On the Existence of Algebraic Natural Proofs

Abstract

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).