Mathematics – Logic
Scientific paper
2012-03-25
Mathematics
Logic
Scientific paper
Unlike Martin-L\"of randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, many equivalent definitions), and further, provides a general method for abstracting "bit-wise" definitions of randomness from Cantor space to arbitrary computable probability spaces. This same method is also applied to give machine characterizations of computable and Schnorr randomness for computable probability spaces, extending the previous known results. This paper also addresses "Schnorr's Critique" that gambling characterizations of Martin-L\"of randomness are not computable enough. The paper contains a new type of randomness---endomorphism randomness---which the author hopes will shed light on the open question of whether Kolmogorov-Loveland randomness is equivalent to Martin-L\"of randomness. It ends with other possible applications of the methods presented, including a possible definition of computable randomness on non-computable probability spaces and a possible definition of K-triviality for computable probability spaces.
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