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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, for mailbox communication, that it is possible to decide if such a bound k exists. [ABSTRACT FROM AUTHOR]