Pseudodeterministic Algorithms and the Structure of Probabilistic Time

TL;DR

This paper explores the connection between pseudodeterministic algorithms and the structural complexity of probabilistic time, offering new constructions, implications for hierarchy theorems, and insights into prime number generation.

Contribution

It introduces the first unconditional pseudorandom generator with polynomial stretch in pseudodeterministic polynomial time and links pseudodeterministic algorithms to hierarchy theorems and complete problems.

Findings

Constructed a pseudorandom generator secure infinitely often against polynomial-time algorithms.

Showed that improvements in certain pseudodeterministic algorithms imply new hierarchy theorems.

Established equivalence between pseudodeterministic string construction and hierarchy theorems for probabilistic time.

Abstract

We connect the study of pseudodeterministic algorithms to two major open problems about the structural complexity of $\mathsf{BPTIME}$: proving hierarchy theorems and showing the existence of complete problems. Our main contributions can be summarised as follows. 1. We build on techniques developed to prove hierarchy theorems for probabilistic time with advice (Fortnow and Santhanam, FOCS 2004) to construct the first unconditional pseudorandom generator of polynomial stretch computable in pseudodeterministic polynomial time (with one bit of advice) that is secure infinitely often against polynomial-time computations. As an application of this construction, we obtain new results about the complexity of generating and representing prime numbers. 2. Oliveira and Santhanam (STOC 2017) established unconditionally that there is a pseudodeterministic algorithm for the Circuit Acceptance Probability Problem ($\mathsf{CAPP}$) that runs in sub-exponential time and is correct with high probability over any samplable distribution on circuits on infinitely many input lengths. We show that improving this running time or obtaining a result that holds for every large input length would imply new time hierarchy theorems for probabilistic time. In addition, we prove that a worst-case polynomial-time pseudodeterministic algorithm for $\mathsf{CAPP}$ would imply that $\mathsf{BPP}$ has complete problems. 3. We establish an equivalence between pseudodeterministic construction of strings of large $\mathsf{rKt}$ complexity (Oliveira, ICALP 2019) and the existence of strong hierarchy theorems for probabilistic time. More generally, these results suggest new approaches for designing pseudodeterministic algorithms for search problems and for unveiling the structure of probabilistic time.