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Graphs Related to Reversibility and Complexity in Cellular Automata

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Encyclopedia of Complexity and Systems Science

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Glossary

Cellular automaton:

is a discrete dynamical system composed by a finite array of cells connected locally, which update their states at the same time using the same local mapping that takes into account the closest neighbors.

Complex automaton:

is a cellular automaton characterized by generating complex structures in its spatial-temporal evolution. For instance, the formation of self-localizations or gliders.

Cycle graph:

is a directed graph in which vertices are finite configurations and edges represent the global mapping between configurations induced by the local evolution rule.

De Bruijn graph:

is a directed graph in which vertices represent partial neighborhoods and edges represent complete neighborhoods obtained by valid overlaps between vertices. Edges are labeled according to the evolution of the neighborhood.

Glider:

is a complex pattern with volume, mass, period, displacement, and direction. Sometimes these nontrivial patterns are referred as particles, waves,...

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Bibliography

Primary Literature

  • Bang-Jensen J, Gutin GZ (2008) Digraphs: theory, algorithms and applications. Springer, London

    Google Scholar 

  • Betel H, de Oliveira PP, Flocchini P (2013) Solving the parity problem in one-dimensional cellular automata. Nat Comput 12(3):323–337

    Article  MathSciNet  MATH  Google Scholar 

  • Bhattacharjee K, Das S (2016) Reversibility of d-state finite cellular automata. J Cell Autom 11:213–245

    MathSciNet  Google Scholar 

  • Bossomaier T, Sibley-Punnett L, Cranny T (2000) Basins of attraction and the density classification problem for cellular automata. In: International conference on virtual worlds, Springer, pp 245–255

    Google Scholar 

  • Boykett T, Kari J, Taati S (2008) Conservation laws in rectangular ca. J Cell Autom 3(2):115–122

    Google Scholar 

  • de Bruijn N (1946) A combinatorial problem. Proc Sect Sci Kon Akad Wetensch Amsterdam 49(7):758–764

    MATH  Google Scholar 

  • Chin W, Cortzen B, Goldman J (2001) Linear cellular automata with boundary conditions. Linear Algebra Appl 322(1–3):193–206

    Article  MathSciNet  MATH  Google Scholar 

  • Chua LO, Pazienza GE (2009) A nonlinear dynamics perspective of wolfram’s new kind of science part xii: period-3, period-6, and permutive rules. Int J Bifurcation Chaos 19(12):3887–4038

    Article  MATH  Google Scholar 

  • Chua LO, Sbitnev VI, Yoon S (2006) A nonlinear dynamics perspective of wolfram’s new kind of science part vi: from time-reversible attractors to the arrow of time. Int J Bifurcation Chaos 16(05):1097–1373

    Article  MATH  Google Scholar 

  • Di Lena P, Margara L (2008) Computational complexity of dynamical systems: the case of cellular automata. Inf Comput 206(9–10):1104–1116

    Article  MathSciNet  MATH  Google Scholar 

  • Garcia GC, Lesne A, Hilgetag CC, Hütt MT (2014) Role of long cycles in excitable dynamics on graphs. Phys Rev E 90(5):052,805

    Article  Google Scholar 

  • Golomb SW et al (1982) Shift register sequences. World Scientific, Singapore

    Google Scholar 

  • Hedlund GA (1969) Endomorphisms and automorphisms of the shift dynamical system. Theor Comput Syst 3(4):320–375

    MathSciNet  MATH  Google Scholar 

  • Hopcroft JE (1979) Introduction to automata theory, languages and computation. Addison-Wesley, Boston

    Google Scholar 

  • Jen E (1987) Scaling of preimages in cellular automata. Complex Syst 1:1045–1062

    MathSciNet  MATH  Google Scholar 

  • Jeras I, Dobnikar A (2007) Algorithms for computing preimages of cellular automata configurations. Phys D 233(2):95–111

    Article  MathSciNet  MATH  Google Scholar 

  • Khoussainov B, Nerode A (2001) Automata theory and its applications, vol 21. Springer, New York

