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arXiv 1601.04743cs.CCcs.CR
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与Merlin和Arthur的强ETH中断：批处理评估的简短非交互式证明

Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation

Ryan Williams

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17 pages
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摘要：

我们提出了一个有效的证明系统多点算术电路评估：对于在字段${\mathbb F}$上的大小为$s$和次数为$d$的每个算术电路$C(x_1,\ldots,x_n)$，以及任何输入$a_1,\ldots,a_K \in {\mathbb F}^n$、$\bullet$，证明器向验证器发送值$C(a_1), \ldots, C(a_K) \in {\mathbb F}$和$\tilde{O}(K \cdot d)$长度的证明，以及$\bullet$，验证器投掷$\textrm{poly}(\log(dK|{\mathbb F}|/\varepsilon))$硬币并且可以在大约$\tilde{O}(K \cdot(n + d) + s)$时间内检查证明，错误概率小于$\varepsilon$。对于小度$d$，这个“Merlin-Arthur”证明系统（a.k.a. MA-证明系统）在近似线性时间内运行，并且具有许多应用。例如，我们获得MA证明系统，运行在$c^{n}$时间（对于各种$c < 2$）的永久，$\#$ Circuit-SAT的所有次线性深度电路，计数哈密顿周期，$0$ - $1$线性程序的不可行性。一般来说，Valiant的类${\sf VP}$中的任何多项式的值都可以比所有可能的赋值的“穷举求和”更快地被证明。这些结果有力地反驳了Russell Impagliazzo等人提出的Merlin-Arthur Strong ETH和Arthur-Merlin Strong ETH。我们还给出了一个三轮（AMA）的证明系统运行在$2^{2n/3+o(n)}$时间的量化布尔公式，近线性时间MA证明系统计数正交向量的集合，并找到最接近的对在汉明度量，和MA证明系统运行在$n^{k/2+O(1)}$ -时间计数$k$ -团图。我们指出了一些潜在的未来方向来反驳不确定性强ETH。

英文摘要：

We present an efficient proof system for Multipoint Arithmetic Circuit Evaluation: for every arithmetic circuit $C(x_1,\ldots,x_n)$ of size $s$ and degree $d$ over a field ${\mathbb F}$, and any inputs $a_1,\ldots,a_K \in {\mathbb F}^n$, $\bullet$ the Prover sends the Verifier the values $C(a_1), \ldots, C(a_K) \in {\mathbb F}$ and a proof of $\tilde{O}(K \cdot d)$ length, and $\bullet$ the Verifier tosses $\textrm{poly}(\log(dK|{\mathbb F}|/\varepsilon))$ coins and can check the proof in about $\tilde{O}(K \cdot(n + d) + s)$ time, with probability of error less than $\varepsilon$. For small degree $d$, this "Merlin-Arthur" proof system (a.k.a. MA-proof system) runs in nearly-linear time, and has many applications. For example, we obtain MA-proof systems that run in $c^{n}$ time (for various $c < 2$) for the Permanent, $\#$Circuit-SAT for all sublinear-depth circuits, counting Hamiltonian cycles, and infeasibility of $0$-$1$ linear programs. In general, the value of any polynomial in Valiant's class ${\sf VP}$ can be certified faster than "exhaustive summation" over all possible assignments. These results strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed by Russell Impagliazzo and others. We also give a three-round (AMA) proof system for quantified Boolean formulas running in $2^{2n/3+o(n)}$ time, nearly-linear time MA-proof systems for counting orthogonal vectors in a collection and finding Closest Pairs in the Hamming metric, and a MA-proof system running in $n^{k/2+O(1)}$-time for counting $k$-cliques in graphs. We point to some potential future directions for refuting the Nondeterministic Strong ETH.

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