Conflict-Free Places and Fireability of a Solution of Matrix Equation in Petri Net

Abstract

Non-negative integer solutions of a matrix equation of a Petri net play an important role in the analysis of the reachability of the net. A solution x of the matrix equation is expressed as x=α+β where α is a non-negative integer solution of the matrix equation and β is a solution of the homogeneous equation. Number of solutions of the matrix equation is, therefore, infinite in general and this causes serious difficulty in examinning whether or not any of the solutions has fireable sequence. The difficulty will be considerably relieved if the conditon is found under which the solution α has fireable sequence when the solution α+β has fireable sequence. This paper gives a sufficient condition, that some places specified by the solutions α and β are to be either forward or backward conflict-free places. This condition, though sufficient, is very much weaker than those previously obtained, that the net is structually conflictfree, in other words, all the places of the net are to be conflict-free.