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Implicit recursion-theoretic characterizations of counting classes

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Abstract

We give recursion-theoretic characterizations of the counting class \(\textsf {\#P} \), the class of those functions which count the number of accepting computations of non-deterministic Turing machines working in polynomial time. Moreover, we characterize in a recursion-theoretic manner all the levels \(\{\textsf {\#P} _k\}_{k\in {\mathbb {N}}}\) of the counting hierarchy of functions \(\textsf {FCH} \), which result from allowing queries to functions of the previous level, and \(\textsf {FCH} \) itself as a whole. This is done in the style of Bellantoni and Cook’s safe recursion, and it places \(\textsf {\#P} \) in the context of implicit computational complexity. Namely, it relates \(\textsf {\#P} \) with the implicit characterizations of \(\textsf {FPTIME} \) (Bellantoni and Cook, Comput Complex 2:97–110, 1992) and \(\textsf {FPSPACE} \) (Oitavem, Math Log Q 54(3):317–323, 2008), by exploiting the features of the tree-recursion scheme of \(\textsf {FPSPACE} \).

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Notes

  1. \(|\bar{x}|=(|x_1|,\dots ,|x_n|)\), where \(\bar{x}=(x_1,\dots ,x_n)\).

  2. “It is well known that \(\textsf {\#P} \) is closed not only under addition and multiplication but also under summation of exponentially many \(\textsf {\#P} \) functions...” [8,  p. 458].

  3. Wagner writes \(k\textsf {\#P} \) for our notation \(\textsf {\#P} _k\).

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Acknowledgements

The first author is supported by the ERC CoG DIAPASoN GA 818616. The research of the third and main author is supported by national funds throught the FCT - Fundação para a Ciência e Tecnologia, I.P., under the scope of the project UIDB/00297/2020 (Center for Mathematics and Applications). The second author is supported by the same project and by the Udo Keller Foundation.

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Authors and Affiliations

  1. University of Bologna, Bologna, Italy

    Ugo Dal Lago

  2. INRIA, Sophia Antipolis, Valbonne, France

    Ugo Dal Lago

  3. Carl Friedrich von Weizsäcker-Center, Universität Tübingen, Tübingen, Germany

    Reinhard Kahle

  4. CMA, FCT, Universidade Nova de Lisboa, Caparica, Portugal

    Reinhard Kahle

  5. CMA and DM, NOVA School of Science and Technology, Universidade NOVA de Lisboa, 2829-515, Caparica, Portugal

    Isabel Oitavem

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  1. Ugo Dal Lago
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  2. Reinhard Kahle
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  3. Isabel Oitavem
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Correspondence to Isabel Oitavem.

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Dal Lago, U., Kahle, R. & Oitavem, I. Implicit recursion-theoretic characterizations of counting classes. Arch. Math. Logic 61, 1129–1144 (2022). https://doi.org/10.1007/s00153-022-00828-4

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