# Mathematical Model of GLUTs and Lookups

From: Lee Corbin (lcorbin@rawbw.com)
Date: Tue Apr 01 2008 - 23:26:35 MDT

Stuart writes

> We got on this point after a long tortuous route: so let's simplify!
>
> S = a finite set of polynomials.

I pretty much have to visualize everything, and probably don't
Here I see (off to the left, actually) a finite set of polynomials
in mathematical platonia (Diagram 1)

> G = a GLUT on S.

Here I see a giant lookup table whose entries actually are
elements of S. But I am maybe guessing at what "on S"
means. Anyway, I visualize this GLUT being at the center
of a whiteboard. (Diagram 2).

> R = a rule that says that any polynomial of n-th degree will be
> mapped, under G, to a polynomial of (n-1)-th degree.

Now I would have thought that this rule existed between
elements of S: a mathematical-platonic rule analogous to
the rule that exists under the name differentiation, but
vastly more complex (I presume). But you say that R
connects n-th degree polynomials *in G* (that is, as
table entries of the GLUT) to particular other polynomials
also in G. Well, I suppose that it amounts to the same
thing. Okay. [ But it sounds a bit odd to refer to G as
a mapping, unless it was (incorrectly) taken to be a
mapping from platonia (Diagram 1) to the GLUT
(Diagram 2), where G(P) is the particular table entry
containing P. <end incorrect interpretation> ]

> f = a hash function from S (to f(S))

Back to Diagram 1 in my mental imagery. Right "underneath"
every polynomial in S written in a differently colored marker
is f(S), a string of apparent gibberish output by f.

> f(G) = the GLUT on f(S) equivalent to G on S.

We may have here Diagram 3, (which I visualize as being
at the extreme right of a whiteboard) and which is very
analogous to Diagram 2 (the GLUT, as you say, "on S").
Here too is a GLUT, but its entries are f(P) for each P in S.

> f(R) = the rule on f(S) equivalent to R on S.

Yes, I think I'm getting it. Just as R was a rule (see Diagram 2)
that connected polynomial entries in the first GLUT, e.g.
GLUTentry(P2) = R(GLUTentry(P1)), we now have in
Diagram 3

GLUT2entry( f(P2) ) = f(R)( GLUT2entry( f(P1) ) )

where I may visualize P1 and P2 as being in S, f(P1) and f(P2)
being their "shadows" still in Diagram 1, and the two GLUT2
entries being connected by the new rule f(R) in Diagram 3.

> Main point: G has very high KC (as does f(G)),

I don't know what that means. (Ah, but see below.)
Perhaps you mean the rule R that functionally connects
two elements of G = GLUT1 has high KC? I think I
know what the KC of a rule or a function would be.

If so, that does seem a bit odd, since where I thought you
were going was to have here an analogy between consecutive
states of a conscious entity. But these latter are causally
connected, hence, I believe have a low KC rule (e.g. Conway's
Life has quite a low KC rule connecting a generation to a
subsequent generation).

> R has low KC, but f(R) has high KC.

Oh, all right---that seems to make perfect sense. But I
guess I still don't know what "G has very high KC" means.

> Theorem (which will be left as as exercise to the reader - i.e. I
> haven't calculated it, but it seems obvious):
>
> KC(f(R)) / KC(f(G)) can be made arbitrarily higher than KC(R) / KC(G) .

Lost here. I understand KC(R), but still don't grok KC(G),
likewise I understand KC(f(R)) but not KC(f(G)).

I'll reply to the remainder here when I've understood better my
problems with the above.

It all does continue to sound quite promising, though.

Lee

> Corollary: We can decrease the KC of G to some extent, and still get
> the inequality above. We want to do this, because we don't want G to
> be a full GLUT but something more "understandable".
>
> Inspiration: If we imagine that G is the rules of consciousness, and R
> is some much simpler property of consciousness (say: it learns form
> experience), then R is only much simpler than G in our current
> description of the universe; in hash equivalent settings, R does not
> qualify as being "much simpler".
>
> Philosophical consequence: different hash equivalent ways of looking
> at problems are not equivalent. Since we essentially cannot deal with
> hash functions in any computable way, the different ways things are
> set up are very important.
>
> Punch line: Let E be an explanation of consciousness, (an explanation
> in the sense we are used to). Let F be a GLUT, hash equivalent to E.
> Then though E and F are hash equivalent, they are not equivalent,
> especially from our point of view.
>
> Hope this helps! I don't know if it's all that useful, though.

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