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On reversible Turing machines and their function universality

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Abstract

We provide a treatment of the reversible Turing machines (RTMs) under a strict function semantics. Unlike many existing reversible computation models, we distinguish strictly between computing the function \(\lambda x.f(x)\) and computing the function \(\lambda x. (x, f(x))\), or other injective embeddings of f. We reinterpret and adapt a number of important foundational reversible computing results under this semantics. Unifying the results in a single model shows that, as expected (and previously claimed), the RTMs are robust and can compute exactly all injective computable functions. Because injectivity entails that the RTMs are not strictly Turing-complete w.r.t. functions, we use an appropriate alternative universality definition, and show how to derive universal RTMs (URTMs) from existing irreversible universal machines. We then proceed to construct a URTM from the ground up. This resulting machine is the first URTM which does not depend on a reversible simulation of an existing universal machine. The new construction has the advantage that the interpretive overhead of the URTM is limited to a (program dependent) constant factor. Another novelty is that the URTM can function as an inverse interpreter at no asymptotic cost.

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Notes

  1. This natural definition of universality for reversible Turing machine was previously suggested by Li and Vitányi, cf. [22, Sect. 8.1.4], although no concrete constructions were given.

  2. However, the reversible machine is not (quasi-)real-time, while the irreversible one is, and it is not obvious that a quasi-realtime RTM should be possible without altering the tape contents. This hints that RTMs and TMs might have different structural complexity, at least fine-grained. For a formal language RTM model with a read-only one-way input tape, Axelsen et al. have recently shown that this is indeed the case, for a hierarchy between real-time and linear time [8].

  3. This can be enforced by tracking the non-blank part of the tape, and ensuring that this contains no blanks at halting.

  4. Should any of these conditions fail, we can rename as necessary to introduce them, without altering the fine-grained behavior of the machines.

  5. Of course, increasing the amount of static resources does not limit expressivity, as the extra resources can simply be ignored.

  6. We could also have defined the semantics for k-tape TMs such that the input was always given on a single tape, i.e., restricted to unary functions, but this would have severely hampered the presentation.

  7. The simulation of symbol rules provided in [25, Lemma 3, p. 227] is unfortunately flawed: translation of the rules (q, (s, u), p) and (r, (t, v), p)) breaks reversibility if the encodings of u and v share a longer prefix than that of the encodings of s and t. Our proof can be adapted for use in the 2-symbol machine of [25], by reading and writing 0’s where our 3-symbol construction uses blanks.

  8. It is not difficult to see that the reliance on legal computation runs turns the definition—which appears to be about the inner workings of the machine—into one about the extensional behavior. For Turing machines this would then fall under Rice’s theorem.

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Acknowledgments

The authors wish to thank Michael Kirkedal Thomsen for help with the figures in Sect. 5, and Tetsuo Yokoyama for inspiring discussions on computability issues.

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Authors and Affiliations

  1. DIKU, Department of Computer Science, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Holger Bock Axelsen & Robert Glück

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  1. Holger Bock Axelsen
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  2. Robert Glück
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Correspondence to Holger Bock Axelsen.

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Revised and extended version of the conference papers [4, 5]. This research was partially supported by Danish Council for Strategic Research under the MicroPower project and the Danish Council for Independent Research | Natural Sciences under the Foundations of Reversible Computing project; the second author was partially supported by the Japan Society for the Promotion of Science. The authors acknowledge partial support from COST Action IC1405 Reversible Computation.

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Axelsen, H.B., Glück, R. On reversible Turing machines and their function universality. Acta Informatica 53, 509–543 (2016). https://doi.org/10.1007/s00236-015-0253-y

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