One more thing... The function:
return :: a -> Beh a
return x t = x
fails to be causal when a is itself a behaviour, since it specializes to
(after a bit of eta-conversion):
return :: Beh a -> Beh (Beh a)
return b t u = b u
which isn't causal. This rules out return, which in turn means that Beh
can't implement the Monad type class.
I don't think this impacts arrowized FRP, but it does mean that any
causal non-arrowized model can't form a monad, which seems unfortunate.
If the "causality" requirement were replaced by "deep causality" then I
believe the model would form a monad (cough cough but in a different
category cough).
A.
On 10/14/2011 05:18 PM, Jeffrey, Alan S A (Alan) wrote:
I should add that I have a pragmatic reason for asking about causality,
which is that over at https://github.com/agda/agda-frp-js I have an
implementation of FRP for Agda running in the browser using an
Agda-to-JS back end I wrote.
In that model, I can see how to implement deep causality, but I can't
see how to implement shallow causality, since the back end interfaces to
the DOM event and time model.
A.
On 10/13/2011 10:43 PM, David Barbour wrote:
On Thu, Oct 13, 2011 at 7:54 AM, Alan Jeffrey<ajeff...@bell-labs.com
<mailto:ajeff...@bell-labs.com>> wrote:
The `problem` such as it exists: you will be unable to causally
construct the argument toith the `weird` function, except by modeling a
nested/simulated world (i.e. modeling one FRP system within another).
This is not an unrealistic endeavor, e.g. one might model the future
position of a thrown baseball in order to predict it. In this sense,
`weird` is not weird.
Ah, I think this is a very good summary. It seems that there's an
implicit shift of worlds when you nest FRP behaviours. The top level
world (the one that reactimate is executing) uses wall-clock time, but
nested behaviours are in a different world, where time is simulated.
Making these worlds explicit (I never met a problem that couldn't use
some more phantom types :-) we have a type Beh W A for a behaviour in
world W of type A, and a definition of causality that's indexed by
worlds. Writing RW for the top-level real world, and SW for a simulated
world, we have:
weird : Beh RW (Beh RW A) -> Beh RW A
weird b t = b t (t + 1) -- not causal
weird : Beh RW (Beh SW A) -> Beh RW A
weird b t = b t (t + 1) -- causal
and:
double : Beh RW A -> Beh RW (Beh RW A)
double b t u = b u -- causal
double : Beh RW A -> Beh RW (Beh SW A)
double b t u = b u -- not causal
[Caveat: details not worked out.]
Making worlds explicit like this I think helps clarify why one person's
"weird" is another person's "perfectly reasonable function" :-)
Does something like this help clarify matters?
A.
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