Fixed-point tile sets and their applications

An aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). We present a new construction of an aperiodic tile set that is based on Kleene’s fixed-point construction instead of geometric arguments. This construction is similar to J. von Neumann self-reproducing automata; similar ideas were also used by P. Gacs in the context of error-correcting computations. This construction it rather flexible, so it can be used in many ways: we show how it can be used to implement substitution rules, to construct strongly aperiodic tile sets (any tiling is far from any periodic tiling), to give a new proof for the undecidability of the domino problem and related results, characterize effectively closed 1D subshift it terms of 2D shifts of finite type (improvement of a result by M. Hochman), to construct a tile set which has only complex tilings, and to construct a “robust” aperiodic tile set that does not have periodic (or close to periodic) tilings even if we allow some (sparse enough) tiling errors. For the latter we develop a hierarchical classification of points in random sets into islands of different ranks. Finally, we combine and modify our tools to prove our main result: there exists a tile set such that all tilings have high Kolmogorov complexity even if (sparse enough) tiling errors are allowed.

💡 Research Summary

The paper revisits the classical problem of aperiodic tilings from a computational‑theoretic perspective and introduces a completely new construction based on Kleene’s fixed‑point theorem. Instead of relying on intricate geometric arguments, the authors encode a self‑referential program into the tiles themselves. Each tile carries a “code” that inspects the configuration of its neighbours and verifies that the local pattern is a correct simulation of a lower‑level tiling. This hierarchical self‑simulation creates a fixed‑point structure reminiscent of von Neumann’s self‑reproducing automata and of Gács’s error‑correcting computations.

The construction proceeds in several layers. The lowest layer consists of data tiles that represent bits of a simulated configuration. Above them sit verification tiles that enforce that the data tiles obey a prescribed substitution rule. Because the verification tiles themselves are subject to the same verification process at the next level, the whole plane becomes a recursive simulation of its own description. Any globally periodic arrangement would inevitably violate the fixed‑point condition at some scale, which yields a strong form of aperiodicity: not only are periodic tilings impossible, but any tiling stays at a bounded distance from every periodic pattern.

Using this framework the authors obtain a suite of applications. First, they show how any substitution system can be turned into an aperiodic tile set, reproducing classical results without geometric constructions. Second, they define “strong aperiodicity” and prove that their tiles enforce a uniform lower bound on the Hamming distance to any periodic pattern, regardless of the size of the region examined. Third, they give a new proof of the undecidability of the domino problem: because the tiles implement a fixed‑point computation, there is no algorithm that can decide whether a given finite set of tiles admits a tiling of the plane.

The paper also bridges symbolic dynamics and computability. By encoding an effectively closed one‑dimensional subshift into the fixed‑point tiling, the authors improve on Hochman’s result, showing that every such subshift is topologically conjugate to a two‑dimensional shift of finite type. This dimensional lift preserves algorithmic complexity and provides a clean reduction from 1‑D to 2‑D dynamics.

A particularly striking contribution is the construction of tile sets whose every valid tiling has high Kolmogorov complexity. The authors attach a complexity constraint to local patterns: a tile may appear only if the surrounding block has a description longer than a prescribed threshold. Combined with the fixed‑point hierarchy, this forces any infinite tiling to contain incompressible information at every scale.

The authors further extend the model to tolerate sparse errors. They introduce a hierarchical classification of error locations into “islands” of increasing rank. Tiles belonging to higher‑rank islands are surrounded by special guard tiles that isolate the error and prevent it from propagating upward in the hierarchy. Even when a sparse set of errors is present, the fixed‑point verification still guarantees strong aperiodicity and high complexity of the resulting tiling.

Finally, the main theorem states that there exists a tile set such that every tiling, even one that contains a sparse set of admissible errors, has high Kolmogorov complexity and remains far from any periodic configuration. The proof assembles all previously developed tools: the fixed‑point construction, the strong aperiodicity bound, the island‑based error‑correction scheme, and the complexity‑threshold enforcement.

In conclusion, the paper provides a versatile, computation‑centric toolkit for constructing aperiodic tilings with a variety of desirable properties—strong aperiodicity, undecidability, dimensional reduction, high algorithmic complexity, and robustness to errors. These results open new avenues for applying tiling theory to areas such as quasicrystals, fault‑tolerant computation, and the study of complex dynamical systems.