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Similarly, in the [[Turing machine]] model of computation, a reversible Turing machine is one whose transition function is invertible, so that each machine state has at most one predecessor.
[[:fr:Yves Lecerf|Yves Lecerf]] proposed a reversible Turing machine in a 1963 paper,<ref>Lecerf (Y.) : [http://vadeker.net/corpus/reversible/lecerf.pdf Logique Mathématique : Machines de Turing réversibles.] Comptes rendus des séances de l'académie des sciences, 257:2597--2600, 1963.</ref> but apparently unaware of Landauer's principle, did not pursue the subject further, devoting most of the rest of his career to ethnolinguistics. In 1973 [[Charles H. Bennett (physicist)|Charles H. Bennett]], at IBM Research, showed that a universal Turing machine could be made both logically and thermodynamically reversible,<ref>C. H. Bennett, "[http://www.dna.caltech.edu/courses/cs191/paperscs191/bennett1973.pdf Logical reversibility of computation]", IBM Journal of Research and Development, vol. 17, no. 6, pp. 525-532, 1973</ref> and therefore able in principle to perform an arbitrarily large number of computation steps per unit of physical energy dissipated, if operated sufficiently slowly. Thermodynamically reversible computers could perform useful computations at useful speed, while dissipating considerably less than ''[[kT (energy)|kT]]'' of energy per logical step. In 1982 [[Edward Fredkin]] and [[Tommaso Toffoli]] proposed the [[Billiard ball computer]], a mechanism using classical hard spheres to do reversible computations at finite speed with zero dissipation, but requiring perfect initial alignment of the balls' trajectories, and Bennett's review<ref>{{cite journal |last1=Bennett |first1=Charles H. |title=The thermodynamics of computation—a review |journal=International Journal of Theoretical Physics |date=December 1982 |volume=21 |issue=12 |pages=905–940 |doi=10.1007/BF02084158 |bibcode=1982IJTP...21..905B }}</ref> compared these "Brownian" and "ballistic" paradigms for reversible computation. Aside from the motivation of energy-efficient computation, reversible logic gates offered practical improvements of [[bit manipulation|bit-manipulation]] transforms in cryptography and computer graphics. Since the 1980s, reversible circuits have attracted interest as components of [[quantum algorithm]]s, and more recently in photonic and nano-computing technologies where some switching devices offer no [[signal gain]].
Surveys of reversible circuits, their construction and optimization, as well as recent research challenges, are available.<ref>Rolf Drechsler, Robert Wille. From Truth Tables to Programming Languages: Progress in the Design of Reversible Circuits. International Symposium on Multiple-Valued Logic, 2011. http://www.informatik.uni-bremen.de/agra/doc/konf/11_ismvl_reversible_circuit_design_tutorial.pdf</ref><ref>{{cite journal |last1=Saeedi |first1=Mehdi |last2=Markov |first2=Igor L. |title=Synthesis and optimization of reversible circuits—a survey |journal=ACM Computing Surveys |date=1 February 2013 |volume=45 |issue=2 |pages=1–34 |doi=10.1145/2431211.2431220 |arxiv=1110.2574 }}</ref><ref>Rolf Drechsler and Robert Wille. Reversible Circuits: Recent Accomplishments and Future Challenges for an Emerging Technology. International Symposium on VLSI Design and Test, 2012. http://www.informatik.uni-bremen.de/agra/doc/konf/2012_vdat_reversible_circuits_accompl_chall.pdf</ref><ref>{{cite journal |last1=Cohen |first1=Eyal |last2=Dolev |first2=Shlomi |last3=Rosenblit |first3=Michael |title=All-optical design for inherently energy-conserving reversible gates and circuits |journal=Nature Communications |date=26 April 2016 |volume=7 |issue=1 |pages=11424 |doi=10.1038/ncomms11424 |pmid=27113510 |pmc=4853429 |bibcode=2016NatCo...711424C }}</ref><ref>{{Cite journal|last =Ang|first = Y. S.|last2 = Yang|first2 = S. A.|last3 = Zhang|first3 = C.|last4 = Ma|first4 = Z. S.|last5 = Ang|first5 = L. K.|date = 2017|title = Valleytronics in merging Dirac cones: All-electric-controlled valley filter, valve, and universal reversible logic gate|journal = Physical Review B|volume = 96|issue = 24|pages = 245410|doi = 10.1103/PhysRevB.96.245410|arxiv = 1711.05906|bibcode = 2017PhRvB..96x5410A}}</ref>
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