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Volume 14 (2018) Article 18 pp. 1-45
Succinct Hitting Sets and Barriers to Proving Lower Bounds for Algebraic Circuits
Received: July 16, 2017
Revised: July 26, 2018
Published: December 18, 2018
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Keywords: algebraic circuits, lower bounds, derandomization, polynomial identity testing, barriers
ACM Classification: F.1.3
AMS Classification: 68Q17, 52C07, 11H06, 11H31, 05B40

Abstract: [Plain Text Version]

$ \newcommand{\poly}{\operatorname{poly}} \newcommand{\polylog}{\operatorname{polylog}} $

We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich (1997) for Boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike in 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 (2016) 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, to the existence of a hitting set for the class of $\poly(N)$-degree $\poly(N)$-size circuits which consists of coefficient vectors of polynomials of $\polylog(N)$ degree with $\polylog(N)$-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 (2001), 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.

A conference version of this paper appeared in the Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC 2017).