Binarization Trees and Random Number Generation

An m-extracting procedure produces unbiased random bits from a loaded dice with m faces. A binarization takes inputs from an m-faced dice and produce bit sequences to be fed into a (binary) extracting procedure to obtain random bits. Thus, binary extracting procedures give rise to an m-extracting procedure via a binarization. An entropy- preserving binarization is to be called complete, and such a procedure has been proposed by Zhou and Bruck. We show that there exist complete binarizations in abundance as naturally arising from binary trees with m leaves. The well-known leaf entropy theorem and a closely related structure lemma play important roles in the arguments.Comment: 8 page

Peres-Style Recursive Algorithms

Peres algorithm applies the famous von Neumann trick recursively to produce unbiased random bits from biased coin tosses. Its recursive nature makes the algorithm simple and elegant, and yet its output rate approaches the information-theoretic upper bound. However, it is relatively hard to explain why it works, and it appears partly due to this difficulty that its generalization to many-valued source was discovered only recently. Binarization tree provides a new conceptual tool to understand the innerworkings of the original Peres algorithm and the recently-found generalizations in both aspects of the uniform random number generation and asymptotic optimality. Furthermore, it facilitates finding many new Peres-style recursive algorithms that have been arguably very hard to come by without this new tool.Comment: 15 pages. Version 2, as of May 22, 2018, includes a proof of Structure Lemma and a few correction

Randomness extraction in computability theory

In this article, we study a notion of the extraction rate of Turing functionals that translate between notions of randomness with respect to different underlying probability measures. We analyze several classes of extraction procedures: a first class that generalizes von Neumann's trick for extracting unbiased randomness from the tosses of a biased coin, a second class based on work of generating biased randomness from unbiased randomness by Knuth and Yao, and a third class independently developed by Levin and Kautz that generalizes the data compression technique of arithmetic coding. For the first two classes of extraction procedures, we identify a level of algorithmic randomness for an input that guarantees that we attain the extraction rate along that input, while for the third class, we calculate the rate attained along sufficiently random input sequences