We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in $\text{Quasi-NP} = \text{NTIME}[n^{(\log n)^{O(1)}}]$ and $\text{NEXP}$ do not have small circuits from various circuit classes ${\cal C}$, by showing that ${\cal C}$ admits non-trivial satisfiability and/or $\#$SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of non-trivial $\#$SAT algorithm for a circuit class ${\mathcal C}$. Say a symmetric Boolean function $f(x_1,\ldots,x_n)$ is sparse if it outputs $1$ on $O(1)$ values of $\sum_i x_i$. We show that for every sparse $f$, and for all "typical" ${\cal C}$, faster $\#$SAT algorithms for ${\cal C}$ circuits actually imply lower bounds against the circuit class $f \circ {\cal C}$, which may be stronger than ${\cal C}$ itself. In particular: $\#$SAT algorithms for $n^k$-size ${\cal C}$-circuits running in $2^n/n^k$ time (for all $k$) imply $\text{NEXP}$ does not have $f \circ {\cal C}$-circuits of polynomial size. $\#$SAT algorithms for $2^{n^{\epsilon}}$-size ${\cal C}$-circuits running in $2^{n-n^{\epsilon}}$ time (for some $\epsilon > 0$) imply $\text{Quasi-NP}$ does not have $f \circ {\cal C}$-circuits of polynomial size. Applying $\#$SAT algorithms from the literature, one immediate corollary of our results is that $\text{Quasi-NP}$ does not have $\text{EMAJ} \circ \text{ACC}^0 \circ \text{THR}$ circuits of polynomial size, where $\text{EMAJ}$ is the "exact majority" function, improving previous lower bounds against $\text{ACC}^0$ [Williams JACM'14] and $\text{ACC}^0 \circ \text{THR}$ [Williams STOC'14], [Murray-Williams STOC'18]. This is the first nontrivial lower bound against such a circuit class.

中文翻译：

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ACC 电路稀疏对称函数的下界：扩展 $\#$SAT 算法的范围

我们继续通过电路可满足性算法证明电路下界的程序。到目前为止，这个程序已经产生了几个具体的结果，证明了 $\text{Quasi-NP} = \text{NTIME}[n^{(\log n)^{O(1)}}]$ 和 $ \text{NEXP}$ 没有来自各种电路类 ${\cal C}$ 的小电路，通过证明 ${\cal C}$ 承认非平凡的可满足性和/或 $\#$SAT 算法击败穷举少量搜索。在本文中，我们为电路类 ${\mathcal C}$ 提出了非平凡 $\#$SAT 算法的新的强下界结果。假设对称布尔函数 $f(x_1,\ldots,x_n)$ 是稀疏的，如果它在 $\sum_i x_i$ 的 $O(1)$ 值上输出 $1$。我们证明，对于每个稀疏的 $f$，以及所有“典型的”${\cal C}$，${\cal C}$ 电路的更快 $\#$SAT 算法实际上意味着电路类 $f \circ {\cal C}$ 的下限，这可能比 ${\cal C}$ 本身更强。特别是：$n^k$-size ${\cal C}$-circuits 的 $\#$SAT 算法在 $2^n/n^k$ 时间内运行（对于所有 $k$）意味着 $\text{NEXP }$ 没有多项式大小的 $f \circ {\cal C}$-电路。$\#$SAT 算法用于 $2^{n^{\epsilon}}$-size ${\cal C}$-circuits 在 $2^{nn^{\epsilon}}$ 时间内运行（对于某些 $\epsilon > 0$) 意味着 $\text{Quasi-NP}$ 没有多项式大小的 $f \circ {\cal C}$ 电路。应用文献中的 $\#$SAT 算法，我们结果的一个直接推论是 $\text{Quasi-NP}$ 没有 $\text{EMAJ} \circ \text{ACC}^0 \circ \text {THR}$ 多项式大小的电路，其中 $\text{EMAJ}$ 是“精确多数”函数，改进先前针对 $\text{ACC}^0$ [Williams JACM'14] 和 $\text{ACC}^0 \circ \text{THR}$ [Williams STOC'14], [Murray-Williams STOC' 的下限18]。这是针对此类电路类的第一个非平凡下界。