Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP *does not* have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size.

In a ... more >>>

We establish nearly tight bounds on the expected shrinkage of decision lists and DNF formulas under the $p$-random restriction $\mathbf R_p$ for all values of $p \in [0,1]$. For a function $f$ with domain $\{0,1\}^n$, let $\mathrm{DL}(f)$ denote the minimum size of a decision list that computes $f$. We show ... more >>>

We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, let $\mathrm{C}$ be a class of polynomial-size concepts, and suppose that $\mathrm{C}$ can be PAC-learned with membership queries under the uniform distribution with error $1/2 - \gamma$ by a time $T$ quantum algorithm. ... more >>>