The Non-hardness of Approximating Circuit Size

The Minimum Circuit Size Problem (MCSP) has been the focus of intense study recently; MCSP is hard for SZK under rather powerful reductions (Allender and Das Inf. Comput. 256, 2–8, 2017), and is provably not hard under “local” reductions computable in TIME(n^0.49) (Murray and Williams Theory Comput. 13(1), 1–22, 2017). The question of whether MCSP is NP-hard (or indeed, hard even for small subclasses of P) under some of the more familiar notions of reducibility (such as many-one or Turing reductions computable in polynomial time or in AC⁰) is closely related to many of the longstanding open questions in complexity theory (Allender and Hirahara ACM Trans. Comput. Theory 11(4), 27:1–27:27, 2019; Allender et al. Comput. Complex. 26(2), 469–496, 2017; Hirahara and Santhanam 2017; Hirahara and Watanabe 2016; Hitchcock and Pavan 2015; Impagliazzo et al. 2018; Murray and Williams Theory Comput. 13(1), 1–22, 2017). All prior hardness results for MCSP hold also for computing somewhat weak approximations to the circuit complexity of a function (Allender et al. SIAM J. Comput. 35(6), 1467–1493, 2006; Allender and Das Inf. Comput. 256, 2–8, 2017; Allender et al. J. Comput. Syst. Sci. 77(1), 14–40, 2011; Hirahara and Santhanam 2017; Kabanets and Cai 2000; Rudow Inf. Process. Lett. 128, 1–4, 2017) (Subsequent to our work, a new hardness result has been announced (Ilango 2020) 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 (KT) and the corresponding decision problem (MKTP). More recently, a new approach for proving improved hardness results for MKTP was developed (Allender et al. SIAM J. Comput. 47(4), 1339–1372, 2018; Allender and Hirahara ACM Trans. Comput. Theory 11(4), 27:1–27:27, 2019), 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 MCSP. In particular, it is known that MKTP is hard for the complexity class DET under nonuniform \(\leq _{\text {m}}^{\mathsf {AC}^{0}}\) reductions, implying MKTP is not in AC⁰[p] for any prime p (Allender and Hirahara ACM Trans. Comput. Theory 11(4), 27:1–27:27, 2019). It was still open if similar circuit lower bounds hold for MCSP (But see Golovnev et al. 2019; Ilango 2020). One possible avenue for proving a similar hardness result for MCSP would be to improve the hardness of approximation for MKTP beyond 1 + o(1) to ω(1), as KT-complexity and circuit size are polynomially-related. In this paper, we show that this approach cannot succeed. More specifically, we prove that PARITY does not reduce to the problem of computing superlinear approximations to KT-complexity or circuit size via AC⁰-Turing reductions that make O(1) queries. This is significant, since approximating any set in P/poly AC⁰-reduces to just one query of a much worse approximation of circuit size or KT-complexity (Oliveira and Santhanam 2017). 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 Allender and Hirahara (ACM Trans. Comput. Theory 11(4), 27:1–27:27, 2019) (for more powerful reductions, but for much worse approximations). This highlights obstacles that would have to be overcome by any proof that MKTP or MCSP is hard for NP under AC⁰ reductions. It may also be a step toward confirming a conjecture of Murray and Williams, that MCSP is not NP-complete under logtime-uniform \(\leq _{\text {m}}^{\mathsf {AC}^{0}}\) reductions.