Garbage collecting the semantics of FRP

Ever since ActiveVRML, the model we’ve been using in functional reactive programming (FRP) for interactive behaviors is (T->a) -> (T->b), for dynamic (time-varying) input of type a and dynamic output of type b (where T is time). In “Classic FRP” formulations (including ActiveVRML, Fran & Reactive), there is a “behavior” abstraction whose denotation is a function of time. Interactive behaviors are then modeled as host language (e.g., Haskell) functions between behaviors. Problems with this formulation are described in Why classic FRP does not fit interactive behavior. These same problems motivated “Arrowized FRP”. In Arrowized FRP, behaviors (renamed “signals”) are purely conceptual. They are part of the semantic model but do not have any realization in the programming interface. Instead, the abstraction is a signal transformer, SF a b, whose semantics is (T->a) -> (T->b). See Genuinely Functional User Interfaces and Functional Reactive Programming, Continued.

Whether in its classic or arrowized embodiment, I’ve been growing uncomfortable with this semantic model of functions between time functions. A few weeks ago, I realized that one source of discomfort is that this model is mostly junk.

This post contains some partially formed thoughts about how to eliminate the junk (“garbage collect the semantics”), and what might remain.

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Thoughts on semantics for 3D graphics

The central question for me in designing software is always

What does it mean?

With functional programming, this question is especially crisp. For each data type I define, I want to have a precise and simple mathematical model. (For instance, my model for behavior is function-of-time, and my model of images is function-of-2D-space.) Every operation on the type is also given a meaning in terms of that semantic model.

This specification process, which is denotational semantics applied to data types, provides a basis for

  • correctness of the implementation,
  • user documentation free of implementation detail,
  • generating and proving properties, which can then be used in automated testing, and
  • evaluating and comparing the elegance and expressive power of design decisions.

For an example (2D images), some motivation of this process, and discussion, see Luke Palmer’s post Semantic Design. See also my posts on the idea and use of type class morphisms, which provide additional structure to denotational design.

In spring of 2008, I started working on a functional 3D library, FieldTrip. I’ve designed functional 3D libraries before as part of TBAG, ActiveVRML, and Fran. This time I wanted a semantics-based design, for all of the reasons given above. As always, I want a model that is

  • simple,
  • elegant, and
  • general.

For 3D, I also want the model to be GPU-friendly, i.e., to execute well on (modern) GPUs and to give access to their abilities.

I hadn’t thought of or heard a model that I was happy with, and so I didn’t have the sort of firm ground I like to stand on in working on FieldTrip. Last February, such a model occurred to me. I’ve had this blog post mostly written since then. Recently, I’ve been focused on functional 3D again for GPU-based rendering, and then Sean McDirmid posed a similar question, which got me thinking again.

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Notions of purity in Haskell

Lately I’ve been learning that some programming principles I treasure are not widely shared among my Haskell comrades. Or at least not widely among those I’ve been hearing from. I was feeling bummed, so I decided to write this post, in order to help me process the news and to see who resonates with what I’m looking for.

One of the principles I’m talking about is that the value of a closed expression (one not containing free variables) depends solely on the expression itself — not influenced by the dynamic conditions under which it is executed. I relate to this principle as the soul of functional programming and of referential transparency in particular.


  • 2009-10-26: Minor typo fix

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Denotational design with type class morphisms

I’ve just finished a draft of a paper called Denotational design with type class morphisms, for submission to ICFP 2009. The paper is on a theme I’ve explored in several posts, which is semantics-based design, guided by type class morphisms.

I’d love to get some readings and feedback. Pointers to related work would be particularly appreciated, as well as what’s unclear and what could be cut. It’s an entire page over the limit, so I’ll have to do some trimming before submitting.

The abstract:

Type classes provide a mechanism for varied implementations of standard interfaces. Many of these interfaces are founded in mathematical tradition and so have regularity not only of types but also of properties (laws) that must hold. Types and properties give strong guidance to the library implementor, while leaving freedom as well. Some of the remaining freedom is in how the implementation works, and some is in what it accomplishes.

To give additional guidance to the what, without impinging on the how, this paper proposes a principle of type class morphisms (TCMs), which further refines the compositional style of denotational semantics. The TCM idea is simply that the instance’s meaning is the meaning’s instance. This principle determines the meaning of each type class instance, and hence defines correctness of implementation. In some cases, it also provides a systematic guide to implementation, and in some cases, valuable design feedback.

