Notes for the Lifebox, the Seashell, and the Soul - Rudy Rucker

Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker
can emulate any system, it can in particular emulate any system that is trying to outrun it.
And from this it follows that nothing can systematically outrun the universal system. For any
system that could would in effect also have to be able to outrun itself.” A New Kind of
Science, p. 742.
Wolfram’s idea seems to be that if we (thesis) have a predictor computation that
emulates our universal computation with faster runtime than that of actually running the
universal computation, then (antithesis) having the universal computation emulate the
predictor should also be faster than the universal computation, and then (synthesis) having
the predictor emulate the universal computation’s emulation of the predictor should be faster
yet, and now we can use the synthesis as a new thesis, and get a yet faster computation, and
so on forever. But it would be impossible to have an endless sequence of smaller and smaller
run times. Therefore a universal computation like can’t be predicted.
But if you think through some specific cases, we can see that this particular argument
doesn’t work. The descending chain will stop as soon as one crosses a cost-benefit condition
whereby the overhead of simulation cost swamps the gain of the successive levels of
simulation. Put differently, the antithetic step won’t work very often.
Let me refer back to the U and DumbU example in section 6.3 of the main text.
If we try and apply Wolfram’s argument, we see that DumbU can emulate U. But,
since DumbU is wired to carry out the wasteful tally doubling operation between each step,
DumbU-emulating-U will run slower than DumbU, so the very first antithetic step fails and
we never even take the first step of Wolfram’s proposed infinite regress.
Falling back, we now ask if there is any kind of counting argument that might suggest
that unpredictable universal computations are more common than predictable ones?
One thinks of Chaitin’s argument that we can find arbitrarily long strings of zeroes
and ones are random in the sense of not having a description shorter than themselves. To be
more precise, Chaitin supposes that we fix a particular universal Turing machine U and that
we say that string NameK is a name for string K if U with input NameK produces the string
K and halts. For any string length n, there are 2 n strings of length n, but only 2 n - 1 strings of
length less than n, so at least one string of length n has no name shorter than n. And if we
say string K of length n is predictable if it has a name of length at most log(n), then very
many strings are unpredictable.
But this style of argument doesn’t apply to universal Turing machines because when I
say U is unpredictable, I’m saying there is no Turing machine V of any size such that V
emulates U with a runtime eventually on the order of the log of U’s runtime. Unlike in the
Chaitin argument, we’re not limited to looking at predictor Turing machines of size smaller
than the object to be predicted.
Nevertheless, with Wolfram, I too, want to believe the Universal Unpredictability
Lemma to be applicable to all of the interesting cases that I’ve discussed ⎯ including the
dynamics of physical systems, the processes of the human mind, and the workings of human
society.
Definition of Oracle
• Definition. The computation OP is an oracle for the computation P iff
(i) OP takes the same inputs as P, is everywhere defined, and has a special extra
output state ó.
(i) If P(a) halts at b, then OP(a) halts at b.
p. 98