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Non-Uniform and Weighted Crossing Gates in Two-Dimensiona...
[Submitted on 25 Jun 2026] · 2026-06-26 · via cs updates on arXiv.org

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Abstract:Determining whether predicting two-dimensional sandpiles lies in $\mathbf{NC}$ or is $\mathbf{P}$-complete has been open for decades. Moore and Nilsson proved $\mathbf{P}$-completeness for the three dimensional case by encoding Boolean circuits into sandpiles, but this method fails in two dimension due to the impossibility of crossing gates.
In this work, we study the existence of crossing gates on non-uniform and weighted grids. We establish an equivalence between uniform weighted crossing gates and a class of simple non-uniform crossing gates, which we call primal. We also exhibit a crossing gate that inherently requires more than one crossing, rather than a single crossing as in standard constructions. Finally, we show that the equivalence between uniform weighted and primal crossings breaks down in more general settings.

Submission history

From: Pablo Concha-Vega [view email]
[v1] Thu, 25 Jun 2026 12:17:09 UTC (30 KB)