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Sufficient conditions for loop resource subsets to derive strict minimal siphons in class of Petri nets Miao Liu, ShouGuang Wang and Zhiwu Li As a structural object of Petri nets, the importance of strict minimal siphons (SMSs) is well recognised in the analysis and control of deadlocks for flexible manufacturing systems. For a class of Petri nets called systems of simple sequential processes with resources (S3PRs), the concepts of critical resource places and their related multi-way holder places are firstly proposed. Next, by analysing the structural properties of the critical resource places and their related loop resource subsets, sufficient conditions for loop resource subsets to derive SMSs are established. Finally, based on the proposed results, all SMSs can be obtained from their related loop resource subsets in an S3PR net.

Introduction: As a structural object of Petri nets, siphons play an important role in the development of deadlock control policies for flexible manufacturing systems (FMSs) [1, 2]. A siphon is a set of places such that their input transition set is included in their output transition set. It is minimal if it does not contain any others as its proper subset. If a siphon is empty, its output transitions become permanently disabled, causing a partial or total system deadlock. Thus, deadlock-freedom and the liveness of a Petri net are closely related to its siphons. Siphon computation has been receiving much attention. However, the number of siphons grows exponentially with the net size and thus the problem of computing all minimal siphons in a net cannot be solved by an algorithm with polynomial complexity. Many efforts have been made to solve the efficiency problem of extracting minimal siphons for ordinary Petri nets. In [3], Li and Zhou first utilise resource circuits to compute strict minimal siphons (SMSs). In their approach, for each resource circuit in the net, they compute its related SMS. By fully utilising the structural information in a net, Li and Zhou [4] proposed a method to compute a set of elementary siphons in systems of simple sequential processes with resource (S3PR) based on resource circuits. Compared with their previous work [3], the method in [4] avoids a complete SMS enumeration, and hence improves the computational efficiency. In [2], a graph-based technique is used to find all elementary resource transition circuits (RTCs) structures. Then, an iterative method is developed to recursively construct all maximal perfect RTCs from the elementary ones. Finally, a one-to-one correspondence between SMSs and maximal perfect RTCs is formulated. A recent effective and computationally efficient SMS computation approach is proposed by Wang et al. [5]. They propose the concept of characteristic resource subnets, based on which, an approach to compute all SMSs is developed. However, the method in [5] needs to generate all the characteristic resource subnets and then to decide the strong connectivity of each characteristic resource subnet, which is a tedious process. This study makes the following contributions. The concept of a critical resource place is proposed, which is important in deciding whether a simple loop resource subset can derive an SMS. Then, sufficient conditions for the simple loop resource subsets to derive SMSs are established, which facilitate deciding whether a composed loop resource subset can generate an SMS. Finally, based on the proposed method, all SMSs can be obtained in an S3PR net. More detailed preliminaries can be found in [1] for Petri nets, S3PRs and SMSs. Sufficient conditions for loop resource subsets to derive SMSs: Without loss of generality, in what follows, N = (PA ∪ P 0 ∪ PR, T, F ) denotes an S3PR net with an acceptable initial marking, where PA, P 0 and PR are the sets of the operation, process idle and resource places, respectively. For r ∈ PR, H(r) = ††r ∩ PA, the operation places that use r, are called the set of holders of r. We use Ω to denote a set of resource places in N, where Ω = {r1, r2, …, rm} ⊆ PR (m ≥ 2). Note that in an S3PR, an SMS contains at least two resource places [6]. We use At and tA to denote the sets of the input and the output operation places of t, respectively, and use Rt and tR to denote the sets of the input and the output resource places of t, respectively. Let Tx ⊆ T be a subset of T. ATx and TxA denote the sets of the input and the output operation places of Tx, respectively, and R Tx and TxR denote the sets of the input and the output resource places of Tx, respectively.

