Reachability in Elementary System Nets

  • Wolfgang Reisig


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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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Computer ScienceHumboldt-Universität zu BerlinBerlinGermany

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