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Turing patterns with Turing machines: emergence and low-level structure formation

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Abstract

Despite having advanced a reaction–diffusion model of ordinary differential equations in his 1952 paper on morphogenesis, reflecting his interest in mathematical biology, Turing has never been considered to have approached a definition of cellular automata. However, his treatment of morphogenesis, and in particular a difficulty he identified relating to the uneven distribution of certain forms as a result of symmetry breaking, are key to connecting his theory of universal computation with his theory of biological pattern formation. Making such a connection would not overcome the particular difficulty that Turing was concerned about, which has in any case been resolved in biology. But instead the approach developed here captures Turing’s initial concern and provides a low-level solution to a more general question by way of the concept of algorithmic probability, thus bridging two of his most important contributions to science: Turing pattern formation and universal computation. I will provide experimental results of one-dimensional patterns using this approach, with no loss of generality to a n-dimensional pattern generalisation.

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Notes

  1. That is, a machine for which a valid program is never the beginning of any other program, so that one can define a convergent probability the sum of which is at most 1.

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Aknowledgments

The author wishes to thank the Foundational Questions Institute (FQXi) for its support (mini-grant No. FQXi-MGA-1212 “Computation and Biology”) and the Silicon Valley Foundation.

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  1. Behavioural and Evolutionary Theory Lab, Department of Computer Science, The University of Sheffield, Sheffield, UK

    Hector Zenil

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  1. Hector Zenil
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Correspondence to Hector Zenil.

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Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Nat Comput 12, 291–303 (2013). https://doi.org/10.1007/s11047-013-9363-z

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