Cellular Automata Hardware Implementation

  • Georgios Ch. SirakoulisEmail author
Reference work entry
Part of the Encyclopedia of Complexity and Systems Science Series book series (ECSSS)


Dynamic System

is a system in which a function describes the time dependence of a point in a geometrical space.

Electronic Hardware

consists of interconnected electronic components which perform analog or logic operations on received and locally stored information to produce as output or store resulting new information or to provide control for output actuator mechanisms.

Field Programmable Gate Array (FPGA)

is an integrated circuit designed to be configured by a customer or a designer after manufacturing.

VHDL [ VHSIC ( Very High-Speed Integrated Circuit) Hardware Description Language]

is a hardware description language (HDL), i.e., a specialized computer language, used to describe the structure and behavior of digital and mixed-signal systems.

VLSI ( Very Large-Scale Integration)

is the level of computer microchip miniaturization and integration which refers to microchips containing in the hundreds of thousands of transistors.

VLSI Architecture

is a set of rules and methods...


Primary Literature

  1. Adamides ED, Iliades P, Argyrakis J, Tsalides P, Thanailakis A (1993) Cellular logic bus arbitration. IEE Proc-E Comput Digit Tech (IEE) 140(6):289–296Google Scholar
  2. Albicki A, Khare M (1987) Cellular automata used for test pattern generation. In: Proceedings of the international conference on computer design. IEEE Computer Society Press, Los Alamitos, pp 56–59Google Scholar
  3. Altera 2007 Designing and using FPGAs for double precision floating-point math. White PaperGoogle Scholar
  4. Amlani I, Orlov AO, Toth G, Bernstein GH, Lent CS, Snider GL (1999) Digital logic gate using quantum-dot cellular automata. Science 284:289–291CrossRefGoogle Scholar
  5. Andreadis I, Karafyllidis I, Tzionas P, Thanailakis A, Tsalides P (1996) A new hardware module for automated visual inspection based on a cellular automaton architecture. J Intell Robot Syst (Springer) 16(1):89–102CrossRefGoogle Scholar
  6. Bak P, Tang C (1989) Earthquakes as a self-organised critical phenomenon. J Geophys Res 94:15635–15637CrossRefGoogle Scholar
  7. Bardell PH (1990) Analysis of cellular automata used as pseudo-random pattern generators. In: Proceedings of the international test conference ’90, pp 762–768CrossRefGoogle Scholar
  8. Bassham L et al. (2010) A statistical test suite for random and pseudorandom number generators for cryptographic applications. NIST.
  9. Bhattacharjee S (1997) Some studies on data compression, error correcting code and boolean function analysis. Ph.D. Thesis, I.I.T., KharagpurGoogle Scholar
  10. Burridge R, Knopoff L (1967) Model and theoretical seismicity. Bull Seismol Soc Am 57(3):341–371Google Scholar
  11. Card HC, Thanailakis A, Pries W, McLeod RD (1986) Analysis of bounded linear cellular automata based on a method of image charges. J Comput Syst Sci (Elsevier) 33(3):473–480MathSciNetCrossRefGoogle Scholar
  12. Chen RJ, Lai JL (2004) VLSI implementation of the universal 2-D CAT/ICAT system. In: Proceedings of the 11th IEEE international conference on electronics, circuits and systems, pp 187–190Google Scholar
  13. Chattopadhyay S (1996) Some studies on theory and applications of additive cellular automata. PhD Thesis, I.I.T., Kharagpur, IndiaGoogle Scholar
  14. Chaudhuri PP, Chowdhury DR, Nandi S, Chattopadhyay S (1997) Additive cellular automata: theory and applications, vol 1. Wiley-IEEE Computer Society Press, Los AlamitoszbMATHGoogle Scholar
  15. Chowdhury DR (1992) Theory and applications of additive cellular automata for reliable and testable VLSI circuit design. Ph.D. Thesis, I.I.T., KharagpurGoogle Scholar
  16. Chowdhury DR, Chaudhuri PP (1989) Parallel memory testing: a BIST approach. In: Proceedings of the 3rd international workshop on VLSI design, Bangalore, pp 373–377Google Scholar
  17. Chowdhury DR, Basu S, Gupta IS, Chaudhuri PP (1994a) Design of CAECC-cellular automata based error correcting code. IEEE Trans Comput (IEEE) 43(6):759–764MathSciNetzbMATHCrossRefGoogle Scholar
