Abstract
The (2w)! reversible transformations on w wires, i.e. reversible logic circuits with w inputs and w outputs, together with the action of cascading, form a group, isomorphic to the symmetric group S 2 w. Therefore, we investigate the group S n as well as one of its subgroups isomorphic to S n/2 × S n/2. We then consider the left cosets, the right cosets, and the double cosets generated by the subgroup. Each element of a coset can function as the representative of the coset. The coset can then be considered as the set of all group elements that differ from the representative by merely multiplying (either to the left or to the right or to both sides) by an arbitrary element of the subgroup. Different choices of the coset space and different choices of the coset representatives lead to six different syntheses for implementing an arbitrary reversible logic operation into hardware. Evaluation of all six methods, by means of three different cost functions (gate cost, switch cost, and quantum cost), leads to a best choice.
Similar content being viewed by others
Bibliography
I. Markov, An introduction to reversible circuits, Proceedings of the 12-th International Workshop on Logic and Synthesis, Laguna Beach, May 2003, pp. 318–319.
M. Frank, Introduction to reversible computing: motivation, progress, and challenges, Proceedings of the 2005 Computing Frontiers Conference, Ischia, May 2005, pp. 385–390.
P. Wayner, Silicon in reverse, Byte 19, August 1994, pp. 67–74.
A. De Vos, Lossless computing, Proceedings of the I.E.E.E. Workshop on Signal Processing, Poznań, October 2003, pp. 7–14.
A. De Vos and Y. Van Rentergem, Reversible computing: from mathematical group theory to electronical circuit experiment, Proceedings of the 2005 Computing Frontiers Conference, Ischia, May 2005, pp. 35–44.
B. Hayes, Reverse engineering, American Scientist 94, March-April 2006, pp. 107–111.
R. Feynman, Quantum mechanical computers, Optics News 11, 11 (1985).
A. De Vos, Computing in finite time, Open Sys. Information Dyn. 13, 179 (2006).
V. Shende, A. Prasad, I. Markov, and J. Hayes, Synthesis of reversible logic circuits, I.E.E.E. Transactions on Computer-Aided Design of Integrated Circuits and Systems 22, 710 (2003).
A. Agrawal and N. Jha, Synthesis of reversible logic, Proceedings of the Design, Automation and Test in Europe Conference, Paris, February 2004, pp. 1384–1385.
P. Kerntopf, A new heuristic algorithm for reversible logic circuit synthesis, Proceedings of the 41-th Design Automation Conference, San Diego, June 2004, pp. 834–837.
P. Kerntopf, Reversible logic circuit synthesis based on a new complexity measure, Proceedings of the 13-th International Workshop on Logic and Synthesis, Temecula, July 2004, pp. 106–113.
D. Maslov, G. Dueck, and D. Miller, Toffoli network synthesis with templates, I.E.E.E. Transactions on Computer-Aided Design of Integrated Circuits and Systems 24, 807 (2005).
D. Maslov, G. Dueck, and D. Miller, Synthesis of Fredkin-Toffoli reversible networks, I.E.E.E. Transactions on Very Large Scale Integration (VLSI) Systems 13, 765 (2005).
Y. Van Rentergem, A. De Vos, and L. Storme, Implementing an arbitrary reversible logic gate, J. Phys. A: Math. Gen. 38, 3555 (2005).
D. Maslov and G. Dueck, Reversible cascades with minimal garbage, I.E.E.E. Transactions on Computer-Aided Design of Integrated Circuits and Systems 23, 1497 (2004).
W. Scott, Group theory, Dover Publications, New York, 1964.
P. Hall, The theory of groups, AMS Chelsea Publishing, Providence, 1968.
A. De Vos and Y. Van Rentergem, Synthesis of reversible circuits, Proceedings of the 14-th International Workshop on Logic and Synthesis, Lake Arrowhead, June 2005, pp. 101–108.
M. Aigner and G. Ziegler, Proofs from The Book, Springer, Berlin, 2001, pp. 7–12.
T. Beth and M. Rötteler, Quantum algorithms: applicable algebra and quantum physics, in: Quantum information: an introduction to basic theoretical concepts and experiments, G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rötteler, H. Weinfurter, R. Werner, and A. Zeilinger, eds., Springer, Berlin, 2001, pp. 96–150.
A. De Vos and Y. Van Rentergem, From group theory to reversible computers, Proceedings of the Workshop ‘From Utopian to Genuine Unconventional Computers’ at the 5-th International Conference on Unconventional Computation, York, September 2006, to be published.
P. Wocjan and M. Horodecki, Characterization of combinatorially independent permutation separability criteria, Open Sys. Information Dyn. 12, 331 (2005).
I. Gradshteyn and I. Ryzhik, Table of integrals, series, and sums, Academic Press, Boston, 1994.
D. Maslov, G. Dueck, and D. Miller, Templates for Toffoli networks synthesis, Proceedings of the 12-th International Workshop on Logic and Synthesis, Laguna Beach, May 2003, pp. 320–325.
D. Maslov, G. Dueck, and D. Miller, Simplification of Toffoli networks via templates, Proceedings of the 16-th Symposium on Integrated Circuits and System Design, Sao Paulo, September 2003, pp. 53–58.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rentergem, Y.V., Vos, A.D. & Keyser, K.D. Six Synthesis Methods for Reversible Logic. Open Syst Inf Dyn 14, 91–116 (2007). https://doi.org/10.1007/s11080-007-9032-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11080-007-9032-8