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Given a set of colors $T = \{1, \ldots, k\}$, a set of horizontal or vertical rules $H, V \subseteq T \times T$ of ordered pairs, and a chosen color $b = 1$, the tiling language is defined as follows:

$$ L = \{n \mid n \text{ is encoded in binary, I can tile an } n \times n \text{ grid with colors of $T$ respecting the horizontal rules } H, \text{ the vertical rules } V, \text{ and placing the boundary conditions } b \text{ at the 4 corners}\} $$

It has been proven that $T$, $b$, $H$, and $V$ exist, such that this language is NEXP-complete (Theorem 2.2 in Section 2 of this paper).

Based on my understanding, in this paper, they take any language $L'\in$ NEXP and reduce it to Tiling problems where the rules are given by the Cook-Levin reduction of the non-deterministic Turing Machine deciding $L'$. This reduction is a bit complicated and poorly interpretable in practice.

My question is: is there another way to prove this? For example, with a natural reduction from one of the succinct languages that is NEXP-complete?

I can't find anything similar in the literature for this specific version of tiling, while for other versions of tiling, such as the one where each tile is divided into 4 quadrants, similar results exist.

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