When designing liveness-enforcing Petri net supervisors, unlike other techniques, Li et al. added control places and arcs to a plant net model for its elementary siphons only, greatly reducing the structural complexity of the controlled system. Their method, however, suffers from the expensive computation of siphons. We propose a new T-characteristic vector z to compute strict minimal siphons (SMS) for S 3 PR (systems of simple sequential processes with resources) in an algebraic fashion. For a special subclass of S 3 PR, called S 4 PR (simple S 3 PR), we discover that elementary siphons can be constructed from elementary circuits where all places are resources. Thus, the set of elementary siphons can be computed without the knowledge of all SMS. We also propose to construct characteristic T-vectors h by building a graph to find dependent siphons without their computations.

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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.

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