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Über dieses Buch

This book constitutes the proceedings of the 11th International Conference on Cellular Automata for Research and Industry, ACRI 2014, held in Krakow, Poland, in September 2014. The 67 full papers and 7 short papers presented in this volume were carefully reviewed and selected from 125 submissions. They are organized in topical sections named: theoretical results on cellular automata; cellular automata dynamics and synchronization; modeling and simulation with cellular automata; cellular automata-based hardware and computing; cryptography, networks and pattern recognition with cellular automata. The volume also contains contributions from ACRI 2014 workshops on crowds and cellular automata; asynchronous cellular automata; traffic and cellular automata; and agent-based simulation and cellular automata.

Inhaltsverzeichnis

Frontmatter

Invited Talks

Conductivity, Memristivity and Creativity in Cellular Automata

Cellular automata is a universal tool in literally any field of sciences and engineering yet more applications and weird discoveries are to come. We show how three barely related natural phenomena — conductivity of an excitable medium, networks of resistors with memory and structure of schizotypy versus cognitive control mental space -— are represented in two-dimensional cellular automata with two or three states, and what discoveries we made using these models. This is an abstract of the talk at the conference on cellular automata, thus we do not provide any basics on cellular automata.

Andrew Adamatzky

Cellular Modeling with Cell-DEVS: A Discrete-Event Cellular Automata Formalism

In recent years, grid-shaped cellular models have gained popularity to understand physical systems. Complex cell spaces can require large amounts of compute time, mainly due to its synchronous nature; the use of a discrete time base also constrains the precision of the model. The Cell-DEVS formalism was defined in order to deal with these issues. We give a brief introduction to the main characteristics of Cell-DEVS, and show how to use the method to model complex cell spaces. We present different examples of application, and show how to integrate cellular models with external data collection and visualization.

Gabriel A. Wainer

Theoretical Results on Cellular Automata

Towards a Comprehensive Understanding of Multi-state Cellular Automata

Motivated by the fact that many cellular automata (CAs) for describing biological, physical or chemical processes are built upon more than two states, whereas most the majority of results on the stability of CAs is restricted to two-state CAs, we show in this paper how non-directional Lyapunov exponents can be used to assess the stability of multi-state CAs. Moreover, we pay particular attention to the different types of defects that may emerge during the evolution of such CAs from a single initial defect of a given type. Numerical results are presented for the family of three-state totalistic CAs.

Jan M. Baetens, Bernard De Baets

Iterative Arrays with Set Storage

Iterative arrays with set storage (SIA) are one-dimensional arrays of interconnected interacting finite automata. The input is supplied sequentially to the distinguished communication cell at the origin. In addition, the communication cell controls a set storage. To this end, it is equipped with a one-way writing tape where strings for the set operations are assembled, and the data storage

set

where words of arbitrary length can be stored. The computational capacity of (real-time) SIA is investigated. It is shown that such devices are strictly stronger than classical iterative arrays and classical set automata. Moreover, the witness languages reveal that the combination of both principles is strictly stronger than just the union of the single principles. Some basic closure properties are studied. Furthermore, in contrast to the situation for classical set automata, it is shown that any constant number of operations on the set cannot increase the computational capacity of classical iterative arrays. Finally, the decidability of the restriction to a finite number of set operations is addressed, where it turns out that the problem is not even semi-decidable.

Martin Kutrib, Andreas Malcher

Isotropic Cellular Automaton for Excitable Media with Random Neighbor Selection

This paper proposes a new isotropic cellular automaton (CA) model for reproducing the Belousov–Zhabotinsky reaction observed in excitable media. Although several CA models have been proposed that exhibit isotropic patterns of the reaction, most of them need complicated rules, a large number of neighboring cells, and multiple thresholds to decide the excitation condition of cells. The proposed model uses only one threshold and simple time-evolution rules on the basis of states of selected neighboring cells; the selected cells are randomly chosen from eight neighboring cells. It is this randomness in selecting neighboring cells that causes the model to generate isotropic patterns. This study shows that patterns generated by the proposed model are highly isotropic. Furthermore, we use simulation results to elucidate how generated patterns are related to the initial states assigned to central cells.

Mio Kobayashi

Power Spectral Analysis of the Computation Process by Rule 110

An elementary cellular automaton rule 110 supports universal computation by emulating cyclic tag system and its evolution starting from random initial configurations exhibits 1/

f

noise. In this research we investigate the power spectra of rule 110 during the computation process emulating cyclic tag system. As a result, 1/

f

-type power spectra are observed in the most actively interacting area among the whole array, while in the less active area the power spectra exhibit Lorentzian or periodic types. These results suggest the relationship between 1/

f

noise and computability in cellular automata.

Shigeru Ninagawa, Genaro J. Martínez

Cellular Automata and Formulae on Monoids

This paper studies cellular automata with binary states on monoids making use of formulae in propositional logic, instead of local functions. Also we prove that the multiplication of formulae, defined by monoid action, determines the composition of transition functions of CA. This result converts the reversibility of transition functions to the reversibility of formulae. Several examples of reversible formulae are illustrated. Finally, introducing the Stone topology on configuration spaces, we give a neat proof of Hedlund’s theorem for CA.

Toshikazu Ishida, Shuichi Inokuchi, Yasuo Kawahara

A Scalable Method for Constructing Non-linear Cellular Automata with Period 2 n  − 1

Non-linear functions are very essential in different crypto primitives as they increase the security of the cipher designs. On the other hand, maximum length sequences help to prevent repeatability of a pseudorandom generator. Linear functions such as LFSR and linear cellular automata are used to generate maximum length sequences. However linear maximum length sequences are not secure. So there is a necessity of a construction that can provide both non-linearity and maximum length sequence for optimized cipher designs. In this work, we propose an algorithm for synthesizing a maximum length non-linear cellular automata to fulfill the requirement. Extensive experimentation on the proposed scheme shows that the construction achieves high non-linearity. Moreover, we have implemented and tested the design in Xilinx Spartan-3 FPGA platform and the hardware overhead is shown to be nominal.

