# Recent results on the complexity of problems related to Petri nets

## Abstract

In this paper, we examine the complexity of the boundedness, containment, equivalence, and reachability problems for certain subclasses of Petri nets (PNs) (equivalently vector addition systems (VASs), vector addition systems with states (VASSs), or vector replacement systems (VRSs)). Specifically, we consider the complexity of the boundedness problem for general VASSs, fixed dimensional VASSs, and conflict-free VRSs. We consider the complexity of the remaining problems for bounded VASSs, 2-dimensional VASSs, and conflict-free VRSs. Instances in each of these classes are known to have effectively computable semilinear reachability sets (SLSs). In each case, our results are derived by showing how to obtain succinct and sometimes special representations of the associated SLSs. The results discussed here constitute a summary of results obtained elsewhere by the authors. No proofs appear in this document, although we do strive to outline the general strategies involved. Readily available sources for the detailed proofs are indicated.

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