Conditional complexity and codes

Let x and y be binary strings. We prove that there exists a program p of size about K(x|y) that maps y tox and has small complexity when x is known (K(p|x)0)*#8776;0). Having in mind the parallelism between Shannon information theory and algorithmic information theory, one can say that this result is parallel to Wolf-Slepian and Korner-Csiszar-Marton theorems, see (I. Csiszar and J. Korner, Information theory, Coding Theorems for Discrete Memoryless Systems, Akademiai Kiado, Budapest, 1981). We show also that for any three stringsx,y,z of length at most n the length of the shortest program p that maps both y and z to x (i.e., p(y)=p(z)=x) equals max(K(x|y),K(x|z)+O(logn.