Abstract
A Petri net is a mathematical method that can be used to represent and analyze discrete event systems. Although research on structural liveness and safety in ordinary free-choice (FC) nets has been reported, analysis methods for weighted Petri nets have not yet been developed. In this study, we propose a method for determining the structural liveness and safety of strongly connected FC nets. The flow rate of tokens for strongly connected marked graphs is defined as the \(circuit\ flow\ value\). In addition, the circuit flow value of a strongly connected FC net is obtained by calculating the superposition of the circuit flow value.
Keywords
1 Introduction
A Petri net is a mathematical method that can be used to represent and analyze discrete event systems [6,7,8]. The properties of Petri nets can be classified into two categories as follows: dynamic properties, which characterize changes in the dynamic behavior of systems, and structural properties, which depend on the structure of the Petri nets. Structural properties include liveness, boundedness, and other properties that can be determined by the calculating an incidence matrix of Petri nets. These conditions can be further examined using linear algebra techniques [1, 5, 6]. Structural analysis cannot be used to analyze the distribution of tokens or the increase or decrease in the number of tokens because the state dynamically changes when tokens are moved between various places by firing. However, examining structural liveness of weighted Petri nets can be performed by behavioral verification. Behavioral verification such as generating state spaces is expensive, however, because it generates the entire state space from the initial marking. Therefore, a flow net has been proposed to view both the structural properties as a graph and the behavior of the tokens [4]. Flow nets are graphs that are converted into weighted directed graphs, with the exception of Petri net transitions.
Workflow modeling and analysis methods for managing tasks and data flow have been proposed as applications of Petri nets [11]. Soundness is a criterion of logical correctness in workflow Petri nets, and marked graph (MG) workflow nets that do not include directed circuits are mathematically guaranteed to be sound [2]. A short-circuit net has been proposed that constitutes a circuit by connecting the start point and end point in a workflow net with a single transition [10]. The soundness decision problem in short-circuit nets can result in liveness and boundedness decision problems [10]. Although workflow nets allow weighted arcs, several conventional studies have targeted nets with input/output conditions or ordinary Petri nets due to the complexity of the analysis [11].
In this study, we propose a weighted net liveness/safety decision method that involves a token flow calculation algorithm using flow net (FN) transformation. The number of input/output locations, which is the configuration requirement of the workflow net, is not one. The token flow of free-choice (FC) net circuits is obtained by calculating the token flow from the start point to the end point of MG circuits and superimposing the token flow for each structure of the MG circuits. This analysis method can be applied to both workflow nets and short-circuit nets.
Figure 1 presents an outline of the proposed method. The net in the Fig. 1 represents a well-formed flow FC net that is covered by two strongly connected MG components, (a) and (b). The net in Fig. 1(a) can be decomposed into two components of the MG circuit to obtain strongly connected MG components in Fig. 1(b) and (c). By applying a flow net transformation to each of the strongly connected MG components in Fig. 1(b) and (c), the circuit flow net value can be calculated by the calculation algorithm. The entire net does not satisfy liveness/boundedness if each circuit does not satisfy liveness/boundedness. The circuit superimposition method does not involve compositing the union as a subgraph structure, but rather, synthesizing the flow values of each circuit. The flow value of the entire net is calculated by superimposing the circuit flow values of the subgraph layer.
This paper is organized as follows. Section 2 introduces a Petri net, while Sect. 3 introduces a flow net. Section 4 presents the calculation of the circuit flow value in an MG circuit, while Sect. 5 presents an analysis of liveness and boundedness by superimposing circuit flow values. Finally, Sect. 6 presents the conclusions of this study.
2 Place/Transition (P/T) Net
The original definitions of Petri nets can be found in previous studies [6,7,8]. A Petri net is denoted \(N =(P\), \(T\), \(F\), \(W\), \(M_0)\), where \(P\) is a finite set of places, \(T\) is a finite set of transitions with \(P \cap T = \phi \), \(F\subseteq (P \times T) \cup (T \times P)\) is a finite set of arcs, \(W\) is the function \(W : F \rightarrow \mathbb {N}\) specifying the arc weights, and \(M_0\) is the initial marking (i.e., a mapping \(M: P \rightarrow \mathbb {N}\), indicating the number of tokens in each place).
