On the Parameterized Complexity of Synthesizing Boolean Petri Nets With Restricted Dependency

Modeling of real-world systems with Petri nets allows to benefit from their generic concepts of parallelism, synchronisation and conflict, and obtain a concise yet expressive system representation. Algorithms for synthesis of a net from a sequential specification enable the well-developed theory of Petri nets to be applied for the system analysis through a net model. The problem of $\tau$-synthesis consists in deciding whether a given directed labeled graph $A$ is isomorphic to the reachability graph of a Boolean Petri net $N$ of type $\tau$. In case of a positive decision, $N$ should be constructed. For many Boolean types of nets, the problem is NP-complete. This paper deals with a special variant of $\tau$-synthesis that imposes restrictions for the target net $N$: we investigate dependency $d$-restricted tau-synthesis (DR$\tau$S) where each place of $N$ can influence and be influenced by at most d transitions. For a type $\tau$, if tau-synthesis is NP-complete then DR$\tau$S is also NP-complete. In this paper, we show that DR$\tau$S parameterized by $d$ is in XP. Furthermore, we prove that it is W[2]-hard, for many Boolean types that allow unconditional interactions set and reset.