See Wikipedia article on Chaitin's construction, referring to e.g.
- Computing a Glimpse of Randomness (written by Cristian S. Calude, Michael J. Dinneen, and Chi-Kou Shu)
- Omega and why math has no TOEs (Gregory Chaitin).
2 Basing it on combinatory logic
See the prefix coding system described in Binary Lambda Calculus and Combinatory Logic (page 20) written by John Tromp:
of course, c, d are meta-variables, and also some other notations are changed slightly.
Having seen this, decoding is rather straightforward. Let us represent it e.g with the following LL1 parser. Of course, we can build it on top of more sophisticated parser libraries (Parsec, arrow parsers)
2.2.1 Decoding module
module Decode (clP) where import Parser (Parser, item) import CL (CL, k, s, apply) import CLExt ((>>^)) import PreludeExt (bool) clP :: Parser Bool CL clP = item (bool applicationP baseP) applicationP :: Parser Bool CL applicationP = clP >>^ clP baseP :: Parser Bool CL baseP = item (bool k s) kP, sP :: Parser Bool CL kP = return k sP = return s
2.2.2 Combinatory logic term modules
module CL (CL, k, s, apply) where import Tree (Tree (Leaf, Branch)) import BaseSymbol (BaseSymbol, kay, ess) type CL = Tree BaseSymbol k, s :: CL k = Leaf kay s = Leaf ess apply :: CL -> CL -> CL apply = Branch
188.8.131.52 CL extension
module CLExt ((>>^)) where import CL (CL, apply) import Control.Monad (Monad, liftM2) (>>^) :: Monad m => m CL -> m CL -> m CL (>>^) = liftM2 apply
184.108.40.206 Base symbol
module BaseSymbol (BaseSymbol, kay, ess) where data BaseSymbol = K | S kay, ess :: BaseSymbol kay = K ess = S
2.2.3 Utility modules
220.127.116.11 Binary tree
module Tree (Tree (Leaf, Branch)) where data Tree a = Leaf a | Branch (Tree a) (Tree a)
module Parser (Parser, item) where import Control.Monad.State (StateT, get, put) type Parser token a = StateT [token]  a item :: Parser a item = do token : tokens <- get put tokens return token
18.104.22.168 Prelude extension
module PreludeExt (bool) where bool :: a -> a -> Bool -> a bool thenC elseC t = if t then thenC else elseC
2.3 Partial function approach
Now, Chaitin's construction will be here
- should denote an unary predicate “has normal form” (“terminates”)
- should mean an operator “decode” (a function from finite bit sequences to combinatory logic terms)
- should denote the set of all finite bit sequences
- should denote the set of syntactically correct bit sequences (semantically, they may either terminate or diverge), i.e. the domain of the decoding function, i.e. the range of the coding function. Thus,
- “Absolute value”
- should mean the length of a bit sequence (not combinatory logic term evaluation!)
2.4 Total function approach
Here, dc is a partial function (from finite bit sequences). If this is confusing or annoying, then we can choose a more Haskell-like approach, making dc a total function:
dc :: [Bit] -> Maybe CL
then, Chaitin's construction will be
where should denote false truth value.
3 Related concepts
4 To do
Writing a program in Haskell -- or in combinatory logic:-) -- which could help in making conjectures on combinatory logic-based Chaitin's constructions. It would make only approximations, in a similar way that most Mandelbrot plotting softwares work: it would ask for a maximum limit of iterations.
chaitin --computation=cl --coding=tromp --limit-of-iterations=5000 --digits=10 --decimal