Let \(a \in P \cup T\), \(\bullet a = \{b \in P \cup T \mid (b, a) \in F \}\) be the pre-set and \(a\bullet = \{b \in P \cup T \mid (a, b) \in F \}\) be the post-set. Let \(t\in T\), \(t\) be the source transition if \(\bullet t = \phi \), and \(t\) be the sink transition if \(t \bullet = \phi \). A transition \(t \in T\) is said to be firable when it satisfies \(\forall p \in \bullet t:M(p)\ge W(p,t)\). Here, \(t\) is firable from marking \(M\) to \(M^{\prime }\), denoted \(M [t> M^{\prime }\). A nite sequence \(\sigma =t_1 \ldots t_n\) is firable at a marking \(M\) if there are markings \(M_1 , M_2 , \ldots , M_n\) such that \(M [t_1> M_1,\) \(M_1 [t_2> M_2,\) \(\ldots ,\) \(M_{n-1} [t_n> M_n\). Here, \(\sigma \) is called a firing sequence. \(M_n\) is reachable from \(M\) when there is a firing sequence that has a marking \(M\) to a marking \(M_n\). We represent the set of all firing sequences from \(M_0\) as \(L (N, M_0)\) or \(L(M_0)\). \(A = \{a_{ij} \}\) is an \(m \times n\) matrix, and \(a_{ij}= a_{ij}^{+} - a_{ij}^{-}\), where \(a_{ij}^{+} = W(t_i, p_j)\), \(a_{ij}^{-} = W(p_j, t_i)\).
Let \(N = (P , T, F, W )\) be a net. A transition \(t\in T\) of \(N\) is live if for every marking \(M\) that is reachable from \(M_0\), there exists a marking \(M^{\prime }\) that be enabled to fire \(t\) and is reachable from \(M\). Here, \(M_0\) is called the live initial marking in \(N\). We use structural liveness to indicate that there exists a live initial marking in \(N\). For some \(k \in \mathbb {N}\), \(\forall M \in L(M_0), p \in P: M (p) \le k\), \(N\) is \(k\)-bounded or bounded. A net structure \(N\) is considered structurally bounded if it is bounded for any finite initial marking \(M_0\). A Petri net is called \(conservative\) if \(Ay =0\) for some \(y >0\), and is called \(consistent\) if \(A^{T}x =0\) for some \(x >0\).
A classification based on conditions regarding the composition of Petri nets is called a subclass. A Petri net \(N = (P, T, F, W)\) is an MG if for all \(p \in P\), \(|\bullet p| =|p \bullet |= 1\). A Petri net \(N = (P, T, F, W)\) is a state machine (SM) if for all \(t \in T\), \(|\bullet t| =|t \bullet |= 1\). A Petri net \(N = (P, T, F, W)\) is a Free choice net (FC) if for all \(p \in P\), \(|p\bullet | = 1\) or \(\bullet \{ p\bullet \} = \{ p \}\).
\(t, t^{\prime }\) are said to be in structural conflict, where \(\bullet t \wedge \bullet t^{\prime } \ne \phi \). A coupled conflict relation is defined as the transitive closure of the structural conflict relation. The equivalence class of transition \(t\) is denoted CSS(\(t\)), and the quotient set is SCCS. The following theorem pertains to the structural properties [3, 8].
The target net in this study is a weighted FC net covered with strongly connected MG components. We define a well formed flow Petri net (WFFP) to allow flow superposition.
Definition 1
Let \(N = (P , T, F, W )\) be a net, and \(X \subseteq P \cup T\) be any strongly connected MG component in N. If an arbitrary path \(\alpha =\{x, y_1, y_2, \ldots , x^\prime \}\) connected to X is a TP- or PP- handle of X, N is called a WFFP.
