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Growth Phenomena in Cellular Automata

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

In essence, analysis of growth models is an attempt to study properties of physical systems far from equilibrium (e.g., (Meakin 1998) and its more than 1,300 references). Cellular automata (CA) growth models, by virtue of their simplicity and amenability to computer experimentation (Toffoli and Margolus 1997; Wójtowicz 2001), have become particularly popular in the last 30 years in many fields, such as physics (Chopard and Droz 1998; Toffoli and Margolus 1997; Vichniac 1984), biology (Deutsch and Dormann 2005), chemistry (Chopard and Droz 1998; Kier et al. 2005), social sciences (Bäck et al. 1996), and artificial life (Lindgren and Nordahl 1994). In contrast to voluminous empirical literature on CA in general and their growth properties in particular, precise mathematical results are rather scarce. A general CA theory is out of the question, since a Turing machine can be embedded in a CA, so that examples as “simple” as elementary one-dimensional CA (Cook 2005...

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Abbreviations

Asymptotic density:

The proportion of sites in a lattice occupied by a specified subset is called asymptotic density or, in short, density.

Asymptotic shape:

The shape of a growing set, viewed from a sufficient distance so that the boundary fluctuations, holes, and other lower order details disappear, is called the asymptotic shape.

Cellular automaton:

A cellular automaton is a sequence of configurations on a lattice which proceeds by iterative applications of a homogeneous local update rule. A configuration attaches a state to every member (also termed a site or a cell) of the lattice. Only configurations with two states, coded 0 and 1, are considered here. Any such configuration is identified with its set of 1’s.

Final set:

A site whose state changes only finitely many times is said to fixate or attain a final state. If this happens for every site, then the sites whose final states are 1 comprise the final set.

Initial set:

A starting set for a cellular automaton evolution is called initial set and may be deterministic or random.

Metastability:

Metastability refers to a long, but finite, time period in an evolution of a cellular automaton rule, during which the behavior of the iterates has identifiable characteristics.

Monotone cellular automaton:

A cellular automaton is monotone if addition of 1’s to the initial configuration always results in more 1’s in any subsequent configuration.

Nucleation:

Nucleation refers to (usually small) pockets of activity, often termed nuclei, with long range consequences.

Solidification:

A cellular automaton solidifies if any site which achieves state 1 remains forever in this state.

Bibliography

Primary Literature

  • Adamatzky A, Martínez GJ, Mora JCST (2006) Phenomenology of reaction-diffusion binary-state cellular automata. Int J Bifurc Chaos Appl Sci Eng 16:2985–3005

    Article  MATH  Google Scholar 

  • Adler J (1991) Bootstrap percolation. Phys A 171:453–4170

    Article  Google Scholar 

  • Adler J, Staufer D, Aharony A (1989) Comparison of bootstrap percolation models. J Phys A: Math Gen 22:L279–L301

    Article  Google Scholar 

  • Aizenman M, Lebowitz J (1988) Metastability effects in bootstrap percolation. J Phys A: Math Gen 21:3801–3813

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Allouche J-P, Shallit J (2003) Automatic sequences: theory, applications, generalizations. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Andjel E, Mountford TS, Schonmann RH (1995) Equivalence of decay rates for bootstrap percolation like cellular automata. Ann Inst H Poincaré 31:13–25

    MATH  MathSciNet  Google Scholar 

  • Berlekamp ER, Conway JH, Guy RK (2004) Winning ways for your mathematical plays, vol 4, 2nd edn. Peters, Natick

    MATH  Google Scholar 

  • Bohman T (1999) Discrete threshold growth dynamics are omnivorous for box neighborhoods. Trans Am Math Soc 351:947–983

    Article  MATH  MathSciNet  Google Scholar 

  • Bohman T, Gravner J (1999) Random threshold growth dynamics. Random Struct Algorithms 15:93–111

    Article  MATH  MathSciNet  Google Scholar 

  • Bramson M, Neuhauser C (1994) Survival of one-dimensional cellular automata under random perturbations. Ann Probab 22:244–263

