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

[jean-christophe mincke]

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
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