Let {r1, r2, …, rm} ⊆ PR (m ≥ 2) be a set of resources in N. An elementary circuit C = r1t1r2t2…rmtmr1 is called a resource circuit C in N. We use CR = {r1, r2, …, rm} to denote the set of resource places in a resource circuit C in N. Definition 1: Let C = {C1, C2 ,…, Cn} (n ≥ 1) be the set of all resource circuits in N. ∀i ∈ {1, 2, …, n}, CiR is called a simple loop resource subset if ∃| j , k ∈ {1, 2, …, n}{i( j ≠ k)} such that CiR = CjR < CkR . The set of all simple loop resource subsets in N is denoted by Θ. It is obvious that one simple loop resource subset may correspond to more than one resource circuit, and hence the number of siphons that correspond to simple loop resource subsets is less than that of resource circuits. As a consequence, we study the relationship between SMSs and loop resource subsets, but not resource circuits. 0 , TV0 , FV0 ) is called a resource subnet Definition 2: NV0 = (PV
 0of Ω if (i) 0 0 † = V; (ii) T = V > V† ; and (iii) FV0 = F > PV × TV0 < P V  V0  0 . TV × PV

Theorem 1: Given an S3PR net N = (P0 ∪ PA ∪ PR, T, F), let Ω = {r1, r2, …, rm}⊆ PR(m ≥ 2) be a loop resource subset. The resource subnet 0 , TV0 , FV0 derived by Ω is strongly connected. NV0 = PV Proof: ∀r, r′ ∈ Ω, (r † ∩ †r′) ⊆ (†Ω ∩ Ω†) and (†r ∩ r′ †) ⊆ (†Ω ∩ Ω†) are necessarily true. That is to say, for each resource place contained in Ω, there always exist its input and output transitions belonging to TV0 and its input (output) transitions are the output (input) transitions for another resource place in Ω. Therefore the resource subnet NV0 is strongly connected. Definition 3: t ∈ †Ω ∩ Ω† is called a Δ-transition with respect to Ω if ∃t′ ∈ †Ω\Ω† such that At ∩ At′ ≠ Ø. The set of all the Δ-transitions related to Ω is denoted by TVD . Definition 4: NΩ = (PΩ, TΩ, FΩ) is called a characteristic resource subnet of Ω if (i) PΩ = Ω; (ii) TV = († V > V† )\TVD ; and (iii) FΩ = F ∩ [(PΩ × TΩ) ∪ (TΩ × PΩ)]. Theorem 2: [5] SΩ = Ω ∪ A(†Ω\Ω†) is a siphon in an S3PR net N = (P0 ∪ PA ∪ PR, T, F), where Ω is a loop resource subset in N. Theorem 3: [5] SΩ is an SMS if Ω is a loop resource subset and its characteristic resource subnet NΩ is strongly connected. This study avoids generating characteristic resource subnets and deciding their strong connectivity. We take full advantage of the structural property of the loop resource subsets and then find sufficient conditions for the loop resource subsets to derive their corresponding SMSs. Definition 5: Let N = (P 0 ∪ PA ∪ PR, T, F ) be an S3PR. r ∈ PR with H(r) = (††r) ∩ PA is called a critical resource place if ∃p ∈ H(r), |p †| ≥ 2. In this case, p is called a multi-way holder place of the critical resource place r. The sets of all critical resource places and their related multiway holder places in N are denoted by PRs and Hs(r), respectively. Theorem 4: Let Ω = {r1, r2, …, rm}⊆PR(m ≥ 2) be a simple loop W PA PR , T , F). SΩ is an resource subset in an S3PR net N = (P SMS if V > PRs = ∅. Proof: Since V > PRs = ∅, ∀r [ PRs and A(†Ω\Ω†) ∩ Hs(r) = Ø. Hence, ∀ ∈ †Ω ∩ Ω†, ∃| t ′ [† V\V† : At ∩ At′ ≠ Ø, i.e. TVD = ∅. Then, by Definitions 2 and 4, NV = NV0 is derived. Based on Theorems 1 and 3, NΩ is strongly connected and SΩ is an SMS. □ Let N = (P 0 ∪ PA ∪ PR, T, F ) be an S3PR net with r′, r ∈ PR. r′ is called an input resource place of r if r′ ∈ R(†r), where R(†r) is the set of all input resource places of the resource place r. Theorem 5: Let Ω = {r1, r2, …, rm} ⊆ PR(m ≥ 2) be a simple loop resource subset in an S3PR net N = (P0 ∪ PA ∪ PR, T, F). SΩ is an SMS if ∀r [ V > PRs , ∀r′ ∈ Ω ∩ R(†r) and ∀p ∈ A(r′† ∩ †r) such that | p †| = |R(p †) ∩ Ω|.

