Given a DPN that works on channels x1,...,xn, the semantics is defined as a state transition system whose states are labeled with n streams, one for each variable x1,...,xn. If a node fires, it consumes values from some of these streams and adds values to other streams. If in some state, more than one node can fire, the question of confluence arises as discussed below.

We may formally define a relation s1->s2 that means that some nodes of the DPN fire and turn state s1 to state s2 by consuming and producing values for the streams in these states. Then, s1->*s2 mean that s2 can be reached from s1 by finitely many transitions, i.e., s1->s3->s4->...->s'->s2, and the epsilon means the reflexive closure of ->, i.e., x->εy means that either x=y or x->y holds.

A DPN is confluent iff for all x,y1,y2 with x->∗y1 and x->∗y2 there is a z with y1->∗ z and y2->∗z. Hence, it does not really matter whether we choose the firings x->∗y1 or the firings x->∗y2 since at the end, they can be joined into z.

See also pages 227 and the following in

https://es.cs.uni-kl.de/publications/datarsg/Schn09.pdf.