Pseudodeterministic Constructions in Subexponential Time

TL;DR

This paper demonstrates the existence of pseudodeterministic algorithms that construct primes and other dense properties in subexponential time, providing new insights into derandomization and complexity theory.

Contribution

It introduces a pseudodeterministic construction of primes in subexponential time and establishes a general theorem about dense property constructions in complexity classes.

Findings

Existence of an infinite sequence of primes constructible in expected sub-exponential time.

A dichotomy theorem for pseudodeterministic and deterministic constructions of dense properties.

Applications to algorithmic problems in complexity theory.

Abstract

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 $\{p_n\}_{n \in \mathbb{N}}$ 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 $p_n$ 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 $\{H_n\}$ of sets, $H_n \subseteq \{0,1\}^n$, such that for each $(1/n^c)$-dense property $Q \in \mathsf{DTIME}(n^c)$ and every large enough $n$, $H_n \cap Q \neq \emptyset$; 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.