Monoid, Functor, Applicative, Monad)

Alternatively, choose a temporally neutral replacement for the name “future values”.

The bug in Z-buffering

The renderD function implements continuous, infinite Z-buffering, with mappend performing the z-compare and conditional overwrite. Z-buffering is the dominant algorithm used in real-time 3D graphics and is supported in hardware on even low-end graphics hardware (though not in its full continuous and infinite glory).

However, Z-buffering also has a serious bug: it is only correct for fully opaque colors. Consider a geometry g and a point p in the domain of the result image. There may be many different points in g that project to p. If g has only fully opaque colors, then at most one place on g contributes to the rendered image at p, and specifically, the nearest such point. If g is the union (mappend) of two other geometries, g == ga `union` gb, then the nearest contribution of g (for p) will be the nearer (mappend) of the nearest contributions of ga and of gb.

When colors may be partially opaque, the color of the rendering at a point p can depend on all of the points in the geometry that get projected to p. Correct rendering in the presence of partial opacity requires a fold that combines all of the colors that project onto a point, in order of distance, where the color-combining function (alpha-blending) is not commutative. Consider again g == ga `union` gb. The contributions of ga to p might be entirely closer than the contributions of gb, or entirely further, or interleaved. If interleaved, then the colors generated from each cannot be combined into a single color for further combination. To handle the general case, replace the single distance/color pair with an ordered collection of them:

type RenderingD d = Image [FutureG d Color]  -- multiple projections, first try

Rendering a union (mappend) requires a merging of two lists of futures (distance/color pairs) into a single one.

More FRP — Events

Sadly, we’ve now lost our monoid morphism, because list mappend is (++), not the required merging. However, we can fix this problem as we did before, by introducing a new type.

Or, we can look for an existing type that matches our required semantics. There is just such a thing in the Reactive formulation of FRP, namely an event. We can simply use the FRP Event type:

type RenderingD d = Image (EventG d Color)

renderD :: (Ord d, Bounded d) => Geometry -> RenderingD d

Spatial transformation

Introducing depths allowed rendering to be defined compositionally with respect to geometric union. Is the depth model, enhanced with lists (events), sufficient for compositionality of rendering with respect to other Geometry operations as well? Let’s look at spatial transformation.

(*%)  :: Transform3 -> Geometry -> Geometry

Compositionally of rendering would mean that we can render xf *% g by rendering g and then using xf in some way to transform that rendering. In other words there would have to exist a function (*%%) such that

forall xf g. renderD (xf *% g) == xf *%% renderD g

I don’t know if the required (*%%) function exists, or what restrictions on Geometry or Transform3 it implies, or whether such a function could be useful in practice. Instead, let’s change the type of renderings again, so that rendering can accumulate transformations and apply them to surfaces.

type RenderingDX = Transform3 -> RenderingD

renderDX :: (Ord d, Bounded d) => Geometry -> RenderingDX d

with or without correct treatment of partial opacity (i.e., using futures or events).

This new function has a simple specification:

renderDX g xf == renderD (xf *% g)

from which it follows that

renderD g == renderDX g identityX

Rendering a transformed geometry then is a simple accumulation, justified as follows:

renderDX (xfi *% g)

  == {- specification of renderDX -}

 xfo -> renderD (xfo *% (xfi *% g))

  == {- property of transformation -}

 xfo -> renderD ((xfo `composeX` xfi) *% g)

  == {- specification of renderDX  -}

 xfo -> renderDX g (xfo `composeX` xfi)

Render an empty geometry:

renderDX mempty

  == {- specification of renderDX -}

 xf -> renderD (xf *% mempty)

  == {- property of (*%) and mempty -}

 xf -> renderD mempty

  == {- renderD is a monoid morphism -}

 xf -> mempty

  == {- definition of pure on functions -}

pure mempty

  == {- definition of mempty on functions -}

mempty

Render a geometric union:

renderDX (ga `mappend` gb)

  == {- specification of renderDX -}

 xf -> renderD (xf *% (ga `mappend` gb))

  == {- property of transformation and union -}

 xf -> renderD ((xf *% ga) `mappend` (xf *% gb))

  == {- renderD is a monoid morphism -}

 xf -> renderD (xf *% ga) `mappend` renderD (xf *% gb)

  == {- specification of renderDX  -}

 xf -> renderDX ga xf `mappend` renderDX gb xf

  == {- definition of liftA2/(<*>) on functions -}

liftA2 mappend (renderDX ga) (renderDX gb)

  == {- definition of mappend on functions -}

renderDX ga `mappend` renderDX gb

Hurray! renderDX is still a monoid morphism.

The two properties of transformation and union used above say together that (xf *%) is a monoid morphism for all transforms xf.

An earlier post introduced functional future values, which are values that cannot be known until the future, but can be manipulated in the present. That post presented a simple denotational semantics of future values as time/value pairs. With a little care in the definition of Time (using the Max monoid), the instances of Functor, Applicative, Monad are all derived automatically.

A follow-up post gave an implementation of Future values via multi threading. Unfortunately, that implementation did not necessarily satisfy the semantics, because it allowed the nondeterminism of thread scheduling to leak through. Although the implementation is usually correct, I wasn’t satisfied.

After a while, I hit upon an idea that really tickled me. My original simple semantics could indeed serve as a correct and workable implementation if I used a subtler form of time that could reveal partial information. Implementing this subtler form of time turned out to be quite tricky, and was my original motivation for the unamb operator described in the paper Push-pull functional reactive programming and the post Functional concurrency with unambiguous choice.

It took me several days of doodling, pacing outside, and talking to myself before the idea for unamb broke through. Like many of my favorite ideas, it’s simple and obvious in retrospect: to remove the ambiguity of nondeterministic choice (as in the amb operator), restrict its use to values that are equal when non-bottom. Whenever we have two different methods of answering the same question (or possibly failing), we can use unamb to try them both. Failures (errors or non-termination) are no problem in this context. A more powerful variation on unamb is the least upper bound operator lub, as described in Merging partial values.

I’ve been having trouble with the unamb implementation. When two (compatible) computations race, the loser gets killed so as to free up cycles that are no longer needed. My first few implementations, however, did not recursively terminate other threads spawned in service of abandoned computations (from nested use of unamb). I raised this problem in Smarter termination for thread racing, which suggested some better definitions. In the course of several helpful reader comments, some problems with my definitions were addressed, particularly in regard to blocking and unblocking exceptions. None of these definitions so far has done the trick reliably, and now it looks like there is a bug in the GHC run-time system. I hope the bug (if there is one) will be fixed soon, because I’m seeing more & more how unamb and lub can make functional programming even more modular (just as laziness does, as explained by John Hughes in Why Functional Programming Matters).

I started playing with future values and unambiguous choice as a way to implement Reactive, a library for functional reactive programming (FRP). (See Reactive values from the future and Push-pull functional reactive programming.) Over the last few days, I’ve given some thought to ways to implement future values without unambiguous choice. This post describes one such alternative.

Edits:

• 2010-08-25: Replaced references to Simply efficient functional reactivity with Push-pull functional reactive programming. The latter paper supercedes the former.
• 2010-08-25: Fixed the unFuture field of FutureG to be TryFuture.

Futures, presently

The current Future type is just a time and a value, wrapped in a a newtype:

newtype FutureG t a = Future (Time t, a)
  deriving (Functor, Applicative, Monad)

Where the Time type is defined via the Max monoid. The derived instances have exactly the intended meaning for futures, as explained in the post Future values and the paper Push-pull functional reactive programming. The “G” in the name FutureG refers to generalized over time types.

Note that Future is parameterized over both time and value. Originally, I intended this definition as a denotational semantics of future values, but I realized that it could be a workable implementation with a lazy enough t. In particular, the times have to reveal lower bounds and allow comparisons before they’re fully known.

Warren Burton explored an applicable notion in the 1980s, which he called “improving values”, having a concurrent implementation but deterministic functional semantics. (See the paper Encapsulating nondeterminacy in an abstract data type with deterministic semantics or the paper Indeterminate behavior with determinate semantics in parallel programs. I haven’t found a freely-available online copy of either.) I adapted Warren’s idea and gave it an implementation via unamb.

Another operation finds the earlier of two futures. This operation has an identity and is associative, so I wrapped it up as a Monoid instance:

instance (Ord t, Bounded t) => Monoid (FutureG t a) where
  mempty = Future (maxBound, undefined)
  Future (s,a) `mappend` Future (t,b) =
    Future (s `min` t, if s <= t then a else b)

This mappend definition could be written more simply:

  u@(Future (t,_)) `mappend` u'@(Future (t',_)) =
    if t <= t' then u else u'

However, the less simple version has more potential for laziness. The time type might allow yielding partial information about a minimum before both of its arguments are fully known, which is the case with improving values.

