Encyclopedia of Complexity and Systems Science

2009 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Reversible Cellular Automata

  • Kenichi Morita
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30440-3_455

Definition of the Subject

Reversible cellular automata (RCAs) are defined as cellularautomata (CAs) with an injective global function. Every configurationof an RCA has exactly one previous configuration, and thus RCAs are“backward deterministic” CAs. The notion of reversibility originally comes from physics. It is one of the fundamentalmicroscopic physical laws of Nature. In this sense, an RCA is thoughtas an abstract model of a physically reversible space as well asa computing model. It is very important to investigate howcomputation can be carried out efficiently and elegantly ina system having reversibility. This is because future computingdevices will surely become those of a nanoscale size.

In this article, we mainly discuss on the properties of RCAsfrom the computational aspects. In spite of the strong constraint ofreversibility, RCAs have very rich ability of computing. We can seethat even very simple RCAs have universal computing ability. We canalso recognize, in some...

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

Bibliography

Primary Literature

  1. 1.
    AmorosoS, Cooper G (1970) The Garden of Eden theorem for finiteconfigurations. Proc Amer Math Soc26:158–164MathSciNetzbMATHGoogle Scholar
  2. 2.
    AmorosoS, Patt YN (1972) Decision procedures for surjectivity and injectivityof parallel maps for tessellation structures. J Comput Syst Sci6:448–464MathSciNetzbMATHGoogle Scholar
  3. 3.
    BennettCH (1973) Logical reversibility of computation. IBM J Res Dev17:525–532zbMATHGoogle Scholar
  4. 4.
    BennettCH (1982) The thermodynamics of computation. Int J Theor Phys21:905–940Google Scholar
  5. 5.
    BennettCH, Landauer R (1985) The fundamental physical limits of computation.Sci Am 253:38–46ADSGoogle Scholar
  6. 6.
    BoykettT (2004) Efficient exhaustive listings of reversible one dimensionalcellular automata. Theor Comput Sci325:215–247MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cook M(2004) Universality in elementary cellular automata. Complex Syst15:1–40zbMATHGoogle Scholar
  8. 8.
    FredkinE, Toffoli T (1982) Conservative logic. Int J Theor Phys21:219–253MathSciNetzbMATHGoogle Scholar
  9. 9.
    GruskaJ (1999) Quantum Computing. McGraw-Hill,LondonGoogle Scholar
  10. 10.
    HedlundGA (1969) Endomorphisms and automorphisms of the shift dynamicalsystem. Math Syst Theor 3:320–375MathSciNetzbMATHGoogle Scholar
  11. 11.
    ImaiK, Morita K (1996) Firing squad synchronization problem in reversiblecellular automata. Theor Comput Sci165:475–482MathSciNetzbMATHGoogle Scholar
  12. 12.
    ImaiK, Morita K (2000) A computation-universal two-dimensional8-state triangular reversible cellular automaton. Theor Comput Sci231:181–191MathSciNetzbMATHGoogle Scholar
  13. 13.
    ImaiK, Hori T, Morita K (2002) Self-reproduction in three-dimensionalreversible cellular space. Artifici Life8:155–174Google Scholar
  14. 14.
    KariJ (1994) Reversibility and surjectivity problems of cellular automata.J Comput Syst Sci 48:149–182MathSciNetzbMATHGoogle Scholar
  15. 15.
    KariJ (1996) Representation of reversible cellular automata with blockpermutations. Math Syst Theor29:47–61MathSciNetzbMATHGoogle Scholar
  16. 16.
    LandauerR (1961) Irreversibility and heat generation in the computing process.IBM J Res Dev 5:183–191MathSciNetzbMATHGoogle Scholar
  17. 17.
    LangtonCG (1984) Self-reproduction in cellular automata. Physica10D:135–144ADSGoogle Scholar
  18. 18.
    MargolusN (1984) Physics-like model of computation. Physica10D:81–95MathSciNetADSGoogle Scholar
  19. 19.
    MaruokaA, Kimura M (1976) Condition for injectivity of global maps fortessellation automata. Inf Control32:158–162MathSciNetzbMATHGoogle Scholar
  20. 20.
    MaruokaA, Kimura M (1979) Injectivity and surjectivity of parallel maps forcellular automata. J Comput Syst Sci18:47–64MathSciNetzbMATHGoogle Scholar
  21. 21.
