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 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:
of course, , are meta-variables, and also some other notations are changed slightly.
Decoding
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)
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, 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
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
where
- 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!)
Approach based on decoding with total function
Here, is a partial function (from finite bit sequences). If this is confusing or annoying, then we can choose a more Haskell-like approach, making a total function:
dc :: [Bit] -> Maybe CL
then, Chaitin's construction will be
where should denote false truth value.
Related concepts
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