Shamit Ghosh, Abhrajit Sengupta, Dhiman Saha, Dipanwita Roy Chowdhury

Systolic Dissemination in the Arrowhead Family

Although cellular automata (CA) are usually driven by a local rule, global communications are often needed either to synchronize a process or to share common data. However, these communications must be carried out from the nearest-neighbor, local transition. Such disseminations are named “systolic” herein: this metaphor is borrowed from the eponymous cellular architectures. The core of this study is the topology of the “

arrowhead

” family underlying the CA network in the hexagonal tessellation. The graphs of this family, directed or undirected, are Cayley graphs, or graphs of groups and are therefore vertex-transitive. As a consequence, the local rule is the same within the whole network. Two types of dissemination are presented: a (one–to–all) broadcasting and a (all–to–all) gossiping. For each type, a 3–port, directed scheme and a 6–port, undirected scheme are derived from construction. It is shown that the complexity of these algorithms is the graph diameter, either directed or undirected, according to the case study.

Dominique Désérable

Cellular Automata Dynamics and Synchronization

On the Dynamics of Multi-information in Cellular Automata

After reviewing a few key quantities of information theory, we investigate in this paper the behaviour of multi-information in elementary cellular automata. It will turn out that the usual classification by Wolfram is not well supported in terms of this information measure, or, more likely, that multi-information is blind to the kind of complexity displayed by those automata.

Gregor Chliamovitch, Bastien Chopard, Alexandre Dupuis

Lyapunov Exponents of One-Dimensional, Binary Stochastic Cellular Automata

In this paper the stability of elementary cellular automata (ECAs) upon introduction of stochasticity, in the form of an update probability for each cell, is assessed. To do this, Lyapunov exponents, which quantify the rate of divergence between two nearby trajectories in phase space, were used. Furthermore, the number of negative Lyapunov exponents was tracked, in order to gain a more profound insight into the interference between the stability and the update probability, and an upper bound on the Lyapunov exponents of stochastic cellular automata (SCAs) was established.

Wouter Van der Meeren, Jan M. Baetens, Bernard De Baets

Synthesis of Non-uniform Cellular Automata Having only Point Attractors

This paper studies a special class of non-uniform cellular automata (CAs) that contain only single length cycle (point) attractors in their state space. These CAs always converge to some point attractors. A number of theorems and lemmas are reported in this paper to characterize this class of CAs.

Reachability tree

, a discrete tool for characterizing 1-d CA, has been utilized to develop theories for these types of CAs. We finally report an algorithm that

synthesizes

a non-uniform cellular automaton having only point attractors.

Nazma Naskar, Sumit Adak, Pradipta Maji, Sukanta Das

Non Uniform Cellular Automata Description of Signed Partition Versions of Ice and Sand Pile Models

This paper reviews the well-known formalisations for ice and sand piles, based on a finite sequence of non-negative integers and its recent extension to signed partitions, i.e. sequences of a non-negative and a non-positive part of integers, both non increasing.

The ice pile model can be interpreted as a discrete time dynamical system under the action of a vertical and a horizontal evolution rule, whereas the sand pile model is characterized by the unique action of the vertical rule.

The signed partition extension, besides these two dynamical evolution rules, also takes into account an annihilation rule at the boundary region between the non-negative and the non-positive regions. We provide an original physical interpretation of this model as a p-n junction of two semiconductors.

Moreover, we show how the sand pile extension of the signed partition environment can be formalized by mean of a non-uniform cellular automaton (CA) since the vertical and the annihilation evolution rules have the formal description of two CA local rules. Finally, we provide a similar construction for the ice pile extension.

Gianpiero Cattaneo, Giampiero Chiaselotti, Alberto Dennunzio, Enrico Formenti, Luca Manzoni

Variable Entangling in a Quantum Battle of the Sexes Cellular Automaton

The effect of variable entangling on the dynamics of a spatial quantum formulation of the iterated battle of the sexes game is studied in this work. The game is played in the cellular automata manner, i.e., with local and synchronous interaction. The effect of spatial structure is assessed when allowing the players to adopt quantum and classical strategies, both in the two and three parameter strategy spaces.

Ramón Alonso-Sanz

Experimental Finitization of Infinite Field-Based Generalized FSSP Solution

In a previous work (see [3]) we presented a general scheme to solve the 1D Generalized Firing Squad Synchronization Problem. We designed it in a modular way using the concept of fields (open CA). The solution was not designed as a finite cellular automaton because we needed unbounded integers as states for distance fields, and the recursive nature of the algorithm leaded to a unbounded number of fields. In this paper, we show as claimed, that this approach does lead to a finite cellular automaton. We exhibit a transformation function from infinite to finite states and write a program that generates the associated finite transition table while checking its validity and the conservation of the input-output behavior of the original cellular automaton.

Luidnel Maignan, Jean-Baptiste Yunès

Cellular Automata (CA) Model for Primality Test

Theory and application of Cellular automata (CA) as a global Transform for detecting compositeness of a number is reported. To test an

n

bit odd valued number

N

in the range 2

n

 − 1

to (2

n

-1), a Compositeness Detecting CA (CDCA) set is designed with

N

=

S

as a Self Loop Attractor (SLA) State, where

S

=

S

×

S

,

S

is the largest factor of

S

,

S

= 3,5,7,⋯. The set has at least one CDCA with the state

S

in its attractor basin; the CA initialized with

S

reaches the attractor

S

after

S

time steps. A number is detected as a prime if no CDCA is synthesized.

Nirmalya Sundar Maiti, Soumyabrata Ghosh, Parimal Pal Chaudhuri

Modeling and Simulation with Cellular Automata

Numerical Modelling of Fracture Based on Coupled Cellular Automata Finite Element Approach

Investigation of failure of Dual Phase steels on the basis of the developed concurrent cellular automata finite element model is the subject of the present paper. Physical background of phenomena responsible for failure in these steels is described first. Then details of the developed random cellular automata model are presented. Particular attention is put on proper definition of the transition rules describing initiation and propagation of fractures across the microstructure. Finally combined cellular automata finite element model is established. Examples of obtained results are also presented within the paper.

Konrad Perzy, Mateusz Sitko, Łukasz Madej

Two-Layer CA Model for Simulating Catalytic Reaction at Dynamically Varying Temperature

A two-layer cellular automata (CA) model of carbon monoxide (CO) oxidation reaction on platinum is proposed and investigated. This reaction in non-equilibrium conditions can be accompanied by appearance of surface waves, spirals and turbulences on the catalyst. A two-layer CA is a parallel composition of the two CA: the main CA simulating the oxidation reaction, and the second layer CA simulating spatio-temporal distribution of the catalyst surface temperature. Using the second layer allows us to take into account changes of surface catalytic properties when temperature changes and investigate the oxidation reaction dynamics for different temperature.