Let \(A\) be the incidence matrix of \(N\). If \(N\) is structurally bounded and live, then \(N\) is consistent and conservative, and \(rank(A) = |SCCS|-1\). In an MG, there is no structural conflict; thus \(n = |T|=|SCCS|\) always holds. With respect to Petri nets, it is known that the rank of the incidence matrix of connected weighted MG nets is \(n-1\) or \(n\) [9]. It is also known that the rank of a neutral weighted MG is \(n-1\). Calculating the circuit flow value to apply our proposed algorithm is equivalent to analyzing the net as neutral.
3 Flow Net Definition and Flow Capacity
3.1 Flow Net
The structural properties of Petri nets are analyzed by solving algebraic equations and inequalities for incidence matrices. Transition firing generates and consumes tokens, and the total number of tokens in the (sub)net changes. We refer to the ratio of token increase and decrease due to transition firing as flow, and propose a flow net that defines the connection weight between nodes by flow [4]. The definition of a flow net \(FT\) is as follows.
Definition 2
A flow net is a 3-tuple, \(FT = (V,E, W_f)\), where \(V = \{ v_1 , v_2 , \cdots v_n \}\) is a finite set of nodes, \(n = |V|=|T|\), \(E\subseteq V \times V\) is a finite set of edges, and \(W_f\) is the function \(W_f : E \rightarrow \mathbb {Q}\). is a rational matrix of \(n \times n\), where each component is given by
Here, for arbitrary nodes \(v_{i}, v_{j} \in V\), .
If the FT transformation is applied to Fig. 2, the directed graph \(FT\) presented in Fig. 3 is obtained as well as the following adjacency matrix .
When the PN structure \(N\) satisfies conditions (i) and (ii), defined below, \(N\) is said to be convertible to graph \(FT\).
Definition 3
Let \(N = (P, T, F, W)\) be a PN structure. N is convertible to graph FT, which signifies that N satisfies conditions (i) and (ii) as follows:
-
(i)
For any \(p \in P\), \(\{\bullet p \} \ne \emptyset \) and \(\{p \bullet \} \ne \emptyset \)
-
(ii)
For different \(p_{1}, p_{2} \in P\) \( \{\bullet p_{1} \} \cap \{\bullet p_{2} \} \ne \emptyset \rightarrow \{p_{1} \bullet \} \cap \{p_{2} \bullet \} = \emptyset \)
We define a bijective function \(h: T \rightarrow V; h(t_i) \,{=}\, v_i \ (t_i \in T)\) that associates the index numbers of transitions with the index numbers of nodes of the flow net. In this paper, flow net nodes are associated with transitions even if \(v_i = h(t_i)\) and \(V_S = h(S) =\{v_i \mid v_i =h(t_i)\), \(\forall t_i \in S\), \(S\subseteq T\}\) are not written.
3.2 Flow Capacity
In a live circuit Petri net, it is important how the token of starting point \(p\) moves as input in the loop, after which it is output to \(p\). Here, we consider the capacity of an MG circuit in terms of the increase and decrease of tokens in the circuit.
Definition 4
Let \(N = (P, T, F, W)\) be a strongly connected MG net, and \(FT = (V,E, W_f)\) be a flow net obtained by converting N, where \(t_{ join} \in T\) is a merging transition. For any natural number \(k = 1,2,\cdots \), we denote the following function sequence \(f_k\) and node set \(S_k\) as the flow product for k in FT and the flow calculation set for k, respectively.
When there exists an upper bound \(\varphi \in \mathbb {R}\), \(\varphi \) is referred to as the flow capacity in flow graph FT. If \(k_1\) is the smallest k where \(S_k = S_1\), then the value of \(f_{k_1}(v) \ (v \in S_{k_1})\) is called the circuit flow value.
Here, the calculation rule of \(\omega \notin \mathbb {Q}\) is defined as \(\forall a \in \mathbb {Q}, \omega \times a = \omega , a\times \omega = \omega , \omega \times \omega = \omega \). The circuit flow value \(f_{k_i}\) contains information regarding whether the token output from the start node set is input to the start node via the a closed loop.
