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 \(\phi ^\mathcal {M}\le r\), 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 \(\Sigma ^0_N\) 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.

我们考虑了可计算呈现的度量结构图的各种量词级别的复杂性（就算术层次而言）。由于连续逻辑语句的真值可能是 [0, 1] 中的任何实数，我们在每个级别引入两种图：闭合图，它封装了 \(\phi ^ \ mathcal { M}\le r\)和打开的图，它封装了形式为\(\phi ^\mathcal {M}< r\)的严格不等式。我们证明了闭图和开图\(\Sigma _N\)分别是\(\Pi ^0_{N+1}\)和\(\Sigma ^0_N\)，闭图和开图\(\Pi _N\)图分别是\(\Pi ^0_N\)和\(\Sigma ^0_{N + 1}\) 。然后，我们引入连续逻辑的有效无限公式，并将我们的结果扩展到超算术层次结构。最后，我们证明我们的结果是最优的。