Which problems have strongly exponential complexity?

For several NP-complete problems, there have been a progression of better but still exponential algorithms. In this paper we address the relative likelihood of sub-exponential algorithms for these problems. We introduce a generalized reduction which we call sub-exponential reduction family (SERF) that preserves sub-exponential complexity. We show that Circuit-SAT is SERF-complete for all NP-search problems, and that for any fixed k, k-SAT, k-Colorability, k-Set Cover Independent Set, Clique, Vertex Cover are SERF-complete for the class SNP of search problems expressible by second order existential formulas whose first order part is universal. In particular, sub-exponential complexity for any one of the above problems implies the same for all others. We also look at the issue of proving strongly exponential lower bounds (that is, bounds of the form 2^Ω(n)) for AC⁰. This problem is even open far depth-3 circuits. In fact, such a bound for depth-3 circuits with even limited (at most n^ε) fan-infer bottom-level gates would imply a nonlinear size lower bound for logarithmic depth circuits. We show that with high probability even degree 2 random GF(2) polynomials require strongly exponential site for Σ₃^k circuits for k=o(loglogn). We thus exhibit a much smaller space of 2(0(ⁿ²)) functions such that almost every function in this class requires strongly exponential size Σ₃^k circuits. As a corollary, we derive a pseudorandom generator (requiring O(n²) bits of advice) that maps n bits into a larger number of bits so that computing parity on the range is hard for Σ₃^k circuits. Our main technical lemma is an algorithm that, for any fixed ε>0, represents an arbitrary k-CNF formula as a disjunction of 2^εn k-CNF formulas that are sparse, e.g., each having O(n) clauses