Non-reducible Descriptions for Conditional Kolmogorov Complexity

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 if ∗Institute of New Technologies; e-mail: [email protected]. The work was supported by RFBR grants 04-01-00427, 06-01-00122a. †The work was supported by CNRS (LIF, Marseille, France; Poncelet laboratory, Moscow), STINT foundation, Uppsala university (Sweden), Royal Holloway College (UK), RFBR (grants 02-01-22001, 03-01-00475, 06-01-00122a) and Scientific schools supporting council (grant NSh-358.2003.1); e-mail: [email protected], [email protected]. ‡Moscow State University, e-mail: [email protected]. The work was supported in part by the RFBR grants 02-01-22001, 03-01-00475, 06-01-00122a, NSh-358.2003.1. §Moscow State University, e-mail: [email protected]. The work was supported in part by the RFBR grants 02-01-22001, 03-01-00475, 06-01-00122a, NSh-358.2003.1. ¶Moscow State University, e-mail: [email protected], [email protected]. The work was supported in part by the RFBR grants 02-01-22001, 03-01-00475, 06-01-00122a, NSh-358.2003.1.