Some complete $\omega$-powers of a one-counter language, for any Borel class of finite rank

TL;DR

The paper demonstrates that for any finite Borel rank, there exists a one-counter automaton language whose infinite concatenation (omega-power) is complete for that Borel class, showing the expressive power of such automata.

Contribution

It constructs specific one-counter automaton languages whose omega-powers are complete for any finite Borel class, extending understanding of automata's descriptive complexity.

Findings

Existence of one-counter automaton languages with omega-powers complete for any finite Borel class

Construction of languages with omega-powers in $ ext{Pi}^0_n$ and $ ext{Sigma}^0_n$ classes

Demonstration of the expressive power of one-counter automata in descriptive set theory

Abstract

We prove that, for any natural number n $\ge$ 1, we can find a finite alphabet $\Sigma$ and a finitary language L over $\Sigma$ accepted by a one-counter automaton, such that the $\omega$-power L $\infty$ := {w 0 w 1. .. $\in$ $\Sigma$ $\omega$ | $\forall$i $\in$ $\omega$ w i $\in$ L} is $\Pi$ 0 n-complete. We prove a similar result for the class $\Sigma$ 0 n .