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The \(\aleph \)-Calculus

A Declarative Model of Reversible Programming

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Part of the Lecture Notes in Computer Science book series (LNCS,volume 13354)


A novel model of reversible computing, the \(\aleph \)-calculus, is introduced. It is declarative, reversible-Turing complete, and has a local term-rewriting semantics. Unlike previously demonstrated reversible term-rewriting systems, it does not require the accumulation of history data. Terms in the \(\aleph \)-calculus, in combination with the program definitions, encapsulate all program state. An interpreter was also written.


  • Reversible Computing
  • Term-Rewriting
  • Declarative Paradigm

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  • DOI: 10.1007/978-3-031-09005-9_11
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  1. 1.

    No such constraint need be applied to invalid computational states [4].

  2. 2.

    The name of the calculus is inspired by the Greek meaning ‘not forgotten’,

  3. 3.

    In the examples in Sect. 2 the reader may notice this is violated. This is for programmer convenience, and must be resolved manually or by the compiler.

  4. 4.

    The requirement of halting, combined with computational inertia, ensures each sub-multiterm takes on a unique state at production and before consumption.

  5. 5.


  1. Abramsky, S.: A structural approach to reversible computation. Theoret. Comput. Sci. 347(3), 441–464 (2005)

    MathSciNet  CrossRef  Google Scholar 

  2. Bennett, C.H.: Logical reversibility of computation. IBM J. Res. Dev. 17(6), 525–532 (1973)

    MathSciNet  CrossRef  Google Scholar 

  3. Di Pierro, A., Hankin, C., Wiklicky, H.: Reversible combinatory logic. Math. Struct. Comput. Sci. 16(4), 621–637 (2006)

    MathSciNet  CrossRef  Google Scholar 

  4. Frank, M.P.: Foundations of generalized reversible computing. In: Phillips, I., Rahaman, H. (eds.) RC 2017. LNCS, vol. 10301, pp. 19–34. Springer, Cham (2017).

    CrossRef  Google Scholar 

  5. Landauer, R.: Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5(3), 183–191 (1961)

    MathSciNet  CrossRef  Google Scholar 

  6. Nishida, N., Palacios, A., Vidal, G.: Reversible term rewriting. In: 1st International Conference on Formal Structures for Computation and Deduction (2016)

    Google Scholar 

  7. Szilard, L.: Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen. Z. Phys. 53(11–12), 840–856 (1929)

    CrossRef  Google Scholar 

  8. Zhang, D.Y., Seelig, G.: Dynamic DNA nanotechnology using strand-displacement reactions. Nat. Chem. 3(2), 103–113 (2011)

    CrossRef  Google Scholar 

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The author would like to acknowledge the invaluable help and support of her PhD supervisor, Gos Micklem. This work was supported by the Engineering and Physical Sciences Research Council, project reference 1781682.

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Correspondence to Hannah Earley .

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Earley, H. (2022). The \(\aleph \)-Calculus. In: Mezzina, C.A., Podlaski, K. (eds) Reversible Computation. RC 2022. Lecture Notes in Computer Science, vol 13354. Springer, Cham.

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  • Print ISBN: 978-3-031-09004-2

  • Online ISBN: 978-3-031-09005-9

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