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TR02-056 | 19th September 2002 00:00

On the Autoreducibility of Random Sequences


Authors: Todd Ebert, Wolfgang Merkle, Heribert Vollmer
Publication: 25th September 2002 21:28
Downloads: 3486


A binary sequence A=A(0)A(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.

We obtain the somewhat counterintuitive result that every Martin-L"of random sequence, in fact even every rec-random or p-random sequence, is i.o. truth-table-autoreducible. Furthermore, we investigate the question of how dense the set of guessed bits can be when i.o. autoreducing a random sequence. We show that rec-random sequences are never i.o. truth-table-autoreducible such that the set of guessed bits has positive constant density in the limit, and that a similar assertion holds for Martin-L"of random sequences and i.o. Turing-autoreducibility. On the other hand, our main result asserts that for any rational-valued computable function r that goes non-ascendingly to zero, any rec-random sequence is i.o. truth-table-autoreducible such that on any prefix of length m at least a fraction of r(m) of the m bits in the prefix are guessed.

We include a self-contained account of the hat problem, a puzzle that has received some attention outside of theoretical computer science. The hat problem asks for guessing bits of a finite sequence, thus illustrating the notion of i.o autoreducibility in a finite setting. The solution to the hat problem is then used as a module in the proofs of the positive results on i.o. autoreducibility.

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