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 τ. 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 tauS) 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 tauS is also NP-complete. In this paper, we show that DR tauS 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.