Anastasia Kireeva

Modelling Ordered Nanopourous Structures by Anodization via Cellular Automata

A cellular automata model to simulate nanostructured alumina creation via anodization is proposed. The model mimics the Field Assisted Dissolution theoretical approach of the anodization process. The key parameters influencing the structures of the layer are identified and an attempt to recreate the two step anodization procedure in simulation conditions is made. The effect of dissolution rate and surface diffusion are considered. Simulation have been run on NVIDIA Tesla cards using the techniques of parallel programming to increase speed of the simulations.

Bartosik Łukasz, Stafiej Janusz, Di Caprio Dung

Overview of Cellular Automaton Models for Corrosion

A review of corrosion process modeling using cellular automata methods is presented. This relatively new and growing approach takes into account the stochastic nature of the phenomena and uses physico-chemical rules to make predictions at a mesoscopic scale. Milestone models are analyzed and perspectives are established.

Cristian Felipe Pérez-Brokate, Dung di Caprio, Damien Féron, Jacques De Lamare, Annie Chaussé

Cellular Automata Finite Element Approach for Modelling Microstructure Evolution under Thermo-Mechanical Processing Conditions

The concurrent cellular automata finite element (CAFE) approach for modelling microstructure evolution under thermo-mechanical processing conditions is the subject of the present work. Particular attention is put on modelling two phenomena, static recrystallization after deformation and phase transformation during heating. Details of the developed models are presented within the paper. Both models are implemented based on the CA Framework, which is also described in the work. Finally cellular automata approaches are combined with the finite element model based on the digital material representation idea. The numerical modelling of complex multistage hot deformation process was selected as a case study to show capabilities of the developed cellular automata finite element model.

Rafal Golab, Mateusz Sitko, Joanna Szyndler, Łukasz Madej

A Preliminary Cellular Model for Secondary Lahars and Simulation of 2005 Case of Vascún Valley, Ecuador

Lahars represent one of the most destructive natural disasters as number of casualties in the world. Secondary lahars are very complex surface flows, which originate from the mobilization of pyroclastic deposits by exceptional heavy rainfalls. Simulation of secondary lahars could be an important tool for risk management in threatened regions. Multicomponent (macroscopic) Cellular Automata (CA) characterize a methodological approach for modelling large scale (extended for kilometers) complex phenomena, that evolve on the basis of local interactions. A preliminary three dimension CA model was developed and partially applied on a real event: the 2005 secondary lahar of Vascún Valley, Ecuador. Simulations are satisfying, a comparison is performed with the previous successful two dimensions model Titan2D, based on PDE, together with simulation results of the same event.

Guillermo Machado, Valeria Lupiano, Maria Vittoria Avolio, Salvatore Di Gregorio

Vulnerability and Protector Control: Cellular Automata Approach

In this work we consider the protector control problem using cellular automata approach. We give some definitions and characterizations of vulnerable zones and protector control for a cellular automaton model. We illustrate this notion through a fire forest example using a developed application with JAVA environment.

Omar Jellouli, Abdessamed Bernoussi, Mina Amharref, Samira El Yacoubi

UNDATA: A Preliminary Cellular Automata Model for Tsunami Simulation

The Cellular Automata (CA) model UNDATA for tsunami simulation is here presented. UNDATA was developed in order to be coupled to SCIDDICA, a CA efficient model for subaerial/subaqueous flow type landslides for cases when a displacement in water of a significant volume could generate a tsunami. This model works also for different generating causes. Applications to theoretic and real cases are satisfying.

Francesco Gullace, Maria Vittoria Avolio, Salvatore Di Gregorio

Modeling Rainfall Features Dynamics in a DEM Satellite Image with Cellular Automata

Cellular automata have been widely used in the field of remote sensing for simulating natural phenomena over two-dimensional satellite images. Simulations on DEM (Digital Elevation Model), three-dimensional satellite images, are very rare. This paper presents a study of modeling and simulation of the weather phenomenon of precipitation over DEM satellite images through a new algorithm, RACA (RAinfall with Cellular Automata). The aim of RACA is to obtain, from the simulation, numerical and 3D results related to the water level that allow us to make decisions on important issues such as avoiding the destruction of human life and property from future natural disasters, establishing future urbanized areas away from locations with high probability of flooding or estimating the future water supply for arid regions.

Moisés Espínola, José Antonio Piedra-Fernández, Rosa Ayala, Luis Iribarne, Saturnino Leguizamón

Cellular Automata Model with Game Theory for Power Management of Hybrid Renewable Energy Smart Grids

In recent years, control of smart grids that match electricity demand in different sites and forms with supply has been considered as one of the most difficult aspect of smart energy grids design. In this paper we present a Cellular Automata (CA) based approach combined with Game Theory for the enhancement of Power Management Strategies (PMSs) of multiple Hybrid Renewable Energy Systems (HYRES) that form a smart grid for the exchange of energy. More specifically, taking advantage of the local interactions of HYRES we coupled CA principles with Public Goods Game (PGG) for modeling. The presented CA model focuses on providing valuable feedback for PMSs of the understudy HYRES connected in a grid. In this manner, a flexible network based HYRES design is considered and applied to specific HYRESs located in Olvio, near Xanthi, Greece, part of SYSTEMS SUNLIGHT facilities. The proposed model can be applied to the understudy HYRESs grid management to enhance and optimize its PMS based on the provided energy prediction scenarios.

Prodromos Chatziagorakis, Constantinos Elmasides, Georgios Ch. Sirakoulis, Ioannis Karafyllidis, Ioannis Andreadis, Nikolaos Georgoulas, Damianos Giaouris, Athanasios I. Papadopoulos, Chrysovalantou Ziogou, Dimitris Ipsakis, Simira Papadopoulou, Panos Seferlis, Fotis Stergiopoulos, Paris Voutetakis

A Novel Algorithm for Coarse-Graining of Cellular Automata

The coarse-graining is an approximation procedure widely used for simplification of mathematical and numerical models of multiscale systems. It reduces superfluous – microscopic – degrees of freedom. Israeli and Goldenfeld demonstrated in [1,2] that the coarse-graining can be employed for elementary cellular automata (CA), producing interesting interdependences between them. However, extending their investigation on more complex CA rules appeared to be impossible due to the high computational complexity of the coarse-graining algorithm. We demonstrate here that this complexity can be substantially decreased. It allows for scrutinizing much broader class of cellular automata in terms of their coarse graining. By using our algorithm we found out that the ratio of the numbers of elementary CAs having coarse grained representation to “degenerate” – irreducible – cellular automata, strongly increases with increasing the “grain” size of the approximation procedure. This rises principal questions about the formal limits in modeling of realistic multiscale systems.

