Awesome

Lambda Cat

This is still heavily work in progress!!!

disclaimer:

This code is based on Philip Zuckers http://www.philipzucker.com/compiling-to-categories-3-a-bit-cuter/

I forked from a specific branch of his repository. All the really complicated stuff is his invention. I just filled the blanks, cleaned up the code a bit and added an interpreter that executes CCC code.

Introduction

Recently I read the very interesting Compiling to Categories paper by Conal Elliot.

He presents the idea to compile haskell programs into expressions of cartesian closed categories by λ-elimination. These expressions can then be used for different purposes like alternative program evaluation, graphic representation of program graphs, designing hardware layouts for algorithms, etc.

Cartesian Closed Categories (CCC)

In his famous paper Compiling to Categories Conal Elliot describes a way to compile from simply typed lambda-calculus terms to cartesian closed categories(CCC).

At the core of his approach sits a transformation from lambda-terms to CCC expressions that are done by eliminating variables by an abstraction function absCCC (again in pseudo-Haskell):

absCCC (\x -> x)   = id
absCCC (\x -> y)   = const y
absCCC (\x -> p q) = apply . ((\x -> p) △ (\x -> q))

Where (△) is introduced by the Cartesian category:

class Category k => Cartesian k where
  (△) :: (a `k` c) -> (a `k` d) -> (a `k` (c, d))

In the (->) instance of Cartesian (△) is defined as:

(△):: (t -> a) -> (t -> b) -> t -> (a, b)
(f △ g) x = (f x, g x)

And where apply is introduced by the Closed category:

class Cartesian k => Closed k where
  apply :: ((a -> b), a) `k` b

In the (->) instance of Closed apply is defined as

apply :: (a -> b, a) -> b
apply (f, x) = f x