  18. Chowdhury DR, Sengupta IS, Chaudhuri PP (1994b) A class of two-dimensional cellular automata and applications in random pattern testing. J Electron Test Theory Appl 5(1):67–82CrossRefGoogle Scholar
  19. Das AK (1990) Additive cellular automata: theory and applications as a built-in self-test structure. Ph.D. Thesis, I.I.T., KharagpurGoogle Scholar
  20. Das AK, Chaudhuri PP (1989) An efficient on-chip deterministic test pattern generation scheme. Microprocess Microprogram (Elsevier) 26(3):195–204CrossRefGoogle Scholar
  21. Das AK, Chaudhuri PP (1993) Vector space theoretic analysis of additive cellular automata and its applications for pseudo-exhaustive test pattern generation. IEEE Trans Comput (IEEE) 42(3):340–352MathSciNetzbMATHCrossRefGoogle Scholar
  22. Das Sukanta (2006) Theory and applications of nonlinear cellular automata in vlsi design. Ph.D. thesis, Bengal Engineering And Science University, Shibpur West BengalGoogle Scholar
  23. Dourvas N, Tsompanas M-AI, Sirakoulis GC, Tsalides P (2015) Hardware acceleration of cellular automata physarum polycephalum model. Parallel Process Lett (World Scientific) 25:1540006. [25 pages]MathSciNetzbMATHCrossRefGoogle Scholar
  24. Feynman RP (1982) Simulating physics with computers. Int J Theor Phys (Springer) 21(6/7):467–488MathSciNetCrossRefGoogle Scholar
  25. Gardner M (1970) The fantastic combinations of John Conway’s new solitaire game “life”. Sci Am (IEEE) 223:120–123CrossRefGoogle Scholar
  26. Georgoudas IG, Sirakoulis GS, Emmanouil MS, Andreadis I (2007) A cellular automaton simulation tool for modelling seismicity in the region of Xanthi. Environ Model Softw (Elsevier) 22(10):1455–1464CrossRefGoogle Scholar
  27. Georgoudas IG, Sirakoulis GC, Andreadis I (2009) On chip earthquake simulation model using potentials. Nat Hazards (Springer) 50(3):519–537CrossRefGoogle Scholar
  28. Georgoudas IG, Koltsidas G, Sirakoulis GC, Andreadis I (2010a) A cellular automaton model for crowd evacuation and its auto-defined obstacle avoidance attribute. In: Proceedings of third international workshop on crowds and cellular automata (C&CA-2010) organized within the 9th international conference on cellular automata for research and industry (ACRI2010), Ascoli-Pizeno, pp 455–464Google Scholar
  29. Georgoudas IG, Kyriakos P, Sirakoulis GC, Andreadis I (2010b) An FPGA implemented cellular automaton crowd evacuation model inspired by the electrostatic-induced potential fields. Microprocess Microsyst (Elsevier) 34(7–8):285–300CrossRefGoogle Scholar
  30. Georgoudas I, Sirakoulis GC, Andreadis I (2011) An anticipative crowd management system preventing clogging in exits during pedestrian evacuation process. IEEE Syst J (IEEE) 5(1):129–141CrossRefGoogle Scholar
  31. Gutenberg B, Richter CF (1944) Frequency of earthquakes in California. Bull Seismol Soc Am 34:185–188Google Scholar
  32. Gutenberg B, Richter CF (1956) Magnitude and energy of earthquakes. Ann Geophys 9:1–15Google Scholar
  33. Halbach M, Hoffmann R (2004) Implementing cellular automata in FPGA logic. In: Proceedings of the 18th international parallel and distributed processing symposium, Santa Fe, pp 3531–3535Google Scholar
  34. Helbing D, Farkas I, Vicsek T (2000) Simulating dynamical features of escape panic. Nature 407:487–490CrossRefGoogle Scholar
  35. Hortensius PD, McLeod RD, Card HC (1989a) Parallel pseudo-random number generation for VLSI systems using cellular automata. IEEE Trans Comput (IEEE) 38(10):1466–1473CrossRefGoogle Scholar
  36. Hortensius PD, McLeod RD, Pries W, Miller DM, Card HC (1989b) Cellular automata based pseudo-random number generators for built-in self-test. IEEE Trans Comput-Aided Des (IEEE) 8(8):842–859CrossRefGoogle Scholar
  37. Hortensius PD, McLeod RD, Card HC (1990) Cellular automata based signature analysis for built-in self-test. IEEE Trans Comput (IEEE) 39(10):1273–1283CrossRefGoogle Scholar
  38. Jendrsczok J, Ediger P, Hoffmann R (2009) A scalable configurable architecture for the massively parallel GCA model. Int J Parallel Emergent Distrib Syst 24(4):275–291MathSciNetzbMATHCrossRefGoogle Scholar
