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There are several papers that propose that the universe itself is analogous to a cellular automata. Obviously, these papers assume that the universe is computable.

But is it there any cosmological cellular automata models that would allow uncomputable laws/processes/functions/numbers/things...etc to exist in that universe? For example, cellular automata universes where there would be truly black hole-singularities (which are uncomputable)?

Maybe Konrad Zuse's cellular automata could do the job?

https://en.m.wikipedia.org/wiki/Calculating_Space

Or maybe some variation of Conway's game of life? https://en.m.wikipedia.org/wiki/Conway%27s_Game_of_Life

Or maybe we could build one using 't Hooft's cellular automata frameworks

https://arxiv.org/abs/gr-qc/9903084 https://arxiv.org/abs/1405.1548

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  • $\begingroup$ CAs already admit several uncomputable problems. For example, the reversibility of 2D (standard) CA is undecidable. What exactly do you wish to achieve? $\endgroup$ – dkaeae Dec 31 '18 at 15:29
  • $\begingroup$ If the "CA Universe" is bounded then everything is computable. If it is infinite then there can be parts of it that behave like a Turing machine and (by the undecidability of the Halting problem) no other part would be able to "calculate" if those parts will stop evolving. $\endgroup$ – Marzio De Biasi Dec 31 '18 at 20:38
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    $\begingroup$ I do not doubt there are research-level papers about mathematical modelling of the universe using cellular automata. What I was trying to communicate is that this particular question is not research-level until it is made sufficiently precise. For instance, you say "true black hole singularities are uncomputable". What is that supposed to mean, precisely? Do you have a precise definition of "black hole" that is applicable outside of the general theory of relativity, so that we can apply it to a model based on cellular automata? $\endgroup$ – Andrej Bauer Jan 13 at 10:44
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    $\begingroup$ To continue: What is the difference between "true black hole" and "black hole"? What does it mean for a black hole to be uncomputable? Can a star, or a galaxy by "uncomputable" and what does that mean? You can't just say these things without being mathematically precise, not on this forum. $\endgroup$ – Andrej Bauer Jan 13 at 10:46
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    $\begingroup$ Consider Conway's game of life on an infinite plane. It is known that it can simulate a Turing machine. On the sixth day of creation, put in the universe a Turing machine whose halting problem is undecidable, and start the universe. Ergo, you have got yourself a game-of-life universe in which a non-computable phenomenon "exists" (namely, we can't compute whether the machine will halt). This quesiton is not research-level because it's either unclear or trivial, depending on which way you want to take it. $\endgroup$ – Andrej Bauer Jan 13 at 10:49

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