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Additive Cellular Automata

  • Reference work entry
Encyclopedia of Complexity and Systems Science

Definition of the Subject

Cellular automata are discrete dynamical systems in which an extended array of symbols from a finite alphabet is iteratively updated according to a specified local rule. Originally developed by John von Neumann [1,2] in 1948, following suggestions from Stanislaw Ulam, for the purpose of showing that self-replicating automata could be constructed. Von Neumann's construction followed a complicated set of reproduction rules but later work showed that self-reproducing automata could be constructed with only simple update rules, e. g. [3]. More generally, cellular automata are of interest because they show that highly complex patterns can arise from the application of very simple update rules. While conceptually simple, they provide a robust modeling class for application in a variety of disciplines, e. g. [4], as well as fertile grounds for theoretical research. Additive cellular automata are the simplest class of cellular automata. They have been extensively...

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Abbreviations

Cellular automata :

Cellular automata are dynamical systems that are discrete in space, time, and value. A state of a cellular automaton is a spatial array of discrete cells, each containing a value chosen from a finite alphabet. The state space for a cellular automaton is the set of all such configurations.

Alphabet of a cellular automaton:

The alphabet of a cellular automaton is the set of symbols or values that can appear in each cell. The alphabet contains a distinguished symbol called the null or quiescent symbol, usually indicated by 0, which satisfies the condition of an additive identity: \( { 0 + x = x } \).

Cellular automata rule :

The rule, or update rule of a cellular automaton describes how any given state is transformed into its successor state. The update rule of a cellular automaton is described by a rule table, which defines a local neighborhood mapping, or equivalently as a global update mapping.

Additive cellular automata:

An additive cellular automaton is a cellular automaton whose update rule satisfies the condition that its action on the sum of two states is equal to the sum of its actions on the two states separately.

Linear cellular automata :

A linear cellular automaton is a cellular automaton whose update rule satisfies the condition that its action on the sum of two states separately equals action on the sum of the two states plus its action on the state in which all cells contain the quiescent symbol. Note that some researchers reverse the definitions of additivity and linearity.

Neighborhood:

The neighborhood of a given cell is the set of cells that contribute to the update of value in that cell under the specified update rule.

Rule table:

The rule table of a cellular automaton is a listing of all neighborhoods together with the symbol that each neighborhood maps to under the local update rule.

Local maps of a cellular automaton:

The local mapping for a cellular automaton is a map from the set of all neighborhoods of a cell to the automaton alphabet.

State transition diagram :

The state transition diagram (STD) of a cellular automaton is a directed graph with each vertex labeled by a possible state and an edge directed from a vertex x to a vertex y if and only if the state labeling vertex x maps to the state labeling vertex y under application of the automaton update rule.

Transient states :

A transient state of a cellular automaton is a state that can at most appear only once in the evolution of the automaton rule.

Cyclic states :

A cyclic state of a cellular automaton is a state lying on a cycle of the automaton update rule, hence it is periodically revisited in the evolution of the rule.

Basins ofattraction :

The basins of attraction of a cellular automaton are the equivalences classes of cyclic states together with their associated transient states, with two states being equivalent if they lie on the same cycle of the update rule.

Predecessor state :

A state x is the predecessor of a state y if and only if x maps to y under application of the cellular automaton update rule. More specifically, a state x is an nth order predecessor of a state y if it maps to y under n applications of the update rule.

Garden-of-Eden:

A Garden-of-Eden state is a state that has no predecessor. It can be present only as an initial condition.

Surjectivity:

A mapping is surjective (or onto) if every state has a predecessor.

Injectivity:

A mapping is injective (one-to-one) if every state in its domain maps to a unique state in its range. That is, if states x and y both map to a state z then \( { x = y } \).

Reversibility :

A mapping \( { \mathcal{X} } \) is reversible if and only if a second mapping \( { \mathcal{X}^{-1} } \) exists such that if \( { \mathcal{X}(x) = y } \) then \( { \mathcal{X}^{-1}(y) = x } \). For finite state spaces reversibility and injectivity are identical.

