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TT-functionals and Martin-Löf randomness for Bernoulli measures. (English) Zbl 1331.03030
Summary: For $$r\in[0,1]$$, the Bernoulli measure $$\mu_r$$ on the Cantor space $$\{0,1\}^{\mathbb{N}}$$ assigns measure $$r$$ to the set of sequences with 1 at a fixed position. In [R. D. Mauldin, “Problems in topology arising from analysis”, in: Open problems in topology. Amsterdam etc.: North-Holland. 617–629 (1990)] it is shown that for $$r,s\in[0,1]$$, $$\mu_s$$ is continuously reducible to $$\mu_r$$ if any only if $$r$$ and $$s$$ satisfy certain purely number theoretic conditions (binomial reduciblility). We bring these results into the context of computability theory and Martin-Löf randomness and show that the continuous maps arising in [loc. cit.] are truth-table functionals (tt-functionals) on $$\{0,1\}^{\mathbb{N}}$$. This allows us extend the characterization of continuous reductions between Bernoulli measures to include tt-functionals. It then follows from the conservation of randomness under tt-functionals that if $$s$$ is binomially reducible to $$r$$, then there is a tt-functional that maps every Martin-Löf random sequence for $$\mu_s$$ to a Martin-Löf random sequences for $$\mu_r$$. We are also able to show using results in [L. Bienvenu and C. Porter, Theor. Comput. Sci. 459, 55–68 (2012; Zbl 1283.68170)] that the converse of this statement is not true.
##### MSC:
 03D32 Algorithmic randomness and dimension
Zbl 1283.68170
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