Aimed at applications of our recently developed rule algebra framework to the study of complex systems, we introduce two novel concepts. Consider a complex system specified in the form of a stochastic rewriting system, in terms of its state space, its initial state and its possible random transitions. Our combinatorial conversion theorem allows to compute for each chosen set of observables of the system an evolution equation for the exponential generating function (EGF) of the statistical moments of these observables. Thus, our theorem permits to convert the problem of analyzing the evolution of the complex system to a problem in the realm of generating functions.
As a second novel concept, we introduce the notion of moment bisimulation, which defines two complex systems to be behaviorally equivalent with respect to two sets of observables (one set per system) if their moment EGF evolution equations agree. The concept will be illustrated in terms of a complete characterization of the bisimilarity class of chemical reaction systems via a standard choice of observables. We conclude with an example of a network model which, for a certain choice of observables, falls within this bisimilarity class.
