The project focuses on the development of a mathematical foundation for interaction-based computer simulations that can be represented as deterministic, discrete-time dynamical systems on finite state sets. Here, an interaction-based simulation is considered to be a collection of variables, each equipped with an update function or a set of rules which is used to compute the state of that variable at the next time step, based on the states of all the other variables. Thus, the description of the system is given via local interactions, and global dynamics is generated through the simultaneous iteration of all local update functions. Cellular automata and Boolean networks are examples such simulations. An important question, that has been studied extensively in the framework of cellular automata is how one can predict global dynamical features of such systems from the structure of the local update functions. By assuming that the state set for each variable is a finite field, such as the field with two elements used typically for cellular automata,this problem can be addressed within the framework of computational algebra. The algorithms developed in this project will become part of a symbolic computation software package for finite dynamical systems, implemented in the computer algebra system Macaulay2.
Interaction-based simulation is becoming increasingly important in the analysis of large biological, epidemiological, and socio-technical networks, such as the immune system, the spread of infectious diseases in urban areas, or road traffic and wireless communications networks. Typically, such networks are understood at the level of individual interactions, but global information tends to be sparse. The software design of such systems is very challenging, and so is the analysis of simulation output, due to the size and complexity of the simulated systems. A mathematical foundation for such simulations will provide tools for the design of large-scale simulations. It will also help to systematically answer questions about biological and other systems, such as optimal ways to treat infectious diseases,or how to contain their spread in large populations.
