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Chaitin's construction

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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. Here is a parser for illustration, but it serves only didactical purposes: it will not be used in the final implementation, because a good term geerator makes parsing superfluous at this task.

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
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 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.

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