Reachability analyses in Petri nets by Groebner bases

Finding a non-negative integer solution x ∉ Z₊^n×1 for Ax = b (A ∈ Z^m×1, b ∈ Z^m×1) in Petri nets is NP-complete. Being NP-complete, even algorithms with theoretically bad worst case and average complexity can be useful for a special class of problems. A Groebner basis approach to integer programming problems was proposed in 1991 and some symbolic computation systems have become useful tools for ideals, varieties, and algorithms for algebraic geometry. In this paper, two kinds of examples are given to show how the Groebner basis approach can be applied to reachability problems in Petri nets.