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Complexity-based induction systems: Comparisons and convergence theorems

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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}\geq CP(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.

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Information Theory, IEEE Transactions on  (Volume:24 ,  Issue: 4 )