On the Undecidability of the Tiling Problem

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Abstract

The tiling problem is the decision problem to determine if a given finite collection of Wang tiles admits a valid tiling of the plane. In this work we give a new proof of this fact based on tiling simulations of certain piecewise affine transformations. Similar proof is also shown to work in the hyperbolic plane, thus answering an open problem posed by R.M.Robinson 1971 [9].