Computability of Algebraic and Definable Closure

Nathanael L. Ackerman, Cameron E. Freer, Rehana Patel

Abstract: We consider computability-theoretic aspects of the algebraic and definable closure operations for formulas. We show that for $$\varphi $$ a Boolean combination of $$\varSigma _n$$-formulas and in a given computable structure, the set of parameters for which the closure of $$\varphi $$ is finite is $$\varSigma ^0_{n+2}$$, and the set of parameters for which the closure is a singleton is $$\varDelta ^0_{n+2}$$. In addition, we construct examples witnessing that these bounds are tight.