Union decomposition of Petri net (Q2726020)

The authors introduce the conception of a union decomposition of Petri nets. For a Petri net \(N= (P,T;F)\), if \(N= N_1\cup N_2\cup\cdots\cup N_m\) holds for some subnets \(N_i= (P_i, T_i; F_i)\) \((i= 1,2,\dots, m)\), where \(m\geq 2\), \(T_i,P_i\) \((i= 1,\dots, m)\) satisfying the conditionNEWLINENEWLINENEWLINE(I) \(T_i\subseteq T\), \(T_1\cup T_2\cup\cdots\cup T_m\), \(P_i= T_i\cup T_i\), orNEWLINENEWLINENEWLINE(II) \(P_i\subseteq P\), \(P_1\cup P_2\cup\cdots\cup P_m\), \(T_i= P_i\cup P_i\), then \(N_1,\dots, N_m\) is called a I-type or II-type union decomposition of the net \(N\), respectively.NEWLINENEWLINENEWLINEThe relationship of structural properties between a net and subnets of its union decomposition, such as repeatability, consistence, boundedness, conservation, fairness and weak fairness, is discussed in the paper.NEWLINENEWLINENEWLINEFifteen theorems are stated without proof (with the exception of Theorem 1 and Theorem 9). The typical result is embodied in the followingNEWLINENEWLINENEWLINETheorem 1. Let \(N= (P,T;F)\) be structural bounded. Let \(N_i= (P_i, T_i; F_i)\) \((i= 1,2,\dots, m)\), \(m\geq 2)\), be connected subnets of a I-type union decomposition of \(N\). Then all subnets \(N_1,\dots, N_m\) are structural bounded.NEWLINENEWLINENEWLINESome illustrating examples are also indicated.