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Making Programs Reversible with Minimal Extra Data

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Abstract

Reversible computing is an unconventional computing paradigm that comes with specific challenges. One of the important questions is the existence of reversible programs with minimal extra output (garbage data). To answer this question for programs that implement partial functions over countable domains, we introduce an order on infinite garbage sets and a notion of minimality. To this end, we present two methods for functions specified by decidable and semi-decidable predicates. Both methods are universal, which means they work for all programs specified by the predicates. They cover Bennett’s classic input-erasing reversible computation of injective functions. Hence, any program written in a Turing-complete programming language can be implemented with g-minimal garbage in an r-Turing-complete reversible programming language. This generality comes at the cost of a considerable runtime due to the generate-and-test approach.

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Notes

  1. The number of garbage lines is minimized to \(\lceil \log _2 k\rceil \), where k is the largest number of identical output bit patterns in the truth table of the given reversible circuit.

  2. We abbreviate argument parentheses, e.g., we write \({ fst }(\texttt {x.y})\) instead of \({ fst }(\texttt {(x.y)})\) [14].

  3. We have \(|\texttt {nil}|=1\), and hence \(|\{\texttt {nil}\}| \ne |\{\}|\). The size differs from [14], where \(|\texttt {nil}|=0\).

  4. In general, \(\texttt {chk2}\) can be any program such that \(\texttt {(x.y)} \in \mathrm {dom}([\![\texttt {chk2}]\!]^\texttt {L}) \Leftrightarrow (\texttt {x},\texttt {y})\in S\).

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Acknowledgements

We thank Professor Masami Hagiya for his continuous encouragement and warm, friendly support for many years. We also thank the anonymous reviewers for their useful feedback. The second author was supported by JSPS KAKENHI Grant no. 18K11250 and Nanzan University Pache Research Subsidy I-A-2 for the 2022 academic year.

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  1. DIKU, Department of Computer Science, University of Copenhagen, 2100, Copenhagen, Denmark

    Robert Glück

  2. Department of Electronics and Communication Technology, Nanzan University, Nagoya, 466-8673, Japan

    Tetsuo Yokoyama

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  1. Robert Glück
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  2. Tetsuo Yokoyama
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Correspondence to Robert Glück.

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Glück, R., Yokoyama, T. Making Programs Reversible with Minimal Extra Data. New Gener. Comput. 40, 467–480 (2022). https://doi.org/10.1007/s00354-022-00169-z

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