# Chaitin's construction

## 1 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.

## 2 Basing it on combinatory logic

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

### 2.1 Coding

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

$\widehat{\mathbf S} \equiv 00$
$\widehat{\mathbf K} \equiv 01$
$\widehat{\left(x y\right)} \equiv 1 \widehat x \widehat y$

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

### 2.2 Decoding

Having seen this, decoding is rather straightforward. Let us represent it e.g. with the following LL1 parser (or maybe is it an LL0 one?). Of course, we can build it on top of more 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.

#### 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

##### 2.2.2.1 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
##### 2.2.2.2 CL extension
 module CLExt ((>>@)) where

import CL (CL, apply)

(>>@) :: Monad m => m CL -> m CL -> m CL
(>>@) = liftM2 apply
##### 2.2.2.3 Base symbol
 module BaseSymbol (BaseSymbol, kay, ess) where

data BaseSymbol = K | S

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

#### 2.2.3 Utility modules

##### 2.2.3.1 Binary tree
 module Tree (Tree (Leaf, Branch)) where

data Tree a = Leaf a | Branch (Tree a) (Tree a)
##### 2.2.3.2 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
##### 2.2.3.3 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

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

where

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

### 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
where the use of
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 dc a total function:

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

where

safe :: MonadPlus m => m a -> m (Maybe a)
safe 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.

## 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