The principle of message counting is used to detect termination of 
distributed computations which consist of processes asynchronously 
communicating over non-FIFO channels. The solution is symmetric and not based 
on a predefined communication structure. An efficient variant of the echo 
algorithm, which dynamically builds a spanning tree, allows a parallel and 
distributed evaluation of the termination predicate in time proportional to 
the diameter of the communication graph. Concurrent and repeated initiation of 
the detection algorithm by different processes is possible at any time without 
prior synchronization due to a subtle method of collision detection and wave 
extinction, which can be regarded as a distributed election scheme where the 
average message complexity increases only logarithmically with the number of 
concurrent initiators. Control messages have a small length and additional 
communication links are not required. Only a fixed number of simple variables 
is needed in every process, global knowledge such as the total number of 
processes or the structure of the network is not used, making the scheme 
useful for dynamic systems. Several variations of the basic principle are 
presented, important issues such as message complexity and fault-tolerance are 
discussed.