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arXiv2020

Formation of morphogenetic patterns in cellular automata

One of the most important problems in contemporary science, and especially in biology, is to reveal mechanisms of pattern formation. On the level of biological tissues, patterns form due to interactions between cells. These interactions can be long-range if mediated by diffusive molecules or short-range when associated with cell-to-cell contact sites. Mathematical studies of long-range interactions in-volve models based on differential equations while short-range interactions are modelled using discrete type models. In this paper, we use cellular automata (CA) technique to study formation of patterns due to short-range interactions. Namely, we use von Neumann cellular automata represented by a finite set of lattices whose states evolve according to transition rules. Lattices can be considered as representing biological cells (which, in the simplest case, can only be in one of the two different states) while the transition rules define changes in their states due to the cell-to-cell contact interactions. In this model, we identify rules result-ing in the formation of stationary periodic patterns. In our analysis, we distin-guish rules which do not destroy preset patterns and those which cause pattern formation from random initial conditions. Also, we check whether the forming patterns are resistant to noise and analyze the time frame for their formation. Transition rules which allow formation of stationary periodic patterns are then discussed in terms of pattern formation in biology.

nlin.CG↗

arXiv2020

Anticipation decides on lane formation in pedestrian counterflow -- a simulation study

Human crowds base most of their behavioral decisions upon anticipated states of their walking environment. We explore a minimal version of a lattice model to study lanes formation in pedestrian counterflow. Using the concept of horizon depth, our simulation results suggest that the anticipation effect together with the presence of a small background noise play an important role in promoting collective behaviors in a counterflow setup. These ingredients facilitate the formation of seemingly stable lanes and ensure the ergodicity of the system.

nlin.CG↗

arXiv2020

Explicit solution of the Cauchy problem for cellular automaton rule 172

Cellular automata (CA) are fully discrete alternatives to partial differential equations (PDE). For PDEs, one often considers the Cauchy problem, or initial value problem: find the solution of the PDE satisfying a given initial condition. For many PDEs of the first order in time, it is possible to find explicit formulae for the solution at the time $t>0$ if the solution is known at $t=0$. Can something similar be achieved for CA? We demonstrate that this is indeed possible in some cases, using elementary CA rule 172 as an example. We derive an explicit expression for the state of a given cell after $n$ iteration of the rule 172, assuming that states of all cells are known at $n=0$. We then show that this expression ("solution of the CA") can be used to obtain an expected value of a given cell after $n$ iterations, provided that the initial condition is drawn from a Bernoulli distribution. This can be done for both finite and infinite lattices, thus providing an interesting test case for investigating finite size effects in CA.

nlin.CG↗

arXiv2020

Fungal Automata

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 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 one dimensional fungal automata is a binary state ternary neighbourhood CA, where every compartment follows one of the elementary cellular automata (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 analyse a structure of the composition space of cell-state transition and pore-state transitions rules, complexity of fungal automata with just few Woronin bodies, and exemplify several important local events in the automaton dynamics.

nlin.CG↗

arXiv2020

Causal Emergence in Discrete and Continuous Dynamical Systems

Emergence, the phenomena where a system's micro-scale dynamics facilitate the development of non-trivial, informative higher scales, has become a foundational concept in modern sciences, tying together fields as diverse as physics, biology, economics, and ecology. Despite it's apparent universality and the considerable interest, historically researchers have struggled to provide a rigorous, formal definition of emergence that is applicable across fields. Recent theoretical work using information theory and network science to formalize emergence in state-transition networks (causal emergence) has provided a promising way forward, however the relationship between this new framework and other well-studied system dynamics is unknown. In this study, we apply causal emergence analysis to two well-described dynamical systems: the 88 unique elementary cellular automata and the continuous Rossler system in periodic, critical, and chaotic regimes. We find that emergence, as well as its component elements (determinism, degeneracy, and effectiveness) vary dramatically in different dynamical regimes in sometimes unexpected ways. We conclude that the causal emergence framework provides a rich new area of research to explore both to theoreticians and natural scientists in many fields.

