Complexity-based induction systems: Comparisons and convergence theorems

In 1964 the author proposed as an explication of {\em a priori} probability the probability measure induced on output strings by a universal Turing machine with unidirectional output tape and a randomly coded unidirectional input tape. Levin has shown that if
tilde{P}'_{M}(x)
is an unnormalized form of this measure, and
P(x)
is any computable probability measure on strings,
x
, then
\tilde{P}'_{M}\geqCP(x)
where
C
is a constant independent of
x
. The corresponding result for the normalized form of this measure,
P'_{M}
, is directly derivable from Willis' probability measures on nonuniversal machines. If the conditional probabilities of
P'_{M}
are used to approximate those of
P
, then the expected value of the total squared error in these conditional probabilities is bounded by
-(1/2) \ln C
. With this error criterion, and when used as the basis of a universal gambling scheme,
P'_{M}
is superior to Cover's measure
b\ast
. When
H\ast\equiv -\log_{2} P'_{M}
is used to define the entropy of a rmite sequence, the equation
H\ast(x,y)= H\ast(x)+H^{\ast}_{x}(y)
holds exactly, in contrast to Chaitin's entropy definition, which has a nonvanishing error term in this equation.