Bitar, N., Goles, E., & Montealegre, P. (2022). COMPUTATIONAL COMPLEXITY OF BIASED DIFFUSION-LIMITED AGGREGATION. SIAM Discret. Math., 36(1), 823–866.
Abstract: Diffusion-Limited Aggregation (DLA) is a cluster-growth model that consists in a set of particles that are sequentially aggregated over a two-dimensional grid. In this paper, we introduce a biased version of the DLA model, in which particles are limited to move in a subset of possible directions. We denote by k-DLA the model where the particles move only in k possible directions. We study the biased DLA model from the perspective of Computational Complexity, defining two decision problems The first problem is Prediction, whose input is a site of the grid c and a sequence S of walks, representing the trajectories of a set of particles. The question is whether a particle stops at site c when sequence S is realized. The second problem is Realization, where the input is a set of positions of the grid, P. The question is whether there exists a sequence S that realizes P, i.e. all particles of S exactly occupy the positions in P. Our aim is to classify the Prediciton and Realization problems for the different versions of DLA. We first show that Prediction is P-Complete for 2-DLA (thus for 3-DLA). Later, we show that Prediction can be solved much more efficiently for 1-DLA. In fact, we show that in that case the problem is NL-Complete. With respect to Realization, we show that restricted to 2-DLA the problem is in P, while in the 1-DLA case, the problem is in L.
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Goles, E., & Montealegre, P. (2014). Computational complexity of threshold automata networks under different updating schemes. Theor. Comput. Sci., 559, 3–19.
Abstract: Given a threshold automata network, as well as an updating scheme over its vertices, we study the computational complexity associated with the prediction of the future state of a vertex. More precisely, we analyze two classes of local functions: the majority and the AND-OR rule (vertices take the AND or the OR logic functions over the state of its neighborhoods). Depending on the updating scheme, we determine the complexity class (NC, P, NP, PSPACE) where the prediction problem belongs. (C) 2014 Elsevier B.V. All rights reserved.
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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., & Montealegre, P. (2020). The complexity of the asynchronous prediction of the majority automata. Inf. Comput., 274(SI).
Abstract: We consider the asynchronous prediction problem for some automaton as the one consisting in determining, given an initial configuration, if there exists a non-zero probability that some selected site changes its state, when the network is updated picking one site at a time uniformly at random. We show that for the majority automaton, the asynchronous prediction problem is in NC in the two-dimensional lattice with von Neumann neighborhood. Later, we show that in three or more dimensions the problem is NP-Complete.
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Goles, E., Maldonado, D., Montealegre, P., & Ollinger, N. (2020). On the complexity of the stability problem of binary freezing totalistic cellular automata. Inf. Comput., 274, 21 pp.
Abstract: In this paper we study the family of two-state Totalistic Freezing Cellular Automata (TFCA) defined over the triangular and square grids with von Neumann neighborhoods. We say that a Cellular Automaton is Freezing and Totalistic if the active cells remain unchanged, and the new value of an inactive cell depends only on the sum of its active neighbors. We classify all the Cellular Automata in the class of TFCA, grouping them in five different classes: the Trivial rules, Turing Universal rules, Algebraic rules, Topological rules and Fractal Growing rules. At the same time, we study in this family the STABILITY problem, consisting in deciding whether an inactive cell becomes active, given an initial configuration. We exploit the properties of the automata in each group to show that: For Algebraic and Topological Rules the STABILITY problem is in NC. For Turing Universal rules the STABILITY problem is P-Complete. (C) 2020 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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