Maki Takata

Reachability analyses in Petri nets by Groebner bases

Finding a non-negative integer solution x /spl notin/ Z/sub +//sup n/spl times/1/ for Ax = b (A /spl isin/ Z/sup m/spl times/1/, b /spl isin/ Z/sup m/spl times/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.

Algorithm for the set of generators of an arbitrary firing count vector in Petri nets

In this paper, an effective method to obtain all nonnegative integer minimal support vectors (U/sub 4/,V/sub 4/) at level 4 for an arbitrary homogeneous/inhomogeneous solution starting from nonnegative rational number minimal support vectors (U/sub 3/,V/sub 3/) at level 3 is proposed. Although V/sub 4/ has been derived from all minimal vectors (U/sub 5/,V/sub 5/) of level 5 which are obtained starting from vectors at level 1, 2, or 3, so far. This proposed method for (U/sub 4/,V/sub 4/) gives us a big shortcut comparing with them. However, it is pointed out that obtaining (U/sub 5/,V/sub 5/) of level 5 from (U/sub 4/,V/sub 4/) of level 4 is not so wise.