Exploration of cellular automata: a comprehensive review of dynamic modeling across biology, computer and materials science

by

Exploration of cellular automata: a comprehensive review of dynamic modeling across biology, computer and materials science

Oleksii Vodka, Mariia Shapovalova

National Technical University “Kharkiv Polytechnic Institute”, 2, Kyrpychova Str. Kharkiv, 61002, Ukraine.

DOI:

https://doi.org/10.7494/cmms.2023.4.0820

Abstract:

This paper delves into the expansive world of cellular automata (CA), abstract models of computation comprised of cells that interact based on predefined rules. Originating from John von Neumann’s work in the 1940s, CA has evolved into a multidisciplinary field with applications ranging from mathematical concepts to complex simulations of biological, physical, computer science, material science, and social systems. The paper reviews its historical development, emphasizing John Conway’s influential Game of Life and Burk’s seminar collection. The authors categorize and explore a myriad of CA topics, including self-replicating automata, the universality of computation, compromises in CA, variants, applications in biological systems, fault-tolerant computation, pattern recognition, CA games, fractals, dynamic properties, complexity, image processing, cryptography, bioinformatics, materials modeling, probabilistic automata, and contemporary research. The significance of cellular automata for materials modeling cannot be overstated and considerable attention has been devoted to the issues of modeling nucleation and recrystallization. The review aims to provide a comprehensive resource for both beginners and experts in the field, shedding light on cellular automata’s dynamic and diverse applications in various aspects of life and scientific inquiry.

Cite as:

Vodka, O., & Shapovalova, M. (2023). Exploration of cellular automata: a comprehensive review of dynamic modeling across biology, computer and materials science. Computer Methods in Materials Science, 23(4), 57–80. https://doi.org/10.7494/cmms.2023.4.0820

Article (PDF):

Keywords:

Cellular automata, Classification, Nucleation modeling, Pattern recognition, Image processing, Cryptography, Recrystallization

References:

Aburas, M.M., Ho, Y.M., Ramli, M.F., & Ash’aari, Z.H. (2016). The simulation and prediction of spatio-temporal urban growth trends using cellular automata models: A review. International Journal of Applied Earth Observation and Geoinformation, 52, 380–389. https://doi.org/10.1016/j.jag.2016.07.007.

Aladyev, V. (1974). Survey of research in the theory of homogeneous structures and their applications. Mathematical Biosciences, 22, 121–154. https://doi.org/10.1016/0025-5564(74)90088-1.

Amoroso, S., & Cooper, G. (1971). Tessellation structures for reproduction of arbitrary patterns. Journal of Computer and System Sciences, 5(5), 455–464. https://doi.org/10.1016/S0022-0000(71)80009-0.

Aruldoss, J.A., & Pricilla, J.A. (2014). Mathematical model of cellular automata and fractals in cancer growth. International Journal of Computing Algorithm, 3(3), 243–245.

Aso, H., & Honda, N. (1985). Dynamical characteristics of linear cellular automata. Journal of Computer and System Sciences, 30(3), 291–317. https://doi.org/10.1016/0022-0000(85)90048-0.

Banks, E.R. (1970). Universality in cellular automata. In 11th Annual Symposium on Switching and Automata Theory (swat 1970). IEEE. 194–215. https://doi.org/10.1109/SWAT.1970.27.

Baran, K., Sitko, M., & Madej, L. (2024). Analysis of the influence of a pseudo-random number generator type on the kinetics of the cellular automata recrystallization model. Journal of Computer Science, 75, 102193. https://doi.org/10.1016/j.jocs.2023.102193.

Bardell, P.H. (1990). Analysis of cellular automata used as pseudorandom pattern generators. In Proceedings. International Test Conference. IEEE. 762–768. https://doi.org/10.1109/TEST.1990.114093.

Bhardwaj, R., & Bhagat, D. (2018). Two level encryption of grey scale image through 2D cellular automata. Procedia Computer Science, 125, 855–861. https://doi.org/10.1016/j.procs.2017.12.109.

Bouchkaren, S., & Lazaar, S. (2014). A fast cryptosystem using reversible cellular automata. International Journal of Advanced Computer Science and Applications, 5(5), 207–210. https://doi.org/10.14569/IJACSA.2014.050531.

Brady, M.L., Raghavan, R., & Slawny, J. (1989). Probabilistic cellular automata in pattern recognition. In International 1989 Joint Conference on Neural Networks, 1. IEEE, 177–182. https://doi.org/10.1109/IJCNN.1989.118577.

