Chaitin's construction

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Revision as of 22:57, 3 August 2006

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.

• 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

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:

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 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

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)
 import Control.Monad (Monad, liftM2)
 
 (>>@) :: 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

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)

should denote the set of all finite bit sequences

Dom_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,

“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)

where

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

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

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

Category: Theoretical foundations