Resource bounded randomness and its applications

Martin-Löf gave the first meaningful definition of a random infinite binary sequence [51]. From a classical standpoint, such a notion seems meaningless, as every sequence has probability zero. Martin-Löf’s insight was to use computability theory to give an effective version of measure theory. With this effectivization, a sequence A is (Martin-Löf) random if the singleton {A} is not of effective measure zero. The field of algorithmic randomness has since grown to include many different notions of randomness, all making essential use of computability [18, 58]. With the prominence of complexity theory in computation, a natural step is to impose resource bounds on the computation in algorithmic randomness. Resource bounded randomness studies the different notions of what it means for a sequence to be “random” relative to a resource bounded observer. Schnorr gave the first definition of resource bounded randomness by imposing time bounds on martingales [65]. However, this work did not have an immediate impact. Instead, resource bounded randomness was largely unexplored until Lutz’s, independent, development of resource bounded measure [46], a resource bounded effectivization of Lebesgue measure theory. As noted by Ambos-Spies, et al. [5], resource bounded measure implicitly redefines Schnorr’s notion of resource bounded randomness. Lutz showed that resource bounded measure, and therefore resource bounded randomness, is a valuable tool in complexity theory. This application ignited interest in resource bounded randomness, resulting in significant growth in the area. Nevertheless, many fundamental problems remain to be explored. One of the great achievements in algorithmic randomness, in the computable setting, is the variety of characterizations of its principal definitions. There are three prominent viewpoints of randomness: statistical tests (Martin-Löf [51]), martingales (Schnorr [65]) and compressibility (Levin [44] and Chaitin [15]). The richness of algorithmic randomness is, in part, due to the fact that most notions can be defined using each paradigm. Unfortunately, this has not yet extended to resource bounded randomness. Thus far, only the martingale approach of Schnorr and Lutz has cemented itself as fundamental. An important goal of resource bounded randomness is to be able to pass freely between the different