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Proposesd is a controllability condition of a siphon that is composed of two elementary siphons in a class of Petri nets, Systems of Simple Sequential Processes with Resources (S 3 PR). It is shown that, under the condition, an S 3 PR admits a maximally permissive liveness-enforcing supervisor that is expressed by a set of monitors if every dependent siphon is 2-composed, which can be decided by an algorithm with polynomial complexity.Introduction: Petri nets [1] are a popular and powerful mathematical tool for handling deadlock problems in flexible manufacturing systems (FMSs) [2]. The methods derived from them to deal with deadlocks either preclude the possibility of deadlock occurrences by preventing some necessary condition or detect and resolve a deadlock when it occurs. They are classified into three strategies: deadlock detection and recovery, deadlock avoidance, and deadlock prevention [3].A deadlock detection and recovery approach permits the occurrence of deadlocks. When a deadlock occurs, it is detected and then the system is put back to a deadlock-free state, by simply reallocating the resources. Deadlock prevention usually uses an off-line computational mechanism to control the request for resources to ensure that deadlocks never occur. The goal of a deadlock prevention approach is to impose constraints on a system to prevent it from reaching deadlock states, which can be achieved by designing a liveness-enforcing supervisor expressed by a set of monitors such that deadlocks are completely eliminated and the resulting net is live. This implies that both a plant model and its supervisor are in a Petri net formalism.Behavioural permissiveness is one of the most important criteria in evaluating the performance of a supervisor. In supervisory control theory of discrete event systems in the framework of formal languages and finite state automata, the existence and synthesis of an optimal, i.e. maximally permissive, supervisor are well addressed. When the control specification languages are controllable, the resulting automaton serves as a maximally permissive supervisor by properly trimming the automaton that represents the behaviour of a plant. The computational complexity depends on the complete state enumeration.By using the theory of regions, an optimal liveness-enforcing Petri net supervisor synthesis method is developed in [4] and [5]. Its advantage is that such a supervisor can be always obtained if it exists. However, the synthesis methods in [4] and [5] need the complete state enumeration of a plant net model, whose computation is expensive or impossible when dealing with either a large-sized net model, or a small-sized one with a large initial marking. It is shown in [5] that, in general, the existence of a maximally permissive liveness-enforcing Petri net supervisor, M P for short, can be determined by first generating the reachability graph, and then solving a linear programming problem (LPP) for each marking/transition separation instance. Unfortunately, the number of LPPs that one needs to s...

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