International Symposium on Nonlinear Theory and its Applications
Isle of Eden in 1D binary cellular automaton as a manifestation of Godel incompleteness and a proposal for a ridge between analytical results and spatial-temporal logic patterns
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The richness of spatial-temporal dynamics is known since the Morphogenesis paper of Turing  and the introduction of the Cellular Automaton by Von Neumann and Ulam . The recent book of S. Wolfram  showcases, by a wealth of examples and one theorem (rule 110), the richness of the simplest spatial temporal binary patterns in the one dimensional (1D) binary cellular automaton, a special case of standard CNN Dynamics . The rigorous study of this model by L. O. Chua and others  led to a surprisingly simple and deep insight in the qualitative behavior of these models. The existence of the so called Isle of Eden, a sequence of states without predecessors and successors, is one interesting phenomenon. The aim of this paper is twofold. (i) We show that the Isle of Eden is a simple manifestation of Godel’s incompleteness theorem by using the way of the original proof of Godel. (ii) We propose a way to generate results on binary spatial-temporal logic patterns via analytical proofs using the binary to continuous transformation introduced by L. O. Chua et. al. . We note that the 1D Cellular Automaton is a simple case of a CNN dynamics programmably embedded in a CNN Universal Machine  and it could be simply implemented on existing cellular camera computers , as well as on different cellular many-core chips like FPGAs.