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Goles, E., Adamatzky, A., Montealegre, P., & Rios-Wilson, M. (2021). Generating Boolean Functions on Totalistic Automata Networks. Int. J. Unconv. Comput., 16(4), 343–391.
Abstract: We consider the problem of studying the simulation capabilities of the dynamics of arbitrary networks of finite states machines. In these models, each node of the network takes two states 0 (passive) and 1 (active). The states of the nodes are updated in parallel following a local totalistic rule, i.e., depending only on the sum of active states. Four families of totalistic rules are considered: linear or matrix defined rules (a node takes state 1 if each of its neighbours is in state 1), threshold rules (a node takes state 1 if the sum of its neighbours exceed a threshold), isolated rules (a node takes state 1 if the sum of its neighbours equals to some single number) and interval rule (a node takes state 1 if the sum of its neighbours belong to some discrete interval). We focus in studying the simulation capabilities of the dynamics of each of the latter classes. In particular, we show that totalistic automata networks governed by matrix defined rules can only implement constant functions and other matrix defined functions. In addition, we show that t by threshold rules can generate any monotone Boolean functions. Finally, we show that networks driven by isolated and the interval rules exhibit a very rich spectrum of boolean functions as they can, in fact, implement any arbitrary Boolean functions. We complement this results by studying experimentally the set of different Boolean functions generated by totalistic rules on random graphs.
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Goles, E., Meunier, P. E., Rapaport, I., & Theyssier, G. (2011). Communication complexity and intrinsic universality in cellular automata. Theor. Comput. Sci., 412(1-2), 2–21.
Abstract: The notions of universality and completeness are central in the theories of computation and computational complexity. However, proving lower bounds and necessary conditions remains hard in most cases. In this article, we introduce necessary conditions for a cellular automaton to be “universal”, according to a precise notion of simulation, related both to the dynamics of cellular automata and to their computational power. This notion of simulation relies on simple operations of space-time rescaling and it is intrinsic to the model of cellular automata. intrinsic universality, the derived notion, is stronger than Turing universality, but more uniform, and easier to define and study. Our approach builds upon the notion of communication complexity, which was primarily designed to study parallel programs, and thus is, as we show in this article, particulary well suited to the study of cellular automata: it allowed us to show, by studying natural problems on the dynamics of cellular automata, that several classes of cellular automata, as well as many natural (elementary) examples, were not intrinsically universal. (C) 2010 Elsevier B.V. All rights reserved.
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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 inter-compartmental and inter-hyphal 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 two-dimensional 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, cross-overs and gates to prove that the fungal automata are P-complete. (C) 2020 Elsevier B.V. All rights reserved.
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