Enumeration of Reversible Functions and Its Application to Circuit Complexity

  • Mathias Soeken
  • Nabila Abdessaied
  • Giovanni De Micheli
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9720)


We review combinational results to enumerate and classify reversible functions and investigate the application to circuit complexity. In particularly, we consider the effect of negating and permuting input and output variables and the effect of applying linear and affine transformations to inputs and outputs. We apply the results to reversible circuits and prove that minimum circuit realizations of functions in the same equivalence class differ at most in a linear number of gates in presence of negation and permutation and at most in a quadratic number of gates in presence of linear and affine transformations.


Reversible function Equivalence class Permutation group Reversible circuit complexity 



This research was supported by H2020-ERC-2014-ADG 669354 CyberCare and by the European COST Action IC 1405 ‘Reversible Computation’.


  1. 1.
    Andrews, G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1984)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ashenhurst, R.L.: The application of counting techniques. In: Proceedings of the ACM National Meeting, pp. 293–305 (1952)Google Scholar
  3. 3.
    Beth, T., Rötteler, M.: Quantum algorithms: applicable algebra and quantum physics. In: Springer Tracts in Modern Physics, vol. 173, pp. 96–150 (2001)Google Scholar
  4. 4.
    De Bruijn, N.G.: Generalization of Pólya’s fundamental theorem in enumerative combinational analysis. Konikl. Nederl. Akademie Van Wetenschappen A 52(2), 59–69 (1959)zbMATHGoogle Scholar
  5. 5.
    Draper, T.G.: Nonlinear complexity of Boolean permutations. Ph.D. thesis, University of Maryland (2009)Google Scholar
  6. 6.
    Golubitsky, O., Maslov, D.: A study of optimal 4-bit reversible toffoli circuits and their synthesis. IEEE Trans. Comput. 61(9), 1341–1353 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Harrison, M.A.: Combinational problems in Boolean algebras and applications to the theory of switching. Ph.D. thesis, University of Michigan (1963)Google Scholar
  8. 8.
    Harrison, M.A.: The number of classes of invertible Boolean functions. J. ACM 10(1), 25–28 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Harrison, M.A.: The number of equivalence classes of Boolean functions under groups containing negation. IEEE Trans. Electron. Comput. 12, 559–561 (1963)CrossRefzbMATHGoogle Scholar
  10. 10.
    Harrison, M.A.: The number of transitivity sets of Boolean functions. J. Soc. Appl. Ind. Math. 11, 806–828 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Harrison, M.A.: On the classification of Boolean functions by the general linear and affine groups. J. Soc. Appl. Ind. Math. 12, 285–299 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lorens, C.S.: Invertible Boolean functions. Technical report 21, Space-General Corporation, El Monte, California, Research Memorandum (1962)Google Scholar
  13. 13.
    Lorens, C.S.: Invertible Boolean functions. IEEE Trans. Electron. Comput. 13, 529–541 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pólya, G.: Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und Chemische Verbindungen. Acta Math. 68, 145–253 (1937)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Primenko, É.A.: Equivalence classes of invertible Boolean functions. Cybernetics 20(6), 771–776 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Slepian, D.: On the number of symmetry types of Boolean functions of \(n\) variables. Can. J. Math. 5, 185–193 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Soeken, M., Thomsen, M.K.: White dots do matter: rewriting reversible logic circuits. In: International Conference on Reversible Computation, pp. 196–208 (2013)Google 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.
    Vatan, F., Williams, C.: Optimal quantum circuits for general two-qubit gates. Phys. Rev. A 69, 032315-1–032315-5 (2004)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Mathias Soeken
    • 1
  • Nabila Abdessaied
    • 2
  • Giovanni De Micheli
    • 1
  1. 1.Integrated Systems Laboratory, EPFLLausanneSwitzerland
  2. 2.Cyber-Physical Systems, DFKI GmbHBremenGermany

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