Narrowing down the Hardness Barrier of Synthesizing Elementary Net Systems

Elementary net system feasibility is the problem to decide for a given automaton A if there is a certain boolean Petri net with a state graph isomorphic to A. This is equivalent to the conjunction of the state separation property (SSP) and the event state separation property (ESSP). Since feasibility, SSP and ESSP are known to be NP-complete in general, there was hope that the restriction of graph parameters for A can lead to tractable and practically relevant subclasses. In this paper, we analyze event manifoldness, the amount of occurrences that an event can have in A, and state degree, the number of allowed successors and predecessors of states in A, as natural input restrictions. Recently, it has been shown that all three decision problems, feasibility, SSP and ESSP, remain NP-complete for linear A where every event occurs at most three times. Here, we show that these problems remain hard even if every event occurs at most twice. Nevertheless, this has to be paid by relaxing the restriction on state degree, allowing every state to have two successor and two predecessor states. As we also show that SSP becomes tractable for linear A where every event occurs at most twice the only open cases left are ESSP and feasibilty for the same input restriction.