Adamatzky, A., Goles, E., Martinez, G. J., Tsompanas, M. A., Tegelaar, M., & Wosten, H. A. B. (2020). Fungal Automata. Complex Syst., 29(4), 759–778.
Abstract: We study a cellular automaton (CA) model of information dynamics on a single hypha of a fungal mycelium. Such a filament is divided in compartments (here also called cells) by septa. These septa are invaginations of the cell wall and their pores allow for the flow of cytoplasm between compartments and hyphae. The septal pores of the fungal phylum of the Ascomycota can be closed by organelles called Woronin bodies. Septal closure is increased when the septa become older and when exposed to stress conditions. Thus, Woronin bodies act as informational flow valves. The onedimensional fungal automaton is a binarystate ternary neighborhood CA, where every compartment follows one of the elementary cellular automaton (ECA) rules if its pores are open and either remains in state 0 (first species of fungal automata) or its previous state (second species of fungal automata) if its pores are closed. The Woronin bodies closing the pores are also governed by ECA rules. We analyze a structure of the composition space of cellstate transition and porestate transition rules and the complexity of fungal automata with just a few Woronin bodies, and exemplify several important local events in the automaton dynamics.

ConchaVega, P., Goles, E., Montealegre, P., & RiosWilson, M. (2022). On the Complexity of Stable and Biased Majority. Mathematics, 10(18), 3408.
Abstract: A majority automata is a twostate 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 timestep. 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 PComplete as a consequence of the capability of the system of simulating Boolean circuits.

ConchaVega, P., Goles, E., Montealegre, P., RiosWilson, M., & Santivanez, J. (2022). Introducing the activity parameter for elementary cellular automata. Int. J. Mod Phys. C, 33(09), 2250121.
Abstract: Given an elementary cellular automaton (ECA) with local transition rule R, two different types of local transitions are identified: the ones in which a cell remains in its current state, called inactive transitions, and the ones in which the cell changes its current state, which are called active transitions. The number of active transitions of a rule is called its activity value. Based on latter identification, a rule R1 is called a subrule of R2 if the set of active transitions of R1 is a subset of the active transitions of R2.
In this paper, the notion of subrule for elementary cellular automata is introduced and explored: first, we consider a lattice that illustrates relations of nonequivalent elementary cellular automata according to nearby subrules. Then, we introduce statistical measures that allow us to compare rules and subrules. Finally, we explore the possible similarities in the dynamics of a rule with respect to its subrules, obtaining both empirical and theoretical results.

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., & 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 nonzero 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 twodimensional lattice with von Neumann neighborhood. Later, we show that in three or more dimensions the problem is NPComplete.

Goles, E., & Moreira, A. (2012). NumberConserving Cellular Automata and Communication Complexity: A Numerical Exploration Beyond Elementary CAs. J. Cell. Autom., 7(2), 151–165.
Abstract: We perform a numerical exploration of numberconserving cellular automata (NCCA) beyond the class of elementary CAs, in search of examples with high communication complexity. We consider some possible generalizations of the elementary rule 184 (a minimal model of traffic, which is the only nontrivial elementary NCCA). as well as the classes of NCCAs which minimally extend either the radius or the state set (with respect to the 2 states and radius 1 of the elementary case). Both for 3 states and radius 1, and for 2 stales and radius 2, NCCA appear that are conjectured to have maximal (exponential) communication complexity. Examples are given also for (conjectured) linear and quadratic behaviour.

Goles, E., & Palacios, A. G. (2007). Dynamical complexity in cognitive neural networks. Biol. Res., 40(4), 479–485.
Abstract: In the last twenty years an important effort in brain sciences, especially in cognitive science, has been the development of mathematical tool that can deal with the complexity of extensive recordings corresponding to the neuronal activity obtained from hundreds of neurons. We discuss here along with some historical issues, advantages and limitations of Artificial Neural Networks (ANN) that can help to understand how simple brain circuits work and whether ANN can be helpful to understand brain neural complexity.

Goles, E., Guillon, P., & Rapaport, I. (2011). Traced communication complexity of cellular automata. Theor. Comput. Sci., 412(30), 3906–3916.
Abstract: We study cellular automata with respect to a new communication complexity problem: each of two players know half of some finite word, and must be able to tell whether the state of the central cell will follow a given evolution, by communicating as little as possible between each other. We present some links with classical dynamical concepts, especially equicontinuity, expansivity, entropy and give the asymptotic communication complexity of most elementary cellular automata. (C) 2011 Elsevier B.V. All rights reserved.

