Lower Bounds for Alternating Online Space Complexity

The notion of online space complexity, introduced by Karp in 1967, quantifies the amount of space required to solve a given problem using an online algorithm, represented by a Turing machine which scans the input exactly once from left to right. In this paper, we study alternating algorithms as introduced by Chandra, Kozen and Stockmeyer in 1976. We devise a lower bound technique relying on boundedly generated lattices of languages, and give two applications of this technique. The first is a hierarchy theorem for languages of polynomial alternating online space complexity, and the second is a linear lower bound on the alternating online space complexity of the prime numbers written in binary. This second result strengthens a result of Hartmanis and Shank from 1968, which implies an exponentially worse lower bound for the same model.