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Abstract

Reversible computing is the study of computation models that exhibit both forward and backward determinism. Understanding the fundamental properties of such models is not only relevant for reversible programming, but has also been found important in other fields, e.g., bidirectional model transformation, program transformations such as inversion, and general static prediction of program properties.

Historically, work on reversible computing has focussed on reversible simulations of irreversible computations. Here, we take the viewpoint that the property of reversibility itself should be the starting point of a computational theory of reversible computing. We provide a novel semantics-based approach to such a theory, using reversible Turing machines (RTMs) as the underlying computation model.

We show that the RTMs can compute exactly all injective, computable functions. We find that the RTMs are not strictly classically universal, but that they support another notion of universality; we call this RTM-universality. Thus, even though the RTMs are sub-universal in the classical sense, they are powerful enough as to include a self-interpreter. Lifting this to other computation models, we propose r-Turing completeness as the ‘gold standard’ for computability in reversible computation models.

References

  1. 1.
    Abramov, S., Glück, R.: Principles of inverse computation and the universal resolving algorithm. In: Mogensen, T.Æ., Schmidt, D.A., Sudborough, I.H. (eds.) The Essence of Computation. LNCS, vol. 2566, pp. 269–295. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Axelsen, H.B.: Clean translation of an imperative reversible programming language. In: Knoop, J. (ed.) CC 2011. LNCS, vol. 6601, pp. 144–163. Springer, Heidelberg (2011)Google Scholar
  3. 3.
    Axelsen, H.B., Glück, R., Yokoyama, T.: Reversible machine code and its abstract processor architecture. In: Diekert, V., Volkov, M.V., Voronkov, A. (eds.) CSR 2007. LNCS, vol. 4649, pp. 56–69. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Bennett, C.H.: Logical reversibility of computation. IBM Journal of Research and Development 17, 525–532 (1973)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Feynman, R.: Quantum mechanical computers. Optics News 11, 11–20 (1985)CrossRefGoogle Scholar
  6. 6.
    Foster, J.N., Greenwald, M.B., Moore, J.T., Pierce, B.C., Schmitt, A.: Combinators for bi-directional tree transformations: A linguistic approach to the view update problem. ACM Trans. Prog. Lang. Syst. 29(3), article 17 (2007)Google Scholar
  7. 7.
    Glück, R., Sørensen, M.: A roadmap to metacomputation by supercompilation. In: Danvy, O., Thiemann, P., Glück, R. (eds.) Partial Evaluation. LNCS, vol. 1110, pp. 137–160. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  8. 8.
    Jones, N.D.: Computability and Complexity: From a Programming Language Perspective. In: Foundations of Computing. MIT Press, Cambridge (1997)Google Scholar
  9. 9.
    Landauer, R.: Irreversibility and heat generation in the computing process. IBM Journal of Research and Development 5(3), 183–191 (1961)MathSciNetCrossRefGoogle Scholar
  10. 10.
    McCarthy, J.: The inversion of functions defined by Turing machines. In: Automata Studies, pp. 177–181. Princeton University Press, Princeton (1956)Google Scholar
  11. 11.
    Morita, K.: Reversible computing and cellular automata — A survey. Theoretical Computer Science 395(1), 101–131 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Morita, K., Shirasaki, A., Gono, Y.: A 1-tape 2-symbol reversible Turing machine. Trans. IEICE, E 72(3), 223–228 (1989)Google Scholar
  13. 13.
    Morita, K., Yamaguchi, Y.: A universal reversible turing machine. In: Durand-Lose, J., Margenstern, M. (eds.) MCU 2007. LNCS, vol. 4664, pp. 90–98. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Mu, S.-C., Hu, Z., Takeichi, M.: An algebraic approach to bi-directional updating. In: Chin, W.-N. (ed.) APLAS 2004. LNCS, vol. 3302, pp. 2–20. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  15. 15.
    Schellekens, M.: MOQA; unlocking the potential of compositional static average-case analysis. Journal of Logic and Algebraic Programming 79(1), 61–83 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Thomsen, M.K., Axelsen, H.B.: Parallelization of reversible ripple-carry adders. Parallel Processing Letters 19(2), 205–222 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Thomsen, M.K., Glück, R., Axelsen, H.B.: Reversible arithmetic logic unit for quantum arithmetic. Journal of Physics A: Mathematics and Theoretical 42(38), 2002 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Toffoli, T.: Reversible computing. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 632–644. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  19. 19.
    van de Snepscheut, J.L.A.: What computing is all about. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  20. 20.
    Van Rentergem, Y., De Vos, A.: Optimal design of a reversible full adder. International Journal of Unconventional Computing 1(4), 339–355 (2005)Google Scholar
  21. 21.
    Yokoyama, T., Axelsen, H.B., Glück, R.: Principles of a reversible programming language. In: Proceedings of Computing Frontiers, pp. 43–54. ACM Press, New York (2008)Google Scholar
  22. 22.
    Yokoyama, T., Axelsen, H.B., Glück, R.: Reversible flowchart languages and the structured reversible program theorem. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 258–270. Springer, Heidelberg (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Holger Bock Axelsen
    • 1
  • Robert Glück
    • 1
  1. 1.DIKU, Department of Computer ScienceUniversity of CopenhagenDenmark

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