On the Existence of Algebraic Natural Proofs

2020-04-29

AI Summary

This work investigates provable lower bounds for the algebraic complexity classes VP and VNP within the algebraic natural proofs framework. The central obstacle—whether VNP admits efficient algebraic equations—is resolved by proving, for the first time, that over fields of characteristic zero, VNP has no efficient equations under the exponential algebraic circuit hardness assumption for the Permanent. It further establishes the essential necessity of coefficient boundedness for the existence of such equations. Methodologically, the proof integrates tools from algebraic complexity theory, pseudorandom extraction, nontrivial hitting set constructions, and an algebraic adaptation of the Heintz–Schnorr paradigm. Key contributions are: (1) the first VNP-equation barrier grounded in a plausible hardness assumption; (2) a proof that every coefficient-bounded VP family admits efficient equations over any field; and (3) a demonstration that the coefficient-boundedness requirement is unnecessary over finite fields. Collectively, these results delineate the technical feasibility frontier for natural proofs of algebraic lower bounds.

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 also extend to the class $ extsf{VNP}$ as is. 2. Over fields of characteristic zero, $mathsf{VNP}$ does not have any efficient equations, if the Permanent is exponentially hard for algebraic circuits. 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).

Problem

• Algebraic Natural Proofs
• Computational Complexity
• VNP Problems

Innovation

• Algebraic Natural Proofs
• VNP Hardness
• Coefficient Constraints