Literature

Primary Literature

  1. von Neumann J (1963) The general and logical theory of automata. In: Taub A (ed) J. von Neumann Collected Works, vol 5. Pergamon, NY, pp 288–328

    Google Scholar 

  2. von Neumann J, Burk AW (ed) (1966) Theory of Self-Reproducing Automata. University of Illinois Press, Urbana

    Google Scholar 

  3. Arbib M (1966) Simple self-reproducing universal automata. Inf Control 9:177–189

    MATH  Google Scholar 

  4. Toffoli T, Margolis N (1987) Cellular Automata Machines: A New Environment for Modeling. MIT Press, Cambridge

    Google Scholar 

  5. Codd EF (1968) Cellular Automata. Academic Press, NY

    MATH  Google Scholar 

  6. Duff MJB, Preston K, Jr. (1984) Modern Cellular Automata: Theory and Applications. Plenum, NY

    MATH  Google Scholar 

  7. Sarkar P (2000) A brief history of cellular automata. ACM Comput Syst 32(1):80–107

    Google Scholar 

  8. Chopard B, Droz M (1998) Cellular Automata Modelling of Physical Systems. Cambridge University Press, Cambridge

    Google Scholar 

  9. Lindenmayer A, Rozenberg G (1976) Automata, Languages, Development. North Holland, Amsterdam

    MATH  Google Scholar 

  10. Mackay AL (1976) Crystal Symmetry. Phys Bull 27:495–497

    Google Scholar 

  11. Langer JS (1980) Instabilities and pattern formation in crystal growth. Rev Mod Phys 52:1

    ADS  Google Scholar 

  12. Lin F, Goldenfeld N (1990) Generic features of late-stage crystal growth. Phys Rev A 42:895–903

    ADS  Google Scholar 

  13. Greenberg JM, Hastings SP (1978) Spatial patterns for discrete models of diffusion in excitable media. SIAM J Appl Math 34(3):515–523

    MathSciNet  MATH  Google Scholar 

  14. Greenberg JM, Hassard BD, Hastings SP (1978) Pattern formation and periodic structures in systems modeled by reaction-diffusion equations. Bull Am Math Soc 84:1296–1327

    MathSciNet  MATH  Google Scholar 

  15. Madore BF, Freedman WL (1983) Computer simulations of the Belousov-Zhabotinsky reaction. Science 222:615–616

    ADS  Google Scholar 

  16. Adamatzky A, Costello BDL, Asai T (2005) Reaction-Diffusion Computers. Elsevier, London

    Google Scholar 

  17. Oono Y, Kohmoto M (1985) A discrete model for chemical turbulance. Phys Rev Lett 55:2927–2931

    MathSciNet  ADS  Google Scholar 

  18. Falk H (1986) Comments on a simple cellular automata in spin representation. Physica D 20:447–449

    MathSciNet  ADS  MATH  Google Scholar 

  19. Canning A, Droz M (1991) A comparison of spin exchange and cellular automata models for diffusion-controlled reactions. In: Gutowitz HA (ed) Cellular Automata: Theory and Experiment. MIT Press, Cambridge, pp 285–292

    Google Scholar 

  20. Vitanni P (1973) Sexually reproducing cellular automata. Math Biosci 18:23–54

    MathSciNet  Google Scholar 

  21. Young D (1984) A local activator-inhibitor model of vertebrate skin patterns. Math Biosci 72:51–58

    MathSciNet  Google Scholar 

  22. Dutching W, Vogelsaenger T (1985) Recent progress in modeling and simulation of three dimensional tumor growth and treatment. Biosystems 18(1):79–104

    Google Scholar 

  23. Moreira J, Deutsch A (2002) Cellular automata models of tumor development: A critical review. Adv Complex Syst 5(2&3):247–269

    MathSciNet  MATH  Google Scholar 

  24. Sieburg HB, McCutchan JA, Clay OK, Cabalerro L, Ostlund JJ (1991) Simulation on HIV infection in artificial immune systems. In: Gutowitz HA (ed) Cellular Automata: Theory and Experiment. MIT Press, Cambridge, pp 208–227

    Google Scholar 

  25. Santos RMZD, Continho S (2001) Dynamics of HIAV approach: A cellular automata approach. Phys Rev Lett 87(16):102–104

    Google Scholar 

  26. Beauchemin C, Samuel J, Tuszynski J (2005) A simple cellular automaton model for influenza A viral infection. J Theor Biol 232(2):223–234

