  The Operational Transformation (OT) approach, used in many collaborative
editors, allows a group of users to concurrently update replicas of a shared
object and exchange their updates in any order. The basic idea of this approach
is to transform any received update operation before its execution on a replica
of the object. This transformation aims to ensure the convergence of the
different replicas of the object, even though the operations are executed in
different orders. However, designing transformation functions for achieving
convergence is a critical and challenging issue. Indeed, the transformation
functions proposed in the literature are all revealed incorrect.
  In this paper, we investigate the existence of transformation functions for a
shared string altered by insert and delete operations. From the theoretical
point of view, two properties - named TP1 and TP2 - are necessary and
sufficient to ensure convergence. Using controller synthesis technique, we show
that there are some transformation functions which satisfy only TP1 for the
basic signatures of insert and delete operations. As a matter of fact, it is
impossible to meet both properties TP1 and TP2 with these simple signatures.
