Causal dynamics of discrete manifolds

05/25/2018

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by   Pablo Arrighi, et al.

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We extend Cellular Automata to time-varying discrete geometries. In other words we formalize, and prove theorems about, the intuitive idea of a discrete manifold which evolves in time, subject to two natural constraints: the evolution does not propagate information too fast; and it acts everywhere the same. For this purpose we develop a correspondence between complexes and labeled graphs. In particular we reformulate the properties that characterize discrete manifolds amongst complexes, solely in terms of graphs. In dimensions n<4, over bounded-star graphs, it is decidable whether a Cellular Automaton maps discrete manifolds into discrete manifolds.