Synthesis of Pure and Impure Petri Nets with Restricted Place-environments: Complexity Issues

Abstract

Petri net synthesis consists in deciding for a given transition system A whether there exists a Petri net N whose reachability graph is isomorphic to A. Several works examined the synthesis of Petri net subclasses that restrict, for every place p of the net, the cardinality of its preset or of its postset or both in advance by small natural numbers ϱ and κ, respectively, such as for example (weighted) marked graphs, (weighted) T-systems and choice-free nets. In this paper, we study the synthesis aiming at Petri nets which have such restricted place environments, from the viewpoint of classical and parameterized complexity: We first show that, for any fixed natural numbers ϱ and κ, deciding whether for a given transition system A there is a Petri net N such that (1) its reachability graph is isomorphic to A and (2) for every place p of N the preset of p has at most ϱ and the postset of p has at most κ elements is doable in polynomial time. Secondly, we introduce a modified version of the problem, namely ENVIRONMENT RESTRICTED SYNTHESIS (ERS, for short), where ϱ and κ are part of the input, and show that ERS is NPcomplete, regardless whether the sought net is impure or pure. In case of the impure nets, our methods also imply that ERS parameterized by ϱ + κ is W[2]-hard.