Hardness Results for the Synthesis of $b$-bounded Petri Nets (Technical Report)

TL;DR

This paper establishes the NP-completeness of the synthesis feasibility problem for various types of bounded Petri nets, including extensions with modular arithmetic, highlighting the computational difficulty of Petri net synthesis.

Contribution

It proves NP-completeness of the synthesis feasibility for pure and extended b-bounded Petri nets, including those with modular group extensions.

Findings

Feasibility is NP-complete for pure b-bounded P/T-nets.

Feasibility remains NP-complete for group extended b-bounded P/T-nets.

Deciding event and state separation properties is NP-complete for these Petri net types.

Abstract

Synthesis for a type $\tau$ of Petri nets is the following search problem: For a transition system $A$, find a Petri net $N$ of type $\tau$ whose state graph is isomorphic to $A$, if there is one. To determine the computational complexity of synthesis for types of bounded Petri nets we investigate their corresponding decision version, called feasibility. We show that feasibility is NP-complete for (pure) $b$-bounded P/T-nets if $b\in \mathbb{N}^+$. We extend (pure) $b$-bounded P/T-nets by the additive group $\mathbb{Z}_{b+1}$ of integers modulo $(b+1)$ and show feasibility to be NP-complete for the resulting type. To decide if $A$ has the event state separation property is shown to be NP-complete for (pure) $b$-bounded and group extended (pure) $b$-bounded P/T-nets. Deciding if $A$ has the state separation property is proven to be NP-complete for (pure) $b$-bounded P/T-nets.