# From Petri Nets to Polynomials and Back: Models, Their Complexity, and New Algorithms

**From Petri Nets to Polynomials and Back: Models, Their Complexity, and New Algorithms**

Petri nets are a widely-used model for parallel and distributed systems of concurrent processes using common resources. They admit a precise algebraic formalization as vector addition or vector replacement systems. If one considers symmetric VRSï¿½s, it turns out that they are equivalent to finitely presented commutative semigroups or to binomial ideals in a multivariate ring over the rationals. We outline and survey the interaction between these domains of computational algebra, system modeling and verification, and, in particular, complexity theory. While many of the fundamental computational problems in these areas turn out to be very complex (i.e., EXPSPACEcomplete or even worse), we also present some new results concerning better complexity for restricted subclasses of the problems.