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Cellular Automata, Universality of

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

Cellular automata (CA) and the subject are briefly defined before two kinds ofuniversality are considered: computational universality and intrinsicuniversality. A more involving section on advanced topics ends this chapter.

Computational universality deals with the capabilityto carry out any computation as defined by Turing machines (in computability Theory) whileintrinsic universalitydeals with the capability tosimulate any other machine of the same class (here cellular automata). This distinction isfundamental here because while computational universality refers to finite inputs and relatesto our understanding of computing with computers, intrinsic universality encompasses infiniteconfigurations and relates to our understanding of the physical world. These universalitiesare presented as simply as possible and an example of universal CA is presented in eachcase. The last section is devoted to the history and advanced topics such as variousdefinitions of...

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Abbreviations

Cellular automata :

They are dynamical systems that are continuous, local, parallel, synchronous and space and time uniform. Cellular automata are used to model phenomena where the space can be regularly partitioned and where the same rules are used everywhere, for example: flow dynamics or percolation in physics, systolic arrays in computer science, epidemics in biology…

The configurations are infinite arrays of cells. Each cell has a state chosen inside a finite set. The dynamics is given by replacing the state of each cell according to it and the states of the cells at a bounded distance. Since there are finitely many neighboring cells, there are finitely many state patterns/inputs. The mapping to the new state is called the local function. The same local function is used for all the cells. They are all updated simultaneously.

Computational universality :

Computability is defined by Turing machine, μ‑recursive functions or λ‑calculus. All these approaches (and many more) end up defining the same set of functions over \( { \mathbb{N} } \) (or on words, i. e. finite sequences over a finite alphabet): the computable functions. They defined, according to the Church–Turing thesis, what can be computed by any reasonable device.

A machine is computation universal if it is able to compute any computable function (indicated as a part of the entry). This corresponds also to the common approach of computer, the hardware is universal and the program to be executed is stored in main memory (like the data to process) and is part of the input as far as the hardware / operating system is concerned.

Intrinsic universality :

It is the capability to simulate any machine in a class of machine. If one think of Turing machines or an equivalent model of computation, this folds back to the classical computational universality. The interest of this notion lays with machines that are not equivalent to Turing machines. This is the case of cellular automata: they update infinite configurations, there are uncountably many possible configurations thus they cannot be encoded in a countable set, say \( { \mathbb{N} } \).

An intrinsically universal CA “represents” all the CA since it can exhibit any phenomenon any other one can.

Bibliography

  1. Adamatzky A (ed) (2002) Collision based computing. Springer

    Google Scholar 

  2. Albert J, Čulik K II (1987) A simple universal cellular automaton and its one-way and totalistic version. Complex Syst 1:1–16

    Google Scholar 

  3. Arbib MA (1966) Simple self‐reproducing universal automata. Inf Control 9(2):177–189

    Google Scholar 

  4. Arrighi P, Fargetton R (2007) Intrinsically universal one‐dimensional quantum cellular automata. arXiv:0704.3961

    Google Scholar 

  5. Banks ER (1970) Universality in cellular automata. In: Eleventh annual symposium on switching and automata theory. IEEE, pp 194–215

    Google Scholar 

  6. Berlekamp E, Conway J, Guy R (1982) Winning ways for your mathematical plays (games in particular), vol 2. Academic Press

    Google Scholar 

  7. Burks A (1970) Essays on cellular automata. Univ of Illinois Press

    Google Scholar 

  8. Codd E (1968) Cellular automata. Academic Press

    Google Scholar 

  9. Cook M (2004) Universality in elementary cellular automata. Complex Syst 15:1–40

    Google Scholar 

  10. Cordero P, Goles E, Hernández G (1992) Q2R + Q2R as a universal billiard. Int J Modern Phys C 3(2):251–266

    Google Scholar 

  11. Čulik K II, Pachl J, Yu S (1990) Computation theoretic aspects of cellular automata. Phys D 45(1–3):357–378

    Google Scholar 

  12. Delorme M, Mazoyer J (2002) Signals on cellular automata. In: Adamatzky A (ed) Collision‐based computing. Springer, pp 234–275

    Google Scholar 

  13. Dubacq J-C (1995) How to simulate Turing machines by invertible 1d cellular automata. Int J Found Comput Sci 6(4):395–402

    Google Scholar 

  14. Durand B, Róka Z (1998) The game of life: Universality revisited. In: Delorme M, Mazoyer J (eds) Cellular Automata: A parallel model. Math Appl, vol 460. Kluwer, Dordrecht, pp 51–74

