Functional interactive behavior

In a previous post, I presented a fundamental reason why classic FRP does not fit interactive behavior, which is that the semantic model captures only the influence of time and not other input. I also gave a simple alternative, with a simple and general model for temporal and spatial transformation, in which input behavior is transformed inversely to the transformation of output behavior.

The semantic model I suggested is the same as used in “Arrow FRP”, from Fruit and Yampa. I want, however, a more convenient and efficient way to package up that model, which is the subject of the post you are reading now.

Next, we took a close look at one awkward aspect of classic FRP for interactive behavior, namely the need to trim inputs, and how trimming relates to comonadic FRP. The trim function allows us to define multi-phase interactive behaviors correctly and efficiently, but its use is tedious and is easy to get wrong. It thus fails to achieve what I want from functional programming in general and FRP in particular, which is to enable writing simple, natural descriptions, free of mechanical details.

The current post hides and automates the mechanics of trimming, so that the intent of an interactive behavior can be expressed directly and executed correctly and efficiently.

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Trimming inputs in functional reactive programming

This post takes a close look at one awkward aspect of classic (non-arrow) FRP for interactive behavior, namely the need to trim (or “age”) old input. Failing to trim results in behavior that is incorrect and grossly inefficient.

Behavior trimming connects directly into the comonad interface mentioned in a few recent posts, and is what got me interested in comonads recently.

In absolute-time FRP, trimming has a purely operational significance. Switching to relative time, trimming is recast as a semantically familiar operation, namely the generalized drop function used in two recent posts.

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Sequences, segments, and signals

The post Sequences, streams, and segments offered an answer to the the question of what’s missing in the following box:

discreteStream Sequence
continuousFunction ???

I presented a simple type of function segments, whose representation contains a length (duration) and a function. This type implements most of the usual classes: Monoid, Functor, Zip, and Applicative, as well Comonad, but not Monad. It also implements a new type class, Segment, which generalizes the list functions length, take, and drop.

The function type is simple and useful in itself. I believe it can also serve as a semantic foundation for functional reactive programming (FRP), as I’ll explain in another post. However, the type has a serious performance problem that makes it impractical for some purposes, including as implementation of FRP.

Fortunately, we can solve the performance problem by adding a simple layer on top of function segments, to get what I’ll call “signals”. With this new layer, we have an efficient replacement for function segments that implements exactly the same interface with exactly the same semantics. Pleasantly, the class instances are defined fairly simply in terms of the corresponding instances on function segments.

You can download the code for this post.


  • 2008-12-06: dup [] = [] near the end (was [mempty]).
  • 2008-12-09: Fixed take and drop default definitions (thanks to sclv) and added point-free variant.
  • 2008-12-18: Fixed appl, thanks to sclv.
  • 2011-08-18: Eliminated accidental emoticon in the definition of dup, thanks to anonymous.

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Sequences, streams, and segments

What kind of thing is a movie? Or a song? Or a trajectory from point A to point B? If you’re a computer programmer/programmee, you might say that such things are sequences of values (frames, audio samples, or spatial locations). I’d suggest that these discrete sequences are representations of something more essential, namely a flow of continuously time-varying values. Continuous models, whether in time or space, are often more compact, precise, adaptive, and composable than their discrete counterparts.

Functional programming offers great support for sequences of variable length. Lazy functional programming adds infinite sequences, often called streams, which allows for more elegant and modular programming.

Functional programming also has functions as first class values, and when the function’s domain is (conceptually) continuous, we get a continuous counterpart to infinite streams.

Streams, sequences, and functions are three corners of a square. Streams are discrete and infinite, sequences are discrete and finite, and functions-on-reals are continuous and infinite. The missing corner is continuous and finite, and that corner is the topic of this post.

discreteStream Sequence
continuousFunction ???

You can download the code for this post.


  • 2008-12-01: Added Segment.hs link.
  • 2008-12-01: Added Monoid instance for function segments.
  • 2008-12-01: Renamed constructor “DF” to “FS” (for “function segment”)
  • 2008-12-05: Tweaked the inequality in mappend on (t :-># a).

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