Embedding arbitrary Boolean circuits into fungal automata

Abstract: Fungal automata are a variation of the two-dimensional sandpile automaton of Bak, Tang, and Wiesenfeld (Phys. Rev. Lett. 1987). In each step toppling cells emit grains only to some of their neighbors chosen according to a specific update sequence. We show how to embed any Boolean circuit into the initial configuration of a fungal automaton with update sequence $HV$. In particular we give a constructor that, given the description $B$ of a circuit, computes the states of all cells in the finite support of the embedding configuration in $O(\log |B|)$ space. As a consequence the prediction problem for fungal automata with update sequence $HV$ is $\mathsf{P}$-complete. This solves an open problem of Goles et al. (Phys. Lett. A, 2020).