But no continuous systems are in the Chomsky hierarchy, right? Isn't that just support for my suggestion that cellular automata are by definition discrete?
The simulation is still using floating point. The fundamental Floating point arithmetic is regular in principle. It's the rules and algorithms that can potentially achieve higher order.
And even classical game of live is turing complete. They built a TM inside GoL!
But it can be modeled by a finite state automaton (if the grid is finite), I suppose, so the rules are just a regular grammar.
A regular machine can output the grammar of a context sensitive language, but not accept the corresponding programms. In the same sense, the turing machine in GoL is ...
PS: I'm not sure how Lenia is implemented. If the model is that the points move, you can nevertheless think of each point as a cell on a regular grid, that holds a n-ary variable, that encodes the color (or hight or what) and the position in a x*y=n-1 dimensional coordinate sytem, with one dimension per cell.
In other words, each cell would encode its distance to all other cells. Which seems a bit redundant.
