  In recent work, the second and third authors introduced a technique for
reachability checking in 1-bounded Petri nets, based on wiring decompositions,
which are expressions in a fragment of the compositional algebra of nets with
boundaries. Here we extend the technique to the full algebra and introduce the
related structural property of decomposition width on directed hypergraphs.
Small decomposition width is necessary for the applicability of the
reachability checking algorithm. We give examples of families of nets with
constant decomposition width and develop the underlying theory of
decompositions.
