The Hardness of Synthesizing Elementary Net Systems from Highly Restricted Inputs

Elementary net systems (ENS) are the most fundamental class of Petri nets. Their synthesis problem has important applications in the design of digital hardware and commercial processes. Given a labeled transition system (TS) $A$, feasibility is the NP-complete decision problem whether $A$ can be equivalently synthesized into an ENS. It is well known that $A$ is feasible if and only if it has the event state separation property (ESSP) and the state separation property (SSP). Recently, these properties have also been studied individually for their practical implications. A fast ESSP algorithm, for instance, would allow applications to at least validate the language equivalence of $A$ and a synthesized ENS. Being able to efficiently decide SSP, on the other hand, could serve as a quick-fail preprocessing mechanism for synthesis. Although a few tractable subclasses have been found, this paper destroys much of the hope that many practically meaningful input restrictions make feasibility or at least one of ESSP and SSP efficient. We show that all three problems remain NP-complete even if the input is restricted to linear TSs where every event occurs at most three times or if the input is restricted to TSs where each event occurs at most twice and each state has at most two successor and two predecessor states.

💡 Research Summary

The paper investigates the computational complexity of synthesizing Elementary Net Systems (ENS) from transition systems (TS) under severe structural restrictions. ENS synthesis is known to be equivalent to checking two properties of a TS: the State Separation Property (SSP) – every pair of distinct states must be distinguishable by a region – and the Event State Separation Property (ESSP) – every event must be inhibitible at states where it does not occur. While the combined feasibility problem (SSP ∧ ESSP) is NP‑complete in general, the authors ask whether the problem becomes easier when the input TS is highly constrained, a question motivated by practical scenarios where events occur only a few times and each state has few incoming/outgoing transitions.

To formalize restrictions, they introduce two parameters: event manifoldness k, the maximum number of edges labeled with the same event (a k‑fold TS has each event occurring at most k times), and state degree g, the maximum number of successors (and predecessors) of any state (a g‑grade TS). A linear TS is a special case with g = 1 and a single initial state without predecessors. The main goal is to determine the complexity of SSP, ESSP, and feasibility for the families of k‑fold, g‑grade, and combined g‑grade k‑fold TSs.

The core technical contribution is a series of polynomial‑time reductions from the well‑known NP‑complete problem cubic monotone one‑in‑three 3‑SAT. The authors construct modular gadgets called unions, which are collections of independent TS components that can be merged while preserving SSP and feasibility (though not necessarily ESSP). By carefully designing these unions and a “joining” operation that stitches them together with auxiliary connector events and states, they embed the SAT instance into a TS with the desired structural bounds.

Key results:

1. Linear k‑fold TS (k ≥ 3) – All three decision problems (SSP, ESSP, feasibility) remain NP‑complete. The reduction first maps the SAT instance to a linear 3‑ESSP instance; because in linear TSs ESSP implies SSP, this also yields NP‑completeness for SSP and feasibility.

2. g‑grade k‑fold TS (g ≥ 2, k ≥ 2) – The same hardness propagates when both event manifoldness and state degree are bounded simultaneously. The authors show that linear 3‑ESSP can be transformed into a 2‑grade 2‑ESSP instance, and similarly for SSP, establishing NP‑completeness for all combinations except the trivial cases.

3. Trivial cases – When each event occurs only once (1‑fold TS), SSP, ESSP, and feasibility are all polynomial (indeed trivial) regardless of state degree. Likewise, for linear 2‑fold TS, the authors present a quadratic‑time algorithm that, given any two states, constructs a separating region, thus placing linear‑2‑fold SSP in P.

4. Open problems – The complexity of linear 2‑ESSP and linear 2‑feasibility remains unresolved. The authors conjecture that these problems are also polynomial, but a formal proof is left for future work.

The paper also provides a comprehensive “complexity map” (Figure 1) that charts the status of each problem across the (k, g) parameter space. Almost every cell is filled with NP‑completeness, highlighting that severe input restrictions do not, in general, alleviate the inherent difficulty of ENS synthesis.

Methodologically, the work showcases the power of modular union constructions for Petri net synthesis reductions. By separating concerns—designing gadgets that enforce specific SSP or ESSP constraints and then merging them without breaking the desired properties—the authors achieve clean, reusable reductions. The joining operation preserves determinism, simplicity, loop‑freeness, reachability, and reducedness, ensuring that the resulting TS remains a valid input for the synthesis problem.

In summary, the paper delivers a strong negative result: even when each event appears at most three times and each state has at most two predecessors and two successors, deciding whether a TS can be realized as an ENS (or whether it satisfies SSP or ESSP) is still NP‑complete. This outcome cautions designers of synthesis tools that preprocessing shortcuts based on event frequency or state degree are unlikely to yield polynomial‑time guarantees for the general case. The remaining open cases (linear 2‑ESSP and linear 2‑feasibility) present an intriguing direction for further research, potentially revealing a narrow island of tractability within an otherwise intractable landscape.