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 describe the seen language with a LL(1) grammar, and let us make use of the lack of backtracking, lack of look-ahead, when deciding which parser approach to use.
Some notes about the used parser library: I shall use the didactical approach read in paper Monadic Parser Combinators (written by Graham Hutton and Erik Meier). The optimalisations described in the paper are avoided here. Of course, we can make optimalisations, or choose sophisticated parser libraries (Parsec, arrow parsers). A pro for this simpler parser: it may be easier to augment it with other monad transformers. But, I think, the task does not require such ability. So the real pro for it is that it looks more didactical for me. Of couse, it may be inefficient at many other tasks, but I hope, the LL(1) grammar will not raise huge problems.
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, runParser, item) where import Control.Monad.State (StateT, runStateT, get, put) type Parser token a = StateT [token]  a runParser :: Parser token a -> [token] -> [(a, [token])] runParser = runStateT item :: Parser token token 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 Approach based on decoding with partial function
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 Approach based on decoding with total function
Seen above, dc was a partial function (from finite bit sequences). We can implement it e.g. as
dc :: [Bit] -> CL dc = fst . head . runParser clP
If this is confusing or annoying, then we can choose a more Haskell-like approach, making dc a total function:
dc :: [Bit] -> Maybe CL dc = fst . head . runParser (neverfailing clP)
neverfailing :: MonadPlus m => m a -> m (Maybe a) neverfailing p = liftM Just p `mplus` return Nothing
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