    Google Scholar 

  15. Durand‐Lose J (1995) Reversible cellular automaton able to simulate any other reversible one using partitioning automata. In: LATIN’95. LNCS, vol 911. Springer, Berlin, pp 230–244

    Google Scholar 

  16. Durand‐Lose J (1997) Intrinsic universality of a 1‑dimensional reversible cellular automaton. In: STACS’97. LNCS, vol 1200. Springer, pp 439–450

    Google Scholar 

  17. Durand‐Lose J (2000) Reversible space-time simulation of cellular automata. Theor Comput Sci 246(1–2):117–129

    Google Scholar 

  18. Durand‐Lose J (2002) Computing inside the billiard ball model. In: Adamatzky A (ed) Collision‐based computing. Springer, pp 135–160

    Google Scholar 

  19. Gardner M (1970) Mathematical games. The fantastic combinations of John Conway’s new solitaire game “life”. Sci Am 223:120–123

    Google Scholar 

  20. Gutowitz H (ed) (1991) Cellular automata, theory and experimentation. MIT/North‐Holland

    Google Scholar 

  21. Hedlund GA (1969) Endomorphism and automorphism of the shift dynamical system. Math Syst Theory 3:320–375

    MathSciNet  Google Scholar 

  22. Herrmann F, Margenstern M (2003) A universal cellular automaton in the hyperbolic plane. Theor Comput Sci 296:327–364

    MathSciNet  Google Scholar 

  23. Hertling P (1998) Embedding cellular automata into reversible ones. In: Calude C, Casti J, Dinneen M (eds) Unconventional models of computation. DMTCS. Springer

    Google Scholar 

  24. Ilachinski A (2001) Cellular automata – A discrete universe. World Scientific

    Google Scholar 

  25. Imai K, Morita K (1998) A computation‐universal two‐dimensional 8-state triangular reversible cellular automaton. In: Margenstern M (ed) Universal machines and computations (UCM 98), vol 2. Université de Metz, pp 90–99

    Google Scholar 

  26. Kari J (2005) Theory of cellular automata: A survey. Theor Comput Sci 334:3–33

    MathSciNet  Google Scholar 

  27. Kutrib M (2001) Efficient universal pushdown cellular automata and their application to complexity. In: Margenstern M, Rogozhin Y (eds) Machines, computations, and universality (MCU’01), Chisinau, Moldavia, 23–27 May 2001. LNCS, vol 2055. Springer, Berlin, pp 252–263

    Google Scholar 

  28. Lindgren K, Nordahl MG (1990) Universal computation in simple one‐dimensional cellular automata. Complex Syst 4:299–318

    MathSciNet  Google Scholar 

  29. Margenstern M (2006) An algorithm for buiding inrinsically universal automata in hyperbolic spaces. In: Arabnia HR, Burgin M (eds) International conference on foundations of computer science (FCS’06). pp 3–9

    Google Scholar 

  30. Margolus N (2002) Universal cellular automata based on the collisions of soft spheres. In: Adamatzky A (ed) Collision‐based computing. Springer, pp 107–134

    Google Scholar 

  31. Martin B (1994) A universal cellular automaton in quasi‐linear time and its S-n-m form. Theor Comput Sci 123:199–237

    Google Scholar 

  32. Mazoyer J (1996) Computations on one dimensional cellular automata. Ann Math Artif Intell 16:285–309

    MathSciNet  Google Scholar 

  33. Mazoyer J, Rapaport I (1998) Inducing an order on cellular automata by a grouping operation. In: 15th annual symposium on theoretical aspects of computer science (STACS’98). LNCS, vol 1373. Springer, Berlin, pp 116–127

    Google Scholar 

  34. Mazoyer J, Rapaport I (1999) Inducing an order on cellular automata by a grouping operation. Discet Appl Math 218:177–196

    MathSciNet  Google Scholar 

  35. Mazoyer J, Terrier V (1999) Signals in one‐dimensional cellular automata. Theor Comput Sci 217(1):53–80

    MathSciNet  Google Scholar 

  36. Minsky M (1967) Finite and infinite machines. Prentice Hall

    Google Scholar 

  37. Morita K (1992) Any irreversible cellular automaton can be simulated by a reversible one having the same dimension. Technical report of the IEICE Comp 92-45(1992-10):55–64

    Google Scholar 

  38. Morita K (1992) Computation‐universality of one‐dimensional one-way reversible cellular automata. Inform Process Lett 42:325–329

    MathSciNet  Google Scholar 

  39. Morita K (1995) Reversible simulation of one‐dimensional irreversible cellular automata. Theor Comput Sci 148:157–163

    Google Scholar 

  40. Morita K, Harao M (1989) Computation universality of one‐dimensional reversible (injective) cellular automata. Trans IEICE E 72(6):758–762

