The dataflow engine I gave in my last post can be seen as an
implementation of self-adjusting computation,
in the style of Acar, Blelloch and Harper's original POPL 2002
paper Adaptive Functional Programming.
(Since then, state of the art implementation techniques have
improved a lot, so don't take my post as indicative of what
modern libraries do.)
Many people have seen resemblances between self-adjusting computation
and functional reactive programming --- a good example of this is Jake
Donham's Froc library for
Ocaml. Originally, I was one of those people, but that's no
longer true: I think SAC and FRP are completely orthogonal.
I now think that FRP libraries can be very minimalistic --- my ICFP
2013 paper
Higher-Order Reactive Programming without Spacetime Leaks
gives a type system, implementation, and correctness proof for
an FRP language with full support for higher order constructions
like higher-order functions and streams of streams, while at the
same time statically ruling out space and time leaks.
The key idea is to distinguish between stable values (like
ints and bools) whose representation doesn't change over time from
dynamic values (like streams) whose representation is
time-varying. Stable values are the usual datatypes, and can be used
whenever we like. But dynamic values have a scheduling constraint: we
can only use them at certain times. For example, with a stream, we
want to look at the head at time 0, the head of the tail at time 1,
the head of the tail of the tail at time 2, and so on. It's a
mistake to look at the head of the tail of a stream at time 0, because
that value might not be available yet.
With an appropriate type discipline, it's possible to ensure
scheduling correctness statically, but unfortunately many people are
put off by modal types and Kripke logical relations. This is a shame,
because the payoff of all this is that the implementation strategy is
super-simple -- we can just use plain-vanilla lazy evaluation to
implement FRP.
Recently, though, I've figured out how to embed this kind of FRP
library into standard functional languages like Ocaml. Since we can't
define modal type operators in standard functional languages, we have
to give up some static assurance, and replace the static
checks of time-correctness with dynamic checks, but we are
still able to rule out space leaks by construction, and still get a
runtime error if we mis-schedule a program. Essentially, we can
replace type checking with contract checking.
As usual, you can find the code on Github here.
Let's look at an Ocaml signature for this library:
module type NEXT =
sig
type 'a t
exception Timing_error of int * int
val delay : (unit -> 'a) -> 'a t
val map : ('a -> 'b) -> 'a t -> 'b t
val zip : 'a t * 'b t -> ('a * 'b) t
val unzip : ('a * 'b) t -> 'a t * 'b t
val ($) : ('a -> 'b) t -> 'a t -> 'b t
val fix : ('a t -> 'a) -> 'a
module Runtime : sig
val tick : unit -> unit
val force : 'a t -> 'a
end
end
The NEXT signature introduces a single type constructor 'a
t, which can be thought of as the type of computations which
are scheduled to be evaluated on precisely the next tick of the
clock. The elements of 'a t are dynamic in the
sense that I mention above: we are only permitted to evaluate it on
the next tick of the clock, and evaluating it at any other time is an
error.
To model this kind of error, we also have a Timing_error
exception, which signals an error whose first argument contains the
time a thunk was scheduled to be evaluated, and whose second argument
contains the actual time.
Elements of 'a t are the only primitive way to create
dynamic values -- other values (like function closures) can be
dynamic, but only if they end up capturing a next-step thunk.
The delay function lets us create a next value from a
thunk, and the map function maps a function over a thunk.
The zip and unzip are used for pairing,
and the $ operation is the McBride/Paterson idiomatic
application operator. (Technically, it's derivable from zip,
but it's easiest to throw it in to the basic API.)
The fix operation is the one that really makes
reactive programming possible -- it says that guarded recursion
is allowed. So if we have a function which takes an 'a next
and returns an 'a, then we can take a fixed point. This
fixed point will always never block the event loop, because its
type ensures that we always delay by a tick before making a recursive
call.
