3

I recently learned in my last question that rule 110, while being Turing-complete, can be built from a set of falsity-preserving functions (such as $\{\lor, \oplus\}$) which isn't functionally complete. I am now wondering, is there a Turing complete cellular automaton (or any other Turing complete system) whose iterations can be built from a set of affine boolean functions (such as $\{\oplus, 1\}$)?

Here is the definition of affine boolean function: let $\mathbb{B} = \{0, 1\}$ be the boolean domain. A boolean function $f : \mathbb{B}^n \rightarrow \mathbb{B}$ is said to be affine when each input variable either always or never affects the final truth value. Affine boolean functions are also the functions of the form $f(x_1, \dots, x_n) = a_1 x_1 \oplus \dots \oplus a_n x_n \oplus a_{n+1}$, where $a_1, \dots, a_{n+1} \in \mathbb{B}$. The set of all affine boolean functions is denoted by $A$ in Post's lattice. This set can be generated from the functions $\{\oplus, 1\}$, where $\oplus$ represents the XOR. Example of affine boolean functions are the constant $0$ function, the constant $1$ function, $\oplus$, $\leftrightarrow$, and $\lnot$.

I would like a Turing complete cellular automaton (or any Turing complete system) for which each iteration can be seen as a circuit of affine boolean functions. The circuit can change over timesteps (just as in Critters which alternates between two rules from one timestep to the next), but this change must not depend on the value of the cells.

Lysandre Terrisse
  • 105
  • There are only 1024 2D affine cellular automata where all cells are updated by themselves and their 8 surrounding neighbors. By removing rotationally/reflectionally equivalent rules, we get only 205 nonequivalent rules. I checked all of them starting from a random state with different population levels, and I also checked with a single cell alive/dead. What happens with a single cell is that the alive/dead cell either disappears immediately (very rarely), or grows infinitely in a Sierpiński-like shape (generally a triangle). There is almost no interesting behavior in random scenarios. – Lysandre Terrisse Jul 10 '25 at 01:13

0 Answers0