Small hitting-sets for tiny arithmetic circuits or: How to turn bad designs into good

TL;DR

This paper explores how designing efficient hitting-sets for specific small-depth arithmetic circuits can lead to breakthroughs in derandomizing polynomial identity testing (PIT) and establishing complexity class separations.

Contribution

It introduces new approaches and measures, like cone-size and cone-closed basis isolation, to develop poly-time hitting-sets for small-depth circuits, advancing the understanding of PIT and circuit lower bounds.

Findings

Poly(s)-time hitting-sets for certain small-depth circuits imply quasipolynomial derandomization of PIT.

New measures like cone-size aid in constructing hitting-sets for specific circuit families.

Arity reduction techniques simplify the general PIT problem to smaller variable cases.

Abstract

We show that if we can design poly($s$)-time hitting-sets for $\Sigma\wedge^a\Sigma\Pi^{O(\log s)}$ circuits of size $s$, where $a=\omega(1)$ is arbitrarily small and the number of variables, or arity $n$, is $O(\log s)$, then we can derandomize blackbox PIT for general circuits in quasipolynomial time. This also establishes that either E$\not\subseteq$\#P/poly or that VP$\ne$VNP. In fact, we show that one only needs a poly($s$)-time hitting-set against individual-degree $a'=\omega(1)$ polynomials that are computable by a size-$s$ arity-$(\log s)$ $\Sigma\Pi\Sigma$ circuit (note: $\Pi$ fanin may be $s$). Alternatively, we claim that, to understand VP one only needs to find hitting-sets, for depth-$3$, that have a small parameterized complexity. Another tiny family of interest is when we restrict the arity $n=\omega(1)$ to be arbitrarily small. We show that if we can design poly($s,\mu(n)$)-time hitting-sets for size-$s$ arity-$n$ $\Sigma\Pi\Sigma\wedge$ circuits (resp.~$\Sigma\wedge^a\Sigma\Pi$), where function $\mu$ is arbitrary, then we can solve PIT for VP in quasipoly-time, and prove the corresponding lower bounds. Our methods are strong enough to prove a surprising {\em arity reduction} for PIT-- to solve the general problem completely it suffices to find a blackbox PIT with time-complexity $sd2^{O(n)}$. We give several examples of ($\log s$)-variate circuits where a new measure (called cone-size) helps in devising poly-time hitting-sets, but the same question for their $s$-variate versions is open till date: For eg., diagonal depth-$3$ circuits, and in general, models that have a {\em small} partial derivative space. We also introduce a new concept, called cone-closed basis isolation, and provide example models where it occurs, or can be achieved by a small shift.