On Thu, Oct 13, 2011 at 7:54 AM, Alan Jeffrey
A function (f : Beh A -> Beh B) is causal whenever it respects =t, i.e. (forall t . a =t b => f a =t f b).
Yes. Function outputs only depend on the past values of the input function. Your solutions for double and weird are accurate. Double is lifting the future at each instant into the present, which is obviously not causal. And the `weird` function presumes you already have a obtained a complete view of a behavior at each instant. The `problem` such as it exists: you will be unable to causally construct the argument to 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. If you want first-class behaviors or behavior transformers, those will need a different abstraction than 'nested' behaviors. Nested != First Class. You'd have special functions to lift a first-class behavior as an argument (e.g. add a phantom type to prohibit non-causal observation), and to lower it into the main system for sampling (e.g. ArrowApply). `Deep` causality is not so much an issue, since it often makes perfect sense to simulate one FRP behavior within another. Regards, Dave