General Computation using Slidable Tiles with Deterministic Global Forces

TL;DR

This paper demonstrates that the Full-Tilt model of motion planning with slidable tiles can simulate space-bounded Turing machines, establishing PSPACE-completeness for several tile reconfiguration problems and exploring computational efficiency in tilt models.

Contribution

It introduces a direct simulation of Turing machines within the Full-Tilt model, proving PSPACE-completeness for key problems and extending results to the Single-Step tilt model and other computational frameworks.

Findings

Simulation of Turing machine steps with O(1) rotations

PSPACE-completeness of occupancy, vacancy, relocation, reconfiguration problems

Implementation of two-tape Turing machines and systolic arrays in tilt models

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.