For a size parameter \(s:\mathbb {N}\to \mathbb {N}\), the Minimum Circuit Size Problem (denoted by MCSP[s(n)]) is the problem of deciding whether the minimum circuit size of a given function f : {0,1}ⁿ →{0,1} (represented by a string of length N := 2ⁿ) is at most a threshold s(n). A recent line of work exhibited “hardness magnification” phenomena for MCSP: A very weak lower bound for MCSP implies a breakthrough result in complexity theory. For example, McKay, Murray, and Williams (STOC 2019) implicitly showed that, for some constant μ₁ > 0, if \(\text {MCSP}[2^{\mu _{1}\cdot n}]\) cannot be computed by a one-tape Turing machine (with an additional one-way read-only input tape) running in time N^1.01, then P≠NP. In this paper, we present the following new lower bounds against one-tape Turing machines and branching programs: (1) A randomized two-sided error one-tape Turing machine (with an additional one-way read-only input tape) cannot compute \(\text {MCSP}[2^{\mu _{2}\cdot n}]\) in time N^1.99, for some constant μ₂ > μ₁. (2) A non-deterministic (or parity) branching program of size \(o(N^{1.5}/\log N)\) cannot compute MKTP, which is a time-bounded Kolmogorov complexity analogue of MCSP. This is shown by directly applying the Nečiporuk method to MKTP, which previously appeared to be difficult. (3) The size of any non-deterministic, co-non-deterministic, or parity branching program computing MCSP is at least \(N^{1.5-o\left (1\right )}\). These results are the first non-trivial lower bounds for MCSP and MKTP against one-tape Turing machines and non-deterministic branching programs, and essentially match the best-known lower bounds for any explicit functions against these computational models. The first result is based on recent constructions of pseudorandom generators for read-once oblivious branching programs (ROBPs) and combinatorial rectangles (Forbes and Kelley, FOCS 2018; Viola, Electron. Colloq. Comput. Complexity (ECCC) 26, 51, 2019). En route, we obtain several related results: (1) There exists a (local) hitting set generator with seed length \(\widetilde {O}(\sqrt {N})\) secure against read-once polynomial-size non-deterministic branching programs on N-bit inputs. (2) Any read-once co-non-deterministic branching program computing MCSP must have size at least \(2^{\widetilde {\Omega }(N)}\).

对于尺寸参数\(s:\mathbb {N}\to \mathbb {N}\)，最小电路尺寸问题（用 MCSP[ s ( n )] 表示）是决定是否最小电路尺寸的问题给定函数f : {0,1} ⁿ →{0,1}（由长度为N := 2 ⁿ的字符串表示）至多为阈值s ( n )。最近的一系列工作展示了 MCSP 的“硬度放大”现象：MCSP 的非常弱的下界意味着复杂性理论的突破性结果。例如，McKay、Murray 和 Williams (STOC 2019) 隐含地表明，对于某个常数μ ₁ > 0，如果\(\text {MCSP}[2^{\mu _{1}\cdot n}]\)无法由及时运行的单带图灵机（带有额外的单向只读输入带）计算N ^1.01，则P≠NP。在本文中，我们提出了以下针对单带图灵机和分支程序的新下界：（1）随机双侧错误单带图灵机（带有附加的单向只读输入带）无法计算\(\text {MCSP}[2^{\mu _{2}\cdot n}]\)在时间N ^1.99中，对于某个常数μ ₂ > μ ₁。(2) 大小为\(o(N^{1.5}/\log N)\)的非确定性（或奇偶性）分支程序无法计算 MKTP，它是 MCSP 的时间限制 Kolmogorov 复杂性模拟。这是通过将 Nečiporuk 方法直接应用于 MKTP 来显示的，这在以前似乎很困难。(3) 任何非确定性、共非确定性或奇偶分支程序计算 MCSP 的大小至少为\(N^{1.5-o\left (1\right )}\). 这些结果是 MCSP 和 MKTP 针对单带图灵机和非确定性分支程序的第一个非平凡下界，并且基本上与针对这些计算模型的任何显式函数的最著名下界相匹配。第一个结果基于最近为只读分支程序 (ROBP) 和组合矩形构建的伪随机生成器（Forbes 和 Kelley，FOCS 2018；Viola，Electron。Colloq。Comput.Complexity (ECCC) 26、51、2019） . 在途中，我们获得了几个相关结果：（1）存在一个（本地）命中集生成器，其种子长度为\(\ widetilde {O}(\sqrt {N})\) N上的确定性分支程序位输入。(2) 任何计算 MCSP 的一次性共同非确定性分支程序的大小必须至少为\(2^{\widetilde {\Omega }(N)}\)。