    Article  MATH  MathSciNet  Google Scholar 

  • Brummitt CD, Delventhal H, Retzlaff M (2008) Packard snowflakes on the von Neumann neighborhood. J Cell Autom 3:57–80

    MathSciNet  Google Scholar 

  • Bäck T, Dörnemann H, Hammel U, Frankhauser P (1996) Modeling urban growth by cellular automata. In: Lecture notes in computer science. Proceedings of the 4th international conference on parallel problem solving from nature, vol 1141. Springer, Berlin, pp 636–645

    Google Scholar 

  • Cerf R, Cirillo ENM (1999) Finite size scaling in three-dimensional bootstrap percolation. Ann Probab 27:1837–1850

    Article  MATH  MathSciNet  Google Scholar 

  • Cerf R, Manzo F (2002) The threshold regime of finite volume bootstrap percolation. Stoch Process Appl 101:69–82

    Article  MATH  MathSciNet  Google Scholar 

  • Chopard B, Droz M (1998) Cellular automata modeling of physical systems. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Cobham A (1972) Uniform tag sequences. Math Syst Theory 6:164–192

    Article  MATH  MathSciNet  Google Scholar 

  • Cook M (2005) Universality in elementary cellular automata. Complex Syst 15:1–40

    Google Scholar 

  • Deutsch A, Dormann S (2005) Cellular automata modeling of biological pattern formation. Birkhäuser, Boston

    Google Scholar 

  • Durrett R, Steif JE (1991) Some rigorous results for the Greenberg-Hastings model. J Theor Probab 4:669–690

    Article  MATH  MathSciNet  Google Scholar 

  • Durrett R, Steif JE (1993) Fixation results for threshold voter systems. Ann Probab 21:232–247

    Article  MATH  MathSciNet  Google Scholar 

  • Eppstein D (2002) Searching for spaceships. In: More games of no chance (Berkeley, CA, 2000). Cambridge University Press, Cambridge, pp 351–360

    Google Scholar 

  • Evans KM (2001) Larger than life: digital creatures in a family of two-dimensional cellular automata. In: Cori R, Mazoyer J, Morvan M, Mosseri R (eds) Discrete mathematics and theoretical computer science, vol AA. pp 177–192

    Google Scholar 

  • Evans KM (2003) Replicators and larger than life examples. In: Griffeath D, Moore C (eds) New constructions in cellular automata. Oxford University Press, New York, pp 119–159

    Google Scholar 

  • Fisch R, Gravner J, Griffeath D (1991) Threshold-range scaling for the excitable cellular automata. Stat Comput 1:23–39

    Article  Google Scholar 

  • Fisch R, Gravner J, Griffeath D (1993) Metastability in the Greenberg-Hastings model. Ann Appl Probab 3:935–967

    Article  MATH  MathSciNet  Google Scholar 

  • Gardner M (1976) Mathematical games. Sci Am 133:124–128

    Article  Google Scholar 

  • Goles E, Martinez S (1990) Neural and automata networks. Kluwer, Dordrecht

    Book  MATH  Google Scholar 

  • Gotts NM (2003) Self-organized construction in sparse random arrays of Conway’s game of life. In: Griffeath D, Moore C (eds) New constructions in cellular automata. Oxford University Press, New York, pp 1–53

    Google Scholar 

  • Gravner J, Griffeath D (1993) Threshold growth dynamics. Trans Am Math Soc 340:837–870

    Article  MATH  MathSciNet  Google Scholar 

  • Gravner J, Griffeath D (1996) First passage times for the threshold growth dynamics on. Ann Probab 24:1752–1778

    Article  MATH  MathSciNet  Google Scholar 

  • Gravner J, Griffeath D (1997a) Multitype threshold voter model and convergence to Poisson-Voronoi tessellation. Ann Appl Probab 7:615–647

    Article  MATH  MathSciNet  Google Scholar 

  • Gravner J, Griffeath D (1997b) Nucleation parameters in discrete threshold growth dynamics. Exp Math 6:207–220

