  A communicating system is $k$-synchronizable if all of the message sequence
charts representing the executions can be divided into slices of $k$ sends
followed by $k$ receptions. It was previously shown that, for a fixed given
$k$, one could decide whether a communicating system is $k$-synchronizable.
This result is interesting because the reachability problem can be solved for
$k$-synchronizable systems. However, the decision procedure assumes that the
bound $k$ is fixed. In this paper we improve this result and show that it is
possible to decide if such a bound $k$ exists.
