assigns low probability to G-describable strings such as the *z* of
Theorem 3.3. However, one might believe in the potential significance of such
constructively describable patterns,
e.g., by accepting their validity as possible pseudorandom perturbations
of a universe otherwise governed by a quickly computable
algorithm implementing simple physical laws -- compare Example 2.1.
Then one must also
look at semimeasures dominating ,
although
the falsifiability problem mentioned above holds for those as well.

The top of the TM dominance hierarchy is embodied by *G* (Theorem 3.3);
the top of our prior dominance hierarchy by *P*^{G},
the top of the corresponding semimeasure dominance hierarchy by .
If Conjecture 5.3
were true, then maximizing *P*^{G}(*xy*) would be equivalent
to minimizing *K*^{G}(*xy*). Even then there would be a fundamental
problem besides lack of falsifiability: Neither *P*^{G} nor
are describable, and not even a ``Great Programmer''
[#!Schmidhuber:97brauer!#] could generally decide whether some GTM output
is going to converge (Theorem 2.1), or whether it actually represents a
``meaningless'' universe history that never stabilizes.

Thus, if one adopts the belief that nondescribable measures do not exist, simply because there is no way of describing them, then one may discard this option.

This would suggest considering semimeasures less dominant than ,
for
instance, one of the most dominant approximable .
According to
Theorem 5.5 and inequality (43),
goes to
zero almost exponentially fast with growing *Km*^{G}(*xy*).

As in the case of ,
this may interest the philosophically inclined
more than the pragmatists: yes, any particular universe history without
short description necessarily is highly unlikely; much more likely are
those histories where our lives are deterministically computed by a short
algorithm, where the algorithmic entropy (compare [#!Zurek:89b!#]) of
the universe does *not* increase over time, because a finite program
conveying a finite amount of information is responsible for everything,
and where concepts such as ``free will'' are just an illusion in a
certain sense. Nevertheless, there may not be any effective way of
proving or falsifying this.

Related links: In the beginning was the code! - Zuse's thesis - Life, the universe, and everything - Generalized Algorithmic Information - Speed Prior - The New AI