Crossing information in two-dimensional Sandpiles

Computational Complexity of the Avalanche Problem on One Dimensional Kadanoff Sandpiles

“…It also remains to classify the computational complexity of avalanches in two dimensions when the parameter p equals 1 (that is, in the classical model introduced by Bak et al [1]). As it is exposed in [6], this question interestingly emphasizes the links between NC, P-completeness, and information crossing.…”

“…In this model AP is P-complete for all parameter p ą 1 [2]. The second definition follows the original model of Bak, Tang and Wiesenfeld [1], and it has been proved that information cannot cross (under reasonable conditions) when p " 1, a strong obstacle for a reduction to a P-complete circuit value problem [6]. -In dimension three or greater: sandpiles are capable of universal computation [7].…”

How Hard is it to Predict Sandpiles on Lattices? A Survey

“…The difficulty in applying Banks' approach here is to overcome planarity imposed by the two-dimensional grid with von Neumann or Moore neighborhood, since the monotone planar circuit value problem (MPCVP) is in NC. In fact, it has been proven to be impossible to perform elementary forms of signal cross-over in this model [32]. The precise statement is a bit technical to state 4 , but corresponds neatly to the intuition of having two potential sequences of cell topplings representing two wires that may convey a bit of information, and cross each other without interacting (i.e.…”

“…[32]). It is impossible to perform elementary forms of signal cross-over in von Neumann and Moore neighborhoods of radius one.…”

Any Shape Can Ultimately Cross Information on Two-Dimensional Abelian Sandpile Models

“…The goal is thus to find conditions on a neighborhood so that it cannot perform a crossing (this requires a precise definition of crossing), and prove that these conditions also imply that the prediction problem is in NC. As an hint for the existence of such a link, it is proven in [7] that crossing information is not possible with von Neumann neighborhood of radius one, for which the computational complexity of the prediction problem has never been proven to be P-complete (neither in NC). The present work continues the study on general uniform neighborhoods, and shows that the conditions on the neighborhood so that it can or cannot perform crossing are intrinsically discrete.…”