    Google Scholar 

  • Leon PA, Martinez GJ (2016) Describing complex dynamics in lifelike rules with de Bruijn diagrams on complex and chaotic cellular automata. J Cell Autom 11(1):91–112

    Google Scholar 

  • Macauley M, Mortveit HS (2009) Cycle equivalence of graph dynamical systems. Nonlinearity 22(2):421

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Macauley M, Mortveit HS (2013) An atlas of limit set dynamics for asynchronous elementary cellular automata. Theor Comput Sci 504:26–37

    Article  MathSciNet  MATH  Google Scholar 

  • Maji P, Chaudhuri PP (2008) Non-uniform cellular automata based associative memory: evolutionary design and basins of attraction. Inf Sci 178(10):2315–2336

    Article  MathSciNet  MATH  Google Scholar 

  • Martínez GJ, McIntosh HV, Seck Tuoh Mora JC, Chapa Vergara SV (2008) Determining a regular language by glider-based structures called phases f(i)1 in rule 110. J Cell Autom 3(3):231

    MathSciNet  MATH  Google Scholar 

  • Martínez GJ, Adamatzky A, Seck-Tuoh-Mora JC, Alonso-Sanz R (2010) How to make dull cellular automata complex by adding memory: rule 126 case study. Complexity 15(6):34–49

    MathSciNet  Google Scholar 

  • Martinez GJ, Mora JC, Zenil H (2013) Computation and universality: class iv versus class iii cellular automata. J Cell Autom 7(5–6):393–430

    MathSciNet  MATH  Google Scholar 

  • Martínez GJ, Adamatzky A, McIntosh HV (2014) Complete characterization of structure of rule 54. Complex Syst 23(3):259–293

    MathSciNet  MATH  Google Scholar 

  • Martínez GJ, Adamatzky A, Chen B, Chen F, Seck JC (2017) Simple networks on complex cellular automata: from de Bruijn diagrams to jump-graphs. In: Swarm dynamics as a complex networks. Springer (To be published), pp 177–204

    Google Scholar 

  • McIntosh HV (1991) Linear cellular automata via de Bruijn diagrams. Webpage: http://delta.cs.cinvestav.mx/~mcintosh

  • McIntosh HV (2009) One dimensional cellular automata. Luniver Press, United Kingdom

    Google Scholar 

  • McIntosh HV (2010) Life’s still lifes. In: Game of life cellular automata. Springer, London, pp 35–50

    Google Scholar 

  • Moore EF (1956) Gedanken-experiments on sequential machines. Autom Stud 34:129–153

    MathSciNet  Google Scholar 

  • Moore C, Boykett T (1997) Commuting cellular automata. Complex Syst 11:55–64

    MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Mortveit H, Reidys C (2007) An introduction to sequential dynamical systems. Springer, New York

    Google Scholar 

  • Nasu M (1977) Local maps inducing surjective global maps of one-dimensional tessellation automata. Math Syst Theor 11(1):327–351

    Article  MathSciNet  MATH  Google Scholar 

  • Nobe A, Yura F (2004) On reversibility of cellular automata with periodic boundary conditions. J Phys A Math Gen 37(22):5789

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Pei Y, Han Q, Liu C, Tang D, Huang J (2014) Chaotic behaviors of symbolic dynamics about rule 58 in cellular automata. Math Probl Eng 2014:Article ID 834268, 9 pages

    Google Scholar 

  • Powley EJ, Stepney S (2010) Counting preimages of homogeneous configurations in 1-dimensional cellular automata. J Cell Autom 5(4–5):353–381

    MathSciNet  MATH  Google Scholar 

  • Rabin MO, Scott D (1959) Finite automata and their decision problems. IBM J Res Develop 3(2):114–125

    Article  MathSciNet  MATH  Google Scholar 

  • Sakarovitch J (2009) Elements of automata theory. Cambridge University Press, New York

    Google Scholar 

  • Seck-Tuoh-Mora JC, Hernández MG, Martínez GJ, Chapa-Vergara SV (2003a) Extensions in reversible one-dimensional cellular automata are equivalent with the full shift. Int J Mod Phys C 14(08):1143–1160