Futures as functions

The Reactive library uses futures to define and implement reactivity, i.e., behaviors specified piecewise. Simplifying away the notion of events for now,

until :: BehaviorG t a -> FutureG t (BehaviorG t a) -> BehaviorG t a

The semantics (but not implementation) of BehaviorG is given by

at :: BehaviorG t a -> (t -> a)

The semantics of until:

(b `until` Future (t',b')) `at` t = b'' `at` t
 where
   b'' = if t <= t' then b else b'

FRP (multi-occurrence) events are then built on top of future values, and reactivity on top of until.

The semantics of until shows what information we need from futures: given a time t, we need to know whether t is later than the future’s time and, if so, what the future’s value is. For other purposes, we’ll also want to know the future’s time, but again, only once we’re past that time. We might, therefore, represent futures as a function that gives exactly this information. I’ll call this function representation “function futures” and use the prefix “S” to distinguish the original “simple” futures from these function futures.

type TryFuture t a = Time t -> Maybe (S.FutureG t a)

tryFuture :: F.FutureG t a -> TryFuture t a

Given a probe time, tryFuture gives Nothing if the time is before or at the future’s time, or Just u otherwise, where u is the simple future.

We could represent F.FutureG simply as TryFuture:

type F.FutureG = TryFuture  -- first try

But then we’d be stuck with the Functor and Applicative instances for functions instead of futures. Adding a newtype fixes that problem:

newtype FutureG t a = Future { unFuture :: TryFuture t a } -- second try

With this representation we can easily construct and try out function futures:

future :: TryFuture t a -> FutureG t a
future = Future

tryFuture :: FutureG t a -> TryFuture t a
tryFuture = unFuture

I like to define helpers for working inside representations:

inFuture  :: (TryFuture t a -> TryFuture t' a')
          -> (FutureG   t a -> FutureG   t' a')

inFuture2 :: (TryFuture t a -> TryFuture t' a' -> TryFuture t'' a'')
          -> (FutureG   t a -> FutureG   t' a' -> FutureG   t'' a'')

The definitions of these helpers are very simple with the ideas from Prettier functions for wrapping and wrapping and a lovely notation from Matt Hellige’s Pointless fun.

inFuture  = unFuture ~> Future

inFuture2 = unFuture ~> inFuture 

(~>) :: (a' -> a) -> (b -> b') -> ((a -> b) -> (a' -> b'))
g ~> h = result h . argument g

These helpers make for some easy definitions in the style of Semantic editor combinators:

instance Functor (FutureG t) where
  fmap = inFuture.fmap.fmap.fmap

instance (Bounded t, Ord t) => Applicative (FutureG t) where
  pure  = Future . pure.pure.pure
  (<*>) = (inFuture2.liftA2.liftA2) (<*>)

Type composition

These Functor and Applicative instances (for FutureG t) may look mysterious, but they have a common and inevitable form. Every type whose representation is the (semantic and representational) composition of three functors has this style of Functor instance, and similarly for Applicative.

Instead of repeating this common pattern, let’s make the type composition explicit, using Control.Compose from the TypeCompose library:

type FutureG t = (->) (Time t) :. Maybe :. S.FutureG t  -- actual definition

Now we can throw out inFuture, inFuture2, (~>), and the Functor and Applicative instances. These instances follow from the general instances for type composition.

Monoid

The Monoid instance could also come automatically from type composition:

instance Monoid (g (f a)) => Monoid ((g :. f) a) where
  { mempty = O mempty; mappend = inO2 mappend }

The O here is just the newtype constructor for (:.), and the inO2 function is similar to inFuture2 above.

However, there is another often-useful Monoid instance:

-- standard Monoid instance for Applicative applied to Monoid
instance (Applicative (g :. f), Monoid a) => Monoid ((g :. f) a) where
  { mempty = pure mempty; mappend = liftA2 mappend }

Because these two instances “overlap” are are both useful, neither one is declared in the general case. Instead, specialized instances are declared where needed, e.g.,

instance (Ord t, Bounded t) => Monoid (FutureG t a) where
  mempty  = (  O .  O ) mempty   -- or future mempty
  mappend = (inO2.inO2) mappend

How does the Monoid instance work? Start with mempty. Expanding:

mempty

  == {- definition -} 

O (O mempty)

  == {- mempty on functions -}

O (O (const mempty))

  == {- mempty on Maybe -}

O (O (const Nothing))

So, given any probe time, the empty (never-occurring) future says that it does not occur before the probe time.

Next, mappend:

O (O f) `mappend` O (O f')

  == {- mappend on FutureG -}

O (O (f `mappend` f'))

  == {- mappend on functions -}

O (O ( t -> f t `mappend` f' t))

  == {- mappend on Maybe -}

O (O ( t -> f t `mappendMb` f' t))
  where
    Nothing `mappendMb` mb'    = mb'
    mb `mappendMb` Nothing     = mb
    Just u `mappendMb` Just u' = Just (u `mappend` u')

The mappend in this last line is on simple futures, as defined above, examining the (now known) times and choosing the earlier future. Previously, I took special care in that mappend definition to enable min to produce information before knowing whether t <= t'. However, with this new approach to futures, I expect to use simple (flat) times, so it could instead be

u@(Future (s,_)) `mappend` u'@(Future (s',_)) = if s <= s' then u else u'

or

u `mappend` u' = if futTime u <= futTime u' then u else u'

futTime (Future (t,_)) = t

or just

mappend = minBy futTime

How does mappend work on function futures? Given a test time t, if both future times are at least t, then the combined future’s time is at least t (yielding Nothing). If either future is before t and the other isn’t, then the combined future is the same as the one before t. If both futures are before t, then the combined future is the earlier one. Exactly the desired semantics!

Relating function futures and simple futures

The function-based representation of futures relates closely to the simple representation. Let’s make this relationship explcit by defining mappings between them:

sToF :: Ord t => S.FutureG t a -> F.FutureG t a

fToS :: Ord t => F.FutureG t a -> S.FutureG t a

The first one is easy:

sToF u@(S.Future (t, _)) =
  future ( t' -> if t' <= t then Nothing else Just u)

The reverse mapping, fToS, is trickier and is only defined on the image (codomain) of sToF. I think it can be defined mathematically but not computationally. There are two cases: either the function always returns Nothing, or there is at least one t for which it returns a Just. If the former, then the simple future is mempty, which is S.Future (maxBound, undefined). If the latter, then there is only one such Just, and the simple future is the one in that Just. Together, (sToF, fToS) form a projection-embedding pair.

We won’t really have to implement or invoke these functions. Instead, they serve to further specify the type F.FutureG and the correctness of operations on it. The representation of F.FutureG as given allows many values that do not correspond to futures. To eliminate these representations, require an invariant that a function future must be the result of applying sToF to some simple future.

We’ll require that each operation preserves this invariant. However, let’s prove something stronger, namely that operations on on F.FutureG correspond precisely to the same operations on S.FutureG, via sToF. In other words, sToF preserves the shape of the operations on futures. For type classes, these correspondences are the type class morphisms. For instance, sToF is a Monoid morphism:

sToF mempty == mempty

sToF (u `mappend` u') == sToF u `mappend` sToF u'

Caching futures

This function representation eliminates the need for tricky times (using improving values and unamb), but it loses the caching benefit that lazy functional programming affords to non-function representations. Now let’s reclaim that benefit. The trick is to exploit the restriction that every function future must be (semantically) the image of a simple future under sToF.

Examining the definition of sToF, we can deduce the following monotonicity properties of (legitimate) function futures:

• If the probe function yields Nothing for some t', then it yields Nothing for earlier times.
• If the probe function yields Just u for some t', then it yields Just u for all later times.

We can exploit these monotonicity properties by caching information as we learn it. Caching of this sort is what distinguishes call-by-need from call-by-name and allows lazy evaluation to work efficiently for data representations.

Specifically, let’s save a best-known lower bound for the future time and the simple future when known. Since the lower bound may get modified a few times, I’ll use a SampleVar (thread-safe rewritable variable). The simple future will be discovered only once, so I’ll use an IVar. I’ll keep the function-future for probing when the cached information is not sufficient to answer a query.

