## Doing more with length-typed vectors

The post Fixing lists defined a (commonly used) type of vectors, whose lengths are determined statically, by type. In `Vec n a`, the length is `n`, and the elements have type `a`, where `n` is a type-encoded unary number, built up from zero and successor (`Z` and `S`).

``infixr 5 :<data Vec ∷ * → * → * where  ZVec ∷                Vec Z     a  (:<) ∷ a → Vec n a → Vec (S n) a``

It was fairly easy to define `foldr` for a `Foldable` instance, `fmap` for `Functor`, and `(⊛)` for `Applicative`. Completing the `Applicative` instance is tricky, however. Unlike `foldr`, `fmap`, and `(⊛)`, `pure` doesn’t have a vector structure to crawl over. It must create just the right structure anyway. I left this challenge as a question to amuse readers. In this post, I give a few solutions, including my current favorite.

You can find the code for this post and the two previous ones in a code repository.

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## The C language is purely functional

There has been a scurry of reaction on twitter and reddit to Robert Fischer’s post saying that Scala is Not a Functional Programming Language. James Iry responded by saying that analogous reasoning leads to the conclusion that Erlang is Not Functional

My first inclination was to suggest that Haskell, as commonly practiced (with monadic IO), is not a functional language either. Instead, I’m going to explain how it is that the C language is purely functional.

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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.

Enjoy, and thanks!

## 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.

Abstract:

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”).

## A handy generalized filter

For quite a while, I’ve been using a handy operation for filtering functional events:

`justE :: Event (Maybe a) -> Event a`

The idea of `justE` is to drop the `Nothing`-valued occurrences and strip off the `Just` constructors from the remaining occurrences. Recently I finally noticed the similarity with a standard function (in `Data.Maybe`):

`catMaybes :: [Maybe a] -> [a]`

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A previous post described future values (or simply “futures”), which are values depend on information from the future, e.g., from the real world. There I gave a simple denotational semantics for future values as time/value pairs. This post describes the multi-threaded implementation of futures in Reactive‘s `Data.Reactive` module.