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The Non-hardness of Approximating Circuit Size

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Computer Science – Theory and Applications (CSR 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11532))

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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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  1. 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. 2.

    The problem \({\epsilon }\mathsf {\text {-}Gap MCSP}\) is defined somewhat differently in [7] than here. See Sect. 2. Thus the form of \(\epsilon (n)\) looks different here than in [7].


  1. Agrawal, M., Allender, E., Rudich, S.: Reductions in circuit complexity: an isomorphism theorem and a gap theorem. J. Comput. Syst. Sci. 57(2), 127–143 (1998)

    Article  MathSciNet  Google Scholar 

  2. Ajtai, M.: \(\varSigma ^1_1\)-formulae on finite structures. Ann. Pure Appl. Log. 24, 1–48 (1983)

    Article  Google Scholar 

  3. Allender, E., Buhrman, H., Kouckỳ, M., van Melkebeek, D., Ronneburger, D.: Power from random strings. SIAM J. Comput. 35(6), 1467–1493 (2006)

    Article  MathSciNet  Google Scholar 

  4. Allender, E., Das, B.: Zero knowledge and circuit minimization. Inf. Comput. 256, 2–8 (2017)

    Article  MathSciNet  Google Scholar 

  5. Allender, E., Grochow, J.A., van Melkebeek, D., Moore, C., Morgan, A.: Minimum circuit size, graph isomorphism, and related problems. SIAM J. Comput. 47(4), 1339–1372 (2018)

    Article  MathSciNet  Google Scholar 

  6. Allender, E., Hellerstein, L., McCabe, P., Pitassi, T., Saks, M.: Minimizing disjunctive normal form formulas and \({\sf AC}^0\) circuits given a truth table. SIAM J. Comput. 38(1), 63–84 (2008)

    Article  MathSciNet  Google Scholar 

  7. Allender, E., Hirahara, S.: New insights on the (non)-hardness of circuit minimization and related problems. In: Proceedings of 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017) (2017)

    Google Scholar 

  8. Allender, E., Holden, D., Kabanets, V.: The minimum oracle circuit size problem. Comput. Complex. 26(2), 469–496 (2017)

    Article  MathSciNet  Google Scholar 

  9. Allender, E., Kouckỳ, M., Ronneburger, D., Roy, S.: The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory. J. Comput. Syst. Sci. 77(1), 14–40 (2011)

    Article  MathSciNet  Google Scholar 

  10. Allender, E., Loui, M.C., Regan, K.W.: Reducibility and completeness. In: Atallah, M.J., Blanton, M. (eds.) Algorithms and Theory of Computation Handbook, pp. 23–23. Chapman & Hall/CRC, New York (2010)

    Google Scholar 

  11. Arora, S.: AC\(^0\)-reductions cannot prove the PCP theorem (1995, unpublished Manuscript)

    Google Scholar 

  12. Furst, M., Saxe, J.B., Sipser, M.: Parity, circuits, and the polynomial-time hierarchy. Math. Syst. Theory 17(1), 13–27 (1984)

    Article  MathSciNet  Google Scholar 

  13. Golovnev, A., Ilango, R., Impagliazzo, R., Kabanets, V., Kolokolova, A., Tal, A.: AC\(^0[p]\) lower bounds against MCSP via the coin problem. Technical report TR19-018, Electronic Colloquium on Computational Complexity (ECCC) (2019). To appear in ICALP 2019

    Google Scholar 

  14. Hatami, P., Kulkarni, R., Pankratov, D.: Variations on the sensitivity conjecture. Theory Comput. Grad. Surv. 4, 1–27 (2011)

    Google Scholar 

  15. Hirahara, S.: Non-black-box worst-case to average-case reductions within NP. In: 59th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 247–258 (2018)

    Google Scholar 

  16. Hirahara, S., Santhanam, R.: On the average-case complexity of MCSP and its variants. In: Proceedings of 32nd Conference on Computational Complexity (CCC). LIPIcs-Leibniz International Proceedings in Informatics, vol. 79. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2017)

    Google Scholar 

  17. Hirahara, S., Watanabe, O.: Limits of minimum circuit size problem as oracle. In: Proceedings of 31st Conference on Computational Complexity (CCC). LIPIcs-Leibniz International Proceedings in Informatics, vol. 50. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2016)

    Google Scholar 

  18. Hitchcock, J., Pavan, A.: On the NP-completeness of the minimum circuit size problem. In: FSTTCS (2015)

    Google Scholar 

  19. Ilango, R.: AC\(^0[p]\) lower bounds and NP-hardness for variants of MCSP. Technical report TR19-021, Electronic Colloquium on Computational Complexity (ECCC) (2019)

    Google Scholar 

  20. Impagliazzo, R., Kabanets, V., Volkovich, I.: The power of natural properties as oracles. In: LIPIcs-Leibniz International Proceedings in Informatics, vol. 102. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2018)

    Google Scholar 

  21. Kabanets, V., Cai, J.Y.: Circuit minimization problem. In: Proceedings of 32nd ACM Symposium on Theory of Computing (STOC), New York, NY, USA, pp. 73–79 (2000)

    Google Scholar 

  22. Murray, C.D., Williams, R.R.: On the (non) NP-hardness of computing circuit complexity. Theory Comput. 13(1), 1–22 (2017)

    Article  MathSciNet  Google Scholar 

  23. Oliveira, I., Pich, J., Santhanam, R.: Hardness magnification near state-of-the-art lower bounds. In: Electronic Colloquium on Computational Complexity 158 (2018)

    Google Scholar 

  24. Oliveira, I., Santhanam, R.: Conspiracies between learning algorithms, circuit lower bounds and pseudorandomness. In: Proceedings of 32nd Conference on Computational Complexity (CCC), vol. 79, pp. 18:1–18:49. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2017)

    Google Scholar 

  25. Oliveira, I.C., Santhanam, R.: Hardness magnification for natural problems. In: Symposium on Foundations of Computer Science (FOCS), pp. 65–76 (2018)

    Google Scholar 

  26. Razborov, A., Rudich, S.: Natural proofs. In: Proceedings of 26th ACM Symposium on Theory of Computing (STOC), New York, NY, USA, pp. 204–213 (1994)

    Google Scholar 

  27. Rudow, M.: Discrete logarithm and minimum circuit size. Inf. Process. Lett. 128, 1–4 (2017)

    Article  MathSciNet  Google Scholar 

  28. Trakhtenbrot, B.: A survey of Russian approaches to perebor (brute-force searches) algorithms. IEEE Ann. Hist. Comput. 6(4), 384–400 (1984)

    Article  MathSciNet  Google Scholar 

  29. Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer, Heidelberg (2013).

    Book  MATH  Google Scholar 

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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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Correspondence to Eric Allender .

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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.

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