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

Abstract

Since their introduction in the 80s, sandpile models have raised interest for their simple definition and their surprising dynamical properties. In this survey we focus on the computational complexity of the prediction problem, namely, the complexity of knowing, given a finite configuration c and a cell x in c, if cell x will eventually become unstable. This is an attempt to formalize the intuitive notion of "behavioral complexity" that one easily observes in simulations. However, despite many efforts and nice results, the original question remains open: how hard is it to predict the two-dimensional sandpile model of Bak, Tang and Wiesenfeld? arXiv:1909.12150v1 [cs.DM] 26 Sep 2019 * → c . However, one can consider also other types updating policies. The sequential policy consists in choosing non-deterministically a cell from the unstable ones and in updating only this chosen cell. Then, repeat the same update policy on the newly obtained configuration and so on. It is clear that the new dynamics might *

Freezing sandpiles and Boolean threshold networks: equivalence and complexity

“…Second, in the reduction of Proposition 11 (and some subsequent ones) it seems important to have many values 4. What is the computational complexity of the weak prediction problem (given a finite stable configuration, plus only one sand grain addition, namely 1 st -col-S-PRED of the survey [4]) in this case?…”

“…The sand pile model, as well as the Boolean threshold automata, have been studied and applied extensively in various domains [4,20,9,14,12,7,13]. The classical sandpile model on the two dimensional grid with von Neumann neighborhood was introduced in the 1980 by Bak, Tang and Wiesenfeld, as a simple and natural model of some physical phenomena [1].…”

“…It turns out that the freezing world breaks this frontier, as majority and other threshold Boolean functions are not sensitive to the order of cells update in this case. Predicting freezing variants of dynamical systems may be thought as a "simplest case" study of their complexity (see [4,Proposition 6] and Remark 1 for a formal discussion).…”

Self-Organised Critical Dynamics as a Key to Fundamental Features of Complexity in Physical, Biological, and Social Networks

“…Another link connects the information complexity [13] and SOC. In the context of data analysis, it concerns the relevance [14] of the information that can be stored in the critical state, as well as the related computational complexity [15,16], the problems of prediction of individual states [17], and mesoscopic avalanche prediction [18]. The idea that complex systems exhibit a hidden order, emerging through self-organised dynamics, links it to information that that ordering patterns may contain [19].…”

Sandpile toppling on Penrose tilings: identity and isotropic dynamics

“…The (im)possibility of building crossing gates in two dimensions is studied in [22,43], and similar issues on closely related threshold automata such as the majority rule have been fruitful [26,28,29,30,31,41]. See [21] for a survey of the results on lattices.…”