Skip to main content

A Classical Propositional Logic for Reasoning About Reversible Logic Circuits

  • Conference paper
  • First Online:
Logic, Language, Information, and Computation (WoLLIC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9803))

  • 628 Accesses


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.

The authors acknowledge support from the Danish Council for Independent Research\(\mid \) Natural Sciences under the Foundations of Reversible Computing project, and partial support from COST Action IC1405 Reversible Computation.

Colors in electronic version.

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

Access this chapter

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions


  1. 1.

    We use the notation \(\varDelta _\mathbb {N}\) for the discrete category specifically to avoid confusion with the ordinal category \(\omega \), which some authors denote \(\mathbb {N}\).


  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. 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)

    Article  Google Scholar 

  3. Buss, S.R.: Handbook of Proof Theory. Elsevier, Amsterdam (1998)

    MATH  Google Scholar 

  4. De Vos, A.: Reversible Computing. Wiley-VCH, Weinheim (2010)

    Book  MATH  Google Scholar 

  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. Feynman, R.P.: Quantum mechanical computers. Found. Phys. 16(6), 507–531 (1986)

    Article  MathSciNet  Google Scholar 

  7. Jacobs, B.: Categorical Logic and Type Theory. Elsevier, Amsterdam (1999)

    MATH  Google Scholar 

  8. Jordan, S.P.: Strong equivalence of reversible circuits is coNP-complete. Quantum Inf. Comput. 14(15–16), 1302–1307 (2014)

    MathSciNet  Google Scholar 

  9. Joyal, A., Street, R.: The geometry of tensor calculus I. Adv. Math. 88(1), 55–112 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102(1), 20–78 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kaarsgaard, R.: Towards a propositional logic for reversible logic circuits. In: Proceedings of the ESSLLI 2014 Student Session, pp. 33–41 (2014).

  12. Landauer, R.: Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5(3), 261–269 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  13. Leinster, T.: Higher Operads, Higher Categories. London Mathematical Society Lecture Note Series, vol. 298. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  15. Polakow, J.: Ordered linear logic and applications. Ph.D. thesis, CMU (2001)

    Google Scholar 

  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. Schellekens, M.P.: MOQA: unlocking the potential of compositional static average-case analysis. J. Log. Algebr. Program. 79(1), 61–83 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Seely, R.A.G.: Linear logic, *-autonomous categories and cofree coalgebras. Contemp. Math. 92, 371–382 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sikorski, R.: Boolean Algebras. Springer, Heidelberg (1969)

    Book  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  23. Wille, R., Drechsler, R.: Towards a Design Flow for Reversible Logic. Springer, Heidelberg (2010)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Robin Kaarsgaard .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Axelsen, H.B., Glück, R., Kaarsgaard, R. (2016). A Classical Propositional Logic for Reasoning About Reversible Logic Circuits. In: Väänänen, J., Hirvonen, Å., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2016. Lecture Notes in Computer Science(), vol 9803. Springer, Berlin, Heidelberg.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-52920-1

  • Online ISBN: 978-3-662-52921-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics