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.

翻译：我们调查了代数关闭和可定义关闭对一组公式的可计算性。我们显示,对于在任何特定的可计算结构中以美元计数的量化方位公式的可计算性集合,任何特定的可计算结构中,该计算方位的代数关闭和可确定性关闭都为$\Sigma ⁇ 0 ⁇ n+2}数据集。我们进一步显示,这些界限是紧凑的。