Understanding Petri Nets pp 141-145 | Cite as

# Reachability in Elementary System Nets

## Abstract

Determining the reachability of an arbitrary marking is one of the interesting, but also one of the most difficult problems of elementary system nets. How to decide whether a marking *M* of an elementary system net *N* is reachable (from the initial marking *M* _{0})? If only a finite number of markings is reachable, it is possible to construct each of them and test whether *M* is among them. If *M* is reachable, it is also possible to incrementally construct the (possibly infinite) marking graph until *M* is eventually encountered.

However, if an infinite number of markings is reachable in *N*, and *M* is not one of them, then this procedure fails. Nevertheless, the problem can be solved: it is possible to construct a finite set *K* of reachable markings of *N* such that *M* is reachable if and only if *M* is an element of *K*. However, this set is incredibly large and it was a long time before the reachability problem was solved.

In this chapter, we will discuss a necessary condition for the reachability and thus a sufficient condition for the unreachability of a marking. At the same time, we will formulate criteria for deciding whether a finite or an infinite number of markings are reachable.

To do this, we will add the *marking equation* and *transition invariants* to our set of linear-algebraic tools. The covering graph, too, yields criteria for the reachability of markings and for the finiteness of the set of reachable markings.

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