Krzysztof Magiera, Witold Dzwinel

Cellular Automata Model for Protein Structure Synthesis (PSS)

This paper is a refinement and extension of the Protein modelling Cellular Automata Machine (PCAM) reported in [1] for prediction of protein structure. An efficient organization of Knowledge Base (KB) is reported in the current paper. The KB is reorganized with emphasis on the residues in the Transition Regions (TRs) between structural regions like alpha helices or beta strands and an unstructured or loop regions. The meta-knowledge derived out of the KB analysis is employed for synthesis of protein structure from the primary chain of amino acid residues. Design of synthesis algorithm ensures incorporation of probable orientation of structural parameters of residues in the TR. A few structures are finally selected based on the computation of exposed surface area and core size. The algorithm is tested for the challenging protein targets from [9] to synthesize structures with reasonable accuracy and low execution time.

Soumyabrata Ghosh, Nirmalya S. Maiti, Parimal Pal Chaudhuri

The Basic Reproduction Number for Chagas Disease Transmission Using Cellular Automata

This paper presents mathematical and numerical results for a cellular automaton model describing the transmission dynamics of Chagas disease in both homogeneous and heterogeneous environments. The basic reproduction number

R

0

which integrates factors that determine whether the pathogen can establish or not will be computed using the next-generation matrix approach. The simulation results show the effect of landscape heterogeneity in the vector transmission.

Baki Cissé, Samira El Yacoubi, Sébastien Gourbière

Modelling Spatial Distribution of the Barents Sea Cod Fishery

The paper presents a cellular automata (CA) model for the growth and spatially distribution of the Northeast Arctic cod including a harvest model based on economical rational behaviour. Rules and range of the CA model are estimated from observations and biological theory, and the environmental conditions are assumed to be in accordance with the IPCC A1B scenario for the following 45 years. The aim of the model developed is to study fleet behaviour based on available management decisions, resource information and economic performance. This paper presents fleet performance in the model under open access conditions, considering two different types of vessels (small and large) placed in four different homeports. Fleet smartness is a key parameter controlling the fish finding ability of each fleet. The study shows that increasing smartness reveals increasing differences between small and large vessels placed in different homeports. While homeport clusters vessels at low levels of smartness, vessel size and range clusters vessels at higher levels of smartness.

Arne Eide

Training Cellular Automata to Simulate Urban Dynamics: A Computational Study Based on GPGPU and Swarm Intelligence

We present some results of a computational study aimed at investigating the relationship between the spatio-temporal data used in the calibration phase and the consequent predictive ability of a Cellular Automata (CA) model. Our experiments concern a CA model for the simulation of urban dynamics which is typically used for predicting spatial scenarios of land-use. Since the model depends on a large number of parameters, we calibrate the CA using Cooperative Coevolutionary Particle Swarms, which is an effective approach for large-scale optimizations. Moreover, to cope with the relevant computational cost related to the high number of CA simulations required by our study, we exploits the computing power of Graphics Processing Units.

Ivan Blecic, Arnaldo Cecchini, Giuseppe A. Trunfio

Cellular Automaton Approach to Arching in Two-Dimensional Granular Media

Clogging of granular materials and jamming of pedestrian crowds occur because of the formation of arches at bottlenecks. We propose a simple microscopic model that is able to reproduce oscillation phenomena due to formation and destabilization of arches in 2-dimensional flows. The dynamics of particles in front of a bottleneck is described by a one-dimensional stochastic cellular automaton on a semicircular geometry. The model predicts the existence of a critical bottleneck size for jamless particle flows and allows to determine the dependence of the jamming probability on the system size. The model can also be studied analytically and the results are in good agreement with simulations.

Takumi Masuda, Katsuhiro Nishinari, Andreas Schadschneider

Modeling of Friction Dynamic Motion by Cellular Automata

Friction vibration in a dynamic system composed of a slider supported by a spring on a belt was modeled and simulated by Cellular Automata (CA). Friction vibration in mechanical systems has been studied for a long time and various models have been proposed to understand its physical phenomena, but dynamical behavior of an elastic body on friction surface has not been revealed yet. This may be caused by the complexity of friction between two surfaces, and CA can be used as a strong tool of modeling physical phenomena introducing local neighbor and transition rules based on observation of phenomena. It may not be easy to derive the governing equation of motion including friction. In this study, a modeling procedure of friction dynamics by CA was discussed. The new model was based on a spring–block model proposed by Burridge and Knopoff, but an additional layer of internal surface was introduced to consider the contact area of sliding surfaces. The self-excited vibration, including the stick-slip vibration and the divergent phenomena could be simulated in the proposed CA model.

Seiya Yamagishi, Shin Morishita

Agent Based Simulation of Spreading in Social-Systems of Temporarily Active Actors

In this work a novel model of information spreading processes in systems of dynamic active-inactive actors is presented. In our model information can flow only through those actors of the system that are currently active. Based on this model we carried out computer simulations showing how the activity of agents affect the process. We also carried out some basic investigation of the effect of inhomogeneous activity. The results of the work can be used to qualitatively predict what would be the effect if the activity of agents would change in a social system.

Gergely Kocsis, Imre Varga

Cellular Automata for Modeling Language Change

This paper describes the use of cellular automata to model dialect feature diffusion as the adaptive aspect of the complex system of speech. We show how a feature, once established, can spread across an area, and how the distribution of a dialect feature as it stands in Linguistic Atlas data could either spread or diminish. Throughout hundreds of iterations, we can watch regional and social distribution patterns emerge as a consequence of update rules. We validate patterns with respect to the linguistic distributions known to occur in the Linguistic Atlas Project.

William A. Kretzschmar, Ilkka Juuso

Sznajd Model with Memory

Modification of the classical Sznajd model, by introducing a probability factor representing persuasibility of the cell on the social pressure has been presented. Two different variants of the factor as a function of the previous cell’s opinion have been investigated. The new model exhibits different and in this context more realistic time of stabilization and probability of the achieved stable points.

Norbert Sendra, Tomasz M. Gwizdałła

Detecting Emergent Phenomena in Cellular Automata Using Temporal Description Logics

Cellular automata are discrete mathematical models that have been proven useful as representations of a wide variety of systems exhibiting emergent behavior. Detection of emergent behavior is typically computationally expensive as it relies on computer simulations. We propose to specify cellular automata using a suitable Temporal Description Logic and we show that we can formulate queries about the evolution of a cellular automaton as reasoning tasks in this logic.

