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 synthesized into an ENS. It is 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.
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