Optimizing CCCs

In the post Overloading lambda, I gave a translation from a typed lambda calculus into the vocabulary of cartesian closed categories (CCCs). This simple translation leads to unnecessarily complex expressions. For instance, the simple lambda term, “λ ds → (λ (a,b) → (b,a)) ds”, translated to a rather complicated CCC term:

apply ∘ (curry (apply ∘ (apply ∘ (const (,) △ (id ∘ exr) ∘ exr) △ (id ∘ exl) ∘ exr)) △ id)

(Recall from the previous post that (∘) binds more tightly than (△) and (▽).)

However, we can do much better, translating to

exr △ exl

which says to pair the right and left halves of the argument pair, i.e., swap.

This post applies some equational properties to greatly simplify/optimize the result of translation to CCC form, including example above. First I’ll show the equational reasoning and then how it’s automated in the lambda-ccc library.

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Overloading lambda

Haskell’s type class facility is a powerful abstraction mechanism. Using it, we can overload multiple interpretations onto a single vocabulary, with each interpretation corresponding to a different type. The class laws constrain these interpretations and allow reasoning that is valid over all (law-abiding) instances—even ones not yet defined.

As Haskell is a higher-order functional language in the heritage of Church’s (typed) lambda calculus, it also supports “lambda abstraction”.

Sadly, however, these two forms of abstraction don’t go together. When we use the vocabulary of lambda abstraction (“λ x → ⋯”) and application (“u v”), our expressions can only be interpreted as one type (constructor), namely functions. (Note that I am not talking about parametric polymorphism, which is available with both lambda abstraction and type-class-style overloading.) Is it possible to overload lambda and application using type classes, or perhaps in the same spirit? The answer is yes, and there are some wonderful benefits of doing so. I’ll explain the how in this post and hint at the why, to be elaborated in futures posts.

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From Haskell to hardware via cartesian closed categories

Since fall of last year, I’ve been working at Tabula, a Silicon Valley start-up developing an innovative programmable hardware architecture called “Spacetime”, somewhat similar to an FPGA, but much more flexible and efficient. I met the founder, Steve Teig, at a Bay Area Haskell Hackathon in February of 2011. He described his Spacetime architecture, which is based on the geometry of the same name, developed by Hermann Minkowski to elegantly capture Einstein’s theory of special relativity. Within the first 30 seconds or so of hearing what Steve was up to, I knew I wanted to help.

The vision Steve shared with me included not only a better alternative for hardware designers (programmed in hardware languages like Verilog and VHDL), but also a platform for massively parallel execution of software written in a purely functional language. Lately, I’ve been working mainly on this latter aspect, and specifically on the problem of how to compile Haskell. Our plan is to develop the Haskell compiler openly and encourage collaboration. If anything you see in this blog series interests you, and especially if have advice or you’d like to collaborate on the project, please let me know.

In my next series of blog posts, I’ll describe some of the technical ideas I’ve been working with for compiling Haskell for massively parallel execution. For now, I want to introduce a central idea I’m using to approach the problem.

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