  39. Kalogeropoulos G, Sirakoulis GC, Karafyllidis I (2013) Cellular automata on FPGA for real-time urban traffic signals control. J Supercomput (Springer) 65:1–18CrossRefGoogle Scholar
  40. Karafyllidis I, Ioannidis A, Thanailakis A, Tsalides P (1997) Geometrical shape recognition using a cellular automaton architecture and its VLSI implementation. Real-Time Imaging (Springer) 3(4):243–254CrossRefGoogle Scholar
  41. Karafyllidis I, Thanailakis A (1997) A model for predicting forest fire spreading using cellular automata. Ecol Modell (Elsevier) 99:87–97CrossRefGoogle Scholar
  42. Karafyllidis I, Andreadis I, Tzionas P, Tsalides P, Thanailakis A (1996) A cellular automaton for the determination of the mean velocity of moving objects and its VLSI implementation. Pattern Recogn (Elsevier) 29(4):689–699CrossRefGoogle Scholar
  43. Karafyllidis I, Andreadis I, Tsalides P, Thanailakis A (1998) Non-linear hybrid cellular automata as pseudorandom pattern generators for VLSI systems. VLSI Des 7(2):177–189CrossRefGoogle Scholar
  44. Katis I, Sirakoulis GC (2012) Cellular automata on fpgas for image processing. In: Proceedings of the 16th panhellenic conference on informatics (PCI 2012), Athens, pp 308–313CrossRefGoogle Scholar
  45. Kotoulas L, Tsarouchis D, Sirakoulis GC, Andreadis I (2006) 1-D cellular automaton for pseudorandom number generation and its reconfigurable hardware implementation. In: Proceedings of 2006 I.E. international symposium on circuits and systems (ISCAS’2006), Island of Kos, pp 4627–4630Google Scholar
  46. Landman BS, RL R (1971) On a pin versus block relationship for partitions of logic graphs. IEEE Trans Comput C – (IEEE) 20(12):1469–1479CrossRefGoogle Scholar
  47. Langhammer, M. 2007. Double precision floating point on FPGAs. In: Proceedings of the 3rd annual reconfigurable systems summer Institute. National Center for Supercomputing Applications, UrbanaGoogle Scholar
  48. Lanzerotti MY, Fiorenza G, Rand RA (2005) Microminiature packaging and integrated circuitry: the work of {E. F. Rent}, with an application to on-chip interconnection requirements. IBM J Res Develop (IBM) 49(4,5):777–803CrossRefGoogle Scholar
  49. Lent CS, Tougaw D (1997) A device architecture for computing with quantum dots. Proc IEEE (IEEE) 85(4):541–557CrossRefGoogle Scholar
  50. Lent CS, Tougaw PD, Porod W, Bernstein GH (1993) Quantum cellular automata. Nanotechnology (IOP) 4(1):49–57CrossRefGoogle Scholar
  51. Mardiris V, Sirakoulis GC, Mizas C, Karafyllidis I, Thanailakis A (2008) A CAD system for modeling and simulation of computer networks using cellular automata. IEEE Trans Syst Man Cybern – Part C (IEEE) 38(2):253–264CrossRefGoogle Scholar
  52. Mardiris V, Sirakoulis GC, Karafyllidis I (2015) Automated design architecture for 1-D cellular automata using quantum cellular automata. IEEE Trans Comput (IEEE) 64(9):2476–2489MathSciNetzbMATHCrossRefGoogle Scholar
  53. Marriot AP, Tsalides P, Hicks PJ (1991) VLSI implementation of smart imaging system using two-dimensional cellular automata. IEE Proc-G Circuits Dev Syst (IEE) 138(5):582–586CrossRefGoogle Scholar
  54. McLeod RD, Hortensius P, Schneider R, Card HC, Bridges G, Pries W (1986) CALBO-cellular automaton logic block observation. In: Proceedings of the Canadian conference on VLSI. IEEE Computer Society Press, Los Alamitos, pp 171–176Google Scholar
  55. Minsky M (1982) Cellular vacuum. Int J Theor Phys (Springer) 21(6/7):537–551zbMATHCrossRefGoogle Scholar
  56. Misra S (1992) Theory and applications of additive cellular automata for easily testable VLSI circuit design. Ph.D. thesis, I.I.T., KharagpurGoogle Scholar
  57. Murtaza S, Hoekstra AG, Sloot PMA (2007) Performance modeling of 2D cellular automata on FPGA. In: Proceedings of the international conference on field programmable logic and applications, pp 74–78Google Scholar
  58. Murtaza S, Hoekstra AG, Sloot PMA (2008) Floating point based cellular automata simulations using a dual FPGA-enabled system. In: Proceedings of the 2nd international workshop on high-performance reconfigurable computing technology and applications, pp 1–8Google Scholar
  59. Murtaza S, Hoekstra AG, Sloot PMA (2011) Cellular automata simulations on a FPGA cluster. Int J High Perform Comput Appl 25(2):193–204CrossRefGoogle Scholar
