Stability of properties of Kolmogorov complexity under relativization

Assume that a tuple of binary strings $\bar a=\langle a_1,\dots,a_n\rangle$ has negligible mutual information with another string $b$. Does this mean that properties of the Kolmogorov complexity of $\bar a$ do not change significantly if we relativize them to $b$? This question becomes very nontrivial when we try to formalize it. In this paper we investigate this problem for a special class of properties (for properties that can be expressed by an $\exists$-formula). In particular, we show that a random (conditional on $\bar a$) oracle b does not help to extract common information from the strings $a_i$.