From: Lee Corbin (lcorbin@rawbw.com)
Date: Sun Mar 23 2008 - 22:13:24 MDT
Stuart writes (welcome back!),
> Some extra precisions on my long lost example (long lost, since it
> happened two long weeks ago...)
>
>> > The answer, whatever it is, will have simpler,
>> > quasi answers - incomplete but informative.
>> > The hash function equivalents will not be
>> > simpler than the hash function equivalent
>> > to the full question.
Let me make sure that I still remember the context. You are giving
a nice, precise, elementary model or example of a normally large
and cumbersome computation carried out by a Giant Look Up Table.
In other words, your "Polynomial Derivative Model" stipulates
A. the analog of a computer or TM's state is a polynomial (over
some field, which we might as well take to be the reals)
B. the analog of the computation of the next state is the construction
of the polynomial's derivative
C. instead of a computation in the ordinary way that polynomials
are differentiated, it would be in principle possible to store all
the results of say, six billion different polynomial differentiations
(one for each person on Earth, or if we need to, 10^15 such
for each computer or person on the Earth)
D. the PDM-GLUT operates by computing a very complicated
hash of each polynomial (no infinite polynomials permitted)
which yields the address within the lookup table of the correct
differentiation (derivative).
E. The process can result in a chain of such derivations, which is
the analog to an ordinary computation, and is an interesting
parallel to a computation that emulates a human consciousness
over some period of time. Indeed, since a polynomial may be
any length whatsoever---that is, of any degree whatsoever---
we might even discover an isomorphism between a sequence
of derivatives and human consciousness, though that may be
too much of a stretch at present.
> Examples of simpler questions: for the derivative, we have interim
> results: the derivative of a monomial is a monomial, the derivative of
> a polynomial of n-th degree is a polynomial of n-1 th degree. These
> are simpler than knowing the full derivative rule, but to list them
> under the hash function is actually more complicated than listing the
> GLUT for the derivative itself (as we first have to define the subsets
> "monomials" and "polynomial of n-th degree").
I don't quite understand, or perhaps it's merely that I don't agree.
The rules for taking derivatives of polynomials have a relatively
small Kolmogorov Complexity. Also, giving rules to determine
the degree of a polynomial has rather low KC. Therefore, it
seems to me that what is in the GLUT may have a lot higher
KC. For example, if we did have 6x10^6 x 10^15 initial
polynomials to try, some of which may be of the octillionth
degree, then clearly the KC of what is stored in the table
exceeds the KC of all our rules and definitions about polynomial
differentiation.
>> > 2) The implicit infinity.
>> > Implicit in the definition of differentiation is the fact that we
>> > could differentiate any polynomial (with, say, rational coefficients).
>> > The definition of differentiation to f(S) does not extend to infinity
>> > in this way; in fact, there is no evident extension of f(S).
>
>> What? Even if you have infinitely many polynomials, each has a
>> derivative, the only difference being that there is no "top dog".
>> Strategically, would it have been better to stick with a crippled
>> definition of derivative, oh, something along the lines of either
>> "we are not going to consider S to be infinite", or "a qDerivative
>> is a derivative of a polynomial function, but the domain of
>> qDerivative is by definition finite"? Or something like that?
>
> The derivative can be defined for any polynomial, and can be defined
> in a finite (and very short) sentence. A finite definition, generating
> an infinite number of relations.
Yes; okay.
> Now try and go the other way; from a GLUT, try and deduce the general
> rule. If we have the GLUT in polynomial form, the universal rule is
> easy to deduce; if we have the GLUT in some hash function equivalent,
> we can't do so.
>
>> I think that the whole infinity digression is a red-herring to your otherwise
>> brilliant analysis (and I don't often call what other posters do "brilliant"),
>> reserving that appellation strictly to apply to my own missives only.
Yes, at least not easily, and, yes, perhaps not at all.
> ...Many thanks. However, the red herring is still, in my view, a
> distinction between a GLUT and a hash-equivalent GLUT.
I probably don't understand that. You did just get through pointing out
a vital difference between the GLUT in polynomial form, and a GLUT
in some hash function equivalent form. So what does the latter mean?
Lee
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