Average-Case Bit-Complexity Theory of Real Functions

Abstract

We introduce, and initiate the study of, average-case bitcomplexity theory over the reals: Like in the discrete case a first, naïve notion of polynomial average runtime turns out to lack robustness and is thus refined. Standard examples of explicit continuous functions with increasingly high worst-case complexity are shown to be in fact easy in the mean; while a further example is constructed with both worst and average complexity exponential: for topological/metric reasons, i.e., oracles do not help. The notions are then generalized from the reals to represented spaces; and, in the real case, related to randomized computation.