Braga, G., Cattaneo, G., Flocchini, P., & Vogliotti, C.Q. (1995). Pattern growth in elementary cellular automata. Theoretical Computer Science, 145(1–2), 1–26. https://doi.org/10.1016/0304-3975(94)00155-C.

Bruno, M. (1994). Inherent generation of fractals by cellular automata. Complex Systems, 8(5), 347–366.

Burks, W. (1970). Essays on Cellular Automata. University of Illinois Press.

Cagigas-Muñiz, D., Diaz-del-Rio, F., Sevillano-Ramos, J.L., & Guisado-Lizar, J.-L. (2022). Efficient simulation execution of cellular automata on GPU. Simulation Modelling Practice and Theory, 118, 102519. https://doi.org/10.1016/j.simpat.2022.102519.

Chang, J.H., Ibarra, O.H., & Vergis, A. (1986). On the power of one-way communication. 27th Annual Symposium on Foundations of Computer Science (sfcs 1986). IEEE. 455–464. https://doi.org/10.1109/SFCS.1986.37.

Chaudhuri, P.P., Chowdhury, D.R., Nandi, S., & Chattopadhyay, S. (1997). Additive Cellular Automata: Theory and Applications (vol. 1). IEEE Computer Society Press.

Chen, L., Qi, J., & Shi, J. (2023). A cellular automata ship traffic flow model considering navigation rules in narrowing channel. Alexandria Engineering Journal, 69, 715–726. https://doi.org/10.1016/j.aej.2023.01.062.

Chen, M.-S., Yuan, W.-Q., Lin, Y.C., Li, H.-B., & Zou, Z.-H. (2017). Modeling and simulation of dynamic recrystallization
behavior for 42CrMo steel by an extended cellular automaton method. Vacuum, 146, 142–151. https://doi.org/10.1016/j.vacuum.2017.09.041.

Codd, E.F. (1968). Cellular Automata. Academic Press.

Connelly, R. (1986). Winning Ways for Your Mathematical Plays, Volumes I & II By Elwyn R. Berlekamp, John H. Conway. The American Mathematical Monthly, 93(5), 411–414, https://doi.org/10.1080/00029890.1986.11971843.

Conway, J.H. (1976). On Numbers and Games. Academic Press.

Cui, J., Li, B., Liu, Z., Xiong, Y., Qi, F., Zhao, Z. & Zhu, S. (2023). Numerical investigation of grain structure under the rotating arc based on cellular automata-finite element method during vacuum arc remelting process. Metallurgical and Materials Transactions B, 54, 661–672. https://doi.org/10.1007/s11663-022-02716-x.

Culik II, K., & Dube, S. (1991). An efficient solution of the firing mob problem. Theoretical Computer Science, 91(1), 57–69. https://doi.org/10.1016/0304-3975(91)90267-6.

Culik II, K., & Yu, S. (1988). Undecidability of CA classification schemes. Complex Systems, 2(2), 177–190.
Culik II, K., Hurd, L.P., & Yu, S. (1990). Computation theoretic aspects of cellular automata. Physica D: Nonlinear Phenomena, 45(1–3), 357–378. https://doi.org/10.1016/0167-2789(90)90194-T.

D’amico, M., Manzini, G., & Margara, L. (2003). On computing the entropy of cellular automata. Theoretical Computer Science, 290(3), 1629–1646. https://doi.org/10.1016/S0304-3975(02)00071-3.

Das, S., Mukherjee, S., Naskar, N., & Sikdar, B.K. (2009). Modeling single length cycle nonlinear cellular automata for pattern recognition. In 2009 World Congress on Nature & Biologically Inspired Computing (NaBIC) (pp. 198–203). https://doi.org/10.1109/NABIC.2009.5393685.

Deng, X., Chen, H., Xu, Q., Feng, F., Chen, X., Lv, X., Lin, X., & Fu, T. (2022). Topology optimization design of three-dimensional multi-material and multi-body structure based on irregular cellular hybrid cellular automata method. Scientific Reports, 12(1), 5602. https://doi.org/10.1038/S41598-022-09249-Y.

Dennunzio, A., Formenti, E., & Margara, L. (2024). An efficient algorithm deciding chaos for linear cellular automata over (Z\mZ)n with applications to data encryption. Information Sciences, 657, 119942. https://doi.org/10.1016/j.ins.2023.119942.

