Compositional analysis of Petri nets

D. A. Zaitsev
2006 Cybernetics and Systems Analysis  
519.74 Foundations of compositional analysis of Petri nets are presented. This analysis consist of the determination of properties of a given Petri net from the properties of its functional subnets. Compositional analysis covers the investigation of behavioral and structural properties of Petri nets with the help of matrix methods that use fundamental equations and invariants. The exponential acceleration of computations as a function of the dimensionality of a net is obtained. INTRODUCTION
more » ... i nets [1, 2] are successfully used in investigating systems and processes in various applied domains [3, 4] . As a rule, a model of a real object has a sufficiently high dimensionality and consists of more than a thousand elements. At the same time, basic methods of analysis of properties of Petri nets except for, perhaps, reduction methods have the exponential computational complexity. Thus, the development of efficient methods for the investigation of properties of Petri nets is a vital scientific issue. The following three groups of methods of analyzing properties of Petri nets are well known [2]: the methods based on the construction of trees of attainable and covering markings, matrix methods using the fundamental equation of a net and invariants, and reduction methods. It is relevant to note that reduction is an auxiliary means of investigation and a special case of equivalent transformations [5] decreasing the dimensionality of a net. Matrix methods are most promising for the analysis of large technical systems [3] . The fundamental equation of a Petri net is a system of linear Diophantine equations [2] . The solutions of such a system are interpreted as vectors of calculation of firings of allowable sequences of transitions and, hence, must be nonnegative integer numbers, which stipulates the specifity of the problem. Methods of solution of such systems are presented in [6] [7] [8] . Unfortunately, all the well-known methods have asymptotically exponential complexity, which complicates their use in analyzing real systems. The objective of this article is the construction of compositional methods that make it possible to analyze properties of Petri nets and to substantially accelerate calculations. In fact, models of complex systems are constructed, as a rule, from models of their components. Thus, it is necessary to formalize this process and to construct methods that allow one to find properties of the entire net on the basis of known properties of its subnets. Moreover, in the cases when the composition of a model in terms of its subnets is not given, one can use the methods of decomposition of Petri nets that are presented in [9, 10] . A decomposition algorithm makes it possible to partition a given Petri net into a set of minimal functional subnets. In this article, it is shown how the properties of functional subnets that specify a partition of the initial Petri net can be used for computation of properties of the entire net. The obtained acceleration of computations is estimated by an exponential function. Since the dimensionality of subnets is, as a rule, substantially less than the dimensionality of the entire net, the actual acceleration of computations can be very sizable, which is substantiated by the results of application of this method to the analysis of net protocols [11, 12] . 126
doi:10.1007/s10559-006-0044-0 fatcat:bjcbeyf7lvezji2izstyl2bine