## 1 Introduction  
Since flexible manufacturing systems (FMSs) use smart robots to achieve their production capacity under the condition of a high degree of resources sharing, deadlock states usually appear unexpectedly. For solving the deadlock problems of FMSs, many pioneers have focused on this issue [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16]. Generally, three kinds of technologies are proposed to solve the deadlock problems of FMSs: deadlock avoidance, deadlock detection and recovery, and deadlock prevention [7]. Firstly, deadlock avoidance is adopted to prevent an FMS from reaching any deadlock state. Although it has a higher efficiency, it always fails to eliminate all deadlock states. The deadlock detection and recovery approach allows existing deadlock states to be reachable. Once deadlock occurs, the system detects and reallocates resources for recovery. Finally, deadlock prevention is proposed to prevent a deadlock situation from being reached. It is important to note that deadlock prevention does not need any run-time cost, since it solves the deadlock of FMSs in the design and planning stages. In the deadlock prevention domain, two main analysis methods are designed to solve FMS deadlock problems: structural and reachability graph analysis, respectively. The former utilizes some structural items, such as all kinds of siphons [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. Some generated siphons are found in uncontrolled models, which may become unmarked or partially marked. A siphon denotes a general set of places. In the case of a deadlock, the siphon cannot regain any new marking from the original ones, i.e., all transitions in this model are disabled, and a deadlock is recognized. To prevent a deadlock state from being reached, no places in these siphons may be empty. The objective is reached by designing and adding a control subnet to the model with its initial marking. [PERSON] et al. [13] develop a formalized and iterative approach with two main stages. These two stages are required in each iteration in order to create new siphons after attaching them to the FMS model. [PERSON] and [PERSON] [18] propose a method with a much smaller control subnet than in previous works, with the same results. [PERSON] et al. [14] introduce an iterative method, adopting two kinds of control places: ordinary and weighted control places. They attempt to prevent siphons from losing their tokens and guarantee more permissive systems. However, structural analysis seems suboptimal and leads to low utilization rates of system resources, although it does lower computational cost. Reachability graph analysis [28, 29, 30, 31, 32, 33, 34, 35, 36, 37] requires the enumeration of all generated markings. The set of these markings is hence referred to as the reachability graph (RG). The RG includes all reachable markings, which can be classified as one of two parts: the live zone (LZ) and deadlock zone (DZ). The DZ contains deadlock markings and critical bad markings, which inevitably lead to deadlock zones. The LZ contains all live markings and is also recognized as a deadlock-free-zone (DFZ). The objective is to forbid all markings in the DZ and to make sure all markings in the LZ are still reachable. According to the characteristics of RG and system complexity, marking explosion problems may occur during the computation of reachability graph analysis, implying that the increase of FMS model size leads to a corresponding increase in computational cost. [PERSON] [28] develops a deadlock prevention policy based on a theory of regions that proves maximal liveness performance. However, the policy fails to determine all sets of event-state separation problems (ESSPs), and its application seems limited to certain special nets only. Therefore, some works [34, 35, 36, 37] aim to develop a more computationally efficient optimal deadlock control policy based on the theory of regions. [PERSON] and [PERSON] [5] propose a reduction technique to simplify large FMS models.