    Google Scholar 

  41. Morita K, Margenstern M, Imai K (1998) Universality of reversible hexagonal cellular automata. In: MFCS’98 Satellite workshop on frontiers between decidability and undecidability

    Google Scholar 

  42. Neary T, Woods D (2006) P‑completeness of cellular automaton rule 110. In: Bugliesi M, Preneel B, Sassone V, Wegener I (eds) International colloquium on automata languages and programming (ICALP’06). LNCS, vol 4051(1). Springer, Berlin, pp 132–143

    Google Scholar 

  43. Nishio H (2007) Changing the neighborhood of cellular automata. In: Durand‐Lose J, Margenstern M (eds) Machine, computations and universality (MCU’07). LNCS, vol 4664. Springer, Berlin, pp 255–266

    Google Scholar 

  44. Ollinger N (2001) Two‐states bilinear intrinsically universal cellular automata. In: FCT’01. LNCS, vol 2138. Springer, pp 369–399

    Google Scholar 

  45. Ollinger N (2002) The quest for small universal cellular automata. In: ICALP’02. LNCS, vol 2380. Springer, pp 318–329

    Google Scholar 

  46. Ollinger N (2003) The intrinsic universality problem of one‐dimensional cellular automata. In: STACS’03. LNCS, vol 2607. Springer, Berlin, pp 632–641

    Google Scholar 

  47. Richard G (2006) A particular universal cellular automaton. oai:hal.ccsd.cnrs.fr:ccsd‐00095821_v1

    Google Scholar 

  48. Richardson D (1972) Tessellations with local transformations. J Comput Syst Sci 6:373–388

    Google Scholar 

  49. Róka Z (1999) Simulation between cellular automata on cayley graphs. Theor Comput Sci 225:81–111

    Google Scholar 

  50. Sarkar P (2000) A brief history of cellular automata. ACM Comput Surv 32(1):80–107

    Google Scholar 

  51. Sipser M (1997) Introduction to the theory of computation. PWS Publishing Co, Boston

    Google Scholar 

  52. Smith AR III (1971) Simple computation‐universal cellular spaces. J ACM 18(3):339–353

    Google Scholar 

  53. Steiglitz K, Kamal I, Watson A (1988) Embedding computation in one‐dimensional automata by phase coding solitons. IEEE Trans Comput 37(2):138–145

    MathSciNet  Google Scholar 

  54. Sutner K (2005) Universality and cellular automata. In: Margenstern M (ed) Machines, computations, and universality (MCU’04). LNCS, vol 3354. Springer, Berlin, pp 50–59

    Google Scholar 

  55. Terrier V (1996) Language not recognizable in real time by one-way cellular automata. Theor Comput Sci 156:281–287

    MathSciNet  Google Scholar 

  56. Theyssier G (2004) Captive cellular automata. In: Fiala J, Koubek V, Kratochvíl J (eds) Mathematical foundations of computer science (MFCS’04), 29th international symposium, Prague, Czech Republic, 22–27 August. LNCS, vol 3153. Springer, Berlin, pp 427–438

    Google Scholar 

  57. Theyssier G (2005) How common can be universality for cellular automata? In: Diekert V, Durand B (eds) 22nd annual symposium on theoretical aspects of computer science (STACS’05), Stuttgart, Germany, 24–26 February. LNCS, vol 3404. Springer, Berlin, pp 121–132

    Google Scholar 

  58. Toffoli T, Margolus N (1987) Cellular automata machine – A new environment for modeling. MIT press, Cambridge

    Google Scholar 

  59. Ulam S (1952) Random processes and transformations. In: International congress of mathematics 1950, vol 2. pp 264–275

    Google Scholar 

  60. von Neumann J (1966) Theory of self‐reproducing automata. University of Illinois Press, Urbana

    Google Scholar 

  61. Wolfram S (2002) A New kind of Science. Wolfram Media

    Google Scholar 

  62. Woods D, Neary T (2007) Small semi‐waekly universal turing machines. In: Durand‐Lolse J, Margenstern M (eds) Machines, computations and universality (MCA’07). LNCS, vol 4664. Springer, Berlin, pp 246–257

    Google Scholar 

  63. Worsch T, Nishio H (2007) Variations on neighborhoods in ca. In: Moreno‐Diaz R (ed) EUROCAST’07. LNCS, vol 4739. Springer, Berlin, pp 581–588

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire d'Informatique Fondamentale d'Orléans, Université d'Orléans, Orléans, France

    Jérôme Durand-Lose

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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Durand-Lose, J. (2009). Cellular Automata, Universality of. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_59

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