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), see the prefix coding system described in Binary Lambda Calculus and Combinatory Logic (page 20) written by John Tromp:
of course, c, d are metavariables, and also some other notations are changed slightly.
Now, Chaitin's construction will be here
- 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
- “Absolut value”
- should mean the length of a bit sequence (not combinatory logic term evaluation!)
Here, 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 dc a total function:
dc :: [Bit] -> Maybe CL
then, Chaitin's construction will be
where 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 use a maximum limit of iterations.
chaitin --computation=cl --coding=tromp --limit-of-iterations=5000 --digits=10 --decimal