A Highly Random Number
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In his celebrated 1936 paper Turing defined a machine to be circular iff it performs an infinite computation outputting only finitely many symbols. We define α as the probability that an arbitrary machine be circular and we prove that α is a random number that goes beyond Ω, the probability that a universal self delimiting machine halts. The algorithmic complexity of α is strictly greater than that of Ω, but similar to the algorithmic complexity of Ω', the halting probability of an oracle machine. What makes α interesting is that it is an example of a highly random number definable without considering oracles.
KeywordsAlgorithmic Complexity Minimal Program Computable Number Program Size Universal Machine
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