Abstract :
A string p is called a program to compute y given x if U(p, x)=y, where U denotes universal programming language. Kolmogorov complexity K(y|z) of y relative to x is defined as minimum length of a program to compute y given x. Let K(x) denote K(x|empty string) (Kolmogorov complexity of x) and let I(x: y)=K(x)+K(y)-K(⟨x, y⟩) (the amount of mutual information in x, y). In the present paper we answer in negative the following question posed previously: Is it true that for any strings x, y there are independent minimum length programs p, q to translate between x, y, that is, is it true that for any x, y there are p, q such that U(p, x)=y, U(q, y)=x, the length of p is K(y|z), the length of q is K(z|y), and I(p:q)=O (where the last three equalities hold up to an additive O(log(K(x|y)+K(y|z))) term)?
