Tracking Down the Bad Guys: Reset and Set Make Feasibility for Flip-Flop Net Derivatives NP-complete

TL;DR

This paper proves that the feasibility problem for certain types of Boolean Petri nets, specifically those involving reset and set interactions, is NP-complete, indicating increased computational complexity.

Contribution

It demonstrates that replacing inp and out with res and set in flip-flop derived nets makes the feasibility problem NP-complete, extending known complexity results.

Findings

Feasibility is polynomial for nets with nop, swap, and some interactions.

Replacing inp/out with res/set increases complexity to NP-complete.

Hardness persists even under low degree state restrictions.

Abstract

Boolean Petri nets are differentiated by types of nets $\tau$ based on which of the interactions nop, inp, out, set, res, swap, used, and free they apply or spare. The synthesis problem relative to a specific type of nets $\tau$ is to find a boolean $\tau$-net $N$ whose reachability graph is isomorphic to a given transition system $A$. The corresponding decision version of this search problem is called feasibility. Feasibility is known to be polynomial for all types of flip flop derivates that contain at least the interactions nop, swap and an arbitrary selection of inp, out, used, free. In this paper, we replace inp and out by res and set, respectively, and show that feasibility becomes NP-complete for the types that contain nop, swap and a non empty selection of res, set and a non empty selection of used, free. The reduction guarantees a low degree for A's states and, thus, preserves hardness of feasibility even for considerable input restrictions.