    MathSciNet  Google Scholar 

  27. Burks C, Farmer D (1984) Towards modeling DNA sequences as automata. Physica D 10:157–167

    MathSciNet  ADS  Google Scholar 

  28. Moore JH, Hahn LW (2002) Cellular automata and genetic algorithms for parallel problem solving in human genetics. In: Merelo JJ, Panagiotis A, Beyer H-G (eds) Lecture Notes in Computer Science. Springer, Berlin, pp 821–830

    Google Scholar 

  29. Gerola H, Seiden P (1978) Stochastic star formation and spiral structure of galaxies. Astrophys J 223:129–135

    ADS  Google Scholar 

  30. Flache A, Hegselmann R (1998) Understanding complex social dynamics: A plead for cellular automata based modeling. J Artif Soc Social Simul 1(3):1

    Google Scholar 

  31. Chen K, Bak P, Jensen MH (1990) A deterministic critical forest-fire model. Phys Lett A 149:207–210

    MathSciNet  ADS  Google Scholar 

  32. Drossel B, Schwabl F (1992) Self-organized critical forest-fire model. Phys Rev Lett 69:1629–1632

    ADS  Google Scholar 

  33. Smith III AR (1972) Real-time language recognition by one‐dimensional cellular automata. J Comput Syst Sci 6:233–253

    MATH  Google Scholar 

  34. Sommerhalder R, van Westrhenen SC (1983) Parallel language recognition in constant time by cellular automata. Acta Inform 19:397–407

    MathSciNet  MATH  Google Scholar 

  35. Ibarra OH, Palis MA, Kim SM (1985) Fast parallel language recognition by cellular automata. Theor Comput Sci 41:231–246

    MathSciNet  MATH  Google Scholar 

  36. Morita K, Ueno S (1994) Parallel generation and parsing of array languages using reversible cellular automata. Int J Pattern Recognit Artif Intell 8:543–561

    Google Scholar 

  37. Jen E (1986) Invariant strings and pattern recognizing properties of 1D CA. J Stat Phys 43:243–265

    MathSciNet  ADS  Google Scholar 

  38. Raghavan R (1993) Cellular automata in pattern recognition. Inf Sci 70:145–177

    Google Scholar 

  39. Chattopadhyay S, Adhikari S, Sengupta S, Pal M (2000) Highly regular, modular, and cascadable design of cellular automata-based pattern classifier. IEEE Trans VLSI Syst 8(6):724–735

    Google Scholar 

  40. Rosenfeld A (1979) Picture Languages. Academic Press, NY

    MATH  Google Scholar 

  41. Sternberg SR (1980) Language and architecture for parallel image processing. In: Gelesma ES, Kanal LN (eds) Pattern Recognition in Practice. North-Holland, Amsterdam, p 35

    Google Scholar 

  42. Hopcroft JE, Ullman JD (1972) Introduction to Automata Theory, Language, and Computation. Addison-Wesley, Reading

    Google Scholar 

  43. Cole SN (1969) Real time computation by n-dimensional iterative arrays of finite state machines. IEEE Trans Comput C-18:349

    Google Scholar 

  44. Benjamin SC, Johnson NF (1997) A possible nanometer-scale computing device based on an additive cellular automata. Appl Phys Lett 70(17):2321–2323

    ADS  Google Scholar 

  45. Carter F (1984) The molecular device computer: point of departure for large scale cellular automata. Physica D 10:175–194

    MathSciNet  ADS  Google Scholar 

  46. Hillis WD (1984) The connection machine: A computer architecture based on cellular automata. Physica D 10:213–228

    ADS  Google Scholar 

  47. Manning FB (1977) An approach to highly integrated computer-maintained cellular arrays. IEEE Trans Comput C-26:536

    Google Scholar 

  48. Atrubin AJ (1965) A one‐dimensional real time iterative multiplier. IEEE Trans Comput EC-14:394

    Google Scholar 

  49. Nishio H (1981) Real time sorting of binary numbers by a 1-dimensional cellular automata. Kyoto University Technical Report

    Google Scholar 

  50. Fischer PC (1965) Generation of primes by a one‐dimensional real time iterative array. J Assoc Comput Machin 12:388

    MATH  Google Scholar 

  51. Pries W, Thanailakis A, Card HC (1986) Group properties of cellular automata and VLSI applications. IEEE Trans Comput C-35:1013–1024

