Normal Numbers and Sources for BPP

In [10], Lutz proposed a notion of source, a nonrandom sequence that can substitute in a certain way for the random bits used by boundederror probabilistic machines. He showed that almost every sequence in DSPACE(2 polynomial ) is a source. We improve this abundance result to PSPACE, by first showing that the sources are exactly the classical normal numbers (or normal sequences) of Borel. There are sequences clearly in P that have long been known to be normal, and we go on to show there are sources in AC 0 : This suggests that alternate notions of source should be explored. 1 Introduction In [10], Lutz examines a particular kind of pseudorandomness useful for simulating the bounded-error probabilistic machines. The pseudorandomness is not in the form of a generator, that expands a short truly random string, but instead is a single computable sequence, called a source, whose elements can substitute for random bits in a repeated simulation of every bounded-error machine. Thus a source..