Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Graphs Related to Reversibility and Complexity in Cellular Automata

  • Juan C. Seck-Tuoh-MoraEmail author
  • Genaro J. Martínez
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_677-1


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.


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

This is a preview of subscription content, log in to check access.


Primary Literature

  1. Bang-Jensen J, Gutin GZ (2008) Digraphs: theory, algorithms and applications. Springer, LondonGoogle Scholar
  2. Betel H, de Oliveira PP, Flocchini P (2013) Solving the parity problem in one-dimensional cellular automata. Nat Comput 12(3):323–337MathSciNetzbMATHCrossRefGoogle Scholar
  3. Bhattacharjee K, Das S (2016) Reversibility of d-state finite cellular automata. J Cell Autom 11:213–245MathSciNetGoogle Scholar
  4. 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 245–255Google Scholar
  5. Boykett T, Kari J, Taati S (2008) Conservation laws in rectangular ca. J Cell Autom 3(2):115–122Google Scholar
  6. de Bruijn N (1946) A combinatorial problem. Proc Sect Sci Kon Akad Wetensch Amsterdam 49(7):758–764zbMATHGoogle Scholar
  7. Chin W, Cortzen B, Goldman J (2001) Linear cellular automata with boundary conditions. Linear Algebra Appl 322(1–3):193–206MathSciNetzbMATHCrossRefGoogle Scholar
  8. 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):3887–4038zbMATHCrossRefGoogle Scholar
  9. 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):1097–1373zbMATHCrossRefGoogle Scholar
  10. Di Lena P, Margara L (2008) Computational complexity of dynamical systems: the case of cellular automata. Inf Comput 206(9–10):1104–1116MathSciNetzbMATHCrossRefGoogle Scholar
  11. Garcia GC, Lesne A, Hilgetag CC, Hütt MT (2014) Role of long cycles in excitable dynamics on graphs. Phys Rev E 90(5):052,805CrossRefGoogle Scholar
  12. Golomb SW et al (1982) Shift register sequences. World Scientific, SingaporeGoogle Scholar
  13. Hedlund GA (1969) Endomorphisms and automorphisms of the shift dynamical system. Theor Comput Syst 3(4):320–375MathSciNetzbMATHGoogle Scholar
  14. Hopcroft JE (1979) Introduction to automata theory, languages and computation. Addison-Wesley, BostonGoogle Scholar
  15. Jen E (1987) Scaling of preimages in cellular automata. Complex Syst 1:1045–1062MathSciNetzbMATHGoogle Scholar
  16. Jeras I, Dobnikar A (2007) Algorithms for computing preimages of cellular automata configurations. Phys D 233(2):95–111MathSciNetzbMATHCrossRefGoogle Scholar
  17. Khoussainov B, Nerode A (2001) Automata theory and its applications, vol 21. Springer, New YorkGoogle Scholar
  18. 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):91–112Google Scholar
  19. Macauley M, Mortveit HS (2009) Cycle equivalence of graph dynamical systems. Nonlinearity 22(2):421ADSMathSciNetzbMATHCrossRefGoogle Scholar
  20. Macauley M, Mortveit HS (2013) An atlas of limit set dynamics for asynchronous elementary cellular automata. Theor Comput Sci 504:26–37MathSciNetzbMATHCrossRefGoogle Scholar
  21. Maji P, Chaudhuri PP (2008) Non-uniform cellular automata based associative memory: evolutionary design and basins of attraction. Inf Sci 178(10):2315–2336MathSciNetzbMATHCrossRefGoogle Scholar
  22. 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):231MathSciNetzbMATHGoogle Scholar
  23. 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):34–49MathSciNetGoogle Scholar
  24. Martinez GJ, Mora JC, Zenil H (2013) Computation and universality: class iv versus class iii cellular automata. J Cell Autom 7(5–6):393–430MathSciNetzbMATHGoogle Scholar
  25. Martínez GJ, Adamatzky A, McIntosh HV (2014) Complete characterization of structure of rule 54. Complex Syst 23(3):259–293MathSciNetzbMATHGoogle Scholar
  26. 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 177–204Google Scholar
  27. McIntosh HV (1991) Linear cellular automata via de Bruijn diagrams. Webpage: http://delta.cs.cinvestav.mx/~mcintosh
  28. McIntosh HV (2009) One dimensional cellular automata. Luniver Press, United KingdomGoogle Scholar
  29. McIntosh HV (2010) Life’s still lifes. In: Game of life cellular automata. Springer, London, pp 35–50Google Scholar
