On Generalized Computable Universal Priors and their Convergence

Solomonoff unified Occam's razor and Epicurus' principle of multiple explanations to one elegant, formal, universal theory of inductive inference, which initiated the field of algorithmic information theory. His central result is that the posterior of the universal semimeasure M converges rapidly to the true sequence generating posterior mu, if the latter is computable. Hence, M is eligible as a universal predictor in case of unknown mu. The first part of the paper investigates the existence and convergence of computable universal (semi)measures for a hierarchy of computability classes: recursive, estimable, enumerable, and approximable. For instance, M is known to be enumerable, but not estimable, and to dominate all enumerable semimeasures. We present proofs for discrete and continuous semimeasures. The second part investigates more closely the types of convergence, possibly implied by universality: in difference and in ratio, with probability 1, in mean sum, and for Martin-Loef random sequences. We introduce a generalized concept of randomness for individual sequences and use it to exhibit difficulties regarding these issues. In particular, we show that convergence fails (holds) on generalized-random sequences in gappy (dense) Bernoulli classes.Comment: 22 page

Arithmetic complexity via effective names for random sequences

We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-L\"{o}f, computably, Schnorr, and Kurtz random sets, weakly 1-generics and their complementary classes, we find that there exist characterizations of the third and fourth levels of the arithmetic hierarchy purely in terms of these notions. More generally, there exists an equivalence between arithmetic complexity and existence of numberings for classes of left-r.e. sets with shift-persistent elements. While some classes (such as Martin-L\"{o}f randoms and Kurtz non-randoms) have left-r.e. numberings, there is no canonical, or acceptable, left-r.e. numbering for any class of left-r.e. randoms. Finally, we note some fundamental differences between left-r.e. numberings for sets and reals

Characterizing the strongly jump-traceable sets via randomness

We show that if a set $A$ is computable from every superlow 1-random set, then $A$ is strongly jump-traceable. This theorem shows that the computably enumerable (c.e.) strongly jump-traceable sets are exactly the c.e.\ sets computable from every superlow 1-random set. We also prove the analogous result for superhighness: a c.e.\ set is strongly jump-traceable if and only if it is computable from every superhigh 1-random set. Finally, we show that for each cost function $c$ with the limit condition there is a 1-random $\Delta^0_2$ set $Y$ such that every c.e.\ set $A \le_T Y$ obeys $c$. To do so, we connect cost function strength and the strength of randomness notions. This result gives a full correspondence between obedience of cost functions and being computable from $\Delta^0_2$ 1-random sets.Comment: 41 page