On computable aspects of algebraic and definable closure
Nathanael Ackerman, Cameron E. Freer, Rehana Patel · Journal of Logic and Computation · 2020
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.