Variations on Muchnik's Conditional Complexity Theorem

Muchnik’s theorem about simple conditional descriptions states that for all strings a and b there exists a short program p transforming a to b that has the least possible length and is simple conditional on b. In this paper we present two new proofs of this theorem. The first one is based on the on-line matching algorithm for bipartite graphs. The second one, based on extractors, can be generalized to prove a version of Muchnik’s theorem for space-bounded Kolmogorov complexity. 1 Muchnik’s Theorem An. Muchnik [8] has proven the following theorem:

• Type: Conference
• Year: 2009
• Where: CSR
• Authors: Daniil Musatov, Andrei E. Romashchenko, Alexander Shen
• Keywords: CSR 2009, On-line Matching Algorithm, Simple Conditional, Simple Conditional Descriptions, Theoretical Computer Science