Uniform formulas for compound siphons, complementary siphons and characteristic vectors in deadlock prevention of flexible manufacturing systems

Unmarked siphons in a Petri net modelling concurrent systems such as those in cloud computing induce deadlocks. The number of siphons grows exponentially with the size of a net. This problem can be relieved by computing compound (or strongly dependent) siphons based on basic siphons. A basic (resp. compound) siphon can be synthesized from an elementary (resp. compound called alternating) resource circuit. It however cannot be extended to cases where two elementary circuits intersect at a directed path rather than a single place (i.e., corresponding to a weakly dependent siphon). This paper develops a uniform formula not only for both cases but also valid for the complementary set of siphon and characteristic vectors. We further propose to generalize it to a compound siphon consisting of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} basic siphons. This helps simplify the computation and the computer implementation to shorten the program size. Also, the formula is easier to be memorized without consulting the references due to the same underlying physics.