    Google Scholar 

  52. Chaudhuri PP, Chowdhury DR, Nandi S, Chattopadhyay S (1997) Additive Cellular Automata: Theory and Applications, vol 1. IEEE Computer Society Press, Los Alamitos

    MATH  Google Scholar 

  53. Bardell PH, McAnney WH (1986) Pseudo-random arrays for built-in tests. IEEE Trans Comput C-35(7):653–658

    Google Scholar 

  54. Hortensius PD, McLeod RD, Card HC (1989) Parallel random number generation for VLSI systems using cellular automata. IEEE Trans Comput 38(10):1466–1473

    Google Scholar 

  55. Tsalides P, York TA, Thanailakis A (1991) Pseudorandom number generation for VLSI systems using cellular automata. IEEE Proc E: Comput Digit Technol 138:241–249

    Google Scholar 

  56. Matsumoto M (1998) Simple cellular automata as pseudorandom m-sequence generators for built-in self-test. ACM Trans Modeling Comput Simul (TOMACS) 8(1):31–42

    MATH  Google Scholar 

  57. Tomassini M, Sipper M, Perrenoud M (2000) On the generation of high-quality random numbers by two-dimensional cellular automata. IEEE Trans Comput 49(10):1146–1151

    Google Scholar 

  58. Das AK, Chaudhuri PP (1989) An efficient on-chip deterministic test pattern generation scheme. Euromicro J, Microprocess, Microprogramm 26:195–204

    Google Scholar 

  59. Serra M (1990) Algebraic analysis and algorithms for linear cellular automata over GF(2) and their applications to digital circuit testing. Congressus Numerantium 75:127–139

    MathSciNet  Google Scholar 

  60. Das AK, Chaudhuri PP (1993) Vector space theoretical analysis of additive cellular automata and its applications for pseudo-exhaustive test pattern generation. IEEE Trans Comput 42:340–352

    MathSciNet  Google Scholar 

  61. Tziones P, Tsalides P, Thanailakis A (1994) A new cellular automaton-based nearest neighbor pattern classifier and its VLSI implementation. IEEE Trans VLSI Implement 2(3):343–353

    Google Scholar 

  62. Mrugalski G, Rajski J, Tyszer J (2000) Cellular automata-based test pattern generators with phase shifters. IEEE Trans Comput-Aided Des 19(8):878–893

    Google Scholar 

  63. Sikdar BK, Ganguly N, Chaudhuri PP (2002) Design of hierarchical cellular automata for on-chip test pattern generator. IEEE Trans Comput Assist Des 21(12):1530–1539

    Google Scholar 

  64. Hortensius PD, McLeod RD, Card HC (1990) Cellular automata based signature analysis for built-in self-test. IEEE Trans Comput C-39:1273–1283

    Google Scholar 

  65. Serra M, Slater T, Muzio JC, Miller DM (1990) Analysis of one dimensional cellular automata and their aliasing probabilities. IEEE Trans Comput-Aided Des 9:767–778

    Google Scholar 

  66. Dasgupta P, Chattopadhyay S, Sengupta I (2001) Theory and application of nongroup cellular automata for message authentification. J Syst Architecture 47(7):383–404

    Google Scholar 

  67. Chowdhury DR, Basu S, Gupta IS, Chaudhuri PP (1994) Design of CAECC cellular automata based error correcting code. IEEE Trans Comput 43:759–764

    MathSciNet  MATH  Google Scholar 

  68. Chowdhury DR, Gupta IS, Chaudhuri PP (1995) Cellular automata based byte error correcting code. IEEE Trans Comput 44:371–382

    MATH  Google Scholar 

  69. Chowdhury DR, Gupta IS, Chaudhuri PP (1995) A low-cost high-capacity associative memory design using cellular automata. IEEE Trans Comput 44:1260–1264

    MATH  Google Scholar 

  70. Nandi S, Kar BK, Chaudhuri PP (1994) Theory and application of cellular automata in cryptography. IEEE Trans Comput 43(12):1346–1357

    MathSciNet  Google Scholar 

  71. Cattell K, Muzio JC (1996) Synthesis of one‐dimensional linear hybrid cellular automata. IEEE Trans Comput-Aided Des 15:325–335