Prefix this caching version with a “C”, to distinguish it from function futures (“F”) and the simple futures (“S”):

data C.FutureG t a =
  Future (SampleVar t) (IVar (S.FutureG t a)) (F.FutureG t a)

Either the simple future or the function future will be useful, so we could replace the second two fields with a single one:

data C.FutureG t a =
  Future (SampleVar t) (MVar (Either (F.FutureG t a) (FF.FutureG t a)))

We’ll have to be careful about multiple independent discoveries of the same simple future, which would correspond to multiple writes IVar with the same value. (I imagine there are related mechanics in the GHC RTS for two threads evaluating the same thunk that would be helpful to understand.) I guess I could use a SampleVar and just not worry about multiple writes, since they’d be equivalent. For now, use the IVar version.

The caching representation relates to the function representation by means of two functions:

dToF :: Ord     t => C.FutureG t a -> F.FutureG t a

fToD :: Bounded t => F.FutureG t a -> C.FutureG t a

The implementation

dToF (C.Future tv uv uf) =
  F.Future $  t' -> unsafePerformIO $
    do mb <- tryReadIVar uv
       case mb of
         j@(Just (S.Future (Max t,_))) ->
           return (if t' <= t then Nothing else j)
         Nothing        ->
           do tlo <- readSampleVar tv
              if t' <= tlo then
                 return Nothing
               else
                 do let mb' = F.unFuture uf t'
                    writeIVarMaybe uv mb'
                    return mb'

-- Perhaps write to an IVar
writeIVarMaybe :: IVar a -> Maybe a -> IO ()
writeIVarMaybe v = maybe (return ()) (writeIVar v)

fToD uf = unsafePerformIO $
          do tv <- newSampleVar t0
             uv <- newIVar
             writeIVarMaybe uv (F.unFuture uf t0)
             return (Future tv uv uf)
 where
   t0 = minBound

It’ll be handy to delegate operations to F.Future:

inF :: (Ord t, Bounded t') =>
       (F.FutureG t a -> F.FutureG t' a')
    -> (  FutureG t a ->   FutureG t' a')
inF = dToF ~> fToD

inF2 :: (Ord t, Bounded t', Ord t', Bounded t'') =>
        (F.FutureG t a -> F.FutureG t' a' -> F.FutureG t'' a'')
     -> (  FutureG t a ->   FutureG t' a' ->   FutureG t'' a'')
inF2 = dToF ~> inF

Then

instance (Ord t, Bounded t) => Monoid (FutureG t a) where
  mempty  = fToD mempty
  mappend = inF2 mappend

instance (Ord t, Bounded t) => Functor     (FutureG t) where
  fmap = inF . fmap

instance (Ord t, Bounded t) => Applicative (FutureG t) where
  pure  = fToD . pure
  (<*>) = inF2 (<*>)

Wrap-up

Well, that’s the idea. I’ve gotten as far as type-checking the code in this post, but I haven’t yet tried running it.

What interests me most above is the use of unsafePerformIO here while preserving referential transparency, thanks to the invariant on F.FutureG (and the consequent monotonicity property). The heart of lazy evaluation of pure functional programs is just such an update, replacing a thunk with its weak head normal form (whnf). What general principles can we construct that allow us to use efficient, destructive updating and still have referential transparency? The important thing above seems to be the careful definition of an abstract interface such that the effect of state updates is semantically invisible through the interface.

In a previous post, I presented a fundamental reason why classic FRP does not fit interactive behavior, which is that the semantic model captures only the influence of time and not other input. I also gave a simple alternative, with a simple and general model for temporal and spatial transformation, in which input behavior is transformed inversely to the transformation of output behavior.

The semantic model I suggested is the same as used in “Arrow FRP”, from Fruit and Yampa. I want, however, a more convenient and efficient way to package up that model, which is the subject of the post you are reading now.

Next, we took a close look at one awkward aspect of classic FRP for interactive behavior, namely the need to trim inputs, and how trimming relates to comonadic FRP. The trim function allows us to define multi-phase interactive behaviors correctly and efficiently, but its use is tedious and is easy to get wrong. It thus fails to achieve what I want from functional programming in general and FRP in particular, which is to enable writing simple, natural descriptions, free of mechanical details.

The current post hides and automates the mechanics of trimming, so that the intent of an interactive behavior can be expressed directly and executed correctly and efficiently.

As before, I’ll adopt two abbreviations, for succinctness:

type B = Behavior
type E = Event

Safe and easy trimming

Previously, I defined an interactive cat behavior as follows:

cat3 :: B World -> B Cat
cat3 world = sleep world `switcher`
               ((uncurry prowl <$> trimf wake   world) `mappend`
                (uncurry eat   <$> trimf hunger world))

I’d really like to write the following

-- ideal:
cat4 = sleep `switcher` ((prowl <$> wake) `mappend` (eat <$> hunger)) 

Let’s see how close we can get.

I can see right off I’ll have to replace or generalize switcher. For now, I’ll replace it:

switcherf :: (B i -> B o)
          -> (B i -> E (B i -> B o))
          -> (B i -> B o)

This function will have to manage trimming:

bf `switcherf` ef =  i ->
  bf i `switcher` (uncurry ($) <$> trimf ef i)

I won’t have to replace mappend, since it’s a method and so can have a variety of types. In this case, mappend applies to a function from behaviors to events. Fortunately, the function monoid is exactly what we need:

instance Monoid b => Monoid (a -> b) where
  mempty        = const mempty
  f `mappend` g =  a -> f a `mappend` f b

or the more lovely standard form for applicative functors:

instance Monoid b => Monoid (a -> b) where
  mempty  = pure   mempty
  mappend = liftA2 mappend

The use of (<$>) (i.e., fmap) in cat4 above won’t work. We want instead to fmap inside the result of a function from behaviors to events. Using the style of semantic editor combinators, we get the following definition, which is fairly close to our ideal:

cat4 = sleep `switcherf`
         ((result.fmap) prowl wake `mappend` (result.fmap) eat hunger)

Generalized switching

To generalize switcher, introduce a new type class:

class Switchable b e where switcher :: b -> e b -> b

The original Reactive switcher is a special case:

instance Switchable (B a) E where switcher = R.switcher

We can switch among tuples and among other containers of switchables. For instance,

instance (Functor e, Switchable b e, Switchable b' e)
      => Switchable (b,b') e where
  (b,b') `switcher` e = ( b  `switcher` (fst <$> e)
                        , b' `switcher` (snd <$> e) )

Temporal values

Looking through the examples above, all we really had to do with the input behavior was to compute all remainders. I used

duplicate :: B a -> B (B a)

More generally,

class Temporal a where remainders :: a -> B a

Behaviors and events are included as a special case,

instance Temporal (B a) where remainders = duplicate

instance Temporal (E a) where remainders = ...

Temporal values combine without losing their individuality, which allows efficient change-driven evaluation as in Simply efficient functional reactivity:

instance (Temporal a, Temporal b) => Temporal (a,b) where
  remainders (a,b) = remainders a `zip` remainders b

Similarly for other triples and other data structures.

Sometimes it’s handy to carry static information along with dynamic information. Static types can be made trivially temporal:

instance Temporal Bool where remainders = pure
instance Temporal Int  where remainders = pure
-- etc

With this Temporal class, the trimming definitions above have more general types.

trim  :: Temporal i => E o -> i -> E (o, i)

trimf :: Temporal i => (i -> E o) -> (i -> E (o, i))

As does function switching:

switcherf :: (Temporal i, Switchable o E) =>
             (i -> o) -> (i -> E (i -> o)) -> i -> o

Types for functional interactive behavior

We’ve gotten almost to my ideal cat definition. We cannot, however, use switcher or (<$>) here with functions from behaviors to behaviors, because the types don’t fit.

To cross the last gap, let’s define new types corresponding to the idioms we’ve seen repeatedly above.

-- First try
type i :~> o = BI (B i -> B o)
type i :-> o = EI (B i -> E o)

Or, using type composition:

-- Second try
type (:~>) i = (->) (B i) :. B
type (:->) i = (->) (B i) :. E

The advantage of type composition is that we get some useful definitions for free, including Functor and Applicative instances.

However, there’s a problem with both versions. They limit us to a single behavior as input. A realistic interactive environment has many inputs, including a mixture of behaviors and events.

In Yampa, that mixture is combined into a single behavior, leading to two difficulties:

• The distinction between behaviors and events gets lost, as well as (I think) accurate and minimal-latency event detection and response.
• The bundled input environment changes whenever any component changes, leading to everything getting recomputed and redisplayed when anything changes.

