Timed Prediction Problem for Sandpile Models
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909660619997184 |
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| author | Concha-Vega, Pablo Perrot, Kévin |
| author_facet | Concha-Vega, Pablo Perrot, Kévin |
| contents | We investigate the computational complexity of the timed prediction problem in two-dimensional sandpile models. This question refines the classical prediction problem, which asks whether a cell q will eventually become unstable after adding a grain at cell p from a given configuration. The prediction problem has been shown to be P-complete in several settings, including for subsets of the Moore neighborhood, but its complexity for the von Neumann neighborhood remains open. In a previous work, we provided a complete characterization of crossover gates (a key to the implementation of non-planar monotone circuits) for these small neighborhoods, leading to P-completeness proofs with only 4 and 5 neighbors among the eight adjancent cells. In this paper, we introduce the timed setting, where the goal is to determine whether cell q becomes unstable exactly at time t. We distinguish several cases: some neighborhoods support complete timed toolkits (including timed crossover gates) and exhibit P-completeness; others admit timed crossovers but suffer from synchronization issues; planar neighborhoods provably do not admit any timed crossover; and finally, for some remaining neighborhoods, we conjecture that no timed crossover is possible. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_21084 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Timed Prediction Problem for Sandpile Models Concha-Vega, Pablo Perrot, Kévin Computational Complexity Cellular Automata and Lattice Gases We investigate the computational complexity of the timed prediction problem in two-dimensional sandpile models. This question refines the classical prediction problem, which asks whether a cell q will eventually become unstable after adding a grain at cell p from a given configuration. The prediction problem has been shown to be P-complete in several settings, including for subsets of the Moore neighborhood, but its complexity for the von Neumann neighborhood remains open. In a previous work, we provided a complete characterization of crossover gates (a key to the implementation of non-planar monotone circuits) for these small neighborhoods, leading to P-completeness proofs with only 4 and 5 neighbors among the eight adjancent cells. In this paper, we introduce the timed setting, where the goal is to determine whether cell q becomes unstable exactly at time t. We distinguish several cases: some neighborhoods support complete timed toolkits (including timed crossover gates) and exhibit P-completeness; others admit timed crossovers but suffer from synchronization issues; planar neighborhoods provably do not admit any timed crossover; and finally, for some remaining neighborhoods, we conjecture that no timed crossover is possible. |
| title | Timed Prediction Problem for Sandpile Models |
| topic | Computational Complexity Cellular Automata and Lattice Gases |
| url | https://arxiv.org/abs/2506.21084 |