Abstract
We analyse two very simple Petri nets inspired by the Oregonator model of the Belousov-Zhabotinsky reaction using our stochastic Petri net simulator. We then perform the Krohn-Rhodes holonomy decomposition of the automata derived from the Petri nets. The simplest case shows that the automaton can be expressed as a cascade of permutation-reset cyclic groups, with only 2 out of the 12 levels having only trivial permutations. The second case leads to a 35-level decomposition with 5 different simple non-abelian groups (SNAGs), the largest of which is A 9. Although the precise computational significance of these algebraic structures is not clear, the results suggest a correspondence between simple oscillations and cyclic groups, and the presence of SNAGs indicates that even extremely simple chemical systems may contain functionally complete algebras.
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Dini, P., Nehaniv, C.L., Egri-Nagy, A., Schilstra, M.J. (2012). Algebraic Analysis of the Computation in the Belousov-Zhabotinksy Reaction. In: Lones, M.A., Smith, S.L., Teichmann, S., Naef, F., Walker, J.A., Trefzer, M.A. (eds) Information Processign in Cells and Tissues. IPCAT 2012. Lecture Notes in Computer Science, vol 7223. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28792-3_27
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DOI: https://doi.org/10.1007/978-3-642-28792-3_27
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