On computable aspects of algebraic and definable closure

01/28/2021

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by   Nathanael Ackerman, et al.

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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 Σ^0_n+2 sets. We further show that these bounds are tight.