  30. Moore EF (1956) Gedanken-experiments on sequential machines. Autom Stud 34:129–153MathSciNetGoogle Scholar
  31. Moore C, Boykett T (1997) Commuting cellular automata. Complex Syst 11:55–64MathSciNetzbMATHGoogle Scholar
  32. Moraal H (2000) Graph-theoretical characterization of invertible cellular automata. Phys D 141(1):1–18ADSMathSciNetzbMATHCrossRefGoogle Scholar
  33. Mortveit H, Reidys C (2007) An introduction to sequential dynamical systems. Springer, New YorkGoogle Scholar
  34. Nasu M (1977) Local maps inducing surjective global maps of one-dimensional tessellation automata. Math Syst Theor 11(1):327–351MathSciNetzbMATHCrossRefGoogle Scholar
  35. Nobe A, Yura F (2004) On reversibility of cellular automata with periodic boundary conditions. J Phys A Math Gen 37(22):5789ADSMathSciNetzbMATHCrossRefGoogle Scholar
  36. 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 pagesGoogle Scholar
  37. Powley EJ, Stepney S (2010) Counting preimages of homogeneous configurations in 1-dimensional cellular automata. J Cell Autom 5(4–5):353–381MathSciNetzbMATHGoogle Scholar
  38. Rabin MO, Scott D (1959) Finite automata and their decision problems. IBM J Res Develop 3(2):114–125MathSciNetzbMATHCrossRefGoogle Scholar
  39. Sakarovitch J (2009) Elements of automata theory. Cambridge University Press, New YorkGoogle Scholar
  40. 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):1143–1160CrossRefGoogle Scholar
  41. 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):7989ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. Seck-Tuoh-Mora JC, Martínez GJ, McIntosh HV (2004) Calculating ancestors in one-dimensional cellular automata. Int J Mod Phys C 15(08):1151–1169zbMATHCrossRefGoogle Scholar
  43. Seck-Tuoh-Mora JC, Vergara SVC, Martínez GJ, McIntosh HV (2005) Procedures for calculating reversible one-dimensional cellular automata. Phys D 202(1):134–141MathSciNetzbMATHCrossRefGoogle Scholar
  44. 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):205–218Google Scholar
  45. 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):941–966ADSMathSciNetCrossRefGoogle Scholar
  46. Shannon CE (2001) A mathematical theory of communication. ACM SIGMOBILE Mobile Comput Commun Rev 5(1):3–55MathSciNetCrossRefGoogle Scholar
  47. Soto JMG (2008) Computation of explicit preimages in one-dimensional cellular automata applying the de Bruijn diagram. J Cell Autom 3(3):219–230MathSciNetzbMATHGoogle Scholar
  48. Sutner K (1991) De Bruijn graphs and linear cellular automata. Complex Syst 5(1):19–30MathSciNetzbMATHGoogle Scholar
  49. Voorhees B (2006) Discrete baker transformation for binary valued cylindrical cellular automata. In: International conference on cellular automata, Springer, pp 182–191Google Scholar
  50. Voorhees B (2008) Remarks on applications of de Bruijn diagrams and their fragments. J Cell Autom 3(3):187MathSciNetzbMATHGoogle Scholar
  51. Wolfram S (1984) Computation theory of cellular automata. Commun Math Phys 96(1):15–57ADSMathSciNetzbMATHCrossRefGoogle Scholar
  52. 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 263–297Google Scholar
  53. 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, BostonGoogle Scholar
  54. 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 324:23–31ADSCrossRefGoogle Scholar
  55. Zamora RR, Vergara SVC (2004) Using de Bruijn diagrams to analyze 1d cellular automata traffic models. In: International conference on cellular automata, Springer, pp 306–315Google Scholar

Books and Reviews

  1. Adamatzky A (ed) (2010) Game of life cellular automata, vol 1. Springer, LondonGoogle Scholar
  2. Gutowitz H (1991) Cellular automata: theory and experiment. MIT Press, Cambridge, MassachuettsGoogle Scholar
  3. Kari J (2005) Theory of cellular automata: a survey. Theor Comput Sci 334(1–3):3–33MathSciNetzbMATHCrossRefGoogle Scholar
  4. Toffoli T, Margolus NH (1990) Invertible cellular automata: a review. Phys D 45(1–3):229–253MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  1. 1.Instituto de Ciencias Básicas e Ingeniería, Área Académica de IngenieríaUniversidad Autónoma del Estado de HidalgoHidalgoMexico
  2. 2.Escuela Superior de Cómputo, Instituto Politécnico Nacional, México Unconventional Computing Center, University of the West of EnglandBristolUK