  In this article we introduce the operations of insertion and deletion working
in a random-context and semi-conditional manner. We show that the conditional
use of rules strictly increase the computational power. In the case of
semi-conditional insertion-deletion systems context-free insertion and deletion
rules of one symbol are sufficient to get the computational completeness. In
the random context case our results expose an asymmetry between the
computational power of insertion and deletion rules: systems of size $(2,0,0;
1,1,0)$ are computationally complete, while systems of size $(1,1,0;2,0,0)$
(and more generally of size $(1,1,0;p,1,1)$) are not. This is particularly
interesting because other control mechanisms like graph-control or matrix
control used together with insertion-deletion systems do not present such
asymmetry.