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Proof: ∀r [ V > PRs , ∀r′ ∈ Ω ∩ R(†r) and ∀p ∈ A(r′ † ∩ †r), if |p †| = |R( p †) ∩ Ω| holds, it follows immediately that ∀t ∈ †Ω ∩ Ω†, ∃| t ′ [† V\V† such that At ∩ At′ ≠ Ø, i.e. TVD = ∅. By Definitions 2 and 4, NV = NV0 is true. By Theorems 1 and 3, NΩ is strongly connected and SΩ is an SMS. □ Theorem 6: Let Ω = {r1, r2, …, rm} ⊆ PR(m ≥ 2) be a simple loop resource subset in an S3PR net N = (P0 ∪ PA ∪ PR, T, F). SΩ is not an SMS if ∀r [ V > PRs , ∀r′ ∈ Ω ∩ R(†r) and ∀p ∈ A(r′† ∩ †r) such that | p †| > |R(p †) ∩ Ω|. Proof: ∀r [ V > PRs , ∀r′ ∈ Ω ∩ R(†r) and ∀p ∈ A(r′ † ∩ †r), if |p †| > |R( p †) ∩ Ω| holds, we conclude that ∀t ∈ †Ω ∩ Ω†, ∃t′ ∈ †Ω\Ω† such that At ∩ At′ ≠ Ø, i.e. TVD = ∅ and TVD $† r. That is to say, r in NΩ has no input transition after we delete the transitions (†r) contained in TVD . Therefore, NΩ is not strongly connected and SΩ is not an SMS. □ Theorem 7: Let Ω = {r1, r2, …, rm} ⊆ PR(m ≥ 2) be a simple loop resource subset in an S3PR net N = (P0 ∪ PA ∪ PR, T, F). SΩ is an SMS if ∀r [ V > PRs , ∀r′ ∈ Ω ∩ R(†r) and ∃p, p′ ∈ A(r′† ∩ †r) such that |p †| > |R(p †) ∩ Ω| and |p′†| = |R(p′†) ∩ Ω|. Proof: ∀r [ V > PRs , ∀r′ ∈ Ω ∩ R(†r), if ∃p, p′ ∈ A(r′ † ∩ †r) such that |p †| > |R( p †) ∩ Ω| and |p′ †| = |R( p′ †) ∩ Ω| hold, TVD = ∅ is obtained. However, by |p′ †| = |R( p′ †) ∩ Ω|, we conclude † r\TVD > TV = ∅. That is to say, even if the input transitions, i.e. p † ∩ †r, of r have been deleted, there still exist other input transitions, i.e. p′ † ∩ †r, of r. Therefore, NΩ is still strongly connected and SΩ is an SMS. □ As described above, we have obtained the relationship between a simple loop resource subset and its related SMS. In the following discussion, based on the previous results, sufficient conditions for all the loop resource subsets to generate SMSs are presented. Definition 6: Let Ωi and Ωj be the resource sunsets in N. Ωi and Ωj are said to be composable if Ωi ∩ Ωj ≠ Ø, Vi  Vj and Vi  Vj . The composition operation ○ on the two resource subsets Ωi and Ωj is defined as follows: (i) Ωi ○ Ωj = Ωi ∪ Ωj if Ωi and Ωj are composable and (ii) Ωi ○ Ωj = Ø if Ωi and Ωj are not composable. In what follows, we use Ωi,j to denote Ωi ○ Ωj if Ωi and Ωj are composable. Trivially, Ωi ○ Ωi = Ø. Definition 7: A resource subset Ω is called a composed loop resource subset if ∃Ω1, Ω2, …, Ωn ∈ Θ such that Ω = Ω1○Ω2○…○Ωn = Ω1 ∪ Ω2 ∪ … ∪ Ωn (n ≥ 2). The set of all composed loop resource subsets in N is denoted by Ξ. Q is used to denote the set of all loop resource subsets in N where Q = Q < J. In principle, Q can be found by using the loop resource subsets composition approach in an iterative way. In the remaining discussion, we assume that Q1 = Q, Q2 = Q1 WQ1 , Q3 = Q1 WQ2 , . . . , Qi−1 = Q1 W Qi−2 and Qi = Q1 W Qi−1 . The composed loop resource subsets Ξ can be obtained by using the simple loop resource subsets composition approach. Since the set of all the loop resource subsets Q in N is composed by Θ and Ξ, by this method, we can obtain all the loop resource sunsets. Based on the previous sufficient conditions for the simple loop resource subsets, it facilitates us to derive the following results. Corollary 1: Let SVi and SVj be two SMSs in an S3PR net N = (P0 ∪ PA ∪ PR, T, F), where Vi , Vj [ Q. If Ωi,j = ΩiWΩj ≠ Ø, then SVi, j is an SMS. Corollary 2: Let N = (P0 ∪ PA ∪ PR, T, F) be an S3PR net. If ∀Vi , Vj [ Q, Ωi,j = ΩiWΩj ≠ Ø and Vi, j > PRs = ∅, then SVi, j is an SMS. Corollary 3: Let N = (P0 ∪ PA ∪ PR, T, F) be an S3PR net. Assume that ∀Ωi ∈ Θ, ∀Vj [ Q, Ωi,j = Ωi W Ωj ≠ Ø, then SVi, j is an SMS if