    Google Scholar 

  72. Cattell K, Zhang S, Serra M, Zmuzio JC (1999) 2-by-n hybrid cellular automata with regular configuration: Theory and applications. IEEE Trans Comput 48(3):285–295

    MathSciNet  Google Scholar 

  73. Hedlund GA (1969) Endomorphisms and automorphisms of the shift dynamical system. Math Syst Theory 4:320–375

    MathSciNet  Google Scholar 

  74. Wolfram S (1983) Statistical mechanics of cellular automata. Rev Mod Phys 55:601–644

    MathSciNet  ADS  MATH  Google Scholar 

  75. Gardner M (1970) The fantastic combinations of John Conway's new solitaire game ‘life’. Sci Am 223:120–123

    Google Scholar 

  76. Gardner M (1971) On cellular automata self-reproduction, the Garden of Eden and the game of ‘life’. Sci Am 224:112–117

    Google Scholar 

  77. Jen E (1988) Cylindrical cellular automata. Commun Math Phys 118:569–590

    MathSciNet  ADS  MATH  Google Scholar 

  78. Tadaki S, Matsufuji S (1993) Periodicity in one‐dimensional finite linear cellular automata. Prog Theor Phys 89(2):325–331

    ADS  Google Scholar 

  79. Nandi S, Pal Chaudhuri P (1996) Analysis of periodic and intermediate boundary 90/150 cellular automata. IEEE Trans Comput 45(1):1–12

    MathSciNet  MATH  Google Scholar 

  80. Chin W, Cortzen B, Goldman J (2001) Linear cellular automata with boundary conditions. Linear Algebra Appl 322:193–206

    MathSciNet  MATH  Google Scholar 

  81. Voorhees BH (1996) Computational Analysis of One Dimensional Cellular Automata. World Scientific, Singapore

    MATH  Google Scholar 

  82. Willson S (1984) Cellular automata can generate fractals. Discret Appl Math 8:91–99

    MathSciNet  MATH  Google Scholar 

  83. Willson S (1984) Growth rates and fractional dimension in cellular automata. Physica D 10:69–74

    MathSciNet  ADS  Google Scholar 

  84. Peitgen H-O, Richter PH (1986) The Beauty of Fractals: Images of Complex Dynamical Systems. Springer, NY

    MATH  Google Scholar 

  85. Willson S (1987) The equality of fractional dimension for certain cellular automata. Physica D 24:179–189

    MathSciNet  ADS  MATH  Google Scholar 

  86. Willson S (1987) Computing fractional dimension of additive cellular automata. Physica D 24:190–206

    MathSciNet  ADS  MATH  Google Scholar 

  87. Culik II K, Dube S (1989) Fractals and recurrent behavior of cellular automata. Complex Syst 3:253–267

    MathSciNet  MATH  Google Scholar 

  88. Willson S (1992) Calculating growth rates and moments for additive cellular automata. Discret Appl Math 35:47–65

    MathSciNet  MATH  Google Scholar 

  89. Voorhees B (1988) Cellular automata, Pascal's triangle, and generation of order. Physica D 31:135–140

    MathSciNet  ADS  MATH  Google Scholar 

  90. von Haeseler F, Peitgen H-O, Skordev G (1992) Pascal's triangle, dynamical systems and attractors. Ergod Theory Dyn Syst 12:479–486

    MATH  Google Scholar 

  91. Allouche JP, von Haeseler F, Peitgen H-O, Skordev G (1996) Linear cellular automata, finite automata and Pascal's triangle. Discret Appl Math 66:1–22

    MATH  Google Scholar 

  92. von Haeseler F, Peitgen H-O, Skordev G (1992) Linear cellular automata, substitutions, hierarchical iterated function systems. In: Encarnacao JL, Peitgen H-O, Sakas G, Englert G (eds) Fractal Geometry and Computer Graphics. Springer, NY

    Google Scholar 

  93. von Haeseler F, Peitgen H-O, Skordev G (1993) Cellular automata, matrix substitution, and fractals. Ann Math Artif Intell 8(3,4):345–362

    MATH  Google Scholar 

  94. von Haeseler F, Peitgen H-O, Skordev G (1995) Global analysis of self-similar features of cellular automata: selected examples. Physica D 86:64–80

