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

Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker
problem. For this reason we can speak of the = e equivalence classes as degrees of
unsolvability. And we can extend the relations ≤ e , = e , and < e to these classes in an obvious
way.
Now suppose that R is some very simple computation which is everywhere defined
⎯ to be specific, suppose that, given any input In, the computation R(In) simply stays in the
In state forever. Any computation at all can emulate R, but we don’t expect that the donothing
R can emulate all the other computations. In symbols, if P is any other computation,
R ≤ e P and possibly R < e P. For this reason, we say that R represents a minimal degree of
unsolvability. Abstracting a bit, we use the symbol R to stand for R’s degree of
unsolvability.
Do note that, due to the fact that the translations used in the definition of emulation
can be any everywhere defined computations at all, it turns that that any everywhere defined
computation will be found in R. Less obviously, any computation that has a solvable halting
problem relative to the default target detector will be in R as well. So the degree R
represents a fairly large class of computations. R is sometimes known as the recursive
degree.
Now suppose that U is a universal computation. Since U can emulate any
computation at all, but some computations, such as R will not be able to emulate U. That is,
if P is any other computation, P ≤ e U and possibly P < e U. In other words, U represents a
maximal degree of unsolvability. Abstracting as before, we use the symbol to stand for
U’s degree of unsolvability. Given that contains all of the various universal computations,
is also a large class of computations. is sometimes known as the universal degree.
Because the three facts that (by Turing’s’ theorem) U has an unsolvable halting
problem, that (since its everywhere defined) R has a solvable halting problem, and that (as
can be proved from the definition of emulation) = e preserves the property of having a
solvable halting problem, we can deduce the following proposition.
Proposition. There are at least two distinct degrees of unsolvability, the recursive
degree and the universal degree. In symbols, < e .
So now we know that we have at least two degrees of computations: the minimal
degree of the computations that are everywhere definable or which have solvable halting
problems, and the maximal degree of the universal computations.
In 1940, the logician Emil Post posed Post’s Problem, asking if there any
intermediate degrees. We can formulate Post’s Problem in various ways.
Are there any intermediate unsolvability degrees such that < e and < e ?
Is there a computation M such that M has an unsolvable halting problem, but M is not
universal?
Why is the second formulation is the same as the first? Assuming is the degree of
M, the condition < e is equivalent to saying that M has an unsolvable halting problem.
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