Variations on Muchnik's Conditional Complexity Theorem

TL;DR

This paper introduces two new proofs of Muchnik's Conditional Complexity Theorem, one using bipartite graph matching and the other based on extractors, with extensions to space-bounded complexity.

Contribution

It provides novel proof techniques for Muchnik's theorem, including a generalization to space-bounded Kolmogorov complexity.

Findings

Two new proofs of Muchnik's theorem are presented.

The extractor-based proof extends to space-bounded complexity.

The bipartite graph approach offers an alternative proof method.

Abstract

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.