Ding, R., & Guo, Z.X. (2002). Microstructural modelling of dynamic recrystallisation using an extended cellular automaton approach. Computational Materials Science, 23(1–4), 209–218. https://doi.org/10.1016/s0927-0256(01)00211-7.

Dioșan, L., Andreica, A., & Enescu, A. (2017). The use of simple cellular automata in image processing. Studia Universitatis Babeș-Bolyai Informatica, 62(1), 5–14. https://doi.org/10.24193/subbi.2017.1.01.

Dobravec, T., Mavrič, B., & Šarler, B. (2017). A cellular automaton – finite volume method for the simulation of dendritic and eutectic growth in binary alloys using an adaptive mesh refinement. Journal of Computational Physics, 349, 351–375. https://doi.org/10.1016/j.jcp.2017.08.011.

Dyer, Ch.R. (1980). One-way bounded cellular automata. Information and Control, 44(3), 261–281. https://doi.org/10.1016/S0019-9958(80)90164-3.

Efstathiou, G., Gkantonas, S., Giusti, A., Mastorakos, E., Foale, C.M., & Foale, R.R. (2023). Simulation of the December 2021 Marshall fire with a hybrid stochastic Lagrangian-cellular automata model. Fire Safety Journal, 138, 103795. https://doi.org/10.1016/j.firesaf.2023.103795.

El Yacoubi, S., & El Jai, A. (2002). Cellular automata modelling and spreadability. Mathematical and Computer Modelling, 36(9–10), 1059–1074. https://doi.org/10.1016/S0895-7177(02)00259-5.

Finelli, M., Manzini, G., & Margara, L. (1998). Lyapunov exponents versus expansivity and sensitivity in cellular automata. Journal of Complexity, 14(2), 210–233. https://doi.org/10.1006/jcom.1998.0474.

Fúster-Sabater, A., & Caballero-Gil, P. (2009). Synthesis of cryptographic interleaved sequences by means of linear cellular automata. Applied Mathematics Letters, 22(10), 1518–1524. https://doi.org/10.1016/j.aml.2009.03.018.

Gács, P. (1986). Reliable computation with cellular automata. Journal of Computer and System Sciences, 32(1), 15–78. https://doi.org/10.1016/0022-0000(86)90002-4.

Garzon, M. (1995). Models of Massive Parallelism: Analysis of Cellular Automata and Neural Networks. Springer Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-77905-3.

Ghosh, P., Mukhopadhyay, A., Chanda, A., Mondal, P., Akhand, A., Mukherjee, S., Nayak, S.K., Ghosh, S., Mitra, D., Ghosh, T., & Hazra, S. (2017). Application of cellular automata and Markov-chain model in geospatial environmental modeling – A review. Remote Sensing Applications: Society and Environment, 5, 64–77. https://doi.org/10.1016/j.rsase.2017.01.005.

Goetz, R.L., & Seetharaman, V. (1998). Modeling dynamic recrystallization using cellular automata. Scripta Materialia, 38(3), 405–413. https://doi.org/10.1016/s1359-6462(97)00500-9.

Green, F. (1987). NP-complete problems in cellular automata. Complex Systems, 1(3), 453–474.

Guan, P. (1987). Cellular automaton public-key cryptosystems. Complex Systems, 1(1), 51–57.

Gutowitz, Н. (1993). Cryptography with dynamical systems. In N. Boccara, E. Goles, S. Martinez, P. Picco (Eds.), Cellular Automata and Cooperative Phenomena (pp. 237–274). Springer Dordrecht. https://doi.org/10.1007/978-94-011-1691-6_21.

Gütschow, J., Nesme, V., & Werner, R.F. (2010). The fractal structure of cellular automata on abelian groups. In Automatam2010 – 16th Intl. Workshop on CA and DCS. Discrete Mathematics & Theoretical Computer Science (pp. 51–70). https://doi.org/10.46298/dmtcs.2759.

Hallberg, H., Wallin, M., & Ristinmaa, M. (2010). Simulation of discontinuous dynamic recrystallization in pure Cu usingma probabilistic cellular automaton. Computational Materials Science, 49(1), 25–34. https://doi.org/10.1016/j.commatsci.2010.04.012.

Harao, M., & Noguchi, S. (1978). On some dynamical properties of finite cellular automata. IEEE Transactions on Computers, C-27(1), 42–52. https://doi.org/10.1109/TC.1978.1674951.

Hatzikirou, H., & Deutsch, A. (2008). Cellular automata as microscopic models of cell migration in heterogeneous environments. Current Topics in Developmental Biology, 81, 401–434. https://doi.org/10.1016/S0070-2153(07)81014-3.

