Time-space tradeoffs in the counting hierarchy

Extends the lower-bound techniques of L. Fortnow (2000) to the unbounded-error probabilistic model. A key step in the argument is a generalization of V.A. Nepomnjasˇcˇii˘´s (1970) theorem from the Boolean setting to the arithmetic setting. This generalization is made possible due to the recent discovery of logspace-uniform TC⁰ circuits for iterated multiplication (A. Chiu et al., 2000). As an example of the sort of lower bounds that we obtain, we show that MAJ-MAJSAT is not contained in PrTiSp(n^1+o(1) , n^ε) for any ε<1. We also extend one of Fortnow´s lower bounds, from showing that S¯A¯T¯ does not have uniform NC¹ circuits of size n^1+o(1), to a similar result for SAC¹ circuits