chapter
17 Pages

Compressibility of Infinite Binary Sequences

WithJosé L. Balcázar, Ricard Gavaldà, Montserrat Hermo

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 both the standard resource-bounded Kolmogorov complexity of infinite strings, and the uniform complexity. Some nontrivial coincidences and disagreements are proved. The resource- unbounded case is also considered.