Triominoes with rotations as universal Wang tiles
Posted: May 17th, 2017, 2:20 pm
TL;DR: Can anyone help me find a minimal set of triominoes that embed a universal computer?
This is not exactly about cellular automata, but it might be of interest to some of the same people (and you can embed 1-D cellular automata as an oriented tiling). I also don't claim to have any new ideas below. Some are definitely old, and some I may have rediscovered.
I had started to think about constraint systems involving mutually adjacent cells on a hex grid and their realization as a tiling, before noticing after a day or two that I had reinvented triominoes (more on that below).
I was originally interested in constraints that were fully symmetric up to rotations and flips, so the combination of colors assigned to the neighborhood would be independent of ordering. I was frustrated by any attempt to embed a universal computer this way and eventually realized why it is impossible: given two adjacent rows of colored hexes satisfying constraints, flip symmetry permits alternating these rows forever, so any 2-row tiling can be extended into a planar tiling. 2-row tilings are (roughly) equivalent to regular expressions, and therefore not universal.
The above observation was kind of a relief, because disallowing flips makes it easier to assign some kind of directionality. My goal is to avoid orienting the tiles by fiat like square Wang tiles (https://en.wikipedia.org/wiki/Wang_tile). I want a set of symmetric tiles that implement hex neighborhood constraints and impose their own orientation. After some trial and error, I found that this could be done with triangular tiles and that they basically match like triomonoes, though I use colors instead of numbers below. Here is an example of a tile set that imposes a (well-known, unique) 3-coloring of a hex grid. Note that in this case, the constraints are flip-symmetric, but you need to put both the tile and its reflection in the set.
Here is another tile set that impose a (well-known) 4-coloring of a hex grid. In this case, the constraints are not flip-symmetric, and the set consists of 4 triangular tiles.
Thinking in terms of embedding a 1-D CA in tiling constraints, I would like to impose a top-to-bottom orientation. I am not very happy with this result, because it requires 18 tiles and 9 "colors" (shown as colors and patterns). However, if you had a deterministic 1-D CA, you could use this to enforce predecessor/successor. This gives a brute-force recipe for embedding a 1D CA that defines a next state based on the two states above it. You just take tiles representing the CA transition rules and form a cartesian product with the tiles above so that predecessor-successor rules always go no-dot/solid-dot/hollow-dot etc.
This is enough to show that you can embed a universal computer by tiling with a set of triominoes with rotations (details left as an exercise). However, it multiplies the set of transition rules by a factor of 18 just to impose an orientation. That's pretty clumsy.
I suspect there is a much small set of triominoes that implement a universal machine, and thus the tiling problem starting with a fixed initial placement is undecidable. Anyone have a reference to results in this area? I do understand that this is equivalent to Wang tiles. My main interest lies in the fact that they are triangular and that rotations are allowed.
This is not exactly about cellular automata, but it might be of interest to some of the same people (and you can embed 1-D cellular automata as an oriented tiling). I also don't claim to have any new ideas below. Some are definitely old, and some I may have rediscovered.
I had started to think about constraint systems involving mutually adjacent cells on a hex grid and their realization as a tiling, before noticing after a day or two that I had reinvented triominoes (more on that below).
I was originally interested in constraints that were fully symmetric up to rotations and flips, so the combination of colors assigned to the neighborhood would be independent of ordering. I was frustrated by any attempt to embed a universal computer this way and eventually realized why it is impossible: given two adjacent rows of colored hexes satisfying constraints, flip symmetry permits alternating these rows forever, so any 2-row tiling can be extended into a planar tiling. 2-row tilings are (roughly) equivalent to regular expressions, and therefore not universal.
The above observation was kind of a relief, because disallowing flips makes it easier to assign some kind of directionality. My goal is to avoid orienting the tiles by fiat like square Wang tiles (https://en.wikipedia.org/wiki/Wang_tile). I want a set of symmetric tiles that implement hex neighborhood constraints and impose their own orientation. After some trial and error, I found that this could be done with triangular tiles and that they basically match like triomonoes, though I use colors instead of numbers below. Here is an example of a tile set that imposes a (well-known, unique) 3-coloring of a hex grid. Note that in this case, the constraints are flip-symmetric, but you need to put both the tile and its reflection in the set.
Here is another tile set that impose a (well-known) 4-coloring of a hex grid. In this case, the constraints are not flip-symmetric, and the set consists of 4 triangular tiles.
Thinking in terms of embedding a 1-D CA in tiling constraints, I would like to impose a top-to-bottom orientation. I am not very happy with this result, because it requires 18 tiles and 9 "colors" (shown as colors and patterns). However, if you had a deterministic 1-D CA, you could use this to enforce predecessor/successor. This gives a brute-force recipe for embedding a 1D CA that defines a next state based on the two states above it. You just take tiles representing the CA transition rules and form a cartesian product with the tiles above so that predecessor-successor rules always go no-dot/solid-dot/hollow-dot etc.
This is enough to show that you can embed a universal computer by tiling with a set of triominoes with rotations (details left as an exercise). However, it multiplies the set of transition rules by a factor of 18 just to impose an orientation. That's pretty clumsy.
I suspect there is a much small set of triominoes that implement a universal machine, and thus the tiling problem starting with a fixed initial placement is undecidable. Anyone have a reference to results in this area? I do understand that this is equivalent to Wang tiles. My main interest lies in the fact that they are triangular and that rotations are allowed.