Pseudodeterministic algorithms and the structure of probabilistic time

We connect the study of pseudodeterministic algorithms to two major open problems about the structural complexity of BPTIME: proving hierarchy theorems and showing the existence of complete problems. Our main contributions can be summarised as follows. A new pseudorandom generator and its consequences. We build on techniques developed to prove hierarchy theorems for probabilistic time with advice (Fortnow and Santhanam [FS04]) 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. For instance, we show unconditionally for each ε > 0 that infinitely many primes p n have a succinct representation in the following sense: there is a fixed probabilistic polynomial time algorithm that generates p n with high probability from its succinct representation of size O(|p n | ε ). This offers an exponential improvement over the running time of previous results, and shows that infinitely many primes have succinct and efficient representations.