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

Abstract

Boolean Petri nets are differentiated by types of nets τ 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 τ is to find a boolean τ -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 defined by {nop, swap} ∪ ω, ω ⊆ {inp, out, used, free}. In this paper, we replace inp, out by res, set and show that feasibility becomes NP-complete for {nop, swap} ∪ ω if ω ⊆ {res, set, used, free} such that ω ∩ {set, res} = ∅ and ω ∩ {used, free} = ∅. The reduction guarantees a low degree for A's states and, thus, preserves hardness of feasibility even for considerable input restrictions. Type of net τ Complexity status # 1 τ = {nop, res} ∪ ω, ω ⊆ {inp, used, free} polynomial time 8 2 τ = {nop, set} ∪ ω, ω ⊆ {out, used, free} polynomial time 8 3 τ = {nop, swap} ∪ ω, ω ⊆ {inp, out, used, free} polynomial time 16 4 τ = {nop} ∪ ω, ω ⊆ {used, free} polynomial time 4 5 τ = {nop, inp, free} or τ = {nop, inp, used, free} NP-complete 2 6 τ = {nop, out, used} or τ = {nop, out, used, free} NP-complete 2 7 τ = {nop, set, res} ∪ ω, ∅ = ω ⊆ {used, free} NP-complete 3 8 τ = {nop, inp, out} ∪ ω, ω ⊆ {used, free} NP-complete 4 9 τ = {nop, inp, res, swap} ∪ ω, ω ⊆ {used, free} NP-complete 4 10 τ = {nop, out, set, swap} ∪ ω, ω ⊆ {used, free} NP-complete 4 11 τ = {nop, inp, set} ∪ ω, ω ⊆ {out,

The Complexity of Synthesizing nop-Equipped Boolean Nets from g-Bounded Inputs (Technical Report)

“…In this paper, we strengthen this results to 1-bounded inputs. The hardness part for the types of §9 origin from [12]. In this paper, we argue that the bound 2 is tight.…”

“…These opposing results motivate the question which interactions of I make the synthesis problem hard and which make it tractable. In our previous work of [12,15,16], we answer this question partly and reveal the computational complexity of 120 of the 128 types that allow nop.…”