Tim Pierson o1lo01ol1o

Fast Discrete Program Search 2

I am investigating how to use Bend (a parallel language) to accelerate Symbolic AI; in special, Discrete Program Search. Basically, think of it as an alternative to LLMs, GPTs, NNs, that is also capable of generating code, but by entirely different means. This kind of approach was never scaled with mass compute before - it wasn't possible! - but Bend changes this. So, my idea was to do it, and see where it goes.

Now, while I was implementing some candidate algorithms on Bend, I realized that, rather than mass parallelism, I could use an entirely different mechanism to speed things up: SUP Nodes. Basically, it is a feature that Bend inherited from its underlying model ("Interaction Combinators") that, in simple terms, allows us to combine multiple functions into a single superposed one, and apply them all to an argument "at the same time". In short, it allows us to call N functions at a fraction of the expected cost. Or, in simple terms: why parallelize when we can share?

A

A Taxonomy of Self-Types

Part I: Introduction to Self-Types

Datatypes in the Formality programming language are built out of an unusual structure: the self-type. Roughly speaking, a self-type is a type that can depend or be a proposition on it's own value. For instance, the consider the 2 constructor datatype Bool:

Alternative to the Church, Scott and Parigot encodings of data on the Lambda Calculus.

When it comes to encoding data on the pure λ-calculus (without complex extensions such as ADTs), there are 3 widely used approaches.

Church Encoding

The Church Encoding, which represents data structures as their folds. Using Caramel’s syntax, the natural number 3 is, for example. represented as:

0 c0 = (f x -> x)
1 c1 = (f x -> (f x))
2 c2 = (f x -&gt; (f (f x)))

	{-# LANGUAGE AllowAmbiguousTypes #-}
	{-# LANGUAGE DataKinds #-}
	{-# LANGUAGE FlexibleContexts #-}
	{-# LANGUAGE FlexibleInstances #-}
	{-# LANGUAGE LambdaCase #-}
	{-# LANGUAGE MagicHash #-}
	{-# LANGUAGE MultiParamTypeClasses #-}
	{-# LANGUAGE NamedWildCards #-}
	{-# LANGUAGE PatternSynonyms #-}
	{-# LANGUAGE PartialTypeSignatures #-}

	{-# LANGUAGE StandaloneDeriving, DataKinds, PolyKinds, GADTs, RankNTypes, TypeOperators, FlexibleContexts, TypeFamilies, KindSignatures #-}

	-- http://www.timphilipwilliams.com/posts/2013-01-16-fixing-gadts.html

	module HRecursionSchemes where

	import Control.Applicative
	import Data.Functor.Identity
	import Data.Functor.Const
	import Text.PrettyPrint.Leijen hiding ((<>))