Pseudodeterministic Constructions in Subexponential Time

We study pseudodeterministic constructions, i.e., randomized algorithms which output the same solution on most computation paths. We establish unconditionally that there is an infinite sequence pn_n ∈ mathbbN of increasing primes and a randomized algorithm A running in expected sub-exponential time such that for each n, on input 1|pn|, A outputs pn with probability 1. In other words, our result provides a pseudodeterministic construction of primes in sub-exponential time which works infinitely often. This result follows from a much more general theorem about pseudodeterministic constructions. A property Q subseteq 0,1* is γ-dense if for large enough n, |Q cap 0,1n| ≥ γ 2n. We show that for each c > 0 at least one of the following holds: (1) There is a pseudodeterministic polynomial time construction of a family Hn of sets, Hn subseteq 0,1n, such that for each (1/nc)-dense property Q ∈ mathsfDTIME(nc) and every large enough n, Hn cap Q neq emptyset; or (2) There is a deterministic sub-exponential time construction of a family H'n of sets, H'n subseteq 0,1n, such that for each (1/nc)-dense property Q ∈ mathsfDTIME(nc) and for infinitely many values of n, H'n cap Q neq emptyset. We provide further algorithmic applications that might be of independent interest. Perhaps intriguingly, while our main results are unconditional, they have a non-constructive element, arising from a sequence of applications of the hardness versus randomness paradigm.