Abstract
We investigate sets that are immune to ACo; that is, sets with no infinite subset in ACo. First we show that such sets exist in PPP. Although this seems like a rather weak result (since ACo is an extremely weak complexity class and PPP contains the entire polynomial hierarchy) we also prove a somewhat surprising theorem, showing that a significant breakthrough will be necessary in order to prove a bound much better than PPP. Namely, we show that any answer to the question:
Are there sets in NP that are immune to ACo? will provide non-relativizable proof techniques suitable for attacking the Ntime vs Dtime question.
We also show the related result that ACC is properly contained in PP, and that there is no uniform family of ACC circuits that computes the permanent of two matrices. This seems to be the first example of a lower bound in circuit complexity where the uniformity condition is essential; it is still unknown if there is any set in Dtime(\documentclass[12pt]{minimal}
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$$2^{n^{o(1)} }$$
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