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 ${pn}{n \in \mathbb{N}}$ of increasing primes and a randomized algorithm $A$ running in expected sub-exponential time such that for each $n$, on input $1^{|p_n|}$, $A$ 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 $\gamma$-dense if for large enough $n$, $|Q \cap {0,1}^n| \geq \gamma 2^n$. 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, $Hn \subseteq {0,1}^n$, such that for each $(1/n^c)$-dense property $Q \in \mathsf{DTIME}(n^c)$ 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, $H'n \subseteq {0,1}^n$, such that for each $(1/n^c)$-dense property $Q \in \mathsf{DTIME}(n^c)$ 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.