Breaking the log-squared barrier in pseudorandom generator constructions for read-once branching programs, namely, achieving seed length $o(\log^2 n)$ for length-$n$ programs, has remained a longstanding open problem since Nisan's seminal construction.

We show that breaking this barrier, even achieving seed length $O(\log^{3/2} n)$ (for, say, constant width), would follow from ... more >>>

We study two conjectures posed in the analysis of Boolean functions $f : \{-1, 1\}^n ? \{?1, 1\}$, in both of which, the Majority function plays a central role: the "Majority is Least Stable" (Benjamini et al., 1999) and the "Non-Interactive Correlation Distillation for Erasures" (Yang, 2004; O'Donnell and Wright, ... more >>>

We study when a sound arithmetic theory $\mathcal S{\supseteq}S^1_2$ with polynomial-time decidable axioms efficiently proves the bounded consistency statements $Con_{\mathcal S{+}\phi}(n)$ for a true sentence $\phi$. Equivalently, we ask when $\mathcal S$, viewed as a proof system, simulates $\mathcal S{+}\phi$. The paper's two unconditional contributions constrain possible characterizations. First, for ... more >>>