Very nice game! I'd like to write something about the game in my snake article, would you mind contacting me? (clicking my name and following some links should get you there)
I didn't find a way to send DM on itch.io so I allow myself to answer you here.
My first inspiration for this game was the cellular automaton like the game of life of conway.
It is something that I was interested in I thought it could fit the 'only one' theme of jam I was participating just by replacing the classic black and white grid with a 0 and 1 grid.
Combining this with a snake game was a way to also have reliable game mechanics (eat to grow, don't eat yourself, ...) that the players will immediately understand.
Looking for idea for having both, I found that using the snake as a list of commands could be nice because it can quickly produce a lot of interesting complexity (where you move affect what actions you will execute that affect where you will move...).
I usually prefer making sandbox game but here it seems to me that having some puzzle levels was the best way to introduce the ideas of the game (but I still added a sandbox mode at the end).
To make the puzzles, I just tried randomly different configurations and chose the one where the snake ends up to a place that will look like unexpected for the players while gradually adding new mechanics.
The Turing machine was not my first inspiration but it ends up looking a little like a 2D Turing machine so I chose this name.
Hope it will be helpful ! Let me know how to contact you if you have more questions.