Fundamental Concepts of Cellular Automata SpringerLink

Abstract

This chapter reviews some fundamental concepts and outcomes of the idea of cellular automata (CA). Topics discussed include classical is a result of the 1960s, relations between various concepts of injectivity and surjectivity, and dynamical system concepts associated with chaos in CA. Most answers are reported without full proofs but may examples are supplied that illustrate the thought of an evidence. The classical results discussed range from the Garden-of-Eden theorem and also the Curtis–Hedlund–Lyndon theorem, along with the balance property of surjective CA. Different variants of sensitivity to initial conditions and mixing qualities are introduced and associated with one another. Also, algorithmic aspects and undecidability answers are pointed out.

References

• Berlekamp ER, Conway JH, Guy RK (1982) Winning methods for your mathematical plays, vol II, chap 25. Academic, LondonGoogle Scholar
• Burks A (1970) Von Neumann's self-reproducing automata. In: Burks A (erectile dysfunction) Essays on cellular automata, College of Illinois Press, Champaign, pp 3–64
• Czeizler E, Kari J (2005) A good straight line bound around the neighborhood of inverse cellular automata. In: Caires L, Italiano GF, Monteiro L, Palamidessi C, Yung M (eds) ICALP, Lecture notes in information technology, vol 3580, Springer, Berlin, pp 410–420Google Scholar
• Devaney RL (1986) Introducing chaotic dynamical systems. Benjamin/Cummings, Menlo Park, CAzbMATH
• Durand B (1998) Global qualities of cellular automata. In: Goles E, Martinez S (eds) Cellular automata and sophisticated systems. Kluwer, Dordrecht
• Durand B, Formenti E, Varouchas G (2003) On undecidability of equicontinuity classification for cellular automata. In: Morvan M, Remila E (eds) Discrete models for complex systems, DMCS’03, DMTCS Conference Volume AB, Lyon, France, pp 117–128Google Scholar
• Gardner M (1970) The great mixtures of John Conway's new solitaire game “; Existence ”. Sci Am 223(4): 120–123CrossRef
• Kari J, Ollinger N (2008) Periodicity and growing old in reversible computing. In: Ochmanski E, Tyszkiewicz J (eds) MFCS, Lecture notes in information technology, vol 5162, Springer, Berlin, pp 419–430Google Scholar
• Kari J, Vanier P, Zeume T (2009) Bounds on non-surjective cellular automata. In: Královic R, Urzyczyn P (eds) MFCS, Lecture notes in information technology, vol 5734, Springer, Berlin, pp 439–450Google Scholar
• Lukkarila V (2010) Sensitivity and topological mixing are undecidable for reversible one-dimensional cellular automata. J Cell Autom 5:241–272MathSciNetzbMATH
• Moore EF (1962) Machine types of self-reproduction. In: Bellman RE (erectile dysfunction) Proceedings of symposia in applied mathematics XIV: “;Mathematical problems within the biological sciences”, AMS, pp 17–33
• von Neumann J (1966) Theory of self-reproducing automata. In: Burks AW (erectile dysfunction) College of Illinois Press, Urbana, London
• Sablik M, Theyssier G (2008) Topological dynamics of 2D cellular automata. In: CiE ’08: Proceedings from the fourth conference on computability in Europe, Springer, Berlin, Heidelberg, pp 523–532
• Wang H (1961) Showing theorems by pattern recognition – ii. Bell Syst Tech J 40:1–42
• Wolfram S (erectile dysfunction) (1986) Theory and applying cellular automata.