Alternating state complexity of the set of primes and squarefree integers

TL;DR

This paper investigates the alternating state complexity of specific number sets, proving exponential complexity for primes and near-maximal complexity for squarefree integers, advancing understanding of automata-based complexity measures.

Contribution

It establishes the exponential alternating complexity of prime numbers and the near-maximal complexity of squarefree integers, confirming a conjecture and extending complexity theory.

Findings

Prime numbers have exponential alternating complexity.

Squarefree integers have essentially maximal alternating complexity.

Confirmed a conjecture by Fijalkow.

Abstract

We show that the set of prime numbers has exponential alternating complexity, proving a conjecture by Fijalkow. We further show that the set of squarefree integers has essentially maximal possible alternating complexity.