Hesselbarth, H.W., & Göbel, I.R. (1991). Simulation of recrystallization by cellular automata. Acta Metallurgica et Materialia,39(9), 2135–2143. https://doi.org/10.1016/0956-7151(91)90183-2.

Holland, J.H. (1976). Studies of the spontaneous emergence of self-replicating systems using cellular automata and formal grammers. In A. Lindenmayer, G. Rozenberg (Eds.), Automata, Languages, Development (pp. 385–404). North-Holland Publishing.

Hurd, L.P., Kari, J., & Culik, K. (1992). The topological entropy of cellular automata is uncomputable. Ergodic Theory and Dynamical Systems, 12(2), 255–265. https://doi.org/10.1017/S0143385700006738.

Ibarra, O.H., & Kim, S.M. (1984). Characterizations and computational complexity of systolic trellis automata. Theoretical Computer Science, 29(1–2), 123–153. https://doi.org/10.1016/0304-3975(84)90015-X.

Ion, I.-A., Moroz-Dubenco, C., & Andreica, A. (2023). Breast cancer images segmentation using fuzzy cellular automaton. Procedia Computer Science, 225, 999–1008. https://doi.org/10.1016/j.procs.2023.10.087.

Isa, M.A.M., Ahmad, M.M., Sani, N.F.M., Hashim, H., & Mahmod, R. (2014). Cryptographic key exchange protocol with message authentication codes (MAC) using finite state machine. Procedia Computer Science, 42, 263–270. https://doi.org/10.1016/j.procs.2014.11.061.

Itô, M., Ôsato, N., & Nasu, M. (1983). Linear cellular automata over Zm. Journal of Computer and System Sciences, 27(1), 125–140. https://doi.org/10.1016/0022-0000(83)90033-8.

Jaiswal, R.K., Singh, D., Sharma, V.M., & Vishwakarma, D. (2023). Performance analysis of full adder and full subtractor using quantum-dot cellular automata. e-Prime – Advances in Electrical Engineering, Electronics and Energy, 6, 100279. https://doi.org/10.1016/j.prime.2023.100279.

Jeon, J.-Ch. (2010). Non-linear and non-group cellular automata chaining technique for cryptographic applications. Mathematical and Computer Modelling, 51(7–8), 995–999. https://doi.org/10.1016/j.mcm.2009.08.033.

Jump, J.R., & Kirtane, J.S. (1974). On the interconnection structure of cellular networks. Information and Control, 24(1), 74–91. https://doi.org/10.1016/S0019-9958(74)80024-0.

Kari, L., Rozenberg, G., & Salomaa, A. (1997). L systems. In Rozenberg, G., Salomaa, A. (Eds.), Handbook of Formal Languages (Vol. 1: Word, Language, Grammar, pp. 253–328). Springer Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59136-5_5.

Khedmati, Y., Parvaz, R., & Behroo, Y. (2020). 2D Hybrid chaos map for image security transform based on framelet and cellular automata. Information Sciences, 512, 855–879. https://doi.org/10.1016/j.ins.2019.10.028.

Kosaraju, S.R. (1975). Speed of recognition of context-free languages by array automata. SIAM Journal on Computing, 4(3), 331–340. https://doi.org/10.1137/0204028.

Krishna, V.M., Mishra, S., & Gurrala, J. (2022). Cellular automata and their applications: a review. International Journal of Computer Applications, 184(5), 13–23. https://doi.org/10.5120/ijca2022922010.

Kurka, P. (1997). Languages, equicontinuity and attractors in cellular automata. Ergodic Theory and Dynamical Systems, 17(2), 417–433. https://doi.org/10.1017/S014338579706985X.

Langton, Ch.G. (1984). Self-reproduction in cellular automata. Physica D: Nonlinear Phenomena, 10(1–2), 134–144. https://doi.org/10.1016/0167-2789(84)90256-2.

Lee, W., Bae, J., Lee, H., Kang, S., Kang, S.-H., & Yoon, J. (2022). Numerical simulation of dendritic growth and porosity evolution in solidification of Al-Cu alloy with lattice Boltzmann – cellular automata method. Journal of Alloys and Compounds, 929, 167233. https://doi.org/10.1016/j.jallcom.2022.167233.

Lewis, A.E., Seckler, M.M., Kramer, H., & van Rosmalen, G. (2015). Industrial Crystallization: Fundamentals and Applications. Cambridge University Press. https://doi.org/10.1017/CBO9781107280427.

