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Goles, E., & Montealegre, P. (2015). The complexity of the majority rule on planar graphs. Adv. Appl. Math., 64, 111–123.
Abstract: We study the complexity of the majority rule on planar automata networks. We reduce a special case of the Monotone Circuit Value Problem to the prediction problem of determining if a vertex of a planar graph will change its state when the network is updated with the majority rule. (C) 2014 Elsevier Inc. All rights reserved.
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Goles, E., & Noual, M. (2012). Disjunctive networks and update schedules. Adv. Appl. Math., 48(5), 646–662.
Abstract: In this paper, we present a study of the dynamics of disjunctive networks under all block-sequential update schedules. We also present an extension of this study to more general fair periodic update schedules, that is, periodic update schedules that do not update some elements much more often than some others. Our main aim is to classify disjunctive networks according to the robustness of their dynamics with respect to changes of their update schedules. To study this robustness, we focus on one property, that of being able to cycle dynamically. (C) 2012 Elsevier Inc. All rights reserved.
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Goles, E., & Salinas, L. (2010). Sequential operator for filtering cycles in Boolean networks. Adv. Appl. Math., 45(3), 346–358.
Abstract: Given a Boolean network without negative circuits, we propose a polynomial algorithm to build another network such that, when updated in parallel, it has the same fixed points than the original one, but it does not have any dynamical cycle. To achieve that, we apply a network transformation related to the sequential update. As a corollary, we can find a fixed point in polynomial time for this kind of networks. (C) 2010 Elsevier Inc. All rights reserved.
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Goles, E., Montalva-Medel, M., Montealegre, P., & Rios-Wilson, M. (2022). On the complexity of generalized Q2R automaton. Adv. Appl. Math., 138, 102355.
Abstract: We study the dynamic and complexity of the generalized Q2R automaton. We show the existence of non-polynomial cycles as well as its capability to simulate with the synchronous update the classical version of the automaton updated under a block sequential update scheme. Furthermore, we show that the decision problem consisting in determine if a given node in the network changes its state is P-Hard.
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Goles, E., Montealegre, P., & Perrot, K. (2021). Freezing sandpiles and Boolean threshold networks: Equivalence and complexity. Adv. Appl. Math., 125, 102161.
Abstract: The NC versus P-hard classification of the prediction problem for sandpiles on the two dimensional grid with von Neumann neighborhood is a famous open problem. In this paper we make two kinds of progresses, by studying its freezing variant. First, it enables to establish strong connections with other well known prediction problems on networks of threshold Boolean functions such as majority. Second, we can highlight some necessary and sufficient elements to the dynamical complexity of sandpiles, with a surprisingly crucial role of cells with two grains. (C) 2021 Elsevier Inc. All rights reserved.
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MacLean, S., Montalva-Medel, M., & Goles, E. (2019). Block invariance and reversibility of one dimensional linear cellular automata. Adv. Appl. Math., 105, 83–101.
Abstract: Consider a one-dimensional, binary cellular automaton f (the CA rule), where its n nodes are updated according to a deterministic block update (blocks that group all the nodes and such that its order is given by the order of the blocks from left to right and nodes inside a block are updated synchronously). A CA rule is block invariant over a family F of block updates if its set of periodic points does not change, whatever the block update of F is considered. In this work, we study the block invariance of linear CA rules by means of the property of reversibility of the automaton because such a property implies that every configuration has a unique predecessor, so, it is periodic. Specifically, we extend the study of reversibility done for the Wolfram elementary CA rules 90 and 150 as well as, we analyze the reversibility of linear rules with neighbourhood radius 2 by using matrix algebra techniques. (C) 2019 Elsevier Inc. All rights reserved.
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