Formenti, E., Goles, E., & Martin, B. (2012). Computational Complexity of Avalanches in the Kadanoff Sandpile Model. Fundam. Inform., 115(1), 107–124.
Abstract: This paper investigates the avalanche problem AP for the Kadanoff sandpile model (KSPM). We prove that (a slight restriction of) AP is in NC1 in dimension one, leaving the general case open. Moreover, we prove that AP is Pcomplete in dimension two. The proof of this latter result is based on a reduction from the monotone circuit value problem by building logic gates and wires which work with an initial sand distribution in KSPM. These results are also related to the known prediction problem for sandpiles which is in NC1 for onedimensional sandpiles and Pcomplete for dimension 3 or higher. The computational complexity of the prediction problem remains open for the Bak's model of twodimensional sandpiles.

Gajardo, A., & Goles, E. (2006). Crossing information in twodimensional Sandpiles. Theor. Comput. Sci., 369(13), 463–469.
Abstract: We prove that in a twodimensional Sandpile automaton, embedded in a regular infinite planar cellular space, it is impossible to cross information, if the bit of information is the presence (or absence) of an avalanche. This proves that it is impossible to embed arbitrary logical circuits in a Sandpile through quiescent configurations. Our result applies also for the nonplanar neighborhood of Moore. Nevertheless, we also show that it is possible to compute logical circuits with a twodimensional Sandpile, if a neighborhood of radius two is used in Z(2); crossing information becomes possible in that case, and we conclude that for this neighborhood the Sandpde is Pcomplete and Turing universal. (c) 2006 Elsevier B.V. All rights reserved.

Goles, E., Tsompanas, M. A., Adamatzky, A., Tegelaar, M., Wosten, H. A. B., & Martinez, G. J. (2020). Computational universality of fungal sandpile automata. Phys. Lett. A, 384(22), 8 pp.
Abstract: Hyphae within the mycelia of the ascomycetous fungi are compartmentalised by septa. Each septum has a pore that allows for intercompartmental and interhyphal streaming of cytosol and even organelles. The compartments, however, have special organelles, Woronin bodies, that can plug the pores. When the pores are blocked, no flow of cytoplasm takes place. Inspired by the controllable compartmentalisation within the mycelium of the ascomycetous fungi we designed twodimensional fungal automata. A fungal automaton is a cellular automaton where communication between neighbouring cells can be blocked on demand. We demonstrate computational universality of the fungal automata by implementing sandpile cellular automata circuits there. We reduce the Monotone Circuit Value Problem to the Fungal Automaton Prediction Problem. We construct families of wires, crossovers and gates to prove that the fungal automata are Pcomplete. (C) 2020 Elsevier B.V. All rights reserved.
