Embedding arbitrary Boolean circuits into fungal automata with arbitrary update sequences

TL;DR

It is proved that the prediction problem for this model with the update scheme $H^4V^4$ is $\textbf{P}$-complete for any update scheme that contains both $H$ and $V$ at least once.

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.

Figures (22)

• Figure 1: Updates according to the word $Z_1 = HVVHHHV \in {H,V}^*$. The initial configuration $c$ is shown in the top-left. The arrows, starting to the right, indicate the next configuration after each corresponding update. The configuration immediately below the initial one corresponds to the state after completing the entire cycle of the word.
• Figure 2: On the left, blocks for $Z = HVVHHHV \in \{H,V\}^{4,3,7}$ divided with thick black lines and with respective positive and negative sources marked with $+,-$. In green are marked diagonally connected blocks (white $Z$-blocks are diagonally connected as well). On the right, letting $B$ be the central block, $\overline{B}$ is marked in blue.
• Figure 3: Source bridges modification. At each source cell, we concatenate cells with value 3 either above or below it. This ensures that a signal reaches the source cell of the block, allowing it to start functioning as intended for the new, non-shifted word.
• Figure 4: Bridge notation for a positive bridge $T = (P, c)$ connecting blocks $B_1$ and $B_2$, which are highlighted in green. Light gray lines indicate the division between $Z$-blocks while the bridge is marked in dark gray. The squares refer to the cells $P(0)$ and $P(k)$. The concavity of the bridge is intentional, highlighting that $T$ starts with a horizontal step and ends with a vertical one. The $Z$-Blocks $B_i$ with $i \in \{3,4,5,6\}$, are relevant for \ref{['remarkfunc']}.
• Figure 5: Functioning of a bridge with a configuration as in \ref{['lem:funcbridge']} (Every cell outside the bridge has value less than 3) for the word $Z_1 = HVVHHHV$. The figure shows how the configuration evolves through each of the transitions until completing a full cycle of $Z_1$. In blue, in the leftmost figure, the cells that are part of the $Z$-path defining the bridge connecting the $Z$-blocks (top-left and bottom-right) are shown. The signal traveling from one block to another is highlighted in red, and the cells outside the light red rectangle remain unchanged with respect to the initial configuration.

Theorems & Definitions (28)

• theorem thmcountertheorem
• definition thmcounterdefinition
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• remark thmcounterremark
• corollary thmcountercorollary
• proof
• lemma thmcounterlemma