On computable aspects of algebraic and definable closure

Nathanael L. Ackerman, Cameron E. Freer, Rehana Patel

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 |$\varSigma ^0_{n+2}$| sets. We further show that these bounds are tight.