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Another Example of Higher Order Randomness

Becher V., Chaitin G.

Fundamenta Informaticae

2002

Vol. 51, nr 4

325-338

We consider the notion of algorithmic randomness relative to an oracle. We prove that the probability b that a program for infinite computations (a program that never halts) outputs a cofinite set is random in the second jump of the halting problem. Indeed, we prove that b is exactly as random as the halting probability of a universal machine equipped with an oracle for the second jump of the halting problem, in spite of the fact that b is defined without considering oracles.