Non-reducible descriptions for conditional Kolmogorov complexity

Abstract

Assume that a program p on input a outputs b. We are looking for a shorter program q having the same property (q(a) = b). In addition, we want q to be simple conditional to p (this means that the conditional Kolmogorov complexity K (q|p) is negligible). In the present paper, we prove that sometimes there is no such program q, 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 arguments, respectively. 1 Definitions and statements Let a and b be binary strings. Consider programs p such that p(a) = b (the program p on input a outputs b). What is the minimal length of such a program? If the programming language is chosen appropriately, this length is close to K (b|a), the conditional Kolmogorov complexity of b given a. We will ignore additive terms of order O(log n) where n is the maximum length of the strings involved. With this precision all the versions of Kolmogorov complexity (the plain one, the prefix one etc.) coincide. For the definition of Kolmogorov complexity K (b) and K (b|a) we refer to the textbook [2]. To avoid references to a specific programming language we will consider “descriptions” instead of programs. A string p is called a conditional description of a string b given a i