Absolute measure

From: Niclas Thisell <niclas.domain.name.hidden>
Date: Mon, 31 Jan 2000 16:55:55 +0100

Here's a simple idea you might find interesting, or at least amusing.

What is the measure of a transcript of The Phantom Menace? We can define
it to be its Solomonoff-Levin probability. The problem is that we must
decide on a UTM model (as opposed to a turing machine, which is its
bitstring + model). The compiler theorem saves the day by declaring that
we can prepend a bitstring to emulate other models. And the complexity
increase can not be greater than the length of this bitstring.

Still, if I cook up a model that outputs the transcript of The Phantom
Menace if the first bit is '1', the measure can be said to be greater
than 1/2 (not with a straight face, though). Any hopes for an absolute
measure of a 'world', 'brain-state' or whatever are crushed. We could,
of course, stick with a 'normal' turing machine model and hope that it's
good enough. But this choice feels a bit arbitrary somehow.

Perhaps one could assign a measure to the models? We could then hope
that models that are prone to output The Phantom Menace are given a low
measure. So, let us try:
O(p,m_A)=Output for model m_A, given a bitstring p.
M(m_A,m_B,N)=(number of bitstrings u~ of length N such that, for an
initial fraction u of u~, and for any bitstring p, O(u+p,m_B)=O(p,m_A)
)/1^2N
M(m_A,m_B)=lim(M(m_A,m_B,N),N->inf)
(1) sum(M(m_A,m_B)M(m_B)-M(m_B,m_A)M(m_A),m_B)=0
M(m_A)>=0 for all m_A
sum(M(m_A),m_A)=1

M(m_A,m_B) is roughly the 'ease' of having model m_B emulate model m_A.
Then we let the models 'rate' each other. That way, hopefully,
complicated models will get low measures. We could then truly define an
absolute measure
M(s,m_A)=Solomonoff-Levin probability of s, using the model m_A.
(2) M(s)=sum(M(s,m_A)M(m_A),m_A)

Ok, but is there a unique non-trivial solution to the system of
equations (1)? It's an infinite order homogenous matrix equation. One
way of solving systems of linear equations is to let them define a set
of linear first order differential equations. A simple picture of this
is a graph with measure 'flowing' between the nodes. The structure of
(1) ensures that total measure is conserved. There's probably a plethora
of theorems on solvability of these types of systems in queue-theory and
even FEM analysis. I think it's a safe guess that there is at least one
solution. The problem is that it sounds plausible that there are several
solutions. Technically, we could still hope that (2) is well defined,
even if there are more than one solution, but it doesn't sound likely.
Also, this definition of measure could perhaps be claimed to be just as
arbitrary as simply sticking with a simple enough turing machine model.

But let us pretend that the equations have a unique solution. Then we
can ask ourselves why we have to stick with universal turing machines.
These equations should work just as well/badly for general operators -
i.e. something that takes something and spits out something else. Well,
to make the definition of measure work without alteration, we should
perhaps stick with input as bitstrings. All we need is an equivalence
operator such that we can compare the output of two models, given a
bitstring. Etc...

Disclaimer: Most of this stuff is probably old news/proven flawed, but I
plead incompetence.

/Niclas
Received on Mon Jan 31 2000 - 07:57:58 PST