## Abstract

The Minimum Circuit Size Problem (\(\mathsf {MCSP}\)) has been the focus of intense study recently; \(\mathsf {MCSP}\) is hard for \(\mathsf {SZK}\) under rather powerful reductions [4], and is provably not hard under “local” reductions computable in \({\mathsf {TIME}}(n^{0.49})\) [22]. The question of whether \(\mathsf {MCSP}\) is \(\mathsf {NP}\)-hard (or indeed, hard even for small subclasses of \(\mathsf {P}\)) under some of the more familiar notions of reducibility (such as many-one or Turing reductions computable in polynomial time or in \(\mathsf {AC}^0\)) is closely related to many of the longstanding open questions in complexity theory [7, 8, 16,17,18, 20, 22].

All prior hardness results for \(\mathsf {MCSP}\) hold also for computing somewhat weak approximations to the circuit complexity of a function [3, 4, 9, 16, 21, 27]. (Subsequent to our work, a new hardness result has been announced [19] that relies on more exact size computations.) Some of these results were proved by exploiting a connection to a notion of time-bounded Kolmogorov complexity (\(\mathsf {KT}\)) and the corresponding decision problem (\(\mathsf {MKTP}\)). More recently, a new approach for proving improved hardness results for \(\mathsf {MKTP}\) was developed [5, 7], but this approach establishes only hardness of extremely good approximations of the form \(1+o(1)\), and these improved hardness results are not yet known to hold for \(\mathsf {MCSP}\). In particular, it is known that \(\mathsf {MKTP}\) is hard for the complexity class \(\mathsf {DET}\) under nonuniform \(\le _{{\text {m}}}^{\mathsf {AC}^0}\) reductions, implying \(\mathsf {MKTP}\) is not in \(\mathsf {AC}^0[p]\) for any prime *p* [7]. It was still open if similar circuit lower bounds hold for \(\mathsf {MCSP}\). (But see [13, 19].) One possible avenue for proving a similar hardness result for \(\mathsf {MCSP}\) would be to improve the hardness of approximation for \(\mathsf {MKTP}\) beyond \(1+o(1)\) to \(\omega (1)\), as \(\mathsf {KT}\)-complexity and circuit size are polynomially-related. In this paper, we show that this approach cannot succeed.

More specifically, we prove that \(\mathsf {PARITY}\) does not reduce to the problem of computing superlinear approximations to \(\mathsf {KT}\)-complexity or circuit size via \(\mathsf {AC}^0\)-Turing reductions that make *O*(1) queries. This is significant, since approximating any set in \(\mathsf {P/poly}\) \(\mathsf {AC}^0\)-reduces to just *one* query of a much worse approximation of circuit size or \(\mathsf {KT}\)-complexity [24]. For weaker approximations, we also prove non-hardness under more powerful reductions. Our non-hardness results are unconditional, in contrast to conditional results presented in [7] (for more powerful reductions, but for much worse approximations). This highlights obstacles that would have to be overcome by any proof that \(\mathsf {MKTP}\) or \(\mathsf {MCSP}\) is hard for \(\mathsf {NP}\) under \(\mathsf {AC}^0\) reductions. It may also be a step toward confirming a conjecture of Murray and Williams, that \(\mathsf {MCSP}\) is not \(\mathsf {NP}\)-complete under logtime-uniform \(\le _{{\text {m}}}^{\mathsf {AC}^0}\) reductions [22].

Supported by NSF grants CCF-1514164 and CCF-1559855. This work was done [in part] while author Eric Allender was visiting the Simons Institute for the Theory of Computing.

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## Notes

- 1.
Although Corollary 6 of [24] does not mention the number of queries, inspection of the proof shows that only one query is performed.

- 2.

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## Acknowledgments

Much of this work was done in the 2018 DIMACS REU, organized by Lazaros Gallos, Parker Hund, and many others. We thank Michael Saks, Shuichi Hirahara, Avishay Tal, and John Hitchcock for helpful discussions.

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Allender, E., Ilango, R., Vafa, N. (2019). The Non-hardness of Approximating Circuit Size. In: van Bevern, R., Kucherov, G. (eds) Computer Science – Theory and Applications. CSR 2019. Lecture Notes in Computer Science(), vol 11532. Springer, Cham. https://doi.org/10.1007/978-3-030-19955-5_2

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