A Is Seth Lloyd's cosmological model background independent?

Summary
Is Seth Lloyd's cosmological model background independent?
I have been interested in Seth Lloyd's cosmological model (which proposes that the universe is a some kind of quantum computer or at least similar to it: https://en.wikipedia.org/wiki/Programming_the_Universe, https://arxiv.org/abs/quant-ph/0501135) since long ago.

I was wondering if his cosmological model was background independent.

Max Tegmark proposed a hypothesis where all mathematical structures would have physical existence.

I found a short review that Lloyd made about Tegmark's book "Our Mathematical Universe":

"*In Our Mathematical Universe, renowned cosmologist Max Tegmark takes us on a whirlwind tour of the universe, past, present—and other. With lucid language and clear examples, Tegmark provides us with the master measure of not only of our cosmos, but of all possible universes. The universe may be lonely, but it is not alone*."

This made think that Lloyd agreed with Tegmark's views and therefore his model was compatible with all Tegmark's universes. I mean, that his model could "produce" all universes proposed by Tegmark, or that his model would be compatible with any universe (mathematical structure) proposed by Tegmark.

But is this right? Could Lloyd's model produce all universes (mathematical structures) proposed by Tegmark? Would this model be compatible with all of them? Even uncomputable ones?
 
A quantum computer has the same set of computable functions as an ordinary digital computer.

The current known laws of physics are all computable with an ordinary digital computer.

What would a non-computable universe be like? There would be processes of nature which cannot be modeled with a conventional digital computer.

Let us imagine a one bit universe with a discrete time. The world history of such an universe is an infinite sequence of binary digits:

...0010110111010...

The sequence of the digits may be a non-computable function from integer numbers f: Z -> {0, 1} to the binary digits.

We cannot simulate such a universe with a finite digital computer.

What if we have an infinite memory in the computer? It does not help unless we allow the program to be infinite. If we allow an infinite program, we can simply store the infinite sequence of digits in the memory.

Conclusion: A finite digital computer can only simulate a countable number of universes. There exist an uncountable number of universes. The computer cannot simulate them all.

If we allow the program to have the same cardinality as the universe, then trivially, we can simulate any universe.
 
A quantum computer has the same set of computable functions as an ordinary digital computer.

The current known laws of physics are all computable with an ordinary digital computer.

What would a non-computable universe be like? There would be processes of nature which cannot be modeled with a conventional digital computer.

Let us imagine a one bit universe with a discrete time. The world history of such an universe is an infinite sequence of binary digits:

...0010110111010...

The sequence of the digits may be a non-computable function from integer numbers f: Z -> {0, 1} to the binary digits.

We cannot simulate such a universe with a finite digital computer.

What if we have an infinite memory in the computer? It does not help unless we allow the program to be infinite. If we allow an infinite program, we can simply store the infinite sequence of digits in the memory.

Conclusion: A finite digital computer can only simulate a countable number of universes. There exist an uncountable number of universes. The computer cannot simulate them all.

If we allow the program to have the same cardinality as the universe, then trivially, we can simulate any universe.
Thank you for your answer!

I suppose that an ordinary computer could not simulate an infinite universe and thus we would also need an unlimited amount of memory, isn't it?

Also, to simulate an infinitely big universe, would we also need to allow the program to be infinite? Would we need to also allow the program to have the same cardinality as the universe?
 
For an infinite universe you may need an infinite memory in the computer. If you have an infinite number of particles, then you in most cases need an infinite memory to store their positions and the velocities.

I started to wonder what happens if we allow an uncountable number of particles. Then an ordinary digital computer would never have time to calculate even a single timestep.

From model theory in mathematical logic we know that if a first order predicate theory has a model, then it always has a countable model. For most physical theories we may then assume that the model is only countably infinite.

Max Tegmark conjectures that every "possible" universe "exists" in some sense. In mathematics, set theory is the branch which studies which sets exist. Most mathematical structures can be represented with sets. We might define a "possible" universe as a set which exists. Then Tegmark's conjecture is a tautology.

The discussion moves to philosophy. Which things "exist"? That is a problem of ontology which people have pondered at least for 2,500 years.
 
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The current known laws of physics are all computable with an ordinary digital computer.

What would a non-computable universe be like? There would be processes of nature which cannot be modeled with a conventional digital computer.
Actually, there are indications that non-computable phenomenon are actually taking place in nature, but this is hardly ever talked about - especially not by physicists - who instead prefer to sweep such kinds of physics beyond the Standard Model carefully under the rug. Constructing arbitrary aperiodic tilings which tile the entire plane or aperiodic crystals which fill the entire space are both classic examples of non-computable problems from mathematics.

There are literally phenomenon taking place in chemistry which seem to do this, e.g. aperiodic crystal formation where aperiodically tiled large crystals are correctly formed, wherein the different largely spaced molecules seem to spontaneously form the correct configuration of the orientations between themselves in order to tile the plane/fill the space aperiodically; this non-computable coordination between distant molecules seems to occur non-locally.
 
'For this reason, I was somewhat doubtful that nature would actually produce such ‘quasi-crystalline’ structures spontaneously’ said Roger Penrose (Thomas, 2011 ▸). ‘I couldn’t see how nature could do it because the assembly requires non-local knowledge’. The growth of quasiperiodic structures is still not fully clarified; however, some realistic models and simulations have been published recently (see Kuczera & Steurer, 2015
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5740452/

There seem to be open questions in quasicrystal formation.
 

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