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Self-Erasing Perturbations of the Abelian Sandpile Automata

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Jeffrey Randolph Moffitt

The early to mid 1980s was the time of the first cellular automata revolution.
Though von Neumann had formulated the concept 30 years earlier, this period
produced the greatest advances in our understanding of cellular automata (CA),
particularly in regards to the modeling of physical systems. It was during
this time that automata whose behavior approximated solutions to certain differential
equations were found. This was also the time of the cellular vacuum and digital mechanics,
suggestions that CA could describe fundamental physics in ways that differential equations
could not. However compelling these proposals may be, we currently do not have the
understanding of general CA to produce automata whose behavior subsume all of modern physics. In short, CA have yet to displace differential equations as the backbone of our understanding of fundamental physics. However, we attribute this fact not to any inherent limitations in CA, but simply to the difference between the centuries of understanding we have of differential equations as opposed to the decades for CA.

We contribute to the growing body of CA knowledge in this thesis by investigating a novel
phenomenon on a certain self-organizing cellular automata, the Abelian scalar sandpile.
We find that there exists a wide class of global perturbations that can be made to
the attracting states of any Abelian scalar sandpile such that the natural local dynamics
of the automaton completely remove these perturbations. We term such perturbations
self-erasing perturbations (SEP) and provide both a general form and an understanding
of why such a phenomenon exists. By using SEP to elucidate certain aspects of the general
Abelian scalar sandpile, we suggest its abilities as a conceptual tool. We also
demonstrate possible applications of this phenomenon in data protection and encryption.
Finally, the generality of this work is suggested by the discovery of a related CA that
demonstrates a SEP phenomenon as well.