The Non-hardness of Approximating Circuit Size

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.