Logic Colloquium 2008
Joseph S. Miller Extracting information is hard
Can randomness---or more technically, ``information''---be effectively extracted from a semi-random source? A special case of this question was answered by von Neumann in 1951. He described a simple method for extracting an unbiased random sequence from the flips of a biased coin. A more general form of the question was asked by Reimann and Terwijn in the context of algorithmic randomness, so we will start with an introduction to Kolmogorov complexity, effective Hausdorff dimension, and Martin-L{\"o}f randomness. Kolmogorov complexity measures the information content of a finite binary string. Informally, the complexity of a string is defined to be the length of the shortest program that generates it. A closely related notion, effective (Hausdorff) dimension, measures the information density of an infinite binary sequences. We can now formulate the question in terms of effective dimension: is there a sequence that has effective Hausdorff dimension 1/2---in other words, a half-random sequence---that does not compute a sequence of higher effective dimension? As it turns out, such sequences exist. We will discuss this and related results.