Niclas Thisell wrote:
>There is no reason to believe that K (representing a fundamental
>constant, I presume) is not compressible. And, presuming the theory
>looks like a linear differential equation, we could iterate the state
>without loss of precision using rational numbers or even integers. The
>problem arises when we need to do calculate or multiply by an irrational
>number, like sqrt(2). They can, of course, be approximated by a power
>series expansion or so. But the series needs a cutoff. I think Hal
>refers to this number - not the actual fundamental constant. Of couse,
>it too can be compressed, but I'm fairly sure his point is that values
>around 5 are still much more likely than values around 100^100. (I
>agree, but I don't agree that this necessarily needs to imply that 'low'
>number dominate).
>Of course, you could write a dovetailing program that iterates through
>all cut-offs simultaneously. And I have no doubt that _you_ could write
>a program that calculates the universe with a seemingly infinite
>precision. The question is if this is automatically given by the
>Schmidhuber plenitude (i.e. including the measure given by your
>dovetailer).
I was just showing I don't even need to invoque cut-off. K could be
compressible of course (my aim, a little like yours was showing Hal is
"bias" against large programs). (BTW do you believe that Planck constant
is compressible ?).
Now, personaly, I will never presume a linear differential equation: that
is really what I try to derive from more reasonable hypothesis like comp.
Arithmetical truth, as the 'Schmidhuber plenitude' is not enough as an
explanation. The explanation will come with the adequacy between the
different type of discourse possible by the 'average' Self-Referentially
Correct Universal Turing machine and our own experience.
I propose and illustrate ways for making this mathematicaly precise in
http://iridia.ulb.ac.be/~marchal.
You can see in the archive that although I agree with Schmidhuber's (and
even has invoked it by myself) computationalist plenitude, I disagree
a lot with Schmidhuber conclusion in his paper.
I don't look at the DU as an explanation for everything. I look at it
as a useful tool to begin to (just) formulate questions like the question
of the origin of the physical laws, or the mind-body problem, the other
mind problems, etc.
With comp we have more a DU problem (more or less equivelent to the
white rabbit problem) than a DU explanation. It is just that we cannot
just drop out the DU.
>> BM: I don't agree. The UD multiplies the executions by
>> dovetailing them (even
>> in an admitedly dummy and ugly ways) on the reals.
>> I'm not sure I understand you when you say "there are more of
>> the former,
>> but they are of lower measure". The measure is defined by the
>> number of
>> (infinite) computations. (This is linked to the old ASSA/RSSA
>> question of
>> course).
>NT: I can sort of accept this point of view as well - i.e. I'm not willing
>to discard it (especially since there are problems with the ASSA as
>well). But I do think that the interpretation of a state of a particular
>universe introduces difficulties. For instance, we can interpret a rock
>to implement a univeral turing-machine. So any universe can be
>interpreted to be equivalent with any other universe.
>Also, the evolution process is not very clear. Relativity teaches us
>that it is usually better to think of time as just another dimension
>with almost the same properties as the spatial dimensions. And
>relativistic QM does indeed not treat time very differently than other
>axes. Whereas these infinite computations more or less assert an
>absolute time.
Not at all. These infinite computations assert no more than the existence
of the natural numbers : the steps of the DU's executions.
Third person physical time and first person subjective time are internal
modalities (infered by machines entangled with deep computations, ...).
They have nothing to do with the 'time step' of the DU. In my 'ultimate
derivation' with the arithmetical provability logics, the DU is
represented
by the \Sigma_1 sentences (that is the arithmetical propositions having
the
form: \Exists n P(n) with P a decidable arithmetical predicate): there is
no need of the time concept there.
Also, with Occam there is no need to run a DU. With a minimal amount of
arithmetical realism the DU exists, like \Pi or \e. The DU need no more
absolute time than any irrational constant like \sqrt(2) or \Pi.
Best regards,
Bruno
Received on Thu Jan 06 2000 - 02:58:01 PST