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Cellular Automata are systems of finite (identical) automata that are connected by edges and take the values (states) of their neighbours as input for a "local transition function". Like in a spreading activation net, the simultaneous execution of all local functions constitute a "global function" defined on the space of configurations of values on the single automata. Cellular automata are a mathematical model of distributed systems and in particular a model of neural networks and associative nets.
Regularities in Time and Space
In my dissertation supervised by Prof. Gy. Targonski (Uni Marburg) I have analyzed regularities of cellular automata in "time" and "space". In "time" refers to the iterative behaviour of the global function on the configuration space. In "space" means with regard to symmetries in the graph defined by the neighbourhood relation between nodes (automata).
Using a generalized definition of cellular automata I have been able to show that cellular automata are the continuous functions on configuration spaces (compare "Cellular Automata are the Continuous Self Mappings of a Configuration Space" (Ferber 1989 [->])). I have analyzed the effect of symmetric operations on the connection graph for the iteration sequence of the global function - in particular for Caley graphs. Further I have given a cellular automata example for a complex limit-set theorem in iteration theory. To visualize examples and to verify specific properties of graphs I have implemented a simulation program for cellular automata on PC.