    MinskyML (1967) Computation: Finite and Infinite Machines. Prentice-Hall,Englewood CliffszbMATHGoogle Scholar
  22. 22.
    MooreEF (1962) Machine models of self-reproduction, Proc Symposia inApplied Mathematics. Am Math Soc14:17–33Google Scholar
  23. 23.
    MoraJCST, Vergara SVC, Martinez GJ, McIntosh HV (2005) Procedures forcalculating reversible one-dimensional cellular automata. Physica D202:134–141MathSciNetADSzbMATHGoogle Scholar
  24. 24.
    MoritaK (1995) Reversible simulation of one-dimensional irreversiblecellular automata. Theor Comput Sci148:157–163zbMATHGoogle Scholar
  25. 25.
    MoritaK (1996) Universality of a reversible two-counter machine. TheorComput Sci 168:303–320zbMATHGoogle Scholar
  26. 26.
    MoritaK (2001) A simple reversible logic element and cellular automatafor reversible computing. In: Proc 3rd Int Conf on Machines,Computations, and Universality. LNCS, vol 2055. Springer, Berlin, pp 102–113Google Scholar
  27. 27.
    MoritaK (2007) Simple universal one-dimensional reversible cellular automata. J Cell Autom 2:159–165MathSciNetzbMATHGoogle Scholar
  28. 28.
    MoritaK, Harao M (1989) Computation universality of one-dimensionalreversible (injective) cellular automata. Trans IEICE JapanE-72:758–762Google Scholar
  29. 29.
    MoritaK, Imai K (1996) Self-reproduction in a reversible cellular space.Theor Comput Sci 168:337–366MathSciNetzbMATHGoogle Scholar
  30. 30.
    MoritaK, Ueno S (1992) Computation-universal models of two-dimensional16-state reversible cellular automata. IEICE Trans Inf SystE75-D:141–147Google Scholar
  31. 31.
    MoritaK, Shirasaki A, Gono Y (1989) A 1-tape 2-symbol reversible Turingmachine. Trans IEICE JapanE-72:223–228Google Scholar
  32. 32.
    MoritaK, Tojima Y, Imai K, Ogiro T (2002) Universal computing in reversibleand number-conserving two-dimensional cellular spaces. In: AdamatzkyA (ed) Collision-based Computing. Springer, London, pp 161–199Google Scholar
  33. 33.
    MyhillJ (1963) The converse of Moore's Garden-of-Eden theorem. Proc Am MathSoc 14:658–686MathSciNetGoogle Scholar
  34. 34.
    RichardsonD (1972) Tessellations with local transformations. J Comput Syst Sci6:373–388zbMATHGoogle Scholar
  35. 35.
    SutnerK (2004) The complexity of reversible cellular automata. Theor ComputSci 325:317–328MathSciNetzbMATHGoogle Scholar
  36. 36.
    ToffoliT (1977) Computation and construction universality of reversiblecellular automata. J Comput Syst Sci15:213–231MathSciNetzbMATHGoogle Scholar
  37. 37.
    ToffoliT (1980) Reversible computing, Automata, Languages andProgramming. In: de Bakker JW, van Leeuwen J (eds) LNCS, vol 85. Springer, Berlin, pp 632–644Google Scholar
  38. 38.
    ToffoliT, Margolus N (1990) Invertible cellular automata: a review.Physica D 45:229–253MathSciNetADSzbMATHGoogle Scholar
  39. 39.
    ToffoliT, Capobianco S, Mentrasti P (2004) How to turn a second-ordercellular automaton into lattice gas: a new inversion scheme. TheorComput Sci 325:329–344MathSciNetzbMATHGoogle Scholar
  40. 40.
    vonNeumann J (1966) Theory of Self-reproducing Automata. Burks AW (ed)University of Illinois Press, UrbanaGoogle Scholar
  41. 41.
    WatrousJ (1995) On one-dimensional quantum cellular automata. In: Proc 36thSymp on Foundation of Computer Science. IEEE, LosAlamitos, pp 528–537Google Scholar

Books and Reviews

  1. 42.
    AdamatzkyA (ed) (2002) Collision-Based Computing. Springer, LondonGoogle Scholar
  2. 43.
    BennettCH (1988) Notes on the history of reversible computation. IBM J ResDev 32:16–23Google Scholar
  3. 44.
    BurksA (1970) Essays on Cellular Automata. University of Illinois Press,UrbanazbMATHGoogle Scholar
  4. 45.
    KariJ (2005) Theory of cellular automata: A survey. Theor Comput Sci334:3–33MathSciNetzbMATHGoogle Scholar
  5. 46.
    MoritaK (2001) Cellular automata and artificial life–computation andlife in reversible cellular automata. In: Goles E, Martinez S (eds)Complex Systems. Kluwer, Dordrecht,pp 151–200Google Scholar
  6. 47.
    WolframS (2001) A New Kind of Science. Wolfram Media, ChampaignGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Kenichi Morita
    • 1
  1. 1.Hiroshima UniversityHigashi-HiroshimaJapan