A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences and the Kolmogorov complexity of a string is proven to be the product of its length and its dimension.Expand

A theory of resource-bounded dimension is developed using gales, which are natural generalizations of martin-gales. When the resource bound /spl Delta/(a parameter of the theory) is unrestricted, the… Expand

It is proved that almost every problem decidable in exponential space has essentially maximum circuit-size and space-bounded Kolmogorov complexity almost everywhere, and it is shown that infinite pseudorandom sequences have high nonuniform complexityalmost everywhere.Expand

The author proves that almost every problem decidable in exponential space has essentially maximum circuit-size and program-size complexity almost everywhere and proves that the strongest relativizable lower bounds hold almost everywhere for almost all problems.Expand

The basic properties of effective strong dimensions are developed and a number of results relating them to fundamental aspects of randomness, Kolmogorov complexity, prediction, Boolean circuit-size complexity, polynomial-time degrees, and data compression are proved.Expand

It is shown that every weakly useful sequence is strongly deep, and this result implies that every high Turing degree contains strongly deep sequences.Expand

The measure structure of these classes is seen to interact in informative ways with bi-immunity, complexity cores, /sub <or=/m/sup P/-reducibility, circuit-size complexity, Kolmogorov complexity, and the density of hard languages.Expand

Under the hypothesis that NP does not have p-measure 0 (roughly, that NP contains more than a negligible subset of exponential time), it is shown that there is a language that is ≤ T P -complete… Expand

Tight upper and lower bounds on the size of complexity cores of hard languages are derived and it is seen that the $\leq P}-hard languages for E are unusually simple, in the sense that they have smaller complexity cores than most languages in E.Expand

A more recent snapshot of resource-bounded measure is given, focusing not so much on what has been achieved to date as on what the authors hope will be achieved in the near future.Expand