To avoid these problems, I’ll take a different approach. Generalize inputs from behaviors to arbitrary Temporal values, which include behaviors, events and tuples and structures of temporal values.

The types for interactive behaviors and interactive events are

type (:~>) i = (->) i :. B
type (:->) i = (->) i :. E

So i :~> o is like i -> B o, and i :-> o is like i -> E o.

Switching for interactive behaviors wraps the switcherf function from above:

instance Temporal i => Switchable (i :~> o) ((:->) i) where
  switcher = inO2 $  bf ef -> bf `switcherf` (result.fmap) unO ef

The inO2 and unO functions from TypeCompose just manipulate newtype wrappers. See Prettier functions for wrapping and wrapping.

This definition is actually more general than the type given here. For instance, it can be used to switch between interactive events as well as interactive behaviors. To see the generalization, first abstract out the commonality between (:~>) and (:->):

type i :->. f = (->) i :. f

type (:~>) i = i :->. B
type (:->) i = i :->. E

The same instance code but with a more general type:

instance (Temporal i, Switchable (f o) E)
      => Switchable ((i :->. f) o) ((:->) i) where
  switcher = inO2 $  bf ef -> bf `switcherf` (result.fmap) unO ef

We can also switch between interactive collections of behaviors and events, though not with the (:->.) wrapping.

Where are we?

Almost all of the pieces are in place now. Another post will relate input trimming to the time transformation of interactive behaviors, as discussed in Why classic FRP does not fit interactive behavior. Also, how interactive FRP relates to Sequences, segments, and signals.

This post takes a close look at one awkward aspect of classic (non-arrow) FRP for interactive behavior, namely the need to trim (or “age”) old input. Failing to trim results in behavior that is incorrect and grossly inefficient.

Behavior trimming connects directly into the comonad interface mentioned in a few recent posts, and is what got me interested in comonads recently.

In absolute-time FRP, trimming has a purely operational significance. Switching to relative time, trimming is recast as a semantically familiar operation, namely the generalized drop function used in two recent posts.

In the rest of this post, I’ll adopt two abbreviations, for succinctness:

type B = Behavior
type E = Event

Trimming inputs

An awkward aspect of classic FRP has to do with switching phases of behavior. Each phase is a function of some static (momentary) input and some dynamic (time-varying) input, e.g.,

sleep :: B World -> B Cat
eat   :: Appetite   -> B World -> B Cat
prowl :: Friskiness -> B World -> B Cat

wake   :: B World -> E Friskiness
hunger :: B World -> E Appetite

As a first try, our cat prowls upon waking and eats when hungry, taking into account its surrounding world:

cat1 :: B World -> B Cat
cat1 world = sleep world `switcher`
               ((flip prowl world <$> wake   world) `mappend`
                (flip eat   world <$> hunger world))

The (<$>) here, from Control.Applicative, is a synonym for fmap. In this context,

(<$>) :: (a -> B Cat) -> E a -> E (B Cat)

The FRP switcher function switches to new behavior phases as they’re generated by an event, beginning with a given initial behavior:

switcher :: B a -> E (B a) -> B a

And the mappend here merges two events into one, combining their occurrences.

When switching phases, we generally want the new phase to start responding to input exactly where the old phase left off. If we’re not careful, however, the new phase will begin with an old input. I’ve made exactly this mistake in defining cat1 above. Consequently, each new phase will begin by responding to all of the old input and then carry on. This meaning is both unintended and is very expensive (the dreaded “space-time” leak).

This difficulty is not unique to FRP. In functional programming, we have to be careful how we hold onto our inputs, so that they can get accessed and freed incrementally. I don’t think the difficulty arises much in imperative programming, because input (like output) is destructively altered, and programs have access only to the current state.

I’ve done it wrong above, in defining cat. How can I do it right? The solution/hack I came up for Fran was to add a function that trims (“ages”) dynamic input while waiting for event occurrences.

trim :: B b -> E a -> E (a, B b)

trim b e follows b and e in parallel. At each occurrence of e, the remainder of b is paired up with the event data from e.

Now I can define the interactive, multi-phase behavior I intend:

cat2 :: B World -> B Cat
cat2 world = sleep world `switcher`
               ((uncurry prowl <$> trim world (wake   world)) `mappend`
                (uncurry eat   <$> trim world (hunger world)))

The event trim world (wake world) occurs whenever wake world does, and has as event data the cat’s friskiness on waking, plus the remainder of the cat’s world at the occurrence time. The “uncurry prowl <$>” applies prowl to each friskiness and remainder world on waking. Similarly for the other phase.

I think this version defines the behavior I want and that it can run efficiently, assuming that trim e b traverses e and b in parallel (so that laziness doesn’t cause a space-time leak). However, this definition is much trickier than what I’m looking for.

One small improvement is to abstract a trimming pattern:

trimf :: (B i -> E o) -> (B i -> E (o, B i))
trimf ef i = trim i (ef i)

cat3 :: B World -> B Cat
cat3 world = sleep world `switcher`
               ((uncurry prowl <$> trimf wake   world) `mappend`
                (uncurry eat   <$> trimf hunger world))

A comonad comes out of hiding

The trim functions above look a lot like snapshotting of behaviors:

snapshot  :: B b -> E a -> E (a,b)

snapshot_ :: B b -> E a -> E b

Indeed, the meanings of trimming and snapshotting are very alike. They both involving following an event and a behavior in parallel. At each event occurrence, snapshot takes the value of the behavior at the occurrence time, while trim takes the entire remainder from that time on.

Given this similarlity, can one be defined in terms of the other? If we had a function to “extract” the first defined value of a behavior, we could definesnapshot via trim.

b `snapshot` e = fmap (second extract) (b `trim` e)

extract :: B a -> a

We can also define trim via snapshot, if we have a way to get all trimmed versions of a behavior — to “duplicate” a one-level behavior into a two-level behavior:

b `trim` e = duplicate b `snapshot` e

duplicate :: B a -> B (B a)

If you’ve run into comonads, you may recognize extract and duplicate as the operations of Comonad, dual to Monad‘s return and join. It was this definition of trim that got me interested in comonads recently.

In the style of Semantic editor combinators,

snapshot = (result.result.fmap.second) extract trim

or

trim = argument remainders R.snapshot

The extract function is problematic for classic FRP, which uses absolute (global) time. We don’t know with which time to sample the behavior. With relative-time FRP, we’ll only ever sample at (local) time 0.

Relative time

So far, the necessary trimming has strong operational significance: it prevents obsolete reactions and the consequent space-time leaks.

If we switch from absolute time to relative time, then trimming becomes something with familiar semantics, namely drop, as generalized and used in two of my previous posts, Sequences, streams, and segments and Sequences, segments, and signals.

The semantic difference: trimming (absolute time) erases early content in an input; while dropping (relative time) shifts input backward in time, losing the early content in the same way as drop on sequences.

What’s next?

While input trimming can be managed systematically, doing so explicitly is tedious and error prone. A follow-up post will automatically apply the techniques from this post. Hiding and automating the mechanics of trimming allows interactive behavior to be expressed correctly and without distraction.

Another post will relate input trimming to the time transformation of interactive behaviors, as discussed in Why classic FRP does not fit interactive behavior.

In functional reactive programming (FRP), the type we call “behaviors” model non-interactive behavior. To see why, just look at the semantic model: t -> a, for some notion t of time.

One can argue as follows that this model applies to interactive behavior as well. Behaviors interacting with inputs are functions of time and of inputs. Those inputs are also functions of time, so behaviors are just functions of time. I held this perspective at first, but came to see a lack of composability.

My original FRP formulations (Fran and its predecessors TBAG and ActiveVRML), as well as the much more recent library Reactive, can be and are used to describe interactive behavior. For simple sorts of things, this use works out okay. When applications get a bit richer, the interface and semantics strain. If you’ve delved a bit, you’ll have run into the signs of strain, with coping mechanisms like start times, user arguments and explicit aging of inputs, as you avoid the dreaded space-time leaks.

Behaviors

Suppose I define an object that pays attention to an input that’s interesting near a certain time. For instance, a cat chasing a bird in the back yard one morning last week. Now shift the behavior a week forward to this morning. The result is just a week-delayed form of the earlier behavior, displaying reactions to week-old input. The original interactive behavior (the cat) was fed input (bird and yard) from a week ago, the result was recorded, and that recording is replayed a week later.

