Resource bounded randomness and computational complexity

The following is a survey of resource bounded randomness concepts and their relations to each other. Further, we introduce several new resource bounded randomness concepts corresponding to the classical randomness concepts, and show that the notion of polynomial time bounded Ko randomness is independent of the notions of polynomial time bounded Lutz, Schnorr and Kurtz randomness. Lutz has conjectured that, for a given time or space bound, the corresponding resource bounded Lutz randomness is a proper re nement of resource bounded Schnorr randomness. This conjecture is answered for the case of polynomial time bound. Moreover, we will show that polynomial time bounded Schnorr randomness is a proper re nement of polynomial time bounded Kurtz randomness. In contrast to this result, we show that the notions of polynomial time bounded Lutz, Schnorr and Kurtz randomness coincide in the case of recursive sets, thus it su ces to study the notion of resource bounded Lutz randomness in the context of complexity theory. The stochastic properties of resource bounded random sequences will be discussed in detail. Schnorr has already shown that the law of large numbers holds for p-random sequences. We will show that another important law in probability theory, the law of the iterated logarithm, holds for p-random sequences too. Hence almost all sets in the exponential time complexity class are “hard” from the viewpoint of statistics. c © 2000 Elsevier Science B.V. All rights reserved.