Advertisement

A class of discrete dynamical systems with properties of both cellular automata and L-systems

  • Roderick EdwardsEmail author
  • Aude Maignan
Article
  • 13 Downloads

Abstract

We introduce and explore a type of discrete dynamic system inheriting some properties of both cellular automata (CA) and L-systems. Originally suggested by Jean Della Dora, and thus called DEM-systems after him and the two current authors, these systems can have the structural flexibility of an L-system as well as algebraic properties of CA. They are defined as sequences on a one-dimensional loop with rules governing dynamics in which new sites can be created, depending on the states of a neighbourhood of sites, and complex behaviour can be generated. Although the definition of DEM-systems is quite broad, we define some subclasses, for which more complete results can be obtained. For example, we define an additive subclass, for which algebraic results on asymptotic growth are possible, and an elementary class of particularly simple rules, for which nevertheless impressive complexity is achievable. Unlike for CA, finite initial sequences can produce positive spatial entropy over time. However, even in cases where the entropy is zero, considerable complexity is possible, especially when the sequence length grows to infinity, and we demonstrate and study behaviours of DEM-systems including fragmentation of sequences, self-reproducing patterns, self-similar but irregular patterns, patterns that not only produce new sites but produce producers of new sites, and sequences whose growth rate is sublinear, linear, quadratic, cubic, or exponential. The most complex behaviour from small finite initial conditions and the simplest class of rules appear to have positive entropy, a suggestion for which we have so far only stong numerical evidence, though we present a proof for these ‘elementary’ DEM-systems that entropy cannot reach the theoretical maximum of 1.

Keywords

Complexity Entropy Self-reproducing systems Self-organizing systems Cellular automata L-systems 

Mathematics Subject Classification

68Q80 68Q70 68Q30 68Q19 

Notes

Funding

Funding was provided to RE by the Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN-2017-04042).

References

  1. Arcuri A, Lanchier N (2017) Stochastic spatial model for the division of labor in social insects. Math Models Methods Appl Sci 27:45–73MathSciNetCrossRefzbMATHGoogle Scholar
  2. Baetens JM, De Baets B (2010) Phenomenological study of irregular cellular automata based on Lyapunov exponents and Jacobians. Chaos 20:033112MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bagnoli F, Rechtman R, Ruffo S (1992) Damage spreading and Lyapunov exponents in cellular automata. Phys Lett A 172:34–38CrossRefzbMATHGoogle Scholar
  4. Berman A, Plemmons RJ (1994) Nonnegative matrices in the mathematical sciences. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  5. Bohun CS, Carruthers SJ, Edwards R, Illner R (2003) Generic emergence of cognitive behaviour in self-generating neural networks. Nonlinear Dyn Syst Theory 3:43–63MathSciNetzbMATHGoogle Scholar
  6. Camarinha-Matos LM, Afsarmanesh H (2004) Emerging behavior in complex collaborative networks. In: Camarinha-Matos LM, Afsarmanesh H (eds) Collaborative networked organizations. Springer, Berlin, pp 229–236CrossRefGoogle Scholar
  7. Dennunzio A (2012) From one-dimensional to two-dimensional cellular automata. Fundam Inform 115:87–105MathSciNetzbMATHGoogle Scholar
  8. Dennunzio A, Di Lena P, Formenti E, Margara L (2013) Periodic orbits and dynamical complexity in cellular automata. Fundam Inform 126:183–199MathSciNetzbMATHGoogle Scholar
  9. Dorogovtsev SN, Goltsev AV, Mendes JFF (2008) Critical phenomena in complex networks. Rev Mod Phys 80:1275–1335CrossRefGoogle Scholar
  10. Dorogovtsev SN, Mendes JFF (2003) Evolution of networks. Oxford University Press, OxfordCrossRefzbMATHGoogle Scholar
  11. Edwards R, Maignan A (2014) Complex self-reproducing systems. In: Sanayei A, Zelinka I, Rossler O (eds) ISCS 2013: interdisciplinary symposium on complex systems. emergence, complexity and computation. Springer, Berlin, pp 65–76Google Scholar
  12. Edwards R, Maignan A (2016) DEM-systems: a new type of adaptive system. In: Exploratory papers of automata 2016, 22nd international workshop on cellular autmomata and discrete complex systems, Zurich, June 2016Google Scholar
  13. Hall ME, Mohtaram NK, Willerth SM, Edwards R (2017) Modeling the behavior of human induced pluripotent stem cells seeded on melt electrospun scaffolds. J Biomed Eng 11:38Google Scholar
  14. Mainzer K, Chua L (2012) The universe as automaton. Springer, BerlinCrossRefzbMATHGoogle Scholar
  15. Martin O, Odlyzko AM, Wolfram S (1984) Algebraic properties of cellular automata. Commun Math Phys 93:219–258MathSciNetCrossRefzbMATHGoogle Scholar
  16. Prusinkiewicz P, Lindenmayer A (1996) The algorithmic beauty of plants. Springer, BerlinzbMATHGoogle Scholar
  17. Samaya H, Pestov I, Schmidt J, Bush BJ, Wong C, Yamanoi J, Gross T (2013) Modeling complex systems with adaptive networks. Comput Math Appl 65:1645–1664MathSciNetCrossRefzbMATHGoogle Scholar
  18. Spicher A, Michel O, Giavitto JL (2011) Interaction-based simulations for integrative spatial systems biology. In: Dubitzky W, Southgate J, Fuss H (eds) Understanding the dynamics of biological systems. Springer, Berlin, pp 195–231CrossRefGoogle Scholar
  19. Spratt ER (1911) Some observations on the life cycle of Anabaena Cycadeae. Ann Bot 25:369–379CrossRefGoogle Scholar
  20. Stauffer A, Sipper M (1998) On the relationship between cellular automata and L-systems: the self-replication case. Phys D 116:71–80MathSciNetCrossRefzbMATHGoogle Scholar
  21. Sutner K (1990) Classifying circular cellular automata. Phys D 45:386–395MathSciNetCrossRefzbMATHGoogle Scholar
  22. Sutner K (2009) Classification of cellular automata. In: Meyers RA (ed) Encyclopedia of complexity and system science. Springer, Berlin, pp 755–768CrossRefGoogle Scholar
  23. Wolfram S (1984) Computation theory of cellular automata. Commun Math Phys 96:15–57MathSciNetCrossRefzbMATHGoogle Scholar
  24. Wolfram S (1984) Universality and complexity in cellular automata. Phys D 10:1–35MathSciNetCrossRefzbMATHGoogle Scholar
  25. Wolfram S (1984) Cellular automata as models of complexity. Nature 311:419–424CrossRefGoogle Scholar
  26. Wolfram S (2002) A new kind of science. Wolfram Media, ChampaignzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada
  2. 2.Univ. Grenoble AlpesGrenoble Cedex 9France

Personalised recommendations