Written functionally,

laterObserveCat :: Behavior World -> Behavior Cat
laterObserveCat = later week . cat

where

world :: Behavior World
cat   :: Behavior World -> Behavior Cat

later :: Duration -> Behavior a -> Behavior a

We can also sit in the living room, five meters north of the back yard, and watch the cat via a camera.

livingRoomObserveCat :: Behavior World -> Behavior Cat
livingRoomObserveCat = north (5 * meter) . cat

Typically recordings are moved both in space and in time. For example, record the cat in the back yard one week and watch the recording in the living room a week later:

replayLRL :: Behavior World -> Behavior Cat
replayLRL = later week . north (5 * meter) . cat

Interactive behavior

I’ve just described transforming a non-interactive behavior (recording) in time and space. That non-interactive behavior resulted applying an interactive behavior (the cat) to some input (bird and yard). An interactive behavior is simply a function from non-interactive behavior (time-varying input) to non-interactive behavior (time-varying output).

Consider what it means to transform an interactive behavior in space. I pick up the cat and carry her five meters north, into the living room. The cat has a different perspective, which is that her environment moves five meters south. For instance, the mouse that was one meter south of her is now six meters sounth.

If I lift the cat four feet above the ground, she sees the ground now four below her. If I spin her clockwise, her environment spins counter-clockwise. If I point my shrinking ray at her, she sees her world as growing larger, and she flees from the giant mice.

We can play the perspective game in time as well as space. I point my speed-up ray at the cat, and she perceives her environment slow down.

When I say I move the cat north and she see her environment move south, I am not suggesting that there are two alternative perspectives and we might take either one. Rather, a complete picture combines both movements. The behavior my cat exhibits when I pick her up is composed of three transformations:

• The world moves south;
• The cat perceives this more southerly world and responds to it; and
• The resulting cat behavior is moved north.

Written functionally,

livingRoomCat :: Behavior World -> Behavior Cat
livingRoomCat = north (5 * meter) . cat . south (5 * meter)

Similarly with other spatial transformation, each with paired inverse transformations. And similarly in time, e.g.,

laterCat :: Behavior World -> Behavior Cat
laterCat = later week . cat . earlier week

So there’s a semantically consistent model of transforming interactive behaviors in time or space, and this model explains the difference between transformation of non-interactive behavior and of interactive behavior.

Aside

Comparing these two definitions with the previous two suggests an alternative implementation. Refactoring,

laterCat :: Behavior World -> Behavior Cat
laterCat = laterObserveCat . earlier week

Instead of carrying the interactive cat a week ahead in time, this definition suggests that we move the world (other than the cat) back one week, record the cat interacting with it, and hold onto that recording to watch a week later. My back-of-the-envelope calculations suggest that this second implementation is less resource-efficient than the first one, so I’ll not consider it further in this post.

Arrow FRP

When we use for an interactive setting the tools suited to non-interactive behaviors, we’re begging for space leaks (filling up recording media) and time leaks (fast-forwarding through old recorded matter to catch up with what we want to see). These problems are what people experience when programming explicitly with might call “classic FRP”, i.e., programming explicitly with (non-interactive) behaviors. Out of this experience was born “Arrow FRP”, as in Fruit and Yampa. The idea of Arrow FRP was to program with a type of “signal functions”, which can be thought of as mappings between behaviors:

type SF a b = Behavior a -> Behavior b

(Behaviors were also renamed to “signals”, but I’ll stick with “behaviors” for now.) We can just look at semantic model for signal transformers and see that they’re about interactive behavior.

A lot of work was done with this new model, mostly growing out of Paul Hudak’s group at Yale. It addressed the inherent awkwardness of the explicit-behavior style.

Despite this fundamental advantage of signal transformers, why am I still fooling around with the “classic” (pre-arrow) style?

• The Arrow style required multiple behaviors to be bunched up into one. That combination appears to thwart efficient, change-driven evaluation, which was the problem I tackled in Simply efficient functional reactivity.
• There was no room for the distinction between behaviors and events. I’d love to see how to combine them without losing expressiveness or semantic accuracy, and I haven’t yet.
• Arrow programming is very awkward without the special arrow notation and has more of a sequential style than I like with the arrow notation. I missed the functional feel of classic FRP.

Coming up

Given the fundamental semantic fit between signal transformers for the problem domain of interactive behavior, I’d like to come up wh convenient and efficient packaging. The next posts will describe the packaging I’m trying out now.

The post Sequences, streams, and segments offered an answer to the the question of what’s missing in the following box:

I presented a simple type of function segments, whose representation contains a length (duration) and a function. This type implements most of the usual classes: Monoid, Functor, Zip, and Applicative, as well Comonad, but not Monad. It also implements a new type class, Segment, which generalizes the list functions length, take, and drop.

The function type is simple and useful in itself. I believe it can also serve as a semantic foundation for functional reactive programming (FRP), as I’ll explain in another post. However, the type has a serious performance problem that makes it impractical for some purposes, including as implementation of FRP.

Fortunately, we can solve the performance problem by adding a simple layer on top of function segments, to get what I’ll call “signals”. With this new layer, we have an efficient replacement for function segments that implements exactly the same interface with exactly the same semantics. Pleasantly, the class instances are defined fairly simply in terms of the corresponding instances on function segments.

You can download the code for this post.

Edits:

• 2008-12-06: dup [] = [] near the end (was [mempty]).
• 2008-12-09: Fixed take and drop default definitions (thanks to sclv) and added point-free variant.
• 2008-12-18: Fixed appl, thanks to sclv.
• 2011-08-18: Eliminated accidental emoticon in the definition of dup, thanks to anonymous.

The problem with function segments

The type of function segments is defined as follows:

data t :-># a = FS t (t -> a)

The domain of the function segment is from zero up to but not including the given length.

An efficiency problem becomes apparent when we look at the Monoid instance:

instance (Ord t, Num t) => Monoid (t :-># a) where
    mempty = FS 0 (error "sampling empty 't :-># a'")
    FS c f `mappend` FS d g =
      FS (c + d) ( t -> if t <= c then f t else g (t - c))

Concatenation (mappend) creates a new segment that chooses, for every domain value t, whether to use one function or another. If the second, then t must be shifted backward, since the function is being shifted forward.

This implementation would be fine if we use mappend just on simple segments. Once we get started, however, we’ll want to concatenate lots & lots of segments. In FRP, time-varying values go through many phases (segments) as time progresses. Each quantity is described by a single “behavior” (sometimes called a “signal”) with many, and often infinitely many, phases. Imagine an infinite tree of concatenations, which is typical for FRP behaviors. At every moment, one phase is active. Every sampling must recursively discover the active phase and the accumulated domain translation (from successive subtractions) to apply when sampling that phase. Quite commonly, concatenation trees get progressively deeper on the right (larger t values). In that case, sampling will get slower and slower with time.

I like to refer to these progressive slow-downs as “time leaks”. There is also a serious space leak, since all of the durations and functions that go into a composed segment will be retained.

Sequences of segments

The problem above can be solved with a simple representation change. Instead of combining functions into functions, just keep a list of simple function segments.

-- | Signal indexed by t with values of type a.
newtype t :-> a = S { unS :: [t :-># a] }

I’ll restrict in function segments to be non-empty, to keep the rest of the implementation simple and efficient.

This new representation allows for efficient monotonic sampling of signals. As old segments are passed up, they can be dropped.

What does it mean?

There’s one central question for me in defining any data type: What does it mean?

The meaning I’ll take for signals is function segments. This interpretation is made precise in a function that maps from the type to the model (meaning). In this case, simply concatenate all of the function segments:

meaning :: (Ord t, Num t) => (t :-> a) -> (t :-># a)
meaning = mconcat . unS

Specifying the meaning of a type gives users a working model, and it defines correctness of implementation. It also tells me what class instances to implement and tells users what instances to expect. If a type’s meaning implements a class then I want the type to as well. Moreover, the type’s intances have to agree with the model’s instances. I’ve described this latter principle in Simplifying semantics with type class morphisms and some other posts.