nlin.CG↗

arXiv2020

A faster algorithm for the FSSP in one-dimensional CA with multiple speeds

In cellular automata with multiple speeds for each cell $i$ there is a positive integer $p_i$ such that this cell updates its state still periodically but only at times which are a multiple of $p_i$. Additionally there is a finite upper bound on all $p_i$. Manzoni and Umeo have described an algorithm for these (one-dimensional) cellular automata which solves the Firing Squad Synchronization Problem. This algorithm needs linear time (in the number of cells to be synchronized) but for many problem instances it is slower than the optimum time by some positive constant factor. In the present paper we derive lower bounds on possible synchronization times and describe an algorithm which is never slower and in some cases faster than the one by Manzoni and Umeo and which is close to a lower bound (up to a constant summand) in more cases.

cs.CC↗

arXiv2020

Rule 184 fuzzy cellular automaton as a mathematical model for traffic flow

The rule 184 fuzzy cellular automaton is regarded as a mathematical model of traffic flow because it contains the two fundamental traffic flow models, the rule 184 cellular automaton and the Burgers equation, as special cases. We show that the fundamental diagram (flux-density diagram) of this model consists of three parts: a free-flow part, a congestion part and a two-periodic part. The two-periodic part, which may correspond to the synchronized mode region, is a two-dimensional area in the diagram, the boundary of which consists of the free-flow and the congestion parts. We prove that any state in both the congestion and the two-periodic parts is stable, but is not asymptotically stable, while that in the free-flow part is unstable. Transient behaviour of the model and bottle-neck effects are also examined by numerical simulations. Furthermore, to investigate low or high density limit, we consider ultradiscrete limit of the model and show that any ultradiscrete state turns to a travelling wave state of velocity one in finite time steps for generic initial conditions.

nlin.PS↗

arXiv2020

Effect of Autonomous Driving on Traffic Breakdown in Mixed Traffic Flow: A Critical Mini-Review

A critical analysis of the effect of autonomous driving vehicles on traffic breakdown in mixed traffic flow is made. Classical adaptive cruise control (ACC) and ACC based on three-phase traffic theory (TPACC) are considered. We show that within a wide range of dynamic parameters of classical ACC, the ACC-vehicles can initiate traffic breakdown and reduce highway capacity. Contrarily, in the same range of parameters of TPACC, the TPACC-vehicles do not initiate traffic breakdown. To understand physical reasons for the effect of classical ACC- and TPACC-vehicles on traffic breakdown, we introduce a model of ACC that can be considered a combination of dynamic features of classical ACC and TPACC. With the use of this model, we find how the amplitude of a local speed disturbance caused by the ACC in a vicinity of a bottleneck and the probability of traffic breakdown depend on the dynamic parameters of the ACC.

physics.soc-ph↗

arXiv2022

Emissions minimization on road networks via Generic Second Order Models

In this paper we consider the problem of estimating emissions due to vehicular traffic on complex networks, and minimizing their effect by regulating traffic at junctions. For the traffic evolution, we consider a Generic Second Order Model, which encompasses the majority of two-equations (i.e. second-order) models available in the literature, and extend it to road networks with merge and diverge junctions. The dynamics on the whole network is determined by selecting a solution to the Riemann Problems at junctions, i.e. the Cauchy problems with constant initial data on each incident road. The latter are solved assuming the maximization of the flow and assigning a traffic distribution coefficient for outgoing roads of diverges, and a priority rule for incoming roads of merges. A general emission model is considered and its parameters are tuned to the NOx emission rate. The minimization of emissions is then formulated in terms of the traffic distribution and priority parameters, taking into account travel times. A comparison is provided between roundabouts with optimized parameters and traffic lights, which correspond to time-varying traffic priorities. Our approach can be adapted to manage traffic in complex networks in order to reduce emissions while keeping travel time at acceptable levels.

