Compressibility and Resource Bounded Measure

Abstract

We give a new definition of resource bounded measure based on compressibility of infinite binary strings. We prove that the new definition is equivalent to the one commonly used. This new characterization offers us a different way to look at resource bounded measure, shedding more light on the meaning of measure zero results and providing one more tool to prove such results. The main contribution of the paper is the new definition and the proofs leading to the equivalence result. We then show how this new characterization can be used to prove that the class of linear autoreducible sets has p- measure 0. We also prove that the class of sets that are truth-table reducible to a p-selective set has p-measure 0 and that the class of sets that Turing reduce to a subpolynomial dense set has p-measure 0. This strengthens various results.