The paper is illustrated with several examples of type, meanings, and morphisms.

You can get the paper and see current errata here.

The submission deadline is March 2, so comments before then are most helpful to me.

Enjoy, and thanks!

What is automatic differentiation, and why does it work?

Bertrand Russell remarked that

Everything is vague to a degree you do not realize till you have tried to make it precise.

I’m mulling over automatic differentiation (AD) again, neatening up previous posts on derivatives and on linear maps, working them into a coherent whole for an ICFP submission. I understand the mechanics and some of the reasons for its correctness. After all, it’s "just the chain rule".

As usual, in the process of writing, I bumped up against Russell’s principle. I felt a growing uneasiness and realized that I didn’t understand AD in the way I like to understand software, namely,

  • What does it mean, independently of implementation?
  • How do the implementation and its correctness flow gracefully from that meaning?
  • Where else might we go, guided by answers to the first two questions?

Ever since writing Simply efficient functional reactivity, the idea of type class morphisms keeps popping up for me as a framework in which to ask and answer these questions. To my delight, this framework gives me new and more satisfying insight into automatic differentiation.

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3D rendering as functional reactive programming

I’ve been playing with a simple/general semantics for 3D. In the process, I was surprised to see that a key part of the semantics looks exactly like a key part of the semantics of functional reactivity as embodied in the library Reactive. A closer look revealed a closer connection still, as described in this post.

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Another lovely example of type class morphisms

I read Max Rabkin’s recent post Beautiful folding with great excitement. He shows how to make combine multiple folds over the same list into a single pass, which can then drastically reduce memory requirements of a lazy functional program. Max’s trick is giving folds a data representation and a way to combine representations that corresponds to combining the folds.

Peeking out from behind Max’s definitions is a lovely pattern I’ve been noticing more and more over the last couple of years, namely type class morphisms.

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Simplifying semantics with type class morphisms

When I first started playing with functional reactivity in Fran and its predecessors, I didn’t realize that much of the functionality of events and reactive behaviors could be packaged via standard type classes. Then Conor McBride & Ross Paterson introduced us to applicative functors, and I remembered using that pattern to reduce all of the lifting operators in Fran to just two, which correspond to pure and (< *>) in the Applicative class. So, in working on a new library for functional reactive programming (FRP), I thought I’d modernize the interface to use standard type classes as much as possible.

While spelling out a precise (denotational) semantics for the FRP instances of these classes, I noticed a lovely recurring pattern:

The meaning of each method corresponds to the same method for the meaning.

In this post, I’ll give some examples of this principle and muse a bit over its usefulness. For more details, see the paper Simply efficient functional reactivity. Another post will start exploring type class morphisms and type composition, and ask questions I’m wondering about.

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Simply efficient functional reactivity

I submitted a paper Simply efficient functional reactivity to ICFP 2008.


Functional reactive programming (FRP) has simple and powerful semantics, but has resisted efficient implementation. In particular, most past implementations have used demand-driven sampling, which accommodates FRP’s continuous time semantics and fits well with the nature of functional programming. Consequently, values are wastefully recomputed even when inputs don’t change, and reaction latency can be as high as the sampling period.

This paper presents a way to implement FRP that combines data- and demand-driven evaluation, in which values are recomputed only when necessary, and reactions are nearly instantaneous. The implementation is rooted in a new simple formulation of FRP and its semantics and so is easy to understand and reason about.

On the road to efficiency and simplicity, we’ll meet some old friends (monoids, functors, applicative functors, monads, morphisms, and improving values) and make some new friends (functional future values, reactive normal form, and concurrent “unambiguous choice”).

Future values

A future value (or simply “future”) is a value that might not be knowable until a later time, such as “the value of the next key you press”, or “the value of LambdaPix stock at noon next Monday” (both from the time you first read this sentence), or “how many tries it will take me to blow out all the candles on my next birthday cake”. Unlike an imperative computation, each future has a unique value — although you probably cannot yet know what that value is. I’ve implemented this notion of futures as part of a library Reactive.


  • 2008-04-04: tweaked tag; removed first section heading.

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