Succinct Hitting Sets and Barriers to Proving Algebraic Circuits Lower Bounds

TL;DR

This paper introduces a framework for algebraically natural lower bounds in algebraic circuit complexity, linking the existence of succinct hitting sets to barriers in proving super-polynomial lower bounds, and explores their implications for algebraic proof complexity.

Contribution

It formalizes algebraically natural lower bounds, connects them to succinct derandomization, and provides explicit constructions supporting the existence of barriers similar to natural proofs.

Findings

Existence of algebraic natural proofs barrier is equivalent to succinct derandomization of polynomial identity testing.

Modified constructions of hitting sets can be made succinct, providing evidence for the barrier.

Implications for the GCT program include lower bounds for the complexity of polynomial defining equations.

Abstract

We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich for boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike the boolean setting, there has been no concrete evidence demonstrating that this is a barrier to obtaining super-polynomial lower bounds for general algebraic circuits, as there is little understanding whether algebraic circuits are expressive enough to support "cryptography" secure against algebraic circuits. Following a similar result of Williams in the boolean setting, we show that the existence of an algebraic natural proofs barrier is equivalent to the existence of succinct derandomization of the polynomial identity testing problem. That is, whether the coefficient vectors of polylog(N)-degree polylog(N)-size circuits is a hitting set for the class of poly(N)-degree poly(N)-size circuits. Further, we give an explicit universal construction showing that if such a succinct hitting set exists, then our universal construction suffices. Further, we assess the existing literature constructing hitting sets for restricted classes of algebraic circuits and observe that none of them are succinct as given. Yet, we show how to modify some of these constructions to obtain succinct hitting sets. This constitutes the first evidence supporting the existence of an algebraic natural proofs barrier. Our framework is similar to the Geometric Complexity Theory (GCT) program of Mulmuley and Sohoni, except that here we emphasize constructiveness of the proofs while the GCT program emphasizes symmetry. Nevertheless, our succinct hitting sets have relevance to the GCT program as they imply lower bounds for the complexity of the defining equations of polynomials computed by small circuits.