What does it mean that a theory admits constructive elimination of quantifiers?
A theory admits elimination of quantifiers when each formula of the theory is equivalent to a quanifier-free formula, right?
But what is meant when we use the term "constructive" ?
I would interpret this to mean that there is a computable procedure to find for any given formula $\varphi$ a quantifier free formula $\varphi^*$ that is equivalent to it modulo the theory.
If the theory $T$ is c.e. axiomatizable, then this is equivalent to ordinary quantifier-elimination, because if a theory $T$ admits elimination of quantifiers, then given any $\varphi$ we can search through all proofs from $T$ until we find a quantifier-free $\varphi^*$ that is $T$-provably equivalent, and then output the first such $\varphi^*$ that we find. So if the theory $T$ is c.e., then QE is the same as constructive QE.
This fact is often important in proving the decidability of a theory. If a c.e. theory admits QE and the quantifier-free sentences have decidable truth values, then it is decidable, because for any sentence, we can search for a quantifier-free equivalent of it, by searching for proofs, and decide the truth of that (trivial) formulation, thereby deciding the theory.
But if the theory $T$ is not c.e., then it could be a stronger requirement that the QE is effective. If you want, I'll try to make an example of a theory that has QE, but not constructive QE.
