This paper presents some structural properties of a generalized Petri net (PN) with an algorithm to determine the (partial) conservativeness and (partial) consistency of the net. A product incidence matrix A=CCT or A~=CTC is defined and used to further improve the relations among PNs, linear inequalities and matrix analysis. Thus, based on Cramer's Rule, a new approach for the study of the solution of a linear system is given in terms of certain sub-determinants of the coefficient matrix and an efficient algorithm is proposed to compute these sub-determinants. The paper extends the common necessary and/or sufficient conditions for conservativeness and consistency in previous papers and some examples are designed to explain the conclusions finally.
This paper introduces a new variant of Petri net controlled grammars, namely a \textit{concurrently controlled grammar}, where the control over the application of the productions of a grammar is realized by a Petri net with different parallel firing strategies. The generative capacity of these grammars is investigated with respect to transition labeling strategies, definitions of final marking sets and parallel transition firing modes. It is shown that the labeling strategies do not effect the computational power whereas the maximal firing modes increase the power of concurrently controlled grammars with erasing rules up to Turing machines.