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

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.

💡 Research Summary

The paper investigates the computational complexity of the feasibility (synthesis) problem for Boolean Petri nets (BPNs) when the set of allowed interactions is extended beyond the well‑studied flip‑flop derivatives. A Boolean Petri net is a model where each place holds at most one token (value 0 or 1) and transitions are governed by one of eight basic interactions: nop, inp, out, set, res, swap, used, and free. A net type τ is defined by a subset of these interactions; there are 256 possible types.

Previous work showed that for all flip‑flop‑derived types τ = {nop, swap} ∪ ω, where ω ⊆ {inp, out, used, free}, the feasibility problem—deciding whether a given transition system A can be realized as the reachability graph of a τ‑net—is solvable in polynomial time. The authors replace the input‑output interactions (inp, out) with reset and set (res, set) and add at least one of the test interactions (used, free). They consider nine new types τ = {nop, swap} ∪ ω with ω ⊆ {res, set, used, free}, ω ∩ {res, set} ≠ ∅ and ω ∩ {used, free} ≠ ∅.

The main result is that feasibility becomes NP‑complete for all these nine types, even when the input transition system is restricted to low degree (g‑grade) where each state has at most g incoming and outgoing edges, for any fixed g ≥ 2. The proof proceeds in two parts:

1. NP‑hardness via reduction – The authors reduce the cubic monotone one‑in‑three 3‑SAT problem (known NP‑complete) to the feasibility problem. They construct, for any 3‑SAT instance φ, a transition system A(φ) whose states encode variable assignments and clause satisfaction. The reset (res) and set (set) interactions are used to force a place to a specific Boolean value, effectively encoding the truth assignment of a variable. The test interactions (used, free) ensure that a clause is satisfied by exactly one true literal. The constructed A(φ) satisfies the τ‑state separation property (τ‑SSP) and the τ‑event state separation property (τ‑ESSP) iff φ is satisfiable. Since τ‑feasibility is equivalent to having both SSP and ESSP, this yields a polynomial‑time many‑one reduction, establishing NP‑hardness.

2. Membership in NP – Given a transition system A and a type τ, a nondeterministic algorithm can guess, for each pair of distinct states (for SSP) and for each state/event pair (for ESSP), a τ‑region (support and signature) that separates them. Verification of each guessed region is linear in the size of A, so the whole verification runs in polynomial time.

To avoid proving nine separate reductions, the authors exploit isomorphisms between net types: swapping set ↔ res and used ↔ free while keeping nop and swap unchanged yields an isomorphic net type. Lemma 1 states that τ‑SSP and τ‑ESSP are preserved under such isomorphisms. Consequently, they prove NP‑completeness for only two representative families (σ₁ and σ₂) and obtain the remaining cases by isomorphism.

The paper also updates the complexity landscape (Table 1), adding the nine new NP‑complete types to the previously known seven, bringing the total to sixteen. The results demonstrate that the presence of reset or set interactions dramatically increases expressive power: they allow direct forcing of a place’s value, which can encode arbitrary Boolean constraints, whereas swap alone only toggles values and does not increase synthesis complexity.

Finally, the authors discuss practical implications. Real‑world transition systems often have low degree (few choices per state), especially in hardware design benchmarks. The reduction preserves low degree, showing that even under realistic input restrictions the synthesis problem remains intractable for the identified net types. This provides a clear guideline for designers: avoiding res/set (or limiting their use) is essential if polynomial‑time synthesis is required.

In summary, the paper establishes that extending flip‑flop nets with any combination of reset/set and at least one test interaction makes the Boolean Petri net synthesis problem NP‑complete, even for bounded‑degree transition systems, and it clarifies the precise boundary between tractable and intractable net types.