Embedding Arbitrary Boolean Circuits into Fungal Automata

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 P-complete. This solves an open problem of Goles et al. (Phys. Lett. A, 2020).