physics.soc-ph↗

arXiv2020

Exploring Semi-bent Boolean Functions Arising from Cellular Automata

Semi-bent Boolean functions are interesting from a cryptographic standpoint, since they possess several desirable properties such as having a low and flat Walsh spectrum, which is useful to resist linear cryptanalysis. In this paper, we consider the search of semi-bent functions through a construction based on cellular automata (CA). In particular, the construction defines a Boolean function by computing the XOR of all output cells in the CA. Since the resulting Boolean functions have the same algebraic degree of the CA local rule, we devise a combinatorial algorithm to enumerate all quadratic Boolean functions. We then apply this algorithm to exhaustively explore the space of quadratic rules of up to 6 variables, selecting only those for which our CA-based construction always yields semi-bent functions of up to 20 variables. Finally, we filter the obtained rules with respect to their balancedness, and remark that the semi-bent functions generated through our construction by the remaining rules have a constant number of linear structures.

nlin.CG↗

arXiv2020

Exploring Millions of 6-State FSSP Solutions: the Formal Notion of Local CA Simulation

In this paper, we come back on the notion of local simulation allowing to transform a cellular automaton into a closely related one with different local encoding of information. This notion is used to explore solutions of the Firing Squad Synchronization Problem that are minimal both in time (2n -- 2 for n cells) and, up to current knowledge, also in states (6 states). While only one such solution was proposed by Mazoyer since 1987, 718 new solutions have been generated by Clergue, Verel and Formenti in 2018 with a cluster of machines. We show here that, starting from existing solutions, it is possible to generate millions of such solutions using local simulations using a single common personal computer.

nlin.CG↗

arXiv2020

Sandpile toppling on Penrose tilings: identity and isotropic dynamics

We present experiments of sandpiles on grids (square, triangular, hexagonal) and Penrose tilings. The challenging part is to program such simulator; and our javacript code is available online, ready to play! We first present some identity elements of the sandpile group on these aperiodic structures, and then study the stabilization of the maximum stable configuration plus the identity, which lets a surprising circular shape appear. Roundness measurements reveal that the shapes are not approaching perfect circles, though they are close to be. We compare numerically this almost isotropic dynamical phenomenon on various tilings.

nlin.CG↗

arXiv2020

Computational model on COVID-19 Pandemic using Probabilistic Cellular Automata

Coronavirus disease (COVID-19) which is caused by SARS-COV2 has become a pandemic. This disease is highly infectious and potentially fatal, causing a global public health concern. To contain the spread of COVID-19, governments are adopting nationwide interventions, like lockdown, containment and quarantine, restrictions on travel, cancelling social events and extensive testing. To understand the effects of these measures on the control of the epidemic in a data-driven manner, we propose a probabilistic cellular automata (PCA) based modified SEIQR model. The transitions associated with the model is driven by data available on chronology, symptoms, pathogenesis and transmissivity of the virus. By arguing that the lattice-based model captures the features of the dynamics along with the existing fluctuations, we perform rigorous computational analyses of the model to take into account of the spatial dynamics of social distancing measures imposed on the people. Considering the probabilistic behavioural aspects associated with mitigation strategies, we study the model considering factors like population density and testing efficiency. Using the model, we focus on the variability of epidemic dynamics data for different countries and point out the reasons behind these contrasting observations. To the best of our knowledge, this is the first attempt to model COVID-19 spread using PCA that gives us both spatial and temporal variations of the infection spread with the insight about the contributions of different infection parameters.

physics.soc-ph↗

arXiv2020

Relationship of Two Discrete Dynamical Models: One-dimensional Cellular Automata and Integral Value Transformations

Cellular Automaton (CA) and an Integral Value Transformation (IVT) are two well established mathematical models which evolve in discrete time steps. Theoretically, studies on CA suggest that CA is capable of producing a great variety of evolution patterns. However computation of non-linear CA or higher dimensional CA maybe complex, whereas IVTs can be manipulated easily. The main purpose of this paper is to study the link between a transition function of a one-dimensional CA and IVTs. Mathematically, we have also established the algebraic structures of a set of transition functions of a one-dimensional CA as well as that of a set of IVTs using binary operations. Also DNA sequence evolution has been modelled using IVTs.