∀r [ Vi, j > PRs , ∀r′ ∈ Ωi,j ∩ R(†r) and ∀p ∈ A(r′† ∩ †r) such that |p †| = |R(p †) ∩ Ωi,j|. Corollary 4: Let N = (P0 ∪ PA ∪ PR, T, F) be an S3PR net. Assume that ∀Ωi ∈ Θ, ∀Vj [ Q, Ωi,j = Ωi W Ωj ≠ Ø and Vi, j > PRs = ∅, then SVi, j is not an SMS if ∀r [ Vi, j > PRs , ∀r′ ∈ Ωi,j ∩ R(†r) and ∀ ∈ A(r′† ∩ †r) such that |p †| > |R(p †) ∩ Ωi,j|. Conclusion: This work firstly proposes the concepts of critical resource places and their related multi-way holder places. Next, by analysing the structural properties of the critical resource places and their related loop resource subsets, sufficient conditions for loop resource subsets to generate SMSs are established. The proposed method avoids finding the characteristic resource subnet for each loop resource subset and deciding the strong connectivity for each characteristic resource subnet, which can help one decide directly whether a loop resource subset can derive an SMS and then compute all SMSs in an S3PR net. The results also reveal the structural properties of the loop resource subsets, which can further help industrial practitioners in developing proper loop resource subset-based deadlock control methods for FMSs. Our future work is to extend this method to more general classes of Petri nets and finding sufficient and necessary conditions for the loop resource subsets corresponding to SMSs. Acknowledgments: This work was supported in part by the National Natural Science Foundation of China under grant nos 61374068 and 61100056, the Fundamental Research Funds for the Central Universities under grant no. 72103326, the Zhejiang Provincial Natural Science Foundation of China under grant no. LY12F03020, the Zhejiang Provincial Education Department Foundation under grant no. Y201018216 and the Opening Project of the Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Southeast University, Nanjing, under grant no. MCCSE2012A05. © The Institution of Engineering and Technology 2014 17 September 2013 doi: 10.1049/el.2013.3095 One or more of the Figures in this Letter are available in colour online. Miao Liu and Zhiwu Li (Xidian University, Xi’an, People’s Republic of China) E-mail: [email protected] ShouGuang Wang (Zhejiang Gongshang University, Hangzhou, People’s Republic of China) References 1 Li, Z.W., and Zhou, M.C.: ‘Deadlock resolution in automated manufacturing systems: A novel Petri net approach’ (Springer, Berlin, 2009) 2 Xing, K.Y., Zhou, M.C., Wang, F., Liu, H.X., and Tian, F.: ‘Resource transition circuits and siphons for deadlock control of automated manufacturing systems’, IEEE Trans. Syst. Man Cybern. A, 2011, 41, pp. 74–84 3 Li, Z.W., Zhou, M.C., and Jeng, M.D.: ‘A maximally permissive deadlock prevention policy for FMS based on Petri net siphon control and the theory of regions’, IEEE Trans. Autom. Sci. Eng., 2008, 5, pp. 182–188 4 Li, Z.W., and Zhou, M.C.: ‘On siphon computation for deadlock control in a class of Petri nets’, IEEE Trans. Syst., Man Cybern. A, 2008, 38, pp. 667–679 5 Wang, S.G., Wang, C.Y., Zhou, M.C., and Li, Z.W.: ‘A method to compute strict minimal siphons in a class of Petri nets based on loop resource subsets’, IEEE Trans. Syst. Man Cybern. A, 2012, 42, pp. 226–237 6 Ezpeleta, J., Colom, J.M., and Martinez, J.A.: ‘Petri net based deadlock prevention policy for flexible manufacturing systems’, IEEE Trans. Robot. Autom., 1995, 11, pp. 173–184

ELECTRONICS LETTERS 2nd January 2014 Vol. 50 No. 1 pp. 25–27