This study proposes a new approach that recovers the system from deadlock states to its former live states, and reaches the same number of states as the original uncontrolled model by adding monitors (and control arcs) with no new problematic siphons. We further propose a lossless approach by coloring some arcs to avoid the material loss.

It has been a hot research topic to compare the effectiveness of new control policies by testing against a wellknown S 3 PR model. So far, only the control policy by Piroddi et al. may reach the optimal number of states among all approaches for a well-known benchmark using a siphon-based approach. The resulting model is a generalized Petri net since some control arcs are weighted, which complicates the system. The only improvement that can be made is to reduce the number of control arcs (by 3), and the number of weighted control arcs (by 9) as we report in this paper. Also the token count is reduced. This is achieved by replacing two monitors with weighted arcs by two new monitors without weighted arcs. INA (Integrated Net Analyzer) analysis indicates that the resulting controlled model is live and reaches the same 21581 states by Piroddi et al. We develop a formal theory for explaining the cause of state losses and providing the foundation for the above improvement model.

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Unlike other techniques, Li and Zhou add control nodes and arcs for only elementary siphons greatly reducing the number of control nodes and arcs (implemented by costly hardware of I/O devices and memory) required for deadlock control in Petri net supervisors. Li and Zhou propose that the number of elementary siphons is linear to the size of the net. An elementary siphon can be synthesized from a resource circuit consisting of a set of connected segments. We show that the total number of elementary siphons, |O E |, is upper bounded by the total number of resource places |P R | lower than that min(|P|, |T|) by Li and Zhou where |P| (|T|) is the number of places (transitions) in the net. Also, we claim that the number of elementary siphons |O E | equals that of independent segments (simple paths) in the resource subnet of an S 3 PR (systems of simple sequential processes with resources). Resource circuits for the elementary siphons can be traced out based on a graph-traversal algorithm. During the traversal process, we can also identify independent segments (i.e. their characteristic T-vectors are independent) along with those segments for elementary siphons. This offers us an alternative and yet deeper understanding of the computation of elementary siphons. Also, it allows us to adapt the algorithm to compute elementary siphons in [2] for a subclass of S 3 PR (called S 4 PR) to more complicated S 3 PR that contains weakly dependent siphons.