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.

中文翻译：

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关于代数和可定义闭包的可计算方面

我们研究了相对于一组公式的代数闭式和可定义闭式的可计算性。我们表明，对于一个可量化的数量级公式的集合，在任何给定的可计算结构中，| $ n $ |⁠相对于该集合的代数和可定义闭包均为| $ \ varSigma ^ 0_ {n + 2} $ | 套。我们进一步证明这些界限是紧密的。