Thus, [[M]] = ∅ if 0 is reachable from itself in N. In a similar way, if M accepts a DAG D, a run of M on D can easily be turned into a nonempty firing sequence of N (under the top-down interpretation of runs) that turns 0 into itself.
we have reduced the emptiness problem for DAG automata to the problem of deciding whether a Petri net is structurally cyclic.