Non-reducible Descriptions for Conditional Kolmogorov Complexity

Abstract

Let a program p on input a outputs b. We are looking for a shorter program p′ having the same property (p′(a)=b). In addition, we want p′ to be simple conditional to p (this means that the conditional Kolmogorov complexity K (p ^′|p) is negligible). In the present paper, we prove that sometimes there is no such program p′, even in the case when the complexity of p is much bigger than K (b|a). We give three different constructions that use the game approach, probabilistic arguments and algebraic (combinatorial) arguments, respectively.