Consider the following model: we work in the $d$-dimensional grid $\mathbb{N}^d$, and we have an alphabet $\Sigma$. The initial cell $(0,\ldots,0)\in \mathbb{N}^d$ is marked with some letter $\Sigma$, and from that we mark each cell with a letter from $\Sigma$ according to its immediate predecessors. That is, we have a function $f:\Sigma^d\to \Sigma$, and for each cell $(a_1,\ldots,a_d)$, we mark it with $f(\sigma_1,\ldots,\sigma_d)$, where $\sigma_i$ is the labeling of the neighbor $(a_1,\ldots,a_i-1,\ldots, a_d)$. This means that the marking is completely determined by $f$ and the initial symbol.

Note that at the edges, where there are fewer then $d$ neighbors, we ignore the missing values (technically, we can assume that anything outside $\mathbb{N}^d$ has some constant fixed marking).

I want to study certain properties of this model (specifically, I want to show that it is semilinear, in a sense). But I couldn't find any resources on it. Specifically, I would like to know if this model is Presburger-definable.

However, it is a very natural restriction of several models:

  • Cellular automata: it can be thought of a single-state cellular automaton, that can only change the value of a cell once (i.e., a cellular transducer).
  • Tile Self-assembly: where the glue relations allow only a deterministic assembly, but the model is not two-dimensional.
  • Multidimensional subshifts: it's not really a shift, but it does have the same flavour of finite rules.
  • Chemical-reaction networks: I'm actually not sure if this is a restriction, but it seems somewhat related.

I'd appreciate any leads on this.


1 Answer 1


If you partition $\mathbb{Z}^d$ into the $(d-1)$-dimensional slices $S(n) := \{ \vec v | \sum_{k=1}^d \vec v_k = n \}$, then your model is essentially a spacetime diagram of the $(d-1)$-dimensional cellular automaton that maps $x|_{S(n)}$ to $x|_{S(n+1)}$. The initial configuration $x|_{S(0)}$ has a special quiescent state everywhere except at the origin. If the CA is linear over a field of characteristic $p$, then these diagrams are $p$-automatic; if not, they can be quite complex, since you don't restrict $f$ in any way. They are not necessarily semilinear in either case. For example, it's pretty easy to embed a computation history of an arbitrary Turing machine in the configuration.

  • $\begingroup$ I'm not so sure about the simulation of a TM. Observe for example that starting from a fixed cell, along every axis direction (e.g., along the $(1,0,0..0)$ direction, the behaviour is semilinear). This is also true for fixed-size blocks. If TMs could be simulated, I wouldn't expect this to hold. $\endgroup$
    – Shaull
    Apr 15, 2021 at 17:02
  • 1
    $\begingroup$ @Shauli The TM can be simulated along the direction $\vec v = (1,1,\ldots,1)$, meaning that one end of its tape is at $n \vec v$ after $n$ simulated computation steps. You're right that each column along a basis vector will be eventually periodic, but other directions can exhibit complex behavior. $\endgroup$ Apr 15, 2021 at 17:58
  • $\begingroup$ I agree with the intuition, but I'm still not convinced. From what I know, we need at least 3-locality for a simulation (in 1D), and here we have only 2-locality. Perhaps in higher dimension this would cause problems, but I don't see how one could use the dimensions to store the tape in a way that would enable a simulation. At any rate, your answer did give me the term "spacetime diagram", which might lead me to a good path. Thanks! $\endgroup$
    – Shaull
    Apr 15, 2021 at 18:10
  • $\begingroup$ @Shauli You can simulate a shifted version of a 3-local CA by a 2-local CA (usually called radius-1 and radius-1/2). In 1D, the alphabet becomes $\Sigma \cup \Sigma^2$. In two steps, the CA maps $\cdots x_0 x_1 x_2 x_3 \cdots \mapsto \cdots (x_0, x_1) (x_1,x_2) (x_2,x_3) \cdots \mapsto \cdots f(x_0, x_1,x_2) f(x_1,x_2,x_3) \cdots$. $\endgroup$ Apr 15, 2021 at 18:38
  • $\begingroup$ Thanks @Ilkka Törmä! I agree this encoding somewhat kills my hope for semilinearity :) $\endgroup$
    – Shaull
    Apr 15, 2021 at 19:03

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