On 21 Oct 2012, at 22:03, Alberto G. Corona wrote:
This does not implies a reality created by an UD algorithm. It may
be a mathematical universe, that is a superset of the computable
universes.
The computable universe is a subset of the mathematical universe.
Just compare: the computable universe correspond to the Sigma_1
sentences. It is already only semi or partially computable, as the
negation of a Sigma_1 sentence can already be not computable (that's
the PI_1 sentence), then you have the sigma_1, still more non
computable, and then the sigma_3, etc... And that is only a tiny part
of arithmetical truth, which is just *much* vaster than the
computable, but can still be considered as a tiny part of the
mathematical truth.
The measure problem in the UD algorith translates to the problem of
the effectivity of the Occam Razor, or the problem of the apparent
simplicity of the phisical laws, or, in other words, their low
kolmogorov complexity, that solomonov translates in his theory of
inductive inference.
This can solve the 3p rabbit problems, but not the 1p rabbit problems.
You will re-awake older discussions.
Kolmogorov complexity can play some role here, but does not solve the
1p-problem, which is transformed into a justification of the stability
of dreams, with still a possibility to define a notion of physical
realm, perhaps by changing some definition.
Complexity exploits the simple/immune complementary in the W_i.
Beliefs, knowledge, even observation, exploits the creativity/
productivity complementarity in the W_i.
Bennett notion of depth should play a role also, to justify a notion
of cosmological history.
The whole problem of the 1p indeterminacy, is that it does give a role
to big programs. The little programs cannot get rid of them so easily
(by just matter of complexity). We are ourselves already relatively
rare *big* relative numbers.
Bruno
2012/10/21 Alberto G. Corona <agocor...@gmail.com>
Ok
I don´t remember the reason why Solomonof reduces the probability of
the programs according with the length in is theory of inductive
inference. I read it time ago. Solomonoff describes in his paper
about inductive inference a more clear and direct solution for the
measure problem. but I though that it was somehow ad hoc.
I tough time ago about the Solomonof solution to the induction
problem, and I though as such: living beings have to find, by
evolution, at least partial and approximate inductive solutions in
order to survive in their environment. This imposes a restriction on
the laws of a local universe with life: It demand a low kolmogorov
complexity for the macroscopical laws. Otherwise these laws would
not be discoverable, there would be no induction possible, so the
living beings could not anticipate outcomes and they woul not survive.
Solomonoff is a living being in a local universe, so shorther
programs are more probable and add more weight for induction.
I´m just thinking aloud. I will look again to the solomonof
inductive inference. I was a great moment when I read it the first
time.
2012/10/20 Russell Standish <li...@hpcoders.com.au>
On Sat, Oct 20, 2012 at 09:16:54PM +0200, Alberto G. Corona wrote:
> This is not a consequence of the shannon optimum coding , in which
the
> coding size of a symbol is inversely proportional to the logaritm
of the
> frequency of the symbol?.
Not quite. Traditional shannon entropy uses probability of a symbol,
whereas algorithmic complexity uses the probability of the whole
sequence. Only if the symbols are independently distributed are the
two the same. Usually, in most messages, the symbols are not id.
>
> What is exactly the comp measure problem?
A UD generates and executes all programs, many of which are
equivalent. So some programs are represented more than others. The
COMP measure is a function over all programs that captures this
variation in program respresentation.
Why should this be unique, independent of UD, or the universal Turing
machine it runs on? Because the UD executes every other UD, as well as
itself, the measure will be a limit over contributions from all UDs.
Cheers
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Prof Russell Standish Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics hpco...@hpcoders.com.au
University of New South Wales http://www.hpcoders.com.au
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