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

[Christopher Lane Hinson]

I had this conversation recently.  My experience with implementing RK4 in 
RSAGL led me to some contrary conclusions:

First, ODEs aren't necessarily useful for interpreting externally sampled 
events, because you don't have the ability to run, i.e. RK4 against a 
mouse cursor position without using time travel to sample past values of 
the mouse position.

So, for example, I might use ODEs to make an object act like it's on a 
spring attached to the mouse pointer, but the the mouse position must be 
treated as a fixed position for each frame interval, while the object 
moves dynamically within that interval.

Second, you still want some kind of recursion/delay/fixed point operator.

For example, ok, I tend to think in terms of monsters with lasers.  The 
monsters point their lasers at each other and run away when they see a 
laser pointed at them.  This might be a classic if involved ODE problem 
until some of the monsters reproduce at random every five to ten minutes 
and some only sense motion while others use echolocation and can be 
distracted by clicking with the mouse, and some only get angry when a CD 
is loaded in the drive tray.  Someone implement *that* without delayed 
recursion.

Friendly,
--Lane

On Mon, 20 Apr 2009, jean-christophe mincke 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
>