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), 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.

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!)

Here, $\mathrm {dc}$ is a partial function (from finite bit sequences). 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

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