General Computation using Slidable Tiles with Deterministic Global Forces
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arXiv
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2025
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| author | Avila-Jimenez, Alberto Barreda, David Evans, Sarah-Laurie Luchsinger, Austin Massie, Aiden Schweller, Robert Tomai, Evan Wylie, Tim |
| author_facet | Avila-Jimenez, Alberto Barreda, David Evans, Sarah-Laurie Luchsinger, Austin Massie, Aiden Schweller, Robert Tomai, Evan Wylie, Tim |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06574 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | General Computation using Slidable Tiles with Deterministic Global Forces Avila-Jimenez, Alberto Barreda, David Evans, Sarah-Laurie Luchsinger, Austin Massie, Aiden Schweller, Robert Tomai, Evan Wylie, Tim Computational Geometry 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. |
| title | General Computation using Slidable Tiles with Deterministic Global Forces |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2512.06574 |