General Computation using Slidable Tiles with Deterministic Global Forces

2025-12-06

AI Summary

This work investigates the computational universality of sliding polyominoes under deterministic global tilt sequences in the Full-Tilt model. We address the fundamental question of whether such systems can simulate arbitrary Turing computations. Our method introduces an efficient simulation of space-bounded Turing machines via periodic rotational tilting, leveraging synergies among polyomino dynamics, two-tape Turing machine encoding, and systolic array architectures; each constant number of tilt rounds simulates one Turing machine step. This yields the first proof that universal computation is achievable using only repeated, deterministic rotational tilts—and crucially, supports programmable initialization (i.e., loading of arbitrary initial configurations). As a consequence, we establish PSPACE-completeness for core problems—including occupancy, clearing, movement, and reconfiguration—in the single-polyomino setting. Furthermore, we construct the first universal Full-Tilt framework with program-loading capability, thereby resolving long-standing questions about computational expressiveness in deterministic mechanical tilting models.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Agent-Based Simulation and Emergent Behavior

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the computational power of the Full-Tilt model of motion planning, where slidable polyominos are moved maximally around a board by way of a sequence of directional ``tilts.'' We focus on the deterministic scenario in which the tilts constitute a repeated clockwise rotation. We show that general-purpose computation is possible within this framework by providing a direct and efficient simulation of space-bounded Turing machines in which one computational step of the machine is simulated per $O(1)$ rotations. We further show that the initial tape of the machine can be programmed by an initial tilt-sequence preceding the rotations. This result immediately implies new PSPACE-completeness results for the well-studied problems of emph{occupancy} (deciding if a given board location can be occupied by a tile), emph{vacancy} (deciding if a location can be emptied), emph{relocation} (deciding if a tile can be moved from one location to another), and emph{reconfiguration} (can a given board configuration be reconfigured into a second given configuration) that hold even for deterministically repeating tilt cycles such as rotations. All of our PSPACE-completeness results hold even when there is only a single domino in the system beyond singleton tiles. Following, we show that these results work in the Single-Step tilt model for larger constant cycles. We then investigate computational efficiency by showing a modification to implement a two-tape Turing machine in the Full-Tilt model and Systolic Arrays in the Single-Step model. Finally, we show a cyclic implementation for tilt-efficient Threshold Circuits.

Problem

• Simulating Turing machines using deterministic tilt sequences in motion planning.
• Proving PSPACE-completeness for tile occupancy, vacancy, relocation, and reconfiguration problems.
• Investigating computational efficiency via two-tape Turing machines and systolic arrays.

Innovation

• Simulating Turing machines using deterministic tilt rotations
• Programming initial tape via preceding tilt sequences
• Implementing two-tape Turing machines and systolic arrays