Li, H., Sun, X., & Yang, H. (2016). A three-dimensional cellular automata-crystal plasticity finite element model for predicting the multiscale interaction among heterogeneous deformation, DRX microstructural evolution and mechanical responses in titanium alloys. International Journal of Plasticity, 87, 154–180. https://doi.org/10.1016/j.ijplas.2016.09.008.

Lindenmayer, A. (1968). Mathematical models for cellular interactions in development. Parts I and II. Journal of Theoretical Biology, 18(3), 280–315. https://doi.org/10.1016/0022-5193(68)90079-9.

Litow, B., & Dumas, Ph. (1993). Additive cellular automata and algebraic series. Theoretical Computer Science, 119(2), 345–354. https://doi.org/10.1016/0304-3975(93)90165-P.

Liu, S., Hong, K., & Shin, Y.C. (2021). A novel 3D cellular automata-phase field model for computationally efficient
dendrite evolution during bulk solidification. Computational Materials Science, 192, 110405. https://doi.org/10.1016/j.commatsci.2021.110405.

Lundine, M.A., Brothers, L.L., & Trembanis, A.C. (2023). Deep learning for pockmark detection: Implications for quantitative seafloor characterization. Geomorphology, 421, 108524. https://doi.org/10.1016/j.geomorph.2022.108524.

Madej, L., & Sitko, M. (2022). Computationally efficient cellular automata‐based full‐field models of static recrystallization:A perspective review. Steel Research International, 94(3), 2200657. https://doi.org/10.1002/srin.202200657.

Maerivoet, S., & De Moor, B. (2005). Cellular automata models of road traffic. Physics Reports, 419(1), 1–64. https://doi.org/10.1016/j.physrep.2005.08.005.

Mahajan, M. (1992). Studies in language classes defined by different types of time-varying cellular automata. Ph.D. Thesis, Indian Institute of Technology, Madras, India.

Maiti, S., Sengupta, M., & Chowdhury, D.R. (2021). Generating nonlinear codes for multi-bit symbol error correction using cellular automata. Physica D: Nonlinear Phenomena, 415, 132758. https://doi.org/10.1016/j.physd.2020.132758.

Maji, P., & Chaudhuri, P.P. (2005). Fuzzy cellular automata for modeling pattern classifier. IEICE Transactions on Information and Systems, E88-D(4), 691–702. https://doi.org/10.1093/ietisy/e88-d.4.691.

Maji, P., Shaw, C., Ganguly, N., Sikdar, B. K., & Chaudhuri, P.P. (2003). Theory and application of cellular automata for pattern classification. Fundamenta Informaticae, 58(3–4), 321–354.

Manzini, G., & Margara, L. (1999). Attractors of linear cellular automata. Journal of Computer and System Sciences, 58(3), 597–610. https://doi.org/10.1006/jcss.1998.1609.

Martin, O., Odlyzko, A.M., & Wolfram, S. (1984). Algebraic properties of cellular automata. Communications in Mathematical Physics, 93, 219–258. https://doi.org/10.1007/BF01223745.

Mastorakos, E., Gkantonas, S., Efstathiou, G., & Giusti, A. (2023). A hybrid stochastic Lagrangian – cellular automata framework for modelling fire propagation in inhomogeneous terrains. Proceedings of the Combustion Institute, 39(3), 3853–3862. https://doi.org/10.1016/j.proci.2022.07.240.

Mi, J., Hou, H., Zhang, S., Hua, Y., Yang, Y., Zhu, Y., & Ding, Z. (2023). Detecting long-term effects of mining-induced ground ndeformation on plant succession in semi-arid areas using a cellular automata model. Ecological Indicators, 151, 110290. https://doi.org/10.1016/j.ecolind.2023.110290.

Mohamed, F.K. (2014). A parallel block-based encryption schema for digital images using reversible cellular automata. Engineering Science and Technology, an International Journal, 17(2), 85–94. https://doi.org/10.1016/j.jestch.2014.04.001.

Morita, K. (1992). Computation-universality of one-dimensional one-way reversible cellular automata. Information Processing Letters, 42(6), 325–329. https://doi.org/10.1016/0020-0190(92)90231-J.

Myhill, J. (1963). The converse of Moore’s Garden-of-Eden theorem. Proceedings of the American Mathematical Society, 14, 685–686. https://doi.org/10.1090/S0002-9939-1963-0155764-9.

