I don't know much about the terminology and the results on cellular automata, but I would like to ask a question about an conjecture I thought.

Consider Turing-complete reversible cellular automata.

For instance, Wolfram's Rule 110 cellular automaton is Turing-complete, but, if I understand correctly, it's not reversible. However, it can be made reversible by converting it into a second-order cellular automaton and then implementing the second-order CA as a first-order CA with a larger cell state space.

My question is whether it's possible to define a finite region of cells that is completely isolated from the outside.

More specifically: if there exist an initial state of a finite connected region of cells (let's call it *Vault*), which is composed of a connected region *Interior* surronded by another region *Wall*, such that the state of the *Interior* at any time *t* depends only on the previous state of *Interior*.

My conjecture is that such isolated regions are impossible.

My intuition stems from the fact that Turing-complete reversible CA can be considered very simple models of physical reality (and proponents of the digital physics hypothesis argue that the physical world is actually a cellular automaton). If I understand correctly, totally isolated physical systems can't exist, thus I conjecture by analogy that this should also apply to all Turing-complete reversible CA.
(Turing-incomplete or irreversible CA can have isolated regions, but they are probably not good models of fundamental physical reality)

Has anybody already proved or disproved this assertion? Or do you have any thoughts about it?

UPDATE:

As Shor pointed out in the comments, the conjecture as stated above is false, since it is possible to use some states which are not required for Turing-completeness to achieve isolation.

I will try to reformulate the question in a way that the conjecture may possibly hold by adding more requirements on the cellular automaton.

I'm thinking of two reformulations based on two different additional requirements:

1. *Minimal universality*: As noted by Grochow, we can define a Turing-complete CA to be "*Minimally universal*" iff the CA obtained by removing one cell state (and the corresponding rules) is no longer Turing-complete.

Thus, the conjecture is: *There does not exist a minimally universal reversible CA such that isolated regions can be defined.*
2. *Total harnessability*: We may say that the computational power of a Turing-complete CA is *totally harnessable* iff there exist undecidable properties regarding all cell states.

More specifically, we define a CA to be *totally harnessable* iff $\forall s \in CellStates \ \exists$ a computable initial configuration $x(t\_0,\,i)$, a finite set of cell indexes $Q$, a vector of cell states $\bar{q}$ containing $s$ and predicate $p(x,\,t)\ \equiv\ \bigwedge\_{i \in Q}x(t,\,i)=q\_i $ such that the question "$\exists t:\ p(x,\,t)=True$" is undecidable, and the question obtained by removing from predicate $p$ the conditions on state $s$ is decidable.

Thus the conjecture is: *There does not exist a totally harnessable Turing-complete reversible CA such that isolated regions can be defined.*