Editing Chaitin's construction (section)

=== More natural norm functions (from CL terms) ===

Question:
If we already move away from the approaches referring to any code concept, then
could we define norm in other ways? E.g.
:<math>\left\Vert\cdot\right\Vert : \mathrm{CL}\to\mathbb N</math>
:<math>\left\Vert\mathbf K\right\Vert = 1</math>
:<math>\left\Vert\mathbf S\right\Vert = 1</math>
:<math>\left\Vert\left(x\;y\right)\right\Vert = 1 + \left\Vert x\right\Vert + \left\Vert y\right\Vert</math>
And is it worth doing it at all? The former one, at leat, had a good theoretical foundation (based on analysis, arithmetic and probability theory). This latter one is not so cleaner, that we should prefer it, so, lacking theoretical grounds.

What I really want is to exclude conceptually the notion of coding, and with it the notion of “syntactically incorrect versus syntactically correct but diverging”. Thus, taking into account only syntactically correct things, seeing only the choice of terminating versus non-terminating. Thus taking only termination vs nontermination into account, when calculating Chaitin's construction.

What I want to preserve:
* it can be interpreted as a probability
* it is a [http://en.wikipedia.org/wiki/Normal_number normal number], as if its digits (in binary representation) were generated by tossing a coin
thus I do not want to spoil these features.

==== Table for simpler CL-terms ====
Let us not take into account coding and thus excluding the notion of “syntactically incorrect coding” even ''conceptually''.
Can we guess a good norm?
{| border="1" cellspacing="0" cellpadding="5" align="center"
! Binary tree pattern
! Maximal depth, vertices, edges
! Leafs, branches
! So many CL-terms = how to count it
! Terminating, ratio
! So many till now, ratio till now
|-
| <math>\cdot</math>
| 0, 1, 0
| 1, 0
| <math>2 = 2</math>
| 2, 1
| 2, 1
|-
| <math>\left(\right)</math>
| 1, 3, 2
| 2, 1
| <math>4 = 2\cdot2</math>
| 4, 1
| 6, 1
|-
| <math>\cdot\left(\right)</math>
| 2, 5, 4
| 3, 2
| <math>8 = 2\cdot2^2</math>
| 8, 1
| 14, 1
|-
| <math>\left(\right)\cdot</math>
| 2, 5, 4
| 3, 2
| <math>8 = 2^2\cdot2</math>
| 8, 1
| 22, 1
|-
| <math>\left(\right)\left(\right)</math>
| 2, 7, 6
| 4, 3
| <math>16 = 2^2\cdot2^2</math>
| 16, 1
| 38, 1
|}