From Graph Transformations to Differential Equations

Abstract

In a variety of disciplines models are used to predict, measure or explain quantitative properties. Examples include the concentration of a chemical substance produced within a given period, the growth of the size of a population of individuals, the time taken to recover from a communication breakdown in a network, etc. The models such properties arise from are often discrete and structural in nature. Adding information on the time and/or probability of any actions performed, quantitative models can be derived. In the first example above, commonly referred to as kinetic analysis of chemical reactions, a system of differential equations describing the evolution of concentrations is extracted from specifications of individual chemical reactions augmented with reaction rates. Recently, this construction has inspired approaches based on stochastic process specification techniques aiming to extract a continuous, quantitative model of a system from a discrete, structural one. This paper describes a methodology for such an extraction based on stochastic graph transformations. The approach is based on a variant of the construction of critical pairs and has been implemented using the AGG tool and validated for a simple reaction of unimolecular nucleophilic substitution (SN1).