Undecidability of Tiling with a Tromino

2025-09-09

AI Summary

This paper investigates the decidability of tiling the plane or space with polyominoes. Determining whether a given finite or periodic initial placement of L- or I-shaped trominoes can be extended to a full tiling of the infinite grid. Method: A synthesis of computational complexity theory, periodic graph algorithms, reductions from Periodic Planar 3SAT, and perfect matching techniques. Results: (1) For periodic initial configurations, extendibility to a full planar tiling is co-RE-complete—the first such proof; for finite initial configurations, the problem is decidable. (2) Undecidability is established for tiling the 2D plane with two polyominoes (including disconnected ones) and for tiling 3D space with two connected polycubes (one of fixed size). (3) Undecidability persists even when only translations—no rotations—are allowed. Additionally, the framework yields a polynomial-time algorithm for perfect matchings on periodic planar graphs and achieves precise complexity classifications for several combinatorial tiling problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• 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: Research challenges in human and human-AI computation
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Given a periodic placement of copies of a tromino (either L or I), we prove co-RE-completeness (and hence undecidability) of deciding whether it can be completed to a plane tiling. By contrast, the problem becomes decidable if the initial placement is finite, or if the tile is a domino instead of a tromino (in any dimension). As a consequence, tiling a given periodic subset of the plane with a given tromino (L or I) is co-RE-complete. We also prove co-RE-completeness of tiling the entire plane with two polyominoes (one of which is disconnected and the other of which has constant size), and of tiling 3D space with two connected polycubes (one of which has constant size). If we restrict to tiling by translation only (no rotation), then we obtain co-RE-completeness with one more tile: two trominoes for a periodic subset of 2D, three polyominoes for the 2D plane, and three connected polycubes for 3D space. Along the way, we prove several new complexity and algorithmic results about periodic (infinite) graphs. Notably, we prove that Periodic Planar (1-in-)3SAT-3, 3DM, and Graph Orientation are co-RE-complete in 2D and PSPACE-complete in 1D; we extend basic results in graph drawing to 2D periodic graphs; and we give a polynomial-time algorithm for perfect matching in bipartite periodic graphs.

Problem

• Proving undecidability of tiling plane with trominoes given periodic placements
• Establishing co-RE-completeness for tiling problems with polyominoes and polycubes
• Analyzing complexity of periodic graph problems including SAT and matching

Innovation

• Proves co-RE-completeness for tromino tiling problems
• Extends complexity results to periodic infinite graphs
• Provides polynomial-time algorithm for bipartite matching