Lower bounds for the state complexity of probabilistic languages and the language of prime numbers

TL;DR

This paper explores the complexity of probabilistic and alternating automata languages, establishing new lower bounds and hierarchy results, including a linear lower bound for the language of prime numbers in binary.

Contribution

It introduces a novel lower bound technique for automata state complexity, demonstrating arbitrarily high complexity for probabilistic languages and a linear lower bound for prime numbers.

Findings

Probabilistic languages can have arbitrarily high deterministic state complexity.

Established a hierarchy theorem for polynomial alternating state complexity.

Proved a linear lower bound on the alternating state complexity of binary prime numbers.

Abstract

This paper studies the complexity of languages of finite words using automata theory. To go beyond the class of regular languages, we consider infinite automata and the notion of state complexity defined by Karp. Motivated by the seminal paper of Rabin from 1963 introducing probabilistic automata, we study the (deterministic) state complexity of probabilistic languages and prove that probabilistic languages can have arbitrarily high deterministic state complexity. We then look at alternating automata as introduced by Chandra, Kozen and Stockmeyer: such machines run independent computations on the word 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 there are languages of arbitrarily high polynomial alternating state complexity, and the second is a linear lower bound on the alternating 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.