Preliminary Version

The main theorem of this paper is that, for every real number < 1 (e.g., = 0:99), only a measure 0 subset of the languages decidable in exponential time are Pn tt-reducible to languages that are not exponentially dense. Thus every Pn tt-hard language for E is exponentially dense. This strengthens Watanabe's 1987 result, that every PO(logn) tt-hard language for E is exponentially dense. The combinatorial technique used here, the sequentially most frequent query selection, also gives a new, simpler proof of Watanabe's result. The main theorem also has implications for the structure of NP under strong hypotheses. Ogiwara and Watanabe (1991) have shown that the hypothesis P 6= NP implies that every Pbtt-hard language for NP is non-sparse (i.e., not polynomially sparse). Their technique does not appear to allow signi cant relaxation of either the query bound or the sparseness criterion. It is shown here that a stronger hypothesis| namely, that NP does not have measure 0 in exponential time|implies the stronger conclusion that, for every real < 1, every Pn tt-hard language for NP is exponentially dense. Evidence is presented that this stronger hypothesis is reasonable. The proof of the main theorem uses a new, very general weak stochasticity theorem, ensuring that almost every language in E is statistically unpredictable by feasible deterministic algorithms, even with linear nonuniform advice.