Stathis Delivorias, Haralampos Hatzikirou, Rafael Peñaloza, Dirk Walther

Cellular Automata-Based Hardware and Computing

Direction-Reversible Self-Timed Cellular Automata for Delay-Insensitive Circuits

We introduce a new Self-Timed Cellular Automaton capable of simulating

reversible delay-insensitive

(DI) circuits. In addition to a number of reversibility and determinism properties, our STCA exhibits

direction-reversibility

, where reversing the direction of a signal and running a circuit forwards is equivalent to running the circuit in reverse. We define also several extensions of the STCA which allow us to realise three larger classes of DI circuits, including

parallel

circuits. We then show which of the reversibility, determinism and direction-reversibility properties hold for these classes of circuits.

Daniel Morrison, Irek Ulidowski

Implementation of a Cellular Automaton with Globally Switchable Rules

Cellular automata represent a discrete model of a computational machine with the inherent concept of totally distributed state transitional function. Previous studies have indicated that well-devised type of a global influence turns out to be an important factor in terms of improving the overall efficiency of a computation process within automata. In this context, polymorphic electronics is an approach that introduces a specific way of a global control to the circuit, not by means of using a dedicated global signal but through employing an inherent environmental variable. In our case the global information is uniformly propagated through the existing voltage supply rail, which is naturally available to all individual cells of a given automaton. It seems that the suggested approach may be very useful for the implementation of enhanced cellular automata. In this paper, the real hardware implementation of a cellular automaton using polymorphic chip and the obtained experimental results are presented together with a subsequent discussion.

Václav Šimek, Richard Růžička, Adam Crha, Radek Tesař

Highly Compact Automated Implementation of Linear CA on FPGAs

The current literature on cellular automata (CA) mostly overlooks the fact that the perceived regularity and locality of interconnects in a CA are often

logical

rather than

physical

, and difficult to achieve in practical implementations. Optimized mapping, placement and routing of circuits are especially challenging for

Field Programmable Gate Array

(FPGA) platforms, which often result in low-performance implementations. We develop a design methodology for the automated implementation of low-resource, high-performance CA circuits, by optimal usage of the underlying FPGA architecture, direct primitive instantiation, and constrained placement. Case study for an 1-D CA circuit reveal higher performance, lower hardware resource requirement (by a factor of 0.5 X), acceptable power-delay product (PDP), and superior design scalability, in comparison to implementations derived by standard FPGA CAD tool design flow.

Ayan Palchaudhuri, Rajat Subhra Chakraborty, Mohammad Salman, Sreemukh Kardas, Debdeep Mukhopadhyay

Shortest Path Computing Using Memristor-Based Circuits and Cellular Automata

This paper addresses Cellular Automata (CA) based algorithm implementations using circuits with memory resistors (memristors). Memristors are two-terminal passive nonvolatile resistance switching devices whose unique adaptive properties are suitable for massively parallel computational purposes. The sparse nature of computations using network configurations of memristors resembles certain operational features and computing capabilities of CA. Here a memristive CA capable of detecting the shortest path between given nodes of a mesh with weighted edges is proposed. Simulation results are in absolute agreement with the solutions given by the corresponding CA-based algorithmic approach. The proposed memristive CA circuit structure is also used for the effective solution of the traveling salesman problem.

Dimitrios Stathis, Ioannis Vourkas, Georgios Ch. Sirakoulis

Cryptography, Networks and Pattern Classification with Cellular Automata

Generation of TPMACA for Pattern Classification

The important prerequisites of designing pattern classifier are high throughput and low cost hardware implementation. The simple, regular, modular and cascadable local neighborhood sparse network of Cellular Automata (CA) suits ideally for low cost VLSI implementation. Thus the multiple attractor CA is adapted for use as a pattern classifier. By concatenating two predecessor multiple attractor CA (TPMACA) we can construct a pattern classifier. In this paper we propose a method for finding dependency vector by using a 0-basic path. Also we propose various methods for generating TPMACA corresponding to a given dependency vector.

Sung-Jin Cho, Han-Doo Kim, Un-Sook Choi, Seok-Tae Kim, Jin-Gyoung Kim, Sook-Hee Kwon, Gil-Tak Gong

Sharing Secrets by Computing Preimages of Bipermutive Cellular Automata

A secret sharing scheme based on one-dimensional bipermutive cellular automata is discussed in this paper. The underlying idea is to represent the secret as a configuration of a bipermutive CA and to iteratively apply a preimage computation algorithm until a sufficiently long configuration to be splitted among the participants is obtained. The scheme is proved to be both perfect and ideal, and a simple extension is shown to induce a sequential access structure which eventually becomes cyclic, where the upper bound on the length of the cycles depends on the radius of the adopted local rule.

Luca Mariot, Alberto Leporati

Inapplicability of Fault Attacks against Trivium on a Cellular Automata Based Stream Cipher

The current work analyses fault attacks on Trivium. These attacks exploit the slow pace of non-linearisation and reversibility of the encryption function. Cellular Automata can be effectively deployed to circumvent these shortcomings. CASTREAM, a CA based stream cipher, is difficult to reverse as well as highly non-linear and the non-linearity is attained very fast. In this paper, we show that CASTREAM is strong against fault attacks for which Trivium is vulnerable.

Jimmy Jose, Sourav Das, Dipanwita Roy Chowdhury

Cellular Automata Approach to Maximum Lifetime Coverage Problem in Wireless Sensor Networks

In this paper, we propose a novel distributed algorithm based on Graph Cellular Automata (GCA) concept to solve Maximum Lifetime Coverage Problem (MLCP) in Wireless Sensor Networks (WSNs). The proposed algorithm possesses all advantages of localized algorithm, i.e. using only some knowledge about the neighbors, WSN is able to self-organize in such a way to prolong its lifetime preserving at the same time required coverage ratio of a target field. The paper presents results of experimental study of the proposed algorithm and comparison of them with a centralized genetic algorithm.