  60. Nagel K, Schreckenberg M (1992) A cellular automaton model for freeway traffic. J Phys I Fr 2(12):2221–2229CrossRefGoogle Scholar
  61. Nakagaki T, Yamada H, Toth A (2000) Intelligence: maze-solving by an amoeboid organism. Nature (Springer Nature) 407(6803):470–470CrossRefGoogle Scholar
  62. Nalpantidis L, Amanatiadis A, Sirakoulis GC, Gasteratos A (2011) An efficient hierarchical matching algorithm for processing uncalibrated stereo vision images and its hardware architecture. IET Image Process (IET) 5(5):481–492CrossRefGoogle Scholar
  63. Nandi S (1994) Additive cellular automata: theory and applications for testable circuit design and data encryption. Ph.D. thesis, I.I.T., KharagpurGoogle Scholar
  64. Ntinas V, Moutafis B, Trunfio GA, Sirakoulis GC (2017) Parallel fuzzy cellular automata for data-driven simulation of wildfire simulations. J Comput Sci (Elsevier) 21:469–485CrossRefGoogle Scholar
  65. Omohundro S (1984) Modelling cellular automata with partial differential equations. Phys D Nonlinear Phenomena (Elsevier) 10:128–134MathSciNetzbMATHCrossRefGoogle Scholar
  66. Pitsianis N, Tsalides P, Bleris GL, Thanailakis A, Card HC (1989a) Deterministic one-dimensional cellular automata. J Stat Phys (Elsevier) 56(1):99–112MathSciNetzbMATHCrossRefGoogle Scholar
  67. Pitsianis N, Tsalides P, Bleris GL, Thanailakis A, Card HC (1989b) Algebraic theory of bounded one-dimensional cellular automata. Complex Syst 3(2):209–227MathSciNetzbMATHGoogle Scholar
  68. Porter R, Frigo J, Conti A, Harvey N, Kenyon G, Gokhale M (2007) A reconfigurable computing framework for multi-scale cellular image processing. Microprocess Microsyst (Elsevier) 31(8):546–563CrossRefGoogle Scholar
  69. Pries W, Thanailakis A, Card HC (1986) Group properties of cellular automata and VLSI applications. IEEE Trans Comput (IEEE) 35(12):1013–1024zbMATHCrossRefGoogle Scholar
  70. Progias P, Sirakoulis GC (2013) An FPGA processor for modelling wildfire spread. Math Comput Model (Elsevier) 57(5–6):1436–1452CrossRefGoogle Scholar
  71. Rukhin Andrew et al (2001) A statistical test suite for random and pseudorandom number generators for cryptographic applications, NIST.
  72. Serra M, Slater T, Muzio JC, Miller DM (1990) Analysis of one dimensional cellular automata and their aliasing probabilities. IEEE Trans Comput-Aided Des (IEEE) 9(7):767–778CrossRefGoogle Scholar
  73. Sirakoulis GC (2004) A TCAD system for VLSI implementation of the CVD process using VHDL. Integr VLSI J (Elsevier) 37(1):63–81CrossRefGoogle Scholar
  74. Sirakoulis GC (2015) The computational paradigm of cellular automata in crowd evacuation. Int J Found Comput Sci (World Scientific) 26(7):851MathSciNetzbMATHCrossRefGoogle Scholar
  75. s N, Thanailakis A (1999) A new simulator for the oxidation process in integrated circuit fabrication based on cellular automata. Model Simul Mater Sci Eng (IOP) 7(4):631–640Google Scholar
  76. Sirakoulis GC, Karafyllidis I, Mardiris V, Thanailakis A (2000a) Study of the effects of photoresist surface roughness and defects on developed profiles. Semicond Sci Technol (IOP Publishing) 15:98CrossRefGoogle Scholar
  77. Sirakoulis GC, Karafyllidis I, Thanailakis A (2000b) A cellular automaton model for the effect of population movement on epidemic propagation. Ecol Model (Elsevier) 133(3):209–223CrossRefGoogle Scholar
  78. Sirakoulis GC, Karafyllidis I, Thanailakis A, Mardiris V (2001) A methodology for VLSI implementation of cellular automata algorithms using VHDL. Adv Eng Softw (Elsevier) 32(3):189–202zbMATHCrossRefGoogle Scholar
  79. Sirakoulis GC, Karafyllidis I, Thanailakis A (2003) A CAD system for the construction and VLSI implementation of cellular automata algorithms using VHDL. Microprocess Microsyst (Elsevier) 27:381–396CrossRefGoogle Scholar
  80. Srisuchinwong B, York TK, Tsalides P, Hicks PJ, Thanailakis A (1992) VLSI implementation of a mod-p multipliers using Homomorphisms and hybrid cellular automaton-based data compression techniques. IEE Proc-E Comput Digit Tech (IEE) 139(6):486–490CrossRefGoogle Scholar
