A State-Equation-Based Backward Approach to a Legal Firing Sequence Existence Problem in Petri Nets
针对Petri网中状态方程有非负整数解但标识不可达的问题，提出基于状态方程的反向算法（SBA），通过识别依赖回路判断合法发射序列是否存在，并在柔性制造系统案例中验证了算法的正确性和有效性。
Reachability is the basis for studying other dynamic properties of Petri nets (PNs). When a state equation is used to determine the reachability of a marking, we need to judge whether there is a corresponding legal firing sequence (LFS) for a non-negative integer solution (NIS), i.e., a firing count vector, of the state equation. The search for an LFS is an NP-hard problem, and previous work cannot always find an LFS for any NISs. This article proposes that transition-dependent circuits or firing-dependent circuits are the root cause that a state equation has an NIS but the marking is nonreachable, i.e., there is no LFS corresponding to an NIS in PNs. Based on this, we propose a state-equation-based backward algorithm (SBA) to determine whether there is an LFS corresponding to an NIS of the state equation in a PN. The correctness and effectiveness of SBA are verified by a case study on a PN-based flexible manufacturing system and through simulation on an S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> PR net. The experimental results show that the time required for SBA to determine the existence of an LFS increases linearly with the transition firing count in NISs. When the number of NISs of a state equation is finite, we can efficiently determine the reachability of a marking. This represents an important result in theory and applications of PNs.