Computability Of Algebraic And Definable Closure

We consider computability-theoretic aspects of the algebraic and definable closure operations for formulas. We show that for phi a Boolean combination of Sigma(n)-formulas and in a given computable structure, the set of parameters for which the closure of phi is finite is Sigma(0)(n+2), and the set of parameters for which the closure is a singleton is Delta(0)(n+2). In addition, we construct examples witnessing that these bounds are tight.