  81. Toffoli T (1984a) Cellular automata as an alternative to (rather than an approximation of) differential equations in modeling physics. Phys D Nonlinear Phenomena (Elsevier) 10(1–2):117–127MathSciNetzbMATHCrossRefGoogle Scholar
  82. Toffoli T (1984b) CAM: a high-performance cellular automaton machine. Phys D Nonlinear Phenomena (Elsevier) 10(1–2):195–204MathSciNetCrossRefGoogle Scholar
  83. Tsalides P (1990) Cellular automata based built-in self-test structures for VLSI systems. IEE Electron Lett (IEE) 26(17):1350–1352CrossRefGoogle Scholar
  84. Tsalides P, Hicks PJ, York TA (1989) Three dimensional cellular automata and VLSI applications. IEE Proc-E Comput Digit Tech (IEE) 136(6):490–495CrossRefGoogle Scholar
  85. Tsalides P, York TA, Thanailakis A (1991) Pseudo-random number generators for VLSI systems based on linear cellular automata. IEE Proc-E Comput Digit Tech (IEE) 138(4):241–249CrossRefGoogle Scholar
  86. Tsalides P, Thanailakis A, Pitsanis N, Bleris GL (1992) Two-dimensional cellular automata: properties and applications of a new VLSI architecture. Comput J (Oxford) 35(4):A377–A386Google Scholar
  87. Tsiftsis A, Georgoudas IG, and Sirakoulis GCh (2016) Real data evaluation of a crowd supervising system for stadium evacuation and its hardware implementation. IEΕE Systems 10(2):649–660CrossRefGoogle Scholar
  88. Tsompanas M-AI, Sirakoulis GC (2012) Modeling and hardware implementation of an amoeba-like cellular automaton. Bioinspir Biomim (IOP) 7:036013. (19 pp.)CrossRefGoogle Scholar
  89. Tsompanas M-AI, Sirakoulis GC, Adamatzky A (2016) Physarum in silicon: the Greek motorways study. Nat Comput (Springer) 15(2):279–295MathSciNetCrossRefGoogle Scholar
  90. Tzionas P, Tsalides P, Thanailakis A (1992) Design and VLSI implementation of a pattern classifier using pseudo 2D cellular automata. IEE Proc-G Circuits Dev Syst (IEE) 139(6):661–668CrossRefGoogle Scholar
  91. Tzionas P, Tsalides P, Thanailakis A (1996) A new-hybrid cellular automaton/neural network classifier for multi-valued patterns and its VLSI implementation. Integr VLSI J (Elsevier) 20(2):211–237zbMATHCrossRefGoogle Scholar
  92. Ulam S (1952) Random processes and transformations. In: Proceedings of the international congress on mathematics, pp 264–275Google Scholar
  93. Vacca M, Wang J, Graziano M, Roch MR, Zamboni M (2015) Feedbacks in QCA: a quantitative approach. IEEE Trans Very Large Scale Integr VLSI Syst (IEEE) 23(10):2233–2243CrossRefGoogle Scholar
  94. Vichniac GY (1984) Simulating physics with cellular automata. Phys D Nonlinear Phenomena (Elsevier) 10:96–116MathSciNetzbMATHCrossRefGoogle Scholar
  95. Viola P, Jones MJ, Snow D (2003) Detecting pedestrians using patterns of motion and appearance. In: 2003 proceedings of IEEE international conference on computer vision, pp 734–741Google Scholar
  96. von Neumann J, Burks AW, and others (1966) Theory of self-reproducing automata. IEEE Trans Neural Netw (IEEE) 5: 3–14Google Scholar
  97. Vourkas I, Sirakoulis GC (2012) FPGA based cellular automata for environmental modeling. In: Proceedings of the 2012 I.E. international conference on electronics, circuits, and systems (ICECS 2012), Seville, pp 308–313Google Scholar
  98. Weston JL, Lee P (2008) FPGA implementation of cellular automata spaces using a CAM based cellular architecture. In: Proceedings of the NASA/ESA conference on adaptive hardware and systems, pp 315–322Google Scholar
  99. Wolfram S (1984) Universality and complexity in cellular automata. Phys D (Elsevier) 10(1–2):1–35MathSciNetzbMATHGoogle Scholar
  100. Wolkow R, Livadaru L, Pitters J, Taucerg M, Piva M, Salomons M, Cloutier M, Martins B (2014) Silicon atomic quantum dots enable beyond-CMOS electronics. In: Field-coupled nanocomputing, Lecture notes in computer science, Springer Berlin Heidelberg, Berlin, Heidelberg. vol 8280, pp 33–58Google Scholar
  101. York TK, Tsalides P, Srisuchinwong B, Hicks PJ, Thanailakis A (1991) Design and VLSI implementation of a mod-127 multiplier using cellular automaton-based data compression techniques. IEE Proc-E Comput Digit Tech (IEE) 138(5):351–356CrossRefGoogle Scholar
  102. Zadeh LA (1965) Fuzzy sets. Inf Control (Elsevier) 8(3):338–353zbMATHCrossRefGoogle Scholar