Neumann, J., von (1951). The general and logical theory of automata. In L.A. Jeffress (Ed.), Cerebral Mechanisms in Behavior. The Hixon Symposium (pp. 1–41). John Wiley & Sons, Chapman & Hall.

Neumann, J., von (1966). Theory of Self Reproducing Automata. University of Illinois Press.

Ni, T. (2003). Some properties of fractals generated by linear cellular automata. Tsinghua Science and Technology, 8(5), 557–563.

Niemi, V. (1997). Cryptology: language-theoretic aspects. In Rozenberg, G., Salomaa, A. (Eds.), Handbook of Formal Languages (Vol. 2: Linear Modeling: Background and Application, pp. 507–524). Springer Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-07675-0_11.

Nishio, H., & Kobuchi, Y. (1975). Fault tolerant cellular spaces. Journal of Computer and System Sciences, 11(2), 150–170. https://doi.org/10.1016/S0022-0000(75)80065-1.

Packard, N.H., & Wolfram, S. (1985). Two-dimensional cellular automata. Journal of Statistical Physics, 38, 901–946. https://doi.org/10.1007/BF01010423.

Pesavento, U. (1995). An implementation of von Neumann’s self-reproducing machine. Artificial Life, 2(4), 337–354. https://doi.org/10.1162/artl.1995.2.4.337.

Popova, E., Staraselski, Y., Brahme, A., Mishra, R.K., & Inal, K. (2015). Coupled crystal plasticity – Probabilistic cellular automata approach to model dynamic recrystallization in magnesium alloys. International Journal of Plasticity, 66, 85–102. https://doi.org/10.1016/J.IJPLAS.2014.04.008.

Popovici, A., & Popovici, D. (2002). Cellular automata in image processing. International Symposium on the Mathematical Theory of Networks and Systems.

Prusinkiewicz, P., & Lindenmayer, A. (1990). The algorithmic beauty of plants. Springer New York, NY. https://doi.org/10.1007/978-1-4613-8476-2.

Qian, M., & Guo, Z.X. (2004). Cellular automata simulation of microstructural evolution during dynamic recrystallization of anvHY-100 steel. Materials Science and Engineering: A, 365(1–2), 180–185. https://doi.org/10.1016/J.MSEA.2003.09.025.

Raabe, D., & Becker, R. (2000). Coupling of a crystal plasticity finite-element model with a probabilistic cellular automaton for simulating primary static recrystallization in aluminium. Modelling and Simulation in Materials Science and Engineering, 8(4), 445. https://doi.org/10.1088/0965-0393/8/4/304.

Reuther, K., & Rettenmayr, M. (2014). Perspectives for cellular automata for the simulation of dendritic solidification – A review. Computational Materials Science, 95, 213–220. https://doi.org/10.1016/j.commatsci.2014.07.037.

Richardson, D. (1972). Tessellations with local transformations. Journal of Computer and System Sciences, 6(5), 373–388. https://doi.org/10.1016/S0022-0000(72)80009-6.

Róka, Z. (1994). One-way cellular automata on Cayley graphs. Theoretical Computer Science, 132(1–2), 259–290. https://doi.org/10.1016/0304-3975(94)90236-4.

Rosin, P.L. (2005). Training cellular automata for image processing. In H. Kalviainen, J. Parkkinen, A. Kaarna (Eds.), Image Analysis 14th Scandinavian Conference, SCIA 2005, Joensuu, Finland, June 19–22, 2005, Proceedings (pp. 195–204). Springer Berlin, Heidelberg. https://doi.org/10.1007/11499145_22.

Roy, S., Karjee, J., Rawat, U.S., Pratik, D.N., & Dey, N. (2016). Symmetric key encryption technique: A cellular automata based approach in wireless sensor networks. Procedia Computer Science, 78, 408–414. https://doi.org/10.1016/j.procs.2016.02.082.

Sabau, A.S., Yuan, L., Fattebert, J.-L., & Turner, J.A. (2023). An OpenMP GPU-offload implementation of a non-equilibrium solidification cellular automata model for additive manufacturing. Computer Physics Communications, 284, 108605. https://doi.org/10.1016/j.cpc.2022.108605.

Sadiq, M.D., & Kumar, B.H. (2015). Efficient cryptography using cellular automata rules. International Journal of Emerging Engineering Research and Technology, 3(12), 18–25.

Salo, V., Theyssier, G., & Törmä, I. (2022). Cellular automata and bootstrap percolation. Theoretical Computer Science, 924, 34–45. https://doi.org/10.1016/j.tcs.2022.04.015.