    MathSciNet  MATH  Google Scholar 

  95. Barbé A, von Haeseler F, Peitgen H-O, Skordev G (1995) Course-graining invariant patterns of one‐dimensional two-state linear cellular automata. Int J Bifurc Chaos 5:1611–1631

    Google Scholar 

  96. von Haeseler F, Peitgen H-O, Skordev G (2001) Self-similar structures of rescaled evolution sets of cellular automata I. Int J Bifurc Chaos 11(4):913–926

    MATH  Google Scholar 

  97. Nagler J, Claussen JC (2005) 1/f\( { ^{\alpha } } \) spectra in elementary cellular automata and fractal signals. Phys Rev E 71:067103

    ADS  Google Scholar 

  98. von Haeseler F, Peitgen H-O, Skordev G (2001) Self-similar structures of rescaled evolution sets of cellular automata II. Int J Bifurc Chaos 11(4):927–941

    MATH  Google Scholar 

  99. Barbé A, von Haeseler F, Peitgen H-O, Skordev G (2003) Rescaled evolution sets of linear cellular automata on a cylinder. Int J Bifurc Chaos 13(4):815–842

    Google Scholar 

  100. Takahashi S (1990) Cellular automata and multifractals: Dimension spectra of linear cellular automata. Physica D 45:36–48

    Google Scholar 

  101. Takahashi S (1992) Self-similarity of linear cellular automata. J Comput Syst Sci 44(1):114–140

    MATH  Google Scholar 

  102. Martin O, Odlyzko A, Wolfram S (1984) Algebraic properties of cellular automata. Commun Math Phys 93:219–258

    MathSciNet  ADS  MATH  Google Scholar 

  103. Jen E (1986) Global properties of cellular automata. J Stat Phys 43(1/2):219–242

    MathSciNet  ADS  MATH  Google Scholar 

  104. Jen E (1988) Linear cellular automata and recurring sequences in finite fields. Commun Math Phys 119:13–28

    MathSciNet  ADS  Google Scholar 

  105. Guan P, He Y (1986) Exact results for deterministic cellular automata with additive rules. J Stat Phys 43(3/4):463–478

    MathSciNet  ADS  MATH  Google Scholar 

  106. Das AK, Sanyal A, Chaudhuri PP (1992) On characterization of cellular automata with matrix algebra. Inf Sci 61:251–277

    MATH  Google Scholar 

  107. Tadaki S (1994) Orbits in one‐dimensional finite linear cellular automata. Phys Rev E 49(2):1168–1173

    ADS  Google Scholar 

  108. Voorhees B (2008) Representations of rule 90 and related rules for periodic, null, and half-infinite boundary conditions. J Cell Autom 3(1):1–25

    MathSciNet  MATH  Google Scholar 

  109. Davis PJ (1979) Circulant Matrices. Wiley-Interscience, NY

    MATH  Google Scholar 

  110. Culik II K (1987) On invertible cellular automata. Complex Syst 1:1035–1044

    MathSciNet  MATH  Google Scholar 

  111. Toffoli T, Margolis N (1990) Invertible cellular automata: A review. Physica D 45:229–253

    MathSciNet  ADS  MATH  Google Scholar 

  112. Kari J (1990) Reversibility of 2D cellular is undecidable. Physica D 45:379–385

    MathSciNet  ADS  MATH  Google Scholar 

  113. Sutner K (1991) De Bruijn graphs and linear cellular automata. Complex Syst 5(1):19–30

    MathSciNet  MATH  Google Scholar 

  114. Morita K (1994) Reversible cellular automata. J Inf Process Soc Japan 35:315–321

    Google Scholar 

  115. Moraal H (2000) Graph-theoretical characterization of invertible cellular automata. Physica D 141:1–18

    MathSciNet  ADS  MATH  Google Scholar 

  116. Voorhees B (1994) A note on injectivity of additive cellular automata. Complex Syst 8(3):151–159

    MathSciNet  MATH  Google Scholar 

  117. Lind DA (1984) Applications of ergodic theory and sofic systems to cellular automata. Physica D 10:36–44

    MathSciNet  ADS  Google Scholar 

  118. Jen E (1989) Limit cycles in one‐dimensional cellular automata. In: Stein DL (ed) Lectures in the Sciences of Complexity. Addison Wesley, Reading