Higher-order wrappers

To keep the code below short and clear, I’ll use some functions for adding and removing the newtype wrappers. These higher-order function apply functions inside of (:->) representations:

inS  :: ([s :-># a] -> [t :-># b])
     -> ((s :->  a) -> (t :->  b))

inS2 :: ([s :-># a] -> [t :-># b] -> [u :-># c])
     -> ((s :->  a) -> (t :->  b) -> (u :->  c))

Using the trick described in Prettier functions for wrapping and wrapping, the definitions are simpler than the types:

inS  = result   S . argument unS
inS2 = result inS . argument unS

Functor

The Functor instance applies a given function inside the function segments inside the lists:

instance Functor ((:->) t) where
    fmap h (S ss) = S (fmap (fmap h) ss)

Or, in the style of Semantic editor combinators,

instance Functor ((:->) t) where
    fmap = inS . fmap . fmap

Why this definition? Because it is correct with respect to the semantic model, i.e., the meaning of fmap is fmap of the meaning, i.e.,

 meaning . fmap h == fmap h . meaning

which is to say that the following diagram commutes:

Proof:

meaning . fmap h 

  == {- fmap definition -}

meaning . S . fmap (fmap h) . unS

  == {- meaning definition -}

mconcat . unS . S . fmap (fmap h) . unS

  == {- unS and S are inverses  -}

mconcat . fmap (fmap h) . unS

  == {- fmap h distributes over mappend -}

fmap h . mconcat . unS

  == {- meaning definition -}

fmap . meaning

Applicative and Zip

Again, the meaning functions tells us what the Applicative instance has to mean. We only get to choose how to implement that meaning correctly. The Applicative morphism properties:

meaning (pure a)    == pure a
meaning (bf <*> bx) == meaning bf <*> meaning bx

Our Applicative instance has a definition similar in simplicity and style to the Functor instance, assuming a worker function appl for (<*>):

instance (Ord t, Num t, Bounded t) => Applicative ((:->) t) where
    pure  = S . pure . pure
    (<*>) = inS2 appl

appl :: (Ord t, Num t, Bounded t) =>
        [t :-># (a -> b)] -> [t :-># a] -> [t :-># b]

This worker function is somewhat intricate. At least my implementation of it is, and perhaps there’s a simpler one.

Again, the meaning functions tells us what the Applicative instance has to mean. We only get to choose how to implement that meaning correctly.

First, if either segment list runs out, the combination runs out (because the same is true for the meaning of signals).

[] `appl` _  = []

_  `appl` [] = []

If neither segment list is empty, open up the first segment. Split the longer segment into a prefix that matches the shorter segment, and combine the two segments with (<*>) (on function segments). Toss the left-over piece back in its list, and continue.

(fs:fss') `appl` (xs:xss')
   | fd == xd  = (fs  <*> xs ) : (fss' `appl`       xss' )
   | fd <  xd  = (fs  <*> xs') : (fss' `appl` (xs'':xss'))
   | otherwise = (fs' <*> xs ) : ((fs'':fss') `appl` xss')
 where
   fd         = length fs
   xd         = length xs
   (fs',fs'') = splitAt xd fs
   (xs',xs'') = splitAt fd xs

A Zip instance is easy as always with applicative functors:

instance (Ord t, Num t, Bounded t) => Zip ((:->) t) where zip = liftA2 (,)

Monoid

The Monoid instance:

instance Monoid (t :-> a) where
  mempty = S []
  S xss `mappend` S yss = S (xss ++ yss)

Correctness follows from properties of mconcat (as used in the meaning function).

We’re really just using the Monoid instance for the underlying representation, i.e.,

instance Monoid (t :-> a) where
    mempty  = S mempty
    mappend = inS2 mappend

Segment

The Segment class has length, take and drop. It’s handy to include also null and splitAt, both modeled after their counterparts on lists.

The new & improved Segment class:

class Segment seg dur | seg -> dur where
    null    :: seg -> Bool
    length  :: seg -> dur
    take    :: dur -> seg -> seg
    drop    :: dur -> seg -> seg
    splitAt :: dur -> seg -> (seg,seg)
    -- Defaults:
    splitAt d s = (take d s, drop d s)
    take    d s = fst (splitAt d s)
    drop    d s = snd (splitAt d s)

If we wanted to require dur to be numeric, we could add a default for null as well. This default could be quite expensive in some cases. (In the style of Semantic editor combinators, take = (result.result) fst splitAt, and similarly for drop.)

The null and length definitions are simple, following from to properties of mconcat.

instance (Ord t, Num t) => Segment (t :-> a) t where
  null   (S xss) = null xss
  length (S xss) = sum (length <$> xss)
  ...

The null case says that the signal is empty exactly when there are no function segments. This simple definition relies on our restriction to non-empty function segments. If we drop that restriction, we’d have to check that every segment is empty:

  -- Alternative definition
  null (S xss) = all null xss

The length is just the sum of the lengths.

The tricky piece is splitAt (used to define both take and drop), which must assemble segments to satisfy the requested prefix length. The last segment used might have to get split into two, with one part going into the prefix and one to the suffix.

  splitAt _ (S [])  = (mempty,mempty)
  splitAt d b | d <= 0 = (mempty, b)
  splitAt d (S (xs:xss')) =
    case (d `compare` xd) of
      EQ -> (S [xs], S xss')
      LT -> let (xs',xs'') = splitAt d xs in
              (S [xs'], S (xs'':xss'))
      GT -> let (S uss, suffix) = splitAt (d-xd) (S xss') in
              (S (xs:uss), suffix)
   where
     xd = length xs

Copointed and Comonad

To extract an element from a signal, extract an element from its first function segment. Awkwardly, extraction will fail (produce ⊥/error) when the signal is empty.

instance Num t => Copointed ((:->) t) where
    extract (S [])     = error "extract: empty S"
    extract (S (xs:_)) = extract xs

I’ve exploited our restriction to non-empty function segments. Otherwise, extract would have to skip past the empty ones:

-- Alternative definition
instance Num t => Copointed ((:->) t) where
    extract (S []) = error "extract: empty S"
    extract (S (xs:xss'))
      | null xs     = extract (S xss')
      | otherwise   = extract xs

The error/⊥ in this definition is dicey, as is the one for function segments. If we allow the same abuse in order to define a list Copointed, we can get an alternative to the first definition that is prettier but gives a less helpful error message:

instance Num t => Copointed ((:->) t) where
    extract = extract . extract . unS

See the closing remarks for more about this diciness.

Finally, we have Comonad, with its duplicate method.

duplicate :: (t :-> a) -> (t :-> (t :-> a))

I get confused with wrapping and unwrapping, so let’s separate the definition into a packaging part and a content part.

instance (Ord t, Num t) => Comonad ((:->) t) where
    duplicate = fmap S . inS dup

with content part:

dup :: (Ord t, Num t) => [t :-># a] -> [t :-># [t :-># a]]

The helper function, dup, takes each function segment and prepends each of its tails onto the remaining list of segments.

If the segment list is empty, then it has only one tail, also empty.

dup []        = []

dup (xs:xss') = ((: xss') <$> duplicate xs) : dup xss'

Closing remarks

• The definitions above use the function segment type only through its type class interfaces, and so can all of them can be generalized. Several definitions rely on the Segment instance, but otherwise, each method for the composite type relies on the corresponding method for the underlying segment type. (For instance, fmap uses fmap, (<*>) uses (<*>), Segment uses Segment, etc.) This generality lets us replace function segments with more efficient representations, e.g., doing constant propagation, as in Reactive. We can also generalize from lists in the definitions above.

• Even without concatenation, function segments can become expensive when drop is repeatedly applied, because function shifts accumulate (e.g., f . (+ 0.01) . (+ 0.01) ....). A more drop-friendly representation for function segments would be a function and an offset. Successive drops would add offsets, and extract would always apply the function to its offset. This representation is similar to the FunArg comonad, mentioned in The Essence of Dataflow Programming (Section 5.2).

• The list-of-segments representation enables efficient monotonic sampling, simply by dropping after each sample. A variation is to use a list zipper instead of a list. Then non-monotonic sampling will be efficient as long as successive samplings are for nearby domain values. Switching to multi-directional representation would lose the space efficiency of unidirectional representations. The latter work well with lazy evaluation, often running in constant space, because old values are recycled while new values are getting evaluated.

• Still another variation is to use a binary tree instead of a list, to avoid the list append in the Monoid instance for (t :-> a). A tree zipper would allow non-monotonic sampling. Sufficient care in data structure design could perhaps yield efficient random access.

• There’s a tension between the Copointed and Monoid interfaces. Copointed has extract, and Monoid has mempty, so what is the value of extract mempty? Given that Copointed is parameterized over arbitrary types, the only possible answer seems to be ⊥ (as in the use of error above). I don’t know if the comonad laws can all hold for possibly-empty function segments or for possibly-empty signals. I’m grateful to Edward Kmett for helping me understand this conflict. He suggested using two coordinated types, one possibly-empty (a monoid) and the other non-empty (a comonad). I’m curious to see whether that idea works out.