Antonina Tretyakova, Franciszek Seredynski, Pascal Bouvry

C&CA - Int. Workshop on Crowds and Cellular Automata

Application of NIST Technical Note 1822 to CA Crowd Dynamics Models Verification and Validation

This paper addresses the issue of application of methodology included in NIST technical note 1822:

The Process of Verification and Validation of Building Fire Evacuation Models

[1] in terms of CA crowd dynamics models. The note is a recently released document (November 2013), that proposes a set of verification and validation (V&V) tests as well as methods for an uncertainty analysis. The main aim of this paper is to investigate these tests and methods applied to CA models by showing results of sample tests and discussing CA specific issues.

Jakub Porzycki, Robert Lubaś, Marcin Mycek, Jarosław Wąs

Effect of Group Behavior on Crowd Dynamics

In recent years, the research on group movement has been drew pedestrian and evacuation dynamicists’ attention. In this paper, we explored to build a group model in a simple way. The group model considered two drift parameters, D and d. Thereinto, D expresses the destination’s attraction and d stands for group leader’s attraction and d is simplified to direct towards the group leader. Meantime, a method of generating groups is proposed. The simulation results show that with the increase of group size, the negative effect of group on crowd becomes greater. And group walking has more obvious effect on group owning large velocity. Besides the increase of group attraction intensity makes the negative effect distinct. In addition, it is found that with the increase of group disperse degree, the negative impact of group walking becomes obvious. What’s more, the effect of group disperse degree will not increase without limitation. The findings in this paper help researchers understand the impact of the presence of groups on crowd.

Wei Xiaoge, Song Weiguo, Xu Xuan, Fang Zhiming, Li Xiaolian

Effects of Boundary Conditions on Single-File Pedestrian Flow

In this paper we investigate effects of boundary conditions on one dimensional pedestrian flow which involves purely longitudinal interactions. Qualitatively, stop-and-go waves are observed under closed boundary condition and dissolve when the boundary is open. To get more detailed information the fundamental diagrams of the open and closed systems are compared using Voronoi-based measurement method. Higher maximal specific flow is observed from the pedestrian movement at open boundary condition.

Jun Zhang, Antoine Tordeux, Armin Seyfried

Simulation Study of the Spiral Motion of Pedestrians: A Cellular Automata Approach

When the pedestrians share the same objectives moving toward the same direction, huge congestion is often created in some regions. The pedestrians lose mobility in the congestion and subsequently create a deadlock phenomenon. This study considers an event that pedestrians rotate around and moves toward a central object. The pedestrian motion is modelled by use of Cellular Automata (CA) to analyse how the congestion at centre region is developed. The model is implemented in hexagonal lattice with static floor fields in polar coordinate. Interaction of rotational and centripetal mobility generates the spiral motions of the pedestrian. In this study, various simulations are carried out to investigate the effect of model parameters and macroscopic properties. The pedestrian motion creates high density area at centre region and the density gradually decreases toward outside. The radial distribution of the circumferential density characterises the pedestrian flow, such that a deadlock occurs at the inner region and a free flow occurs at the outer region. Moreover a possible solution for easing the deadlock is also suggested in this paper.

Kenichiro Shimura, Stefania Bandini, Katuhiro Nishinari

CA Crowd Modeling for a Retirement House Evacuation with Guidance

This paper studies the impact of guidance on evacuation processes. A Cellular Automata (CA) based model has been, therefore, elaborated in order to introduce group categorization and guidance attributes. The crowd is categorized according to motional skills and a special group is assigned leadership features. The presented scenario includes the evacuation of a retirement house with the help of the nursing staff. Simulation results prove the significance of proper guidance. The latter optimizes the response of the model by activating alternative routes that decrease congestion levels in front of exits.

Eleftherios Spartalis, Ioakeim G. Georgoudas, Georgios Ch. Sirakoulis

Multiscale Simulation of Pedestrians for Faster Than Real Time Modeling in Large Events

The Hermes project [1] demonstrated the usefulness of on site faster than real time simulations of probable evacuation scenarios for security personnel. However, the hardware needed was prohibitively expensive [2]. The present paper shows that a multiscale approach can perform the simulation in a fraction of time without loss of useful information. The main problem is the correct passing of agents from a coarse scale model to a fine scale model, here from a CA model to a force based model. This will be achieved by inserting agents into the force based model at positions and speeds optimized for smooth walking either by a priori information or using Voronoi cells. Connecting a Queue model to a continuous model has already been done successfully [3].

We also show that a slightly modified CA method can address the problem, too, at even less computational cost, with some possible loss of accuracy.

Bernhard Steffen, Mohcine Chraibi

Cellular Automata Pedestrian Movement Model SIgMA.CA: Model Parameters as an Instrument to Regulate Movement Regimes

In the paper a connection of model parameters with movement regimes in different geometrical conditions are considered for the cellular automata floor field pedestrian movement model SIgMA.CA. Evacuation time is considered as measure of influence.

Ekaterina Kirik, Tat’yana Vitova

Case Study of Phase Transition in Cellular Models of Pedestrian Flow

One room with one exit and one multiple entrance is modelled using 32 different settings and modifications of floor field model. The influence of following aspects are investigated in the scope of the transition from free flow to congestion phase with respect to the inflow rate: Heterogeneity/Homogeneity; With/Without bounds; Moore/von Neumann neighbourhood; Synchronous/Asynchronous update; High/Low friction. Considering the average travel time through the room and average room occupancy the settings incorporating the bounds and synchronous update seems to match the experimental data from the qualitative point of view.

Marek Bukáček, Pavel Hrabák

Simulation of Public Opinion with Ideas of Cellular Automata

Cellular automata approach for public opinion modeling is considered. Aiming the possibility of a practical use, we extend standard peer effects model by introducing the idea of states dependence and irregular lattice into it. The applicability of proposed approach was examined by modeling the parliamentary elections in Ukraine.

Terpil Ievgen, Makarenko Alexander

Estimating Speeds of Pedestrians in Real-World Using Computer Vision

This paper proposes a novel approach to a computer vision based automatic system for the estimation of pedestrian velocity in real world traffic systems in which a fixed camera is available. The paper will introduce the adopted framework, which includes a preprocessing phase, an identification and tracking phase, and a speed estimation final phase. Speed estimation, implying a conversion from image to real world coordinates, can be carried out with two different techniques that will be discussed in details and evaluated with reference to achieved results.

Sultan Daud Khan, Fabio Porta, Giuseppe Vizzari, Stefania Bandini

ACA - Int. Workshop on Asynchronous Cellular Automata and Asynchronous Discrete Models

Geometric Characterization of Hereditarily Bijective Boolean Networks

The study of relationships between structure and dynamics of asynchronous Boolean networks has recently led to the introduction of hereditarily bijective maps and even or odd self-dual networks. We show here that these two notions can be simply characterized geometrically: through orthogonality between certain affine subspaces. We also use this characterization to provide a construction of the class of hereditarily bijective maps, and to study its stability properties.