4 Circuit-Flow
4.1 Primitive Circuit Net
In simple flow nets, edges between nodes are considered to be the ratio of the number of tokens moved between the places of Petri nets. As an example, Fig. 4 consists of \(P \,{=}\, \{p_1\), \(p_2\), \(\cdots \), \(p_n \}\) and \(T \,{=}\, \{t_1\), \(t_2\), \(\cdots \), \(t_{n-1} \}\). Letting \(s_1\), \(s_2\), \(\cdots \), \(s_{n-1}\in \mathbb {N}\), there are \(T_1 \,{=}\, s_1 w(p_1,t_1)s_2 w(p_2, t_2)\cdots s_{n-1} w(p_{n-1}, t_{n-1})\) tokens in place \(p_1\) as the initial marking. When transition \(t_1\) fires \(s_1 s_2 w(p_2, t_2) \cdots s_{n-1} w(p_{n-1}, t_{n-1})\), \(s_1 w(t_1, p_2)s_2 w(p_2, t_2) \cdots s_{n-1} w(p_{n-1}, t_{n-1})\) tokens move to place \(p_2\). Similarly, transitions \(t_2 \cdots t_{n-1}\) are fired sequentially to cause the movement of tokens. Finally, there are \(T_n \,{=}\, s_1 w(t_1, p_2)\cdots s_{n-1}w(t_{n-1}, p_n)\) tokens in place \(p_n\). Comparing \(T_1\) and \(T_n\), the following can be written:
If \(p_1 = p_n\), then \(N\) is a simple circuit net. If
is satisfied, \(N\) is consistent because the total number of tokens does not change. \(T_1\) takes the minimum marking when \(s_1 = s_2 = \cdots s_ {n-1} = 1\). The component of (5) corresponds to the edge weight \(W_f (v_i, v_ {i + 1})\) from node \(v_i\) to \(v_ {i + 1}\) in the flow net. If \(t_ {n-1}\) is \(t_ {join}\), then \(S_0 = {p_1}\), and the circuit flow value in \(FT\) obtained by converting \(N\) satisfies \(1\).
Theorem 1
Let \(N = (P, T, F, W)\) be an MG simple circuit, and FT be a flow net obtained by converting N. Here, \(| P | = | T | = m\). If \(v_1 = v_{m + 1}\), then the following are equivalent:
-
(i)
\(\prod _{v_i, v_i\in V}^{} w_f(v_i, v_{i+1}) =1\ (i = 1, 2, \cdots , m)\).
-
(ii)
The circuit flow value of FT is 1.
-
(iii)
N is conservative and repetitive.
Proof
It is clear that \((\mathrm{i})\) and \((\mathrm{ii})\) are equivalent, because N is a simple circuit. Statements \((\mathrm{iii})\) and \((\mathrm{ii})\) can be proven by solving the determinant of the incidence matrix because N is conservative. To prove \((\mathrm{i})\)–\((\mathrm{iii})\), if \(\prod _ {v_i, v_j \in V}^{} w_f(v_i, v_{i + 1}) = 1\ (i = 1, 2, \cdots , m)\) is satisfied, then N is consistent. This has already been demonstrated. Here, we can demonstrate that the simple circuit N is conservative. Let \(A = A^+ - A^-\) be an incidence matrix of N. From the definition of conservative, we can demonstrate that there exists \(y> 0\) such that \(Ay = 0\). If \(f_k =W_f(v_k, v_{k+1}) = \frac{w_(t_k. p_{k+1})}{w(p_k ,t_k )}\) is satisfied, then
holds for all \(c \in \mathbb {N}\). If \(c \,{=}\, w(p_1, t_1) w(p_2,t_2) \cdots w(p_{n-1}, t_{n-1})\) is satisfied, then y consists of all integer solutions. Equation (2) can be proven inductively; however, it is omitted due to space limitations.
