Developing Mathematical Formalisms for Cellular Automata in Modelling and Simulation

Cellular Automata are being applied as a method for mathematical modelling and simulation in a variety of scientific areas. Usually a cellular automaton constitutes a microscopic description or abstraction of a natural system. The tools used for a mathematical derivation and validation of the cellular automaton modelling approach are however often optimised for specific natural systems. As a consequence there exists neither a dedicated mathematical formalism for cellular automata as a method for modelling and simulation nor general methods for model analysis and validation. This thesis regards cellular automata as a distinct mathematical ansatz for modelling and contains a rigorous and systematic definition of various types of cellular automata. The aim is to develop methods and approaches for a general discussion and comparison of this modelling approach. Thereby the focus is less the introduction of a classification of cellular automata than the deduction and simplification of methods for analysis, validation and comparison. Additionally a systematic description of cellular automata can ease the communicability of the chosen modelling approach and provides a basis for further and more specific investigations. The mathematical description of cellular automata in this thesis is based on locally defined update functions that operate in an iterative manner on graphs respectively regular lattices. A strong focus is put on the characteristic topological structure, which also delineates cellular automata from other concepts like agentbased modelling approaches and arbitrary network models. Furthermore a modification of the concept of cellular automata with continuous time and space is introduced under the paradigm of evolution systems, which ultimately yields a connection to strongly continuous semigroups, abstract Cauchy problems and parabolic partial differential equations. A stochastic formalism of cellular automata finally discloses this modelling approach as a very specific variant of Markov processes and allows efficient deterministic methods for analysis and simulation.