Paul Ruet

Inner-Independent Radius-Dependent Totalistic Rule of Universal Asynchronous Cellular Automaton

We propose a model of a 2-dimensional 2-state asynchronous updating cellular automaton with inner-independent radius-dependent totalistic rule. An inner-independent rule is such that the cell’s updating does not depend on the state of the center cell. A radius-dependent totalistic rule is a totalistic rule which the neighborhood is an extended Moore neighborhood that consists of cells at orthogonal or diagonal distances 1, 2, 3, 4 and 5 from the center cell, taking summations of the living cells in their domain individually. The rule set designed in this paper is universal for computation, that is, any delay-insensitive circuit can be constructed. We also show the algorithm to prove the correct operations.

Susumu Adachi

Bifurcations of Local Structure Maps as Predictors of Phase Transitions in Asynchronous Cellular Automata

We show that the local structure approximation of sufficiently high order can predict the existence of second order phase transitions belonging to the directed percolation university class in

α

-asynchronous cellular automata.

Henryk Fukś, Nazim Fatès

Computing Symbolic Steady States of Boolean Networks

Asymptotic behavior is often of particular interest when analyzing asynchronous Boolean networks representing biological systems such as signal transduction or gene regulatory networks. Methods based on a generalization of the steady state notion, the so-called symbolic steady states, can be exploited to investigate attractor properties as well as for model reduction techniques conserving attractors. In this paper, we propose a novel optimization-based method for computing all maximal symbolic steady states and motivate their use. n particular, we add a new result yielding a lower bound for the number of cyclic attractors and illustrate the methods with a short study of a MAPK pathway model.

Hannes Klarner, Alexander Bockmayr, Heike Siebert

Equivalences in Multi-valued Asynchronous Models of Regulatory Networks

Multi-valued network models can be described by their topology and a set of parameters capturing the effects of the regulators for each component. Dynamics can then be derived and represented as state transition systems. Different network models may lead to the same transition system, meaning dynamics analysis of a representative model covers a larger class of models. While rather clear in the Boolean case, the properties contributing to this effect become more involved for multi-valued models. We analyze these properties and present a mathematical description of the resulting model equivalence classes.

Adam Streck, Heike Siebert

Effective Parallelism Rate by Reversible PCA Dynamics

Probabilistic Cellular Automata generalise CA by implementing an updating rule defined through a probability. It means a

synchronous

updating of the constituting cells/sites’ states is possible. PCA differ from the

interacting particle systems

where in general at most one site is possibly updated at a time. For a family of reversible (in a stochastic sense) PCA dynamics, we study through numerical simulations the effective flips occurring. When infinitely many sites are considered, there are two regime: an ergodic one and a phase transition regime. When finitely many interacting sites are considered, these regimes corresponds to very different effective parallelism rate. We quantify these changes. When phase transition holds, PCA dynamics is in fact an

α

-asynchronous one.

Pierre-Yves Louis

Quick Convergence to a Fixed Point: A Note on Asynchronous Elementary Cellular Automata

This note describes a small step in the analysis of the fully asynchronous cellular automata (i.e., the cells are updated uniformly at random at each time step). We establish the rapid convergence of fifteen minimal Elementary Cellular Automata, showing that their average convergence time to a fixed point scales logarithmically with the size of the automaton. Techniques involve the use of Markov chain analysis and the construction of adequate potential functions. The problem is however left open for twelve other minimal rules, which shows the need to develop this analysis further.

Nazim Fatès

TCA - Int. Workshop on Traffic and Cellular Automata

A Study of Aggregated Speed in Road Networks Using Cellular Automata

Several recent works have focused on studying the relationship between the aggregated flow and density in arterial road networks. Analogous studies involving aggregated speed appear not to have been yet undertaken, however. Here we study and compare such relations for arterial road networks controlled by different types of adaptive traffic signal systems, under various boundary conditions. To study such systems we simulate stochastic cellular automaton models. Our simulation results suggest that network speed could be used as a surrogate for density, due to a strong anticorrelation between these two network observables. Since speed estimates can be more easily obtained than density estimates, e.g. from probe vehicle data, this suggests that Macroscopic Fundamental Diagrams relating aggregated flow with speed might be a practically useful alternative to those relating flow to density.

Lele Zhang, Somayeh Shiri, Timothy M. Garoni

A New Cellular Automaton Model with Spatiotemporal Process of Lane Changing Execution

In this paper, the spatiotemporal process of lane changing is considered in the cellular automaton models for traffic flow. The lane-changing time and the space required depend on the instantaneous velocity of the vehicle and following vehicle in the destination lane. The simulation is carried out in a two-lane homogeneous system. The speed change of the lane changing vehicle accords with the fundamental diagram and a 2D region is found in the lane changing frequency-velocity plane.

Hui-xuan Li, Chun-fu Shao, Hao-ling Wu, Jun-fang Tian, Xun Ji

Cellular Automaton Model with Non-hypothetical Congested Steady State Reproducing the Three-Phase Traffic Flow Theory

A new assumption is assumed to explain the mechanisms of traffic flow that in the noiseless limit, vehicles’ space gap will oscillate around the desired space gap, rather than keep the desired space gap, in the homogeneous congested traffic flow. It means there are no steady states of congested traffic and contradicts with the fundamental diagram approach and three-phase traffic flow theory both of which admit the existence of steady states of congested traffic. In order to verify this assumption, a cellular automaton model with non-hypothetical congested steady state is proposed, which is based on the Nagel-Schreckenberg model with additional slow-to-start and the effective desired space gap. Simulations show that this new model can produce the synchronized flow, the transitions from free flow to synchronized flow to wide moving jams, and multiple congested patterns observed by the three-phase theory.

Junfang Tian, Martin Treiber, Chenqiang Zhu, Bin Jia, HuiXuan Li

Asymmetric Lane Change Rules for a Microscopic Highway Traffic Model

For simulating multi-lane highway traffic with cellular automata (CA) traffic models, realistic lane change rules are required. In many countries, legal regulations distinguish between driving lanes and overtaking lanes. Therefore, asymmetric lane change rules are needed. In this contribution, the CA traffic model by Lee

et al

(Phys. Rev. Lett.