4.2 MG Circuits Excluding a Simple Circuit
Before introducing a general MG circuit, we discuss the relationship between a conditional net and flow product. First, we consider a net in which the number of merging transitions included in the MG circuit path is at most one. Let such a net be \(N_1 =(P_1, T_1, F_1, W_1)\). The number of degrees of freedom for the incidence matrix \(A_1\) of \(N_1\) matches the number of input places for the merging transition \(t_{last}\in T_1\). Let \(t \in T_1\), where \(t\) is not a merging transition. There is only one input place for \(t\) because the subclass of \(N_1\) is an MG and does not overlap with the input places for other transitions. Therefore, \(| P_1 | = |T_1| + | \bullet t_{last} | -1\) holds. If \(| P_1 | = m\), then the number of degrees of freedom of \(A_1\) is \(m- rank(A) = m-(n-1) = m-((| P_1 |-| \bullet t_{last}| +1) -1) = | \bullet t_{last} |\). This fact can be used to demonstrate the following.
Theorem 2
Let N be an MG circuit net N, where it is assumed that there is only one merging transition. The circuit flow value of the converted flow net N is 1 if and only if N is structurally live and bounded.
Theorem 3
Let N be an MG circuit net, and \(FT = (V, E, W_f)\) be a flow net converted from N, where the number of merging transitions is greater than one, and there is one merging transition that appears in all circuits in N. Assuming that the circuit flow is \(f= f_k (v) \ (v \in S_k) \) of FT, the following conditions are required for N to be conservative and consistent.
-
(i)
The following equation holds for any \({v_x, v_y {\in } \bullet v}\): \(j \,{<}\, k, v \,{\in }\, {S_j}^{\prime }, {|\bullet v|} \,{>}\,1\). Then, the following equation holds: \(f_{j-1}(v_x)w_f(v_x,v ) \,{=}\, f_{j-1}(v_y)w_f(v_y,v )\)
-
(ii)
\(f_k(v) = 1 , v\in S_k\)
When multiple merging transitions appear in all circuits in \(N\), the total product of the flow products for each subgraph can be expressed as follows.
Theorem 4
Let \(N = (P, T, F, W)\) be an MG circuit net, and \(FT = (V, E, W_f)\) be a net obtained by flow transformation of N, where \(T_{join} \subseteq T\) is the set of merging transitions that appear in all circuits, consisting of \(\{t_{{join}_1}\), \(\cdots \), \(t_{{join}_i}\), and \(\cdots \), \(t_{{join}_l} \}\ (i = 1\), 2, \(\cdots \), l). Given \(t_{{join}_i} \bullet = P_{{join}_i}\), N can be expressed as a union of the following subnets: \(P = \bigcup _{i=1}^{l} P_i\), \(T = \bigcup _{i=1}^{l} T_i\), \(\bullet T_i =\{ p \mid p \in \bullet t\), and \(t\in T_i\}\subset P_i\). Here, for j and k, if \(k = j + 1\), then \(P_k \cap P_j = P_{{join}_j}\), where \(P_{l + 1} = P_1\). Otherwise, \(P_j \cap P_k = \phi \). If f is a circuit flow value of FT, then \(f = \prod {f_i}_{k_i} (v)\) holds. If \(N_i = (P_i, T_i, F_i, W_i)\) is a partial net of N consisting of arc set \(F_i\) and weighted function \(W_i\) for \(P_i\) and \(T_i\), the function sequence \(f_i\) is a flow product of flow net \(FP_i\), where \({S_i}_{0}= P_{{join}_{i-1}},{S_i}_{k_i} = P_{{join}_i}\). Subscript \(k_i\) is the smallest integer for which the value of \({f_i}_{k_i}(v) \ (v \in {S_i}_{k_i})\) is updated from 0.
5 Flow Calculation
5.1 Synthesis of Circuit-Flow Value
In this section, we discuss nets that are strongly connected structures in FC subclasses. The token flow of an FC circuit is calculated by superimposing the token flow of an MG circuit. When the union of two MG circuits is constructed by the structure of the handle for each circuit, the calculation of the circuit flow value is defined for each type of handle. The TT-handle is not discussed because it is already included in the calculation of the circuit flow value.
