A Classical Propositional Logic for Reasoning About Reversible Logic Circuits

  • Holger Bock Axelsen
  • Robert Glück
  • Robin Kaarsgaard
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9803)

Abstract

We propose a syntactic representation of reversible logic circuits in their entirety, based on Feynman’s control interpretation of Toffoli’s reversible gate set. A pair of interacting proof calculi for reasoning about these circuits is presented, based on classical propositional logic and monoidal structure, and a natural order-theoretic structure is developed, demonstrated equivalent to Boolean algebras, and extended categorically to form a sound and complete semantics for this system. We show that all strong equivalences of reversible logic circuits are provable in the system, derive an equivalent equational theory, and describe its main applications in the verification of both reversible circuits and template-based reversible circuit rewriting systems.

References

  1. 1.
    Arabzadeh, M., Saeedi, M., Zamani, M.S.: Rule-based optimization of reversible circuits. In: Proceedings of the ASP-DAC 2010, pp. 849–854. IEEE (2010)Google Scholar
  2. 2.
    Bérut, A., Arakelyan, A., Petrosyan, A., Ciliberto, S., Dillenschneider, R., Lutz, E.: Experimental verification of Landauer’s principle linking information and thermodynamics. Nature 483(7388), 187–189 (2012)CrossRefGoogle Scholar
  3. 3.
    Buss, S.R.: Handbook of Proof Theory. Elsevier, Amsterdam (1998)MATHGoogle Scholar
  4. 4.
    De Vos, A.: Reversible Computing. Wiley-VCH, Weinheim (2010)CrossRefMATHGoogle Scholar
  5. 5.
    De Vos, A., Burignat, S., Glück, R., Mogensen, T.Æ., Axelsen, H.B., Thomsen, M.K., Rotenberg, E., Yokoyama, T.: Designing garbage-free reversible implementations of the integer cosine transform. ACM J. Emerg. Tech. Com. 11(2), 11:1–11:15 (2014)Google Scholar
  6. 6.
    Feynman, R.P.: Quantum mechanical computers. Found. Phys. 16(6), 507–531 (1986)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Jacobs, B.: Categorical Logic and Type Theory. Elsevier, Amsterdam (1999)MATHGoogle Scholar
  8. 8.
    Jordan, S.P.: Strong equivalence of reversible circuits is coNP-complete. Quantum Inf. Comput. 14(15–16), 1302–1307 (2014)MathSciNetGoogle Scholar
  9. 9.
    Joyal, A., Street, R.: The geometry of tensor calculus I. Adv. Math. 88(1), 55–112 (1991)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102(1), 20–78 (1993)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kaarsgaard, R.: Towards a propositional logic for reversible logic circuits. In: Proceedings of the ESSLLI 2014 Student Session, pp. 33–41 (2014). http://www.kr.tuwien.ac.at/drm/dehaan/stus2014/proceedings.pdf
  12. 12.
    Landauer, R.: Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5(3), 261–269 (1961)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Leinster, T.: Higher Operads, Higher Categories. London Mathematical Society Lecture Note Series, vol. 298. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  14. 14.
    Orlov, A.O., Lent, C.S., Thorpe, C.C., Boechler, G.P., Snider, G.L.: Experimental test of Landauer’s principle at the sub-\(k_{\rm {b}} t\) level. Japan. J. Appl. Phys. 51, 06FE10 (2012)CrossRefGoogle Scholar
  15. 15.
    Polakow, J.: Ordered linear logic and applications. Ph.D. thesis, CMU (2001)Google Scholar
  16. 16.
    Rendel, T., Ostermann, K.: Invertible syntax descriptions: unifying parsing and pretty printing. ACM SIGPLAN Notices, vol. 45, No. 11, pp. 1–12 (2010)Google Scholar
  17. 17.
    Schellekens, M.P.: MOQA: unlocking the potential of compositional static average-case analysis. J. Log. Algebr. Program. 79(1), 61–83 (2010)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Seely, R.A.G.: Linear logic, *-autonomous categories and cofree coalgebras. Contemp. Math. 92, 371–382 (1989)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Sikorski, R.: Boolean Algebras. Springer, Heidelberg (1969)CrossRefMATHGoogle Scholar
  20. 20.
    Soeken, M., Thomsen, M.K.: White dots do matter: rewriting reversible logic circuits. In: Dueck, G.W., Miller, D.M. (eds.) RC 2013. LNCS, vol. 7948, pp. 196–208. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  21. 21.
    Thomsen, M.K., Glück, R., Axelsen, H.B.: Reversible arithmetic logic unit for quantum arithmetic. J. Phys. A Math. Theor. 43(38), 382002 (2010)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Toffoli, T.: Reversible computing. In: de Bakker, J., van Leeuwen, J. (eds.) Automata, Languages and Programming. LNCS, vol. 85, pp. 632–644. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  23. 23.
    Wille, R., Drechsler, R.: Towards a Design Flow for Reversible Logic. Springer, Heidelberg (2010)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Holger Bock Axelsen
    • 1
  • Robert Glück
    • 1
  • Robin Kaarsgaard
    • 1
  1. 1.DIKU, Department of Computer ScienceUniversity of CopenhagenCopenhagenDenmark

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