nlin.CG↗

arXiv2020

Drone swarms in fire suppression activities

Recent huge technological development of Unmanned Aerial Vehicles (UAVs) can provide breakthrough means of fighting wildland fires. We propose an innovative forest firefighting system based on the use of a swarm of hundreds of UAVs able to generate a continuous flow of extinguishing liquid on the fire front, simulating rain effect. Automatic battery replacement and refilling of the extinguishing liquid ensure the continuity of the action, and fire-resistant materials protect drones exposed to possible high temperatures. We demonstrate the validity of the approach in Mediterranean scrub first computing the critical water flow rate according to the main factors involved in the evolution of a fire, then estimating the number of linear meters of active fire front that can be extinguished depending on the number of drones available and the amount of extinguishing fluid carried. A fire propagation cellular automata model is also employed to study the evolution of the fire. Simulation results suggest that the proposed system can successfully integrate, or in case of low-intensity and limited extent fires completely replace, current forest firefighting techniques.

cs.RO↗

arXiv2021

Hierarchically Organized Latent Modules for Exploratory Search in Morphogenetic Systems

Self-organization of complex morphological patterns from local interactions is a fascinating phenomenon in many natural and artificial systems. In the artificial world, typical examples of such morphogenetic systems are cellular automata. Yet, their mechanisms are often very hard to grasp and so far scientific discoveries of novel patterns have primarily been relying on manual tuning and ad hoc exploratory search. The problem of automated diversity-driven discovery in these systems was recently introduced [26, 62], highlighting that two key ingredients are autonomous exploration and unsupervised representation learning to describe "relevant" degrees of variations in the patterns. In this paper, we motivate the need for what we call Meta-diversity search, arguing that there is not a unique ground truth interesting diversity as it strongly depends on the final observer and its motives. Using a continuous game-of-life system for experiments, we provide empirical evidences that relying on monolithic architectures for the behavioral embedding design tends to bias the final discoveries (both for hand-defined and unsupervisedly-learned features) which are unlikely to be aligned with the interest of a final end-user. To address these issues, we introduce a novel dynamic and modular architecture that enables unsupervised learning of a hierarchy of diverse representations. Combined with intrinsically motivated goal exploration algorithms, we show that this system forms a discovery assistant that can efficiently adapt its diversity search towards preferences of a user using only a very small amount of user feedback.

cs.LG↗

arXiv2023

The sandpile identity element on an ellipse

We consider certain elliptical subsets of the square lattice. The recurrent representative of the identity element of the sandpile group on this graph consists predominantly of a biperiodic pattern, along with some noise. We show that as the lattice spacing tends to 0, the fraction of the area taken up by the pattern in the identity element tends to 1.

math.AP↗

arXiv2020

Research on PM emission and energy consumption of mixed traffic flow based on cellular automata FI model

Traffic congestion leads to a sharp increase in fuel consumption and exhaust emissions, which easily leads to air pollution and fuel waste. Based on Fukui-Ishibashi (FI) model, this paper studies particulate matter(PM) emission and energy dissipation of traffic flow by considering the maximum speed, mixing ratio and random slowing probability. Simulation results show that the energy dissipation of the traffic flow consisting of the same length and different maximum velocity is associated with the maximum velocity. The greater the maximum velocity, the more the energy consumes. When the maximum velocity is different, the PM emission is similar to the change of energy consumption. Moreover, for the same maximum velocity and different mixing ratio, the energy dissipation and the PM emission related to the mixing ratio. There are the maximum fuel consumption and emission value in the maximum flow. For the same maximum velocity and length of vehicles, their energy consumption and emission are determined by random slowing probability.

nlin.CG↗