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My question is pure mathematics when restricted to the cellular automata theory.

John von Neumann got the grasp of and defined life. Many years later biologists supported von Neumann's definition of life by discovering DNA and modern genetics.

Life is possible and is based on the following premises:

  1. the environment is flexible and rich into simple elements (building blocks) which randomly (yes, randomly) show up in the neighborhood of the automaton, and can be matched up (recognized and a kind of absorbed or processed);

  2. the universal Turing machines (among all possible Turing machines) happen randomly with a great probability--in a purely mathematical sense;

  3. a construction (life) can happen randomly which includes a universal Turing machine, and which includes as one of its macro-parts a description of itself and of its reproduction behavior (self-reference).

REMARK   The requirement of exact reproduction is an elegant mathematical goal while less than exact reproduction is of even greater importance. This vastly increases the chance of life. However, the chance of encountering life is still very small.

QUESTION   What are the intermediate constructions/notions between the lifeless configurations $\ A,\ $ and configurations which include life $\ L$? The point is to define intermediate constructions $\ i_k\ $ such that the step by step process:

$$ A\rightarrow i_1\rightarrow\ \ldots\ \rightarrow i_n\rightarrow L $$

would have a clearly higher probability than a direct transformation:

$$ A\ \rightarrow\ L $$

Perhaps the notion of intermediacy should be robust in the sense of its longevity (of existing in an intermediate or higher state)

References (...probability of Turing machines approaches 1...):

The classic:

http://cba.mit.edu/events/03.11.ASE/docs/VonNeumann.pdf

Also:

https://www.quora.com/Among-all-possible-Turing-machines-how-common-are-universal-Turing-machines

or better(?), one may go directly to:

            https://arxiv.org/abs/1510.01671

(Sorry, that's what I have at this moment and not more yet).

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  • $\begingroup$ it would be useful, to add some references. $\endgroup$ – Konstantinos Kanakoglou Feb 11 '17 at 23:28
  • $\begingroup$ To the right of this post, I see a list of 5 MathOverflow links related to Conway's life "game". $\endgroup$ – Włodzimierz Holsztyński Feb 11 '17 at 23:43
  • $\begingroup$ @KonstatinKanakoglou, yes. I've added 3 links from outside MO. ***I am in a rare situation where for various reasons my access to literature is very limited. I'll google though.The key names are John von Neumann, Stan Ulam, perhaps John Horton Conway, ... (the topic of life is in my opinion among the most important in mathematics). – $\endgroup$ – Włodzimierz Holsztyński Feb 12 '17 at 0:37

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