    Google Scholar 

  119. Elspas B (1959) The theory of autonomous linear sequential networks. TRE Trans Circuits CT-6:45–60

    Google Scholar 

  120. Stevens JG, Rosensweig RE, Cerkanowicz AE (1993) Transient and cyclic behavior of cellular automata with null boundary conditions. J Stat Phys 73(1,2):159–174

    MathSciNet  ADS  MATH  Google Scholar 

  121. Stevens JG (1999) On the construction of state diagrams for cellular automata with additive rules. Inf Sci 115:43–59

    MathSciNet  MATH  Google Scholar 

  122. Thomas DM, Stevens JG, Letteiri S (2006) Characteristic and minimal polynomials of linear cellular automata. Rocky Mountain J Math 36(3):1077–1092

    MathSciNet  MATH  Google Scholar 

  123. Sutner K (1988) On σ-automata. Complex Syst 2(1):1–28

    MathSciNet  MATH  Google Scholar 

  124. Sutner K (1988) Additive automata on graphs. Complex Syst 2:649–661

    MathSciNet  MATH  Google Scholar 

  125. Sutner K (2000) Sigma-automata and Chebyschev polynomials. Theor Comput Sci 230:49–73

    MathSciNet  MATH  Google Scholar 

  126. Sutner K (2001) Decomposition of additive CA. Complex Syst 13(2):245–270

    MathSciNet  ADS  MATH  Google Scholar 

  127. Lidl R, Pilz G (1984) Applied Abstract Algebra. Springer, NY

    MATH  Google Scholar 

  128. Voorhees B (1993) Predecessors of cellular automata states: I. Additive automata. Physica D 68:283–292

    MathSciNet  ADS  MATH  Google Scholar 

  129. Bulitko V, Voorhees B, Bulitko V (2006) Discrete baker transformation for linear cellular automata analysis. J Cell Autom 1:40–70

    MathSciNet  Google Scholar 

  130. Arnold V, Aviz A (1968) Ergodic Problems of Classical Mechanics. Benjamin, NY

    Google Scholar 

  131. Illichinsky A (2001) Cellular Automata: A Discrete Universe. World Scientific, Singapore

    Google Scholar 

  132. Wuensche A, Lesser M (1992) The Global Dynamics of Cellular Automata. Addison Wesley, Reading

    MATH  Google Scholar 

  133. Chaitin G (2006) Meta Math! Vintage, NY

    Google Scholar 

Books and Reviews

  1. Wolfram S (1994) Cellular Automata and Complexity. Addison Wesley, Reading

    MATH  Google Scholar 

  2. Wolfram S (2002) A New Kind of Science. Wolfram Media, Champaign

    Google Scholar 

  3. Legendi T (1987) Parallel Processing by Cellular Automata and Arrays. Kluwer, Dordrecht

    MATH  Google Scholar 

  4. Rietman E (1989) Exploring the Geometry of Nature: Computer Modeling of Chaos, Fractals, Cellular Automata, and Neural Networks. McGraw-Hill, NY

    Google Scholar 

  5. Manneville P, Boccara N, Vishniac GY, Bidaux R (1990) Cellular Automata and Modeling of Complex Physical Systems. Springer, NY

    Google Scholar 

  6. Goles E, Martinez S (eds) (1992) Statistical Physics, Automata Networks and Dynamical Systems. Kluwer, Dordrecht

    MATH  Google Scholar 

  7. Boccara N, Goles E, Martenez S (1993) Cellular Automata and Cooperative Systems. Kluwer, Dordrecht

    MATH  Google Scholar 

  8. Goles E, Martinez S (1994) Cellular Automata, Dynamical Systems and Neural Networks. Kluwer, Dordrecht

    MATH  Google Scholar 

  9. Adamatzky A (1994) Identification of Cellular Automata. Taylor & Francis, London

    MATH  Google Scholar 

  10. Perdang JM, Lejeune A (eds) (1994) Cellular Automata: Prospects in Astronomy and Astrophysics. World Scientific, Singapore