Sapino, F., Haer, T., Saiz-Santiago, P., & Pérez-Blanco C.D. (2023). A multi-agent cellular automata model to explore water trading potential under information transaction costs. Journal of Hydrology, 618, 129195. https://doi.org/10.1016/j.jhydrol.2023.129195.

Sarkar, P. (2000). A brief history of cellular automata. ACM Computing Surveys, 32(1), 80–107. https://doi.org/10.1145/349194.349202.

Sarkar, P., & Barua, R. (1998). Multidimensional σ-automata, π-polynomials and generalised S-matrices. Theoretical Computer Science, 197(1–2), 111–138. https://doi.org/10.1016/S0304-3975(97)00160-6.

Seiferas, J.I. (1974). Observations on nondeterministic multidimensional iterative arrays. In Proceedings of the ACM Symposium on the Theory of Computing (pp. 276–289). ACM Press. https://doi.org/10.1145/800119.803905.

Sethi, B., & Das, S. (2016). On the use of asynchronous cellular automata in symmetric−key cryptography. Computing and Communications, 30–41.

Smith III, A.R. (1972). Real-time language recognition by one-dimensional cellular automata. Journal of Computer and System Sciences, 6(3), 233–253.

Smith III, A.R. (1976). Introduction to and Survey of Polyautomata Theory. In A. Lindenmayer, G. Rozenberg (Eds.), Automata, Languages, Development (pp. ). North-Holland Publishing.

Sree, P.K., & Babu, I.R. (2008). Identification of protein coding regions in genomic DNA using unsupervised FMACA based pattern classifier. IJCSNS International Journal of Computer Science and Network Security, 8(1), 305–309. https://doi.org/10.48550/arXiv.1401.6484.

Sree, P.K., & Babu, I.R. (2010). Identification of promoter region in genomic DNA using cellular automata based text clustering. The International Arab Journal of Information Technology, 7(1), 75–78.

Sree, P.K., Babu, I.R., & Devi, N.U. (2013). PSMACA: An automated protein structure prediction using MACA (Multiple Attractor Cellular Automata). Journal of Bioinformatics and Intelligent Control, 2(3), 211–215. https://doi.org/10.1166/jbic.2013.1052.

Sree, P.K., Babu, I.R., & Devi, N.U. (2014). Cellular automata and its applications in bioinformatics: A review. Global Perspectives on Artificial Intelligence, 2(2), 16–22. https://doi.org/10.48550/arXiv.1404.0453.

Sutner, K. (1989). A note on the Culik-Yu classes. Complex Systems, 3(1), 107–115.

Sutner, K. (1990). Classifying circular cellular automata. Physica D: Nonlinear Phenomena, 45(1–3), 386–395. https://doi.org/10.1016/0167-2789(90)90196-V.

Svyetlichnyy, D. (2023). Frontal cellular automata for modelling microstructure evolution: Computational complexity analysis. Computational Materials Science, 230, 112478. https://doi.org/10.1016/j.commatsci.2023.112478.

Tang, J., Kumar, S., De Lorenzis, L., & Hosseini, E. (2023). Neural cellular automata for solidification microstructure modelling. Computer Methods in Applied Mechanics and Engineering, 414, 116197. https://doi.org/10.1016/j.cma.2023.116197.

Terry-Jack, M., & O’Keefe, S. (2023). Classifying 1D elementary cellular automata with the 0–1 test for chaos. Physica D: Nonlinear Phenomena, 453, 133786. https://doi.org/10.1016/j.physd.2023.133786.

Tezuka, S., & Fushimi, M. (1994). A method of designing cellular automata as pseudorandom number generators for built-in self-test for VLSI. Contemporary Mathematics, 168, 363–367. https://doi.org/10.1090/conm/168/01713.

Tomassini, M., & Perrenoud, M. (2001). Cryptography with cellular automata. Applied Soft Computing, 1(2), 151–160. https://doi.org/10.1016/S1568-4946(01)00015-1.

Ushmaev, D., Liao, Z., Notron, A., & Axinte, D. (2022). On the importance of interface stability in cellular automata models: planar and dendritic solidification in laser melted YSZ. Materials & Design, 219, 110823. https://doi.org/10.1016/j.matdes.2022.110823.

Vispoel, М., Daly, A. J., & Baetens, J.M. (2022). Progress, gaps and obstacles in the classification of cellular automata. Physica D: Nonlinear Phenomena, 432, 133074. https://doi.org/10.1016/j.physd.2021.133074.

