SIAM Journal on Computing, Ahead of Print.
We prove that if every problem in ${NP}$ has $n^k$-size circuits for a fixed constant $k$, then for every ${NP}$-verifier and every yes-instance $x$ of length $n$ for that verifier, the verifier's search space has an $n^{O(k^3)}$-size witness circuit: A witness for $x$ that can be encoded with a circuit of only $n^{O(k^3)}$ size. An analogous statement is proved for nondeterministic quasi-polynomial time, i.e., ${NQP} = {NTIME}[n^{\log^{O(1)} n}]$. This significantly extends the Easy Witness Lemma of Impagliazzo, Kabanets, and Wigderson [J. Comput. System Sci., 65 (2002), pp. 672--694] which only held for larger nondeterministic classes such as ${NEXP}$. As a consequence, the connections between circuit-analysis algorithms and circuit lower bounds can be considerably sharpened: Algorithms for approximately counting satisfying assignments for given circuits which improve over exhaustive search can imply circuit lower bounds for functions in ${NQP}$, or even ${NP}$. To illustrate, applying known algorithms for satisfiability of ${ACC} \circ {THR}$ circuits [R. Williams, New algorithms and lower bounds for circuits with linear threshold gates, in Proceedings of the 46th Annual ACM Symposium on Theory of Computing, ACM, New York, 2014, pp. 194--202] we conclude that for every fixed $k$, ${NQP}$ does not have $n^{\log^k n}$-size ${ACC} \circ {THR}$ circuits.

中文翻译：

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从新的简易见证引理中不确定的准多重时间的电路下界

《 SIAM计算杂志》，预印本。
我们证明，如果$ {NP} $中的每个问题都具有固定常数$ k $的$ n ^ k $大小的电路，则对于每个$ {NP} $验证者和长度为$的每个yes实例$ x $对于该验证者，如果$ n $，则验证者的搜索空间具有一个$ n ^ {O（k ^ 3）} $大小的见证电路：$ x $的见证人只能用$ n ^ {O（ k ^ 3）} $大小。对于不确定的拟多项式时间，证明了一个类似的陈述，即$ {NQP} = {NTIME} [n ^ {\ log ^ {O（1）} n}] $。这大大扩展了Impagliazzo，Kabanets和Wigderson的Easy Witness Lemma [J. 计算 System Sci。，65（2002），pp。672--694]，仅适用于较大的不确定类，例如$ {NEXP} $。因此，电路分析算法和电路下限之间的联系可以大大提高：对给定电路的满意分配进行近似计数的算法比穷举搜索有所改进，该算法可能暗示$ {NQP} $甚至$ {NP} $中函数的电路下限。为了说明，应用已知的算法来满足$ {ACC} \ circ {THR} $电路的需求。Williams，具有线性阈值门的电路的新算法和下界，在第46届ACM计算理论年会论文集中，ACM，纽约，2014年，第194--202页]，我们得出结论，每固定$ k $ ，$ {NQP} $没有$ n ^ {\ log ^ kn} $大小的$ {ACC} \ circ {THR} $电路。