In recent implementations of Classic FRP, for instance reactive-banana, there are event streams and signals, which are step functions (reactive-banana calls them behaviours but they are nevertheless step functions). I've noticed that Elm only uses signals, and doesn't differentiate between signals and event streams. Also, reactive-banana allows to go from event streams to signals (edited: and it's sort of possible to act on behaviours using reactimate' although it not considered good practice), which kind of means that in theory we could apply all the event stream combinators on signals/behaviours by first converting the signal to event stream, applying and then converting again. So, given that it's in general easier to use and learn just one abstraction, what is the advantage of having separated signals and event streams ? Is anything lost in using just signals and converting all the event stream combinators to operate on signals ?
edit: The discussion has been very interesting. The main conclusions that I took from the discussion myself is that behaviours/event sources are both needed for mutually recursive definitions (feedback) and for having an output depend on two inputs (a behaviour and an event source) but only cause an action when one of them changes (<@>).
(Clarification: In reactive-banana, it is not possible to convert a Behavior back to an Event. The stepper function is a one-way ticket. There is a changes function, but its type indicates that it is "impure" and it comes with a warning that it does not preserve the semantics.)
I believe that having two separates concepts makes the API more elegant. In other words, it boils down to a question of API usability. I think that the two concepts behave sufficiently differently that things flow better if you have two separate types.
For example, the direct product for each type is different. A pair of Behavior is equivalent to a Behavior of pairs
(Behavior a, Behavior b) ~ Behavior (a,b)
whereas a pair of Events is equivalent to an Event of a direct sum:
(Event a, Event b) ~ Event (EitherOrBoth a b)
If you merge both types into one, then neither of these equivalence will hold anymore.
However, one of the main reasons for the separation of Event and Behavior is that the latter does not have a notion of changes or "updates". This may seem like an omission at first, but it is extremely useful in practice, because it leads to simpler code. For instance, consider a monadic function newInput that creates an input GUI widget that displays the text indicated in the argument Behavior,
input <- newInput (bText :: Behavior String)
The key point now is that the text displayed does not depend on how often the Behavior bText may have been updated (to the same or a different value), only on the actual value itself. This is a lot easier to reason about than the other case, where you would have to think about what happens when two successive event occurrences have the same value. Do you redraw the text while the user edits it?
(Of course, in order to actually draw the text, the library has to interface with the GUI framework and does keep track of changes in the Behavior. This is what the changes combinator is for. However, this can be seen as an optimization and is not available from "within FRP".)
The other main reason for the separation is recursion. Most Events that recursively depend on themselves are ill-defined. However, recursion is always allowed if you have mutual recursion between an Event and a Behavior
e = ((+) <$> b) <@> einput
b = stepper 0 e
There is no need to introduce delays by hand, it just works out of the box.
Something critically important to me is lost, namely the essence of behaviors, which is (possibly continuous) variation over continuous time.
Precise, simple, useful semantics (independent of a particular implementation or execution) is often lost as well.
Check out my answer to "Specification for a Functional Reactive Programming language", and follow the links there.
Whether in time or in space, premature discretization thwarts composability and complicates semantics.
Consider vector graphics (and other spatially continuous models like Pan's). Just as with premature finitization of data structures as explained in Why Functional Programming Matters.
I don't think there's any benefit to using the signals/behaviors abstraction over elm-style signals. As you point out, it's possible to create a signal-only API on top of the signal/behavior API (not at all ready for use, but see https://github.com/JohnLato/impulse/blob/dyn2/src/Reactive/Impulse/Syntax2.hs for an example). I'm pretty sure it's also possible to write a signal/behavior API on top of an elm-style API as well. That would make the two APIs functionally equivalent.
WRT efficiency, with a signals-only API the system should have a mechanism where only signals that have updated values will cause recomputations (e.g. if you don't move the mouse, the FRP network won't re-calculate the pointer coordinates and redraw the screen). Provided this is done, I don't think there's any loss of efficiency compared to a signals-and-streams approach. I'm pretty sure Elm works this way.
I don't think the continuous-behavior issue makes any difference here (or really at all). What people mean by saying behaviors are continuous over time is that they are defined at all times (i.e. they're functions over a continuous domain); the behavior itself isn't a continuous function. But we don't actually have a way to sample a behavior at any time; they can only be sampled at times corresponding to events, so we can't use the full power of this definition!
Semantically, starting from these definitions:
Event == for some t ∈ T: [(t,a)]
Behavior == ∀ t ∈ T: t -> b
since behaviors can only be sampled at times where events are defined, we can create a new domain TX where TX is the set of all times t at which Events are defined. Now we can loosen the Behavior definition to
Behavior == ∀ t ∈ TX: t -> b
without losing any power (i.e. this is equivalent to the original definition within the confines of our frp system). Now we can enumerate all times in TX to transform this to
Behavior == ∀ t ∈ TX: [(t,b)]
which is identical to the original Event definition except for the domain and quantification. Now we can change the domain of Event to TX (by the definition of TX), and the quantification of Behavior (from forall to for some) and we get
Event == for some t ∈ TX: [(t,a)]
Behavior == for some t ∈ TX: [(t,b)]
and now Event and Behavior are semantically identical, so they could obviously be represented using the same structure in an FRP system. We do lose a bit of information at this step; if we don't differentiate between Event and Behavior we don't know that a Behavior is defined at every time t, but in practice I don't think this really matters much. What elm does IIRC is require both Events and Behaviors to have values at all times and just use the previous value for an Event if it hasn't changed (i.e. change the quantification of Event to forall instead of changing the quantification of Behavior). This means you can treat everything as a signal and it all Just Works; it's just implemented so that the signal domain is exactly the subset of time that the system actually uses.
I think this idea was presented in a paper (which I can't find now, anyone else have a link?) about implementing FRP in Java, perhaps from POPL '14? Working from memory, so my outline isn't as rigorous as the original proof.
There's nothing to stop you from creating a more-defined Behavior by e.g. pure someFunction, this just means that within an FRP system you can't make use of that extra defined-ness, so nothing is lost by a more restricted implementation.
As for notional signals such as time, note that it's impossible to implement an actual continuous-time signal using typical programming languages. Since the implementation will necessarily be discrete, converting that to an event stream is trivial.
In short, I don't think anything is lost by using just signals.
I've unfortunately have no references in mind, but I distinctly remember
different reactive authors claiming this choice is just for efficiency. You expose
both to give the programmer a choice in what implementation of the same idea is
more efficient for your problem.
I might be lying now, but I believe Elm implements everything as event streams under the
hood. Things like time wouldn't be so nice like event streams though, since there are an
infinite amount of events during any time frame. I'm not sure how Elm solves this, but I
think it's a good example on something that makes more sense as a signal, both conceptually
and in implementation.
