Chaitin's construction

Introduction

Are there any real numbers which are defined exactly, but cannot be computed? This question leads us to exact real arithmetic, foundations of mathematics and computer science.

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).

Basing it on combinatory logic

Some more direct relatedness to functional programming: we can base $\Omega$ on combinatory logic (instead of a Turing machine).

Coding

See the prefix coding system described in Binary Lambda Calculus and Combinatory Logic (page 20) written by John Tromp:

${\widehat {\mathbf {S} }}\equiv 00$

${\widehat {\mathbf {K} }}\equiv 01$

${\widehat {\left(xy\right)}}\equiv 1{\widehat {x}}{\widehat {y}}$

of course, $c$, $d$ are meta-variables, and also some other notations are changed slightly.

Decoding

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.

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

Combinatory logic term modules

CL

 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

CL extension

 module CLExt ((>>@)) where

 import CL (CL, apply)
 import Control.Monad (Monad, liftM2)

 (>>@) :: Monad m => m CL -> m CL -> m CL
 (>>@) = liftM2 apply

Base symbol

 module BaseSymbol (BaseSymbol, kay, ess) where

 data BaseSymbol = K | S

 kay, ess :: BaseSymbol
 kay = K
 ess = S

Utility modules

Binary tree

 module Tree (Tree (Leaf, Branch)) where

 data Tree a = Leaf a | Branch (Tree a) (Tree a)

Parser

 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

Prelude extension

 module PreludeExt (bool) where

 bool :: a -> a -> Bool -> a
 bool thenC elseC t = if t then thenC else elseC

Approach based on decoding with partial function

Now, Chaitin's construction will be here

$\sum _{p\in \mathrm {Dom} _{\mathrm {dc} },\;\mathrm {hnf} \left(\mathrm {dc} \;p\right)}2^{-\left|p\right|}$

where

$\mathrm {hnf}$

should denote an unary predicate “has normal form” (“terminates”)

$\mathrm {dc}$

should mean an operator “decode” (a function from finite bit sequences to combinatory logic terms)

$2\!\;^{*}$

should denote the set of all finite bit sequences

$\mathrm {Dom} _{\mathrm {dc} }$

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, $\left\{00,01,1\;00\;00,1\;00\;01,1\;01\;00,1\;01\;01,\dots \right\}=\mathrm {Dom} _{\mathrm {dc} }=\mathrm {Rng} _{\widehat {\ }}$

“Absolute value”

should mean the length of a bit sequence (not combinatory logic term evaluation!)

Approach based on decoding with total function

Seen above, $\mathrm {dc}$ was a partial function (from finite bit sequences). We can implement it e.g. as

dc :: [Bit] -> CL
dc = fst . head . runParser clP

where the use of head reveals that it is a partial function (of course, because not every bit sequence is a correct coding of a CL-term).

If this is confusing or annoying, then we can choose a more Haskell-like approach, making $\mathrm {dc}$ a total function:

 dc :: [Bit] -> Maybe CL
 dc = fst . head . runParser (neverfailing clP)

where

 neverfailing :: MonadPlus m => m a -> m (Maybe a)
 neverfailing p = liftM Just p `mplus` return Nothing

then, Chaitin's construction will be

$\sum _{p\in 2^{*},\;\mathrm {maybe} \;\downarrow \;\mathrm {hnf} \;\left(\mathrm {dc} \;p\right)}2^{-\left|p\right|}$

where $\downarrow$ should denote false truth value.

Related concepts

• Exact real arithmetic
• Combinatory logic
• Computer science
• Mathematics

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