What kind of thing is a movie? Or a song? Or a trajectory from point A to point B? If you’re a computer programmer/programmee, you might say that such things are sequences of values (frames, audio samples, or spatial locations). I’d suggest that these discrete sequences are representations of something more essential, namely a flow of continuously time-varying values. Continuous models, whether in time or space, are often more compact, precise, adaptive, and composable than their discrete counterparts.

Functional programming offers great support for sequences of variable length. Lazy functional programming adds infinite sequences, often called streams, which allows for more elegant and modular programming.

Functional programming also has functions as first class values, and when the function’s domain is (conceptually) continuous, we get a continuous counterpart to infinite streams.

Streams, sequences, and functions are three corners of a square. Streams are discrete and infinite, sequences are discrete and finite, and functions-on-reals are continuous and infinite. The missing corner is continuous and finite, and that corner is the topic of this post.

You can download the code for this post.

Edits:

• 2008-12-01: Added Segment.hs link.
• 2008-12-01: Added Monoid instance for function segments.
• 2008-12-01: Renamed constructor “DF” to “FS” (for “function segment”)
• 2008-12-05: Tweaked the inequality in mappend on (t :-># a).

Streams

I’ll be using Wouter Swierstra’s Stream library. A stream is an infinite sequence of values:

data Stream a = Cons a (Stream a)

Stream is a functor and an applicative functor.

instance Functor Stream where
    fmap f (Cons x xs) = Cons (f x) (fmap f xs)

instance Applicative Stream where
    pure  = repeat
    (<*>) = zipWith ($)

repeat :: a -> Stream a
repeat x = Cons x (repeat x)

Comonads

Recently I’ve gotten enamored with comonads, which are dual to monads. In other words, comonads are like monads but wearing their category arrows backwards. I’ll be using the comonad definitions from the category-extras library.

The most helpful intuitive description I’ve found is that comonads describe values in context.

The return method injects a pure value into a monadic value (having no effect).

return  :: Monad m     => a -> m a

The dual to monadic return is extract (sometimes called “counit” or “coreturn“), which extracts a value out of a comonadic value (discarding the value’s context). category-extras library splites this method out from Comonad into the Copointed class:

extract :: Copointed w => w a -> a

Monadic values are typically produced in effectful computations:

a -> m b

Comonadic values are typically consumed in context-sensitive computations:

w a -> b

(Kleisli arrows wrap the producer pattern, while CoKleisli arrows wrap the consumer pattern.)

Monads have a way to extend a monadic producer into one that consumes to an entire monadic value:

(=<<) :: (Monad m) => (a -> m b) -> (m a -> m b)

We more often see this operation in its flipped form (obscuring the conceptual distinction between Haskell arrows and arbitrary category arrows):

(>>=) :: (Monad m) => m a -> (a -> m b) -> m b

Dually, comonads have a way to extend a comonadic consumer into one that produces an entire comonadic value:

extend :: (Comonad w) => (w a -> b) -> (w a -> w b)

which also has a flipped version:

(=>>) :: (Comonad w) => w a -> (w a -> b) -> w b

Another view on monads is as having a way to join two monadic levels into one.

join      :: (Monad   m) => m (m a) -> m a

Dually, comonads have a way to duplicate one level into two:

duplicate :: (Comonad w) => w a -> w (w a)

For a monad, any of join, (=<<), and (>>=) can be used to define the others. For a comonad, any of duplicate, extend, and (=>>) can be used to define the others.

The Stream comonad

What might the stream comonad be?

The Stream library already has functions of the necessary types for extract and duplicate, corresponding to familiar list functions:

head :: Stream a -> a
head (Cons x _ ) = x

tails :: Stream a -> Stream (Stream a)
tails xs = Cons xs (tails (tail xs))

where

tail :: Stream a -> Stream a
tail (Cons _ xs) = xs

Indeed, head and tails are just what we’re looking for.

instance Copointed Stream where extract   = head
instance Comonad   Stream where duplicate = tails

There is also a Monad instance for Stream, in which return is repeat (matching pure as expected) and join is diagonalization, producing a stream whose n^th element is the n^th element of the n^th element of a given stream of streams.

Adding finiteness

Lists and other possibly-finite sequence types add an interesting new aspect over streams, which is concatenation, usually wrapped in a Monoid instance.

class Monoid o where
    mempty  :: o
    mappend :: o -> o -> o

Lists also have take and drop operations, which can undo the effect of concatenation, as well as a notion of length (duration). Let’s generalize these three to be methods of a new type class, Segment, so that we can defined continuous versions.

class Segment seg len where
    length :: seg -> len
    drop   :: len -> seg -> seg
    take   :: len -> seg -> seg

For lists, we can use the prelude functions

instance Segment [a] Int where
    length = Prelude.length
    drop   = Prelude.drop
    take   = Prelude.take

Or the more generic versions:

instance Integral i => Segment [a] i where
    length = genericLength
    drop   = genericDrop
    take   = genericTake

These three functions relate to mappend, to give us the following “Segment laws”:

drop (length as) (as `mappend` bs) == bs
take (length as) (as `mappend` bs) == as

t <= length as ==> length (take t as) == t
t <= length as ==> length (drop t as) == length as - t

Adding continuity

Streams and lists are discrete, containing countably many or finitely many elements. They both have continuous counterparts.

When we think of a stream as a function from natural numbers, then John Reynolds’s alternative arises: functions over real numbers, i.e., a continuum of values. If we want uni-directional streams, then stick with non-negative reals.

Many stream and list operations are meaningful and useful not only for discrete sequences but also for their continuous counterparts.

The infinite (stream-like) case is already handled by the class instances for functions found in the GHC base libraries (Control.Functor.Instances and Control.Applicative).

instance Functor ((->) t) where
    fmap = (.)

instance Applicative ((->) t) where
    pure = const
    (f <*> g) x = (f x) (g x)

instance Monad ((->) t) where
    return = const
    f >>= k =  t -> k (f t) t

As a consequence,

  join f == f >>= id ==  t -> f t t

Assume a type wrapper, NonNeg, for non-negative values. For discrete streams, r == NonNeg Integer, while for continuous streams, r == NonNeg R, for some type R representing reals.

The co-monadic instances from the category-extras library:

instance Monoid o => Copointed ((->) o) where
    extract f = f mempty

instance Monoid o => Comonad ((->) o) where
    duplicate f x =  y -> f (x `mappend` y)

Finite and continuous

Functions provide a setting for generalized streams. How do we add finiteness? A very simple answer is to combine a length (duration) with a function, to form a “function segment”:

data t :-># a = FS t (t -> a)

The domain of this function is from zero to just short of the given length.

Now let’s define class instances.

Exercise: Show that all of the instances below are semantically consistent with the Stream and ZipList instances.

Monoid

Empty function segments have zero duration. Concatenation adds durations and samples either function, right-shifting the second one.

instance (Ord t, Num t) => Monoid (t :-># a) where
    mempty = FS 0 (error "sampling empty 't :-># a'")
    FS c f `mappend` FS d g =
      FS (c + d) ( t -> if t <= c then f t else g (t - c))

Segment

The Segment operations are easy to define:

instance Num t => Segment (t :-># a) t where
    length (FS d _) = d
    drop t (FS d f) = FS (d - t) ( t' -> f (t + t'))
    take t (FS _ f) = FS t f

Notice what’s going on with drop. The length gets shortened by t (the amount dropped), and the function gets shifted (to the “left”) by t.

There’s also a tantalizing resemblance between this drop definition and duplicate for the function comonad. We’ll return in another post to tease out this and

I’ve allowed dropping or taking more than is present, though these cases can be handled with an error or a by taking or dropping fewer elements (as with the list drop and take functions).

Functor, Zip and Applicative

fmap applies a given function to each of the function values, leaving the length unchanged.

instance Functor ((:->#) t) where
    fmap h (FS d f) = FS d (h . f)

zip pairs corresponding segment values and runs out with the shorter segment. (See More beautiful fold zipping for the Zip class.)

instance Ord t => Zip ((:->#) t) where
    FS xd xf `zip` FS yd yf = FS (xd `min` yd) (xf `zip` yf)

pure produces a constant value going forever. (<*>) applies functions to corresponding arguments, running out with the shorter.

instance (Ord t, Bounded t) => Applicative ((:->#) t) where
    pure a = FS maxBound (const a)
    (<*>)  = zipWith ($)

Copointed and Comonad

extract pulls out the initial value (like head).

instance Num t => Copointed ((:->#) t) where
    extract (FS _ f) = f 0

duplicate acts like tails. The generated segments are progressivly dropped versions of the original segment.

instance Num t => Comonad ((:->#) t) where
    duplicate s = FS (length s) (flip drop s)

Monad

I don’t know if there is a monad instance for ((:->#) t). Simple diagonalization doesn’t work for join, since the n^th segment might be shorter than n.