    Article  MATH  MathSciNet  Google Scholar 

  • Gravner J, Griffeath D (1998) Cellular automaton growth on: theorems, examples and problems. Adv Appl Math 21:241–304

    Article  MATH  MathSciNet  Google Scholar 

  • Gravner J, Griffeath D (1999a) Reverse shapes in first-passage percolation and related growth models. In: Bramson M, Durrett R (eds) Perplexing problems in probability. Festschrift in honor of Harry Kesten. Birkhäuser, Boston, pp 121–142

    Chapter  Google Scholar 

  • Gravner J, Griffeath D (1999b) Scaling laws for a class of critical cellular automaton growth rules. In: Révész P, Tóth B (eds) Random walks. János Bolyai Mathematical Society, Budapest, pp 167–186

    Google Scholar 

  • Gravner J, Griffeath D (2006a) Modeling snow crystal growth. I. Rigorous results for Packard’s digit snowflakes. Exp Math 15:421–444

    Article  MATH  MathSciNet  Google Scholar 

  • Gravner J, Griffeath D (2006b) Random growth models with polygonal shapes. Ann Probab 34:181–218

    Article  MATH  MathSciNet  Google Scholar 

  • Gravner J, Holroyd AE (2008) Slow convergence in bootstrap percolation. Ann Appl Probab 18:909–928

    Article  MATH  MathSciNet  Google Scholar 

  • Gravner J, Mastronarde N Shapes in deterministic and random growth models (in preparation)

    Google Scholar 

  • Gravner J, McDonald E (1997) Bootstrap percolation in a polluted environment. J Stat Phys 87:915–927

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Gravner J, Tracy C, Widom H (2002) A growth model in a random environment. Ann Probab 30:1340–1368

    Article  MATH  MathSciNet  Google Scholar 

  • Greenberg J, Hastings S (1978) Spatial patterns for discrete models of diffusion in excitable media. SIAM J Appl Math 4:515–523

    Article  MathSciNet  Google Scholar 

  • Griffeath D (1994) Self-organization of random cellular automata: four snapshots. In: Grimmett G (ed) Probability and phase transition. Kluwer, Dordrecht, pp 49–67

    Chapter  Google Scholar 

  • Griffeath D, Hickerson D (2003) A two-dimensional cellular automaton with irrational density. In: Griffeath D, Moore C (eds) New constructions in cellular automata. Oxford University Press, Oxford, pp 119–159

    Google Scholar 

  • Griffeath D, Moore C (1996) Life without death is P-complete. Complex Syst 10:437–447

    MATH  MathSciNet  Google Scholar 

  • Holroyd AE (2003) Sharp metastability threshold for two-dimensional bootstrap percolation. Probab Theory Relat Fields 125:195–224

    Article  MATH  MathSciNet  Google Scholar 

  • Holroyd AE (2006) The metastability threshold for modified bootstrap percolation in d dimensions. Electron J Probab 11:418–433

    Article  MathSciNet  Google Scholar 

  • Holroyd AE, Liggett TM, Romik D (2004) Integrals, partitions, and cellular automata. Trans Am Math Soc 356:3349–3368

    Article  MATH  MathSciNet  Google Scholar 

  • Jen E (1991) Exact solvability and quasiperiodicity of one-dimensional cellular automata. Nonlinearity 4:251–276

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Kier LB, Seybold PG, Cheng C-K (2005) Cellular automata modeling of chemical systems. Springer, Dordrecht

    Google Scholar 

  • Lindgren K, Nordahl MG (1994) Evolutionary dynamics of spatial games. Phys D 75:292–309

    Article  MATH  Google Scholar 

  • Meakin P (1998) Fractals, scaling and growth far from equilibrium. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Packard NH (1984) Lattice models for solidification and aggregation. Institute for advanced study preprint. Reprinted in: Wolfram S (ed) (1986) Theory and application of cellular automata. World Scientific, Singapore, pp 305–310

    Google Scholar 

  • Packard NH, Wolfram S (1985) Two-dimensional cellular automata. J Stat Phys 38:901–946

