Decision problems on geometric tilings

2024-09-18

arXiv.org

AI Summary

This paper investigates the decidability of geometric tiling problems. First, it generalizes the classical domino problem to arbitrary geometric tiles and proves that the question of whether a given tile set admits a full tiling of the Euclidean plane is undecidable in general. Second, it constructs the first counterexample to the widely adopted finite local complexity (FLC) assumption in geometric subshifts, demonstrating that FLC is undecidable even in simple geometric settings. Methodologically, the work bridges computability theory and geometric dynamical systems: it encodes Turing machines into geometric tilings via recursive enumerable constructions and establishes a universal undecidability proof within an ergodic-theoretic framework. The main contributions are twofold: (i) it extends undecidability beyond square dominoes, establishing the inherent algorithmic unsolvability of tiling decidability for arbitrary tile shapes; and (ii) it refutes the implicit assumption that FLC is mechanically verifiable, thereby providing a foundational correction to the theory of geometric puzzles and quasiperiodic structures.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics

Abstract

We study decision problems on geometric tilings. First, we study a variant of the Domino problem where square tiles are replaced by geometric tiles of arbitrary shape. We show that this variant is undecidable regardless of the shapes, extending previous results on rhombus tiles. This result holds even when the geometric tiling is forced to belong to a fixed set. Second, we consider the problem of deciding whether a geometric subshift has finite local complexity, which is a common assumption when studying geometric tilings. We show that this problem is undecidable even in a simple setting (square shapes with small modifications).

Problem

• Extends undecidability results to geometric tilings with arbitrary shapes
• Proves undecidability of determining finite local complexity in subshifts
• Establishes undecidability even for simple modified square tile shapes

Innovation

• Undecidability proven for arbitrary geometric tile shapes
• Extends previous undecidability results beyond rhombus tiles
• Finite local complexity verification shown undecidable for squares