Nearly Optimal Language Compression using Extractors

. We show two sets of results applying the theory of extractors to resource-bounded Kolmogorov complexity: Most strings in easy sets have nearly optimal polynomial-time CD complexity. This extends work of Sipser [Sip83] and Buhrman and Fortnow [BF97]. We use extractors to extract the randomness of strings. In particular we show how to get from an arbitrary string, an incompressible string which encodes almost as much polynomial-time CND complexity as the original string. 1 Introduction The Kolmogorov complexity of a string x, denoted C(x), is the length of the shortest program which prints out that string. Researchers have used Kolmogorov complexity in extremely powerful ways to help prove results in many areas of theoretical computer science (see the textbook by Li and Vit#nyi [LV97]). One of the most important tools is the following lemma that allows us to measure sizes of sets with Kolmogorov complexity. Lemma 1. Let A be a recursive set. For all n and for all x 2 A " \..