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Chaitin's construction

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m (I had confused e.g. (exempla gratia) with i.e. (idem esse). Now I have corrected it)
m (The set of syntactiaclly correct bitstring codings of combinatory logic is exactly both the domain of decoding and range of coding function)
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Now, Chaitin's construction will be here
 
Now, Chaitin's construction will be here
:<math>\sum_{p\in \mathrm{Rng}_\mathrm{dc},\;\mathrm{hnf}\left(\mathrm{dc}\;p\right)} 2^{-\left|p\right|}</math>
+
:<math>\sum_{p\in \mathrm{Dom}_\mathrm{dc},\;\mathrm{hnf}\left(\mathrm{dc}\;p\right)} 2^{-\left|p\right|}</math>
 
where
 
where
 
;<math>\mathrm{hnf}</math>
 
;<math>\mathrm{hnf}</math>
Line 24: Line 24:
 
;<math>2\!\;^{*}</math>
 
;<math>2\!\;^{*}</math>
 
:should denote the set of all finite bit sequences
 
:should denote the set of all finite bit sequences
;<math>\mathrm{Rng}_\mathrm{dc}</math>
+
;<math>\mathrm{Dom}_\mathrm{dc}</math>
:should denote the range of decoding function, i.e. the syntactically correct bit sequences (semantically, they may either terminate or diverge),
+
: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”
 
;“Absolut value”
 
:should mean the length of a bit sequence (not [[combinatory logic]] term evaluation!)
 
:should mean the length of a bit sequence (not [[combinatory logic]] term evaluation!)

Revision as of 11:39, 3 August 2006

Contents


1 Introduction

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), 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 metavariables, 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

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
“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

\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