Measure, stochasticity, and the density of hard languages

Abstract

The main theorem of this paper is that, for every real number α < 1 (e.g., α = 0.99), only a measure 0 subset of the languages decidable in exponential time are < 1, every ≤Pnα-tt-hard language for NP is exponentially dense. Evidence is presented that this stronger hypothesis is reasonable. Also presented here (and used in proving the main theorem) is a weak stochasticity theorem, ensuring that almost every language in E is statistically unpredictable by feasible deterministic algorithms, even with linear nonuniform advice.