# Is Steiner symmetrization "Turing complete"?

This question stems from intuition so it is a little soft. It concerns performing computation using transformations on sets. The idea is that a rearrangement like Steiner symmetrization might be "computationally strong enough" to be used to execute arbitrary programs.

First we probably should pick an appropriate set $$\mathcal{B} \subset \mathcal{P}(\mathbb{R}^d)$$, maybe the Lebesgue or Borel measurable sets. Then consider a set of transformations $$(T_i)_{i \in I}$$ where $$T_i: \mathcal{B} \longrightarrow \mathcal{B}$$ for every $$i \in I$$.

We also need a program function $$f: \mathcal{B} \longrightarrow I$$. Lastly we need some condition that decides when the program finishes running, formally $$E: \mathcal{B} \longrightarrow \{\mathit{True}, \mathit{False}\}$$.

Then computation is done as follows:

1. Some $$M_0 \in \mathcal{B}$$ is the input

2. $$M_{n+1} = T_{f(M_n)}(M_n)$$

3. If $$E(M_n) = \mathit{True}$$ then the program stops running and the output is $$M_n$$

1. Question: How are such systems of computation formalized? (i.e. does there exist a "name" for this)

2. Question: Can such systems be "Turing complete" in some sense? How would one show this?

I will also give the specific example that made me wonder: $$\mathcal{B}$$ are the Lebesgue measurable sets and $$(T_i)_{i \in I}$$ are the Steiner symmetrizations in all directions. Then $$f$$ is something I don't know and $$E$$ could be a condition like the set being convex.

I am sorry for this question being a little vague and would appreciate if someone could improve it.