Lower Bounds for Alternating Online State Complexity

TL;DR

This paper extends online state complexity to alternating machines, establishing an infinite hierarchy and proving a linear lower bound for recognizing prime numbers, thus advancing understanding of computational limits in this model.

Contribution

It introduces a new lower bound technique for alternating online state complexity and demonstrates its power through hierarchy and prime number recognition results.

Findings

The polynomial hierarchy of alternating online state complexity is infinite.

Linear lower bound on the complexity of recognizing binary prime numbers.

Strengthens previous exponential lower bounds for the same model.

Abstract

The notion of Online State Complexity, introduced by Karp in 1967, quantifies the amount of states required to solve a given problem using an online algorithm, which is represented by a deterministic machine scanning the input from left to right in one pass. In this paper, we extend the setting to alternating machines as introduced by Chandra, Kozen and Stockmeyer in 1976: such machines run independent passes scanning the input from left to right and gather their answers through boolean combinations. 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 , stating that the polynomial hierarchy of alternating online state complexity is infinite, and the second is a linear lower bound on the alternating online state 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.