5.2 PP Handle
Let two distinct strongly connected MG circuits be denoted \(c_1\) and \(c_2\), \(c_1 \cup c_2 = c_1 \cup h_1\) or \(c_2 \cup h_2\), where \(h_1\), \(h_2\) is the PP handle. Let \(f_1 , f_2\) represent \(c_1\), \(c_2\) of the circuit flow value, respectively. Using natural numbers \(a\) and \(b\), we can represent the circuit flow value of \(c_1 \cup c_2\) as \({f_1}^a + {f_2}^b\). This signifies that \(c_1\) and \(c_2\) token flows synchronously through the transition, which is the start point of the TP-handle. In addition, it signifies that the output transition of the place that is the starting point of the PP-handle is selective and that the tokens are selectively transferred \(a\) times on the \(c_1\) circuit or \(b\) times on the \(c_2\) circuit.
5.3 TP Handle
Let two distinct strongly connected MG circuits be denoted \(c_1, c_2\), \(c_1 \,\cup \, c_2 = c_1 \,\cup \, h_1\) or \(c_2 \cup h_2\), where \(h_1\), \(h_2\) is a TP-handle. Let \(f_1\) and \(f_2\) represents \(c_1\) and, \(c_2\) of the circuit flow value, respectively. The circuit flow value of \(c_1 \cup c_2\) is represented by \(f_1 + f_2\); this signifies that \(c_1\) and \(c_2\) token flows synchronously through the transition that is the start point of the TP-handle.
5.4 Examples of Strongly Connected MG Closed Token Flow Synthesis
Figure 2 displays a strongly connected weighted FC WFFP net, that is covered by MG circuits \(c_1 = \{p_1\), \(p_2\), \(p_3\), \(p_6\), \(p_7\), \(t_1\), \(t_3\), \(t_7\}\), \(c_2=\{p_1\), \(p_4\), \(p_5\), \(p_6\), \(p_7\), \(t_5\), \(t_6\), \(t_7 \}\). The circuit flow values of \(c_1\) and \(c_2\) are \(f_1 = 1\) and \(f_2 = 1\), respectively, and \(c_1 \circ c_2 = f_1^a f_2^b = 1^a \times 1^b = 1\). Therefore, this net satisfies structural boundedness and liveness by the circuit calculation of PP-handles.
Figure 5 also displays a strongly connected weighted FC WFFP net, that is structurally bounded and live, however, it becomes unbounded when the weights are ordinary. This net is covered by MG circuits \(c_1 = \{p_1\), \(p_2\), \(p_3\), \(p_4\), \(p_6\), \(t_1\), \(t_2\), \(t_3\) ,\(t_4\}\), \(c_2=\{p_2\), \(p_5\), \(p_6\), \(t_1\), \(t_3\), \(t_5 \}\). The circuit flow values of \(c_1\) and \(c_2\) are \(f_1 = \frac{1}{2}\) and \(f_2 = \frac{1}{2}\), respectively, and \(c_1 \circ c_2 = f_1 + f_2 = 1\). Therefore, this net satisfies structural boundedness and liveness by the circuit calculation of TP-handles.
6 Conclusion
Up to now, analysis of the structural liveness and boundedness of strongly connected weighted FC nets has not been performed. It has been demonstrated that analysis of the structural liveness and safety of Petri nets can be performed by net circuit flow value calculation using flow net transformation for a strongly connected MG. In this paper, we assume that there are merging transitions included in all circuits, and we propose that this precondition can be eliminated by systematizing the circuit flow value calculation. In future work, we plan to define the composition of circuits related to PT-handles.
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Harie, Y., Wasaki, K. (2020). Analysis of Structural Liveness and Boundedness in Weighted Free-Choice Net Based on Circuit Flow Values. In: Arai, K., Kapoor, S., Bhatia, R. (eds) Intelligent Computing. SAI 2020. Advances in Intelligent Systems and Computing, vol 1230. Springer, Cham. https://doi.org/10.1007/978-3-030-52243-8_41
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