Goles, E., Lobos, F., Montealegre, P., Ruivo, E. L. P., & de Oliveira, P. P. B. (2020). Computational Complexity of the Stability Problem for Elementary Cellular Automata. J. Cell. Autom., 15(4), 261–304.
Abstract: Given an elementary cellular automaton and a cell v, we define the stability decision problem as the determination of whether or not the state of cell v will ever change, at least once, during the time evolution of the rule, over a finite input configuration. Here, we perform the study of the entire elementary cellular automata rule space, for the two possible decision cases of the problem, namely, changes in v from state 0 to 1 (0 > 1), and the other way round (1 > 0). Out of the 256 elementary cellular automata, we show that for all of them, at least one of the two decision problems is in the NC complexity class.

Goles, E., Maldonado, D., & Montealecre, P. (2021). On the complexity of asynchronous freezing cellular automata. Inf. Comput., 281, 104764.
Abstract: In this paper we study the family of freezing cellular automata (FCA) in the context of asynchronous updating schemes. A cellular automaton is called freezing if there exists an order of its states, and the transitions are only allowed to go from a lower to a higher state. A cellular automaton is asynchronous if at each timestep only one cell is updated. We define the problem ASYNCUNSTABILITY, which consists in deciding there exists a sequential updating scheme that changes the state of a given cell.
We begin showing that ASYNCUNSTABILITY is in NL for any onedimensional FCA. Then we focus on the family of lifelike freezing CA (LFCA). We study the complexity of ASYNCUNSTABILITY for all LFCA in the triangular and square grids, showing that almost all of them can be solved in NC, except for one rule for which the problem is NPComplete. (C) 2021 Elsevier Inc. All rights reserved.

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 twostate 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 PComplete. (C) 2020 Elsevier Inc. All rights reserved.

Goles, E., Meunier, P. E., Rapaport, I., & Theyssier, G. (2011). Communication complexity and intrinsic universality in cellular automata. Theor. Comput. Sci., 412(12), 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 spacetime 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.

Goles, E., MontalvaMedel, M., MacLean, S., & Mortveit, H. (2018). Block Invariance in a Family of Elementary Cellular Automata. J. Cell. Autom., 13(12), 15–32.
Abstract: We study the steady state invariance of elementary cellular automata (ECA) under different deterministic updating schemes. Specifically, we study a family of eleven ECA whose steady state invariance were left under conjecture in [2].

Goles, E., MontalvaMedel, M., Mortveit, H., & RamirezFlandes, S. (2015). Block Invariance in Elementary Cellular Automata. J. Cell. Autom., 10(12), 119–135.
Abstract: Consider an elementary cellular automaton (ECA) under periodic boundary conditions. Given an arbitrary partition of the set of vertices we consider the block updating, i.e. the automaton's local function is applied from the first to the last set of the partition such that vertices belonging to the same set are updated synchronously. The automaton is said blockinvariant if the set of periodic configurations is independent of the choice of the block updating. When the sets of the partition are singletons we have the sequential updating: vertices are updated one by one following a permutation pi. In [5] the authors analyzed the piinvariance of the 2(8) = 256 possible ECA rules (or the 88 nonredundant rules subset). Their main result was that for all n > 3, exactly 41 of these nonredundant rules are piinvariant. In this paper we determine the subset of these 41 rules that are block invariant. More precisely, for all n > 3, exactly 15 of these rules are block invariant. Moreover, we deduce that block invariance also implies that the attractor structure itself is independent of the choice of the block update.

Goles, E., Montealegre, P., Perrot, K., & Theyssier, G. (2018). On the complexity of twodimensional signed majority cellular automata. J. Comput. Syst. Sci., 91, 1–32.
Abstract: We study the complexity of signed majority cellular automata on the planar grid. We show that, depending on their symmetry and uniformity, they can simulate different types of logical circuitry under different modes. We use this to establish new bounds on their overall complexity, concretely: the uniform asymmetric and the nonuniform symmetric rules are Turing universal and have a Pcomplete prediction problem; the nonuniform asymmetric rule is intrinsically universal; no symmetric rule can be intrinsically universal. We also show that the uniform asymmetric rules exhibit cycles of superpolynomial length, whereas symmetric ones are known to have bounded cycle length. (C) 2017 Elsevier Inc. All rights reserved.

