Embedding arbitrary Boolean circuits into fungal automata with arbitrary update sequences

TL;DR

This paper proves that predicting the evolution of fungal automata with any update sequence containing both horizontal and vertical updates is computationally hard (P-complete), extending previous results to all such schemes.

Contribution

It establishes that the prediction problem remains P-complete for any update scheme containing both H and V, generalizing prior specific cases.

Findings

Prediction problem is P-complete for any update scheme with both H and V.

Extends previous results from specific schemes to all schemes containing H and V.

Shows computational complexity of fungal automata evolution is generally hard.

Abstract

The sandpile automata of Bak, Tang, and Wiesenfeld (Phys. Rev. Lett., 1987) are a simple model for the diffusion of particles in space. A fundamental problem related to the complexity of the model is predicting its evolution in the parallel setting. Despite decades of effort, a classification of this problem for two-dimensional sandpile automata remains outstanding. Fungal automata were recently proposed by Goles et al. (Phys. Lett. A, 2020) as a spin-off of the model in which diffusion occurs either in horizontal $(H)$ or vertical $(V)$ directions according to a so-called update scheme. Goles et al. proved that the prediction problem for this model with the update scheme $H^4V^4$ is $\textbf{P}$-complete. This result was subsequently improved by Modanese and Worsch (Algorithmica, 2024), who showed the problem is $\textbf{P}$-complete also for the simpler updatenscheme $HV$. In this work, we fill in the gaps and prove that the prediction problem is $\textbf{P}$-complete for any update scheme that contains both $H$ and $V$ at least once.