    Google Scholar 

  11. Goles E, Martenez S (1996) Dynamics of Complex Interacting Systems. Kluwer, Dordrecht

    MATH  Google Scholar 

  12. Sipper M (1997) Evolution of Parallel Cellular Machines: The Cellular Programming Approach. Springer, NY

    Google Scholar 

  13. Delorme M, Mazoyer J (eds) (1998) Cellular Automata: A Parallel Model. Kluwer, Dordrecht

    Google Scholar 

  14. Goles E, Martinez S (eds) (1999) Cellular Automata and Complex Systems. Kluwer, Dordrecht

    MATH  Google Scholar 

  15. Crutchfield JP, Hanson JE (1999) Computational Mechanics of Cellular Processes. University of California Press, Berkeley

    Google Scholar 

  16. Wolf-Gladrow DA (2000) Lattice-Gas Cellular Automata and Lattice Boltzmann Models. Springer, NY

    MATH  Google Scholar 

  17. Lafe O (2000) Cellular Automata Transformations: Theory and Applications in Multimedia Compression, Encryption and Modeling. Kluwer, Dordrecht

    Google Scholar 

  18. Yang T (2001) Cellular Image Processing. Nova, NY

    Google Scholar 

  19. Griffeath D, Moore C (eds) (2003) New Constructions in Cellular Automata. Oxford University Press, NY

    MATH  Google Scholar 

  20. Deutsch A, Dormann S (2004) Cellular Automata Modeling of Biological Pattern Formation: Characterization, Applications, and Analysis. Springer, Berlin

    Google Scholar 

  21. Amos M (2004) Cellular Computing. Oxford University Press, NY

    MATH  Google Scholar 

  22. Rothman DH, Zaleski S (2004) Lattice-Gas Cellular Automata: Simple Models of Complex Hydrodynamics. Cambridge University Press, Cambridge

    Google Scholar 

  23. Batty M (2005) Cities and Complexity: Understanding Cities with Cellular Automata, Agent-Based Models, and Fractals. MIT Press, Cambridge

    Google Scholar 

  24. Kier LB, Seybold PG, Cheng C-K (2005) Cellular Automata Modeling of Chemical Systems: A Testbook and Laboratory Manual. Springer, NY

    Google Scholar 

  25. Schiff JL (2007) Cellular Automata: A Discrete View. Wiley-Interscience, NY

    Google Scholar 

  26. Gros C (2007) Complex and Adaptive Dynamical Systems: A Primer. Springer, NY

    Google Scholar 

  27. Bandini S, Moroni L (eds) (1996) ACRI `96: Proceedings of the 2nd International Conference on Cellular Automata for Research and Industry. Springer, NY

    Google Scholar 

  28. Bandini S, Serra R, Liverani FS (eds) (1998) Cellular Automata: Research Towards Industry, ACRI `98: Proceedings of the 3rd International Conference on Cellular Automata for Research and Industry. Springer, NY

    Google Scholar 

  29. Bandini S, Worsch T (eds) (2000) Theory and Practical Issues on Cellular Automata: Proceedings of the 4th International Conference on Cellular Automata for Research and Industry. Springer, NY

    Google Scholar 

  30. Tomassini M, Chopard B (eds) (2002) Cellular Automata: Proceedings of the 5th International Conference on Cellular Automata for Research and Industry. Springer, NY

    Google Scholar 

  31. Sloot PMA, Chopard B, Hoekstra AG (eds) (2004) Cellular Automata: Proceedings of the 6th International Conference on Cellular Automata for Research and Industry. Springer, NY

    Google Scholar 

  32. El Yacoubi S, Chopard B, Bandini S (eds) (2006) Cellular Automata: Proceedings of the 7th International Conference on Cellular Automata for Research and Industry. Springer, NY

    Google Scholar 

Websites

  1. http://cell-auto.com/links/ Gives many links to other sites on cellular automata

  2. http://www.theory.org/complexity/cdpt/html/node4.html Provides reviews of theoretical aspects of cellular automata

  3. http://www.ddlab.com An excellent site; it provides access to the Discrete Dynamics Lab program, a valuable asset in work on cellular automata and random Boolean networks

  4. http://cellular.ci.ulsa.mx Provides access to a number of worthwhile unpublished papers and a number of useful references

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Authors and Affiliations

  1. Center for Science, Athabasca University, Athabasca, Canada

    Burton Voorhees

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  1. Burton Voorhees
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Voorhees, B. (2009). Additive Cellular Automata. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_4

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