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 PG, the top of the corresponding semimeasure dominance hierarchy by . If Conjecture 5.3 were true, then maximizing PG(xy) would be equivalent to minimizing KG(xy). Even then there would be a fundamental problem besides lack of falsifiability: Neither PG 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 KmG(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.