On infinite sequences (almost) as easy as pi

Abstract

It is known that infinite binary sequences of constant Kolmogorov complexity are exactly the recursive ones. Such a kind of statement no longer holds in the presence of resource bounds. Contrary to what intuition might suggest, there are sequences of constant, polynomial-time bounded Kolmogorov complexity that are not polynomial-time computable. This motivates the study of several resource-bounded variants in search for a characterization, similar in spirit, of the polynomial-time computable sequences. We propose some definitions, based on Kobayashi's notion of compressibility, and compare them to the standard resource-bounded Kolmogorov complexity of infinite strings. Some nontrivial coincidences and disagreements are proved and, in particular, they coincide for all usual bounds that are at least logarithmic. The resource-unbounded case is also considered.