Almost Everywhere High Nonuniform Complexity

We investigate the distribution of nonuniform complexities in uniform complexity classes We prove that almost every problem decidable in exponential space has essentially maximum circuit size and space bounded Kolmogorov complexity almost everywhere The circuit size lower bound actually exceeds and thereby strengthens the Shannon n n lower bound for almost every problem with no computability constraint In exponential time complexity classes we prove that the strongest relativizable lower bounds hold almost everywhere for almost all problems Finally we show that in nite pseudorandom sequences have high nonuniform complexity almost everywhere The results are uni ed by a new more powerful formulation of the underlying measure theory based on uniform systems of density functions and by the introduction of a new nonuniform complexity measure the selective Kolmogorov complexity This research was supported in part by NSF Grants CCR and CCR and in part by DIMACS where the author was a visitor while part of this work was performed