    Article  Google Scholar 

  • Seck-Tuoh-Mora JC, Hernández MG, Vergara SVC (2003b) Reversible one-dimensional cellular automata with one of the two welch indices equal to 1 and full shifts. J Phys A Math Gen 36(29):7989

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Seck-Tuoh-Mora JC, Martínez GJ, McIntosh HV (2004) Calculating ancestors in one-dimensional cellular automata. Int J Mod Phys C 15(08):1151–1169

    Article  MATH  Google Scholar 

  • Seck-Tuoh-Mora JC, Vergara SVC, Martínez GJ, McIntosh HV (2005) Procedures for calculating reversible one-dimensional cellular automata. Phys D 202(1):134–141

    Article  MathSciNet  MATH  Google Scholar 

  • Seck-Tuoh-Mora JC, Hernández MG, Chapa Vergara SV (2008) Pair diagram and cyclic properties characterizing the inverse of reversible automata. J Cell Autom 3(3):205–218

    Google Scholar 

  • Seck-Tuoh-Mora JC, Medina-Marin J, Martínez GJ, Hernández-Romero N (2014) Emergence of density dynamics by surface interpolation in elementary cellular automata. Commun Nonlinear Sci Numer Simul 19(4):941–966

    Article  ADS  MathSciNet  Google Scholar 

  • Shannon CE (2001) A mathematical theory of communication. ACM SIGMOBILE Mobile Comput Commun Rev 5(1):3–55

    Article  MathSciNet  Google Scholar 

  • Soto JMG (2008) Computation of explicit preimages in one-dimensional cellular automata applying the de Bruijn diagram. J Cell Autom 3(3):219–230

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Voorhees B (2006) Discrete baker transformation for binary valued cylindrical cellular automata. In: International conference on cellular automata, Springer, pp 182–191

    Google Scholar 

  • Voorhees B (2008) Remarks on applications of de Bruijn diagrams and their fragments. J Cell Autom 3(3):187

    MathSciNet  MATH  Google Scholar 

  • Wolfram S (1984) Computation theory of cellular automata. Commun Math Phys 96(1):15–57

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Wuensche A (2005) Discrete dynamics lab: tools for investigating cellular automata and discrete dynamical networks, updated for multi-value, section 23, chain rules and encryption. In: Adamatzky A, Komosinski M (eds) Artificial life models in software, Springer-Verlag, London, pp 263–297

    Google Scholar 

  • Wuensche A, Lesser M (1992) The global dynamics of cellular automata: an atlas of basin of attraction fields of one-dimensional cellular automata. Addison-Wesley, Boston

    Google Scholar 

  • Yang B, Wang C, Xiang A (2015) Reversibility of general 1d linear cellular automata over the binary field z2 under null boundary conditions. Inf Sci 324:23–31

    Article  ADS  Google Scholar 

  • Zamora RR, Vergara SVC (2004) Using de Bruijn diagrams to analyze 1d cellular automata traffic models. In: International conference on cellular automata, Springer, pp 306–315

    Google Scholar 

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Authors and Affiliations

  1. Instituto de Ciencias Básicas e Ingeniería, Área Académica de Ingeniería, Universidad Autónoma del Estado de Hidalgo, Ciudad del Conocimiento, Carretera Pachuca- Tulancingo Km. 4.5, Col. Carboneras, Mineral de la Reforma, C. P. 42184, Hidalgo, Mexico

    Juan C. Seck-Tuoh-Mora

  2. Escuela Superior de Cómputo, Instituto Politécnico Nacional, México Unconventional Computing Center, University of the West of England, Bristol, UK

    Genaro J. Martínez

Authors
  1. Juan C. Seck-Tuoh-Mora
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  2. Genaro J. Martínez
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Correspondence to Juan C. Seck-Tuoh-Mora .

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  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

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Seck-Tuoh-Mora, J.C., Martínez, G.J. (2017). Graphs Related to Reversibility and Complexity in Cellular Automata. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_677-1

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  • DOI: https://doi.org/10.1007/978-3-642-27737-5_677-1

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  • Print ISBN: 978-3-642-27737-5

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