  Given a set X of finite strings, one interesting question to ask is whether
there exists a member of X which is simple conditional to all other members of
X. Conditional simplicity is measured by low conditional Kolmogorov complexity.
We prove the affirmative to this question for sets that have low mutual
information with the halting sequence. There are two results with respect to
this question. One is dependent on the maximum conditional complexity between
two elements of X, the other is dependent on the maximum expected value of the
conditional complexity of a member of X relative to each member of X.