What’s ahead?

The instances above remind me strongly of type class instances for several common types. Another post will tease out some patterns and reconstruct (t :-># a) out of standard components, so that most of the code above can disappear.

Another post incorporates (t :-># a) into a new model and implementation of relative-time, comonadic FRP.

In 1989, I was a grad student nearing completion at Carnegie Mellon. Kavi Arya gave a talk on “functional animation”, using lazy lists. I was awe-struck with the elegance and power of that simple idea, and I’ve been hooked on functional animation ever since.

At the end of Kavi’s talk, John Reynolds offered a remark, roughly as follows:

You can think of sequences as functions from the natural numbers. Have you thought about functions from the reals instead? Doing so might help with the awkwardness with interpolation.

I knew at once I’d heard a wonderful idea, so I went back to my office, wrote it down, and promised myself that I wouldn’t think about Kavi’s work and John’s insight until my dissertation was done. Otherwise, I might never have finished. A year or so later, at Sun Microsystems, I started working on functional animation, which then grew into functional reactive programming (FRP).

In the dozens of variations on FRP I’ve played with over the last 15 years, John’s refinement of Kavi’s idea has always been the heart of the matter for me.

Lately, I’ve been rethinking FRP yet again, and I’m very excited about where it’s leading me.

The semantic model of FRP has been based on behaviors of infinite duration and, mostly on absolute time. Recently I realized that some problems of non-modular interaction could be elegantly addressed by switching to finite duration and relative time, and by adopting a co-monadic approach.

Upcoming post will explore these ideas.

When I first started playing with functional reactivity in Fran and its predecessors, I didn’t realize that much of the functionality of events and reactive behaviors could be packaged via standard type classes. Then Conor McBride & Ross Paterson introduced us to applicative functors, and I remembered using that pattern to reduce all of the lifting operators in Fran to just two, which correspond to pure and (<*>) in the Applicative class. So, in working on a new library for functional reactive programming (FRP), I thought I’d modernize the interface to use standard type classes as much as possible.

While spelling out a precise (denotational) semantics for the FRP instances of these classes, I noticed a lovely recurring pattern:

The meaning of each method corresponds to the same method for the meaning.

In this post, I’ll give some examples of this principle and muse a bit over its usefulness. For more details, see the paper Simply efficient functional reactivity. Another post will start exploring type class morphisms and type composition, and ask questions I’m wondering about.

Behaviors

The meaning of a (reactive) behavior is a function from time:

type B a = Time -> a

at :: Behavior a -> B a

So the semantic function, at, maps from the Behavior type (for use in FRP programs) to the B type (for understanding FRP programs)

As a simple example, the meaning of the behavior time is the identity function:

at time == id

Functor

Given b :: Behavior a and a function f :: a -> b, we can apply f to the value of b at every moment in (infinite and continuous) time. This operation corresponds to the Functor method fmap, so

instance Functor Behavior where ...

The informal description of fmap on behavior translates to a formal definition of its semantics:

  fmap f b `at` t == f (b `at` t)

Equivalently,

  at (fmap f b) ==  t -> f (b `at` t)
                == f . ( t -> b `at` t)
                == f . at b

Now here’s the fun part. While Behavior is a functor, so is its meaning:

instance Functor ((->) t) where fmap = (.)

So, replacing f . at b with fmap f (at b) above,

  at (fmap f b) == fmap f (at b)

which can also be written

  at . fmap f == fmap f . at

Keep in mind here than the fmap on the left is on behaviors, and on the right is functions (of time).

This last equation can also be written as a simple square commutative diagram and is sometimes expressed by saying that at is a “natural transformation” or “morphism on functors” [Categories for the Working Mathematician]. For consistency with similar properties on other type classes, I suggest “functor morphism” as a synonym for natural transformation.

The Haskell wiki page on natural transformations shows the commutative diagram and gives maybeToList as another example.

Applicative functor

The fmap method applies a static (not time-varying) function to a dynamic (time-varying) argument. A more general operation applies a dynamic function to a dynamic argument. Also useful is promoting a static value to a dynamic one. These two operations correspond to (<*>) and pure for applicative functors:

infixl 4 <*>
class Functor f => Applicative f where
  pure  :: a -> f a
  (<*>) :: f (a->b) -> f a -> f b

where, e.g., f == Behavior.

From these two methods, all of the n-ary lifting functions follow. For instance,

liftA3 :: Applicative f =>
          (  a ->   b ->   c ->   d)
       ->  f a -> f b -> f c -> f d
liftA3 h fa fb fc = pure h <*> fa <*> fb <*> fc

Or use fmap h fa in place of pure h <*> fa. For prettier code, (<$>) (left infix) is synonymous with fmap.

Now, what about semantics? Applying a dynamic function fb to a dynamic argument xb gives a dynamic result, whose value at time t is the value of fb at t, applied to the value of xb at t.

at (fb <*> xb) ==  t -> (fb `at` t) (xb `at` t)

The (<*>) operator is the heart of FRP’s concurrency model, which is determinate, synchronous, and continuous.

Promoting a static value yields a constant behavior:

at (pure a) ==  t -> a
            == const a

As with Functor, let’s look at the Applicative instance of functions (the meaning of behaviors):

instance Applicative ((->) t) where
  pure a    = const a
  hf <*> xf =  t -> (hf t) (xf t)

Wow — these two definitions look a lot like the meanings given above for pure and (<*>) on behaviors. And sure enough, we can use the function instance to simplify these semantic definitions:

at (pure a)    == pure a
at (fb <*> xb) == at fb <*> at xb

Thus the semantic function distributes over the Applicative methods. In other words, the meaning of each method is the method on the meaning. I don’t know of any standard term (like “natural transformation”) for this relationship between at and pure/(<*>). I suggest calling at an “applicative functor morphism”.

Monad

Monad morphisms are a bit trickier, due to the types. There are two equivalent forms of the definition of a monad morphism, depending on whether you use join or (>>=). In the join form (e.g., in Comprehending Monads, section 6), for monads m and n, the function nu :: forall a. m a -> n a is a monad morphism if

nu . join == join . nu . fmap nu

where

join :: Monad m => m (m a) -> m a

For behavior semantics, m == Behavior, n == B == (->) Time, and nu == at.

Then at is also a monad morphism if

at (return a) == return a
at (join bb)  == join (at (fmap at bb))

And, since for functions f,

fmap h f == h . f
join f   ==  t -> f t t

the second condition is

at (join bb) == join (at (fmap at bb))
             ==  t -> at (at . bb) t t
             ==  t -> at (at bb t) t
             ==  t -> (bb `at` t) `at` t

So sampling join bb at t means sampling bb at t to get a behavior b, which is also sampled at t. That’s exactly what I’d guess join to mean on behaviors.

Note: the FRP implementation described in Simply efficient functional reactivity does not include a Monad instance for Behavior, because I don’t see how to implement one with the hybrid data-/demand-driven Behavior implementation. However, the closely related but less expressive type, Reactive, has the same semantic model as Behavior. Reactive does have a Monad instance, and its semantic function (rats) is a monad morphism.

Other examples

The Simply paper contains several more examples of type class morphisms:

• Reactive values, time functions, and future values are also morphisms on Functor, Applicative, and Monad.
• Improving values are morphisms on Ord.

The paper also includes a significant non-example, namely events. The semantics I gave for Event a is a time-ordered list of time/value pairs. However, the semantic function (occs) is not a Monoid morphism, because

occs (e `mappend` e') == occs e `merge` occs e'

and merge is not (++), which is mappend on lists.

Why care about type class morphisms?

I want my library’s users to think of behaviors and future values as being their semantic models (functions of time and time/value pairs). Why? Because these denotational models are simple and precise and have simple and useful formal properties. Those properties allow library users to program with confidence, and allow library providers to make radical changes in representation and implementation (even from demand-driven to data-driven) without breaking client programs.

When I think of a behavior as a function of time, I’d like it to act like a function of time, hence Functor, Applicative, and Monad. And if it does implement any classes in common with functions, then it had better agree the function instances of those classes. Otherwise, user expectations will be mistaken, and the illusion is broken.

I’d love to hear about other examples of type class morphisms, particularly for Applicative and Monad, as well as thoughts on their usefulness.