On the complexity of the theory of a computably presented metric structure

We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in $[0,1]$, we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form $\phi^\mathcal{M} \leq r$, and the open diagram, which encapsulates strict inequalities of the form $\phi^\mathcal{M} < r$. We show that the closed and open $\Sigma_N$ diagrams are $\Pi^0_{N+1}$ and $\Sigma_N$ respectively, and that the closed and open $\Pi_N$ diagrams are $\Pi^0_N$ and $\Sigma^0_{N + 1}$ respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal.