On computable aspects of algebraic and definable closure

TL;DR

This paper explores the computational complexity of algebraic and definable closures in structures, establishing tight bounds on their classification within the arithmetical hierarchy based on the quantifier rank of formulas.

Contribution

It provides a precise characterization of the computability levels of algebraic and definable closures relative to collections of formulas, extending understanding of their complexity.

Findings

Algebraic and definable closures are $ ext{Sigma}^0_{n+2}$ sets for formulas of quantifier rank at most n.

The bounds on the complexity of these closures are proven to be tight.

The results link the complexity of closures to the quantifier rank of formulas.

Abstract

We investigate the computability of algebraic closure and definable closure with respect to a collection of formulas. We show that for a computable collection of formulas of quantifier rank at most $n$, in any given computable structure, both algebraic and definable closure with respect to that collection are $\Sigma^0_{n+2}$ sets. We further show that these bounds are tight.