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DY: Fachverband Dynamik und Statistische Physik

DY 16: Pattern Formation

DY 16.9: Vortrag

Mittwoch, 13. März 2013, 12:00–12:15, H46

Origin of Complexity in Cellular Automata — •Vladimir Garcia-Morales — Technische Universität München - Institute for Advanced Study, Lichtenbergstr. 2a, D-85748 Garching, Germany

Cellular automata (CA) constitute paradigmatic models of complexity in nature, from snowflakes, patterns in mollusc seashells and spiral waves in the Belousov- Zhabotinsky reaction to neural networks and the fundamental physical reality. A universal map encompassing all 1D deterministic first-order in time CA has been very recently derived [1]. This map is to be considered as the discrete counterpart of partial differential equations in continuum systems. The map does not depend on freely adjustable parameters and is valid for any neighborhood and alphabet size. It can be easily extended to an arbitrary number of dimensions and topologies and higher orders in time. Symmetry arguments applied to the map allow to classify all dynamical CA rules into equivalence classes and a theorem can be proved which establishes how a CA rule is constructed in terms of rules of lower range [2]. The crucial result is that the most complex CA rules can be found with a simple prescription, starting from rules possessing the symmetry upon addition modulo an integer number p, and weakly breaking this symmetry through an additional degree of freedom. It is illustrated how this mechanism is the origin of complexity in 1D CA.

[1] V. Garcia-Morales, Phys. Lett. A 376 (2012) 2645.

[2] V. Garcia-Morales, Phys. Lett. A (2012, in press),

http://dx.doi.org/10.1016/j.physleta.2012.11.052