This raw interface is, honestly, not so useful as is, but the slightly
miraculous fact is that this is the complete API we need to build all
the higher-level abstractions --- like events and streams --- that we
need to do real reactive programming.
Now, let's see what an implementation of this library could look like.
module Next : NEXT = struct
let time = ref 0
We can keep track of the current time in a reference cell.
type 'a t = {
time : int;
mutable code : 'a Lazy.t
}
The type of a thunk is a record consisting of a lazy thunk, and the time when it is
safe to force it.
type s = Hide : 'a t -> s
let thunks : s list ref = ref []
We also have a list that stores all of the references that we've
allocated. We'll use this list to enforce space-safety, by mutating
any thunk that gets too old.
exception Timing_error of int * int
let delay t =
let t = { time = 1 + !time; code = Lazy.from_fun t} in
thunks := (Hide t) :: !thunks;
t
When we create a thunk with the
delay function, we are
creating a thunk to be forced on the next time tick. So we can dererefence
time in order to find out the current time, and add 1 to
get the scheduled execution time for the thunk. We also add it to the
list
thunks, so that we can remember that we created it.
let force t =
if t.time != !time then
raise (Timing_error(t.time, !time))
else
Lazy.force t.code
Forcing a thunk just forces the code thunk, if the current time
matches the scheduled time for the thunk. Otherwise, we raise a
Timing_error. Note that memoization is handled by Ocaml's
built-in 'a Lazy.t type.
let map f r = delay (fun () -> f (force r))
let zip (r, r') = delay (fun () -> (force r, force r'))
let unzip r = (map fst r, map snd r)
let ($) f x = delay (fun () -> force f (force x))
The
map,
zip,
unzip, and
($)
operators just force and delay things in the obvious places.
let rec fix f = f (delay (fun () -> fix f))
The fixed point operation looks exactly like the standard lazy fixed point.
module Runtime = struct
let force = force
The runtime exposes the
force function to the event loop.
let cleanup (Hide t) =
let b = t.time < !time in
(if b then t.code <- lazy (raise (assert false)));
b
let tick () =
time := !time + 1;
thunks := List.filter cleanup !thunks
end
end
The tick function advances time by doing two things. First,
it increments the current time.
Then, it filters the list of thunks using the cleanup
function, which does two things. First, cleanup returns
true if its argument is older than the current time. As a result, we
only retain thunks in thunks which can be forced now or in
in the future.
Second, if the argument thunk to cleanup is old, it
replaces the code body with an assertion failure, since no
time-correct program should ever force this thunk. Updating the code
ensures that by construction next-step thunks always lose
their reference to their data once they age out, because every
thunk is placed onto thunks when it is created, and
when the clock is ticked past its time, it is guaranteed to drop
its references to its data.
This guarantees that spacetime leaks are impossible, since
we dynamically zero out any thunks that get too old! So here we see
how essential data abstraction is for imperative programming,
and not just functional programming.
As you can see, the implementation of the Next library is
pretty straightforward. The only mildly clever thing we do is to keep
track of the next-tick computations so we can null them out when they
get too old.
You should be wondering now how we can actually write reactive
programs, when the primitive the API provides only lets you schedule a
computation to run on the next tick, and that's it. The answer is
datatype declarations. Now that we have a type that lets
us talk about time, We can re-use our host language's facility to
define types which say more interesting things about time.
Let's start with the classic datatype of functional reactive
programming: streams. Streams are a kind of lazy sequence, which
recursively give you a value now, and a stream starting tomorrow,
thereby giving you a value on every time step.
module Stream :
sig
type 'a stream = Cons of 'a * 'a stream Next.t
val head : 'a stream -> 'a
val tail : 'a stream -> 'a stream Next.t
val unfold : ('a -> 'b * 'a Next.t) -> 'a -> 'b stream
val map : ('a -> 'b) -> 'a stream -> 'b stream
val zip : 'a stream * 'b stream -> ('a * 'b) stream
val unzip : ('a * 'b) stream -> 'a stream * 'b stream
end = struct
We give a simple signature for streams above. They are
a datatype exactly following the English description above,
as well as a collection of accessor and constructor functions,
like head, tail, map,
unfold and so on. All of these have pretty much
the expected types.
