Modelling Cellular Automata with Temporal Description Logic

Cellular automata are discrete mathematical models that have been proven useful as representations of a wide variety of systems that feature non-linear dynamics, in many scientific fields. They are composed of an enumerable set of homogenous cells which can take on a finite number of states. Rules of local interaction are then conditioning the global behavior which evolves in discrete time steps. What makes cellular automata interesting is the fact that by employing relatively simple rules, one can model complex processes that appear in nature but also in computer science, economics and social sciences. Because of the complex and chaotic nature of the observed properties, there is a broad spectrum of different approaches when it comes to research in cellular automata. Our interest is in translating the cellular automata to temporal description logic languages. This will enable us to have an immediate estimate of the computational complexity of the questions we might have regarding the properties and the evolution of a cellular automaton. Furthermore, this is a first class opportunity to look deeper into the mathematics and get an alternative understanding of attributes that are inherent and structural. This endeavor is divided in three parts. On the first section we give the basic definitions of cellular automata and the properties which will be modeled, on the second section we define the specific temporal description logic language that we will use and its semantics and in the third (and arguably core) section we make the translation and prove several statements that illustrate the desired correspondence and indicate the prospect of related research in the future.