Concha-Vega, P., Goles, E., Montealegre, P., & Rios-Wilson, M. (2022). On the Complexity of Stable and Biased Majority. Mathematics, 10(18), 3408.
Abstract: A majority automata is a two-state cellular automata, where each cell updates its state according to the most represented state in its neighborhood. A question that naturally arises in the study of these dynamical systems asks whether there exists an efficient algorithm that can be implemented in order to compute the state configuration reached by the system at a given time-step. This problem is called the prediction problem. In this work, we study the prediction problem for a more general setting in which the local functions can be different according to their behavior in tie cases. We define two types of local rules: the stable majority and biased majority. The first one remains invariant in tie cases, and the second one takes the value 1. We call this class the heterogeneous majority cellular automata (HMCA). For this latter class, we show that in one dimension, the prediction problem for HMCA is in NL as a consequence of the dynamics exhibiting a type of bounded change property, while in two or more dimensions, the problem is P-Complete as a consequence of the capability of the system of simulating Boolean circuits.
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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., Montealegre, P., & Rios-Wilson, M. (2020). On The Effects Of Firing Memory In The Dynamics Of Conjunctive Networks. Discret. Contin. Dyn. Syst., 40(10), 5765–5793.
Abstract: A boolean network is a map F : {0, 1}(n) -> {0, 1}(n) that defines a discrete dynamical system by the subsequent iterations of F. Nevertheless, it is thought that this definition is not always reliable in the context of applications, especially in biology. Concerning this issue, models based in the concept of adding asynchronicity to the dynamics were propose. Particularly, we are interested in a approach based in the concept of delay. We focus in a specific type of delay called firing memory and it effects in the dynamics of symmetric (non-directed) conjunctive networks. We find, in the caseis in which the implementation of the delay is not uniform, that all the complexity of the dynamics is somehow encapsulated in the component in which the delay has effect. Thus, we show, in the homogeneous case, that it is possible to exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in, given an initial condition, determinate if a fixed coordinate will eventually change its state. We find again that in the non-homogeneous case all the complexity is determined by the component that is affected by the delay and we conclude in the homogeneous case that this problem is PSPACE-complete.
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Goles, E., Montealegre, P., Salo, V., & Torma, I. (2016). PSPACE-completeness of majority automata networks. Theor. Comput. Sci., 609, 118–128.
Abstract: We study the dynamics of majority automata networks when the vertices are updated according to a block sequential updating scheme. In particular, we show that the complexity of the problem of predicting an eventual state change in some vertex, given an initial configuration, is PSPACE-complete. (C) 2015 Elsevier B.V. All rights reserved.
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