Complexity-based induction systems: Comparisons and convergence theorems

Abstract

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 iftilde{P}'_{M}(x)is an unnormalized form of this measure, andP(x)is any computable probability measure on strings,x, then\tilde{P}'_{M}\geqCP(x)whereCis a constant independent ofx. 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 ofP'_{M}are used to approximate those ofP, 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 measureb\ast. WhenH\ast\equiv -\log_{2} P'_{M}is used to define the entropy of a rmite sequence, the equationH\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.