The only difference from the usual stream types is that sometimes
we need a Next.t to tell us when a value
needs to be available. Now let's look at the implementation.
open Next
type 'a stream = Cons of 'a * 'a stream Next.t
let head (Cons(x, xs)) = x
let tail (Cons(x, xs)) = xs
We can write accessor functions for streams, for convenience.
let map f = fix (fun loop (Cons(x, xs)) -> Cons(f x, loop $ xs))
The
map function uses the
fix fixed point
operator in our API, because we want to call the recursive function at
a later time.
let unfold f = fix (fun loop seed ->
let (x, seed) = f seed in
Cons(x, loop $ seed))
The unfold function uses a function
f and an initial
seed value to incrementally produce a sequence of values. This is
exactly like the usual unfold, except we have to use the applicative
interface to the
'a Next.t type to apply the function.
let zip pair =
unfold (fun (Cons(x, xs), Cons(y, ys)) -> ((x, y), Next.zip (xs, ys)))
pair
let unzip xys = fix (fun loop (Cons((x,y), xys')) ->
let (xs', ys') = Next.unzip (loop $ xys') in
(Cons(x, xs'), Cons(y, ys')))
xys
end
zip and
unzip work about the way we'd expect,
in that we use
Next.zip and
Next.unzip to
put together and take apart delayed pairs to build the ability to
put together and take apart streams.
This is all very nice, but the real power of giving a reactive API based on a
next-step type is that we can build types which aren't streams. For
example, let's give a datatype of events, which is the type
of values which will become available at some point in the future,
but we don't know exactly when.
module Event :
sig
type 'a event = Now of 'a | Wait of 'a event Next.t
val map : ('a -> 'b) -> 'a event -> 'b event
val return : 'a -> 'a event
val bind : 'a event -> ('a -> 'b event) -> 'b event
val select : 'a event -> 'a event -> 'a event
end
= struct
open Next
type 'a event = Now of 'a | Wait of 'a event Next.t
We represent this with a datatype
'a event, which has
two constructors. We say that an
'a event is either a value
of type
'a available
Now, or we have to
Wait to get another event tomorrow. So this is a
single value of type
'a that could come at any time ---
and we don't know when!
let map f = fix (fun loop e ->
match e with
| Now x -> Now (f x)
| Wait e' -> Wait (loop $ e'))
We can map over events, by waiting until the value becomes
available and then applying a function to the result.
let return x = Now x
let bind m f =
fix (fun bind m ->
match m with
| Now v -> f v
| Wait e' -> Wait (bind $ e'))
m
Events also form a monad, which corresponds to the
ability to sequence promises or futures in the promises libraries
you'll find in Javascript or Scala. (The
bind here
is a bit like the code
promise.then() method in JS.
let select e1 e2 =
fix (fun loop e1 e2 ->
match e1, e2 with
| Now a1, _ -> Now a1
| _, Now a2 -> Now a2
| Wait e1, Wait e2 -> Wait (loop $ e1 $ e2))
e1
e2
end
The really cool thing is that we can also join on two events to
wait for the first one to complete! This can be extended to lists
of events, if desired, but the pattern is easiest to see in the
binary case.
Of course, here's a small example of how you can actually put
this together to actually run a program. The run
function gives an event loop that runs for k
steps and halts, and prints out the first k
elements of the stream it gets passed as an argument.
module Test =
struct
open Next
open Stream
let ints n = unfold (fun i -> (i, delay(fun () -> i+1))) n
let rec run k xs =
if k = 0
then ()
else
let (x, xs) = (head xs, tail xs) in
Printf.printf "%d\n" x;
Runtime.tick();
run (k-1) (Runtime.force xs)
end
