论代数自然证明的存在性
On the Existence of Algebraic Natural Proofs
AI总结：
本文利用代数硬度与伪随机性的联系，证明了有界系数VP子类具有高效方程，并指出在Permanent具有指数级硬度的假设下VNP不存在高效方程，揭示了代数自然下界证明的障碍。
摘要：
代数自然证明框架由 Forbes、Shpilka 和 Volk（2018）以及 Grochow、Kumar、Saks 和 Saraf（2017）独立提出，旨在研究代数复杂性中常用下界证明技术的有效性。我们利用代数硬度与伪随机性之间的已知联系，进一步阐明与该框架相关的问题，具体如下。\n1. 包含有界系数多项式族的 $\mathsf{VP}$ 子类具有高效方程。在有限域上，该结果对系数没有任何限制。此外，这两个结果均可扩展至任何具有低变量、低度通用映射（即该类中所有多项式的生成器）的类。大多数被充分研究的类都具有此性质，例如 $\mathsf{VNP}$、$\mathsf{VBP}$、$\mathsf{VF}$。\n2. 在特征为零的域上，如果 Permanent 对代数电路具有指数级硬度，则 $\mathsf{VNP}$ 不具备任何高效方程。此外，Permanent 在近似意义下的指数级硬度，甚至排除了大度数的高效方程。这提供了“自然”下界技术唯一已知的障碍（该障碍源于可信的硬度假设），同时也表明第一类关于 $\mathsf{VNP}$ 的结果中对系数的限制是必要的。\n第一组结果本质上是通过将利用非平凡命中集生成硬度的著名方法（如 Heintz 和 Schnorr 1980）进行代数化而得出的。$\mathsf{VNP}$ 方程的条件性硬度利用了以下事实：可以从对该类具有（充分）硬度的多项式中提取出针对该类的伪随机性（Kabanets 和 Impagliazzo, 2004）。
英文摘要：
The framework of algebraically natural proofs was independently introduced in the works of Forbes, Shpilka and Volk (2018), and Grochow, Kumar, Saks and Saraf (2017), to study the efficacy of commonly used techniques for proving lower bounds in algebraic complexity. We use the known connections between algebraic hardness and pseudorandomness to shed some more light on the question relating to this framework, as follows. 1. The subclass of $\mathsf{VP}$ that contains polynomial families with bounded coefficients, has efficient equations. Over finite fields, this result holds without any restriction on coefficients. Further, both these results extend to \emph{any} class that admits a low-variate, low-degree universal map: a generator for all polynomials in the class. Most well-studied classes have this property, e.g. \textsf{VNP}, \textsf{VBP}, \textsf{VF}. 2. Over fields of characteristic zero, $\mathsf{VNP}$ does not have any efficient equations, if the Permanent is exponentially hard for algebraic circuits. Moreover, exponential hardness of the Permanent in the approximative sense, even rules out efficient equations of large degree. This gives the only known barrier to ``natural'' lower bound techniques (that follows from believable hardness assumptions), and also shows that the restriction on coefficients in the first category of results about $\mathsf{VNP}$ is necessary. The first set of results follows essentially by algebraizing the well-known method of generating hardness from non-trivial hitting sets (e.g. Heintz and Schnorr 1980). The conditional hardness of equations for $\mathsf{VNP}$ uses the fact that pseudorandomness against a class can be extracted from a polynomial that is (sufficiently) hard for that class (Kabanets and Impagliazzo, 2004).
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