Monday, December 12, 2016

Comment on the kolmogorov complexity wiki vs the finitude of the possible



Uncomputability of Kolmogorov complexity


Proof: Otherwise all of the infinitely many possible finite strings could be generated by the finitely many[note 2] programs with a complexity below n bits.-wiki


Comment:


I guess this goes to the what doe we mean by compute issue.   Some people would say if you generate something, say a particular number by an algorithm, you computed such even if the computer does not halt and keeps on churning.   It is true, that if we're using deterministic systems, it is basically impossible for a finite program to code for an infinite number of strings as output when it halts, as of course it will halt deterministically if it halts at all,  it is also easily deducible from this that even an arbitrary yet finite group of programs will of course fail to code for an infinite number of  possible outputs.  

But what we're talking about here is systems that halt, systems that do not halt will trivially generate all possible sequences throughout their evolution, even with trivial finite designs.    They will just keep working past the production of any particular result.

And we again have to talk about whether this infinity of outcomes truly carries infinite information.   Already some physicists have said that the number of possible arrangements of the visible universe is vast yet finite, and if there is variation and a truly infinite universe, these finite set of possibilities would endlessly repeat.    That is there is a finite set of possible arrangements for the visible universe, and they will repeat at intervals.    Would we say that if we reordered these entire unchanged universes in different sets of repetitions we're adding meaningful information?   Or can it be said that once you've got the finite set of possible arrangements you've basically gathered it all, at least as pertains to this particular ruleset?

Suppose you rearrange words, sure additional meaning can be derived, rearrange sentences additional meaning can be derived, rearrange paragraphs additional meaning can be derived, rearrange pages, additional meaning can be derived, rearrange the position of entire books without changing any of their contents... clearly the contents of a bookshelf, rearranging the books in the bookshelf does not seem to add meaningful information, at least to the contents of the books, and whatever information may be added seems like it would be symbolically in terms of the arrangement of the books, which would be a minute quantity of information.   Note this does not mean that the order of reading does not matter, but that if you already know all the books reading them in different orders will not give you meaningfully more information if anything at all..   It seems like the more complex you get, the more is already contained within the complex entities being rearranged, and less and less is added.



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