# Domino Problem for Pretty Low Complexity Subshifts

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Part of the Emergence, Complexity and Computation book series (ECC,volume 42)

## Abstract

Given a set P of allowed $$n\times m$$ rectangular patterns of colors, a coloring of the grid $$\mathbb {Z}^2$$ is called valid if every $$n\times m$$ pattern in the coloring is in P. It is known that if the number of allowed $$n\times m$$ patterns is at most nm and if there exists a valid coloring of $$\mathbb {Z}^2$$ then there exists a valid periodic coloring, and consequently there is an algorithm to determine for given nm patterns if they admit any valid coloring. If the number of allowed patterns is higher the situation changes: We prove that for every $$\varepsilon >0$$ it is undecidable for given dimensions nm and a given set of at most $$(1+\varepsilon )nm$$ patterns of size $$n\times m$$ whether they admit any valid coloring. In other words, the upper bound nm is multiplicatively optimal for the number of allowed patterns that guarantees decidability and periodicity. The undecidability result that we prove is actually slightly better: it holds in the square case $$n=m$$, and instead of bound $$(1+\varepsilon )n^2$$ we can use $$n^2+f(n)n$$ for any computable function $$f:\mathbb {N}\longrightarrow \mathbb {N}$$ that is not bounded above by a constant.

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## Acknowledgements

The research was supported by the Academy of Finland grant 296018.

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Correspondence to Jarkko Kari .

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