Abstract
A set A ⊆ 0,1* is called i.o. Turing-autoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don’t-know symbol on any given input x, and outputs A(x) for infinitely many x. If in addition the oracle Turing machine terminates on all inputs and oracles, A is called i.o. truth-table autoreducible. Ebert and Vollmer obtained the somewhat counterintuitive result that every Martin-Löf random set is i.o. truth-table-autoreducible and investigated the question of how dense the set of guessed bits can be when i.o. autoreducing a random set.
We show that rec-random sets are never i.o. truth-table-autoreducible such that the set of guessed bits has strictly positive constant density in the limit, and that a similar assertion holds for Martin-Löf random sets and i.o. Turing-autoreducibility. On the other hand, our main result asserts that for any computable function r that goes non-ascendingly to zero, any rec-random set is i.o. truth-table-autoreducible such that the set of guessed bits has density bounded from below by r(m).
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Ebert, T., Merkle, W. (2002). Autoreducibility of Random Sets: A Sharp Bound on the Density of Guessed Bits. In: Diks, K., Rytter, W. (eds) Mathematical Foundations of Computer Science 2002. MFCS 2002. Lecture Notes in Computer Science, vol 2420. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45687-2_18
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DOI: https://doi.org/10.1007/3-540-45687-2_18
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