On computable aspects of algebraic and definable closure

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