92

(23) (2004) 238702) is extended with those rules. The presented ruleset is then studied in simulations of two-lane and three-lane highways.

Lars Habel, Michael Schreckenberg

Interactions between Multiple Junctions

We consider a simple TASEP (Totally Asymmetric Simple Exclusion Process) network model with an aggregation point and a branching point. Generally speaking, the aggregation point behaves as a bottleneck and the branching point enables particles to encourage their velocity. However, the correlation among multiple junctions in TASEP network is not known so much. In order to investigate the correlation, we consider a simple TASEP network which including two junctions and discuss the network with an aggregation point and a branching point. From our theoretical analysis and numerical results, it is shown that aggregations become bottlenecks in TASEP networks and that branches enable flow of particles to be larger in many cases.

Takahiro Tannai, Katsuhiro Nishinari

Modeling Disruption and Recovery of Traffic in Road Networks

We study the impact of disruptions on traffic networks, and the relaxation of the system after the removal of the disruption. We model the steady-state density along the disrupted route using a simple phenomenological model. We then combine this model with domain wall theory to analyze the transient behavior of the system. We compare the predictions produced by these macroscopic models with simulations of a stochastic cellular automaton model.

Lele Zhang, Timothy M. Garoni

ABSim and CA - Int. Workshop on Agent-Based Simulation and Cellular Automata

Simulation of Pedestrians Behavior in a Shopping Mall

The knowledge of phenomena connected with pedestrian movement and behavior is important in retail and service sectors. Such a belief finds confirmation in numerous researches, which proved true a strong relationship between shops’ profitability and the way customers move inside them. The goal of the article is to work out a model of pedestrian behavior in a shopping mall based on a multi-agent approach on the basis of some already-existing successful solutions, namely we combine PED4 and Social Distances algorithms into one simulation framework.

Paweł Kłeczek, Jarosław Wąs

How Agents Can Form a Specific Pattern

A multi-agent system is considered, comprised of a square 2D cell field of cells with uniform agents controlled by finite state machines (FSMs). Each cell contains a particle with one out of four colors, which can be changed by the agents. Initially the agents and colors are randomly distributed. The objective is to form a specific target pattern belonging to a predefined pattern class. The target patterns (path patterns) shall consist of preferably long narrow paths with the same color. The quality of the path patterns is measured by a degree of order, which is computed by counting matching 3 x 3 patterns (templates). The used agents can perform 32 actions, combinations of moving, turning and coloring. They react on the own color, the color in front, and blocking situations. The agents’ behavior is determined by an embedded FSM with 6 states. For a given 8 x 8 field, near optimal FSMs were evolved by a genetic procedure separately for

k

 = 1 .. 48 agents. The evolved agents are capable to form path patterns with a high degree of order. Agents, evolved for a 8 x 8 field, are able to structure a 16 x 16 field successfully, too. The whole multi-agent system was modeled by cellular automata. In the implementation of the system, the CA-w model (cellular automata with write access) was used in order to reduce the implementation effort and speed up the simulation.

Rolf Hoffmann

An Integrated Model for the Simulation of Pedestrian Crossings

The present paper represents an approach to the modeling of pedestrians and vehicles interaction in the area of a zebra crossing, either signalised or not, employing two existing models devoted to the simulation of the specific pedestrian and vehicular sub-systems and integrating them in a comprehensive agent

environment

. The latter acts as a bridge allowing mutual perception of the different heterogeneous agents that cooperate to avoid accidents: vehicles give way to perceived pedestrians whenever they can safely brake to let them pass, pedestrians yield whenever they perceive cars that would not be able to stop before the zebra crossing. The paper presents the model and shows results in simple crossing scenarios.

Luca Crociani, Giuseppe Vizzari

Agent-Based Pedestrian Activity Simulation in Shopping Environments Using a Choice Network Approach

Most of current approaches for processing agent-based pedestrian activity simulations propose movement choice networks. Choice mechanisms include where to stop, in what order, and which overall route to take. In our network approach, the movement choice network is approximated using a lattice of irregular cells representing streets and shops. In this approach, cell centroids are considered the nodes of an implicit movement network. A pedestrian agent is located in a node and can move on the implicit movement network to other nodes and is situated randomly in the cell related to that node. In this paper, the focus is on the generation of the movement network and the underlying behavioral rules that conducts the activation of pedestrians on the network representing a shopping environment.

Jan Dijkstra, A. Joran Jessurun

Software Implementation of Population of Cognitive Agents Learning to Cross a Highway

We describe the model and the software implementation of population of simple cognitive agents, naïve creatures experiencing fear and/or desire while learning to cross a highway. The creatures use an observational learning mechanism for adoption or rejection of a strategy to cross the highway. Presented simulation results are consistent with the fact that crossing a highway becomes more difficult with increase of cars density and it is affected by the creatures’ fears and desires. The transfer the knowledge base acquired in one environment to another one combined with creatures ability to change a crossing point improves creatures success of crossing a highway.

Anna T. Lawniczak, Bruno N. Di Stefano, Jason B. Ernst

The Effects of Supraregional Innovation and Production Collaboration on Technology Development in a Multiregional World: A Spatial Agent-Based Model Study

With globalization, firms acquire locally unavailable inputs from and collaborate in innovation with firms in other regions. We contend that, depending on the collaboration distances feasible and spatial layout of regions, a core-periphery structure of regions emerges, in which core regions produce more advanced and complex products. We develop a spatial agent-based model of (supraregional) firm collaboration in production and innovation to study technological progress. We find that when collaboration is possible over greater distances, agents produce more advanced and more complex products. Moreover, we find that, in general, the core-periphery structure indeed emerges. However, for some layouts, the core-periphery structure vanishes almost immediately, while for others first becomes stronger, peaks and then vanishes with an increase in collaboration distance. Moreover, we find that the properties of the technology structure play a prominent mediating role, e.g. the effect of supraregional collaboration on technological progress may be strong for some and relatively weak for other structures.

Ben Vermeulen, Andreas Pyka

Erratum: Modelling Ordered Nanopourous Structures by Anodization via Cellular Automata

The acknowledgement text of the initially published paper was incomplete. It should have been as follows:

The work of Ł. B. was realized within the International PhD Projects Programme of the Foundation for Polish Science, cofinanced from European Regional Development Fund within Innovative Economy Operational Programme “Grants for innovation”.

Bartosik Łukasz, Stafiej Janusz, Di Caprio Dung

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