To decide whether a propositional formula $P$ is classically provable, the completeness theorem gives an easy algorithm : simply test all finite boolean combinations of the variables of $P$. It is exponential in the number of variables, but it does finish in all cases.
Now completeness for constructive proofs means that we must test all evaluations of the variables of $P$ into all Heyting algebras, not just $\{0,1\}$. The open sets of $\mathbb{R}$ form such a Heyting algebra, which is infinite. So is there a constructive decision algorithm ?
