Computation of Minimal Siphons in Petri Nets Using Problem Partitioning Approaches Dan You, Member, IEEE, Oussama Karoui, Associate Member, IEEE, and Shouguang Wang, Senior Member, IEEE Abstract—A large amount of research has shown the vitality of siphon enumeration in the analysis and control of deadlocks in various resource-allocation systems modeled by Petri nets (PNs). In this paper, we propose an algorithm for the enumeration of minimal siphons in PN based on problem decomposition. The proposed algorithm is an improved version of the global partitioning minimal-siphon enumeration (GPMSE) proposed by Cordone et al. (2005) in IEEE Transactions on Systems, Man, and Cybernetics—Part A: Systems and Humans, which is widely used in the literature to compute minimal siphons. The experimental results show that the proposed algorithm consumes lower computational time and memory compared with GPMSE, which becomes more evident when the size of the handled net grows. Index Terms—Petri nets (PNs), problem decomposition, resource- allocation systems, siphons.    I. Introduction P ETRI nets (PNs) [1]–[4] are a well-recognized mathe- matical tool suitable for the modeling, analysis, and control of resource-allocation systems [5], [6]. The analysis of PNs may be performed in different ways such as structural analysis and reachability analysis. A lot of effort has been devoted to structural analysis due to its advantage of low computational cost. Siphons [7], which are particular sets of places in a PN, are often utilized to detect the liveness of a considered net. More specifically, the liveness of a PN is closely related to whether siphons are sufficiently marked [8]–[14]. For ordinary PNs, the occurrence of deadlocks is due to the emptiness of some siphon, which leads to some transitions disabled permanently [15]–[17]. A seminal work to prevent deadlocks in automated manufacturing systems was established by Ezpeleta et al. [18] for a class of ordinary PNs called systems of simple sequential processes with resources (S 3 PR). In more detail, they add a monitor (i.e., a control place) for each unmarked strict minimal siphon such that all output arcs of the monitor are linked with source transitions of the net. When all siphons are controlled, the augmented net is live. Note that, in this method, complete siphon enumeration is required since a monitor is added for each siphon in the net. As we know, the number of siphons grows exponentially with respect to the net size [19], [20], the method thus suffers from high computa- tional complexity. Indeed, deadlock resolution policies based on siphon control all require the complete or partial siphon enumeration [21], [22]. To improve the computational efficiency of such methods, it is of great importance to improve the efficiency of siphon computation.    A. Literature Review There are a variety of methods in the literature that compute siphons. These methods can be classified into two categories in terms of application scope. One applies to special classes of PNs. For instance, methods based on parallel algorithms [23], resource circuits [24], [25], pruning-graphs [26], genetic algorithms [27], [28] and loop resource subsets [12]. The other applies to arbitrary classes of PNs such as linear integer programming methods [16], [29], integrated net analyzer (INA)-based methods [30], methods based on semi-tensor product of matrices [31] and problem decomposition [32]. Methods in the first category are basically much more efficient in computation than those in the second category, whereas methods in the second category have the advantage of having no restriction on the class of PN. This work focuses on siphon computation methods that are applicable to arbitrary classes of PNs. To our best knowledge, among all the methods in the literature applicable to arbitrary classes of PNs, the methods proposed by Cordone et al. [32] are the most efficient ones. Their methods are based on the idea of problem decomposition. To put it simply, the idea is that a problem is decomposed into multiple sub-problems so that the solution of the original problem can be deduced by combining the solutions of all the sub-problems. Cordone et al. [32] developed two siphon enumeration approaches based on the technique of problem decomposition, namely, global partitioning minimal-siphon enumeration (GPMSE) and local partitioning minimal-siphon enumeration (LPMSE). They have been used extensively in the literature due to their low computational complexity. The main difference between them is that GPMSE allows a current problem to be decomposed using a siphon that is already found before, Manuscript received April 16, 2021; revised May 28, 2021; accepted June 22, 2021. This is an extended version of our previous paper that was presented in IEEE Conference on Decision and Control. Compared with the conference paper, the main differences of this paper lie in: Formal proofs of all the theoretical results, complexity analysis of the proposed method, and more detailed explanations of the proposed method. This work was supported in part by the Zhejiang Natural Science Foundation (LQ20F020009), the Zhejiang Provincial Key Laboratory of New Network Standards and Technologies (2013E10012), and the Public Technology Research Plan of Zhejiang Province (LGJ21F030001). Recommended by Associate Editor Jun Zhang. (Corresponding author: Oussama Karoui.) Citation: D. You, O. Karoui, and S. G. Wang, “Computation of minimal siphons in Petri nets using problem partitioning approaches,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 2, pp. 329–338, Feb. 2022. The authors are all with the School of Information and Electronic Engineering (Sussex Artificial Intelligence Institute), Zhejiang Gongshang University, Hangzhou 310018, China (e-mail: youdan000@hotmail.com; rtkaroui@gmail.com; wsg5000@hotmail.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JAS.2021.1004326 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 9, NO. 2, FEBRUARY 2022 329