Vitányi, P. (1973). Sexually reproducing cellular automata. Mathematical Biosciences, 18(1–2), 23–54. https://doi.org/10.1016/0025-5564(73)90018-7.

Vodka, O. (2019). Analysis of quantitative characteristics of microstructures that are generated by the probabilistic cellular automata method. In 2019 IEEE 2nd Ukraine Conference on Electrical and Computer Engineering, UKRCON 2019 – Proceedings (pp. 990–994). https://doi.org/10.1109/UKRCON.2019.8879959.

Vodka, O. (2020). Computer modeling of microstructures with probabilistic cellular automata method using different nucleation rate functions. In S. Subbotin (Ed.), International Workshop on Computer Modeling and Intelligent Systems (CMIS-2020). Zaporizhzhia, Ukraine, April 27 – May 1, 2020 (pp. 450–461). https://doi.org/10.32782/cmis/2608-34.

Vollmar, T. (1977). Cellular spaces and parallel algorithms, an introductory survey. In M. Feilmeier (Ed.), Parallel Computation-Parallel Mathematics. Proceedings of the IMACS (AICA)-GI Symposium, March 14–16, 1977, Technical University of Munich (pp. 49–58). North-Holland Publishing.

Wang, L., Fang, G., & Qian, L. (2018). Modeling of dynamic recrystallization of magnesium alloy using cellular automata considering initial topology of grains. Materials Science and Engineering: A, 711, 268–283. https://doi.org/10.1016/j.msea.2017.11.024.

Wang, W. (2003). A Mathematical Model of Dendritic Microstructures in Nickel-Based Superalloys [Doctoral thesis, Imperial College London]. Search Spiral. Imperial College London’s repository. http://hdl.handle.net/10044/1/11524.

Wikipedia (n.d.). Dendrite (metal), https://en.wikipedia.org/wiki/Dendrite_%28metal%29.

Willson, S.J. (1984). Cellular automata can generate fractals. Discrete Applied Mathematics, 8(1), 91–99. https://doi.org/10.1016/0166-218X(84)90082-9.

Wolfram, S. (1983). Statistical mechanics of cellular automata. Reviews of Modern Physics, 55(3), 601–644. https://doi.org/10.1103/RevModPhys.55.601.

Wolfram, S. (1984). Computation theory of cellular automata. Communications in Mathematical Physics, 96, 15–57. https://doi.org/10.1007/BF01217347.

Wolfram, S. (1986). Theory and Applications of Cellular Automata: Including Selected Papers 1983–1986. World Scientific.

Wolnik, B., Dziemiańczuk, M., & De Baets, B. (2023). Non-uniform number-conserving elementary cellular automata on the infinite grid: A tale of the unexpected. Information Sciences, 649, 119680. https://doi.org/10.1016/j.ins.2023.119680.

Wongthanavasu, S., & Ponkaew, J. (2013). Cellular Automata for Pattern Recognition. In A. Salcido (Ed.), Emerging Applications of Cellular Automata (pp. 53–67). IntechOpen. https://doi.org/10.5772/52364.

Yamada, H., & Amoroso, S. (1969). Tessellation Automata. Information and Control, 14(3), 299–317. https://doi.org/10.1016/S0019-9958(69)90090-4.

Zhang, Ch., Zhang, L., Xu, Q., Xia, Y., & Shen, W. (2016). The kinetics and cellular automaton modeling of dynamic recrystallization behavior of a medium carbon Cr-Ni-Mo alloyed steel in hot working process. Materials Science and Engineering: A, 678, 33–43. https://doi.org/10.1016/j.msea.2016.09.056.

Zhao, Y., Qin, R.S., & Chen, D.F. (2013). A three-dimensional cellular automata model coupled with finite element method and thermodynamic database for alloy solidification. Journal of Crystal Growth, 377, 72–77. https://doi.org/10.1016/j.jcrysgro.2013.05.006.

Zhao, Y., Chen, D., Long, M., Arif, T.T., & Qin, R. (2014). A three-dimensional cellular automata model for dendrite growth with various crystallographic orientations during solidification. Metallurgical and Materials Transactions: B, 45, 719–725. https://doi.org/10.1007/s11663-013-9960-3.

Zhao, Y., Qin, R., Chen, D., Wan, X., Li, Y., & Ma, M. (2015). A three dimensional cellular automata model for dendrite growth in non-equilibrium solidification of binary alloy. Steel Research International, 86(12), 1490–1497. https://doi.org/10.1002/srin.201400318.