  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 his 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. We investigate the
existence and convergence of computable universal (semi)measures for a
hierarchy of computability classes: finitely computable, estimable, enumerable,
and approximable. For instance, M is known to be enumerable, but not finitely
computable, and to dominate all enumerable semimeasures. We define seven
classes of (semi)measures based on these four computability concepts. Each
class may or may not contain a (semi)measure which dominates all elements of
another class. The analysis of these 49 cases can be reduced to four basic
cases, two of them being new. The results hold for discrete and continuous
semimeasures. We also investigate 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.
