Where does all the randomness come from? Many physicists would be content with a statistical theory of everything (TOE) based on simple probabilistic physical laws allowing for stochastic predictions such as "We do not know where this particular electron will be in the next nanosecond, but with probability 0.04 we will find it in a certain volume V".

## Advertising

Any source of randomness, however, leaves us with an unsatisfactory TOE, since randomness does not have a compact or simple explanation, by definition. Where does the enormous information conveyed by a particular history of random events come from? A TOE that cannot explain this is incomplete. The in hindsight obvious solution is an "ensemble TOE" which covers all possible universe histories. The ensemble conveys less information than most particular histories - one main motivation of this mailing list. Which are the possible histories? Let us focus on well-defined ensembles only, and ignore those that cannot be sufficiently specified to permit reconstruction through a formally describable computer. In particular, we may ignore uncountable ensembles such as continua, or other ensembles including histories without finite descriptions. Is there an optimally efficient way of computing all the "randomness" in all the describable (possibly infinite) universe histories? Yes, there is. There exists a machine-independent ensemble-generating algorithm called FAST that computes any history essentially as quickly as this history's fastest algorithm. Somewhat surprisingly, FAST is not slowed down much by the simultaneous computation of all the other histories. It turns out, however, that for any given history there is a speed limit which greatly depends on the history's degree of randomness. Highly random histories are extremely hard to compute, even by the optimal algorithm FAST. Each new bit of a truly random history requires at least twice the time required for computing the entire previous history. As history size grows, the speed of highly random histories (and most histories are random indeed) vanishes very quickly, no matter which computer we use (side note: infinite random histories would even require uncountable time, which does not make any sense). On the other hand, FAST keeps generating numerous nonrandom histories very quickly; the fastest ones come out at a rate of a constant number of bits per fixed time interval. Now consider an observer evolving in some universe history. He does not know in which, but as history size increases it becomes less and less likely that he is located in one of the slowly computable, highly random universes: after sufficient time most long histories involving him will be fast ones. Some consequences are discussed in http://www.idsia.ch/~juergen/toesv2/node39.html Juergen Schmidhuber