Book & Reviews

  1. Adamatzky A (2010a) Physarum machines: computers from slime mould, vol 74. World Scientific, Singapore/HackensackCrossRefGoogle Scholar
  2. Adamatzky A (2010b) Game of life cellular automata. Springer, LondonzbMATHCrossRefGoogle Scholar
  3. Chopard B, Droz M (1998) Cellular automata modeling of physical systems. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  4. Hurst SL (1998) VLSI testing: digital and mixed analogue/digital techniques. The Institution of Electrical Engineering (IEE), LondonCrossRefGoogle Scholar
  5. Knuth DE (1981) The art of computer programming-seminumerical algorithms. Addison-Wesley, ReadingzbMATHGoogle Scholar
  6. Maraglia George (1995) The Marsaglia random number CDROM including the Diehard battery of tests of randomness. Florida State University. Archived from the original on 25 Jan 2016
  7. Pettey C (1997) Diffusion (cellular) models. In: Handbook of evolutionary computation. Oxford University PressGoogle Scholar
  8. Preston Kendall Jr, M.J.B. Duff. 1984. Modern cellular automata. Theory and applications SpringerGoogle Scholar
  9. Rosin P, Adamatzky A, Sun X (2014) Cellular automata in image processing and geometry. Springer, ChamzbMATHCrossRefGoogle Scholar
  10. Sirakoulis GC, S Bandini (2012) Cellular automata – proceedings of 10th international conference on cellular automata for research and industry, ACRI 2012, SpringerGoogle Scholar
  11. Toffoli T, Margolus N (1987) Cellular automata machines: a new environment for modeling. MIT Press, CambridgezbMATHGoogle Scholar
  12. Was J, Sirakoulis GC, Bandini S (2014). Cellular automata – proceedings of 11th international conference on cellular automata for research and industry, ACRI 2014. SpringerGoogle Scholar
  13. Wolfram S (1994) Cellular automata and complexity: collected papers. Westview Press, BoulderzbMATHGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.School of Engineering, Department of Electrical and Computer EngineeringDemocritus University of ThraceXanthiGreece

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