Goles, E., Montealegre, P., & Vera, J. (2016). Naming Game Automata Networks. J. Cell. Autom., 11(56), 497–521.
Abstract: In this paper we introduce automata networks to model some features of the emergence of a vocabulary related with the naming game model. We study the dynamical behaviour (attractors and convergence) of extremal and majority local functions.

Goles, E., Moreira, A., & Rapaport, I. (2011). Communication complexity in numberconserving and monotone cellular automata. Theor. Comput. Sci., 412(29), 3616–3628.
Abstract: One third of the elementary cellular automata (CAs) are either numberconserving (NCCAs) or monotone (increasing or decreasing). In this paper we prove that, for all of them, we can find linear or constant communication protocols for the prediction problem. In other words, we are able to give a succinct description for their dynamics. This is not necessarily true for general NCCAs. In fact, we also show how to explicitly construct, from any CA, a new NCCA which preserves the original communication complexity. (C) 2011 Elsevier B.V. All rights reserved.

Lobos, F., Goles, E., Ruivo, E. L. P., de Oliveira, P. P. B., & Montealegre, P. (2018). Mining a Class of Decision Problems for Onedimensional Cellular Automata. J. Cell. Autom., 13(56), 393–405.
Abstract: Cellular automata are locally defined, homogeneous dynamical systems, discrete in space, time and state variables. Within the context of onedimensional, binary, cellular automata operating on cyclic configurations of odd length, we consider the general decision problem: if the initial configuration satisfies a given property, the lattice should converge to the fixedpoint of all 1s ((1) over right arrow), or to (0) over right arrow, otherwise. Two problems in this category have been widely studied in the literature, the parity problem [1] and the density classification task [4]. We are interested in determining all cellular automata rules with neighborhood sizes of 2, 3, 4 and 5 cells (i.e., radius r of 0.5, 1, 1.5 and 2.5) that solve decision problems of the previous type. We have demonstrated a theorem that, for any given rule in those spaces, ensures the non existence of fixed points other than (0) over right arrow and (1) over right arrow for configurations of size larger than 2(2r), provided that the rule does not support different fixed points for any configuration with size smaller than or equal to 2(2r). In addition, we have a proposition that ensures the convergence to only (0) over right arrow or (1) over right arrow of any initial configuration, if the rule complies with given conditions. By means of theoretical and computational approaches, we determined that: for the rule spaces defined by radius 0.5 and r = 1, only 1 and 2 rules, respectively, converge to (1) over right arrow or (0) over right arrow, to any initial configuration, and both recognize the same language, and for the rule space defined by radius r = 1.5, 40 rules satisfy this condition and recognize 4 different languages. Finally, for the radius 2 space, out of the 4,294,967,296 different rules, we were able to significantly filter it out, down to 40,941 candidate rules. We hope such an extensive mining should unveil new decision problems of the type widely studied in the literature.

MacLean, S., MontalvaMedel, M., & Goles, E. (2019). Block invariance and reversibility of one dimensional linear cellular automata. Adv. Appl. Math., 105, 83–101.
Abstract: Consider a onedimensional, 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.

MontalvaMedel, M., de Oliveira, P. P. B., & Goles, E. (2018). A portfolio of classification problems by onedimensional cellular automata, over cyclic binary configurations and parallel update. Nat. Comput., 17(3), 663–671.
Abstract: Decision problems addressed by cellular automata have been historically expressed either as determining whether initial configurations would belong to a given language, or as classifying the initial configurations according to a property in them. Unlike traditional approaches in language recognition, classification problems have typically relied upon cyclic configurations and fully paralell (twoway) update of the cells, which render the action of the cellular automaton relatively less controllable and difficult to analyse. Although the notion of cyclic languages have been studied in the wider realm of formal languages, only recently a more systematic attempt has come into play in respect to cellular automata with fully parallel update. With the goal of contributing to this effort, we propose a unified definition of classification problem for onedimensional, binary cellular automata, from which various known problems are couched in and novel ones are defined, and analyse the solvability of the new problems. Such a unified perspective aims at increasing existing knowledge about classification problems by cellular automata over cyclic configurations and parallel update.