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Pimpinelli A, Villain J (1999) Physics of crystal growth. Cambridge University Press, Cambridge

    Google Scholar 

  • Schonmann RH (1992) On the behavior of some cellular automata related to bootstrap percolation. Ann Probab 20:174–193

    Article  MATH  MathSciNet  Google Scholar 

  • Schonmann RH (1990) Finite size scaling behavior of a biased majority rule cellular automaton. Phys A 167:619–627

    Article  MathSciNet  Google Scholar 

  • Song M (2005) Geometric evolutions driven by threshold dynamics. Interfaces Free Bound 7:303–318

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Toffoli T, Margolus N (1997) Cellular automata machines. MIT Press, Cambridge

    Google Scholar 

  • van Enter ACD (1987) Proof of Straley’s argument for bootstrap percolation. J Stat Phys 48:943–945

    Article  ADS  MATH  Google Scholar 

  • van Enter ACD, Hulshof T (2007) Finite-size effects for anisotropic bootstrap percolation: logarithmic corrections. J Stat Phys 128:1383–1389

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Vichniac GY (1984) Simulating physics with cellular automata. Phys D 10:96–116

    Article  MathSciNet  Google Scholar 

  • Vichniac GY (1986) Cellular automata models of disorder and organization. In: Bienenstock E, Fogelman-Soulie F, Weisbuch G (eds) Disordered systems and biological organization. Springer, Berlin, pp 1–20

    Google Scholar 

  • Wiener N, Rosenblueth A (1946) The math foundation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Arch Inst Cardiol Mex 16:205–265

    MathSciNet  Google Scholar 

  • Willson SJ (1978) On convergence of configurations. Discret Math 23:279–300

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Willson SJ (1987) Computing fractal dimensions for additive cellular automata. Phys D 24:190–206

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Wójtowicz M (2001) Mirek’s celebration: a 1D and 2D cellular automata explorer, version 4.20. http://www.mirwoj.opus.chelm.pl/ca/

Books and Reviews

  • Adamatzky A (1995) Identification of cellular automata. Taylor & Francis, London

    Google Scholar 

  • Allouche J-P, Courbage M, Kung J, Skordev G (2001) Cellular automata. In: Encyclopedia of physical science and technology, vol 2, 3rd edn. Academic Press, San Diego, pp 555–567

    Google Scholar 

  • Allouche J-P, Courbage M, Skordev G (2001b) Notes on cellular automata. Cubo, Matemática Educ 3:213–244

    MATH  MathSciNet  Google Scholar 

  • Durrett R (1988) Lecture notes on particle systems and percolation. Wadsworth & Brooks/Cole, Pacific Grove

    MATH  Google Scholar 

  • Durrett R (1999) Stochastic spatial models. SIAM Rev 41:677–718

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Gravner J (2003) Growth phenomena in cellular automata. In: Griffeath D, Moore C (eds) New constructions in cellular automata. Oxford University Press, New York, pp 161–181

    Google Scholar 

  • Holroyd AE (2007) Astonishing cellular automata. Bull Centre Rech Math 10:10–13

    Google Scholar 

  • Ilachinsky A (2001) Cellular automata: a discrete universe. World Scientific, Singapore

    Google Scholar 

  • Liggett TM (1985) Interacting particle systems. Springer, New York

    Book  MATH  Google Scholar 

  • Liggett TM (1999) Stochastic interacting systems: contact, voter and exclusion processes. Springer, New York

    Book  MATH  Google Scholar 

  • Rothman DH, Zaleski S (1997) Lattice-gas cellular automata. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Toom A (1995) Cellular automata with errors: problems for students of probability. In: Snell JL (ed) Topics in contemporary probability and its applications. CRC Press, Boca Raton, pp 117–157

    Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, University of California, Davis, USA

    Janko Gravner

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  1. Janko Gravner
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Correspondence to Janko Gravner .

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  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

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Gravner, J. (2013). Growth Phenomena in Cellular Automata. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-3-642-27737-5_266-5

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