[Haskell-cafe] FRP, integration and differential equations.

[Paul L]

Trying to give different semantics to the same declarative definition based
on whether it's recursively defined or not seems rather hack-ish, although
I can understand what you are coming from from an implementation angle.

Mathematically an integral operator has only one semantics regardless
of what's put in front of it or inside. If our implementation can't match this
simplicity, then we got a problem!

The arrow FRP gets rid of the leak problem and maintains a single definition
of integral by using a restricted form of recursion - the loop operator.
If you'd rather prefer having signals as first class objects, similar technique
existed in synchronous languages [1], i.e., by using a special rec primitive.

Disclaimer: I was the co-author of the leak paper [2].

[1] A co-iterative characterization of synchronous stream functions, P
Caspi, M Pouzet.
[2] Plugging a space leak with an arrow, H. Liu, P. Hudak

-- 
Regards,
Paul Liu

Yale Haskell Group
http://www.haskell.org/yale

On 4/20/09, jean-christophe mincke <jeanchristophe.mincke at gmail.com> wrote:
> In a post in the *Elerea, another FRP library *thread*,* Peter Verswyvelen
> wrote:
>
> *>I think it would be nice if we could make a "reactive benchmark" or
> something: some tiny examples that capture the essence of reactive systems,
> and a way to compare each solution's >pros and cons.* *
> *
> *>For example the "plugging a space leak with an arrow" papers reduces the
> recursive signal problem to
> *
> *
> *
> *>e = integral 1 e*
> *
> *
>  *>Maybe the Nlift problem is a good example for dynamic collections, but I
> guess we'll need more examples.*
> *
> *
> *>The reason why I'm talking about examples and not semantics is because the
> latter seems to be pretty hard to get right for FRP?*
>
> I would like to come back to this exemple. I am trying to write a small FRP
> in F# (which is a strict language, a clone of Ocaml) and I also came across
> space and/or time leak. But maybe not for the same reasons...
>
> Thinking about these problems and after some trials and errors, I came to
> the following conclusions:
>
> I believe that writing the expression
>
>       e = integral 1 *something*
>
>       where e is a Behavior (thus depends on a continuous time).
>
> has really two different meanings.
>
> 1. if *something *is independent of e, what the above expression means is
> the classical integration of a time dependent function between t0 and t1.
> Several numerical methods are available to compute this integral and, as far
> as I know, they need to compute *something *at t0, t1 and, possibly, at
> intermediate times. In this case, *something *can be a Behavior.
>
> 2. If *something *depends directly or indirectly of e then we are faced with
> a first order differential equation of the form:
>
>        de/dt = *something*(e,t)
>
>     where de/dt is the time derivative of e and  *something*(e,t) indicates
> that *something* depends, without loss of generality, on both e and t.
>
> There exist specific methods to numerically solve differential equations
> between t0 and t1. Some of them only require the knowledge of e at t0 (the
> Euler method), some others needs  to compute *something *from intermediate
> times (in [t0, t1[ ) *and *estimates of e at those intermediary times.
>
> 3. *something *depends (only) on one or more events that, in turns, are
> computed from e. This case seems to be the same as the first one where the
> integrand can be decomposed into a before-event integrand and an after-event
> integrand (if any event has been triggered). Both integrands being
> independent from e. But I have not completely investigated this case  yet...
>
> Coming back to my FRP, which is based on residual behaviors, I use a
> specific solution for each case.
>
> Solution to case 1 causes no problem and is similar to what is done in
> classical FRP (Euler method, without recursively defined behaviors). Once
> again as far as I know...
>
> The second case has two solutions:
> 1. the 'integrate' function is replaced by a function 'solve' which has the
> following signature
>
>        solve :: a -> (Behavior a -> Behavior a) -> Behavior a
>
>       In fact,  *something*(e,t) is represented by an integrand function
> from behavior to behavior, this function is called by the
> integration           method. The integration method is then free to pass
> estimates of e, as constant behaviors, to the integrand function.
>
>       The drawbacks of this solution are:
>       - To avoid space/time leaks, it cannot be done without side effects
> (to be honest, I have not been able to  find a solution without
> assignement). However these side effects are not visible from outside of the
> solve function. ..
>       - If behaviors are defined within the integrand function, they are not
> accessible from outside of this integrand function.
>
> 2. Introduce constructions that looks like to signal functions.
>
>       solve :: a -> SF a a -> Behavior a
>
>    where a SF is able to react to events and may manage an internal state.
>    This solution solves the two above problems but make the FRP a bit more
> complex.
>
>
> Today, I tend to prefer the first solution, but what is important, in my
> opinion, is to recognize the fact that
>
>     e = integral 1 *something*
>
> really addresses two different problems (integration and solving of
> differential equations) and each problem should have their own solution.
>
> The consequences are :
>
>    1. There is no longer any need for my FRP to be able to define a Behavior
>    recursively. That is a good news for this is quite tricky in F#.
>    Consequently, there is no need to introduce delays.
>    2. Higher order methods for solving of diff. equations can be used (i.e.
>    Runge-Kutta). That is also good news for this was one of my main goal in
>    doing the exercice of writing a FRP.
>
> Regards,
>
> J-C
>