Abstract
I prove that the natural multicellular environment requires random scheduling of cell activity in order to successfully execute arbitrary amorphous programs. This proof uses an unusual new method of constructing amorphous programs. Amorphous processes model natural multicellular information processing and so it is finite in most respects. Thus it is surprising that a non-computable operation (random choice for cell activity) is an essential part of the formalism’s interpretation. The necessary randomness could originate in thermodynamic events. To the extent that amorphous processes successfully model multicellular biological information processing, this result suggests that there is something essential to the processes of life and evolution that lies just beyond Turing computability.
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Notes
For example, if \(\sigma =\{0,3,4\}\) in a subset of cells 0,1,2,3,4,5,6,7, then \(\langle \sigma \rangle =2^1*7^1*11^1 = 154\).
\(\sigma =\urcorner f(n)\ulcorner \) is the set satisfying \(\ulcorner \sigma \urcorner =f(n)\).
A generic schedule would be unstructured and so typical of \({\mathcal{S}} . \) It would be generated, not by successive calls to f, but by successive random choices from \({\mathcal{P}}(C).\) Given any \(\sigma _m,\ldots \sigma _{m+n}\) the next \(\sigma _{n+m+1}\) is random if it is independent of those that came before — i.e., \(P{\{\sigma _{n+m+1} | \sigma _m,\ldots \sigma _{m+n}\}}=2^{-||C||}.\)
This property is enough to guarantee that a random schedule is not computable in bounded space, and so not pseudorandom—see the discussion of I,J in the next section for proof.
If the product space \({\mathcal{S}}={\mathcal P}^{\mathbb{N}}\) is given the usual [0,1]-valued product measure μ, then \(\mu ({\mathcal{S}})=1\) and the countable subset \(\mu ({{\mathcal{S}}_f})=0 .\)
2PartitioN is ({0,1}, {0,1,10}, α, input) where input(M) = 0 if M contains no 1, = 1 if M contains no 0, and = 10 otherwise; α(v, M) = v if M = (1− v and \(\alpha (v,M)=(1-v)\) otherwise. A couple of dozen other amorphous processes are defined and explored in Stark (2013).
Over non-bipartite networks other than K3, we may need to go a bit beyond \(s_{M+1}\) to get a constant state.
References
Hodges A (2012) Beyond turing machines. Science 336(6078):163–164
Linz P (2001) Introduction to formal languages and automata. Jones and Bartlet, Sudbury
Stark WR (2013) Amorphous computing: examples, mathematics and theory, natural computing, also available free from Springer or from the Wikipedia (item 14 under Amorphous Computing)
Stark WR (2015) Amorphous processes cast into Kleene’s partial recursive functions, notes available from the author
Wiedermann J (2012) Computability and non-computability issues in Amorphous computing, vol 7604. IFIP TCS, pp 1–9
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Stark, W.R. Random choice in amorphous processes. Nat Comput 14, 649–653 (2015). https://doi.org/10.1007/s11047-014-9461-6
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DOI: https://doi.org/10.1007/s11047-014-9461-6