What is the Σ32-circuit complexity (depth 3, bottom-fanin 2) of the 2n-bit inner product function? The complexity is known to be exponential 2αnn for some αn=Ω(1). We show that the limiting constant α≔limsupαn satisfies0.847...≤α≤0.965.... Determining α is one of the seemingly-simplest open problems about depth-3 circuits. The question was recently raised by Golovnev, Kulikov, and Williams (ITCS 2021) and Frankl, Gryaznov, and Talebanfard (ITCS 2022), who observed that α∈[0.5,1]. To obtain our improved bounds, we analyse a covering LP that captures the Σ32-complexity up to polynomial factors. In particular, our lower bound is proved by constructing a feasible solution to the dual LP.

OS design is traditionally heavily intertwined with protection mechanisms. OSes statically commit to one or a combination of (1) hardware isolation, (2) runtime checking, and (3) software verification early at design time. Changes after deployment require major refactoring; as such, they are rare and costly. In this paper, we argue that this strategy is at odds with recent hardware and software trends: protections break (Meltdown), hardware becomes heterogeneous (Memory Protection Keys, CHERI), and multiple mechanisms can now be used for the same task (software hardening, verification, HW isolation, etc). In short, the choice of isolation strategy and primitives should be postponed to deployment time.